Mathematics
OF
Accounting and Finance
By
SEYMOUR WALTON, A.B., C.P.A.
Dean of the Walton School of Commerce, Chicago And
H. A. FINNEY, Ph.B., C.P.A.
Professor of Accounting, Northwestern University Chicago
Third Printing
NEW YORK
THE RONALD PRESS COMPANY
1922
Copyright, 1921, by The Ronald Press Company
All Rights Reserved
PREFACE
This book has been prepared as a manual of business calcu- lation, for use in connection with the problems which continually arise in the course of business as conducted today.
In deciding what subjects to include the authors have been guided throughout by the desire to make the book directly useful in business offices, particularly to persons working in accounting and in the various lines of finance. There are many school texts on commercial arithmetic, but they are necessarily too rudi- mentary to be of much aid to readers of mature experience. No space has been given here to a consideration of the elementary and fundamental processes of mathematics.
The effort has been, instead, to present material of more advanced nature which has not been generally available. This material falls into three general classes.
The earlier chapters explain in considerable detail a number of short processes and practical suggestions that may be applied in routine computations of any sort. Particular attention has been given to the matter of adequate checks upon calculations.
The central portion of the book treats of the special appli- cations of arithmetical principles and short methods to the problems of individual lines of business.
In the final chapters an attempt has been made to explain, in simple terms, convenient ways of using logarithmic and ac- tuarial methods in the solution of business problems relating to compound interest, investments, annuities, bond discount and premium, effective bond rates, leaseholds and depreciation.
During the last months of his life, Mr. Walton devoted most of his strength to this book, and it is a source of profound personal
iv PREFACE
regret that he did not live to see its publication. Few men were his equal as an accountant, a teacher and a writer. None surpassed him as a friend.
H. A. Finney.
EvANSTON, III. August 15, 1921
CONTENTS
Chapter Page
I Short Methods and Practical Suggestions ... 3
Short Methods of Computation
Balancing an Account
Combined Addition and Subtraction
Adding a Part of a Column
Deducting Several vSubtrahends from one Minuend
Table with Net Decrease
Complements
Short Methods of Multiplication
Multiplying by 1 1
Multiplying by 11 1
Tabulating Multiples of a Multiplier
Tabulating Multiples of a Divisor
Division by Use of Reciprocals
II Fractions and Proof Figures 16
Addition and Subtraction of Fractions Cross Multiplication
Equivalent Common and Decimal Fractions Proof Figures
III Arithmetical Progression 22
Elements in Arithmetical Progression Analysis of Simple Progression Computing Total Simple Interest C. P. A. Problem
IV Average 29
Utility of Average Simple Average Moving Average Progressive Average Periodic Average Weighted Average
V Averaging Accounts 42
Settling an Account Calculating Interest Items of Varying Amounts Focal Date Rules Applied
VI CONTENTS
Chapter Page
Reducing Days to Months Compound Average Example of Compound Average Another Illustration
VI Percentage 50
Percentage
Terms Used in Percentage
Fundamental Processes
Percentage of Increase and Decrease
Some Applications of Percentage in Business Per Cent of Goods Sold — Various Manufacturers Monthly Sales Compared on a Percentage Basis Sales for the Week Ending December 18, 1920 Comparison of Sales by Departments Individual Sales Compared with Average Individual Sales Compared with Maximum
Apportionment Division of Profits Distribution of Factory Overhead
Gross Profit Method of Approximating Inventory
Analysis of Statements
Percentage Analysis to Determine Causes of Variation in Profits
VII Equation in the Solution of Problems .... 66
Solving Equations Illustration I Illustration 2 Illustration 3 Illustration 4
VIII Trade and Cash Discount ......... 76
Trade Discount
Cumulative Trade Discounts
Methods of Finding Net Price
Cash Discount
Discount as a Protection Against Loss
Cash Discount Regarded as an Expense
IX Turnover 83
Indefinite Meaning of "Turnover" Normal Inventories Necessary Different Bases of Comparison Working Capital as Basis of Turnover De'in'tion of Working Capital Need of Exact Definitions
CONTENTS vii
Chapter Page
X Partnerships 89
Division of Profits
Liquidation of Partnerships
Periodical Distributions
Reducing Capitals to Profit and Loss Ratio
XI The Clearing House 08
Principle of the Clearing House
Debits and Credits with Clearing House
Clearing House Transactions
Manager's Sheet
Economy of System
Application of Principle Extended
XII Building and Loan Association 103
General Characteristics of Building and Loan Associations
Terminating Plan
Practicabihty of Plan
Serial Plan
Distribution by Partnership Plan
Distribution by Dexter's Rule
Withdrawal of Shares
Sources of Income
Premiums
Individual Plan
Dayton or Ohio Plan
XIII Good-Will and Consolidation 116
Purchasing a Business with Stock
Allocation of Net Earnings
Good-Will
Appraising Good- Will — Year's Purchase Method
Capitalizing Gross Income
Issue of Two Classes of Stock
XIV Foreign Exchange 123
Conversion of Foreign Coinage Reverse Conversion Dealing in Foreign Exchange Average Date of Current Account Conversion of Foreign Branch Accounts Reconciliation of Accounts
XV Logarithms i^p
Use
Multiplication
Division
Vlll
CONTENTS
Chapter
Page
Calculating Powers
Roots
Nature of Logarithms
The Characteristic and the Mantissa
Tables of Logarithms
Characteristics of Logarithms of Numbers Between i and lO
Use of the Characteristic in Pointing off Results
Logarithms of Numbers Smaller Than i
Use of Negative Characteristics in Pointing oflF Answers
Computing with Logarithms having Negative Characteristics
Determining Mantissa by Interpolation
Determining Numbers by Interpolation
XVI Simple and Compound Interest 154
Simple Interest — Methods of Calculating
Rates Other Than 6%
365-day Basis
Partial Payments
Compound Interest
Symbols
Amount of Principal
Frequency of Compounding
Determining the Amount
Determining Interest
Determining Present Worth
Determining the Compound Discount
Summary
XVII Annuities 171
Definition of Annuities
Symbols
Amount of an Annuity
Sinking Fund Contribution
Present Worth of an Annuity
Rent of an Annuity
Equal Periodical Payments on Principal and Interest
Annuities Due
To Find the Amount of an Annuity Due
Sinking Fund
Required Annual Contribution
Present Worth of an Annuity Due
Rents
XVIII Logarithms in Compound Interest and Annuity Com- putations
Calculating Compound Interest and Annuities by Logarithms To Find the Compound Interest on i To Find the Principal To Find the Rate To Find the Time
190
CONTENTS IX
Chapter Page
To Find the Present Value of i
To Find the Compound Discount on i
To Find the Amount of an Annuity
To Find the Amount of Sinking Fund Contributions
To Find the Present Worth of an Annuit}'
XIX Bond Discount and Premium 194
Bonds Purchased Below and Above Par
Discount
Scientific Method of Amortization
Income Rates
Bond Premium
Computing the Premium and the Price
First Method
Second Method
Computing the Discount and the Price
First Method
Purchases at Intermediate Date
Serial Bonds
XX Leaseholds 217
Commuted Rents
XXI Depreciation Methods 221
Annual Depreciation
Appendix A — Values of Foreign Coins 231
B — Logarithms of Numbers 233
C — Compound Interest and Other Computations . 254
Mathematics of Accounting and Finance
CHAPTER I
SHORT METHODS AND PRACTICAL SUGGESTIONS
Short Methods of Computation
Since the work of an accountant necessarily involves a great deal of computation, it is desirable that he be familiar with cer- tain labor-saving devices. Those which are presented in this chapter have been selected because of their simplicity and because an accountant has occasion to apply them so frequently as to make their use habitual. No attempt is made here to describe all the innumerable "short methods" which, however ingenious, are difficult to remember and rarely available.
Balancing an Account
In striking a balance in an account, the proper method for determining and inserting the balance is to add the larger column and enter its total in both columns; then to add the smaller column and insert each figure of the balance necessary to produce the total. The following example is given to illustrate the method.
Debits
Credits
$1,846.22
$ 126.13
2,913.68
248.71
4,327-11
1,635.48
2,319.11
4,757-58
$9,087.01 $9,087.01
The balance of $4,757.58 in this account is found and entered in the following way. First, the larger side of the account, which happens to be the debit side, is added for a total of $9,087.01 and the amount is placed on both sides; then the items on the credit
3
4 MATHEMATICS OP ACCOUNTING AND FINANCE
side are added and the figure in each digit column necessary to give the figure in the corresponding column of the total is in- serted in the place reserved for the balance.
Thus, adding the first column, 3 + i + 8 + i, gives 13, to which 8 is added to make 21 and produce the i in the total. The
8 is inserted in the balance as shown, while the 2 of 21 is carried. The second column of digits and the 2 carried forward are added to 15, and to this a 5 is added to make 20 and give o in the total. The 5 is entered in the same column of the balance and the 2 of 20 is carried. In the same way the third column is added to 30, 7 is inserted in the balance to make 37, and the 3 of 37 is carried. The fourth column is added to 13, 5 is inserted to produce 18, and I is carried. The fifth column is added to 13, 7 is inserted to make 20, and 2 is carried. The last column requires a 4 to make
9 in the total, and the entire balance is found to be $4,757.58.
Combined Addition and Subtraction
It often happens that columns of figures are given in which are included both positive and negative values, that is, plus and minus numbers. It may be necessary, for example, to compare the figures of two years to show the increase or decrease of certain items and to determine the net increase or decrease for the second year. If the figures are written in ink, the minus quantities may be inserted in red. In a newspaper or book the minus numbers are usually printed in italics or indicated by a star.
Two methods may be followed in adding a column of such figures. The first method is the ordinary way of adding the positive numbers and subtracting from their sum the sum of the negative numbers. The other method is to add the numbers continuously, the negative digits being in each case added nega- tively, that is subtracted. Thus, in adding the third column of the following table, which is a comparison of the manufacturing costs of 19 1 6 with those of 191 5, the procedure is as explained below:
SHORT METHODS AND PRACTICAL SUGGESTIONS
Account 1915
Material used $12,638.13
Direct labor 16,469.42
Indirect labor 5,827.59
Other manufacturing expense 7,962.28
Incre.\se or
I9I6
Decrease
$14,228.64
$1,590.51
19,672.18
3,202.76
4,713-92
1,113.67
8,242.37
280.09
$46,857.11
$3,959.69
^2,897•42
The net increase of $3,959.69 for all four items is found by adding the increases and decreases in the following manner. The first column of digits on the right is added thus, 1+6 = 7, 7 — 7 = o, o + 9 = 9. The second column is added thus, 5 + 7 = 12, 12 — 6 = 6. In the third column 3 is subtracted from 2 to make — I, and the i is subtracted from 10 borrowed from the next column. The addition is therefore, 2 — 3 + 10 = 9. In the fourth column the borrowed 10 is represented by — i, and the column adds thus, —1+9 — 1+8 = 15. The i of 15 is carried, as in ordinary addition, to the next column, which added is, 1 + 5 + 2 — 1 + 2 = 9. The final column adds, i + 3 —
I = 3-
In practice, items of the same value but of opposite signs may be offset at once. Thus, a glance at the cents column in the fore- going example shows that i and 6 offset — 7, and the remaining 9 can, therefore, be put down at once in the result.
Adding a Part of a Column
This manner of addition is useful when it is desired to find the sum of all but the last few numbers of a long column whose total is known. For instance, a page may have an amount on each of 40 lines, and the addition of the entire column has been verified. It is now necessary to ascertain the total of the first 35 fines, as in the illustration below, in which these first 35 lines are omitted to save space.
6 MATHEMATICS OF ACCOUNTING AND FINANCE
36th line $ 2,382.46
37th " 1,896.28
38th " 1,237.61
39th " 2,068.42
40th " 1,923.16
Total of 40 amounts on page $82,642.57
The total of the items to and including the 35th may be found by adding the last five items on a memorandum paper and deduct- ing their sum from the total of the page. A much quicker and more workmanlike procedure, however, is to subtract them by addition as explained in the previous section. Adding the first column of digits, 6 + 8 + I + 2 + 6 gives 23. As 4 is required to make the
7 in the grand total, 4 must be the last digit in the footing at the 35th hne. Carrying 2 and proceeding in the same way with the second and the other columns, the required footing is found to be $73,134.64. The entire operation involves no more writing than the insertion of the figures under the 35th line.
Deducting Several Subtrahends from One Minuend
It is sometimes necessary to subtract several amounts from one amount, as, for example, to make various deductions from total sales. If there is room enough the various items may be listed in an inside column, their total inserted under gross sales, and the difference carried out. If only one column is possible, the subtraction can be made by addition, as shown in the follow- ing table, in which the items between the lines are deductions from the total sales, and the remainder is the amount of the net sales.
Total sales $132,629.14
Less:
Cash discount $ 1,436.27
Freight 2,389.16
Breakages 948.63
Sundry allowances 769.42
Net sales $1 27,085.66
SHORT METHODS AND PRACTICAL SUGGESTIONS 7
In performing the operation it should be remembered that the digits of the remainder are always those necessary in each column to make the digit of the same column in the total.
Table with Net Decrease
At times the amounts to be deducted are greater than the amount from which they are to be deducted. The table below is an example of this. It represents a comparison between two years to show the increaseor decrease of business in each of seven depart- ments and the total net result of the whole business. The table might have been arranged in four columns, one for the figures of 1920, one for the figures of 1921 , one for the increases, and one for the decreases; or the departments might have been listed in order, with the changes appearing in one column but the decreases indicated by red ink or italics. In the table as given the depart- ments showing an increase are listed first, and those showing a decrease are listed underneath.
Dept. 1920 1 92 1 Increase or Decrease
I $168,242.19 $174,629.46 Inc. $ 6,387.27
3 192,429.36 195,716.83 " 3,287.47
4 156,283.27 160,142.76 " 3,859-49 7 84,619.43 86,223.62 " 1,604.19
Total $601,574.25 $616,712.67 Tot. Inc. $15,138.42
2 $165,328.46 $158,693.82 Dec. $ 6,634.64
5 212,642.88 204,976.38 " 7,666.50
6 198,731.54 189,218.23 " 9.513-31
Total $576,702.88 $552,888.43 Tot. Dec. $23,814.45 Grand total $1,178,277.13 $1,169,601.10 Net Dec. $8,676.03
The net decrease in the foregoing table is the sum of the de- creases less the sum of the increases. The total of the incieases is, therefore, added negatively to the total of the decreases, that is, deducted digit by digit.
8 MATHEMATICS OF ACCOUNTING AND FINANCE
Complements
The arithmetical complement of a number is the difference between that number and the next higher power of lo. Thus the complement of 7 is 3, or the difference between 10 and 7, and the complement of 81 is 19, or the difference between 100 and 81.
If instead of subtracting a number less than 10, its comple- ment is added, the result is 10 larger than the result of the sub- traction. Thus 8 subtracted from 15 gives 7; while the com- plement of 8, which is 2, added to 15, gives 17. Hence to subtract digits, in adding a column of numbers negatively, their complements may be added to the other digits and the sum reduced by ten times the number of complements added.
The following table represents an addition of positive and negative quantities:
929 248
- 1,086
- 74 3,126
- 174 306
3,275
In the unit column of this table, 9 -]- 8 -f 4 (complement of 6) + 6 (complement of 4) + 6 + 6 (complement of 4) + 6 = 45, and 45 ~" 30 = 15; 30 being subtracted because three complements have been added. The 5 of the 15 is entered in the footing and the I is carried. In the tens column i (carried) + 2 -f 4 + 2 (complement of 8) +3 (complement of 7) + 2 -f 3 (complement of 7) = 17. The 7 is entered in the footing, and since three complements were added the i is dropped and — 2 carried to the hundreds column. In the hundreds column 8 (complement of the — 2 carried) + 9 + 2-f-i + 9 (complement of i) + 3 = 32, and 32 — 20 (there being only two complements) = 12. The 2 is entered and the i is carried. In the thousands column i + 9
SHORT METHODS AND PRACTICAL SUGGESTIONS 9
(complement of i) + 3 = 13, and 13 — 10 (there being only one complement) = 3, which is entered in the footing. The result of the addition is, therefore, 3,275.
Short Methods of Multiplication
To multiply numbers ending in zeros, both of which are integers, the significant figures of the numbers are multiplied and to the product are annexed as many zeros as there are final zeros in both the multiplicand and the multiplier. Thus, 3,400 times 1,200 equals 34 times 12, or 408, with four zeros annexed, or 4,080,000.
When one of the two numbers multiplied is a decimal fraction the significant figures in the two numbers are multiplied and the decimal point in the product is moved as many places to the right as there are final zeros in the integer. This may necessitate annex- ing zeros. In the example .486X300, .486 is first multiplied by 3, to make 1.458. As there are two zeros in the multiplier, the deci- mal point in this product is moved two places to the right, making it 145.8.
To multiply by 9, 99, or any number that contains no other figure than 9, as many zeros are annexed to the multiphcand as the multipher has 9's, and from the result the multiplicand is deducted. Thus, in multiplying 293 by 99, two zeros are annexed to 293, making 29,300, and from this 293 is deducted. The result is 29,007.
Multiplying by 1 1
To multiply an amount by 1 1 it is not necessary to put down the figures, but only the result, which is secured by addition. The procedure is as follows :
Beginning with the right-hand or unit figure in the multipli- cand, add to each digit the digit next to the right of it. Put the last digit of this sum down in the result, carrying the other digit of the sum if there is one. Continue this until each figure in the
10 MATHEMATICS OF ACCOUNTING AND FINANCE
multiplicand has been used twice. The method is illustrated in the following example :
Illustration
Multiply 1,342 by 11
Solution:
1,342
Product 14,762
The first digit, 2, at the right of the multiplicand, has no figure to the right of it. Therefore nothing is added and it remains 2
The second digit, 4, is added to the first digit, 2, making 6
The third digit, 3, is added to the second digit, 4, making 7
The fourth digit, i, is added to the third digit, 3, making 4
Since the fourth digit has been used only once it is necessary to use it a second time, thus, i
The product is, therefore, 14,762
An analysis of the ordinary method of multiplication shows that the procedure is exactly the same in both cases. Looking at the example when multiplied in the usual way, it is seen that,
1342 II
1342 1342
[4.762
As in the above explanation, the product is formed by putting down 2 alone, then adding 4 to 2, 3 to 4. i to 3 and repeating the i. It is, however, unnecessary to write out these figures, as the whole operation can be performed mentally and the product put down at once.
SHORT METHODS AND PRACTICAL SUGGESTIONS 1 1
Multiplying by 1 1 1
To multiply by 1 1 1 the same procedure is followed as when the multiplier is ii, except that each figure in the multiplicand is added to the two figures to the right, and is, therefore, used three times, as shown in the following illustration:
Illustration
Multiply 1,342 by 11 1
Solution:
1,342
Product 148,962
The first digit in the multiplicand is put down alone, thus, 2
The second digit, 4, is added to the first digit, 2, making 6
The third digit, 3, is added to the second digit, 4, and the
first digit, 2, making 9
The fourth digit, i, is added to the third digit, 3, and the
second digit, 4, making 8
The fourth digit, i, is added to the third digit, 3, making 4
The fourth digit, i, is put down alone, thus, i
148,962
If the same method is applied in multiplying 1,796 by in, it will be found necessary to carry over i, 2 and i, in adding the digits.
In multiplying by any number that consists of a series of i 's each digit in the multiplicand is used as many times in the way explained above as there are digits in the multiplier; twice when the multipher is 1 1 , three times when it is in, four times when it is 1,1 1 1, and so on.
Tabulating Multiples of a Multiplier
When a number containing several digits is to be used re- peatedly as a multiplier, it saves time and promotes accuracy to make a table of its multiples. The best way of compiling the
12 MATHEMATICS OF ACCOUNTING AND FINANCE
table is to determine each multiple by adding the number to the preceding multiple. The addition is, of course, made on the table itself and not on a separate sheet of paper. The table is carried out to ten multiples in order to prove the work, as the tenth line must be the same as the first with one zero annexed. This is proof only in case the multiples are built up by addition.
A table of this sort constructed for 683,947 as a multiplier is the following:
ULTIPLIER
Multiple
I
683,947
2
1,367,894
3
2,051,841
4
2,735,788
5
3,419,73s
6
4,103,682
7
4,787,629
8
5,471,576
9
6,155,523
10
6,839,470
Supposing it is desired to multiply 683,947 by 3,469, the solu- tion is as follows :
683,947 multiplied by 9 = 6155523
" 6 = 4103682
" " 4 = 2735788
" 3 = 2051841
Product 237261 2143
Tabulating Multiples of a Divisor
When a number containing several figures is to be used re- peatedly as a divisor, a table of multiples of the divisor may be prepared, to show at a glance how many times the divisor is contained in successive remainders and do away with the necessity of performing the multiplication. This method may also save considerable labor by reason of the fact that instead of writing
SHORT METHODS AND PRACTICAL SUGGESTIONS 1 3
down the multiples to make each subtraction, they may be written on a card and the particular multiple to be deducted placed alongside the amount from which it is to be subtracted. The subtraction is made from the card across to the amount, and only the remainder is set down.
The directions for the use of this method of division may be given as follows :
Placing the card at the left of the dividend, find by inspection which multiple is to be used; sHde the card until that multiple is on a line with the dividend; deduct the multiple from the corre- sponding first figures of the dividend, putting the remainder under the dividend figures; then bring down the next figure and proceed in the same way with the new amount to be divided. The follow- ing example, in which 872,976,654 is divided by 683,947, illus- trates this method.
Card
OF Multiples
Dividend
Quotient
(as slid for each division)
I
683,947
872976654
I
2
1,367,894
1890296
2
7
4,787,629
5224025
7
6
4,103,682
4363964
6
3
2,051,841
2602820
3
8
5,471,576
5509790 38214
8
Quotient is 1,276.38, and the remainder 382.14.
Division by Use of Reciprocals
Two numbers are reciprocal when their product is i. For instance, 2 and .5 are reciprocal numbers, because 2 X .5 = i. It is evident from this that the reciprocal of a number is found by dividing i by the number. The quotient obtained by dividing by a divisor is the same as the product obtained by multiplying by the reciprocal of the divisor. For example, 12 -^ 2 and 12 X -5 are each equal to 6.
14 MATHEMATICS OF ACCOUNTING AND FINANCE
Repeated division by the same divisor can be made easier, particularly if an adding machine is available, by determining the reciprocal of the number, tabulating the multiples of the recipro- cal, and proceeding as in multiplication. In dividing, for example, 872,976,654 by 683,947, the first step is to divide i by the divisor in order to obtain its reciprocal, which is .0000014621. The multiples of the reciprocal are then tabulated, as in the table below. The zeros in the reciprocal are not repeated in the table, as it is only necessary to remember the number of places to be pointed off in the product, ten in the present case. Having the multipHers, the procedure thenceforth is precisely as in multipli- cation, as shown in the second of the subjoined tables.
ULTiPLEs OF Reciprocal
I
14,621
14621
2
29,242
872976654
3
43,863
58484
4
58,484
73105
5
73,105
87726
6
87,726
87726
7
102,347
102347
8
116,968
131589
9
131,589
29242
10
146,210
102347 1 16968
12763791658134
Pointing off ten decimals, the result is 1,276.38
Since the reciprocal is approximate only, the last figures of the product must be ignored, as they do not show the exact remainder.
The work may be shortened considerably by beginning at the left and ignoring all multiples and portions thereof beyond the first, second, or third place to the right of the last place desired in the product, as follows:
SHORT METHODS AND PRACTICAL SUGGESTIONS 15
14621 872976654
8
I 16968
7
102347
2
29242
9
131589
7
10235
6
877
6
88
5
7
4
I
127637917 or 1,276.38
CHAPTER II
FRACTIONS AND PROOF FIGURES
Addition and Subtraction of Fractions
Fractions having the same numerator may be added by multiplying the sum of the denominators by the common numera- tor to obtain the numerator of the result, and by multiplying together the denominators of the fractions to obtain the de- nominator of the result which is then reduced to its simplest form. To add 1/9 and 1/7 by this method the procedure is as follows:
9 + 7 = 16; 16 X I = 16, the numerator of sum 9X7 =63, the denominator of stun
Sum = 16/63
The solution of 3/1 1 + 3/16 is as follows:
II -|- 16 = 27; 27 X 3 = 81, the numerator of sum II X 16 = 176, the denominator of sum
Sum = 81/176
To subtract fractions when the numerators are the same, the difference between the denominators is first multiplied by the common numerator. This gives the numerator of the result. Then the denominators are multipHed to obtain the denominator of the result, which is reduced to its simplest terms. This method is illustrated in the solution of the following subtraction, 3/1 1 — 3/16:
16— 11=515X3= 15, the numerator of the difference 16 X II =176, the denominator of the difference
Difference = 15/176
Cross Multiplication
Two fractions may be added by the method of cross multipli- cation. The numerator of each is multiplied by the denominator
16
FRACTIONS AND PROOF FIGURES l^
of the other, and the two products are added to form the numera- tor of the result. The denominators of the fractions are multi- pHed to form the denominator of the result, which is then reduced to its simplest terms. This method is illustrated in the following example, in which 5/8 and 7/9 are added.
5X9= 45 7X 8= j6
loi, numerator of the sum 8X9= 72, denominator of the sum
Sum = 101/72 or I 29/72
Cross multiplication may also be used in subtracting fractions. The numerator of each fraction is multiplied by the denominator of the other, and the smaller product is subtracted from the larger to form the numerator of the result. The two denominators are multiplied to form the denominator of the result. Reduction to the simplest form then follows. In the following example 5/8 is subtracted from 7/9 by this method:
7 X 8 = 56 5 X 9 = 45
II, numerator of the difference 9X8= 72, denominator of the difference Difference = 11/72
The approximate product of two mixed numbers may be found by multiplying first the integers, then each integer by the other fraction to the nearest unit, and finally by adding the three products. The following illustration in which 1 7 1/5 is multiphed by 14 1/3 shows the application of the method :
14 X 17 = 238
1/5 of 14 = 3 (nearest unit) i/3 0fi7 = __6
Approximate product 247 The exact product of this multiplication is 246 8/15.
If the multiplication of the fractions is carried out to at least one decimal place, the result is more exact, as the following shows:
i8
MATHEMATICS OF ACCOUNTING AND FINANCE
14X 17 = 238
1/5 of 14 = 2A 1/3 of 17= 5.;
Result 246.5
Equivalent Common and Decimal Fractions
The following is a list of the most frequently used decimal fractions and their equivalent common fractions.
Equivalent
Decimal
Common
Fractions
Fractions
5
y'^
■33 1/3
1/3
662/3
2/3
25
1/4
75
3/4
16 2/3
1/6
^3^/3
5/6
125
1/8
375
3/8
625
5/8
87s
7/8
081/3
1/12
41 2/3
5/12
581/3
7/12
91 2/3
11/ 1 2
0625
1/16
1875
3/16
3125
5/16
4375
7/16
5625
9/16
6875
11/16
8x25
13/16
9375
15/16
These figures can be used in a variety of ways. One of the most obvious is to substitute the common fraction for its equiva- lent decimal, as it is usually easier to multiply or divide by the
FRACTIONS AND PROOF FIGURES I9
common fraction than by the decimal. Thus, if it is desired to find 12 1/2% of a number, it is far easier to divide the number by 8 than to multiply it by the decimal .125.
This substitution of a common fraction for the equivalent decimal is simple enough when the numerator of the common fraction is i. When, however, the numerator is greater than i, it is necessary to split up the decimal into such parts as will per- mit the substitution of equivalent common fractions with a nu- merator of I . The number taken is multiplied by each of these parts and the products are added to find the result. The fewer the parts into which the decimal is split, the simpler is the opera- tion. The appHcation of the rule is illustrated in the following problem.
Example
How much will 3,648 yards of cloth cost at 43 3/4 cents per yard? Solution: The answer can be found in three ways:
1. By multiplying 3,648 by $.4375.
2. By substituting for $.4375 its equivalent, 7/16 of $1, and mul-
tiplying 3,648 by the latter.
3. By spHtting up the common fraction of 7/16 into 1/4, 1/8, and
1/16, or the decimal of .4375 into .25, .12 1/2, and .06 1/4, and substituting the equivalent fractions, and then multiplying 3,648 by each of these fractions and adding the products, as shown below:
At $1.00 per yard the price would be $3,648.00
At 1/4 or .25 per yard the price would be g 12.00
At 1/8 or .12 1/2 (1/2 of above) 456.00
At 1/16 or .06 1/4 " " " 228.00
At 7/16 or .43 3/4 the price is $1,596.00
The operation can be performed much more quickly if it is recognized that 7/16 is the same as 8/16 minus 1/16, that is, 1/2 minus 1/8 of 1/2, and the calculation is made as shown below:
20
MATHEMATICS OF ACCOUNTING AND FINANCE
1/2 of $3,648 = $1,824.00 1/8 of 1,824= 228.00
7/16 of $3,648 = $1,596.00
It is always advisable to prove all computations when it does not involve too much work. This is easily done in most cases where a fraction can be split up. Thus, in the above example it has been found that 7/16 of $3,648 is $1,596. If the figure for 1/16 is added to instead of subtracted from that for 8/16 the result is 9/16, and the calculation of $1,596 can be proved as follows :
8/16 minus 1/16 is 7/16 or 8/16 plus 1/16 is 9/16 or
$1,596.00
2,052.00
16/16 or original amount $3,648.00
Proof Figures
There are two systems of proof figures for testing the accuracy of mathematical computations. One is based on the casting out of 9's and the other on the casting out of ii's. Many bookkeepers use one or the other system to check all their work. Both systems are used in the illustration below to check an addition, the figure at the right of each amount being the excess over the 9's or ii's in the amount, or the remainder left after the digits are added and the largest multiple of 9 or 11 is subtracted.
365,592
286,548
64,320
94,094
810,554
Excess
OVER 9's
3 6 6 8
_ Total 23, or an 5 excess of 5
Excess
OVER ii's
365,592
7
286,548
9
64,320
3
94,094
° Total 19, or an
810,554
8 excess of 8
To cast out 9's from a number, the digits are added ignoring 9's, zeros and combinations of digits which add to 9. From the
FRACTIONS AND PROOF FIGURES 21
sum thus obtained the largest multiple of 9 contained therein is subtracted. Thus, the digits in the first number in the above example are added as follows, beginning at left: Ignore the 3 and 6 because they add to 9; 5 + 5 = 10; ignore the 9; 10 + 2 = 12. 12 — 9 = 3, th^ excess.
To cast out 1 1 's begin with the first figure at the right and add to it the third, fifth and so on, and take the excess of their sum over ii's. Then add together the second, fourth, sixth figures, etc., and find the excess of their sum over i I's. Subtract this excess from the first excess and the result is the check num- ber. If the first excess is smaller than the second, add 11 to it before subtracting. Thus, in the first number of the foregoing example, 2 + 5 + 6 = 13, or an excess of 2 over 11. Similarly, 9 + 5 + 3 = 17, or an excess of 6 over 11. Adding 11 to the 2 and subtracting 6 gives 7 as the check number.
Addition is proved if the excess over 9's or ii's in the sum of the individual check numbers is the same as the check number of the total of the numbers added, as shown in the preceding example. This is not an absolute proof, as an error of 9 or 1 1 or multiples thereof may be made.
Multiplication is proved by multiplying the check figure of the multipHcand by that of the multiplier. The excess over 9's or ii's in the result should be the check figure of the product, as is seen from the following:
Excess Excess
Numbers over 9's overii's
4,621 4 I
3,274 7 7
15,129,154 28 or I 7
CHAPTER III ARITHMETICAL PROGRESSION
Elements in an Arithmetical Progression
An arithmetical progression is a series of numbers increasing or decreasing by a common difference. The numbers in the series are called the terms; the first and last terms are called the extremes, and the intermediate terms the means. An in- creasing or ascending series is formed by adding the common difference to each preceding term. For example, 7, 12, 17, 22, 27, is an ascending series with a common difference of 5. A decreasing or descending series is formed by subtracting the com- mon difference from each preceding term. Thus, 26, 23, 20, 17, 14, is a descending series with a common difference of 3.
There are five elements in an arithmetical progression, which in the formulas to be presently derived are represented by the following symbols :
First term /
Last term /
Common difference d
Number of terms n
Sum of series 5
When any three of these elements are known, the other two can be computed.
Analysis of Simple Progression
The following is an example of a short progression:
ist term 3
2nd " 6
3d " 9
4th " 12
Sth " 15
6th or last term 18
r 22
ARITHMETICAL PROGRESSION 23
The common difference here is 3, the number of terms is 6, and the sum of the terms is 63.
It will be seen that, although the number of terms is 6, the common difference is added only five times. This explains why in Cases I and 2 considered below, i is subtracted, and why in Case 3, I is added. It will also be seen that the series consists of a number of pairs, as follows:
ist and last terms make a pair the sum of which is 21 2nd " 5th " " " " " " " " " 21 3d " 4th " " " " " " " " " 21
The sum of series is 63
It is evident from this that the sum of a series is the sum of the first and last terms multiplied by half of the number of terms. It is also evident that since the sum of each pair of terms is 21, the average single term is 10^, which multiplied by 6, or the number of terms, gives 63, or the sum of the series.
If the number of terms is uneven, the number of pairs includes a half pair. Thus, if the series given above is extended to seven terms, the last term is 21, and the sum of the terms is 84, which is 3>2 times 24, the sum of the first and last terms.
In the following four cases the derivation of the formulas for computing the various elements of an arithmetical progression is explained.
Case I. Given the first term, common difference and the num- ber of terms, to find the last term.
If the series is ascending, the common difference must be added as many times, less one, as there are terms in the series. Hence the formula is:
/ = /+(«- i)^
Example
The first term is 9, the common difference is 3, and the number of terms is 5. Find the last term.
24 MATHEMATICS OF ACCOUNTING AND FINANCE
Solution:
/= 9+(5- 1)3 = 9+ 12 = 21
If the series is descending, the common difference must be deducted as many times, less one, as there are terms in the series. Hence the formula is :
/ = /- („- i)d
Example
The first term is 21, the common difference is 3, and the number of terms is 5. Find the last term.
Solution:
/= 21 - (5 - i) 3 = 21—12 = 9
Case 2. Given the extremes and the number of terms, to find the common difference.
The number of common differences is one less than the num- ber of terms; and the sum of the common differences is the differ- ence between the extremes. Hence in an ascending series the formula for finding the common difference is :
.= ^
w — I
and in a descending series it is :
Example
The first term is 9, the last term is 21, and the number of terms is 5. Find the common difference.
ARITHMETICAL PROGRESSION 25
Solution:
21 — Q d=
s- I 12
4 = 3
Ca^e J. Given the extremes and the common difference, to find the number of the terms.
The difference between the extremes is the sum of the common differences, and the number of the common differences is one less than the number of terms. Hence in an ascending series the formula for calculating the number of terms is:
l-f
d and in a descending series it is :
f-l
+ I
+ I
Example
The first term is 9, the last term is 21, and the common difference is 3. Find the number of terms:
Solution:
21 — 9
n =
12 = -+ I 3
= 4+1
= 5
Case 4. If the extremes and the number of terms are given, the sum of the terms is found by the following formula :
s = X n
26 MATHEMATICS OP ACCOUNTING AND FINANCE
or by the following:
5= (/+/)X- 2
Example
The extremes are g and 21, and the number of terms is 5. Find the sum of the terms.
Solution:
9+21 5
X 5 = 75 or 5 = (9 + 21) X - = 75
2 2
Computing Total Simple Interest
The most important applications of arithmetical progression with which accountants are concerned fall under Case 4. They are made in computing the total simple interest on a principal which constantly increases or decreases from period to period by a common difference.
For example, a $1 ,000,000 bond issue bearing interest at 5% is to be repaid in forty equal annual instalments, and it is necessary to compute the total interest to be paid during that period. Since 1/40 of the loan is to be paid annually, the principal during the last year will be 1/40 of $1,000,000 or $25,000. The interest for the first year will be 5% of $1,000,000, or $50,000, which is the first term of an arithmetical progression containing forty terms; the interest for the last year will be 5% of $25,000, or $1,250, which is the last term of the progression. Since
2
the total interest, which is s, will be
$50,000 + $1,250
X 40, or $1,025,000
ARITHMETICAL PROGRESSION 27
C. P. A. Problem
The application of the principle of arithmetical progression to the computation of interest on a given principal for an increas- ing or decreasing series of time periods, may be illustrated by a problem similar to one given in an Illinois C. P. A. examination. The problem is as follows :
Problem
Upon the death of a retired business man in June, 1910, a will is found conveying real and personal property aggregating $300,000 to the widow, who is his second wife, for her life, and upon her death to four children in equal shares. It is discovered after his death that his first wife had left to her two children, Henry and Emma, $20,000, consisting of securities for $10,000 bearing 6% interest, and uninvested cash of $10,000. The father had regularly collected the semi-annual interest on the investment, but there was no evidence as to his disposition of the cash portion of the be- quest. Exactly ten years elapsed between the death of his first wife and his own death, so that he had collected twenty items of interest, the last one just before he died. Henry and Emma were of age at the time of their father's death, and had never been informed of their legacy.
Prepare a statement showing what would accrue to each of the four children at the death of the widow, who died immediately after her hus- band, including the amounts to which Henry and Emma would be entitled on account of their mother's estate. Exclude and do not consider any accrued income of the estate unexpended.
In Illinois the legal rate of interest on undisclosed debts is 5%.
Solution: The proceeds of the property that belonged to his first wife constitute a trust fund belonging to the two children, Henry and Emma; the total of this fund on June i, 1900, comprises:
Open account for cash collected by decedent, June,
1890 $10,000.00
Interest thereon at 5% (legal rate on undisclosed debt)
for ten years 5,000.00
Principal of securities 10,000.00
Interest collected on securities 6,000.00
Each collected coupon increased the undisclosed debt of the father as guardian. The first coupon was collected
28 MATHEMATICS OF ACCOUNTING AND FINANCE
91/2 years or 19 half years before the father's death. The twentieth coupon had just been collected. The computa- tion of the interest on these coupons can be accomplished thus:
First $300 earns interest at 5% for 9 1/2 years. .$142.50 Last $300 " " " " " o " . . o
Sum of extremes $142.50
Number of terms (coupons) 20
Applying the formula:
s = X n
2
or
5= (/+/)X - 2
The total interest = 142.50 X 10 or 1,425.00
Total due Henry and Emma from mother's estate. . . $32,425.00
The division of the estate would be as follows:
Total real and personal property $300,000.00
Amount due Henry and Emma from mother's
estate 32,425.00
Balance divided among four children $267,575.00
Of which one-fourth is $ 66,893.75
The allowance of $1,425 for interest on the coupons collected must not be confused with compound interest. The item of $1,425 is interest on actual cash received by the father, for which he did not account. How he came into possession of the cash is immaterial.
CHAPTER IV AVERAGE
Utility of Average
The principle of average may be used for determining proba- bilities, for comparing numbers with a standard or with each other, or for the purpose of simplifying calculations by using an average instead of a number of related values. Where it is pos- sible to collate statistics covering extensive and varied observa- tions, it is possible to determine an average which may be assumed, from the law of averages, to be standard. Reliance can then be placed on the probability that other similar cases, though individually at great variance from the average, will in the aggregate closely approximate the average. Mortality tables, for instance, are averages determined by exhaustive investigation. While they do not determine probabiHties for individuals, they do determine probabilities for large groups of individuals.
When the principle of average is utilized, the basis of com- parison may be a simple average, a moving average, a progressive average, a periodic average, or a weighted average as the case may be. The choice of a base depends on the information desired and, in the case of weighted average, on the necessity imposed by the facts themselves.
Simple Average
The process of determining a simple average consists merely of adding the units to be averaged and dividing the sum by the number of units. If the sum of the units is known, the process requires division only.
29
30 MATHEMATICS OF ACCOUNTING AND FINANCE
Suppose the daily sales for a week are as follows :
Monday. $3,560.35; Tuesday, $3,115.95; Wednesday, $2,946.86; Thursday, $2,868.79; Friday, $3,269.87; Saturday, $3,896.43. Total sales, $19,658.25.
The daily average of the sales for the week is found by divid- ing the total of $19,658.25 by the number of days taken, or 6, which gives a result of $3,276.37^.
It must be remembered in figuring average that the divisor is the number of units in the dividend, and not necessarily the num- ber of items added to obtain the dividend. For example, suppose it is desired to determine the average contribution made by a number of persons to a benevolent fund, where
2 men each contributed '
I5OO,
totaling :
$1,000.00
4 "
a (I
250.
"
1,000.00
10 "
u u
100,
"
1,000.00
16 "
a li
50,
u
800.00
18 "
U ((
25>
"
450.00
so " " " a total of $4,250.00
The number of items added here to find the dividend of $4,250 is only 5, but the number of units to be used as the divisor is 50, making the average contribution equal to $4,250 divided by 50 or
$85.
Moving Average
When it is desired to compare a series of numbers relating to units of time of uniform duration and of uninterrupted sequence, a moving average of a number of these units may be used as a basis of comparison. This kind of average is determined by tak- ing a simple average of the numbers to serve as the starting point ^or the moving series, and dropping after the lapse of each time unit the first number of the series and adding the number of the next time unit in order to form a new series and obtain a new average.
AVERAGE 31
Assume, for example, the following conditions :
1. The values averaged are monthly sales approximated to
the nearest $1,000.
2. The time units are months.
3. The number of units is 12, making up one year.
4. The first month of each series is dropped, as a new month
is included in the next series.
In an example of this sort each month's sales can be compared with any one of the twelve averages in the calculation of which the month's sales are included. For instance, the sales of Decem- ber, 1914, can be compared with the twelve monthly averages for the years ending with each month from December, 1914, and to November, 191 5, inclusive.
Some of the possible comparisons in which the moving aver- age may be used are illustrated in the table below. Their value depends on the nature of the business whose figures are used. The most significant fact shown in the illustration is the almost uninterrupted increase in the moving average, although the monthly sales show wide variation due to the seasonal nature of the business. The decline in the moving average during the last months of 19 14 and the first months of 191 5, shows the effects of the war.
Following is an explanation of the methods by which the figures in the various columns are determined :
Moving average:
Total sales for 1 2 months of 1913 335
Moving average including December, 1913, 335 -r- 12... 27. q
335 — 20 (January, 1913) + 53 (January, 1914) 368
Moving average including January, 1914, 368 -^ 12 ... . 30.66
Increase or decrease* in moving average — i month:
Moving average including January, 1914 30.7
" " " December, 1913 27.9
Increase 2.8
32 MATHEMATICS OF ACCOUNTING AND FINANCE
More or less* than annual average:
December, 1913, sales 27
Moving average including December 27.9
Decrease .9*
January, 1914, sales 53
Moving average including January 30.7
Increase 22.3
Increase in moving average — i year:
Moving average including December, 1914 35.4
" " " December, 1913 27.9
Increase 7.5
Increase in moving average — 2 years:
Moving average including January, 1916 48
" " " January, 1 914 30.7
Increase 1 7.3
Moving Average
Sales
Moving
I9I3
(000 omitted)
average
Jan.
20
Feb.
17
Mar.
21
Apr.
17
May
26
June
40
July
28
Aug.
21
Sept.
36
Oct.
55
Nov.
27
Dec.
27
27.9
More or less* than
annual average
AVERAGE
33
Moving Average — Continued
I9I4
Sales (000 omitted)
Moving average
Increase or decrease* in
moving average — I month
More or
less* than
annual
average
Increase in mov- ing aver- age— I year
Jan.
S3
30.7
2.8
22.3
Feb.
47
33-2
2-5
13-8
Mar.
47
35-3
2.1
II. 7
Apr.
45
37-7
2.4
7-3
May
?>2
38.2
•5
6.2*
June
3,2
37-5
•7*
5.5*
July
24
37-2
■3*
13.2*
Aug.
17
36.8
.4*
19.8*
Sept.
39
37-1
•3
1.9
Oct.
42
36.0
I.I*
6.0
Nov.
20
35-4
.6*
IS.4*
Dec.
27
35-4
.0
8.4*
7-S
Increase
I9IS
Sales (000 omited)
Moving average
Increase or decrease* in
moving average — I month
More or
less* than
aimual
average
or
decrease*
in moving
average —
I year
Jan.
45
34-8
.6*
10.2
4.1
Feb.
43
34-4
.4*
8.6
1.2
Mar.
39
33.8
.6*
5-2
i-S*
Apr.
38
33-2
.6*
4.8
4.5*
May
32
33-2
.0
1.2*
5-0*
June
25
32.6
.6*
7.6*
4.9*
July
27
32.8
.2*
5.8*
4.4*
Aug.
32
34-1
1-3
2.1*
2.7*
Sept.
49
34-9
.8
14.1
2.2*
Oct.
85
38.5
3-6
46.5
2-5
Nov.
67
42.4
3-9
24.6
7.0
Dec.
47
44.1
1-7
2.9
8.7
34
MATHEMATICS OF ACCOUNTING AND FINANCE
Moving Average — Continued
I9I6
Sales (000 omitted)
Moving average
Increase or decrease* in
moving average— I month
More or
less* than
annual
average
Increase
in
moving
average
— I year
Increase
in
moving
average
—2 years
Jan.
92
48.0
3-9
44.0
13.2
^7-3
Feb.
97
52.5
4
5
44-5
18.I
19-3
Mar.
83
56.2
3
7
26.8
22.4
20.9
Apr.
44
56.7
5
12.7*
23-5
19.0
May
33
56.8
I
23.8*
23.6
18.6
June
42
58.2
I
4
16.2*
25.6
20.7
July
38
59-1
9
21. I*
26.3
21.9
Aug.
55
61.0
I
9
6.0*
26.9
24.2
Sept.
80
63.6
2
6
16.4
28.7
26.5
Oct.
97
64.6
I
0
32.4
26.1
28.6
Nov.
84
66.0
I
4
18.0
23.6
30.6
Dec.
S8
66.9
9
8.9*
22.8
3^-5
Progressive Average
Progressive average is cumulative, a new unit being added to form each successive dividend, and the divisor being constantly increased in an arithmetical progression of i . In the table given below the figures used are those of the preceding illustration. The first average is that of the first two months' sales; the second, of the first three months' sales, etc.
The table shows the continuous growth of the business; but the differences between the successive progressive averages are not so significant as the differences between the successive mov- ing averages, because the former are borne down by the smaller sales of the first months and years. Moreover, an increase of sales in an early month increases the progressive average for that month to a greater extent than the same increase in sales in a later month will increase the progessive average of that month, because the total sales in the latter case are divided by a larger number of months. The last column of the table indicates the months in which the sales run above or below the average. It
AVERAGE
35
will be noted that the current month is not included in the aver- age used as a base. If it were included, any increase or decrease in the month's sales would affect the standard as well as the month compared.
Following is an explanation of the methods of determining the figures appearing in the various columns :
Progressive average:
February line (20 + 1 7) -J- 2 18.5
March line (20 + 17 + 21) -r- 3 1Q.3
Increase or decrease"^ in progressive average:
Average for first three months 19.3
" " " two months 18.5
Increase .8
More or less* than progressive average:
Sales of March, 1913 21
Progressive average for prior months 18.5
Increase 2.5
Progressive Average
Increase or
More or
DECREASE* IN
LESS* THAN
Progressive
PROGRESSIVE
PROGRESSIVE
I9I3
Sales
AVERAGE
AVERAGE
AVERAGE
January
20
February
17
18.5
March
21
19-3
.8
2-5
April
17
18.8
.5*
2.3*
May
26
20.2
1.4
7.2
June
40
23-5
3-3
19.8
July
28
24.1
.6
4-5
August
21
23-8
•3*
3-1*
September
36
25.1
1-3
12.2
October
55
28.1
30
29.9
November
27
28.0
.1*
I.I*
December
27
27.9
.1*
I.O*
36
MATHEMATICS OF ACCOUNTING AND FINANCE
Progressive Average — Continued
Increase or
More or
decrease* in
less* than
Progressive
progressive
progressive
I9I4
Sales
average
average
average
January
S3
29.8
1.9
25-1
February
47
3I-I
1-3
17.2
March
47
32.1
I.O
15-9
April
45
32.9
.8
12.9
May
32
32.9
.0
.9*
June
32
32.8
.1*
.9*
July
24
32.4
.4*
8.4*
August
17
31.6
.8*
15-4*
September
39
32.0
•4
7-4
October
42
32.4
•4
10. 0
November
20
319
•5*
12.4*
December
27
31-7
.2*
4.9*
1915
January
45
32-2
•S
133
February
43
32.7
•s
10.8
March
39
32.9
.2
6.3
April
38
33.0
.1
5-1
May
32
33-0
.0
I.O*
June
25
32.7
.3*
8.0*
July
27
32.5
.2*
S-7*
August
32
32.S
.0
•S*
September
49
330
•5
16.S
October
85
34-6
1.6
52.0
November
67
35-5
•9
32.4
December
47
35.8
•3
II-5
1916
January
92
37-3
1-5
56.2
February
97
38.9
1.6
59-7
March
83
40.0
I.I
44.1
April
44
40.1
.1
4.0
May
33
40.0
.1*
7.1*
June
42
40.0
.0
2.0
July
38
40.0
.0
2.0*
AVERAGE
37
Progressive Average — Continued
Increase or
More or
DECREASE* IN
LESS* THAN
Progressive
PROGRESSIVE
PROGRESSIVE
19 1 6 — Cont.
Sales
average
AVERAGE
AVERAGE
August
55
40-3
•3
15.0
September
80
41.2
•9
39-7
October
97
42.4
1.2
SS-8
November
84
43-3
•9
41.6
December
58
43-6
•3
14.7
Periodic Average
In order to show the variation in the volume of business be- tween seasons, periodic average may be utilized, as in the illus- tration below, in which a simple average is taken of the figures for the same month in the years 1913-1916 inclusive:
Month
1913
1914
1915
1916
Total
Average
January
20
53
45
92
210
52.5
February
17
47
43
97
204
51.0
March
21
47
39
83
190
47-5
April
17
45
38
44
144
36.0
May
26
32
32
33
123
30.7s
June
40
32
25
42
139
34.75
July
28
24
27
38
117
29.25
August
21
17
32
55
125
31-25
September
36
39
49
80
204
5I-0
October
55
42
85
97
279
69-75
November
27
20
67
84
198
49-5°
December
27
27
47
58
159
39-75
The principle of progressive average may be utilized in con- nection with the principle of periodic average, as indicated in the following table :
i^EAR
January
Progressive average
More or
LESS* than preceding PROGRESSIVE AVERAGE
I913
20
I914
53
36.5
I915
45
39-3
8.5
I916
92
52.5
52.7
38 MATHEMATICS OF ACCOUNTING AND FINANCE
Each progressive average in the foregoing illustration is also a periodic average. This organization of the numbers and aver- ages makes possible a comparison of the sales of each month with the average sales of the same month in all preceding years. The figures in the final column show the difference between the Janu- ary sales of the year and the progressive average of the January sales of the preceding years. Thus the figure 45 for 191 5 is 8.5 greater than the 36.5 progressive average for January, 191 3, and 1914.
Weighted Average
When the values entering into the computation of an average differ in two or more particulars, a simple average is impossible. Take the following case, for example :
3 men earn $5.00 per day S " " 6.00 " "
4 " " 7-00 " "
In computing the average daily wage of these twelve men, the fact must be recognized that the wage payments differ in two particulars:
1 . The daily wage
2. The number of men receiving each wage
The daily average is found by dividing the aggregate of their daily wages by their number, as shown below :
Men
Wage
Product
3
$5
$15
5
6
30
4
7
28
12 $73
$73 -^ 12 = $6Vi2, the average daily wage
This example serves to illustrate the principle that each value must be weighted by multiplying it by the number of units to
AVERAGE $9
which the value is applicable. Additional illustrations will make the principle clearer.
Illustration i
What is the average rate of interest earned on the following investments made for one year?
$100 at
7%
$5,000 at
6%
$15,000 at
5%
Solution:
Principal
Rate
Product
$ lOO
7%
$ 7.00
5,000
6%
300.00
15,000
5%
750.00
$20,100
$1,057.00
i)057 ^ 20,100 = 5.258%, the average rate of interest
Illustration 2
What is the average life of a plant, the various fixed assets of which have the following costs and estimated lives?
(
Class of
Asset
Cost
Life
A
$ 3,000.00
5 years
B
15,000.00
10 "
C
35.000.00
20 "
Solution:
Depreclation
Annual
Class
Cost
Life Rate
D
EPRECIATIC
A
$ 3,000.00
5 years 20%
$ 600.00
B
15,000
.00
10 " 10%
1,500.00
C
35,000,
.00
-^0 •' 5%
1,750.00
$53,000.
00
$3,850.00
53,000-^ 3.850= 13.76+ ^o the average life
40
MATHEMATICS OF ACCOUNTING AND FINANCE
Illustration 3
What is the average per cent of increase in the cost of manufacture under the following conditions: Where one pound of each item is used in the manufacture of each completed article, and the cost of each iiem has increased by the per cent shown in the last column of the first of the sub- joined tables?
Material Cost per Pound Per cent Increase
1917
918
A
$4
200
B
5
10
C
8
6X
D
IS
10
E
18
20
Solution i :
No.
Cost per
Per cent
Material
Pounds
Pound
Increase I
NCREASE
.1917
1918
1918
A
$ 4
200
$8.00
B
5
10
•50
C
8
ty^
•50
D
IS
10
1.50
E
18
20
3.60
$14.10
14.10 -^ 50 = 28.2%, average increase
Solution 2:
Cost per
Per cent
Per cent
Weighted
Material
Pound
OF
Total
Increase
Per cent
1917
1918
1918
A
$4
8
200
16.0
B
S
10
10
I.O
C
8
16
6K
I.O
D
IS
30
10
30
E
18
36
20
7.2
$50
28.2
AVERAGE
41
Illustration 4
The same conditions are taken here as in the preceding illustration, except that the items composing the finished article are each of a different weight, as indicated in the second column of the following two solutions:
Solution
i:
No.
%
Cost per
Weighted
%
Weighted
Material
Pounds
Total
Pound
Product
Increase
%
1917
1917
1918
1918
A
2
10
$ 4
$ 40
200
80
B
8
40
5
200
10
20
C
4
20
8
160
(^Va
10
D
4
20
15
300
10
30
E
2
10
iS
180
20
36
20 100 fooo
176 -^ 880 = 20%, weighted average percent
176
Solution 2:
No.
Cost per
Total
%
Material
Pounds
Pound
Cost
Increase
Increase
1917
1917
1918
1918
A
2
$ 4
$ 8
200
$16.00
B
8
5
40
10
4.00
C
4
8
^2
6>^
2.00
D
4
15
60
10
6.00
E
2
18
36
20
7.20
$176
$35-20
35.20 -j- 176 = 20%, weighted average per cent
CHAPTER V AVERAGING ACCOUNTS
Settling an Account
The object of averaging an account is to determine a single date, known as the average date, on which the account may be settled with fairness to both debtor and creditor.
As a simple illustration of how an account is averaged, sup- pose B's account with A, to whom he is indebted, is as follows:
March i — 60 days $1,000
March 31 — 60 days 1,000
The first item of this account is due on April 30 and the second on May 30. B may pay each amount at its maturity, or the entire $2,000 at the average maturity, which is May 15. He can make an equitable settlement on this average date because the time he gains in deferring payment of the first $1,000 for fifteen days is exactly offset by the time he loses in paying the second $1,000 the same number of days in advance of the due date.
It may be desirable to compute the average date for the purpose of dating or determining the maturity of a note or other document given in settlement of an account. For instance, B might cover his account by giving A a single non-interest bearing note for $2,000, due May 15, instead of two non-interest bearing notes of $1,000 each, one due April 30, and the other due May 30.
Calculating Interest
The average date may also be desired for the purpose of com- puting interest on the balance instead of the individual items of an account. For instance, if A should settle the account on June 14 by a single payment of $2,000, equity would require that he pay interest, say, at 6% as follows:
42
AVERAGING ACCOUNTS 43
On $i,ooo from April 30 to June 14 — 45 days $ 7.50
On $1,000 from May 30 to June 14 — 15 days 2.50
Total $10.00
The interest might, however, be computed on the entire bal- ance of $2,000 for a period of thirty days from the average date, May 15 to June 14. Its amount in this case would also be $10.
Assume that instead of paying cash, B, on April 14 gave A a note for $2,000 due in two months. The maturity of this note would be June 14, or thirty days after the average date. An equitable settlement would require the addition of $10 interest to the face of the note, which would make it $2,010.
Items of Varying Amounts
In the foregoing illustrations the amounts are the same and only the varying number of days has had to be considered in arriving at the average date. If, however, the amounts are not the same, they must also be considered. Suppose, for example, that B's account with A was as follows:
March i — 60 days $2,000
" 31 — 60 days 1,000
The $2,000 item is due April 30 and the $1,000 item is due May 30. In this case the average date of maturity for the total of $3,000 would not be May 15 as in the previous example, since the payment of $2,000 fifteen days after maturity would not be off- set by the payment of $1,000 fifteen days before maturity. It would be May 10, because $2,000 paid ten days after maturity would be counterbalanced by $1,000 paid twenty days before maturity. It is evident from this that in averaging accounts due consideration must be given to amounts as well as to dates. The dates to be used in averaging an account are those at which the items may be assumed to have a cash value equivalent to the amount at which they appear in the accounts. Thus, sales on cash terms, being due on the day of sale, take their invoice
44 MATHEMATICS OF ACCOUNTING AND FINANCE
dates in averaging; sales with credit terms take the dates on which the invoices are due; returns and allowances take the dates when the invoices to which they apply are due; an interest-bear- ing note takes the date of the note since the face is the cash value at that date; a non-interest bearing note is not worth its face until due and hence its maturity is used in the calculation of the average date.
Focal Date
The average dates in the preceding illustrations were deter- mined by inspection. When an account is not so simple the com- putation of its average date requires a method involving the principle of weighted average and the selection of a basic date for calculating the time of each item. This basic date is called the focal date and the one most advantageously employed is the last day of the month preceding the earliest date used in averaging.
Take, for example, the following account:
Date of Terms of Date Used
Transaction Payment in Averaging Amount
March i 60 days April 30 $2,000.00
" 31 60 " May 30 1,000.00
The first date to be used in averaging the account is April 30. Hence March 31, the last day of the preceding month, is selected for the focal date. The subsequent steps in the process of averag- ing the account may be enumerated as follows:
1. Assume that each item is paid on the focal date.
2. Determine the number of days each item would be pre-
paid if it were paid on the focal date.
3. Multiply its amount by this number of days.
4. Add the products thus obtained.
5. Divide this sum by the total of the items; the quotient
represents the number of days the focal date precedes the average date.
AVERAGING ACCOUNTS 45
Rules Applied
The application of these rules to the foregoing account is as follows :
1. The two items are assumed to be paid on March 31, the
focal date taken.
2. The first item is therefore assumed to be prepaid 30 days,
and the second 60 days.
3. Paying $2,000 30 days before it is due is equivalent to
paying $1, 60,000 (2,000 X 30) days before it is due; and paying $1 ,000 60 days before it is due is equivalent to paying $1 , 60,000 (i ,000 X 60) days before it is due.
4. Hence the two assumed prepayments are equivalent to a
prepayment of $1 by 120,000 days.
5. One dollar prepaid 120,000 days is equivalent to $3,000
prepaid 1/3 ,000 of 1 20,000 days, or 40 days. The aver- age date is therefore 40 days forward from March 31. Forty days are taken as the equivalent of i month and 10 days, making the average date May 10. The computation of the average date is shown in tabular form
as follows :
Date of Terms of Date in Time from Focal Transaction Payment Average to Maturity Date Amount Product
March i 60 days April 30 30 days $2,000 $60,000
March 31 60 " May 30 60 " 1,000 60,000
$3,000 $120,000
Dividing the sum of the products by the sum of the amounts (120,000 -^ 3,000) gives 40, or the number of days the average date follows the focal date.
Reducing Days to Months
When the time between the focal date and the date of any of the items in the account is more than one month, the reduction of this time to days may be avoided by the method outlined below :
46 MATHEMATICS OF ACCOUNTING AND FINANCE
1 . Express the time in months and days.
2. Multiply the amount of each item by the number of
months to obtain a product of months; and by the number of days to obtain a product of days.
3. Add the products of months; also the products of days.
4. Reduce the sum of the products of months to days by
multiplying by 30.
5. To the product of days thus obtained, add the product of
days previously obtained.
6. Divide this sum by the balance of the account to find
the time in days between the focal and the average dates.
7. Reduce this time to months and days on the basis of
thirty days to a month.
This method is illustrated in the following example, in which the focal date taken is February 28, being the last day of the month preceding the earliest date in the average, which is March 3.
Date of
Terms of
Date in
Time
Product
Transaction
Payment
Average
Mos
1. Days Amount Mos. Days
March 3
cash
March 3
0
3
$250 $ 0 $ 750
March 18
I month
April 18
I
18
500 500 9,000
April 10
30 days
May 10
2
10
200 400 2,000
May 8
cash
May 8
2
8
400 800 3,200
$1,350 $1,700 $14,950 30 X 1,700=51,000
$65,950
65,950 -T- 1,350 =48 115/135. or 49 days 49 days = I month and 19 days
One month and nineteen days forward from February 28, the focal date, is April 19, which is the average date.
This computation is based on the assumption that there are 30 days in each month and 360 days in a year. Strictly speaking, however, the item due May 10 runs for seventy-one days instead
AVERAGING ACCOUNTS 47
of seventy, and the item due May 8 runs for sixty-nine days in- stead of sixty-eight. The result, however, would not be ma- terially modified if the exact number of days were taken. The other method is, therefore, sufficiently accurate for all ordinary commercial transactions.
Compound Average
The average date of an account is computed by simple aver- age, when the account contains either debit or credit items, but not both. When both debits and credits are included, the aver- age date is found by means of compound average, which involves the following steps:
1. Determine the products of months and days for the debits
and credits.
2. Determine the diff'erence between the sum of the debit
products and the sum of the credit products.
3. Divide this difference by the balance of the account.
4. If the difference of the products is on the same side of the
account as its balance, the average date is forward from the focal date; but if the difference of the products and the balance of the account are on different sides, the average date is backward from the focal date. This latter condition rarely occurs if the focal date selected is prior to the dates of the items in the account.
Example of Compound Average
In illustrating compound averaging the following account is taken :
Debit Credit
June I $500 July 5 Note (2 mo. with-
" 20, I mo 400 out int.) $500
July 10 600 " 10 Returns (Inv.
August 5 500 June 20) 50
August I Cash 300
48 MATHEMATICS OF ACCOUNTING AND FINANCE
As the July 5 th note on the credit side of the account is non- interest bearing, it does not have a cash value of $500 until its maturity on September 5. It therefore takes this date in the average. The July loth credit, being an offset to the debit of June 20, takes the same maturity date in the average as the debit item, or July 20th. As June i is the earliest date on which any of the items in the account has a cash value equal to the face of the item, the most convenient focal date is May 31.
The sum of the products of months and days for the debit items of the account is found as follows :
n Date of Terms of Date in Transaction Payment Average Mos.
June I June i o
" 20 I month July 20 i
July 10 July 10 I
August 5 August 5 2
Total debits $2,000 $2,000 $17,000
30 X 2,000 = 60,000
Sum of the debit products $77,000
The sum of the products of months and days for the credit items is found by the following computation.
Date of Terms of Date in ^ime Products
Transaction Payment Average Mos. Days Amount Mos. Days
^500 $1,500 $ 2,500
SO 50 1,000
300 600 300
T
Products
)ays
Amount Mos. Days
I
$500 $ 0 $ 500
20
400 400 8,000
10
600 600 6,000
5
500 1,000 2,500
July 5
2 months Sept. 5
3
5
July 10
offset
June 20
Dr. July 20
I
20
August I
August I
2
I
Total credits
50 $2,150 $ 3,800 2,150 X 30 = 64,500
Sum of the credit products $68,300
The account has a debit balance of $1,150 and the sum of its debit products exceeds the sum of its credit products by 8,700.
AVERAGING ACCOUNTS 49
The average date is forward from May 3 1 , the focal date, as many days as the number of times 1,150 is contained in 8,700, or ap- proximately 8. June 8 is, therefore, the focal date. The account could be averaged by deducting the $50 credit from the June 20th debit of $400 and dealing only with the net debit of $350.
Another Illustration
To illustrate the conditions under which the average date is backward from the focal date, assume that the July 5th non- interest bearing note was due in six months. All of the debit items are due on or before August 5, but since the creditor would have to wait until January 5 for the $500 payable on the note, which in the meantime would earn no interest, the balance of the account should carry an early average maturity. The sum of the debit products of days would be computed as in the preceding example and would total 77,000, while the sum of the credit pro- ducts of days would be computed as follows:
Date of Terms of Date in
Time Product
Trans.\ction Payment Average Mos. Days Amount Mos. Days
July 5 6 months January 57 5 $500 $3,500 $ 2,500
July 10 offset July 20 i 20 50 50 1,000
August I August 12 I 300 600 300
Total credits $850 $4,150 $ 3,800
30 X 4,150 = 124,500 Sum of the credit products $128,300
The balance of the account would still be a debit of $1,150, but the difference of the products of days would now be on the credit side, and would amount to 51,300. The average date would, therefore, be backward from June i, the focal date, by forty-five days, or the number of times $1,150 is contained in 51,300. Counting thirty days to the month, the average date would be one month and fifteen days backward from the focal date and would, therefore, be April 15.
CHAPTER VI PERCENTAGE
Percentage
Percentage is a method of computing by hundredths. The symbol % means per cent or hundredths. A rate per cent is equivalent to a common fraction the numerator of which is ex- pressed and the denominator of which is indicated by the symbol % as being loo. Thus the same facts may be stated in the form of a common fraction, a decimal fraction or a per cent. The following are equivalent:
Common Fractions Decim.\l Fractions %
17/100 .17 17
iX 1. 25 125
iH 7-So 750
12/4 3.00 300
Terms Used in Percentage
Base. The number of which a given per cent is to be taken is called the base.
Rate. The per cent of the base to be taken is called the rate.
Percentage. The result obtained by taking a certain per cent of the base is called the percentage.
Fundamental Processes
All mathematical computations involving percentage may be grouped under three headings :
I. To find a given per cent of a number; that is, to find the percentage.
Rule: Multiply the base by the rate Example: $60 X 20% = $12
Base X Rate = Percentage
50
PERCENTAGE 5 1
2. To find what per cent one number is of another; that is, to
find the rate.
Rule: Divide the percentage by the base Example: $12 -r- $60 = 20%
Percentage -H Base = Rate
3. To find a number when a certain per cent of it is known;
that is, to find the base.
Rule: Divide the percentage by the rate Example: $12 -7- 20% = $60
Percentage -^ Rate = Base
Percentage of Increase and Decrease
Percentage is frequently employed to compare numbers and to show how much larger or smaller one number is than the other. No new mathematical principles are involved in such computa- tions, as may be shown by the following illustrations:
1 . Percentage of Increase. The sales of a certain business in May, 1919, were $16,000 while the sales in May, 1920, were $18,000.
The smaller number is taken as the base; the difference be- tween the two numbers is the percentage of increase. Then
2,000 -f- 16,000 = 12^%
Percentage of increase -^ Base = Per cent of increase
2. Percentage of Decrease. The profits of a business for the year 1919 were $20,000 while the profits for 1920 were $17,000.
The larger number is taken as the base; the difference be- tween the two numbers is the percentage of decrease. Then
3,000 -7- 20,000 = 15%
Percentage of decrease 4- Base = Per cent of decrease
Some Applications of Percentage in Business
Comparisons
Business statistics may be tabulated and compared on a per- centage basis to determine the relative effectiveness, desirabihty
Number
Per Cent
Sold
Sold
293
79.6
316
64.1
582
91. 1
416
79-5
52 MATHEMATICS OF ACCOUNTING AND FINANCE
or productivity of similar factors. The illustrations given in this chapter are intended to be suggestive but not exhaustive.
The following tabulation shows the number of units of a cer- tain commodity purchased from various manufacturers during a year, the number sold, and the per cent sold. A comparison of the per cents indicates the relative salability of the goods pur- chased from the various manufacturers.
Per Cent of Goods Sold — Various Manufacturers
Number Manufacturer Purchased
Walker & Co 368
White & Dudley 493
Davis Mfg. Co 639
Barton-Walsh 523
The following tabulation compares the sales of various sales- men during a month. Each man's sales (as a percentage) divided by the total sales (as a base) produces a rate which measures his portion of the total.
Monthly Sales Compared on a Percentage Basis
Name Sales Per Cent of Total
Arthur Bradley $ 1,264.90 7.8
J.B.Henderson 1,913.52 11. 8
Fred Bates 1,732.69 10.6
Arthur Dutton 2,213.72 13.6
J. L. Weston 1,963.45 12.1
Carter Doane 1,627.32 lo.o
Walter S. Waite 1,692.18 10.4
Frank Chesley 2,138.45 13. i
Harold Peters 1,728.46 10.6
Total $16,274.69 loo.o
The statistics may be so arranged as to obtain two percentage analyses, as illustrated in the following tabulation which shows
PERCENTAGE
53
what per cent of the sales of the week was made by each salesman, and what per cent was made each day.
Sales for the Week Ending December i8, 1920
D\\^
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Salesmen's totals
Per cents
Smith
I362.50 415-75 396.21 472.96 387.29 493-89
$2,528.60
Brown
J562.83 475-92 415-60 516.29 42936 562.64
32,962.64
Jones
I862.94 732.83 769-42 640.20 721.32 816.25
$4,542.96
White
5126.39 143-62 129.38 145-17 96.27 163.92
504.75
Daily Totals
$1,914.66 1,768.12 1,710.61 1,774-62 1,634-24 2,036.70
$10,838.95
Per Cents
17-6 16.3 15.8 16.4 15-1
The following tabulation illustrates the use of percentage of increase and decrease as a means of comparing the sales of each department of a store on the corresponding days of two years.
Comparison of Sales by Departments
Department
Sales-Thursday December is. 1919
Sales-Thursday December 16, 1920
Increase Decrease*
% Increase — Decrease*
A
$ 826.95 1.034-78 1.237.62 2,643.80 1,413.80 962.40 2,642.16 1.964-39 1,636.48 1,213.42
$ 914-32 1,231.64 1,196.14 2.843-27 1.376.29 1,235.96 2,927.92 2,129.80 1,596.27 1,723-96
$ 87-37 196.86
41.48* 199.47
37-51* 273-56 285-76 165.41
40.21* 510.54
10.57 19.02 3-35* 7-54 2.6s* 28.42 10.81 8.42 2.46* 42.07
B
C
D
E
f
G
H
I
J
Total
$15,575-80
S17. 175-57
Si, 599-77
10.27
The average of a number of quantities may be accepted as the basis of comparison, the relation of each quantity to the average
54 MATHEMATICS OF ACCOUNTING AND FINANCE
being shown in terms of per cents. The figures in the preceding tabulation of "Monthly Sales Compared on a Percentage Basis" are used for the following illustration:
Individual Sales Compared with Average
Name Sales Per Cent of Average
Arthur Bradley $1,264.90 69.95
J.B.Henderson .... 1,913.52 105.82
Fred Bates 1,732.69 95.82
Arthur Button 2,213.72 122.42
J.L.Weston 1,963.45 108.58
Carter Doane 1,627.32 89.99
Walter' S. Waite 1,692.18 93.58
Frank Chesley 2,138.45 118.26
Harold Peters 1,728.46 95-58
Average $1,808.30 100.00
Or the maximum may be accepted as the basis of comparison, the relation of all quantities to the maximum being shown in terms of per cents. Using the same statistics for an illustration:
Individual Sales Compared with Maximum
Name Sales Per Cent of Maximum
Arthur Bradley $1,264.90 57-i4
J.B.Henderson 1,913.52 86.44
Fred Bates 1,732.69 78.27
Arthur Button 2,213.72 100.00
J. L. Weston 1,963.45 88.69
Carter Boane 1,627.32 73-5i
Walter S. Waite 1,692.18 76.44
Frank Chesley 2 138.45 96.60
Harold Peters 1,728.46 78.08
The following tabulation is suggestive of the use which may be made of percentage in comparing quantities with two or more similar quantities and with the average thereof. In this case the average is a progressive one.
PERCENTAGE
Comparison of Sales of Successive Years
55
Year
Sales
Inc. or Dec*
FROM Preceding
Year
Inc. or Dec*
from First
Year
Progressive
Average Prior Years
Inc. or Dec*
FROM Progressive
Average
Amount
Amount
rf
/O
Amount
%
1917 1918 1919 1920
$200,000 238,000 190,000 285,000
$38,000 48,000* 95.000
19.00
20.17* 50.00
$38,000 10,000* 85,000
19.00 s-oo* 42.50
$219,000 209,333
$29,000* 75.667
13.24* 36.15
Apportionment
When a quantity is to be divided or partitioned, the basis of the partition may be expressed in rates per cent. The partition is then accompHshed by applying to the base a number of rates, the total of which is 100%. The division of partnership profits is a familiar illustration.
Division of Profits
Partners P. & L. Ratio Profits
A 20% $3,200.00
B 35% s,6oo.oo
C 45% 7,200.00
Total 100% $16,000.00
Or the apportionment may be accomphshed by applying the same rate to a number of bases to obtain the desired percentages. In this case the rate is computed by dividing the total percentage by the total of the bases. The distribution of factory overhead is illustrative.
Distribution of Factory Overhead
(Direct Labor Cost Basis)
Total direct labor, all departments $5,295.00 (base)
Total factory overhead 3,460.00 (percentage)
Then 3,460 -^ 5,295 = 65.34+ % (rate)
56 MATHEMATICS OF ACCOUNTING AND FINANCE
Since the rate is approximate only, the distribution will not be exact; a remainder of 24 cents will be undistributed.
Process or Proportion
Department Direct Labor of Overhead
I $2,140.00 $1,398.28
2 1,965-00 1,283.93
3 1,190.00 777-55
Total $5.295-00 $3,459.76
Gross Profit Method of Approximating Inventory
The rate of gross profit of prior periods may be used to ap- proximate an inventory when it is impracticable or impossible to take a physical inventory. This is accompHshed by utilizing the elements involved in the computation of gross profits. In a bal- anced table, when all but one element is known, the unknown element is found as the amount necessary to balance the table. The gross profit on sales may be computed by setting up a mer- chandise account as follows :
Merchandise
Inventory, Jan. i . . .$100,000.00 Sales $350,000.00
Purchases 300,000.00 Inventory, Dec. 31 105,000.00
The gross profit would be $55,000.00, the amount necessary to bring the account into balance. Now if the inventory were not known, but the gross profit could be estimated at $55,000.00, the inventory could be determined thus:
Merchandise
Inventory, Jan. i $100,000.00 Sales $350,000.00
Purchases 300,000.00
Gross profit 55,000.00
The inventory would be $105,000, the amount necessary to bring the account into balance.
Of course the gross profit could not be definitely ascertained without an inventory, but it could be approximated by using the
PERCENTAGE 57
average rate of gross profit on sales of former years, if no radical variations have occurred in this rate and if there is no reason to believe that the rate of the current period has been radically different from the average rate of the past.
To illustrate, let us assume that the sales and gross profits of the business whose merchandise account appears above, were as follows :
Year S.vles Gross Profit Per Cent
Third preceding .... $200,000.00 $31,400.00 15.7
Second " .... 310,000.00 48,050.00 15.5
First " .... 385,000.00 60,170.00 15.6
,000.00 $139,620.00 15.6
The annual rates are computed to determine whether there has been any considerable variation in the rates of gross profit. Then, on the assumption that the rate of gross profit for the cur- rent period was the same as the average of the rates of the three last preceding years:
15.6% of $350,000.00 (sales) = $54,600.00, approximate gross profit
Then:
Inventory, January i $100,000.00
Add purchases 300,000.00
Total $400,000.00
Deduct cost of goods sold (approximate) :
Sales $350,000.00
Less estimated gross profit 54,600.00 295,400.00
Inventory, Dec. 31 (approximate) $104,600.00
The inventory thus computed is $400 less than that shown by the merchandise account.
The following problem from a C. P. A. examination will fur- ther illustrate the method, the principal uses of which are in
58 MATHEMATICS OF ACCOUNTING AND FINANCE
approximating the value of merchandise destroyed by fire and in applying the gross profit test to the verification of an inventory.
Problem
The accountant is called on to confirm the inventory of a mercantile establishment. Investigation shows that inventories have been incor- rectly taken and are padded. It is mutually agreed that all inventories, except the first one, which is to be used as the basis, shall be entirely ignored.
The accountant is to ascertain, on a fixed percentage of profit which it is decided shall be 33yj% of sales, what stock should be on hand December 31, 1914, with the following data obtained from the various books:
Inventory referred to as a basis, January i, 191 1 $47,350.29
Gross purchases for the year ending December 31, 1911 . . . 76,320.15
Returned purchases 4,350.16
Freight and drayage on purchases 325.14
Gross sales 115,469.31
Returned sales 1,317.12
Gross purchases for the year ending December 31, 191 2 . . . $65,506.80
Returned purchases 3,715.16
Freight and drayage on purchases 41 7- 15
Gross sales 105,716.10
Returned sales 1,215.84
Gross purchases for the year ending December 31, 1913 . . . $62,517.10
Returned purchases 1,314.17
Freight and drayage on purchases 316.17
Gross sales 101,317.18
Returned sales 1,216.06
Gross purchases for the year ending December 31, 1914. . . . $58,715.16
Returned purchases 287.50
Freight and drayage on purchases 290.10
Gross sales 95,371.16
Returned sales 41 7- n
Solution: Since the rate of gross profit was constant throughout the four years, and since only the final inventory is required by the problem, the data can be summarized and the four years' totals used in the inventory calculation.
PERCENTAGE
59
Summary 1911-1914
Returned
Returned
Year
Purchases
Purchases
Freight
Sales
Sales
1911
$76,320.15
$4,350.16
$325-14
$115,469.31
$1,317.12
1912
65,506.80
3,715-16
417-15
105,716.10
1,215.84
1913
62,517.10
1,314-17
316.17
101,317.18
1,216.06
1914
58,715-16 $263,059.21
287.50
290.10
95,371-16
417. II
Total
$9,666.99
$1,348.56
$417,873.75
$4,166.13
Statement of Approximation of Inventory At December 31, 1914
Inventory, January i, 19 11 $47,350.29
Add cost of goods purchased, 191 1-1914:
Purchases $263,059.21
Z,e55 returned purchases 9,666.99 $253,392.22
Add freight.
1,348.56 254,740.78 $302,091.07
Deduct cost of goods sold, 1911-1914:
Sales
Less returned sales
^17,873-75 4,166.13 $413,707.62
Less gross profit (s^HVo of sales) 137,902.54 275,805.08
Inventory, December 31, 19 14 (estimated) $26,285.99
Analysis of Statements
The following statements indicate the use which may be made of percentage in analyzing the financial statements of a business to show such facts as the ratio of cost of sales, expenses and profits to sales; the relative cost of the various elements of manufactured goods, and the variation in operating costs of different years.
Problem
From the following data obtained from the books of Johnson and Com- pany, construct a profit and loss statement showing cost of goods manufac- tured, and cost and gross profit of the goods sold. Also show percentage of each eli^ment based upon cost of manufacture and based upon sales.
6o
MATHEMATICS OF ACCOUNTING AND FINANCE
Raw material, January i, 1915 $42,000,00
" " December 31, 1915 45,000.00
" " purchases during 1915 130,000.00
Freight inward 4,218.00
Wages (productive) 70,000.00
Sundry manufacturing expenses 3,500.00
Sales 280,000.00
Finished goods, January i, 1915 18,000.00
" " December 31, 191 5 22,000.00
Selling expenses 22,000.00
Administrative expenses 20,000.00
Solution:
Johnson and Company
Profit and Loss Statement Year Ending December 31, 1915
Sales 1280,000.00
Deduct:
Cost of goods sold: Raw material:
Inventory, Jan. i, 1915 . . $ 42.000.00 Purchases, 1915 130,000.00
Total $172,000.00
Inventory, Dec. 31, 1915. 45,000.00 $127,000.00
Freight inward 4,218.00
Productive labor 70,000.00
Manufacturing expense. . .. 3,500.00
Cost of goods
manufactured $204,718.00
Deduct:
Inventory variation — finished goods:
December 31, 1915 $22,000.00
January i, 1915 i8,ooo.f)0 4,000.00
Cost of goods sold 200,718.00
Gross profit on sales 179,282.00
Deduct selling expenses 22,000.00
Net profit on sales $57,282.00
Deduct administrative expenses. 20,000.00
Net profit on operations $37,282.00
% OF % OF
Cost Sales
62.0
2.1
34-2
1.7
71-7
28.3 7.9
PERCENTAGE 6l
This illustration, showing the per cent of net profit and gross profit on sales raises the question whether sales or cost of sales should be used as the base in the computation of rates of gross profit. In common parlance, when a statement is made that a sale has resulted in realizing a certain rate of profit, the rate is understood to have been applied to the cost. Thus, if it is said that an article costing $2 was sold at a 10% profit, one assumes that the profit was 20 cents and the selling price $2.20. But in percentage analyses of revenue statements it is much more con- venient to use the net sales as the base. Selling expenses nor- mally are proportionate to sales and the per cent of selling expense is computed on the basis of sales. By computing the cost of goods sold and afl other deductions from sales as percentages of the net sales, the statement begins with 100% as the base and continues on the same basis throughout. But if cost were taken as the base, the sales would be represented by a rate exceeding 100%, and the selling expenses, including such items as advertising, salesmen's commissions, freight out and store expense, would hsive to be rated on the illogical basis of cost.
Therefore the percentage analysis of the revenue statement should properly be made on two bases : elements of manufacturing cost, including material, labor and factory overhead, should be considered as percentages of cost; and all deductions from sales, including the cost of goods sold, the selling expenses and the administrative expenses, should be rated as percentages of the net sales. This method was followed in the preceding illustration.
As an illustration of the use of percentage in the comparison of successive revenue statements, the following condensed state- ments of the J. E. Smith Wire and Iron Company are presented, together with comparative percentage analyses thereof. The analytical statement shows that the increase in sales and the increase in cost of sales have not been proportionate, as the proportion of cost to sales has steadily increased, causing a corresponding decrease in the percentage of profit.
62
MATHEMATICS OF ACCOUNTING AND FINANCE
H
O
o
o
o
o o
o
o
o
o o
a,
o
o
o
o o
in
o
in
o o
H
CO
•* in
u
Vi
J2
u
>
t: «
5 w
o
o
o
o o
« z
o
o
o
o o
H H
o
o
o
o o
S2 a-
1/5
o
in
o m
Z ^
N
N
ro M
s w
^
Q
<
M
(«
z
w
o
o
o
o o
X
o
o
o
o o
w
o
o
o
o o
o
o
o
O in
z
w»
vO
<^
CO t^
J
>J
(d
W
u
u
<;
o
o
n
o o
o
o
O
o o
o
o
o
o o
0
o
o
o
o o
o
o
O vO
m
Tt- "5
w
0
u
c/)
(d
o
o
o
o o
o
o
o
o o
<
o
o
o
o
o
o
o o
lO
o
in
in o
2:
^
^
in in
«
<
o.
o
M
N n
l>i
o
a
a
a
o o
M W
rR ^
^ «
IIJOHJ
00 O M t~ O ct in tt n o ■4 t^ PI r-- d
N M M W
XldOHtJ xajvi
$85,000 70,000 55,000 40,000 50,000
saivs do %
(> 0 vO in 0 0) 0 in 't q Tt in in in rn
asNadxa a.MivHXsiNiKav
00000 00000 q q q q q
in d in 0' in
« P) N ro w 6^
saivg NO xiJOHd xaN do %
t^ 0 00 N 0
m m t~ t- 0 CO ci t.1 N ro
W W M W M
saivs
NO XIdOH(J xaM
00000 00000
0 (» q 0 0 d d 0" d in
0 Oi 00 t^ 0
sasNadxg ONmag HO %
0> 0 10 m 0 M q in in q Tt in in ^ in
sasNadxg ONmas
00000 00000 q q q q q d 0" 0" d in m -0 t~ 00 t^
xidoaj SSOHO -iio %
0 0 0 t^ 0 00 in r-p N q ri t^ fO t^ 06
•<t (^ fO M M
saavg NO xidOHd ssoao
00000 00000
0 0 0 q q d d d d d in in m in -^
sanvs ox XS03 JO %
^0 1^ ro 0
(H in 0 t^ 0
m -0 0 i^ t~
saivg HO XS03
00000 00000 q q q q q d d d d d 0 m 0 0 vo tN n ro •* rn
6%
saivs xaist
00000 00000
q q q q q 6 6 6 6 6 in 0 m m 0 rn -t rf m m
avaA
g 2 : :: :?
Os o\ 0 0 Oi
PERCENTAGE
63
As another illustration of the utility of percentage as a means of measuring variations, the following condensed statement of comparative manufacturing costs is given.
Condensed Statement of Manufacturing Costs
Year 1916
Year 1917
D0LL.\RS
% of Total
Dollars
% of Total
$320,000.00 441,500.00 268,500.00
31-07 42.86 26.07
$287,600.00 301,300.00
187.100.00
37.06 38.83
24.11
Manufacturing expense
Total
$1,030,000.00
100.00
$776,000.00
While this statement shows the variation in the rate per cent of each item to the total cost, it does not give an adequate idea of the variation in unit costs. The following table shows this variation.
Comparative Statement of Unit Costs
Units Produced: 1916 1917
10.000 8,000
Year 1916
Year 1917
In'crease or Decrease*
Total
Unit
Total
Unit
Dollars
Per cent
$320,000.00 441,500.00 268,500.00
$32.00 44-15 26.85
$287,600.00 301,300.00 187,100.00
$35-95 37-66 23-39
$3 95 6.49* 3-46*
12.34%
14.70*
12.89*
Manufacturing expense
Total
$1,030,000.00
S103.00
$776,000.00
$97-00
$6.00*
S-83*
Percentage Analyses to Determine Causes of Variation in Profits
Revenue statements may be compared on a percentage basis to show the cause of the increase or decrease in net profit, but in order to arrive at accurate results it is necessary to know the per
64
MATHEMATICS OF ACCOUNTING AND FINANCE
cent of increase or decrease in the unit selling price of the second year as compared with the unit selling price of the first year. To illustrate, let us assume the following statements each of which shows a percentage analysis based on sales:
The Wharton Manufacturing Company
Comparative Profit and Loss Statements
Year Ending Year Ending
December 31, 1916 December 31, 1917
Sales $400,000 100.00% |6oo,ooo 100.00%
Deduct: Cost of sales:
Material Iioo.ooo 25.0 $150,000 25.0
Labor 200.000 50.0 330,000 55.0
Manufacturing
expense 50,000 12.5 60,000 10. o
Total 350,000 87.50 540,000 00.0
Gross profit on sales ... . $50,000 12.50 $60,000 10. o
DedMci selling expense . io,ooo 2.50 IS, 000 2.5
Net profit on sales $40,000 10.00 $45,000 7.5
/JeiiMc/ general expenses 15,000 3.75 15,000 2.5
Net profit $25,000 6.25 $30,000 5.0
This comparative statement has very little meaning until the further fact is known that there was a 20% advance in the selling price of all goods in 191 7. With this additional information, the sales of 191 7 can be reduced to the value which they would have brought in 19 16, thus:
$600, ooo-^ 1 20%= $500,000, the price which the same goods would have
sold for in iqi6
A supposititious profit and loss statement for 191 7 may now be drawn up, beginning with sales of $500,000 (volume of 191 7 at 19 1 6 prices) and replacing the actual figures of 191 7 with amounts obtained by multiplying $500,000 by the various rates per cent shown in the 1916 statement. This supposititious statement will show what the figures for 191 7 would have been if there had been merely a change in volume of business but no change in the rates of the various expenses to sales. A comparison of these sup- posititious figures with the actual figures for 191 7 will show the
PERCENTAGE
65
effect on 191 7 profits of fluctuations in cost of manufacture and of expenses.
Profit and Loss Statement of 191 7 Reduced to 1916 Rates and Compared with Actual Statement for 191 7
Statement on Basis
OF 1916 Rates to Sales ,, _
Variations of Profit Amounts Rates of at 1916 Actual Decreases Increases
1916 Rates 1917 Dollars % Dollars %
Sales 100.00% Jsoo, 000 S6oo,ooo Jioo.ooo 20.0
Deduct:
Cost of goods sold:
Material 23.00 125,000 150,000 $25,000 20.0
Labor 50.00 250,000 330,000 80,000 32.0
Manufacturing expense. . . 12.50 62,500 60,000 2,500 4.0
Total 87.50 $437,500 J540.000 $102,5002343
Gross profit 12.50 862,500 56o,ooo $2,500 4.00
Z?e<iuc< selling e.xpenses .. . 2.50 12,500 15,000 2.500 20.00
Net profit on sales 10.00 $50,000 $45,000 $5,000 10.00
Deduct geneval expenses . . 3.75 18,750 15,000 3,750 200
Net profit 6.25 $31250 $30,000 $1,250 4.00
The additional profit of 191 7 can now be accounted for as follows :
i2>^% gross profit (1916 rate) on $100,000.00 ad- ditional business done in 191 7 $1 2,500.00
Deduct excess of additional cost to manufacture over increase in selling prices:
Additional cost of material $25,000.00
Additional cost of labor 80,000.00
Total $105,000.00
Less decrease in cost of manufacturing
expense 2,500.00
Net excess in manufacturing cost $102,500.00
Less increase in selling prices 100,000.00 2,500.00
Excess of gross profit of 191 7 over 191 6 $10,000.00
Deduct addhional selling expenses of 191 7. . . 5,000.00
Additional net profit of 191 7 $5,000.00
CHAPTER VII
EQUATIONS IN THE SOLUTION OF PROBLEMS
Solving Equations
Problems may often be solved by stating the conditions in the form of an equation and solving the equation by applying one or more of the following processes :
1. Multiplying both sides of the equation by the same
number
2. Dividing both sides of the equation by the same number
3. Adding the same number to both sides of the equation
4. Subtracting the same number from both sides of the equa-
tion
Illustration i
A manufacturer produced a certain commodity which he sold the first year at a certain price; he raised the price 25% the second year; increased that price 20% the third year; and in the fourth year he increased the third-year price by 1673%. The price the fourth year was $35. What was the price the first year?
Solution: In order to obtain an equation it is necessary to represent some value by 100%. The value chosen to be represented by 100% will depend on the conditions of the problem; where convenient, 100% should represent the value required by the problem. In this case —
Let 100% = the selling price the first year
then 100% X 125% = 125% the selling price the second year and 125% X 120% =150% " " " " third "
and 150% X ii6V3% = i75% " " " " fourth "
Since the selling price the fourth year was $35 we obtain the equation:
175% = l3S 66
EQUATIONS IN THE SOLUTION OF PROBLEMS 67
But it is desired to determine 100%, which is accomplished by dividing both ?ides of the equation by 1.75:
100% = $20, the selling price the first year
Illustration 2
A entered into partnership with B and was to act as manager of the business. Before dividing profits equally with B, A was to receive a special bonus of 25% of the net profit. Before calculating A's commission, the profits were shown by the revenue statement to be $5,000. How should the $5,000 be divided between A and B?
Solution: This problem illustrates the difficulty which frequently arises in interpreting contracts which provide that commissions and bonuses shall be determined as percentages of the net profit. The difii- culty arises from the uncertainty as to what is the amount of the net profit; for, if the bonus is to be considered as an expense of the business, the net profits are less than $5,000; if the bonus is not to be considered as an expense but as part of the distribution of profits, the net profit is $5,000. From the statement of the problem, it is impossible to tell whether or not the commission is to be considered as an expense; hence it is necessary to give two solutions.
Assuming the bonus is not an expense:
Since the bonus is 25% of the net profit, the net profit must be 100%. Since the bonus is not an expense, the $5,000 is all to be considered net profit —
Then, 100% = $5,000, the net profit and 25% = 1,250, the bonus
and 75% = $3,750, the remaining profit to be divided equally. 37^2% = $1,875, share to A and B each
Therefore, the division is as follows:
A B Total Rate
Bonus $1,250 $1,250 25%
Remainder, K each 1,875 $1,875 3,75° 75%
Total $3,125 $1,875 $5,000 100%
68 MATHEMATICS OF ACCOUNTING AND FINANCE
Assuming that the bonus is an expense to be deducted from the $5,000 to obtain the net profit:
25% = bonus 100% = net profit
125% = bonus plus profit, or $5,000
Then 100% = $4,000, net profit and 25% = $1,000, bonus
Proof 125% = $5,000
The division of the $5,000 on this assumption would be:
A B Total Rate
Gross profit $5,000 125%
Deduct bonus $1,000 1,000 25%
Net profit, y2 each 2,000 $2,000 $4,000 100%
Total distribution $3,000 $2,000
The net difference of $125 in the distribution under the two interpretations emphasizes the necessity for care in drawing up contracts of this character. In such cases an accountant should always be consulted as to the wording of the contract, as any competent accountant would recognize the danger of there being two constructions placed on the contract, with a consequent dispute.
Illustration 3 (From Ohio C. P. A. Examination, October, 1919)
The American Manufacturing Company commenced business on January i, 1918, with a paid-up cash capital equal to the sales for the year 1918.
The net profits for the year 1918 were $26,100.
Of the total charges to manufacturing during the year, 40% was for materials, 30% for productive labor, and 30% for manufacturing ex-
EQUATIONS IN THE SOLUTION OF PROBLEMS 69
penses (including 5% depreciation on plant and machinery, amounting to $3,000).
The value of the materials used was 80% ot the amount purchased, and 90% of the amount purchased was paid during the year.
The inventory value of finished goods on hand at December 31, 1918, was 10% of the cost of finished units delivered to the warehouse, and the work in process at that date was equal to 50% of the cost of units delivered to the warehouse.
The selling and administrative expenses were equal to 20% of the sales; also to 40% of the cost of goods sold. Ninety per cent of these ex- penses were paid during the year 1918. Plant and machinery purchased during the year were paid for in cash.
All labor and manufacturing expenses (exclusive of depreciation) were paid in full up to and including December 31, 1918.
Of the total sales for the year, 80% was collected and 1% charged ofT as worthless.
From the given data you are required to prepare a balance sheet and a profit and loss statement, showing cost of goods delivered to the warehouse, cost of goods sold, and net profit for the year.
Solution: Let sales equal 100%
Then (since selling and administrative expense is 20% of sales or 40% of cost of goods sold) the cost of goods sold is half of the sales or 50%
and the gross profit is 50%
(That is, if a is 20% of x and is also 40% of y, then y
must be half of x.) Selling and administrative expenses are equal to 20%
and bad debts equal 1% 21%
Hence the net profit is 29%
Then, 29% = $26,100
and 100% = $90,000, sales for the year, and the cash capital at the beginning of the year.
Cost of goods sold = 50% of $90,000 $45,000
Gross profit 45,000
Selling and administrative expense 18,000
Bad debts 900
70 MATHEMATICS OF ACCOUNTING AND FINANCE
Since the inventory of finished goods at December 31, i9i8,wasio%of the cost of finished units delivered to the warehouse, the cost of goods sold was 90% of the finished goods manufactured during the year.
Then, $45,000-7-90% = $50,000, cost of finished goods manufactured And $50,000 — $45,000 = $5,000, inventory of finished goods at Decem- ber 31, 1918
Since the work in process at December 31, 191 8, was 50% of the cost of finished goods delivered to the warehouse,
50% of $50,000 = $25,000, Work in process inventory
$50,000, Goods finished during 1918 25,000, Work in process at Dec. 31, 1918
$75,000, Total manufacturing cost of 1918
40% of $75,000.00 = $30,000, Cost of materials used 30% of 75,000.00 = 22,500 " " productive labor 30% of 75,000.00 = 22,500 " " manufacturing expense
Of this manufacturing expense, $3,000 was depreciation on plant and machinery; hence the manufacturing expense paid in cash was $19,500. Since the rate of depreciation was 5%, the cost of plant and machinery was $60,000, all of which was paid for in cash.
Since 80% of the material purchased was used in manufacturing, $30,000-^- 80% = $37,500, the cost of the material purchased; and $7,500 is the inventory of raw material at December 31, 191 8. Also 90% of $37,500, or $33,750, is the amount of cash paid for purchases; and $37,500 ~~ ^33-75° = $3,750- the accounts payable at December 31, 1918, for purchases.
The selling and administrative expenses were $1 8,000. Of this amount, 90%, or $16,200, was paid in cash. The remainder, $1,800, is an addition to the accounts payable.
80% of the sales of $90,000 were collected. 1% was written off. Hence,
Sales $90,000
Less:
Cash collections $72,000
Bad debts 900 72,900
Balance of accounts receivable. . . . $17,100
EQUATIONS IN THE SOLUTION OF PROBLEMS 71
The cash summary is :
Cash capital paid in $90,000
Collections on accounts receivable 72,000 $162,000
Deduct:
Plant and machinery $60,000
Materials 33-75°
Productive labor 22,500
Manufacturing expense IQ.500
Selling and administrative expense 16.200
Total disbursements 151,950
Balance $ 10,050
American Manufacturing Company
Trial Balance December 31, 191S
Capital stock $90,000
Plant and machinery $60,000
Reserve for depreciation, plant and machinery .... 3,000
Sales 90,000
Purchases 3 7,500
Productive labor 22,500
Manufacturing expense 19,500
Depreciation, plant and machinery 3,000
Selling and administrative expense 18,000
Bad debts 900
Accounts receivable 17,100
Accounts payable (3,750 + 1,800) 5, 550
Cash 10,050
$188,550 $188,550
Inventories: raw material, $7,500; goods in process, $25,000; finished goods, $5,000.
72
MATHEMATICS OF ACCOUNTING AND FINANCE
American Manufacturing Company
Profit and Loss Statement Year Ending December 31, 1918
Sales $90,000.00
Deduct:
Cost of goods sold : Material:
Purchases $37,500.00
Less inventory, Dec. 31,
igi8 7,500.00 $30,000.00
Productive labor 22,500.00
Manufacturing expense 19,500.00
Depreciation — plant and
machinery 3,000.00
Total manufacturing cost . $75,000.00
Deduct goods in process —
Dec. 31, 1918 25,000.00
Cost of finished goods manu- factured $50,000.00
Deduct inventory finished
goods — Dec. 31, 1918 5.000.00 45,000.00
Gross profit on sales $45,000.00
Deduct:
Selling and administrative ex- $18,000.00
pense
Bad debts 900.00 18,900.00
Net profit $26,100.00
EQUATIONS IN THE SOLUTION OF PROBLEMS
73
American Manufacturing Company
Assets
Plant and machinery . . Less depreciation . . . .
Balance Shej December 31, i
. 160,000.00
3,000.00 $ 57,000.00
ET 918
Liabilities
Accounts payable . . .
Raw material
Goods in process
7,500.00 25,000.00
5.000.00 17,100.00 10,050.00
5,550.00
Accounts receivable . . . . Cash
$121,650.00
$121,650.00
Illustration 4
As another illustration of the use of percentage in the solution of prob- lems, the following C. P. A. problem is given.
The Orinoco Coal Company was incorporated under the laws of the state of Illinois, with an authorized capital of $8,000, divided into eighty shares of the par value of f 100 each, which were subscribed for as follows:
Samuel Black 60 shares
William Green 10 "
John White 10 "
Samuel Black was elected president; George Brown, vice-president and manager, and Charles Pinck, secretary and treasurer. Neither Brown nor Pinck held any stock, but were to receive in addition to their salaries a percentage of the profits after charging off all losses from whatever source — Brown 15% and Pinck 10%. These shares in the profits were not to be considered an expense deductible to obtain the basis of the bonuses.
At the end of the year a meeting of the stockholders was held and the following balance sheet was presented:
Assets
Liabilities
Cash
$ 8,031.12
Accounts payable ....
$ 740.22
Accounts receivable . .
817-32
Reserve for bad debts
376.05
Coal
6,644-15
Capital stock
8,000.00
Charles Pinck
2,264.14
Undivided profits ....
8,640.46
$17,756.73
$17,756.73
74 MATHEMATICS OP ACCOUNTING AND FINANCE
It was announced that Treasurer Pinck, who was not financially responsible and was not bonded, had disappeared and his account was uncollectible, but would be reduced by crediting the account with his share of the net profits.
What amounts should Pinck and Brown receive as salary addition?
What is the amount of Pinck's defalcation?
If the remaining profit is divided among the stockholders, what divi- dend should each receive?
Solution: Since the bonuses allowable to Pinck and Brown are to be calculated on the net profit after deducting all losses, the defalcation must be deducted from the undivided profits of $8,640.46 to obtain the net profit.
Let 100% = the net profit,
then $8,640.46 — defalcation = 100%.
But the amount of the defalcation is not known, except that it is the amount of Pinck's debit balance, $2,264.14, minus his 10% of the net profit ; hence —
$2,264.14 — 10% = defalcation
Substituting the first term of this equation for "defalcation" in the first equation, we obtain —
$8,640.46 — ($2,264.14 — 10%) = 100%
Removing the parentheses and changing signs —
$8,640.46 — $2,264.14+ 10% = 100%
Subtracting 10% from both sides of the equation —
$8,640.46 — $2,264.14 = 90% or $6,376.32 = 90%
Dividing both sides by 90% —
$7,084.80 = 100%, the net profit Then 708.48 = 10%, Pinck's bonus and 1,062.72 = 15%, Brown's bonus
$2,264.14, Pinck's debit balance 708.48 " bonus credited
'1)555-66 " defalcation
EQUATIONS IN THE SOLUTION OF PROBLEMS 75
$8,640.46 undivided profit per balance sheet 1,555.66 Pinck's defalcation
$7,084.80 net profit — basis of Brown's and Pinck's bonus $708.48 Pinck's bonus
1,062.72 1,771.20 Brown's bonus and total
$5,313.60 profits remaining for dividends
Vs of $5,313.60 = $3,985.20, Black's dividend yg " " = 664.20, Green's "
1/8 " " = 664.20, White's "
i/g" " = $5,313.60, total (as above)
CHAPTER VIII TRADE AND CASH DISCOUNT
Trade Discount
There are two kinds of discount affecting the amount received for goods sold or the amount paid for goods bought. The first of these is trade discount, which is a device for varying prices without interfering with basic or *'hst" prices, often called retail prices. The convenience of its use arises from the fact that an expensive catalogue can be made permanent by recording only the list prices. The real or trade prices are determined by the trade discounts offered by the seller, which are usually contained in confidential letters or circulars sent to customers.
A great saving of expense is effected by not having to issue a new catalogue whenever market prices are modified. Moreover the printing and circulating of large catalogues would take con- siderable time so that it would be impossible to give effect to price revisions until weeks after the necessity for them had arisen. But a circular altering the trade discount and thus raising or lowering the actual prices, can be prepared on a reproducing ma- chine and sent out to customers in one or two days.
The way in which a trade discount operates is as follows: A manufacturer or wholesaler sells to a retailer at 75 cents an article the list or gross price of which is $1.25. He bills the article to the buyer at the gross price but then deducts the discount of 40%, thus:
16 dozen of article I15.00 $240.00
Less 40% 96.00 $144.00
In this way the actual or net price of the article isestabhshed at 75 cents. This actual price is the amount entered in the books
76
TRADE AND CASH DISCOUNT 77
of both seller and purchaser, neither of whom makes any record of the list price or the trade discount.
Cumulative Trade Discounts
In order to provide for fluctuations in price the device of cumulative trade discounts has been adopted. Thus the dis- counts quoted may be 30, 20, 10, and 5. This does not mean a total discount of 65, because each successive rate is calculated on the amount left after deducting the discount at the preceding rate. Thus, if the list price is $240, and the discount is 30, 20, 10, and 5, the computation is as follows:
List price $240.00
Less 30% of $240.00 72.00 $72.00
$i6S.oo Less 20% of $168.00 3360 33.60
$134.40 Less 10% of $134.40 13.44 13-44
$120.96 Less 5% of $1 20.96 6.05 6.05
$114-91 $125.09
The total discount of $125.09 is almost exactly 52 3^ % of the gross price of $240. The net or real price is $1 14.91 .
If the wholesaler wishes to advance the price, he notifies the trade that the last discount of 5% is discontinued, which will make the amount $120.96, or that the last two discounts of 10 and 5 are replaced by 5 alone, raising the price to $127.68 ($134.40 - $6.72).
If the price is to be lowered, it is done by adding a further dis- count. If another 5 is added the net price becomes $109.16 ($114.91 - $5.75).
Methods of Finding Net Price
In order to avoid the necessity of making a separate computa- tion for each discount, it is well to know how to find one rate that
78 MATHEMATICS OF ACCOUNTING AND FINANCE
will give the same result as that reached by the successive steps of the combination rate. The rule for this is :
Add the first two discounts; multiply the two discounts; sub- tract the second result from the first. With this result as one dis- count combine the third in the same manner; and take up in turn each of the other discounts.
Thus in the illustration —
The sum of .30 and .20 is .50
The product of .30 and .20 is .06 .44
The sum of .44 and .10 is .54
The product of .44 and .10 is .044 .496
The sum of .496 and .05 is .546
The product of .496 and .05 is .0248 .5212
Therefore the combination rate is .5212
It is not necessary to remember any rule, because this same result can be reached by computing the discount on $100 thus:
List price $100.00
Less 30% 30.00 .30
$70.00 Less 20% 14.00 .14
$56.00 Less 10% 5.60 .056
$50.40 Less s% 2-52 .0252
Net price and total discount rate. . $47.88 .5212
This latter method has the further advantage of being sus- ceptible of easy proof, since the net price and the discount must add to 100.
A still better way of reaching the same result is to compute the net amount at once, instead of finding the discount and deduct-
TRADE AND CASH DISCOUNT 79
ing it. Thus, instead of ascertaining that 30% of $240 is $72 and deducting it to find $168, it is shorter to multiply 240 by 70%, or rather by .7, to get the same result, and then to multiply 168 by .8 to get 134.40, and so on. In other words, mul- tiply each successive amount by the complement of its discount rate, and the final result will be the net price, with all discounts deducted.
In order to find a single rate for the net amount, multiply all the complements of the discount rates. Thus, the product of .7 X .8 X .9 X .95 is .4788, the single rate for the net price when the discounts are 30, 20, 10, and 5. Applying this single rate to $240 will give $114.91, the same result as by the other method.
It will save a great deal of labor, as well as insure greater ac- curacy, if tables are made of the net amounts resulting from the application of the discounts usually given by the concern. These tables should be made for amounts of $1 to $100 and should be built up by successive additions and not multiplications. They will prove at every tenth amount, since if the net for $1 is $.4788, it will be $4.7880 for $10, thus proving every intermediate amount, whereas there is no proof if each amount is found by independent multiplication.
It is not necessary to carry the tables beyond $100 if all the decimals are used. It is necessary only to be careful to move the decimal point enough spaces to the right to represent the higher numbers. Thus, if the list price is $4,836 and the net rate .4788, the table will show on line 36 the amount of 17.2368 and on line 48 the amount of 22.9824. Therefore:
The net of $ 36.00 is $ 17.2368 " " " 4,800.00 " 2,298.24
" " " $4,836.00 " $2,315.48
The same result could be reached with a table of only ten lines, from $1 to $10, but a table of 100 lines shortens the computa- tions and is still easily contained on a comparatively small card.
8o MATHEMATICS OP ACCOUNTING AND FINANCE
Cash Discount
In order to induce prompt payment of accounts merchants frequently offer to deduct a certain per cent from bills if they are paid within a fixed number of days. This deduction is called a cash discount and is always applied to the net or trade price reached after all trade discounts have been deducted.
The discount terms are expressed by the figure of the per cent followed by the number of days in which the discount is allowed and then by the total number of days that may elapse before the bill becomes due, thus:
2/ 10 ; 1/30 ; N/60
which means that if payment is made in 10 days 2% may be de- ducted; if paid in 30 days 1%; and that the bill is due in 60 days net, that is, without any discount.
The advantages of giving cash discount are usually said to be that they decrease :
1. Loss from bad debts
2. Cost of collecting accounts
3. Amount of capital tied up in outstanding accounts
As only those who are comparatively strong financially are able to avail themselves of discounts offered, the first advantage would not appear to be very often realized.
The advantage to the purchaser is that he makes much more than normal interest by taking his discount. Thus, in the case of 2/10; 1/30; N/60, if the 2% is taken the purchaser gains 2% for 50 days' use of the money, which is at the rate of 14.6 per cent per annum, while if only 1% is taken he gains 1% for 30 days, or 1 2% per annum. If he has sufficient bank credit, he can well afford to borrow at 6 or 7% in order to take his discounts.
Discount as a Protection against Loss
The term "cash discount" is usually understood to refer to de- ductions that amount to a rather heavy interest. In some cases,
TRADE AND CASH DISCOUNT 8l
however, the deduction allowed is far more than this. For in- stance, one concern manufacturing electrical apparatus sells on terms of 40% discount if paid within 30 days. If not paid within the time the price is at list. So large a discount is not usually considered to come within the definition of a cash discount, al- though it is such, strictly speaking, since it is dependent upon the payment of cash. It would perhaps be better to call it a trade discount with a time limit.
A discount of this kind is adopted as a protection against loss in case of the bankruptcy of a customer. If a discount is given purely as a trade discount, a price is established that remains the same whether a bill is paid at maturity or not. If the customer becomes bankrupt, the claim filed with the receiver must be the net with all trade discounts deducted. If the list price is $1,000 and the unconditional trade discount is 40%, the claim must be filed for $600. If the final settlement is for 50%, the creditor loses $300. But if the discount has a time limit of 30 or 60 days, the time will have expired and a claim can be filed for the list price of $1 ,000, on which the dividend will be $500. The creditor will lose only $100, instead of $300.
Cash Discount Regarded as an Expense
A view of cash discount not very generally accepted is that the net price is the real price and that if the bill is not paid in time the discount is added as a penalty. This, of course, reverses the usual understanding of the subject, which is that discount taken is a profit. If this view is adopted, cash discount taken is eliminated entirely, and discount not taken becomes an ex- pense.
To illustrate, if the trade price is $1,000 with an option of a cash discount of 2%, the entries would be as follows:
Purchases $980
Cash discount 20
Creditor $1,000
82 MATHEMATICS OF ACCOUNTING AND FINANCE
Then if discount is taken,
Creditor $i,ooo
Cash $980
Cash discount 20
In this case the discount disappears entirely. If the discount is not taken the entry would be :
Creditor $1 ,000
Cash $1 ,000
This leaves the charge of $20 in the cash discount account as an expense.
CHAPTER IX
TURNOVER
Indefinite Meaning of " Turnover"
The principal difficulty in discussing turnover is to obtain a clear idea of what is meant by the term.
In spite of the efiforts of several committees of the American Association of Public Accountants, we are no nearer an authori- tative standard of accounting terminology than we were ten years ago. The principal difficulty in arriving at correct definitions is that few authors on accounting subjects attempt any definitions at all. They seem to take for granted that everyone else attaches the same meaning to a term as they do themselves and that a definition of it is, therefore, unnecessary. There does not seem to be any formal definition of "turnover" in any standard work on accounting, and the word does not appear to be used with any clearcut meaning.
R. H. Montgomery comes nearest to defining the word, but even he only suggests a definition. After stating that authorities differ greatly as to what the term means, he says: "Uniformity is desirable in accounting terminology, so the author suggests this definition: The turnover of a merchant or manufacturer repre- sents the number of times his capital in the form of stock-in- trade is re-invested in stock-in-trade during a given period."^
This is the generally accepted definition of turnover; that is, the number of times the merchandise is turned over. Mr. Mont- gomery further says: "To ascertain the turnover, take the start- ing inventory, add the purchases or cost of manufactured goods, and deduct the inventory at the end; divide the total by the start-
' R. H. Montgomery, Auditing Theory and Practice, 1919, p. 455.
83
84 MATHEMATICS OF ACCOUNTING AND FINANCE
ing inventory. The result will be the number of times the capital invested in stock-in-trade has been turned over during the period. The calculations are based upon a normal inventory."^
Normal Inventories Necessary
The application of this rule to two successive years of a business will exhibit the importance of the qualification that the inventories must be normal. It will also show the difficulty of determining in the case of the majority of concerns how many times the stock-in-trade is turned over.
If the starting inventory of the first year is $25,000, the pur- chases $200,000, and the inventory at the end is $50,000, the turnover during the year is 7, since $25,000 -j- $200,000 — $50,000, or $1 75,000, is seven times the first inventory. If in the following year the starting inventory is $50,000, the purchases $175,000, and the ending inventory $25,000, the turnover is 4, since $50,000 + $175,000 — $25,000, or $200,000, is four times the first inven- tory. Thus by the mere accident of a difference in the amount of the inventories at the beginning of the two years, the second year, which did the larger business, shows only a little more than half of the turnover of the first year. As the number of turnovers in the year is supposed to be a measure of the prosperity of the busi- ness, this method of determining it is evidently unsatisfactory.
The difficulty is greatly reduced if it is possible to determine the average, normal quantity of stock-in-trade carried. This normal quantity may or may not be the same as the inventory at the beginning of the year as there is no necessary connection between the two. Usually, however, we only know that there was a total turnover of $175,000 or $200,000 during the year. Not knowing the amount of the concern's normal inventory from any figures contained in either its revenue statement or balance sheet, we have no means of finding how many times the stock has been
^ R. H. Montgomery "Auditing Theory and Practice," 1919. P- 455-
TURNOVER 85
turned over. This we can ascertain only it inventories are taken monthly or at other intervals throughout the year, the average of which may be regarded as the normal inventory. In any event it is necessary to deline the word "normal" as it may mean the average inventory that should be carried or the inventory that is customarily carried.
Different Bases of Comparison
Even if this difficulty is met, another presents itself when choosing a basis for comparing the turnovers of two concerns. Mr. Montgomery probably represents at least the majority of American accountants in considering turnover to be the number of times the normal stock-in-trade is reinvested in the goods sold. British accountants, on the other hand, following Lisle, say that it should be the relation between the inventory and the sales. It can readily be seen that a serious misunderstanding results in the comparison of the turnovers of two concerns if one is calculated on the basis of cost and the other on the basis of sales, especially if high selling expenses make the cost of the goods a compara- tively small part of the selling price.
This difference of opinion as to the proper basis for comparing turnovers has reference to trading businesses. In the case of manufacturing businesses the disagreement is even greater, as three bases of comparison may be used, raw material, raw ma- terial plus labor, and the total cost of the goods sold.
Working Capital as Basis of Turnover
Owing to this wide divergence of opinion, it has been sug- gested, and it would seem justifiably, that a better basis for cal- culating the turnover is the working capital of the business. If this method is adopted, most of the difficulties encountered in computing the turnover disappear, because the amount of work- ing capital is indicated in the balance sheet, being the excess of current assets over current liabilities. Except as affected by the
86 MATHEMATICS OF ACCOUNTING AND FINANCE
slight increase due to undistributed profits, it remains the same throughout the entire period, and, therefore, affords a stable basis of comparison.
Every business, whether it is engaged in trading or manufac- turing, requires capital for two purposes: first, to provide the necessary fixed assets to carry on the operations; and second, to furnish sufficient funds to carry the stock-in-trade and accounts receivable until cash is realized on them and more goods or mate- rial for manufacture are bought. The point that interests the proprietor is how many times a year he can invest his available floating capital by repeating the process of buying the goods, selling them, and collecting the proceeds. If one person with a working capital of $40,000 is able to sell in a year goods costing $200,000, while another with the same amount of working capital is able to sell goods costing only $160,000, the first person has turned over his capital five times to the other person's four times.
Some exception is taken to this definition of turnover. It is said that working capital is used for many purposes, including advertising campaigns and similar items of expense that do not affect the cost of manufacture. It is also objected that accounts receivable vary to a great extent between different concerns, as some sell on short time and others on long time, and some keep a big stock of raw materials, while others do not consider it neces- sary, or cannot afford to do so.
The question of what constitutes the proper basis for reckon- ing the turnover seems to turn on the object for which the turn- over is used. If it is merely for the purpose of showing that one manager can handle more goods than another with the same average amount of stock on hand, the proper test is the relation between the normal inventory and the cost of the goods sold. But a manager could establish a record on this basis by following the foohsh policy of buying in small quantities at retail prices and paying the high expressage instead of the low freight rates. His turnover would be large, but would lead to disaster.
TURNOVER 87
If a profitable business is the object sought for, the use the manager makes of the working capital would seem to be one of the best measures of his success. If too much of the working capital is diverted from the production or purchase of goods for sale to carrying on an extensive advertising campaign, or if it is tempora- rily locked up in long-time accounts receivable, the quantity of goods sold is apt to decrease. This will show itself in a lessened turnover of working capital. On the other hand, if either the beginning or the normal inventory is made the measure, the turnover will remain the same even though the business done is smaller, because the inventory will decline with the volume of business. An inventory of $25,000 and cost of sales totaling $100,000 will show the same turnover on this basis as $50,000 in- ventory and cost of sales totaling $200,000. When the yardstick varies in length, comparative measurements are of little value.
The use of working capital as the basis of turnover is logical, first, because the capital is put in the business for the purpose of being turned over as rapidly as possible; second, because it is virtually constant; and third, because it presents all the elements concerned in the turnover, not only the stock-in-trade, but also the accounts and notes receivable, by means of which the turnover is effected. The turnover of working capital also furnishes a better criterion of the excellence of the management. With the inventory as the only standard a manager can make an apparently good record by starving his stock-in-trade. If, however, he uses working capital as the standard, he makes his best record by dihgence in collecting outstanding accounts, and increasing the supply of cash for the development and handling of a more extensive business.
Definition of Working Capital
Even if this is granted, the difficulty does not come to an end,
because there is no authoritative detinition of working capital.
H. R. Hatfield says: "Working capital has long had a specific
88 MATHEMATICS OF ACCOUNTING AND FINANCE
meaning as a collective term for what are often called quick assets, e. g., cash, accounts receivable, perhaps merchandise, etc." ^
H. C. Bentley says: "Working capital is the excess of quick assets over quick liabilities."'*
Each of these authorities has his followers, as was shown in a recent discussion of the subject in the Journal of Accountancy. The definitions differ because in one the notes and accounts pay- able are considered to be borrowed capital, while in the other only the amount contributed by the proprietor is treated as capi- tal. The first is the economic view of what constitutes capital, while the second is the business and general accounting view. If the ordinary business man is asked how much capital he has in his business, he will always state the amount of his proprietary interest. Unlike Micawber, he does not think he has added to his capital whenever he issues a note payable.
Need of Exact Definitions
The final lesson to be learned from the consideration of this subject is that in preparing comparative statements of turnover, accountants should first define the terms used and should plainly state the basis of the calculations. Otherwise the conclusions reached will be entirely misleading to persons whose conception of the subject is dift'erent. It is to be hoped that the Institute of Accountants will eventually end the present ambiguity by formu- lating an authoritative definition of turnover.
Having settled upon the basis to be used, the accountant de- termines the turnover by ascertaining how many times the normal inventory will go into the cost of the goods sold, or into the sales, according to which view is adopted, or how many times the work- ing capital has been reinvested in the purchase or production of goods sold. The greater the quotient in each case, the more pros- perous and better managed the business is supposed to be.
3 H. R. Hatfield, Modern Accounting, 1909, p. 179.
4 H. C. Bentley, Science of Accounts, 191 1.
CHAPTER X
PARTNERSHIPS
Division of Profits
Profits may be divided by partners in any proportions to which they agree; if they malce no express agreement the law im- plies an agreement to divide the profits equally, regardless of the capital or services contributed. The customary methods of dividing profits are:
1. In the ratio of the capital balances at the beginning of the
period.
2. In the ratio of the average capitals for the period.
3. In an arbitrary ratio, usually expressed in terms of frac-
tions or per cents.
4 . In an arbitrary ratio after allowing interest on the capitals.
To illustrate these methods, assume the following facts:
Illustration
A and B are in partnership and the profits for division at the end of the year are $12,000.
A's capital account during the year undergoes the following changes:
Credit balance, January i $50,000
Investment, March i 2,000
" November i 3,000
Total credits $55,000
Withdrawal, May i $500
" December i 1,000
Total debits 1,500
Credit balance, December 31 $53,500
90
MATHEMATICS OF ACCOUNTING AND FINANCE
B's capital account changes as follows:
Credit balance, January i $25,000
Investment, February i 10,000
" June 1 5,000
Total credits
Withdrawal, October i
Credit balance, December 31
j.0,000 1,000
59, 000
Solution i: Division of profits in the ratio of the capital balances at the beginning of the year:
This ratio is A, 50, and B, 25; or 2 to i. A, therefore, is credited with % of $12,000, or |8,ooo; and B with }4 of $12,000, or $4,000.
Solution 2: Division of profits in the ratio of the average capital for the period:
This ratio may be computed in either of two ways. The first is as follows: Multiply each capital account credit by the number of months or days from the date of the credit until the end of the period, and find the sum of these products. Multiply each capital account debit by the number of months or days from the date of the debit until the end of the period, and find the sum of these products. Find the difference between the credit products and the debit products.
Do this with each capital account and determine what fraction each difference is of the sum of the differences. The difference between the credit and debit products of A's capital account is found thus:
Credits
Date Amount Time
January i $50,000 12 mo.
March i 2,000 10 "
November i 3,000 2 "
Debits
May I $ 500 8 mo.
December i i ,000 r "
Difference
Product
$600,000
20,000
6,000
$626,000
$ 4,000
1,000
5,000
$621,000
PARTNERSHIPS
91
The difference between the credit and debit products of B's capital account is found thus:
Credits
Date Amount Time
January i $25,000 12 mo.
February i 10,000 11 "
June 1 5.000 7 "
Debits October i $1,000 3 mo.
Difference
Product
$300,000 110,000 35,000 $445, 00c
3,000
|.2,000
The partnership profits of $12,000 are accordingly divided in the follow- ing ratios and amounts:
Ratios Expressed in Fractions
A 621/1063
B 442/1063
Division of Profits
$ 7,010.35 4,989.65
$12,000.00
The second method by which profits may be divided in the ratio of the average capitals for the period is as follows:
Multiply the opening balance of each account by the number of months or days it remained unchanged.
Multiply each new balance resulting from investments or withdrawals by the number of months or daj's it remained unchanged.
Find the sum of these products for each capital account.
Find the ratio of each sum to the total for all.
The products for A's capital account are computed in the following way:
92
MATHEMATICS OF ACCOUNTING AND FINANCE
Balance
From
To
Time
Product
$50,000
January
March
2 mos.
$100,000
52,000
March
May
2 "
104,000
51,500
May
November
6 "
309,000
54,500
November
December
I mo.
54,500
53,500
December
December
31
I " 12 mos.
53,500
$621,000
The products for B's capital account are figured as follows:
Balance
From
To
Time
Product
$25,000
January i
February
I
I mo.
$ 25,000
35'000
February i
June
I
4 mos.
140,000
40,000
June I
October
I
4 "
160,000
39,000
October i
December
31
3 "
117,000
This method results in the same ratios as the first method, and it has the advantage that the final balances shown in the computation ($53,500 and $39,000) are the same as the balances of the accounts; and that the time numbers used as multipliers add to a full year. Checks on the accuracy of the computations are thus provided.
It must be understood that these computations determine the average capital ratios but not the average capitals. To compute the average capitals it would be necessary to divide by 12, thus:
$621,000 -^ 12 = $51,750.00, A's average capital $442,000 H- 12 = $36,833.33, B's "
Since only the ratio between the average capitals is required, the division by 12, or by 365 if the numbers of days have been used as multipliers, is unnecessary.
Solution 3 : Division of profits in an arbitrary ratio. No limit can be placed on the variety of arbitrary ratios which can be agreed upon and illustrations are unnecessary.
Solution 4: Division of profits in an arbitrary ratio after allowing interest on capital. No interest can be allowed unless there is a specific agreement to do so, and in that event the rate should be agreed upon.
PARTNERSHIPS 93
Assuming a rate of 6% on the opening balances of the capital accounts, the distribution, if the remaining profits are divided equally, would be as follows :
A B Total
6% of $50,000 $3,000
6% of 25,000 $1,500
Total interest $4,500
Balance equally 3,750 3,750 7,500
$6,750 $5,250 $12,000
If the agreement provides for interest on partners' capitals, it must be credited to them even though it exceeds the total profits. In that event the resulting debit balance in the profit and loss account is charged to the partners in the agreed ratio.
Assuming that the profits were only $4,000, the division would be as follows:
A B T0T.VL
Credits for interest (as above) $3,000 $1,500 $4,500
Debits for excess of interest over profits 250 250 500
Net credits $2,750 $1,250 $4,000
If interest is provided for in the agreement, it must be credited to the partners even though a loss has been incurred instead of a profit earned. Assuming a loss of $1,000, the division would be as follows:
A B Total Debits for sum of loss of $1 ,000 and in- terest of $4,500 $2,750 $2,750 $5,500
Credits for interest 3,000 1,500 4,500
Net credit $250
Net debit $1,2^0 $1,000
The result is that B bears all of the loss from operations as well as the $250 net credit to A.
94 MATHEMATICS OF ACCOUNTING AND FINANCE
Liquidation of Partnerships
When a partnership is terminated, the procedure to be fol- lowed in realizing the assets, liquidating the liabilities and dis- tributing the partners' capitals, depends on whether all losses on realization have been ascertained before payments are made to the partners. If they have been, the losses are deducted from the partners' capitals in the profit and loss ratio, and the remaining assets, after paying the outside creditors, are distributed to the partners in amounts sufficient to pay off the capitals.
To illustrate, assume that all partnership debts have been paid, and that the capital accounts are:
A $10,000 B $8,000
There must be assets of $18,000. These are sold for $15,000. With losses divided equally, the division of cash proceeds is as
follows :
A B Total
Capitals $10,000 $8,000 $18,000
Losses on realization i,Soo 1,500 3,000
Balances paid in cash $8,500 $6,500 $15,000
Periodical Distributions
If periodical distributions to the partners are made before all assets are realized and all losses ascertained, they should be made, if possible, in such a way as to reduce the balances of the capital accounts to the profit and loss ratio existing between the partners, so that if all remaining assets are lost each partner's capital account will be exactly sufficient to cover his share of the loss.
To illustrate, assume that all liabilities are paid and the part- ners' capitals are as follows :
A $15,000 B 20,000 C 25,000
PARTNERSHIPS 95
The assets total $60,000. In realizing on $i8,oco worth of assets a loss of $3,000 is incurred, so that there is $15,000 in cash to divide. The division of cash should be made as follows, as- suming that the partners share profits and losses equally.
ABC Total
Capitals $15,000 $20,000 $25,000 $60,000
Loss 1,000 1,000 1,000 3,000
Balance before dividing cash $14,000 $19,000 $24,000 $57,000
Cash 5.000 10,000 15,000
Balances left in P. & L. ratio $14,000 $14,000 $14,000 $42,000
It should be noted that the cash is not distributed in the capi- tal ratio, the profit and loss ratio, nor any other ratio, but in arbitrary amounts sufficient to reduce the capitals to the profit and loss ratio.
Reducing Capitals to Profit and Loss Ratio
It is not always possible to bring the capital account balances to the profit and loss ratio at the first distribution of cash. This is the case if the capital of any partner after charging off all ascer- tained losses is less than his profit and loss ratio of the assets which will remain after making the proposed distribution.
To illustrate, assume the capitals to be as follows:
A $10,000
B 20,000
C 30,000
Losses are to be shared equally. The assets total $60,000 and all Habilities are paid. Assets carried on the books at $30,000 are sold for $24,000, the loss of $6,000 being divided as follows:
ABC
Total
Capitals before dividing loss
Loss
Capitals before distribution of cash
$10,000 $20,000 $30,000 2,000 2,000 2,000
$60,000 6,000
$8,000 $18,000 $28,000
$54,000
96 MATHEMATICS OP ACCOUNTING AND FINANCE
After the $24,000 is divided there will remain a total capital of $30,000. If possible the $24,000 cash should be divided in such a way as to leave each partner with a balance of $10,000; but this is clearly impossible since A's balance is already reduced to $8,000. Since A's capital is not sufficient to bear his share of the total possible loss of the $30,000 of assets which remain after the distribution, nothing should be paid to A. In the event that a total loss of $30,000 is incurred the charge of $10,000 to A would leave his account with a debit balance of $2,000. If he could not pay in the $2,000 it would have to be charged against B and C in their profit and loss ratios, which in this case happen to be equal. Therefore, B and C may possibly lose the following amounts :
B C
One-third each of total possible $30,000 loss $10,000 $10,000 Excess of A's share of possible loss over his
capital 1 ,000 1 ,000
Total possible loss $11,000 $11,000
The $24,000 cash should be divided between the two partners in such a way as to reduce their balances to these amounts, as follows :
ABC Total Capitals before distribu- tion of cash $8,000 $r8,ooo $28,000 $54,000
Cash distributed 7,000 17,000 24,000
Balances $8,000 $11,000 $11,000 $30,000
In the next realization, assets carried at $15,000 are realized at $12,000, the loss being $3,000. After dividing the loss the cash can be distributed in amounts which will reduce the capital balances to the profit and loss ratio of equality.
PARTNERSHIPS
ABC Total
Balances (as above) $8,000 $11,000 $11,000 $30,000
°^^ i-ooo 1,000 1,000 3,000
Balances before distribut-
^"S cash $7,000 $10,000 $10,000 $27,000
^^ 2,000 5,000 5,000 12,000
Balances (reduced to P. &
^- ^^^^°) $5,000 $ 5,000 $5,000 $15,000
The remaining assets are sold for $9,000, the di- vision of loss and cash being as follows:
^°^^ 2,000 2,000 2,000 6,000
Balances paid in cash ... . $3,000 $3,000 $ 3,000 $ 9,000
97
CHAPTER XI
THE CLEARING HOUSE
Principle of the Clearing House
The general principle on which a clearing house operates is that of offsetting debits and credits and dealing only with net differences. Its simplest form is illustrated when two persons buy from and sell to each other. If instead of each paying his bill to the other in full, the one who owes the larger sum deducts the other's debt to him and pays the difference, he has to that extent adopted the clearing house principle.
In the fullest application of the principle, an outside party is introduced to act as settling or clearing agent. Each unit in the combination that forms the clearing house has dealings with every other unit, but instead of settling its dealings with each of the other units individually it charges the total of all its debits and credits the total of all its credits to the clearing agent, with which it settles the net difference of the totals. In this way a single settlement takes the place of a large number of settlements. An example will make this clear.
Debits and Credits with Clearing House
Suppose there are six banks in a city. Every day they re- ceive on deposit and otherwise, checks on each other. If there is no clearing house, each bank presents to each of the others the checks drawn on it and collects or pays the difference due to or from it, according as the amount of the checks it pre- sents is greater or smaller than the amount of those presented to it.
If the banks form a clearing house, each provides itself with a blank form, in which it inserts the amount of checks it holds
98
THE CLEARING HOUSE
99
against each of the others. The total it charges to clearing house checks on the teller's blotter. This form and the checks themselves are sent to the clearing house. The checks are de- livered to the representatives of the banks on which they are drawn, with a memorandum of the amount for each bank. As each bank receives the checks drawn on it, it enters the amounts in the second column of the form and the total becomes the credit to clearing house checks. If a bank's credits to the clearing house are greater than its debits, it pays the difference to the clearing house. The amounts thus paid in are afterwards paid to those banks whose debits to the clearing house are greater than their credits.
Clearing House Transactions
The following are assumed to be the transactions on a certain day of the six banks in the clearing house, it being remembered that the first column of figures on each blank is made up before the checks leave the bank and the second column at the clearing house.
First National Bank
Atlas Nat ^27,819.32
Merchants St 12,948.24
Traders St 22,687.19
State Tr. Co 18,729.63
Union Nat 14,963.48
Atlas National Bank
$16,248.76 First Nat $16,248.76
14,629.83 Merchants St 11,283.42
26,963.42 Traders St 12,732.63
21,246.38 State Tr. Co 15,376.28
IS. 782. 16 Union Nat 9.659.87
C. H. owes .
197,147-86 $94,870.55
$ 2,277-31 Owes C. H.
$27,819.32
8,664.19
10,497.26
11.958.47
7,326.58
$65,300.96 $66,265.82 $ 964.86
Merchants State Bank
i^irst Nat
. . $14,629.83
$12,948.24
Atlas Nat
8.664.19
11,283.42
Traders St
9.586.28
10,392.16
State Tr. Co . . .
11,473.47
9.627.59
Union Nat
10,827.42
11.432.27
$55,181.19
$55,683.68
Owes C. H
$ 502. 49
Traders State Bank
First Nat $26,963.42
Atlas Nat 10,497.26
Merchants St 10,392.16
State Tr. Co 11,788.39
Union Nat 9.673.57
C. H.
$22,687.19 12,732.63
9,586.28 13.249.36
8,865.71
$69,314-80 $67,121.17 $ 2,193.63
100 MATHEMATICS OF ACCOUNTING AND FINANCE
s
First Nat .
TATE
Trust Compan
Y
$18,729.63 15,376.28 11.473-47 11,788.39
8,842.74
Union
First Nat
Atlas Nat
National Ban
. . $15,782-16
7,326.58
11,432.27
8,865.71
8,842.74
K
Atlas Nat .
St...
11,958.47 9.627.59 13,249.36
7.982.25
9,659.87
Merchants Traders St. Union Nat .
Merchants St. . .
Traders St
State Tr. Co
Owes C. H
10,827-42 9.673.57 7,982.2s
Owes C. H.
$64,064.05 $ 2,146.46
$66,210.51
$52,249.46 . . $ 857-13
$53,106.59
Debit
Credit
Debits
Credits
Balances
Balances
$ 94.870.SS
$ 97.147.86
$
$2,277-31
66,265.82
65.300.96
964.86
55.683-68
SS,l8l-I9
502.49
67.121. 17
69.314-80
2,193.63
66,210.51
64.064.05
2,146.46
S3. 106.59
52,249-46
857.13
$403,258.32
$403,258.32
$4,470.94
$4,470-94
Manager's Sheet
The representatives of the different banks are allowed a cer- tain number of minutes in which to finish the preparation of their respective sheets. They then in turn call out their figures which are registered by the manager of the clearing house on his own sheet in the following manner:
Banks
First National Bank Atlas National Bank Merchants State Bank Traders State Bank State Trust Company Union National Bank
If the manager's sheet balances, he rings his bell to signify that the session is over. If it does not balance he announces the fact. A certain number of minutes is allowed for finding the error. If found within that time the one who made the error is fined a certain sum. The fine increases progressively with any added time taken to reach a balance.
Later, the banks with debit balances make their payments to the clearing house manager, who shortly after pays off the banks with credit balances.
The total figures are kept by the manager for statistical pur- poses. The sum of the checks presented ($403,258.32) is the clearing house movement for the day, and the sum of the balances ($4,470.94) is the amount of the balances reported in the financial
THE CLEARING HOUSE lOI
news. The corresponding figures for the week, month and year are published periodically in the financial columns of the news- papers, and are considered a barometer of business activity.
Economy of System
The convenience and the economy of the clearing house sys- tem may be demonstrated by an analysis of the transactions of the Merchants State Bank. This bank has collected checks aggregating $55,181.19 and has paid checks totaling $55,683.68, or a total settlement of $1 10,864.87, by a payment of only $502.49, or less than one half of 1%. Without the clearing house, it would have been compelled, even by offsetting with each of the other banks, to make the following payments :
Atlas National Bank $2,619.23
Traders State Bank 805.88
Union National Bank 604.85 $4,029.96
It would have received the following amounts:
First National Bank $1,681.59
State Trust Company 1,845.88 3,527.47
Total cash movement in the bank would have been. . . $7,557.43
One payment of $502.49 takes the place of five transactions aggregating fifteen times as much.
Each bank must settle on the basis of the clearing house re- turns. The corrections of any errors in addition, the listing of checks, either for wrong amounts or on the wrong bank, etc., and the settlement for checks returned for lack of funds or on account of missing endorsements, are all matters left to the individual banks to adjust with each other.
Application of Principle Extended
The clearing house principle is applied to the settlement of accounts whenever several parties have reciprocal relations, in-
102 MATHEMATICS OF ACCOUNTING AND FINANCE
volving the transfer of value in the form of money, securities, or merchandise. It is applied to the settlement of accounts between brokers on boards of trade and other exchanges, and also to the regulation of sales of merchandise when an agreement exists among concerns in the same line of business restricting the output of each to an agreed percentage. At present there is very little of the latter use of the method, owing to the danger of violating the law which prohibits agreements in restraint of trade.
CHAPTER XII
BUILDING AND LOAN ASSOCIATIONS
General Characteristics of Building and Loan Associations
A building and loan association is founded on the general principle of co-operation among a large number of small investors and a smaller number of intending borrowers, who wish to get advances for the purpose of erecting buildings on land they own, or of paying off existing loans on real estate that has been partially paid for. There are several different plans on which such associa- tions are organized, but the idea of co-operation is present in them all.
In some foreign countries, notably Italy, this principle of co- operation has been extended to cover loans to workmen for the purchase of tools or machinery and even to the furnishing of large sums to working contractors for building railroads and other public works.
The usual characteristics of a building and loan association are as follows: Each member subscribes to a definite number of shares, on which he pays monthly instalments of one half of i% of the par of the stock. If, for example, the par is fixed at $ioo, the monthly payment on each share is 50 cents. When these instalments together with the profits earned by each share amount to $100, the stock is said to have matured and is paid off.
Terminating Plan
The terminating plan of organization, the first to be adopted, is the simplest of all. It involves the subscription by a limited num- ber of persons to a certain number of shares which are to be paid in instalments, and the dissolution of the association upon the
103
I04 MATHEMATICS OF ACCOUNTING AND FINANCE
maturity of the shares. As money is paid by subscribers it is loaned from time to time. As the interest collected is also loaned, the association is in theory receiving compound interest. For instance, if 2,000 shares are taken and 50 cents per share is paid monthly, a loan of $1,000, say at 6%, can be made at the end of the first month. At the close of the second month there is another $1,000 to lend, together with $5 interest collected on the first loan, and this is repeated from month to month. The com- pound interest theory, however, does not hold strictly here, as it is difficult to lend odd amounts, unless other security than real estate is taken.
The objection to this plan is that it is impossible to employ the money profitably in real estate loans, because during the latter part of the association's life loans have to be made for so short a time that great difficulty is experienced in making them; and if the later loans should be made for the usual term of five years the dues and profits on the stock could not be paid when they matured. It can, however, be arranged that after the stock reaches par no further payments should be made on the shares and no shares should be withdrawn, but the money should be kept invested and cash dividends paid out of the interest collected.
Practicability of Plan
The terminating plan can be used to best advantage when a large number of persons wish to pay in a given time a large sum of money with interest, each person paying a small amount monthly for the given time. This plan, for example, is extremely useful in paying off a church debt. If a church without any rich members owes $10,000 on which it is paying 6% interest, it might seem impossible to raise so large an amount. If a building and loan association is formed with 100 shares of $100 each, to run for five years, on which the monthly payments are fixed at $1.93 per month, it will usually not be found difficult to persuade the con
BUILDING AND LOAN ASSOCIATIONS I05
gregation to take the whole 100 shares. This will not only pay off the entire debt in five years, but will also pay the interest on the diminishing amount of the note, as the holder of the note will be willing to accept partial payments of even hundreds of dollars.
The plan works out in this way — Each holder of a share agrees to pay $100 and interest in monthly instalments of $1.93. If he makes his payments regularly, his account for the first two months runs as follows :
Original agreement $100.00
One month's interest at 6% .50
$100.50 First month's payment 1.93
$ 98.57 One month's interest at 6% .49
$ 99.06 Second month's payment 1.93
$ 07-13
The treasurer credits the interest account each month with the interest paid that month, and the principal with the balance of the payments, applying the money to the payment of the inter- est and principal of the note.
The amount of the monthly payments theoretically necessary is found from the compound interest table headed, "Amount per annum necessary to pay a debt of $1 now due and the interest thereon." In order to reduce it to monthly payments it is neces- sary to use the table of ^ % for the number of months in the life of the association. In our illustration we find the amount necessary, or $1.93, in the Yi % column on the line of the 60th period. For six years or 72 months the amount is $1.65^. As all the amounts will not be paid promptly, it is better to make the monthly payments $2 and $1.70 respectively.
io6
MATHEMATICS OF ACCOUNTING AND FINANCE
Serial Plan
The serial plan is virtually a succession of terminating plans combined in one association. A series is started at regular inter- vals of three, six or twelve months, with whatever number of shares may be subscribed. In some states the par of a share is $200 and the monthly payment per share is $1 ; but in most of the states the par is $100 and the payment is 50 cents per month, sometimes 25 cents per week. After a new series is begun no more shares can be issued in any previous series, although the holder of stock in any series can sell it or otherwise transfer it.
A serial association is a partnership with limited liability, the subscribers to the different series being the partners. The profits are divided among the series in proportion to their capitals which are the amounts paid in plus the accumulated profits to date. A simple illustration will make the process clear.
Suppose that an association is started on January i, 191 7, and that on September 30, 1918, the shares outstanding are as fol- lows:
Series
Date of
Issue
nu.mber of Shares
Paid per Share
Profit per Share
V\\LUE PER
Share
I
2
3
January, 1917
April,
July,
October,
January, 1018
April,
July,
800 600 700 500 400 600 500
Jio.so 9.00 7. so 6.00 450 3 00 1.50
$2
I I
29
71
25
88 56 27 02
$12.79
10.71
8.7S
6.88
s
6
7
5.06 3-27
1-52
During the cjuarter beginning October i, 1918, the eighth series of 700 shares is sold. The net profits for the quarter from interest, fines, etc., less the secretary's salary and other expenses, are $489.92. The undivided profits on September 30, 1918, are
BUILDING AND LOAN ASSOCIATIONS 107
$4,571.28. There are two plans by which the profits may be divided, the partnership plan and the Dexter rule.
Distribution by Partnership Plan
In the so-called partnership plan all previous profits are ig- nored and the capital of each series is taken as the money actually paid in multiplied by the equated number of months during which it has been left in the business. As the dues are payable on the first of each month, the equated time is found by adding i to the number of months the series has run and dividing the sum by 2. Thus, on December 31, 1918, the first series has run 24 months. Adding i and dividing by 2 gives 12^^ months as the equated time. As there are 800 shares in that series and as each share has paid in $12 the capital is $12 multiplied by 800 or $9,600 and it has been in an equated time of 12^ months. Multiplying $9,600 by 12^ gives $120,000 as the equated capital of Series i. Pursuing the same method the equated capitals of the several series are calculated as follows:
Series
I $12.00 X i2>^ mos. X 800 shares $120,000
2 10.50 X II "X 600 " 69,300
3 9.00 X 9/2 " X 700 " 59,850
4 7-5° X 8 "X 500 " 30,000
5 6.00 X 6>2 " X 400 " 15,600
6 4-5° X 5 " X 600 " 13,500
7 3-00 X 3>2 " X 500 " 5,250
8 1-50 X 2 "X 700 " 2,100
Total of equated capitals $315,600
The total profits to date, amounting to $5,061.20, are divided among the series in the proportion that each one's equated capital bears to the total equated capital. Reducing to the least com- mon denominator of 2,104, we have the following division of the profits among the several series.
io8
MATHEMATICS OF ACCOUNTING AND FINANCE
Series
I .
2.
3-
soo/ .
... 462/ • • ■ 399/
4-
200/
5- 6.
7- 8.
■ ■ ■ 104/ 90/ 35/
. . . 14/
800/2104 of $5,061.20 equals $1,924.40
,924.40
$2.41
Der share
,111-34
1.85
959-79
1-37
481.10
.96
250.17
.62
216.50
-36
84.19
-17
33-68
•OS
2,104
;,o6i.i7
Of course it is impossible to balance exactly the total profits with the sum of the profits assigned to each series.
Distribution by Dexter's Rule
The first step in applying the Dexter rule is to calculate the capitals of the series. The capital of each series on September 30 is the value per share as shown in the table on page 106 [of this chapter] multipHed by the number of shares outstanding. To find the earning capital on December 31, it is necessary to add the contribution of 50 cents per share for each of the three months of the current quarter. This is an average of $1 per share, cal- culated thus :
50 cents paid October i was in 3 months or $1.50 for i month
50 " " November i " "2 " " i.oo " i
50 " " December i " " i month " .50 " i "
Total $3.00
This is an average of $1 for three months. One dollar must, therefore, be added to the value per share shown in the first table. Bearing this in mind, the earning capital of the different series is found as follows:
BUILDING AND LOAN ASSOCIATIONS
109
Series
I 800 shares at $13.79 P^r share
" II. 71
600 700 500 400 600 500 700
9-75 7.88 6.06 4.27 2.52 1. 00
)II,032
7,026
6,825 3,940 2,424 2,562
1,260 700
Total capital $35,769
This rule leaves undisturbed the previous distributions of profits and adds to them the profits of the current quarter on the basis of the present earning capitals. The current profits of $489.92 are 1.37 per cent of the capital of $35,769. Hence the profit per share for each series is calculated as follows :
Series
Capital
Current Profit
Previous Profit
Total Profit
Shares
Profit per Share
$11,032 7,026 6,825 3.940 2,424 2,562 1,260 700
$151.14 96.26 93-50 53-98 33-21 35-10 17.26
9-59
$1,832 1,026 875 440 224 162 10
$1,983.14 1,122.26 968.50 493-98 257.21 197-10 27.26
9-59
800 600 700 500 400 600 500 700
$2.48
1.87
1.38
-99
.64
6
■ 33
• 05
8
.01
$35,769
$490.04
$4,569
$5,059-04
A comparison of the profits per share shows that the partner- ship plan gives larger profits to the later at the expense of the earlier series. This is a great injustice, as the earlier series are made to share the profits they have accumulated, with the later series, which had no part in producing those profits. This is not done in an ordinary commercial business when new partners are admitted. Dexter's rule is the true partnership plan and is the one usually adopted.
no MATHEMATICS OF ACCOUNTING AND FINANCE
Withdrawal of Shares
In case any shares are withdrawn during the quarter the holders are given the amount they have paid in and interest for the equated time at 3 or 4 %. Thus, if fifty shares in the third series are withdrawn on December i, 191 8, the amount paid in is $8.50 per share or $425, the interest at 3% for the equated time of nine months is $9.56, and the total amount paid is $434.56. The book value of this stock is $9.75 per share or $487.50. There is a book profit of $5 2. 94 on the transaction. No entry is made for this profit, but it is added to the cash profits of $489.92, which makes the divisible profits for the quarter $542.86.
Sources of Income
The profits of building and loan associations arise from inter- est paid by the borrowers and also from fees, fines and premiums.
The fees are membership fees, usually 25 cents per share, paid when the stock is originally subscribed. Sometimes these fees are capitalized in the series in which they are paid, making the value per share 25 cents more. Otherwise they are treated as a general profit, as in our example. In some associations a transfer fee is paid when stock changes hands.
The fines are levied on stockholders who are delinquent in their monthly payments, usually 5 cents per share on stock of non-borrowers (investment stock), and 10 cents per share on stock of borrowers, for each month the delinquency continues.
Premiums
The premiums are either payable monthly or are deducted from the loan when made. They arise from the bidding for loans by those desiring to borrow. When the association is in posses- sion of loanable funds, bids are invited. The person offering to pay the highest premium for the money receives the loan, if his
BUILDING AND LOAN ASSOCIATIONS III
real estate security is found to be adequate. On the cash pre- mium plan, 1% of the premium is paid monthly in cash. On the deducted premium plan, the whole premium is deducted from the loan in advance and only the net amount is paid.
If the loanable funds are $2,000 and the cash premium bid is 25%, the borrower makes his note for $2,000 and his monthly payments are: Dues $10, interest at 6% $10, and premium amounting to 1% of $500 premium or $5, making a total cash payment of $25.
For the same amount of money on the deducted premium plan, the bid may be for $2,500 at a premium of 20%. The borrower receives $2,000 cash, but his note is made for $2,500 and his monthly payments are: Dues $12.50 and interest $12.50, a total cash payment of $25. The premium of $500 is credited to unearned premiums, and at the end of each quarter profit and loss is credited and unearned premiums charged with $15.
Each borrower is obliged to subscribe for stock the par value bf which is equal to the face of his loan. When the dues paid on this stock and the profits credited to it cause it to be worth par, it is paid off by cancelling the loan. No payments are ever applied to the loan, which remains in full force until cancelled by the matured stock.
It will be noted that in both the serial plans the treatment is entirely on the basis of the several series being fully paid to date. If stock in any series is delinquent, it still receives its share of the profit, the only offset being the small fine, which in the older series is much less than the earnings for the quarter.
Individual Plan
To meet this objection to the serial plan, some associations have adopted the individual plan, which really makes each stock- holder a series by himself. The capital is then the actual amount paid in plus accumulated profits. The procedure as to earning capital and profits is otherwise the same as in the serial plan.
112 MATHEMATICS OF ACCOUNTING AND FINANCE
Dayton, or Ohio Plan
One great objection to the plans hitherto considered, or to any modification of them, is that they place the borrowers in an uncertain position with regard to the time at which their loans will be cancelled, and consequently with regard to the amount of principal and interest they will be called upon to pay. Their loans are not cancelled until their stock matures. If the associ- ation proves to be very prosperous, the profits will be large and the stock will mature in a comparatively short time. On the other hand, in a poorly managed association, it may take a long time to mature the stock, and in the meantime the borrowers will have to continue the payment of their instalments and interest, making their loans much more expensive than they have anticipated.
To meet this objection various forms of the "Dayton Plan" have been adopted, all of which agree in this — that the borrower is given a definite contract. His loan is to be repaid in a fixed number of months by the regular payment of a definite amount, which will cover both principal and interest. The calculation of the amount to be paid is based on the same principle as the one already shown for liquidating a church debt. The amount applicable to the principal is credited to the loan each month by the association, as the borrower does not have to own any stock. The loanable funds are obtained from the instalments paid by investing stockholders, with whom the accounts are kept on the individual plan.
This system is really that of a co-operative savings bank, and is the fairest method of all. When properly managed, it is very successful.
The two following problems illustrate principles dealt with in this chapter.
Problem i
A building and loan association published the following statement on December 31, iqi8:
BUILDING AND LOAN ASSOCIATIONS
Age and Condition of Shares
113
Series
Date of
Issue
Number of Shares
Paid per
Share
Profit per Share
Value per Share
I
January, 191 7 April, July,
October, " January, 1918 April, July. October, "
800 600 700 Soo 400 600 500 600
$12.00 10.50 9.00 7.50 6.00 4-50 3.00 I. so
$2.56
1.97
ISI
1. 14
.82
• 53
.26
.02
$14.56 12.47 10.51
3
4---
s
8.64 6.82
6
3.26
8
The balance sheet showed that the profits to date were $5,647.82. On March 31, 1919, the net profits from interest, premiums, fines and fees, less salary and expenses, were found to be $483.16. There had been with- drawn during the quarter 50 shares of the 4th series on which $10.75 ^^- terest had been paid, not included in the expenses previously mentioned. The Qth series was opened and 400 shares were subscribed and paid for the quarter.
Distribute the profits March 31, 1919, on the partnership plan.
Solution:
First find the equated time; which is ascertained by adding i to the number of months in the age of each series and dividing by 2. Next multi- ply the equated time by the amount paid and then by the number of shares outstanding in each series, as follows:
Series
I . .
•• 27
months old.
equated
14 mos. X 3
13.50 X 800 si
1. $151,200
2. .
.. 24
I2>2
' X
12.00 X 600
90,000
3--
21
II
' X
10.50 X 700
80,850
4--
.. 18
9>4
' X
9.00 X 450
38,475
S--
• 15
8
' X
7.50 X 400
24,000
6..
12
6>^
' X
6.00 X 600
23,400
7-
9
5
" X
4.50 X 500
11,250
8..
6
s'A
" X
3.00 X 600
6,300
9..
•• 3
2
' X
1.50 X 400
1,200
$426,675
114
MATHEMATICS OF ACCOUNTING AND FINANCE
The profits to be divided are found as follows:
Undivided profits, December 31, 1918
Profits of quarter, net
Less interest paid on withdrawal
Profits to be divided
The profits are divided in the following manner: Series
;,647. 82
483-16
),i30.98
IO-75
3,120.23
2oi6/5689*of|6,i2o.23equals$2,i68.8i,orpershare $
1200 1078
513 320 312
150
84
16
5,689
,290.96 '
,159-71 '
551-89'
344.26 '
335-65 '
u u
161.37 '
u u
90.37 '
17.22 '
2.71
2-15
1.66
1.23
.86
•56
•32
•15 .04
),I20.24
*i5i, 200/426,675 = 2016/5689
Problem 2
Using the same figures as in the previous problem, distribute the profits in accordance with Dexter's rule.
Solution:
First find the capitals of the series by multiplying the value per share plus$i by the number of shares outstanding, as follows: Series
800 shares by $15.56 equals $12,448
600 700
450 400 600 500 600 400
13-47
8,082
II. 51
8,057
9-64
4,338
7.82
3,128
6.03
3,618
4.26
2,130
2.52 '
1. 512
1. 00
* 400
$43,713
BUILDING AND LOAN ASSOCIATIONS
115
The profits for the quarter are $483.16
Plus profit on withdrawn shares 46.25
Total to be divided $529.41
The withdrawal profits are ascertained thus:
Book profit for 4th series is $1.14 per share, and for 50 shares. Actually paid
Retained profit on withdrawal
>S7-oc 10.75
'■25
The divisible profit of $529.41 is a trifle more than 1.21% of the capital of $43,713. Calculated for each series it is as follows:
Series
Shares
Old Profit PER Share
Old Profit OF Series
New Profit OF Series
Total Profit per Share
New Profit PER Share
I
800
J2.S6
$2,048.00
J150.62
$2,198.62
$2.75
2
600
1-97
1,182.00
97
79
1,279.79
2.13
3
700
l-Sl
1,057-00
97
49
1. 154-49
1-65
4
450
1. 14
51300
52
49
565-49
1.26
S
400
.82
328.00
37
85
365-85
-91
6
600
.53
318.00
43
78
361.78
.60
7
500
.26
130.00
25
77
155-77
• 31
8
600
.02
12.00
18
30
30-30
■OS
9
400
0
4
84
4.84
.02
$5. 588. 00
S528
93
$6,116.93
The last series is usually given 2 cents as i cent is so very small. With the profit per share known the statement of age and condition of shares can now be made up if asked for.
CHAPTER XIII
GOOD-WILL AND CONSOLIDATIONS
Purchasing a Business with Stock
When two or more concerns are merged, or when a business is taken over by a holding company, or when a business is bought by a corporation already in existence, the payment for the busi- ness taken over is frequently made in the stock of the corporation acquiring the business.
If the amount of stock to be given for the business is to be determined on an equitable basis, two elements in the business must be taken into consideration. These are the fair value of the net assets of the business as a going concern and the earning ca- pacity of the business.
The fair value of the net assets may be reached by mutual agreement between the parties, or by an appraisal of the fixed assets by an appraisal company, and an estimate of the value of the active assets by an accountant.
When the value is determined, stock of a corresponding amount in face value is allotted as the price paid for the net assets.
Allocation of Net Earnings
To ascertain the earning power for which further stock should be issued, it is first necessary to establish a normal, or standard rate of earnings. This can be done only by agreement between the parties concerned. When the rate is agreed upon, it is ap- plied to the stock already allotted for net assets. The result is the amount of net earnings to be devoted annually as dividends on the stock given in payment of the net assets.
The net earnings remaining after deducting dividends at the agreed rate on the stock issued for net assets, is the basis for
Ii6
GOOD-WILL AND CONSOLIDATIONS II7
calculating the amount of stock to be allotted for the excess earning capacity of the business acquired.
The theory on which stock is allotted for excess earnings is based on the fact that one of the principal reasons for investing in a stock is that it will produce an income that is considered adequate return on the money invested. What the rate of return should be is determined by the conditions of the enterprise. An investment in a business whose conditions are stable and which may be depended upon to maintain a steady earning power, or even to increase it, yields a much lower rate of return than one in a speculative and unreliable business.
Good-Will
The name given to the valuecreatedby the excess earnings of a business is "good- will." Consideration of good- will is involved in the discussion of a stock distribution on the basis of earnings. The stock allotment is partly based on the value of the net assets acquired and partly on the good-will, or excess earning power.
Since good-will is the measure of earning capacity, it follows that only an established business can possess it. It resides usually in the reputation the business has built up in consequence of the excellence of its product, the practice of fair dealing or any other characteristics of its management giving it an advantage over its competitors. The existence of good-will cannot be proved except by a more or less prolonged experience. There- fore, when a new business is started it is false accounting to issue any stock for good-will, because its promoters expect it to earn more than the normal rate of profit. A new company, however, which has taken over an old business with a developed good-will may carry good-will among its assets and issue stock therefor.
Appraising Good- Will — Years' Purchase Method
In allotting stock the good-will may be valued in several different ways. It may be expressed as a given number of years'
Il8 MATHEMATICS OF ACCOUNTING AND FINANCE
purchase of the total profits. This means that the purchaser is wilHng to forego profits from the business for the time agreed upon. Each condition of the terms must be agreed upon between the buyer and seller. Thus, if it is agreed that the basis shall be two and one half years' purchase of the average profits for the last ten years, it will be necessary to ascertain the total profits for ten years. One tenth of this sum will be the average yearly profit, and two and one half times the average profit will be two and one half years' purchase.
If the total profit for lo years is $256,232.00
the average annual profit is $ 25,623.20
and 23^ times the average is $ 64,058.00
which is the value placed upon the good-will on the basis of two and one half years' purchase.
It is better to ascertain the average of a number of years than to take the figures for the last two and one half years, as the latter may not be normal.
This method is not as logical as allowing a certain number of years' purchase of the profits in excess of a rate agreed upon as normal. The latter calculation is the same as that given above, except that the normal profits are first deducted. Thus, if the invested capital is $200,000 and the normal rate agreed upon is 10%, the value of the good-will is figured as follows:
Average profits (as shown above) .... $25,623.20 Normal profit 20,000.00
Excess profit $ 5,623.20
8-years' purchase $44,985.60
Capitalizing Gross Income
A third method is to capitalize the gross income. If the capital is known and the per cent of income is also known, the income is found by multiplying the capital by the rate per cent. Thus, if the capital is $200,000 and the rate of profit is
GOOD-WILL AND CONSOLIDATIONS 119
15%, the income is $30,000. On the other hand, if it is desired to ascertain what capital will produce $30,000 income if the rate per cent is 15, it is necessary to divide the income by the per cent expressed as a decimal, thus,
$30,000 divided by .15 equals $200,000.
Therefore, to capitalize a given income at a given rate per cent, the income is divided by the decimal expressing the rate.
This method, like the first, is objectionable because it does not take into consideration the fact that part of the income must be used to furnish a return on the stock already allotted for net assets. That amount of the income is in fact duplicated, and the duplication reduces the amount of the return to the concern with the largest rate of profit and increases the return to the concern with the smallest rate.
Suppose the following companies wish to combine on the basis of net assets and good-will representing gross income capitalized at 15%.
ABC
Net assets and capital $200,000 $300,000 $500,000
Average annual income 40,000 45,000 65,000
Rate of profit 20% 15^^ 13%
If it is proposed to form a new company with a capital of $2,000,000, the distribution of the stock is as follows:
For net assets $200,000 $300,000 $500,000
For income capitalized at 15%. . 266,667 300,000 43i^3,.^3
Total $466,667 $600,000 $933,333
The total income being $150,000, the rate of profit is 7>^ %, which will have to be distributed as follows:
A receives y}4 % of $466,667 $35,000
B " " " " 600,000 45,000
C " " " " 933^333 70.000
Total $2,000,000 $150,000
120 MATHEMATICS OF ACCOUNTING AND FINANCE
A would thus receive $5,000 less per year than when operating alone and will certainly object to the plan. B would not be affected because he is already receiving 15%, while C would gain the $5,000 lost by A.
Therefore, a fourth plan should be adopted by which the duplication of profits will be avoided. After allowing for a fixed rate of return on the capital issued for net assets to each of the merging concerns, the amount of profits necessary to provide the dividend at that rate is deducted from the total profits previously earned by the concern and the capitalization of the remaining profits is made by employing the same rate. This amounts to fixing the rate desired on the new stock and issuing stock to each concern sufficient to earn the profits previously enjoyed. If 8% is fixed upon as the rate desired, the first allotment of stock for net assets is the same as before — A $200,000, B $300,000, and C $500,000. The remaining profits are then calculated as follows :
A
B
C
Total
$ 40,000 16,000
$ 45,000 24,000
$ 65,000 40,000
$
150,000
80,000
$ 24,000
$ 21,000
$ 25,000
$
70,000
Capitalizing at 8%
$300,000 200,000
$262,500 300,000
$312,500 500,000
$ ]
875,000
,000,000
$500,000
$562,500
$812,500
$
,875,000
Income at 8%
$ 40,000
$ 45,000
$ 65,000
$
150,000
In this way each concern will receive the same earnings after the merger as before, and none will suffer.
Issue of Two Classes of Stock
We have thus far considered the issue of only one kind of stock. This necessitates the use of only one rate of return on the stock
GOOD-WILL AND CONSOLIDATIONS 121
issued. A more usual procedure when a combination of this kind is effected is the issuance of two classes of stock, preferred and common. It is an almost universal practice to issue preferred stock for the net assets and common stock for the good-will, or excess earnings.
The objection to calculating the stock given for good-will on the basis of the total earnings is the same as in the third plan. If the preferred stock is issued at 6% and the good-will is capital- ized on the total profits at 20%, the common stock issued for good-will is as follows:
A $200,000
B 225,000
C 3 25^000
Total $750,000
As it takes $60,000 to pay the preferred dividend, there will be left $90,000 for the common stock, which will allow a 12% divi- dend. The profits will therefore be divided as follows:
A $200,000 preferred at 6% $12,000
200,000 common at 12% 24,000 $36,000
B 300,000 preferred at 6%
225,000 common at 12%
C 500,000 preferred at 6%
325,000 common at 12%
Total
$18,000
27,000
45,000
$30,000
30,000
6g,ooo
$:
[50,000
In this way A makes a loss and C a gain of $4,000.
The proper method is to issue the common by capitalization of the excess profits after the preferred dividends have been provided for, thus:
A
Original profits $40,000
Preferred dividends 6% 12,000
Excess remaining $28,000 $27,000 $35,000
Capitalized at 20% $140,000 $135,000 $175,000
B
C
$45,000
$65,000
18,000
30,000
122 MATHEMATICS OF ACCOUNTING AND FINANCE
It is evident that dividends of 6% on the preferred and 20% on the common stock will give the same return as formerly.
After the preferred dividends are deducted it does not make any difference what basis is adopted for capitalizing, because whatever rate is used the relative proportion remains the same. Thus, if the excess is capitalized at four years' purchase the result- ant common stock is as follows: A $112,000, B $108,000, and C $140,000, a total of $360,000, on which the $90,000 remaining profits will provide a dividend rate of 25%, and the distribution will be thus:
A for preferred $12,000
" common, 25% of $112,000 28,000 $40,000
B " preferred $18,000
" common, 25% of $108,000 27,000 45,000
C " preferred $30,000
" common, 25% of $140,000 35, 000 65,000
The practice of starting a new business with a capital stock of a par value greater than the total of the net assets is often in- dulged in. In such a case a debit must be made to some account, in order to balance the books. The account debited is often called good-will. This is altogether wrong, since, as we have seen, good- will is a matter of growth and cannot be possessed by a business which has had no time to develop it.
CHAPTER XIV FOREIGN EXCHANGE
Conversion of Foreign Coinage
In order to provide a basis for the comparative values of United States and foreign coinage, the Director of the United States Mint periodically estimates the values of foreign coins, which are proclaimed by the Secretary of the Treasury. The values thus fixed are called the mint pars of exchange, and repre- sent the intrinsic or bullion values of foreign coins in terms of United States money. A table of values recently proclaimed appears in the Appendix.
To reduce a value in foreign coinage to United States money, multiply the value of the coin in United States money by the number of coins.
Illustration
What will a draft for £410 'i9/'6 cost in dollars at the mint par of exchange?
Solution: There are two ways to calculate this:
1. Reduce shillings and pence to decimals of the pound, thus:
£410.
ig shillings are 19/20 of a pound .95
6 pence are 6/240 or 1/40 of a pound .025
£410.975 £410.975 multiplied by $4.8665 is $2,000.01.
2. Reduce pence, shillings and pounds to value in dollars, thus:
6 pence are 1/40 of $4.8665 $ .12
19 shillings are 19/20 of $4.8665 4.62
410 pounds at $4.8665 1,995.27
£410/19/6 at $4.8665 $2,000.01
123
124 MATHEMATICS OF ACCOUNTING AND FINANCE
Reverse Conversion
To reduce a value in United States coinage to a foreign money value, divide the value in United States money by the value of the foreign unit.
Illustration
What is the value in pounds of $2,000 at the mint par of exchange?
Solution: There are again two ways to calculate this, which are as follows :
1. 2,000 -^ 4.8665 = 410.973, number of pounds
.973 of 20 (shillings in the pound) = 19.46 .46 of 12 (pence in the shilling) = 5.52
Hence $2,000 = £410/19/6.
2. To find the number of pounds:
4.8665 1 2,000.00 I 4io£
1,946.60
53-400
48.665
Remainder
4-735°
Reduce to shillings
20
48665 947000 1 19 s
48665
460350
437985
Remainder
22365
Reduce to pence
12
48665I 268380 1 5.52 d
243325
250550
243325
72250
Result £ 4 1 0/19/6. In this particular case there is a much shorter method, which is as follows :
FOREIGN EXCHANGE 1 25
£410/19/6 is the same as £411 minus 6d, or £411 minus 1/40 of £1. We can, therefore, obtain the result by performing the following sub- traction :
£4ii/oo'o X $4.8665 $2,000.13
— 6 or 1/40 of $4.8665 —.12
£410/19/6 $2,000.01
Current rates of exchange vary from the unit par rate, because the supply and demand for foreign drafts are afifected by the balance of trade between countries. Transactions involving foreign exchange are made at current instead of mint par rates.
Exchange is quoted at the current value in cents and con- versions are made as illustrated in the conversions at mint rates.
Illustration
What is the cost of a 300 franc draft on Paris, purchased at 8.49?
Solution: $.0849 X 300 = $25.47
A United States merchant sends $5,000 to Paris when the rate is 8.49. What is the value in francs?
Solution: 5,000 -^ .0849= 58,892.8, the number of francs.
Dealing in Foreign Exchange
There are two methods of recording transactions in foreign exchange. When the first method is used, the charges and credits are entered in two values, domestic and foreign, the conversion of each item being made at the rate applying to the transaction When the account is closed the balance in foreign money is valued by converting either at cost or current rate, and the profit or loss on exchange is ascertained.
The other method attempts to ascertain the profit or loss on each item, by computing the difference between the conversion value at par and current rates. Both methods are illustrated below :
126 MATHEMATICS OF ACCOUNTING AND FINANCE
Illustration
A banking concern deals in foreign exchange and the following are the transactions with a London correspondent for one month:
Debits
Sept. 1 Remittance, 30-day bill £400 at $4.86
10 " sight bill £100/10 " 4-87
"15 " " " £200/0/6 " 4-86K
Credits Sept. 2 Draft, sight £300 at $4.87^
12
£200/12/5 " 4-87
20 Cable £100 " 4.88
1. Ascertain the profit or loss in the account for the month.
2. State the balance of the account at the end of the month in foreign and domestic currency, the current rate of sterling exchange for cable transfers being $4.89.
Solution i : Although the problem states that the current rate of exchange for cable transfers is $4.89, it seems incorrect to inventory the balance of the account at this rate, since the rate is not only higher than cost but higher than any of the prices at which sales have been made. On the theory that the first items purchased are the first sold we find that the first two remittances are ofTset by the first two credits. Applying this theory, the balance of the account must have cost $4.86^, the rate apply- ing to the remittance of September 15. Setting up the account and valu- ing the balance at $4.86^, we have the following:
Account with London Correspondent Debits
Foreign Rate Domestic
Sept. I, Remittance, 30-day bill £400/00/0 $4.86 $1,944.00
" xo, " sight bill loo/io/o 4.87 489.44
" 15, " " " 200/00/6 4.86K 973.62
" 30, Profit 6.74
£700/10/6 $3,413.80
Oct. I, Balance — Inventory £99/18/1 $4.86^ $486.28
FOREIGN EXCHANGE 127
Credits
Sept. 2, Draft, sight £300/00/0 $4,871^ $1,462.50
" 12, " " 200/12/5 4.87 977-02
" 20, Cable 100/ 00/0 4.88 488.00
" 30, Balance 99/18/1 4-86^ 486.28
£700/10/6 $3,413.80
While $6.74 is shown as profit it must be remembered that the rate of 4.86^ is used because the current rate for checks is unknown, that interest has been disregarded in the calculation, and that the profit as shown is nominal only and may be increased or decreased when realized by a sale.
If the balance were inventoried at $4.89, its domestic value would be $488.53, and the profit would be $8.gg.
It would be correct to value the balance at the current buying rate on September 30, if that were known. This is the usual practice.
In calculating the dollar value of the balance, the shortest way is to take £99/18/1 as I s 11 d less than £100/00/0, as follows:
I shilling is 1/20 of $4.8675 or $0.24337
II pence is 1/12 less than i shilling, that is, $.24337
— .02027 .22310
5.47
£100 $486.75
- i/ii --47
£ 99/18/1
Solution 2: The above is the usual method of keeping a foreign ex- change account in this country. In Canada each transaction as recorded in the buying and selling registers is compared with the par of exchange and an entry made debiting or crediting Exchange, as the case may be. The par of sterling is $4.8665 or $4.86?/^. The registers, in the example taken, would show the following:
Bought Date Time Sterling R.ate Paid Par Dr. Ex. Cr. Ex.
Sept. 1 30 d £40000/0 $4.86 $1,944.00 51,946.67 S $2.67
10 St. loo/io/o 4.87 489.44 489.10 .34
IS " 200/00/6 4-8634 973.62 973-45 -'7
£700/10/6 J3,409-22 $.51 $2.67
128 MATHEMATICS OF ACCOUNTING AND FINANCE
Sold
Sept. 2 .. . . 12 ... . 20.. . . 30
St.
. . . . ca. .... balance
£300/00/0
200/12/5
loo/oo/o
99/18/1
4.87 4.88
$1,462.50
977-02 488.00
$1,460.00 976.35 486.67 486.20
£700/10/6
$3,409.22
*2.S0
.67 1.33
$4-50
The ledger would contain an account with the London correspondent charged and credited with the entries in the Par columns, and an account with Exchange, showing a debit of $0-51 and credits of $2.67 and $4.50. The sterling column is a memorandum inventory account only. The account with Exchange would indicate a profit of $6.66, the diflference from the profit of $6.74 shown by the United States method being caused by inventorying the balance at par instead of at cost. Of course, the postings would be made in daily totals in a bank in which a daily trial balance is taken off.
The balance may be entered at cost of $4.86K, or $486.28 in the sold column and 8 cents in the credit exchange column, which will make the profit $6.74, the same as by the other method.
Average Date of Current Account
The principles already explained for determining the average date of an account apply when the account is kept in a foreign coinage. The following is an illustration.
Illustration
A of London, in current account with B of New York, engages an account- ant to prepare a statement, to be mailed to B, based on the following data:
1914 Debits:
May 12 £750
May 30 117
June 12 340
July 1 150
Total debits £1,357
FOREIGN EXCHANGE 129
Credits:
June 10 ^Soo
June 30 300
Total credits 800
Balance i 557
Find the average due date of the account and the interest at 5% to July I, taking 365 days to the year.
Solution:
Focal date, April 30 —
Debits
May 12 £ 750 12 days £9,000
" 30 117 30 " 3,510
June 12 340 43 " 14,620
July 1 150 62 " 9,300
£1,357 £36,430
Credits
June 10 £ 500 41 days £20,500
" 30 300 61 " 18,300
£ 800 38,800
Balances £ 557 Debit £2,370 Credit
The amount, 2,370 divided by 557 equals 4- But since the balance of the account is a debit while the balance of the products is a credit, the average date is four days backward from April 30, or April 26.
From April 26 to July i is 66 days. Interest on £557 for 66 days at 5% may be calculated in either of the following ways:
I. £557 X .05 X ^ = £5-57 X - = ^5-036
£1 X .036= 2osX.o36= .72s isX -72= 12 dX .72 = 8.64 d Total interest £5/0/9
130 MATHEMATICS OF ACCOUNTING AND FINANCE
2. At 6% •fS-S? is int. for 6o days on 360 days basis
" 6% .557
£6.127 " " " 66 " " 1% 1.021
n u u a II u ii u
" 5% £5.106 " "
Reducing to 365 day basis .07 Subtract 1/73
a a ic II (I
£5.036 which is reduced to £5/0/9 as above
The decimal .036 can also be reduced as follows:
£1 = 240 d
£ .036 = 240 X .036, or 9 d
Conversion of Foreign Branch Accounts
The following example shows how foreign branch account balances are converted to domestic values, and how the home office and foreign branch accounts are consolidated in the period- ical statements.
Illustration
A New York corporation builds a plant and establishes a branch in Liverpool, England. At the expiration of its fiscal period, a trial balance is forwarded to the New York ofhce, as follows:
Plant £250,000
Accounts receivable 187,500
Expenses 25,000
Inventory (end of fiscal period) 50,000
Remittance account 150,000
Cash 12,500
Accounts payable £ 87,500
Income from sales 250,000
New York office 33 7,500
£675,000 £675,000
FOREIGN EXCHANGE 131
A trial balance of the New York books on the same date is as follows:
Capital stock $2,500,000.00
Patents $1,500,000.00
Liverpool account 1,640,250.00
Remittance account 729,281.25
Expenses at New York 25,000.00
Cash 64,031.25
$3,229,281.25 $3,229,281.25
The remittance account is composed of four 60 -day drafts on Liverpool for £37,500 each, which were sold in New York at $4.85^^, $4.86, $4.86^^ and $4.86^ respectively.
Prepare a balance sheet of the New York books after closing and a statement of assets and liabilities of the Liverpool branch consolidated with the New York books. Close the books at the rate of exchange on the last day of the fiscal period, which is $4.87 1<4, conversion of remittances to be made at the average rate for the four bills.
Solution: Since the fixed assets are not subject to revaluation at current exchange rates, the plant cost must be taken out of the New York office account, which is subdivided to show indebtedness to the main office for fixed assets and for current assets. Since the cost of the fixed assets is not given, we must assume that it is at the same rate as the rest of the New York account. Dividing the Liverpool account balance of $1,640,250 by the New York office account balance of £337,500 we obtain the ex- change rate of 4.86, at which rate the plant has a value in United States money of $1,215,000. The reciprocal accounts are then subdivided as follows :
On Liverpool books:
New York office — plant £250,000
" " " — current account 87,500
On New York books:
Liverpool account — plant $1,215,000.00
" current account 425,250.00
The next step is to close the Liverpool books into the New York office current account as follows:
132 MATHEMATICS OF ACCOUNTING AND FINANCE
New York Office Current Account
Expenses £ 25,000 Balance £ 87,500
Remittance account . . . 150,000 Income from sales 250,000
Balance 162,500
£337,500 £337,500
Balance. . . .• £162,500
The balance sheet of the Liverpool branch is now as follows: (Plant valued at cost, $4.86; current assets and liabilities valued at current ex- change rate of $4.87}^).
Assets
Plant $4-86
Accounts receivable 4-^7/4
Inventory 4-^7/^
Cash 4.87M'
£250,000
$1,215,000.00
187,500
913,503-75
50,000
243,625.00
12,500
60,906.25
£500,000 $2,433,125.00
Liabilities
Accounts payable $4-87K £87,500 $ 426,343.75
New York office — plant account .... 4.86 250,000 1,215,000.00
New York office — current account — A-^7% 162,500 791,781.25
£500,000 $2,433,125.00
On the New York books, the Liverpool current account is credited with the remittances, and a charge is made to the account (offset by a credit to profit and loss) sufficient to leave the account with a balance of $791,781.25, the balance of the net current assets in Liverpool at $4.87>^. It is not necessary to find the average rate of the remittance account since the whole account is used. It would be found, by dividing $729,- 281.25 by 150,000, to be $4.86 3/16. The figures to be used are then found by multiplying £150,000 by $4.86 3/16, which gives $729,281.25, an amount which we already have.
FOREIGN EXCHANGE
133
Liverpool Current Account
Balance $ 425,250.00 Remittances... $ 729,281.25
Profit 1,095,812.50 Balance 791,781.25
M, 521,062. 50
51,521,062.50
Balance $ 791,781.25
The New York profit and loss account will appear as follows:
Profit and Loss
New York expenses.. .$ 25,000.00 Liverpool Branch $1,095,812.50
Net profit to surplus. . 1,070,812.50 $1,095,812.50
$1,095,812.50
This profit is subject to a further charge for reduction of the patents account.
The New York office trial balance is as follows:
Capital stock
Surplus
Patents
Liverpool plant account . . Liverpool current account Cash
$2,500,000.00 1,070,812.50 $1,500,000.00 1,215.000.00 791,781.25
64,031-25
$3,570,812.50 $3,570,812.50
The balance sheet is as follows:
Assets Fixed assets:
Patents
Liverpool Plant
Current assets:
Accounts receivable,
Liverpool
Inventory
Cash — Liverpool $60,906.25
Cash — New York 64,031.25
51,500,000.00 1,215,000.00
5 013.593-75 243,625.00
52.715,000.00
124,937-50 1.282,156.25 $3,997,156.25
134 MATHEMATICS OF ACCOUNTING AND FINANCE
Liabilities
Capital stock $2,500,000.00
Surplus 1,070,812.50 $3,570,812.50
Accounts payable, Liverpool 426,343.75
$3,997,156.25
The Liverpool branch profit of $1,095,912.50 is made up as follows:
Income from sales £250,000
Less expenses 25,000
Net profit on sales £225,000, at $4.87/,^ $1,096,312.50
Increase of current exchange rate ($4. 8714^) over original valuation ($4.86) of opening balance of current account:
£87,500 at $4.87^ $426,343.75
£8/, 500 at $4.86 425,250.00 1,093.75
$1,097,406.25
Less exchange loss on remittances:
£150,000 at $4.87^ (current rate) .... $730,875.00 £150,000 at $4.861875 (average con- version rate) 729,281.25 i, 593-75
Net profit $1,095,812.50
Even in this explanation it is not necessary to use the average con- version rate.
CHAPTER XV
LOGARITHMS
Use
Logarithms are a special compilation of figures, used to reduce the labor of multiplication, division, computation of powers and extraction of roots. After explaining the method of employing logarithms, their theory will be briefly discussed.
Every number has its corresponding logarithm — but a knowl- edge of the method of determining the logarithms of various numbers is not essential, as they may be obtained from prepared tables. The logarithms of certain numbers are stated below for use in illustrations. For a more complete table, see Appendix.
Table of Logarithms of Selected Numbers
Number
Logarithm
2
•30103
3
.47712
4
.60206
5
.69897
6
•7781S
7
.84510
8
.90309
9
•95424
lO
1. 00000
12
1. 07918
15
I.I 7609
16
1. 20412
30
1.47712
100
2.00000
160
2.20412
300
2.47712
1,000
3.00000
1,600
3.20412
3,000
3-47712
135
136
MATHEMATICS OF ACCOUNTING AND FINANCE
Multiplication
Multiplication is accomplished by the addition of logarithms, as follows:
1. In the table, find the logarithms of the numbers to be
multipHed.
2. Add the logarithms; the sum is the logarithm of the
desired product.
3. In the table, find the number corresponding to the
logarithm obtained in 2.
The following examples, though more easily performed with- out logarithms, will illustrate the steps of the process;
Example
2X3=?
Solution:
1. Table shows
2. Sum of logarithms
Number Logarithm
2 .30103
3 -47712
.77815 log of product
3. Table shows .77815 is log of 6, the product
Example
2X3X2=?
Solution:
1. Table shows
<( a
2. Sum of logarithms
Number Logarithm
2 -30103
3 -47712 2 -30103
1.07918 log of product
3. Table shows 1.07918 is log of 12, the product
LOGARITHMS 137
Without going beyond the selected table, the following multipli- cations may be performed by using logarithms:
5X6=? 4X3=? 3X2X5=?
Division
Division is accomplished by the subtraction of logarithms as follows:
1. In the table find the logarithms of the dividend and the
divisor.
2. Subtract the logarithm of the divisor from the logarithm
of the dividend; the difference is the logarithm of the desired quotient.
3. In the table, find the number corresponding to the loga-
rithm obtained in 2.
Example 6-^2=?
Solution:
Number Logarithm
1. Table shows:
Dividend 6 -77815
Divisor 2 -30103
2. Difference of logarithms -47712 log of quotient
3. Table shows .47712 is log of 3, the quotient
The following divisions may be performed by logarithms with the aid of the selected table:
8-i-2 10 4-5 16-J-4 300 -T- 30
Calculating Powers
A power of a number i3 the product obtained by using that number repeatedly as a factor. The square of a. number is the
138 MATHEMATICS OF ACCOUNTING AND FINANCE
second power; the cube, the third power, etc. The exponent of the power indicates the number of times the factor is to be used.
For instance 3^ means the second power of 3; or 3 squared; the number used as a factor is 3. The superscript number "" is the exponent of the power, showing that 3 is to be used twice as a factor.
Thus, 3' = 3 X 3 = 9
A number is raised to a power by logarithms as follows:
1 . In the table find the logarithm of the number to be raised
to a power.
2. Multiply the logarithm by the exponent of the power; the
product is the logarithm of the power.
3. In the table find the number corresponding to the loga-
rithm obtained in 2.
Example i
3'= ?
Solution:
Number Logarithm
1. Table shows 3 -47712
2. Multiply by exponent of power 2
.95424 log of power
3. Table shows .95424 is log of 9, the second power of 3.
Example 2
2" = ?
Solution:
Number Logarithm
1 . Table shows 2 -30103
2. Multiply by exponent of power 4
1. 204 1 2 log of power
3. Table shows 1.204x2 is log of 16, the fourth power of 2.
LOGARITHMS ^39
Roots
A root is a number which, used repeatedly as a factor, will
produce a given power.
The square root of 9 is 3, because 3 used twice as a factor
produces 9. , • r ^f^^
The fourth root of 16 is 2, because 2 used 4 times as a factor
^'""ThTroot of a number is expressed by writing the number under the radical sign; the index figure indicates the root to be
extracted.
The square root of 9 is expressed thus:
The superscript ^ is the index of the root.
Any root of a number may be extracted by logarithms as
follows:
I. In the table find the logarithm of the number, the root of which is to be extracted.
2 Divide the logarithm by the index of the root; the quo- tient is the logarithm of the desired root.
3. In the table find the number corresponding to the loga- rithm obtained in 2 .
Example i
Solution:
I. Table shows log of 9 to be .95424
2 Divide by index of root: .954^4 - ^ = .47712, log of root
;. Table shows .47712 is log of 3, the square root of 9
Example 2
<r^= ?
140 MATHEMATICS OF ACCOUNTING AND FINANCE
Solution:
1. Table shows log of i6 to be i. 2041 2
2. Divide by index of root: i. 20412 -^ 4 = .30103, log of root
3. Table shows .30103 is log of 2, the fourth root of 16
Nature of Logarithms
Logarithms are exponents of the base 10. The logarithm of any number shows how many times 10 must be used as a factor to produce that number. This is easily seen in the case of num- bers which are exact powers of 10. The selected table shows the following :
Number Logarithm 10 1. 00000
100 2.00000 (10 X 10 = 100)
1000 3.00000 (10 X 10 X 10 = 1000)
Numbers which are the exact powers of 10 have logarithms which are whole numbers.
Most numbers which are not exact powers of 10 have loga- rithms which are mixed numbers, composed of two parts: an integer, and a decimal fraction.
The following are taken from the selected table:
Dumber
Logarithm
10
1. 00000
12
1.07918
15
I.I 7609
16
1. 20412
30
1.47712
100
2.00000
160
2.20412
1,000
3.00000
1,600
3.20412
The Characteristic and the Mantissa
The two parts of a logarithm are: the characteristic, which is an integer, and the mantissa, which is the decimal part.
LOGARITHMS I4I
The mantissa is determined by the figures of the number; (initial and final zeroes in the number do not affect the mantissa). Thus the logarithms of the numbers 16, 160, 1600, etc., all have the same mantissa, .20412.
The characteristic indicates the exponent of the largest power of 10 contained in the number.
To illustrate:
Largest Power
OF
10
Exponent
VT UMBER
CONT.
\INED IN N
UMBER
OF
Power
Logarithm
16.
10
I
I. 20412
160.
100
2
2.20412
1600.
1000
3
3.20412
16000.
1 0000
4
4.20412
It will be noted that the characteristic of the logarithm is always one less than the number of figures at the left of the decimal point in the number.
Figures .a.t Left Number of Decimal Point Characteristic Logarithm
16 2 I 1. 20412
160 3 2 2.20412
1600 4 3 3.20412
16000 5 4 4.20412
Tables of Logarithms
Referring to the table of logarithms in the Appendix it will be noted that only the mantissas are given; in forming logarithms the mantissa is taken from the table, and the characteristic is determined by the location of the decimal point in the number. The following lines are taken from the table in the Appendix:
No. 01 23456789
160 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656
161 20683 20710 20737 20763 20790 20817 20844 20871 20898 20925
The heavy figures at the left and at the top indicate the se- quence of figures in the number.
142 MATHEMATICS OF ACCOUNTING AND FINANCE
The smaller numbers in the body of the table indicate the mantissas. For instance :
Line i6o, Column o, shows 20412, which is the mantissa of 16, 160, 1,600, 16,000, etc.
Line 160, Column 4, shows 20520, which is the mantissa of 16.04, 160.4, i)6o4, 16,040, etc.
Line 161, Column 9, shows 20925, which is the mantissa of 16.19, 161. 9, 1,619, 16,190, etc.
Applying the method explained in the preceding section and using the sequence of figures 1,619 as an illustration, the char- acteristic is ascertained and prefixed as follows:
Figures at Left Number of Decimal Point Characteristic Logarithm
16,190 5 4 4-20925
1,619 4 3 3-20925
161. 9 3 2 2.20925
16.19 2 I 1.20925
Characteristics of Logarithms of Numbers between i and 10
From the preceding illustration it will be noted that as the decimal point is moved one place to the left the characteristic decreases by i .
The table is continued in the following:
Figures at Left Number of Decimal Point Characteristic Logarithm
16.19 2 I 1.20925
1. 619 I o 0.20925
or 20925
Thus it is seen that the characteristic is o when the number lies between i and 10, which is in accordance with the general rule that the characteristic is one less than the number of figures at the left of the decimal point.
LOGARITHMS I43
Examples
Determine the characteristics and state the logarithms of the follow- ing numbers:
25,000 2.5 492
2,500 37,010 7,635
250 370.1 200
25 3.701 4,005
Use of the Characteristic in Pointing Off Results
If the characteristic of a logarithm is one less than the number of figures at the left of the decimal point in the number, then the figures at the left of the decimal point in the number must be one more than the characteristic.
Referring to the brief table on page 141, 20790 is the mantissa of the number 1,614; the characteristic preceding this mantissa would determine the location of the decimal point. To illustrate :
Places
at
Left
3GARITHM
Character:
STIC
OF Decimal
Point
Number
4.20790
4
5
16,140.
3.20790
3
4
1,614.
2.20790
2
3
161. 4
1.20790
I
2
16.14
.20790
0
I
1. 614
The following examples are given to illustrate the relation between the characteristic and the position of the decimal point.
Illustration
24 X 3-6 = ? Solution (Refer to table for mantissas) : I. Determine logarithms:
log 24: mantissa .38021
characteristic i logarithm i. 38021
log 3.6: mantissa 55630
characteristic o logarithm 0.55630
144 MATHEMATICS OP ACCOUNTING AND FINANCE
2. Add logarithms:
1. 3802 1 log 24 0-55630 " 3-6
1.9365 1 log of product
3. Determine number corresponding to the logarithm 1.93651. Mantissa .93651 is found in the table thus:
No.
0
864
93651
and represents sequence 8640.
Characteristic i indicates that there are to be two places at the left of the decimal point in the product, 86.4.
Illustration 9.46 ^ 4.3 =
Solution:
I. log 9.46: mantissa -97589
characteristic o logarithm 0.97589
log 43:
mantissa -63347
characteristic o logarithm 0.63347
0.97589 log of dividend
0-63347 " " divisor
0.34242 " " quotient
.34242 is mantissa of figures 2200 Characteristic o; one figure at left of decimal point; quotient 2.2
LOGARITHMS
145
Examples
ind the value of the following:
I. 23 X IS
7-
8^
2- 35 X 2.5
8.
4^
3. II. 4 X 7
9-
1.2^
4. 392 -^ 8
10.
V 729
5. 952 -^ 28
II.
^64
6. 9.38 -J- 1.4
12.
V^ 5-29
Logarithms of Numbers Smaller than i
The following numbers with their logarithms have been used to show that moving the decimal point in a number one place to the left reduces by i the characteristic of the logarithm.
Number 16,190. 1,619. 161. 9 16.19 1. 619
Figures at Left OF Decimal Point
S 4 3
2
Characteristic 4 3
2
I
Logarithm 4.20925 3.20925 2.20925 1.20925 0.20925
If the decimal point is again moved one place to the left, we have:
.1619
1.20925
In other words, if the number is less than i , the characteristic of the logarithm is a negative quantity. The minus sign is placed above the negative characteristic to show that it applies to the characteristic only, the mantissa being a positive quantity. The logarithm 1.20925 may also be written 9.20925-10.
The negative characteristic is one greater than the number of zeroes between the decimal point and the first significant figure in the number.
In the following, the table of numbers smaller than i and their logarithms is continued :
146 MATHEMATICS OP ACCOUNTING AND FINANCE
^J UMBER
Zeros Between
Decimal Point and
First Significant
Figure
Negative Characteristic
Logarithm
.1619
0
— I
1.20925 or 9.20925 — 10
.O161Q
I
— 2
2.20925 or 8.20925 — 10
.00 1 6 1 9
2
-3
3.20925 or 7.20925 — 10
etc.
Examples
Determine the characteristics by inspection; refer to the table for the mantissas; and state the logarithms of the following numbers:
.346 .0346 .00346 .000346
.0201 .0031 .29 .0101 .000003
Use of Negative Characteristic in Pointing Off Answers
When the logarithm obtained by a computation has a nega- tive characteristic, it indicates that the corresponding number is smaller than i ; that is, it is a decimal fraction. Point off the number by placing between the decimal point and the first sig- nificant figure one fewer zeros than the number of the character- istic thus :
Negative
ogarithm
Characteristic
Zeros
Number
3.20925
-3
2
.001619
2.20925
— 2
I
.01619
1.20925
— I
0
.1619
Examples
Determine the numbers corresponding to the following logarithms: 2.98005 3.88969 1.99782 0.71642
Computing with Logarithms Having Negative Characteristics
The following are illustrations of the use of logarithms with negative characteristics, in multiplication, division, raising to powers, and extraction of roots.
Multiplication
Illustration
021 X .000451 =
?
Solution:
Number
Logarithm
.021
2.32222 or
8.32222 —
-10
.00045 1
4.65418 or
6.65418-
- 10
Sum of logs
6.97640 or
14.Q7640-
- 20
147
.97640 is the mantissa of the figures 9471
Characteristic 6; five zeroes in answer; product, .000009471
Division
Illustration i
.0341 -f- .31 = ?
Solution :
Number Logarithm
Dividend 0341 2.53275
Divisor 31 i-49i36
Logarithm of quotient i. 04139
.04139 is the mantissa of the figures 11 Characteristic 7; no zeroes; quotient .11
Illustration 2
.341 -j- .00031 = ?
Solution:
Number Logarithm
Dividend 34i 1-53275
Divisor 00031 4-4QI36
Logarithm of quotient 3.04139
.04139 is mantissa of 11
Characteristic 3 is positive; point off four places from the left; quotient 1,100
148 MATHEMATICS OF ACCOUNTING AND FINANCE
Illustration 3
34.1 -f- .31 = ?
Solution:
Number Logarithm
Dividend 34.1 1-53275
Divisor 031 2.49136
Logarithm of quotient 3.04130
(Note: subtracting — 2 is the same as adding
+ 2) .04139 is the mantissa of 11 Characteristic 3 ; put decimal point four places from left; quotient 1,100
Powers
Illustration
.034^ = ?
Solution:
Number Logarithm
.034 2.53i4«
Multiply by exponent 2
Logarithm of power 3.06296
Note: the 3 characteristic is the result of the following:
Characteristic Mantissa
Logarithm 2 -53148
Multiply by 2 2
4 1.06296
The difference between 4 and i is 3, as above
.06296 is mantissa of 1156
Characteristic 3; two zeroes required; hence .034^ = .001156
LOGARITHMS
149
Roots
Illustration
-V/.000729
= 7
Solution:
Number
.000729
Logarithm
4.86273
This logarithm is to be divided by 3; but if —4 is divided by 3, the quotient is — i, and — 2>2>?) "I" to carry. This negative remainder causes confusion in the division of the positive mantissa. Hence the procedure below is followed:
Add and subtract 20* Divide by 3
Number Logarithm 000729 4.86273 or 6.86273 — 10 20. — 20
3 ) 26.86273 - 30 8.95424 — 10 )r 2.95424
.95424 is the mantissa of 9
Characteristic 2; one zero; hence -y^ .000729 = .09
*2o is used to obtain the —30, which is divisible by 3
Examples
Compute the following:
I. .031 X .3
2
.0045 X .00021
3
.0036 X 2.2
4
.000011 X 56000
5
.0023 X 42000
6
.006 X 6.5
7
.038 X .0003
8
504 -f- 24
9
.368 -r 23
10
.00351 -^ .027
II.
390 H-
•15
12. 13-
.0994 - 28^
- 71
14.
1.3^
15- 16.
17-
.9^ .003 <^
18. V.0625
19. V. 000343
20. 4^ X \/.ooi6
150 MATHEMATICS OF ACCOUNTING AND FINANCE
Determining Mantissas by Interpolation
Tables of logarithms are limited in scope and do not contain the mantissas of all possible numbers. For numbers beyond the scope of the table, the mantissa may be approximated by interpolation.
Illustration
What is the logarithm of 18565?
Solution: Number Mantissa
The table shows 18570 .26S81
" 18560 .26858
Difference 10 .00023
If a difference of 10 in the numbers causes a change of .00023 in the mantissa, a difference of 5 will cause a change of approximately V.o of .00023 in the mantissa.
Now, .5 of .00023 = .000115
Then .26858, mantissa of 18560
plus .000115
sum .268695, mantissa of 18565 (approximate)
Logarithm, 4.268695
Examples
By interpolation determine the logarithms of the following numbers:
3642 1209 47629 758263 25.69 3.479 .004682 32.0046
Determining Numbers by Interpolation
When a computation results in a mantissa not to be found in the table, the number corresponding to the mantissa may be determined approximately by interpolation.
Solution:
LOGARITHMS 151
Illustration 15^= ?
Number Logarithm IS I. 17600
Multiply by exponent of power 4
Logarithm of 1 5 "* 4. 70436
The mantissas larger and smaller than .70436 shown in the table are:
Number Mantissa
5063 5062
I
.70441 •70432
Differences ....
.00009
The mantissa oils'* is
" 5,062 is ... .
.70436 •70432
Difference
.00004
If an increase of .00009 in the mantissa represents an increase of i in the number, an increase of .00004 in the mantissa represents an increase of approximately 4/9 of i, or .444 + in the number.
Hence .70432 Mantissa of 5062
add interpolation for .00004 444 +
Result .70436 mantissa of 5062444 +
4.70436 is the logarithm of 50624.44 or 15'' Exact answer is 50,625.
Examples
What are the approximate numbers corresponding to the following logarithms?
2.76354 4^94965 3.42345 2.4166572 1.82629 3.57927
152
MATHEMATICS OF ACCOUNTING AND FINANCE
Illustration
What is the value of 1856^?
Solution:
The logarithm of 1,856, is 3.26858
Multiply by exponent of power 2
Logarithm of 1856^ 6.53716
Table shows: Number
Larger mantissa 3445
Smaller " 3444
Differences i
M.-VNTISSA
53719 53706
Mantissa of.
1856^ 3444
Difference.
00013
53716 53706
10/13 of I = .769 approximately
Hence 3444
Plus 760
•53706 .00010
Sum 3444769 .53716
Then 3444769 is the number represented by 6.537148
1856^ is accurately 3444736
Approximated by logarithms 3444769
Error 3^
If the problem had been 18.56''
The accurate power would be 344.4736
Approximate power (by logs) 344.4769
Error .
■0033
This example shows that fairly accurate results may be obtained by interpolation, but absolutely accurate work requires tables extended to show the mantissas of all numbers required by the problem.
LOGARITHMS 153
Examples
What are the values of:
1. 24 X 18
2. 360 X 800
3. 32.6 X 3
4. .27 X .71
5. 864 X .00321
6. 2462 X 3278
7. .00964 X 3425
8. 34400 -^ 43 21. </ .001833316
9. 9280 -J- 232 22. -y^ .0000000032
10. 1 1000 -T- 12 23. '/ 768
11. 5893 -^ 830 M 12
12. 754.314- 6234 24. -' 81^
13. 17^ ^ ^
CHAPTER XVI SIMPLE AND COMPOUND INTEREST
Simple Interest — Methods of Calculating
Interest is the income or expense arising from the use of money or credit and the increase in investment or indebtedness resulting therefrom.
When only the original indebtedness bears interest, the prin- ciples of simple interest apply; when the interest increase of the indebtedness also bears interest, the principles of compound interest apply.
There are a number of methods of computing simple interest. Those given below are among the easiest. These methods are based upon a rate of 6%, since that is a common rate and is besides a factor of 1 2, the number of months in a year, and of 30, the number of days in a month. Adjustments may be easily made for other rates. The methods described are also based on 360 days to a year. Corrections may be made to adjust the result to a basis of 365 days.
A rate of 6% (.06) per year is equivalent to a rate of 1% (.01) for 2 months or 60 days (one sixth of a year). As .01 of a number is computed by moving the decimal point two places to the left, the interest on any principal at 6% for 2 months or 60 days may be computed by moving the decimal point two places to the left. Since 6 days are one tenth of 60 days, interest for 6 days may be computed by moving the decimal point three places to the left. And since 600 days are ten times 60 days, interest at 6% for 600 days may be computed by moving the decimal point one place to the left.
154
SIMPLE AND COMPOUi\D INTEREST 155
Thus, to find the interest at 6% on any principal for
6 days, point off in the principal 3 decimal places to the left 60 " " " " " " 2 " " " " "
600 " " " " " " I " place " " "
6000 " the interest is the same as the principal
For instance, the interest at 6% on $8,245.75 for
6 days is $8.24575 or $8.25 60 " " $82.4575 or $82.46 600 " " $824,575 or $824.58 6,000 " " $8,245.75
The time is frequently a number of days which maybe separated into parts, each of which is a fraction or multiple of 6, 60, 600, or 6,000. In such cases the interest may be computed by finding the total of the interest for the component time parts.
Illustration
Find the interest on $726.32 for 88 days.
Solution:
$ 7.2632 = interest for 60 days
2.4211 = " " 20 " (Mof 60) .72632 = " " 6 " .24211 = " " 2 " {Hoi 6)
$10.65273
If the days are not convenient fractions or multiplies of 6, 60, etc., the principal may be such a fraction or multiple, in which case principal and time may be interchanged thus:
Illustration Find the interest on $3,600 for 37 days.
Solution: This is equivalent to interest on $37 for 3600 days. Then $3.70 (pointing off one place; the interest for 600 days, multiplied by 6 equals $22.20 the interest on $37 for 3,600 days, or the interest on $3 ,600 for 37 days.
156 MATHEMATICS OF ACCOUNTING AND FINANCE
When neither of these methods can be utiHzed, the procedure is as follows :
Point off three places in the principal — Result: Interest for
six days. Multiply by number of days — Result: Interest for six times
the number of days. Divide by 6 — Result: Interest for given number of days.
Illustration
Find the interest on $827.52 for 119 days at 6%.
Solution:
$ .82752 interest for 6 days 119
$98.47488 " " 6 X 119 days
$98.47488 -^ 6 = $16.41248 " " 119 days
Rates Other than 6%
To calculate simple interest when the rate is other than 6%, the interest at 6% is found and the result is adjusted to the given rate.
Illustration
What is the interest on $827.52 for 119 days at 4>^%?
Solution: The interest at 6% has already been computed being $16.41248. Since 4>2% is yi of 6%, the interest at 4^% is found as follows :
$16.41248 interest at 6 %
4.. 03 1 2 " " iK%(><of6%)
$12.30936 " " Ay2%
SIMPLE AND COMPOUND INTEREST 157
365 Day Basis
To adjust to a basis of exact interest, 365 days to a year, the interest is computed on a 360 day basis as explained, and the result is decreased y^j of itself. The 5 days difference is 773 of 365 days, and since interest is a computation which may be
expressed
Principal X Rate X Days
Number of days in Year the use of 360 (a denominator which is ^j ^^ too small) will result in the interest's being y.3 too large.
Illustration
What is the exact interest on $827.52 at 4>^% for 119 days?
Solution: The interest at ^Yi^o on a 360 day basis has already been
computed, being $12.30936
Deduct 7^3 of $12.30936 .16862
Exact interest on 365 day basis $12.14074
Partial Payments
There are two methods of calculating interest on a debt on which partial payments are made. The one in common use among business men and known as the Merchants' Rule, consists in calculating the interest on the principal sum from its date to the date of settlement and a similiar calculation on the partial pay- ments, the difference between the total interest on the respective sides being the net interest due. On small sums and for short periods this is accurate enough for all practical purposes.
The other, called the United States Rule, gives precedence to the interest due at the time of each payment, and requires that each payment shall be first applied to the liquidation of the inter- est then due, only the remainder after the interest is deducted being applicable to the reduction of the principal. If the pay- ment is not equal to the interest then due, it is applied in reducing the interest, but the excess interest is not added to the principal,
158 MATHEMATICS OF ACCOUNTING AND FINANCE
as this would be compounding it, but it is carried down and added to the interest to be deducted from the next payment. This method is made legal by statute in nearly every state.
An example of the two methods will show the difference between them, 30 days to the month being used for convenience.
Illustration
Jan. I. Original amount $600 March i, payment $200
May I, " 200
June I, " 100
Interest is to be charged at 6%. Required — amount due July i Solution:
By the Merchants' Rule :
Interest on $600 for 6 months is $18.00
" " 200 "4 " " $4.00
" " 200 "2 " " 2.00
" " 100 " I " " 50 6.50
Interest due July i $11.50
Unpaid principal 100.00
Total due July i $111.50
By the United Stales Rule:
Original debt $600.00
Payment March i $200.00
Less interest on $600 for 2 months 6.00 194.00
$406.00
Payment May i $200.00
Less interest on $406 for 2 months 4.06 195.94
$210.06
Payment June i $100.00
Less interest on $210.06 for i month. . . 1.05 98.95
$111. II
Interest on $111. 11 for i month .56
Total due July i $111.67
SIMPLE AND COMPOUND INTEREST 159
While the difference of 1 7 cents in the results obtained by the two methods is negligible, it is otherwise when the amounts are large and the time is long. The following example is taken from the accounts of an estate that was not settled for thirty years. It was virtually as follows :
Illustration
An amount of $36,000 was due one of the heirs of an estate, who was to receive 6% interest until paid.
January i, 1886, legacy $36,000
" I, 1 89 1, payment 12,000
" I, 1896, " 12,000
" I, 1906, " 12,000
" I, 1911, " 19,000
The administrator, using the commercial rule, agreed to settle the account as of January i, 1916, as follows:
Original amount $ 36,000
Interest for 30 years at 6% 64,800
$100,800
Less total cash paid $55,000
Interest on $12,000 for 25 years 18,000
Interest on $12,000 for 20 years 14,400
Interest on $12,000 for 10 years 7,200
Interest of $19,000 for 5 years 5, 700 100,300
Amount due January i, 1916 $500
The heir refused the settlement and made his claim under the law of the state (Illinois) as follows:
Original amount $36,000.00
January i, 1891, payment $12,000.00
Interest 5 years on $36,000.00. . . 10,800.00 1,200.00
$34,800.00
January i, 1896, payment $12,000.00
Interest 5 years on $34,800.00. . . 10,440.00 1,560.00
$33,240.00
I60 MATHEMATICS OF ACCOUNTING AND FINANCE
January i, 1906, payment $12,000.00
Interest 10 years on $33,240.00.. 19,944.00
Interest carried forward $ 7,944.00
January i, 191 1, payment $19,000.00
Interest 5 years on $33,240.00. . . $9,972.00
Interest brought forward 7,944.00 17,916.00 1,084.00
$32,156.00
January i, 1916, interest 5 years
on $32,156.00 9,646.80
Amount due January i, 1916.. $41,802.80
It can readily be seen that a little knowledge of correct principles was a valuable asset to the heir.
Compound Interest
The principles of interest are involved in the computations which an accountant may be required to make in connection with such matters as bond premium and discoimt, leasehold premiums, depreciation and sinking funds. These calculations involve not only compound interest and the amount of a given principal at compound interest, but also the more complex problems of annuities.
All scientific computations of interest on an indebtedness or an investment extending over more than one period of time must be based on compound interest. This is because the indebtedness or investment increases with the lapse of time. If only simple interest is charged, interest is earned only on the original invest- ment instead of on the investment for each period. For instance, if $100 is invested at 6%, the investment at the beginning of the first year is $100. It increases $6 during the first year and amounts at the end of the first year to $106.
Since the investment has now increased to $106, interest
SIMPLE AND COMPOUND INTEREST l6l
should be earned thereon. If simple interest only is charged, $6 will be earned during the second year on an investment of $io6. This is at the rate of only 5.66 +%.
Scientific computations of interest are based on the sup- position that the increase arising from interest is re-invested. This is equitable in theory, for if the interest of one period in- creases the investment at the close of that period, the investor should, during the next period, earn interest on the increased investment. To maintain the agreed interest rate during a series of periods, computations must be made on the basis of periodical compounding.
Symbols
The following standard symbols by Sprague and Perrine, will be used throughout this book:
I = $1, £1, or any other unit of value
i = the rate of interest for a single period
« = an indefinite number of periods
a = the amount of $1 for a given time at a given rate
/ = the compound interest on $1 for a given time at a given rate
p = the present worth of $1 for a given time at a given rate
D = the discount on $1 for a given time at a given rate
r = {i -\- i), the periodic ratio of increase'
Amount of Principal
In finding the amount of a given principal at compound inter- est for a given number of years at a given rate per year com- pounded annually, it is customary to compute the amount of $1 for the given time at the given rate, and multiply this result by the number of dollars in the principal.
When i = 6% or .06, the accumulation of the amount may be computed by either of the following methods:
-C. E. Sprague and L. L. Perrine, The Accountancy of Investment, 1914.
l62 MATHEMATICS OF ACCOUNTING AND FINANCE
First Method Second Method
Dollars End of Dollars Symbols
Principal i.oo i.oo I
Interest on Ji. 00 .06 Multiply by ... . 1.06 (i + i)
1.06 I year 1.06 (i + «')
" " 1.06 0636 " " .... 1.06 (i + »')
1. 1236 2 " 1. 1236 (i + «')'
" " 1. 1236 067416 " " .... 1.06 (i + »)
1.191016 3 " 1.191016 (i +»)^
1.191016 071461 " " .... 1.06 (i + »)
1.262477 4 " 1.262477 (i +')*
The ratio of increase is, in symbols, (i+i), and in figures 1.06. The investment at the beginning of each year must, therefore, be multiplied by (i+i) to obtain the investment, or amount, at the end of that year. Hence i invested at the rate i becomes at the end of
1 year (i + O
2 " d + O'
3 " (i-hi)'
4 " d+O'
n " (i + O"
Thus the formula is obtained
a= (1+ i)"
If it is desired to find the amount of i at 5% for 20 years, compounded annually, the formula becomes
Frequency of Compounding
In the preceding illustration the interest was compounded annually. But the period of compounding may be shorter than one year, with the result that the compounding occurs with greater frequency than once a year. In such cases the formula stated above still applies. Although compounding may occur
SIMPLE AND COMPOUND INTEREST 163
semiannually, quarterly, monthly or even daily, the rate is usually stated as a certain per cent per year, but i in the formula is the annual rate divided by the number representing the periods in a year. For instance, if the rate is 6% per year, iunder various conditions would be as follows:
Frequency of
Compounding Value of i
Annually .06
Semiannually (.06 -^ 2 ) .03
Quarterly (.06 -^ 4 ) .015
Monthly (.06 -^ 12 ) .005
Daily (.06 -^ 365) -^
The number of periods, represented in the formula by n, will be the number of years times the number of periods per year. For instance n, under various conditions, is as follows:
Frequency of Value of n
Compounding One Year Two Years Three Years
Annually i 2 3
Semiannually. . . 2 4 6
Quarterly 4 8 12
Monthly 12 24 36
Daily 365 730 1095
Illustration
What is the amount of $1 at 6% interest for 20 years, compounded quarterly?
Solution:
i = .06 -4- 4 = .015, the rate for one period
n = 20 X 4 = 80, the number of periods
a = (i + i)** or (1.0x5) *°, which amounts to 3.29066279
Determining the Amount
Interest tables show the amount of $1 at various rates for various periods. Such a table appears in the Appendix of this
1 64
MATHEMATICS OF ACCOUNTING AND FINANCE
book. When a table is not available, the amount may be computed with a table of logarithms. When neither an inter- est table nor a table cf logarithms is available, it is necessary to compute the amount by repeated multiplication, thus:
1.03 1-03
1. 0609 1.03
1.092727 1.03
a = (1.03)'= ?
= (1.03)^ = (1.03)^
1.12550881 = (1.03)"
At this point the number of decimal places becomes so large as to make the computation too laborious. The number may be reduced to six places, by approximation, thus:
1.12550881 becomes 1. 125509
The multiplication continued is as follows:
1.125509 = (i.03)'* 1.03
159274 03
(1-03) ■
194052 = (1.03)' 03
229874 = (1.03)7 03
266770 = (1.03)
This last amount is the amount of $1 at 6% interest for 4 years, compounded semiannually
SIMPLE AND COMPOUND INTEREST 165
But when the number of periods is large, as will be the case when the interest must be calculated over a long term of years, these repeated multiplications become irksome. It is possible, however, to cut down on the number of operations necessary. A short method may be utilized which should be readily understood if it is remembered that (14-^)" means that (i +i) is used n times as a factor. If we know the value of (i +i) ^ we have the amoimt obtained by using (i+i) twice as a factor. By squaring this amount we get (1+^)'', which is the product resulting from the use of (1+/) four times as a factor. Squaring this amount in turn gives us (i +i) ^, the product obtained by using (i +/) eight times as a factor.
By the application of this principle, a very material reduction may be effected in the number of multiplications, as is made clear in the example given below :
1.03 1.03
1.0609 1.0609
= (i + i) = (i + i)
1. 125509 1. 125509
= (1 + i)'
1.266770 = (i + i)
The quantity 1.03'° can be determined in the following manner :
1.26677 (i + «)^ 1.0609 (i "f" ^)^
1. 34391 6 (i + i)^"
The general application of this principle may be made clearer by another illustration.
I66 MATHEMATICS OF ACCOUNTING AND FINANCE
Illustration
Required — the value of (i + i)^°:
Solution:
(I + i) (I + i)
: +
i)'
+
iy
+
i)'
+
i)'
+
i)'
+
iV
+
■yU
+
i)'
+
iy
+
iy
+
iV
+
iy
Thus, only seven multiplications are required to obtain the 30th power.
The 80th power could be obtained thus :
(i + 7) '^ (obtained by four multiplications)
(1+ i)"
(1+ i)'' (1 + i)''
u + iy''
SIMPLE AND COMPOUND INTEREST 167
The same principle may be utilized in the use of compound interest tables which do not extend to the desired number of periods.
Illustration
Given a table of 40 periods, required — the value of (i + i) '^.
Solution:
( I + /) '^ " shown by table
(I + /V " " "
(i + i)"
Determining Interest
The amount of $1 at compound interest is composed of two elements: the original investment of $1 and the accumulated interest. Hence, to find the compound interest, apply the follow- ing formula:
/ = c — I or / = (i + J-)" - I
If $1.266770 is the amount of $1 at 6% interest, compounded semiannually for four years, $1 .000000 is the original investment, and $.266770 is the compound interest.
Determining Present Worth
The present worth of a sum due at a fixed future date is a smaller sum which with interest will amount to the future sum. For instance, the present value of $1 due in one year at 6% is a sum, smaller than $1, which with interest at 6% for one year will amount to $1. Representing the present value of this $1 by p
p X 1.06 = $1
or in symbols, pX (i + i) = i when the dollar is
due in one year.
I68 MATHEMATICS OF ACCOUNTING AND FINANCE
Since pX {i + i) = i,
it follows that i -r- (i + t) = /»
(Since p X 1.06 = i, it follows that I-^ 1.06 = .943396.) If the dollar is due two periods hence,
PX (i-i- i)'= I and it follows that i -^ (i + f)^ = />
If the dollar is due three periods hence
pX (i-\- i)'= I and it follows that i -^ {i -\- i)^ = p
If the dollar is due ?i periods hence
px {i-\- ir = I
and it follows that i -^ (i -\- i)" = p
But since (i + i)" = a
the formula p = i -^ (i + i)"
may also be stated p = i -^ a
In determining the present value at 6% of $1 due in four years, interest compounded semiannually, several methods are available.
1 . It may be possible to refer to a table of present values, similar to the one in the Appendix.
2. If a table of present values is not available, the present value may be computed by using the formula
p = I -T- a It will first be necessary to obtain the value of a. When in- terest at 6% per annum is compounded semiannually, 7 = .03 ; with semiannual compounding for four years, « = 8. Then, a = (1.03)^. The value of a may be found by any of the methods previously explained; it is 1.266770. Then
$1 -V- 1.266770 = $.789409, the present value
SIMPLE AND COMPOUND INTEREST
169
3. Instead of dividing $1 by (1.03) ^ i.e., by 1.266770, the same result can be obtained by dividing by 1.03 eight times, using I as the first dividend, each succeeding dividend being the quo- tient resulting from the preceding division, thus
$1.000000 -
- 1.03 = $.
.970874 -
- 1.03 = .
.942596 -
- 1.03 = .
.915142 -
- 103 = .
.888487 -
- I-03 = .
.862609 -
- 1.03 = .
.837484 -
- 1.03 = .
.813092 -
- 1.03 = .
970874, present value of $
942596,
915142,
862609,
837484, 813092, 789409,
due in i period
2 periods
3
4
5
6
7
Determining the Compound Discount
As shown in the preceding section, one would, in exchange for a promise to pay $1 a given number of periods hence, loan the present value of $1. Present values for various numbers of periods, and the discount earned, are shown below (discounted at 3% per period) :
Due Periods Hence
Amount Loaned Present Value
I $.970874
2 942596
3 915142
4 888487
8 .789409
Discount
Earned
(i-p)
$.029126 .057404 .084858
•111513 .210591
The discount for two or more periods is called compound discount. Compound discount may be computed by the formula
D = I - p
It may also be computed by the formula
D= I -^ a
170 MATHEMATICS OF ACCOUNTING AND FINANCE
This formula requires explanation. Compound discount is really compound interest, deducted in advance; but it is the compound interest on the money actually loaned.
For instance, if $.78940915 loaned on a promise to pay$i eight periods hence at 3%, the discount $.210591 is 3% compound interest on $.789409, the principal actually loaned. This fact can be demonstrated thus :
The amount of $1 at 3% compound interest for eight periods is $1.266770. Hence the compound interest on $1 is $.266770 and the compound interest on $.789409 is the result in the fol- lowing multiplication:
.789409 p
Multiplied by .266770 i
Product .210591 d
Since the compound discount {D) is really the compound interest (/) on the actual loan {p) ,
D= IX p And since /> = i -^ a,
we can substitute (i H-a) for p, and the formula becomes
D= IX I or D = I -h a
This formula may also be explained thus : The present value of $1 due in eight periods at 3% is $.789409, or $1 -^(1.03)^, or$i-^a. If the loan were i, the compound interest would be /, or $.266770; but since the loan is i-i-a, the compound discount is / -7-a, or $.266770^ 1. 266770=$. 210590.
Summary
In this chapter the following formulas have been derived:
a = (i + ir
I = a — I
p = 1 -r- a; ori-T-(i + t)''
D = 1 — p, or / -7- a
CHAPTER XVII
ANNUITIES
Definition of Annuities
A series of equal payments, due at regular intervals, is an an- nuity. Although the word "annuity" suggests annum and year, the interval may be any period, as a month, quarter or half-year.
Symbols
In the discussion of annuities the following symbols will be employed :
A = the amount of an annuity of $i for a given time at a given
rate P = the present value of an annuity of |i for a given time at a
given rate
Amount of an Annuity
Let us assume that a contract provides for payments as in the following :
Example
January i, 191 5 $100
" I, 1916 100
" I, 1917 100
" I, 1918 100
Total $400
Required — the accumulated value of this annuity at January i, 1918, interest at 5%.
Solution: It is assumed that each payment is put at interest, in which case the amount of each payment computed separately, is as follows:
171
172
MATHEMATICS OP ACCOUNTING AND FINANCE
Payment Made
January i, 19 is
I, 1916
I, 1917
" I, 1918
Amount of the annuity
Periods at Interest
Payment
$100 100 100 100
Amount
Symbol
(I +i)' (I +.)' (I + 0
115.7625 110.2500 105.0000 100.0000
431.0125
Although the amount of an annuity may be computed by determining the amount of each payment, it is unnecessary to resort to this labor, as the following short method may be used:
To find the amount of an annuity of $i for a given number of periods at a given rate, divide the compound interest on $i for the number of periods at the given rate, by the interest rate.
Or, in symbols, A = I -^ i
Applying this formula to the illustration above, /, the compound interest on $i for four periods at 5%, is shown by an interest table to be $.215506; then
$ .215506 -J- .05 = $4.31012, amount of an annuity of $1 $4.31012 X 100 = $431,012, amount of an annuity of $100
This formula requires explanation. Let us suppose that $1 is loaned on January i, 1914, the contract requiring payment of simple interest annually at 5% as follows:
D.\TE Interest
January i, igi5 $.05
I, 1916 .05
I, 1917 05
1,1918 05
These payments, being equal in amount and made at regular intervals, constitute an annuity of $.05 per year. If put at 5% compound interest, the amount of each payment computed separately would be as follows:
ANNUITIES
173
Payment Made
January i, ipis I, 1916 I. 1917 I, 1918
Totals
Periods at Interest
Payment
$.05
• OS
• OS
• OS
f.20
Amount
Symbols
(l + i)i (I +i}' (I + i)
.057881 •055125 .052500 .050000
.215506
The total of the annuity payments is $.20, the simple interest on $1 for four years ; and the amount of the annuity, $.215 506, is the compound interest on $1 for four years, or /. Hence, compound interest is merely the amount of an annuity. If the compound interest $.215506 is the amount at 5% of four annual payments of 5 cents,
$.215506 -^ 5 = $.0431012, the amount at 5% of four annual pay- ments of $.01
or $.215506 -^ .05 =
or in symbols
I..31012, the amount at 5% of four annual payments of $1.
1 ^ i= A
Illustration
Required the amount of an annuity of $25 per month for five years at 6% per annum, interest compounded monthly.
Solution:
12 (payments per year) X 5 = 60, number of terms 6% -^ 12 = K% or -ooS) the rate
Now, I = a — 1
And a = (1.005)^°= 1.348849
Then \ = I -^ i
= .348849 ^ .005
= 69.7698, amount of an annuity of $1 $69.7698 X 25 = $1,744.25, amount of an annuity of $25
174 MATHEMATICS OF ACCOUNTING AND FINANCE
Sinking Fund Contribution
The preceding section developed a method of determining the amount which results from the accumulation of a known annuity. It is the purpose of this section to discuss the converse problem, of finding an unknown annuity which will produce a required amount.
This problem finds application among accountants in com- puting the periodical contribution necessary to accumulate a required sinking fund. Let us assume that a sinking fund of $100,000 is to be accumulated in five years by equal instalments made at the end of each year; what is the required contribution, assuming 4% interest compounded annually?
The annual contributions constitute an annuity and the accumulated fund is the amount of the annuity. We shall first find what fund would be accumulated by a contribution of $1, applying the formula
A = I ^ i
To find I:
1.04 Multiplied by 1.04
1.0816 " 1.0816
= (1 + i)'
1.16Q85856 " " 1.04
= (1+ /)^
1. 2166529024 Deduct I .
= (i + f)5
.2166529024 = /, compound interest on $1 for five periods at 4%
To find A :
.2166529024 -^ .04 = 5.41632256, amount of an annuity of $1
ANNUITIES
175
Since annual contributions of $1 will produce a fund of $5.41632256, to find the contributions necessary to produce a fund of $100,000:
$100,000 -r- 5.41632256 = $18,462.71, required contribution
The computation was performed without using an interest table. If tables are available, the work may be materially decreased.
For instance, a table of amounts of $1 per annum shows the amount of $1 for live periods at 4% to be $5.416323. Only one computation is necessary :
$100,000 -j- 5.416323 = $18,462.71
If only a compound interest table is available, it will show the amount of $1 in five periods at 4% to be $1.216653. The following computations will be necessary:
$.216653 -^ -04 = $5.416325, amount of annuity of $1
$100,000 -^ 5.416325 = $18,462.71
The following tabulation of sinking fund accumulations shows the accumulation of the fund from the two elements, annual contributions and interest:
End of Ye.-vr
Contribution
Interest
Total Fund
$18,462.71 18,462.71 18,462.71 18,462.71 18,462.71
$ 738.51 1,506.56 2,305.33 3.136.05
$18,462.71 37.663.93 57.633.20 78,401.24
Totals
$92,313.55
$7,686.45
Present Worth of an Annuity
Let us assume that a contract, made on January i, 1914, provides for the following payments :
176
MATHEMATICS OP ACCOUNTING AND FINANCE
January i, 1915 $100
" I, 1916 100
" I, 1917 100
" I, 1918 100
Total $400
We are to find the present value of this series of payments, or annuity, on January i, 1914, discounted at 5%.
The present value of each payment, computed separately, is as follows:
Payment Due
Periods Hence
Payment
Present Value
Symbol
$
I
2 3 4
lioo 100 100 100
I -=- Cr 4- i)
95.2381 90.7029
86.3838
82.2702
I, 1916 .
I - I -
- (I +i)'
- (i +i)^
- ri 4- .-14
I, 1917
I, 1918
Totals
J400
3S4.S9S0
The present value of each payment, and the present value of the annuity, may be computed thus:
5100 95.2381 90.7029 86.3838
-T- 1.05 = $ 95.2381 present value of $100 due in i year
1.05 = 90.7029, 1.05 = 86.3838, 1.05 = 82.2702,
$100 $100 $100
2 years
3
4
$354.5950, present value of the annuity.
When interest tables are not available, this method is satis- factory; but w^ith interest tables available a short method may be used which is similar to the short method of computing the amount of an annuity.
To find the present value of an annuity of $1 for a given
ANNUITIES
177
number of periods at a given rate, divide the compound discount on $1 for the given number of periods by the interest rate. Or in symbols,
P = D -^ t
Applying this formula to the illustration above, the present value of $1 due four periods hence at 5%, is shown by an interest table to be $.822702;
D, the compound discount, is $.177298.
$.177298 -4- .05 = $3.54596, the present value of an annuity of $1
$3-54595 X 100 = $354,596, " " " " " " " $100
This formula also requires explanation. Let us suppose that a loan of $1 is made on January i, 1914, the contract requiring the payment of simple interest annually at 5% as follows :
Date Interest
January i, 191 5 $.05
" I, 1916 05
I, 1917 05
" I, 1918 05
These payments, being equal in amount, and made at regular intervals, constitute an annuity of five cents per year. If dis- counted at 5%, the present value of each payment on January I, 1 9 14, computed separately, is as follows:
Payment Due
Periods Henxe
Payment
Present Value
Symbols
$
January i, 1915
I, 1916
I 2 3
4
S.os
• 05
• OS ■ 05
I - I - I -
- (1 + .)
- (i +.)'
- (i +.-)3
- Ct -I- ,U
.0476191 •0453514 .0431919 .0411351
Present value of the annuity ....
•177297s
178 MATHEMATICS OF ACCOUNTING AND FINANCE
Now, instead of paying five cents interest each year, and pay- ing the $1 at maturity, the present value ($.1772975) of the four interest payments might be paid, or deducted, in advance, thus:
$1.0000000 payment to be made at end of four years .1772975 present value of interest payments
$ .8227025 present value of $1
In other words, the simple interest on $1 at 5% for four years is an annuity of five cents; and $. 1772975, the present value of these in- terest payments, is the present value at 5% of an annuity of four five cents payments; and$.i 772975 is also the compound discount on $1 due four periods hence at 5%. Therefore the compound discount on $1 , due in four periods at 5%, is the present value of an annuity of five cents in four periods at 5%.
If the compound discount $.1772975 is the present value at 5% of four annual payments of 5 cents, then
$.1772975 H- 5 = $.0354595, the present value at 5% of four annual
payments of i cent
or $.1772975 -4- .05 = $3.54595, the present value at 5% of four
annual payments of $1
or in symbols D -^ i = P
Illustration
Required the present value of an annuity of $25 per month for five years at 6% per annum, compounded monthly.
Solution: 12 (payments per year) X 5 = 60, number of periods 6% -i- 12 = K% or .005, the rate
D = 1 - p and /> = I -^ (1.005)^"= .741372
Then D = i — .741372 = .258628, the compound discount on $1
due in sixty periods at >2 %
ANNUITIES 179
and
D ^ i= P
$ .258628 -^ .005 = $ 51.7256, present value of an annuity
of $1 $51.7256 X 25 = $1,293.14, the present value of an an- nuity of $25
Rent of an Annuity
The preceding section, containing an explanation of the method of determining the present worth of a known annuity, developed the formula, P = D-^i. Each periodical instalment of an annuity is known as the rent of an annuity. This section will deal with the question of determining what rent will be produced by a known present worth.
It has been shown that the present value of an annuity of $1 for four periods at 5% is $3.54595,
Since a present value of $3.54595 will produce four annual rents of $1 at 5%, a present value of $1 will produce four
annual rents of of $1, or $.282012.
3-54595
Therefore, to find the rent of an annuity with a present value of $1 for a given number of periods at a given rate:
1. Find the present value of an annuity of $1 for the given
time and rate
2. Divide $1 by this present value of an annuity of $1
Or expressed in symbols,
R= I -^ P
Illustration
A man invests $3,000 in an annuity to be repaid to him in five annual instalments, interest computed at 6% annually. What annual rent will be produced?
l8o MATHEMATICS OF ACCOUNTING AND FINANCE
Solution: Formula — R = i -^ P
The present value of $i due five periods hence at 6% may be com- puted thus:
I -=- (i.o6)s
or it may be determined from interest tables. The compound interest table shov/s
1. 06^ = 1.338226 then
I -^ 1.338226 = .747258, the present value of $1 due
in five periods
I — .747258 = .252742, the compound discount on $1
due in five periods .252742 -T- .06 = 4.21236, the present value of an annuity
of $1 for five periods at 6%
then
R = 1 -7- 4.21236 = .237396, rent produced by present worth of
$1 $.237396 X 3,000 = $712.19, rent produced by present worth of
$3,000
The following schedule shows the reduction of the investment due to the excess of the periodical payments over the interest earned on the decreasing investment.
Original investment $3,000.00
Interest ist year, 6% of $3,000 180.00
$3,180.00 Deduct ist rent 712.19
Balance, end of ist year $2,467.81
Interest 2d year, 6% of $2,467.81 148.07
$2,615.88 Deduct 2d rent 712.19
Balance, end of 2d year $1,903.69
Interest 3d year, 6% of $1,903.69 114.22
$2,OI7.Qi
Deduct 3d rent 712.19
ANNUITIES l8l
Balance, end of 3d year $1,305.72
Interest 4th year, 6% of $1,305.72 78.34
$1,384.06 Deduct 4th rent 712.19
Balance, end of 4th year $671.87
Interest, 5th year, 6% of $671.87 40.31
$712.18 Deduct 5th rent 712.18
$ .00
The schedule could also be arranged as follows, showing that each payment is composed of two parts:
1 . Interest on the diminishing principal
2. Repayment of the principal
Original investment $3,000.00
First rent:
Interest on $3,000 at 6% $180.00
Repayment of principal 532.19 532.19
$712.19 $2,467.81
Second rent:
Interest on $2,467.81 at 6% $148.07
Repayment of principal 564.12 564.12
$712.19 $1,903.69
Third rent:
Interest on $1,903.69 at 6% $114.22
Repayment of principal 597-97 597-97
$712.19 $1,305.72
Fourth rent:
Interest on $1,305.72 at 6% $ 78.34
Repayment of principal 633.85 633.85
$712.19 $ 671.87 Fifth rent:
Interest on $671.87 at 6% $ 40.31
Repayment of principal 671.87 671.87
$712.18 $ .00
l82 MATHEMATICS OF ACCOUNTING AND FINANCE Or the schedule may be shown thus: Payment Rent
I $ 712.19
2 712.19
3 712.19
4 712.19
5 712.18
Totals $3,560.94
[ntere-st
Reduction of Investment
Diminishing Investment
$3,000.00
$180.00
$
532.19
2,467.81
148.07
564.12
1,903.69
114.22
78.34
40.31
$3
597-97 633-85 671.87
,000.00
1,305-72
671.87
0
$560.94
Equal Periodical Payments on Principal and Interest
When a debt together with the interest thereon is to be paid in equal periodical instalments, the principal of the debt is the pres- ent value of an annuity, and the periodical payments are rents, to be computed by the method already explained.
To illustrate, if $3,000 bearing 6% interest compounded annually, is to be paid in five annual instalments, each to include the accrued interest and a portion of the principal, $3,000 is the present value of five unknown rents. These rents were computed in the preceding illustration , being $712.19. The reduction of the debt may be tabulated thus :
Original principal $3,000.00
First payment $712.19
Less I year's interest on $3,000 180.00
Payment on principal 532.19
Balance of principal $2,467.81
Second payment $712.19
Less I year's interest on $2,467.81 .... 148.07
Payment on principal 564.12
Balance of principal $1,903.69
Third payment $712.19
Less I year's interest on $1,903.69 114.22
Payment on principal 597-97
ANNUITIES 183
Balance of principal $1,305.72
Fourth payment $712.19
Less I year's interest on $1,305.72 78.34
Payment on principal 633,85
Balance of principal $671.87
Fifth payment $712.18
Less I year's interest on $671.87 40.31
Payment on principal 671.87
Balance of principal $ .00
Annuities Due
In the foregoing discussion of annuities, rents, and sinking fund contributions, the formulas and methods described apply to the ordinary form of annuities, in which the payments are made at the end of the periods.
When the payments are made at the beginning of the periods, the annuity is called an annuity due. Changing the payment from the end to the beginning of the period affects the compound interest, thus changing both the amount and the present value of the annuity.
To Find the Amount of an Annuity Due
To arrive at a method of computing the amount of an annuity due, let us compare the amounts of:
1. An ordinary annuity (A) of six periods (due at end of
period)
2. An annuity due (B) of live periods (due at beginning of
period)
In each annuity, the interest is compoimded annually at 4%. A requires the payment of $1 on December 31 for each of six years, beginning December 31, 191 1.
1 84
MATHEMATICS OF ACCOUNTING AND FINANCE
B requires the payment of $i on January i for each of five years,
beginning January i, 191 2. It will be noted that the first five payments of annuities A and B are made on practically identical dates; but annuity A has one more payment than annuity B ; hence the amount of annuity A will exceed the amount of annuity B by one payment of $1. This may be more clearly shown by the following table which shows the amount of each annuity payment and the amount of the annuity.
Annuity A
Annuity B
Date
Payment
Amount at Dec. 31. 1916
Date
Payment
Amount at Dec. 31. 1916
Dec. 31, 1911
•' 31. 1912
" 31. 1913
" 31. 1914
" 31. 191S
" 31, 1916
Ji.oo 1. 00 1. 00 1. 00 1. 00 1. 00
Si. 216653 1. 169859 1. 124864 1. 081600 1.040000 1. 00
Jan. I, 1912
" I. 1913
" I. 1914
" I. 191S
" I, 1916
$1.00 1. 00 1. 00 1. 00 r.oo
Si. 216653 1. 169859 1. 1 24864 1. 081600 1.040000
$6.632976
55-632976
Annuity B is an annuity due of five rents; annuity A is an or- dinary annuity of six rents; the periodical payments and interest rate are the same in each case. The difference between the two amounts is $1, or one periodical payment.
Hence we can find the amount of annuity B (annuity due) for five periods by ascertaining the amount of an ordinary annuity (A) of the same payment and rate for six periods, and deducting $1 . The amount of annuity A (ordinary) may be computed thus:
$1.265319, amount of $1 for six periods 1. 00
$ .265319, compound interest for six periods
$.265319 -^ .04 = $6.632975, amount of an ordinary annuity of six periods
ANNUITIES 185
To compute the amount of B, the annuity due:
$6.632975, amount of ordinary annuity of six periods Deduct 1. 000000, one rent
$5.632975, amount of annuity due of five periods
Sinking Fund
Contributions to the sinking fund of a bond issue are ordin- arily made at the end of the period. If a bond issue is to run twenty years and provision is to be made for it by twenty con- tributions to a sinking fund, the first contribution is usually made at the end of the first year, thus allowing time in which to acquire cash from profits. Such contributions to a fund constitute an ordinary annuity.
When contributions to a fund are made at the beginning of the period, the payments become an annuity due. To find the periodical contribution necessary to accumulate the required fund by such payments :
1. Find the amount of an ordinary annuity of $1 for one
more than the number of periods
2. Deduct $1 in order to find the amount of an annuity
due of $1 for the given number of periods
3. Divide the required fund by the amount of an annuity due
of $1.
Required Annual Contribution
In the discussion of sinking fund contributions in the early part of this chapter it was required to find the sinking fund contribution necessary to invest at 4% at the end of each of five years to pay an obligation of $100,000. This was found to be $18,462.71. We shall now calculate the required annual contri- bution if made at the beginning of each year.
These payments constitute an annuity due of five periods. The calculation will proceed by the following steps:
1 86
MATHEMATICS OF ACCOUNTING AND FINANCE
I. Find the amount of an ordinary annuity of six periods: !i. 265319, amount of $1 at compound interest for six periods at 4%
.265319, compound interest on $1 for six periods at 4%
.265319 -r- .04 = $6.632975, amount of ordinary annuity of six periods
2. Find the amount of an annuity due of five periods:
$6.632975
.632975, amount of annuity due of five periods
3. Divide total fund by amount of annuity of $1 :
$100,000 -H 5.632975 = $17,752.61, required annual contribution
The following sinking fund table shows the accumulation of the fund due to the two elements of contributions and compound interest.
Ye.\r
Contribution
Fund Beginning Year
Interest
Fund End of Ye.\r
I 2 3 4 5
$17,752.61 17,752.61 17,752.61 17,752.61 17,752.61
517.752-61 36,215.32 55.416.54 75,385.81 96,153.85
$ 710.10 1,448.61 2.216.66 3.015.43 3.846.15
$ 18,462.71
37,663.93
57,633.20
78,401.24
100,000.00
$88,763.05
Six, 236. 95
Present Worth of an Annuity Due
When the annuity payments are due at the beginning of the period, the present value of the annuity is composed of two elements :
1. A rent, or payment, now due (a)
2. The present value of an ordinary annuity of one less than
the given number of periods (b)
ANNUITIES 187
For instance, $5.451822 deposited at 4% interest compounded annually permits the immediate withdrawal of:
(a) $1 rent now due, and leaves a balance of
(b) $4.451822, which is the present value of live annual pay-
ments of $1, the first of which is due one year hence.
Therefore, to find the present value of an annuity due, proceed as in the following illustration.
Illustration
What is the present value at 4% per annum of an annuity of six pay- ments of $1 , the first of which is due?
Solution: i. Find the present value of an ordinary annuity for one less than the given number of periods:
$1.000000
.821927, present value of $1 due in five years at 4%
$ .178073, compound discount on $1 due in five years at 4% $ .178073 -^ .04 = $4.451825, present value of an ordinary annuity of five payments
2. To this present value add one rent:
$4.451825, present value of ordinary annuity of five payments 1. 00
$5.451825, present value of an annuity due of six payments
The reduction of this present value, due to the accumulation of interest and the payment of rents is, tabulated below.
Present value of annuity due of six rents $5.451825
First rent i .
Balance $4.451825
Interest earned ist period .178073
Balance end of " " $4.629898
Second rent i .
I88 MATHEMATICS OF ACCOUNTING AND FINANCE
Balance beginning 2d period $3.629898
Interest earned " " .145196
Balance end of " " $3.775094
Third rent i.
Balance beginning 3d period $2.775094
Interest earned " " .111004
Balance end of " " $2.886098
Fourth rent i.
Balance beginning 4th period $1.
Interest earned " " .075444
Balance end of " " $1.961542
Fifth rent i .
Balance beginning 5th period $ .961542
Interest earned " " .038461
Balance end of " " $1.000003
Sixth rent i .
Since the sixth rent is withdrawn at the beginning of the sixth period, there is no balance at the beginning of, nor interest earned during, the sixth period.
Rents
The converse of the above problem is to determine what periodical payment or rent a known present value will produce, when the first payment is due immediately.
For instance, what rent will an investment of $1,000 produce, the first of six annual rents being due at once and the interest being 4%?
1. Find what present value will produce a rent of $1
This was found above to be $5.451825
2. Divide the known present values by the present value of
an annuity of $1
$1000.00 -j- 5.451825 = $183.42
ANNUITIES 189
The reduction of this present value may be tabulated thus:
Period
Rent
Balance
Interest
Balance
$1,000.00
I
3183.42
$816.58
I32.66
849.24
2
183.42
665.82
26.63
692.4s
3
183.42
sog.03
20.36
529.39
4
183.42
345.97
13.84
3S9.8I
S
183.42
176.39
7.06
183.4s
6
183.4s
0.
Si. 100.55
$100.55
CHAPTER XVIIi
LOGARITHMS IN COMPOUND INTEREST AND ANNUITY COMPUTATIONS
Calculating Compound Interest and Annuities by Logarithms
The following examples illustrate the methods of utilizing logarithms in compound interest and annuity computations.
Examples
I. To find the amount of i. Required — the amount of $4,500 at 4% per annum for twenty years, compounded semiannually
Solution:
Amount = $4,500 X 1.02''°
Log 1.02 = .00860
Multiply by 40
Log 1.02''" = .34400, the log of 2.208
Hence 1.02''''= 2.208
Amount = $4500 X 2.208 = $9936
Amount computed with aid of interest table = $9936.18
Logarithms may be used more extensively in this solution, as follows:
Log 4,500 = 365321
Log 1.02 = .00860 Multiply by 40
Log 1.02''" = .34400
Log 4,500+ log 1.02'"'= 3.99721, log of 9,936 Hence $4,500 X 1.02'"'= $9,936
2. To find the compound interest on i.
Solution: Compute the amount as above and deduct the principal
Amount $9,936.
Principal 4.500.
Compound interest $5,436. iqo
LOGARITHMS AND ANNUITY COMPUTATIONS
191
3. To find the principal. What sum invested at 4% compounded quarterly will amount to $5000 in eight years? Solution:
Principal = 5000 -7- (i.oi)-*^
Log 5000 = 3.69897
Log 1. 01 = .00432 Multiply by 32
Log i.oi^^ = .13824
Log of principal = 3.56073, the log of 3636.92
Principal = $3636.92
Principal computed by interest table = $3636.52.
4. To find the rate. If $1,125 is to be returned at the expiration of seven years for a loan of $800, what rate of interest compounded an- nually is earned? Solution:
Since 1,125 = 800 X (i + tV
(i + i)^= 1,125 -J- 800 Log 1,125 = 305115
Log 800
Log (i +1)7 Log I + i Hence \ -\- i
i
90309
14806
14806 -7- 7 = .021151, the log of 1.0499
0499
0499, or nearly 5%
5. To find the time. For how many years should $2,000 be placed at 5% interest compounded annually to produce $5,054?
Solution:
Since $5,054 = $2,000 X 1.05" 5,054 -^ 2,000 = 1.05" Hence 3.70364, which is log 5054 Minus 3.30103, " " " 2,000
Equals .40261 '' " " 1.05" Since .40261 = log 1.05 X n
n = .40261 -T- log 1.05 Since log 1.05 = .02119
n = .40261 -^ .02119 — 19) the number of years
192 MATHEMATICS OF ACCOUNTING AND FINANCE
6. To find the present value of i. What is the present value of $15,000 due in five years at 4%, interest compounded quarterly?
Solution:
Present value = $15,000 -^ (i.oi)^"
Log 15,000 = 4.17609
Log 1. 01 = .00432
Multiply by 20
Log i.oi^" = .08640
Log of present value = 4.08969
Present value = $12,293.89
Present value by compound interest table = $12,293.17
7. To find the compound discount on i.
Solution: Compute the present value as in the preceding example: deduct this present value from the amount due at maturity. Amount due at maturity $15,000.00
Present value 12,293.89
Compound discount $ 2,706.11
8. To find the amount of an annuity. What is the amount of an ordinary annuity of $25 for twenty periods at 4% per period?
Solution:
1.04'"- I
Amount of an annuity of $1 =
.04
Log 1.04 = .01703
Multiply by 20
Log 1.04^" = .34060, the log of 2.1908
1.04^" = 2.1908
Deduct 1. 0000
Compound interest = 1.1908
Amount of annuity of $ i = $ 1.1908 ^ .04 = $ 29.77 " " " " $25 = 29.77 X 25 = $744-25
" " " " " by interest tables = $744-45 Logarithms may be used more extensively in these computations, but it is desired to make the solutions as simple as possible.
9. To find the amount of sinkiui^ fund contributions. What annual contribution must be made at the end of each of twenty years to amount to $50,000 at 4K%?
LOGARITHMS AND ANNUITY COMPUTATIONS I93
Solution:
$50,000 -r- amount of annuity of $1 = contribution
Log 1.045 . = .01912
Multiply by 20
Log 1.045^" = .38240, the log of 2.41 21 1 1
Hence 1.045^° = 2.41 21 11
Deduct 1. 000000
1.412111
Compound interest
Amount of annuity of $1 = 1.412111 -^ .045 = 31.38
$50,000-=- 31.38 =$1593-37
10. To find the present worth of an annuity. What is the present value of an annuity of $50 per year for ten years at 6%?
Solution: The present value of the annuity = 50 X I>
D =.- ^
1.06
Log 1.06 = .02531 Multiply by 10
Log of 1. 06'° = •-?53io, the k-gof amount uf i .00
Determine the present value as follows:
Log I. = 0.00000 or 10.00000 — 10
Log 1.06'" = .25310
Hence log—— = g. 74690 - 10, the log of .5583375
1.06 1. 0000000 Minus .5583375
Equals .4416625, the compound discount
.4416625 -^ .06 = 7.36104, present value of annuity of $ i $7.36104 X 50 = $368,052, " " " " " $50
CHAPTER XIX BOND DISCOUNT AND PREMIUM
Bonds Purchased below and above Par
Bonds are frequently purchased at prices either below or above par; that is, at a discount or a premium. When a bond purchased at a discount is held until maturity and paid at par, the owner makes a profit amounting to the difference between the cost and par; that is, to the discount. If the bond is purchased at a premium and repaid at par, the owner incurs a loss amount- ing to the premium. Conversely, one who issues a bond below par and repays it at par loses the discount, while one who issues a bond at a premium and repays it at par gains the premium. The question arises as to when such profit shall be credited, or such loss be charged, to profit and loss. The following discussion considers the subject from the point of view of the investor; the same principles apply to writing off premium or discount on the books of the business issuing the bonds.
Discount
Let us assume that a $ioo bond, due in two years and bearing 4% interest payable semiannually, is bought for $96.28. When paid at par there will be a gain of $3.72, the amount of the dis- count. The discount may be taken up by the following methods, which are, however, unscientific:
I. By an immediatecredit to income of $3.72, thus raising the investment account on the books to the par of $100 and taking credit at once for the discount. This method is clearly wrong because it takes up at the time of purchase an income which is earned gradually as the bond approaches maturity.
194
BOND DISCOUNT AND PREMIUM
195
2. By carrying the investment at its cost of $96.28 until it is paid, at which time the difference between the cash received and the cost of the bond is credited to income. This method also is erroneous, although the error is not so apparent. In the first place, while the earning is made gradually as the bond approaches maturity, it would appear from the above treatment that none of the discount was earned until the very day of maturity; and in the second place the bond increases in value as the day of matur- ity approaches, when it is paid at par.
3. By considering the $3.72 as extra interest earned, divid- ing the amount by 4, the number of semiannual interest periods, charging }i of $3.72 each six months to the bond account and crediting it to interest. This has the effect of raising the invest- ment account gradually to par, while spreading the earning over the four periods in which the bond is held. The effect of this treatment is shown by the following schedule:
Period
Bond account
beginning of
6 months period
Portion of discou.nt
TAKEN UP during ONE PERIOD
Int. collected
Total credit to interest
I 2 3
4 Maturity
$ 96.28 97-21 98.14 99.07 100.00
% .93 .93 .93 ■93
$2.00
2.00
2.00 2.00
J 2.93 2.93 2.93 2.93
Total
I3.72
$8.00
$1 1.72
It may appear that this method answers all requirements, since it raises the investment account gradually to par, taking up the discount periodically. But it is subject to the criticism that, while the investment is gradually increasing, the interest remains constant. This means that the rate of interest is gradu- ally decreasing, as shown in the following table:
196
MATHEMATICS OF ACCOUNTING AND FINANCE
Period
Asset value in- bond ACCOUNT
Credit to Income
Semiannual rate of income
I 2 3
4
$96.28 97-21 98.14 99-07
f2.93 2.93 2.93 2.93
3.043 +%
3-014 + %
2.98s + %
2.957 + %
Since the investment or asset value is increasing as the bond approaches maturity, the credits to income should also increase, so that the same rate of income will be maintained each period. When the amounts are small, the error arising from this method is immaterial and the discount may be written off in equal amounts, as above. But when the amounts are large, this method may result in serious unfairness to some parties in interest. Assuming, for instance, that $96,280 is invested in the purchase of $100,000 of bonds for a trust and that the bene- ficiary changes at the expiration of the second six months' period, the first beneficiary would receive over 3% income semiannually on the assets of the trust, while the second beneficiary would receive less than 3%. It is not sufficient that they each receive the same number of dollars. The property, or investment, held for the second beneficiary is of greater value than that held for the first, and hence the second should receive more income per period, since each is entitled to the same rate of income on the investment of the trust.
Scientific Method of Amortization
Any method of writing off discount, to be scientific, must take the following into consideration :
1. The investment increases in value as the bond approaches maturity, and at each interest date the investment value should be increased on the books, thus raising it gradually to par.
2. The purchase of a bond at a discount results in the rate of interest earned on the investment being higher than the nominal
BOND DISCOUNT AND PREMIUM 197
rate paid on the par of the bond. Hence, the amount added to the asset value of the bond at each interest date should also be credited to interest, together with the cash collected.
3. The total credit to interest (cash collected plus portion of discount) at the end of each period should be an increasing amount but always the same per cent of the carrying or asset value of the bond at the beginning of the period.
When a bond is bought at a price other than par, there are two interest rates:
1 . The nominal or cash rate paid on the par of the bond
2. The effective, basic, or income rate actually earned on the
investment
When the bond is bought at a discount, the income rate is greater than the nominal rate for two reasons :
1. The investment is less than par, although it gradually
increases to par.
2. The income is more than the coupons collected, since the
income is composed of two parts :
(a) The coupons collected
(b) The periodical portion of the discount
When the discount is scientifically amortized or written off the method is as follows:
1 . Determine the effective rate earned on the investment, as
explained later.
2. At each interest date multiply the carrying value of the
investment by the effective rate, to determine the amount of revenue earned during the period.
3. Make the following entry:
Debit cash for coupon collected
Debit investment for portion of discount
Credit interest for total income computed in (2) above.
198 MATHEMATICS OF ACCOUNTING AND FINANCE
The amount charged to investment will be the difference between the total income and the cash collected.
To illustrate, the price of $96.28 in the example above, was chosen because it was known that a 4% bond due in two years, with semiannual interest, would yield an effective rate of income of 6% per year or 3% semiannually if purchased at this price. The following shows the scientific amortization of the discount :
Cost of bond $96.28
First period:
Income, 3% of $96.28 $2.89
Coupon, 2% of $100.00 2.00
Balance — portion of discount charged to
investment .89
Second period:
Carrying value of bond $97- 1 7
Income, 3% of $97.17 $2.92
Coupon 2.00
Balance — portion of discount charged to in- vestment -92
Third period:
Carrying value of bond $98.09
Income, 3% of $98.09 $2.94
Coupon 2.00
Balance — portion of discount charged to
investment -94
Fourth period:
Carrying value of bond $99-03
Income, 3% of $99-03 $2.97
Coupon 2.00
Balance — portion of discount charged to in- vestment -97
Par of bond $100.00
BOND DISCOUNT AND PREMIUM
199
It will be noted that this process conforms to the three require- ments of the scientific method as outlined above.
The investment is increased gradually to par by writing off a portion of the discount at each interest period, thus:
Beginning of
Period
Investment
I
$96.28
2
97.17
3
98.09
4
Q9-03
Par
100.00
The amount added to the investment value at each interest period is also credited to interest, as follows:
Credits to Interest
End of Period
For C.\sh
For Portion' of Discount
TOT.\L
I 2 3
4
$2.00 2.00 2.00 2.00
$ .89 .92 •94 • 97
$2.89 2.92 2.94 2.97
The total credit to interest increases each period and is always the same per cent of the increased investment at the beginning of each period, as shown below:
Period
Investment, beginning of period
'Income for period
R.\TE
I 3 3 4
$96.28 97.17 98.09 99-03
$2.89 2.92 2.94 2.97
•i /o 3% 3% 3%
200
MATHEMATICS OF ACCOUNTING AND FINANCE
While, as stated before, the difference is immaterial when the investment is small, the following table comparing results by the equal instalment method and the scientific amortization method will serve to show the injustice which might be caused by the improper method.
Investment Value
Income
Equal
Scientific
Differ-
Equal
Scientific
Differ-
INSTAL. PLAN
AMORTZN. PLAN
ence
INSTAL. PLAN
AMORTZN. plan
ence
I
$96.28
I96.28
i.oo
52. 93
$2.89
S+-04
2
97-21
97-17
.04
2.93
2.92
+.01
3
98.14
98.09
-05
2-93
2.94
— -01
4
99-07
99-03
-04
2.93
2.97
-.04
Assuming that the bond was held as an investment for a trust and that the beneficiary changed at the end of the second period, the column of income differences shows that, by the equal instal- ment method, the first beneficiary would receive 5 cents too much income, while the second beneficiary would receive 5 cents less than was rightfully his.
Assuming that the bond was held for an estate for the benefit of a life tenant and a remainderman, that the life tenant died at the end of the second half year, and that the estate reverting to the remainderman, the equal instalment method would work detriment to the remainderman. Let us assume that the estate originally consisted of $100. By the equal instalment method the estate reverting to the remainderman would be less than $100.
The income represented by the portion of discount written off would be paid to the life tenant in cash and would be added to the carrying value of the bond.
The following would be a statement of the assets of the estate:
BOND DISCOUNT AND PREMIUM
20I
Equal Instalment
Method
1
Scientific Amortiza- tion Method
Cash
Bond true value
Total
Cash
Bond TRUE value
Total
$100.00
96.28 $ 3.72
2.00
J96.28 97-17 98.09
$100.00 100.00
9996
99-95
$100.00
96.28
J96.28 97.17 98.09
First period:
Cash collected ...
$ 3-72 2.00
$100.00
$ 5-72 2.93
$ 5.72 2.89
$ 2.79 2.00
S 2.83 2.00
100.00
Second period: Cash collected . .
S 4-79 2.93
$ 4-83 2.92
S 1.86
S 1-91
100.00
It is seen that if the equal instalment method was followed, the assets of the estate would be impaired 5 cents by paying the life tenant in cash 5 cents erroneously computed as income. The result would be that the remainderman would be defrauded, the injustice being concealed by turning the bond back to him at a value of $98.14 when it really had a value of only $98.09.
Income Rates
Bond dealers frequently offer bonds at prices which net the investor a rate other than the cash or nominal rate. In the illustration above, where a 4% bond was purchased to yield an income rate of 6%, the quotation might be stated, "4% bond paying 6%," "4% bond yielding 6%," "4% bond to net 6%," or ' ' 4% bond on a 6% basis. ' ' To avoid the necessity of computing the price at which each sale must be made, bond tables have
202 MATHEMATICS OF ACCOUNTING AND FINANCE
This price varies
been prepared showing the price to be paid, with
1 . The nominal rate
2. The income rate
3. The number of periods mi til maturity
Since bonds usually bear semiannual interest there are two periods per year, and bond tables usually show a price based on the assumption that the interest is paid semiannually.
The following portion of a page in a bond table shows various prices to be paid for a 2-year bond, depending on the nominal and effective rates. The rates at the head of the columns are nominal rates; those at the side are the effective rates. For instance, in the 4% nominal rate column on the 6% effective rate line is found the price of $96.28 used in the preceding illustrations of a 2-year 4% bond netting 6%.
2 Years
Interest Payable Semiannually
Per Cent Per Annum
3%
3 1/2%
4%
4 1/2%
5%
6%
7%
4.80
96.61
97.55
98.49
99.43
100.38
102.26
104.IS
4 7/8
96.47
97.41
98.3s
99.29
100.24
102.12
104.00
4.90
96.42
97.36
98-31
99-25
100.19
102.07
103.95
S
96.24
97.18
98.12
99 06
100.00
101.88
103.76
5-10
96.05
96.99
97-93
98.87
99.81
101.69
103.57
S 1/8
96.01
96.95
97.89
98.83
99.77
101.64
103.52
S-20
95.87
96.81
97-75
98-69
99.62
101.50
103.38
5 1/4
95.78
96.72
97-66
98.59
99-53
101.41
103.28
5.30
95.69
96.63
97-56
98.50
99-44
101.31
103.19
5 3/8
95-55
96.49
97.43
98.36
99-30
101.17
103.04
S-40
95.51
96.44
97.38
98.32
99-25
I0I.I2
103.00
SI/2
95-33
96.26
97-20
98.13
99-07
100.93
102.80
5 5/8
95-10
96.03
96.97
97.90
98.83
100.70
102-57
53/4
94-87
95.81
96.74
97-67
98.60
100.47
102.33
5 7/8
94-65
95-58
96.51
97-44
98.37
100.23
102.09
6
94.42
95-35
96.28
97-21
98.14
100.00
101.86
BOND DISCOUNT AND PREMIUM 203
Bond Premium
When a bond is bought at a price above par, the effective rate is less than the nominal rate, for two reasons:
1 . The investment is more than the par to which the nominal
rate applies.
2. The coupons collected are not all income; since only par
will be repaid, a portion of the cash received at each interest date must be considered as a return of the premium (which is principal) and only the balance as income.
Bond premium should be scientifically written off as follows:
1 . Determine the effective rate.
2. Multiply the gradually diminishing investment by the
effective rate to compute the income earned during the period.
3. Make the following entry: Debit cash for coupon collected
Credit income for amount computed in (2) above
' ' investment for difference between cash and income.
To illustrate, the bond table shows that a 2-year $100 6% bond to net 4% should be purchased for $103.81 . The following shows the scientific amortization of the premium:
Cost of bond $103.81
First period:
Coupon, 3% of $100.00 $3.00
Income, 2% of $103.81 2.08
Balance, premium written off .92
Second period:
Carrying value of bond $102.89
Coupon $3.00
Income, 2% of $102.89 2.06
Balance, premium written off .94
204
MATHEMATICS OP ACCOUNTING AND FINANCE
Third period:
Carrying value of bond $101.95
Coupon $3.00
Income, 2% of $101.95 2.04
Balance, premium written off .96
Fourth period:
Carrying value of bond $100.99
Coupon $3.00
Income, 2% of $100.99 2.02
Balance, premium written off .98
Par of bond $100.01
The error of i cent arises from repeated approximations of interest to the nearest cent.
The following table shows the periodical entries :
Period
Debit Cash
Credit Interest
Credit Investment (or Premium)
Investment
Cost
$ 3.00 300 300 3- 00
$2.08 2.06 2.04 2.01*
1 .92
.94 .96 .99*
$103.81
I 2 3
4
102.89 101.95 101.99 100.00
Total
$12.00
18. 19
$3.81
' Adjustment of i cent to correct discrepancy.
The second column shows the total cash collected; the third, the amount of cash taken as income; the fourth, the portion of the coupon applied to repayment of the premium ($3.81) which, with the final repayment of the bond, completely realizes the invest- ment. Since a portion of the investment is realized periodically, the investment gradually diminishes in value. Hence the credits to interest decrease with each successive period.
BOND DISCOUNT AND PREMIUM 205
Computing the Premium and the Price
While the price above par to be paid for a bond to yield an effective rate less than the nominal rate may be found in a bond table, such tables are not always available. If a table of com- pound interest, present values or annuities is at hand, it will be of assistance, but the price may be computed without tables of any kind. Two methods are explained, as follows:
First Method
An interest bearing bond comprises two promises, as follows:
1. To pay the par at maturity — in the last preceding
illustration, $ioo at the expiration of four six-months periods.
2. To pay a stipulated amount of interest periodically — in
the illustration, $3 at the end of each of four six-months periods.
The price to be paid for the bond is the present value, discounted at the effective rate, of all cash payments promised. In the illustration, the price is the sum of
1. The present value of $100 due four periods hence, dis-
counted at 2% per period.
2. The present value of an annuity of $3 for four periods,
discounted at 2% per period.
These present values may be found in the interest table; thus the table shows that:
Present value of $1 due four periods hence at 2% = $.923845
$.923845 X 100 = $92.3845, P. V. of principal Present value of an annuity of $1 for four periods at 2% = $3.807729
$3.807729 X 3 = 11.4232, P. V. of coupons
Total $103.8077
or $103.81, price
206 MATHEMATICS OF ACCOUNTING AND FINANCE
If the table does not show present values but does show com- pound interest, the computation may be made as follows:
The table shows the amount of $i at compound interest for four periods at 2% to be $1.082432.
Then, $1.00 -J- 1.082432= $ .923845, P. V.of$i
$.923845 X 100 = $ 92.3845, P. V. of principal
Also, $1.00 — .923845 = $ .076155, compound discount, four periods $.076155 -^ .02 = $ 3.8077,
P. V. of annuity of $1 for four periods $3.8077 X3 = 1 1. 423 1, P. V. of coupons
Total $103.8076, price
When no interest table of any kind is available, recourse may be had to the methods explained in the chapter on compound interest and annuities. Perhaps the easiest method would be to compute the amount of $1 for four periods at 2% thus:
1.02 1.02
1.0404 1.0404
1.082432 The procedure following would be as shown above.
Second Method
This method determines only the premium to be paid to reduce the income from the nominal rate to the effective rate. The premium so determined, added to the par of the bond, com- prises the total price. The method is based on the following reasoning :
Let us assume that a 4% annual, or 2% semiannual, income is required. If the bond bore 4% interest, the price would be par. Hence the payment of $100, or par, entitles the holder of the bond to receive:
BOND DISCOUNT AND PREMIUM 20/
1. Principal, at maturity
2. Interest, $2 at the end of each six months
But if, as in the preceding illustration, the bond pays 3% semi- annually, each semiannual coupon collected will be $3. Of this, the payment of par entitles the holder to receive $2. And the payment of a premium entitles him to receive the remain- ing $1. Hence, the premium is the sum which must be paid to entitle the holder to collect that portion of the periodical coupon which is in excess of the product obtained by multiplying the par of the bond by the effective periodical rate.
This excess interest is an annuity. In the case of a 2-year 6% bond bought to net 4%, the annuity is $1 for four periods. The premium is the present value of this annuity, discounted at the effective rate, and may be computed as follows:
$100 X 3% (semiannual cash rate) = $3.00 coupon
$100 X 2% ( " effective rate) = 2.00 effective income on par
Excess $1.00
The present value at 2% of an annuity of $1 for four periods is shown by an interest table to be $3.807729, to which is added the par, $100, the total being the price of the bond or $103.81.
When a table showing the present value of an annuity is not at hand, one of the methods already explained may be used to find the present value. Perhaps the easiest method is by successive divisions, as follows:
Example
What is the premium to be paid on a 3-year 7% bond, interest payable semiannually, bought to net 5%; par $100,000?
Solution:
$100,000 X 3K% (cash rate per period) =$3,500
$100,000 X 2>^% (effective rate per period) = 2,500
Excess $1,000
208
MATHEMATICS OF ACCOUNTING AND FINANCE
The required premium, therefore, is the present value at 2^% of an annuity of $1,000 for six periods.
975.610 P. V. of $1000 due I period hence
Now,
$1,000.00 -
- 1.025 =
$
975.610
975.610-
- 1.025 =
951.814
951.814 -
- 1.025 =
928.599
928.599 -
- 1-025 =
905-950
905-950 -
- 1.025 =
883.854
883.854 -
- 1-025 =
862.297
$1000 $1000 $1000 $1000 $1000
2 periods
3 "
4 " 5
6
$ 5,508.124, premium 100,000.000, par
$105,508.12, price
The premium could also be computed as follows: Find amount of $1 for six periods at 2^%, as follows:
1.025 1.025
1.050625 amount of $1 for 2 periods 1.050625
1.103813 " " I " 4 " 1.050625
1.159693 " " I " 6 "
Find compound discount on $1 due six periods hence at 2^ % thus:
$1 -^ 1. 159693 = $.862297 P. V. of $1 due six periods hence $1 — $ .862297 = .137703 compound discount or $.159693 (comp. int.) -^ 1. 159693 (amt.) = $. 137703 compound discount
Find present value of annuity of $1 ,000 for six periods at 2% %, thus:
% -137703 -^ -025 = $5.50812 P. V. of annuity of $1 $5.50812 X 1000 = $5,508.12 P. V. of annuity of $1,000
BOND DISCOUNT AND PREMIUM
209
The following table shows the periodical entries and the diminishing balance of the investment.
Period
Debit Cash
Credit Interest 2j^% OF Investment
Credit
Investment
Investment
Cost
$ 3,500 3.500 3.500 3.500 3,500 3,500
$ 2,637.70 2,616.15 2,59405 2,571-40 2,548-19 2,524-38
$ 862.30 883.85 905.95 928.60 951-81 975-62
•?I05,508.I3 104.645.83 103,761.98 102,856.03 101.927.43 100,975-62 100,000.00
6
Total
|2I,000
115,491.87
55,508.13
Computing the Discount and the Price
When bond tables cannot be consulted to determine the price to pay for a bond to net an income rate higher than the nominal rate, the price may be computed by methods similar to those described for determining a premium.
First Method. The first method consists of the following three steps:
(a) Compute the present value of the par, discounted at
the effective rate.
(b) Compute the present value at the effective rate of all
coupons to be collected.
(c) Add the foregoing two items, the sum being the price to
be paid.
In the case of the $100 2 -year 4% bond bought to net 6%, at a price of $96.28, the first item (a) is the present value of $100 due four periods hence at 3% (the effective rate) per period. The second item (b) is the present value of an annuity of $2 for four periods discounted at 3%.
These present values may be found in a book of tables, thus :
210 MATHEMATICS OP ACCOUNTING AND FINANCE
Present value of par:
P. V. of $1 due in four periods at 3% is $ .888487
$.888487 X 100 = $88.8487
Present value of coupons:
P. V. of annuity of $1 for four periods at 3% is $3.717098 $3.717098X2= 7-4342
Total price $96.2829
or as shown by the bond table $96.28
If an interest table is not available, the methods already described for determining present values may be used.
2. Second Method. This method determines the discount to be deducted from par. Since the cash rate does not produce the required income, the seller permits the deduction of an amount which, invested at the effective rate, will produce the extra periodical income required. The discount is the present value of an annuity of the extra income.
For instance, in the case of the $100 2-year 4% bond bought to net 6%, at a price of $96.28, the 3% effective rate is equivalent to $3 per period, while the 2% coupon produces only $2 per period, the required excess being $1 per period for four periods. The present value of 3% (the effective rate) of an annuity of $1 for four periods is shown by an interest table to be $3.717098.
Then $100.00 par Less 3.72 discount
$96.28 price
When neither a bond table nor an interest table can be used, one must resort to the previously described method of comput- ing the present value of the annuity.
Illustration
What are the discount and the purchase price of a 3-year 5% bond, interest payable semiannually, bought to net 6%; par $100,000?
BOND DISCOUNT AND PREMIUM
211
Solution:
$100,000 X 3% (effective rate per period) $100,000 X 23^% (cash " " " )
)3,ooo 2,500
5 500
Deficient interest per period
The required discount, therefore, is the present value at 3% of an annuity of $500 per period.
-- $485,437 P. V. of $500 due I period
Now, $500
- I.
Hence, 485-437 "
- i.(
47I.2Q8-
- i.<
457-571-
- I.
444.244-
- I.
431.305 -
- I.
Discount
03 = 471
03 = 457
03 = 444
03 = 431
03 = 418
298 571 244
305 742
500 500 500 500 500
2 periods
3
4 "
5 6
$2,708,597
Then par Less discount
Price
$100,000.00 2,708.60
$ 97,291.40
The discount can also be computed as follows:
Find the amount of $1 for six periods at 3%, thus:
1.03 1.03
1.0609 amount of $1 for 2 periods 1.0609
1. 125509 " •• I " 4 1.0609
1. 194052 " " I " 6
Find the compound discount on$i due six periods hence at 3%, thus: $1.00 -^ 1. 194052 = $.837484 P. V. of $1 due six periods hence $1.00— .837484= $.162516 compound discount
Find the present value of an annuity of $500 for six periods at 3%
thus:
$.162516 (comp. dis.) -J- .03 = $5.4172 P. V. of annuity of $1
$5.4172 X 500 = $2708.60 present value of annuity of $500, or discount
212
MATHEMATICS OF ACCOUNTING AND FINANCE
The following table, or schedule of amortization, shows the periodical entries and the remaining balance of the investment.
Period
Credit Income 3% OF Investment
Debit Cash
Debit Investment
Investment
I
2
3 4 5 6
$ 2,918.74 2,931-31 2,944-24 2,957-57 2,971-30 2,985-44
1 2,500 2,500 2,500 2,500 2.500 2,500
$ 418.74 431-31 444-24
457-57 471-30 485-44
$ 97.291-40 97.710.14 98,141.45 98,585.69 99.043.26 99.514-56 100,000.00
$17,708.60
$15,000.00
$2,708.60
Purchases at Intermediate Date
In the preceding explanations of methods for computing prices for bonds either at a premium or at a discount, it has been assumed that the purchase occurred on an interest date. When this is not the case, the customary method of determining the price is as follows :
1. Compute the price as if the purchase had been made at the next preceding interest date; also the price as if the purchase were to be made at the next succeeding interest date. The difference is the portion of premium or discount to be amortized during the period.
2. Such a proportion of this premium or discount is amortized as the elapsed time between the preceding interest date and the date of purchase bears to the total interest period.
3. To the amortized value thus obtained add the accrued interest at the nominal rate.
For instance, referring to the table on page 209 let us assume that the interest dates are January i and July i ; and that $105,508.13 is the value on Jan. i, 1918, on a 5% basis
104,645-83 " " " " July i> 1918, " " " "
$ 862.30 " " premium to be amortized during the period
BOND DISCOUNT AND PREMIUM 213
If the purchase is made on February i, 19 18, one-sixth of the interest period has elapsed, hence one-sixth of $862.30 should be amortized, thus:
Value at January i, 1918 $105,508.13
Deduct Ye of $862.30 i43-72
$105,364.41 Add accrued interest: ^/g of $3500 583.33
Price $105,947.74
The price may also be computed by adding to the price on January i the accrued interest at the effective rate, thus:
Price on January i, 1918 $105,508.13
Effective interest for six months:
2H% of $105,508.13 = $2,637.70
Ve of $2,637.70 439-6i
Price on February i, 1918 $105,947.74
Of this amount $105,364.41 is charged to investment and $583.33 to accrued interest. When the coupon is collected on July i, it is applied as follows:
Cash $3500.00
Accrued interest $ 583.33
Interest earned 2,198.09
Investment amortization ($862.30 — $143.72). .. 718.58
The investment of $105,364.41, reduced by the amortization of $718.58, is novi^ carried at the true value (seepage 209) on July i, $104,645.83.
While this is the customary method of computing a flat price, it is unfair to the buyer, since he advances to the seller $583.33 accrued interest five months before it is due.
When the bond is to be purchased at a discount, the propor- tion of discount to be amortized for the fractional period should be added. Referring to page 212, let us assume that interest is payable on January i and July i, that $97,291.40 is the value at
214 MATHEMATICS OF ACCOUNTING AND FINANCE
January i, 1918, on a 6% basis, and that the transfer is to be made on May i, 1918.
Then, $97,710.14 is the value on July i, igi8 97,291.40 " " " " January i, 1918
$ 418.74 " " discount to be amortized during the period % of $418.74 = $ 279.16 the discount to be amortized during two-thirds of the period
$97,291.40 value at Jan. i, 1918
279.16 two-thirds of discount amortized in six months 1,666.67 accrued interest — two-thirds of $2,500
$99,237.23 price on May i, 1918
Or, value at January i, 1918 $97,291.40
Add two-thirds of 3% (effective rate) on $97,291.40 1,945.83
$99,237.23
Of this amount $97,570.56 is charged to investment and $1,666.67 to accrued interest. On July i, the investment is adjusted to its true value (see table on page 212) by the following entries :
Cash $2500.00
Investment (discount $418.74 — $279.16) 139-58
Accrued interest $1666.67
Interest 972.91
All entries after July i, 1918, are as indicated in the schedule on page 212.
Serial Bonds
Instead of providing a sinking fund for the eventual redemption of a bond issue, the bonds may be retired gradually by serial redemption. In computing the price at which the entire issue may be purchased to net an effective rate other than the cash rate, the bonds maturing at each redemption date must be considered separately, the several values so obtained being added to fmd the total price.
BOND DISCOUNT AND PREMIUM 215
As a simple illustration assume that five bonds of $100 each, bearing 6% interest payable semiannually, are to be retired in amounts of $100 at the end of each of five years. Required — the price to net 5%. The following values are taken from a bond table, though they could be computed by the methods already explained.
Maturities Value
Bond due in i year $100.96
" " " 2 years 101.88
" "3 " 102.75
4 103.59
5 " 104-38
(( u
Total value of issue $513.56
The following schedule shows the reduction of the premium and the serial redemption of the bonds:
Cost $513-56
First period:
Coupons 3% of $500 $15.00
Income 2 1/2% of $513.56 12.84 2.16
Second period:
Carrying value $511.40
Coupons 3% of $500 $15.00
Income 2 1/2% of $511. 40 12.78 2.22
.i« First redemption 100.00
Third period:
Carrying value $409.18
Coupons 3% of $400 $1 2.00
Income 2 1/2% of $409.18 10.23 1.77
Fourth period:
Carrying value $407.41
Coupons 3% of $400 $12.00
Income 2 1/2% of $407.41 10.18 1.82
$405.59
216 MATHEMATICS OF ACCOUNTING AND FINANCE
Second redemption loo.oo
Fifth period:
Carrying value $305.59
Coupons 3% of $300 $ Q.oo
Income 2 1/2% of $305.59 7.64 1.36
Sixth period:
Carrying value $304. 23
Coupons 3% of $300 $ 9.00
Income 2 1/2% of $304.23 7.61 1.39
$302.84 Third redemption 100.00
Seventh period:
Carrying value $202.84
Coupons 3% of $200 $ 6.00
Income 2 1/2% of $202.84 5.07 .93
Eighth period:
Carrying value $201.91
Coupons 3% of $200 $ 6.00
Income 2 1/2% of $201.91 5.05 .95
$200.96 Fourth redemption 100.00
Ninth period:
Carrying value $100.96
Coupons 3% of $100 $ 3.00
Income 2 1/2% of $100.96 2.53 ^ .47
Tenth period:
Carrying value $100.49
Coupons 3% of $100 $ 3.00
Income 2 1/2% of $100.49 2.51 .49
$100.00 Fifth redemption 100.00
o
CHAPTER XX
LEASEHOLDS
Commuted Rents
The problem of determining the present value of an annuity arises when real estate is leased and an advance payment is made covering, or applying on, a series of rents which would otherwise be paid at regular intervals in the future.
Let us assume that it is proposed to lease certain property on January i, 1918, for five years at an annual rental of $1,000, payable on January i of each year. This contract would require the following payments :
Date Rent
January i I I I I
1 91 8 $1,000
1919 1,000
1920 1,000
1 92 1 1,000
1922 1,000
The lessee desires to make one payment on January i, 1918, covering the entire rental; and it is agreed between the parties to discount at 5% the payments which would otherwise be made in 1919, 1920, 192 1 and 1922. The single payment to be made (i.e., the present value of the annuity) may be computed thus:
Date
Due
Rent
Symbols
Present Value
January r, 1918
I. 1919
" I, 1920
1, 1921
I, 1922
Now
1 year hence
2 years
3 "
4 "
li.ooo 1,000 1,000 1,000 1,000
J -
- (i + <•)
- (r + <•)'
- ( r + i)i
- (i + i)^
Si. 000. 000 952.381 907.029 S63.838 822.702
Single payment
Jj. 545.950
217
2l8
MATHEMATICS OF ACCOUNTING AND FINANCE
The single payment can also be computed thus :
Present value of first payment
Present value of future payments:
Present value of an annuity of $i,ooo for four periods at 5% Present value of $1 due in four years at
5% $.822702
Compound discount $.177298
Present value annuity of $1
= $.177298 ^ .05 $3-54596
$3.54596 X 1,000
Single payment
3,545-96 54,545-96
Although the single payment of $4,545.96 pays the rent for five years, the annual entries in the accounts must show the following :
1. Annual rental of $1,000
2. Annual interest earning on the advance payment
3 . Application of advance payment to annual rent
4. Reduction of value of advance payment
The following schedule shows the figures used in the annual journal entries.
Date
Debit Rent
Credit Interest
Credit Leasehold
Debit balance of Leasehold Account
li.ooo 1,000 1,000 1,000 1,000
I177.30 136.16 92.97 47.62
$1,000.00
822.70 863.84 907.03 952.38
$4. 545. 95
3,545.95
2,723.2s
1,859.41
952.38
I, 1920
Totals
S.5,000
1454-05
$4,545.95
It is manifest that the value of a leasehold depends largely on the rate of interest used. If the above lease had been com-
LEASEHOLDS
219
muted on a basis of 4% its value would have been $4,629.90, or $83.95 more than it is worth at 5%. On a very short lease the difiference is not very great, but on a 99-year lease it amounts to a very large sum.
Sublease
A similar problem arises when property is subleased. Let us assume that A leases certain property on January i, 191 2, for a period of ten years at an annual rental of $3,000, and that he occupies the property until December 31, 191 7, at which time he assigns the lease to B. Due to a rise in land values, B is willing to assume the lease at an annual rental of $5,000. B may, therefore, make the following payments:
Date
Payment to Owner of Fee
Payment to A
$3,000 3.000 3.000 3.000 3.000
$2,000 2,000
I. 1919
Or B may, with A's consent, pay A a lump sum on January i, 1918, instead of annual payments. If the future payments of this annuity of $2,000 are discounted at 5%, the lump sum is com- puted as follows :
Present value of first payment
Present value of an annuity of $2,000 for four
periods at 5%: Present value of an annuity of $1,000 (as
computed above) $3,545.95 Then $3,545-95 X 2 =
Single payment
i2,000.00
7,091.90
,091.90
220 MATHEMATICS OF ACCOUNTING AND FINANCE
The following schedule shows the journal and cash entries to be made annually by B.
Date
Debit
Credit
Credit
Credit Cash
Balance of
Rent
Interest
Leasehold
(paid owner)
Leasehold Account
Cost of leasehold . . .
$9,091-90
January i, 1918 ....
$ 5.000
$2,000.00
$ 3,000.00
7,091.90
I. 1919
5.000
1354-60
1,645.40
3,000.00
5.446.50
I, 1920 ....
5.000
272.33
1.727.67
3,000.00
3.718.83
I, 1921 ....
5.000
185.94
1,814.06
3,000.00
1,904.77
I, 1922 ....
5.000
95-24
1,904.76
3,000.00
.01
Total
$25,000
J908.11
$9,091-89
$15,000.00
CHAPTER XXI
DEPRECIATION METHODS
Annual Depreciation
There are six methods in more or less general use for the de- termination of the amount of depreciation which business en- terprises should provide annually on their fixed assets. Writing of^ an arbitrary amount at any convenient time is not included, because it is too haphazard and unscientific to be called a method. The six recognized methods are as follows:
1. Straight line method
2. Diminishing value, using a fixed per cent
3. Diminishing value method based on the sum of the years'
digits
4. Annuity method
5. Sinking fund method
6. Production method
It is not within the province of this book to examine the merits of the different methods. The object is only to indicate how the amount to be written off annually is ascertained when the managers of the business have decided upon the method to be used.
The illustration to be used throughout is that of an asset costing $5 ,000 which is estimated to have a life of six years and a residual or scrap value of $200, if any.
I . The straight line method has the advantage of simplicity, as the annual amount is determined by the simple process of dividing the total depreciation by the number of years the asset is expected to remain effective, as shown below.
222
MATHEMATICS OF ACCOUNTING AND FINANCE
Year
Annual Depreciation
Carrying Value
I
$ 8oo 800 8oo
800
800
8oo
Is, 000 4,200 3,400
2
4
s
6
1,000 200
Total
S4,8oo
2. The diminishing value method, when a fixed per cent is applied, is a desirable one to use, but it has the disadvantage that the rate on the diminishing value is difficult to compute. The process requires the extraction of a root the index of which is the same as the number of years covered by the depreciation.
The formula for this method is
In this formula r represents the desired rate; w the number of periods ; 5 the scrap value ; and c the cost. The formula is applied to the illustration as follows :
6' 200
\ 5,ooo
6/
= I — V -04
Log .04 is 8.60206 — 10
Add 50. ~ 50
58.60206 — 60
Divide by 6 9.76701 — 10 log of .5848
r= I- .5848 = .4152 = 41.52%
The following table shows the depreciation charges and the diminishing value in the example taken.
DEPRECIATION METHODS
223
Year
Depreciation
Diminishing value
I 2 3 4 5 6
41.52% of Is, 000. 00
" 2,924.00
" 1,709.96
99998
584-79
341-99
12,076.00 1,214.04 709-98 415-19 242-80 141.99
$5,000.00
2,924.00
1,709.96
999-98
584-79
341-99
200-00
Total
$4,800.00
The fixed per cent of diminishing value is very frequently used, but because of the difficulty or ignorance of the method of computing it, the rate is guessed at, or a purely arbitrary one is used, which, however, is usually too small. Thus, it might seem to many factory managers that 25% would be too large a rate, in the preceding illustration but it really is not, as the fol- lowing shows :
Year
25% OF Diminishing value
Diminishing value
I
2
3
4 ,
S
6
$1,250-00
937-50 703-13 527-34 395-51 296.63
$5,000.00
3.750.00 2,812.50 2,109.37 1,582.03 1,186.52 889.89
Total
$4,110.1 1
3. The diminishing value method based on the sum of the years' digits is one which is often used to avoid the difficulties of the percentage method described above. To find the sum of the years' digits the figures representing the successive years are added together to form a denominator, in our illustration, 21
224 MATHEMATICS OF ACCOUNTING AND FINANCE
(1 + 2+3+4+5 + 6). Each year a numerator is used represent- ing the successive number of years the asset is expected to Hve, and the fraction thus obtained is apphed to the estimated total depreciation to determine the amount to be charged off that year. The result of applying this method in our illustration is as follows:
Year
Fraction
Annual Charge
Carrying Value
$5. 000. 00
I
6/21
$1,371.43
3.628.57
2
S/21
1. 142. 86
2,485.71
3
4/21
914.29
1,571.42
4
3/21
685.71
885.71
5
2/21
457-14
428.57
6
1/21
228.57
200.00
Total . . .
21/21
$4,800.00
As is readily seen, this method spreads the depreciation with less difference between the early and the late years than does the second method.
4. The annuity method, to quote Hatfield
. . . rests upon the assumption that the cost of production includes not only repairs and the depreciation of machinery, but as well interest on the amount of capital invested in the machine. Depreciation, on this theory, should be a sum figured as a constant annual charge, sufficient not only to write off the decline in value but also to write off annual interest charges on its diminishing value. '
In other words, this method treats the cost of machinery as an investment earning interest. Hence the cost of machinery is dealt with as the present value of an annuity; the depreciation to be written off periodically is an equal amount, and the credit to interest decreases periodically because of the diminishing value of the asset.
' H. R. Hatfield, Modern Accounting, 191 1. P- 131.
DEPRECIATION METHODS
225
If there were no scrap value, the computation of the annual depreciation charge would be as follows: The cost of the machine is the present value of an annuity of unknown rents, or depreci- ation charges; this cost is divided by the present value of an annuity with rents of $1. If it is assumed that the asset in our illustration will have no scrap value and that the annuity is to be based on an interest rate of 6%, the computation is as follows:
Present value of $1 due six periods hence. ... $ .70406054
Compound discount ($1 — $.70496054) .29503946
Present value of annuity of $1 ($.29503946
■^ .06) 4.917324
Annual amount required ($5,000 -J- 4.917324). 1,016.81
The following is a table of depreciation for our example, based on the annuity method, assuming the asset has no scrap value.
Year
Debit Depreciation
Credit Interest
Credit
Depreciation
Reserve
Balance-carrying value Basis of interest
I 2 3 4 5 6
I1.016.81 i,oi6.8r 1. 016. 81 1. 016. 81 r.oi6.8i 1,016.81
$ 300.00 256.99 21 r.40 163.08 111.8s 57-56
$ 716.81 759.82 80s. 41 853-73 904.96 959-25
Is. 000.00 4.283.19 3,52337 2,717.96 1,864.23 959.27 .02
Total
16,100.86
Si, 100. 88
14,999-98
When the asset has a scrap value, the conditions are more complicated, because the cost of the asset consists of two elements :
(a) The present worth of an annuity of the depreciation
charges, and
(b) The present worth of the scrap value.
226 MATHEMATICS OF ACCOUNTING AND FINANCE
Referring to our illustration, if the asset has a scrap value of $200, the $5,000 invested in it is the sum of the following two items :
(a) The present value at 6% of $200 remaining after six
years
(b) The present value on the basis of 6% of an annuity of
six rents of unknown amounts
The rents, or depreciation charges, are computed thus:
Cost of asset $5,000.00
Deduct present value of $200 due in six years
(.70496054 X 200) 140.99
Present value of six depreciation charges $4,859.01
In the preceding illustration the present value of an annuity of $1 for six periods at 6% was found to be $4.917324.
Then $4,859.01 -r- 4.917324 = $988.14 annual depreciation
The following is the table of depreciation based on the an- nuity method, when the asset has a scrap value of $200.
Debit
Credit
Credit
Year
Depreciation
Interest
Depreciation Reserve
Balance
Is, 000. 00
I
1 988.14
J 300.00
$ 688.14
4. 311. 86
2
988.14
258.71
729-43
3,582.43
3
988.14
214.9s
773.19
2,809.24
4
988.14
168.55
819.59
1,989.6s
S
988.14
119.38
868.76
1,120.89
6
988.14
67.25
920.89
200.00
Total
JS.928.84
Ji, 128.84
14,800.00
It is not pertinent here to enter into a discussion of the propriety of including interest on fixed assets among the manu- facturing expenses.
DEPRECIATION METHODS 227
5. The sinking fund method is based on the assumption that a fund is set aside to accumulate at compound interest with which to acquire a new asset when the old one is discarded. It assumes that the funds for the purchase of the asset will be pro- vided from two sources :
(a) The scrap value of the old asset
(b) The sinking fund
Since the fund accumulates at compound interest, it is a sinking fund in the mathematical sense, but not in the accounting sense, which limits the term " sinking fund " to a fund accumu- lated to pay a definite hability. It would be preferable to call this fund a replacement fund accumulated on the sinking fund principle.
At the end of the life of the asset the fund should equal the amount of the depreciation. The annual contributions to the fund will be the rents of an annuity which will produce this total depreciation fund.
In our illustration the total depreciation is $4,800, or cost minus scrap value (c—s). Hence the annual contribution to the fund {SFC) will be $4,800 divided by the amount of an annuity of $1 for the given time and at the given rate. This is found by dividing the compound interest on $1 for the given time by the given rate of interest (I-^i). The formula is:
SFC = {c- s)^r t
or SFC = (c- s)X-
Assuming that a fund is to be accumulated on a 4% basis, the compound interest (/) on $1 at 4% for six years being $0.265319, the formula is applied thus:
SFC = ($5,000 - $200) X '^"^
.265319
22« MATHEMATICS OF ACCOUNTING AND FINANCE
= $4,800 X = $192
.04
•265319
.265319 ^723-66
If a fund is established, the entries therefor will be a debit to fund and a credit to cash each year for $723.66. When the interest is collected, the cash goes into the fund by an entry debiting fund and crediting interest. In addition there must be entries for depreciation, debiting depreciation and crediting reserve for depreciation with an amount equal to the sum of the cash contributed and the interest earned each year.
The following table shows the operation of the sinking fund method :
End of
Credit
Credit
Debit
Total Fund
Carrying
Year
Cash
Interest
Fund
and Reserve
Value
$5,000.00
I
1 723-66
$ 723-66
J 723-66
4.276.34
3
723-66
1 28.9s
752-61
1,476.27
3.523.73
3
723-66
59-05
782.71
2,258.98
2,741.02
4
72366
90.36
814.02
3.073-00
1,927.00
5
723.66
122.92
846.58
3.919-58
1,080.42
6
72366
156.78
880.44
4,800.02
199-98
Total ...
$4,341-96
I458.06
I4.800.02
The annual charges to depreciation are the amounts in the column headed "Debit Fund." Thus the reserve for depreci- ation is always equal to the fund. The charge to operations on account of depreciation increases annually, but this increase is offset by the credit to interest, making the net expense the same each year.
6. Concerning the production method ^lontgomery says:
A method of making depreciation allowances which has its advantages under certain conditions is that of charging an established rate per unit of
DEPRECIATION METHODS
229
output. This is especially applicable in the case, say, of a blast furnace where the frequency with which the linings will need to be renewed de- pends on the extent to which the furnace is being used. If it is being run at full capacity night and day, the wear on the linings is obviously much greater than if the furnace had not been in continual use during the entire fiscal period. ^
Mr. Montgomery applies this method with perfect justice to the lessening in value of a wasting asset, such as timber, or the coal or ore in a mine, but this lessening in value is not caused by depreciation, but by an actual consumption, or removal and conversion of the asset. In fact, this is the only possible method to be applied to those assets which diminish in exact ratio to the amount used.
There can be no rules formulated for the determination of the amount to be written ofif against each unit of production. That is a matter that must be left to the judgment of the factory managers, guided by experience.
The following is a comparative table of depreciation charges, made according to the first five methods :
Per Cent of
Sum of
Straight
Diminishing
Years'
Sinking
Year
Line
Value
Digits
Annuity
Fund
I
$ 800
$2,076.00
$1,371-43
1 988.14
$ 723.66
2
800
1,214.04
1,142.86
988
14
752.61
3
800
709-98
914.29
988
14
782.71
4
800
415-19
685.71
988
14
814.02
S
800
242.80
4S7.I4
988
14
846.58
6
800
141.99
228. S7
988
14
880.44
Totals
I4.800
$4,800.00
$4,800.00
$5,928.84
$4,800.02
Credit to interest
$1,128.84
Net
' R. H. Montgomery, Auditing Theory and Practice, 1919, p. 550.
230
MATHEMATICS OF ACCOUNTING AND FINANCE
The comparative carrying valuts in the same example under the five methods are as follows:
End of Year
Straight Line
Per Cent of
Diminishing
Value
Sum of Years' Digits
Annuity
Sinking Fund
$S,ooo
I5.000.00
$5,000.00
$5,000.00
$5,000.00
I
4,200
2,924.00
3.628.57
4.3 1 1 86
4.276.34
2
3.400
1,709.96
2,485-71
3.582.43
3.523-73
3
2,600
999.98
1.571-42
2,809.24
2.74102
4
1,800
584-79
885-71
1,989.65
■ 1,927-00
S
1. 000
341-99
428.57
1,120.89
1,080.42
6
200
200.00
200.00
200.00
199.98
APPENDIX A
VALUES OF FOREIGN COINS
Following is a list of foreign monetary units and their values, representing the pars of exchange, as estimated by the United States Director of the Mint.
Value in
COUNTRY
Legal Standard
Monetary Unit
Terms of
U. S.
Money
Argentine Republic
Gold
Peso
S0.9648
Gold
.1930 .3893
Bolivia
Gold
Boliviano
Brazil
Gold
Milreis
• 5462
British Colonies in Austral-
asia and Africa
Gold
Pound sterling
4.866s
Gold
Dollar
Central American States:
Costa Rica
Gold
Colon
.4653
British Honduras
Gold
Dollar
1. 0000
Gold
1. 0000
Guatemala \
Honduras /
Silver
Peso
.4403
Gold Gold
Peso Peso
.5000
Chile
■3650
Amoy
.7219
Canton
■ 7197
Cheefoo
.6904
Chin Kiang
.7052
Fuchau
.6678
Haikwan
• 7345
(customs)
Hankow
■6754
Tael ■
Kiaochow
Nankin
Niuchwang
Ningpo
Peking
.6995 .7143 .6770 .6940 ■ 7037
China
Silver J
Shanghai Swatow Takau ^Tientsin Yuan
■ 6594 .6668 .7264 .6995 •4730
Dollar
Hongkong
British
Mexican
.4748 .4748 .4783
231
232
VALUES OF FOREIGN COINS
Country
Legal Standrd
Colombia
Cuba
Denmark
Ecuador
Egypt
Finland
France
Germany
Great Britain
Greece
Haiti
India [British]
Indo-China
Italy
Japan
Liberia
Mexico
Netherlands
Newfoundland
Norway
Panama
Paraguay
Persia
Peru
Philippine Islands . .
Portugal
Roumania
Russia
Santo Domingo . . . .
Serbia
Siam
Spain
Straits Settlements .
Sweden
Switzerland
Turkey
Uruguay
Venezuela
Gold
Gold Gold Gold Gold
Gold
Gold and silver
Gold
Gold
Gold and silver
Gold
Gold
Silver
Gold
Gold
Gold
Gold Gold Gold Gold Gold Gold
(Gold \ Silver
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold and silver
Gold Gold Gold
Gold
Gold Gold
Monetary Unit
(Dollar) Peso
Dollar
Krone
Sucre
Pound (lOO piasters)
Finmark
Franc
Mark
Pound sterling
Drachma
Gourde
Rupee
Piaster
Lira
Yen
Dollar
Peso
Guilder (Florin)
Dollar
Krone
Dollar
Peso (Argentine)
Ashrafi
Kran
Libra
Peso
Escudo
Leu
Ruble
Dollar
Dinar
Tical
Peseta
Dollar Krona Franc
Turkish Pound
Peso Bolivar
Value in
Terms of
U. S.
Money
1. 0000
.12680
.4867
4-94 J I
• igjo .1930 .2382
4.866s
• 1930 .2500 •3244
• 4755 .1930 .4985
1. 0000
■ 4985 .4020
1. 0000 .2680
1. 0000 .9648
■ 0959 .0811
4.866s .5000
1.080S .1930 .5146
r.oooo .1930
• 3709 .1930
.5678 .2680 .1930
.0440
10342 .1930
APPENDIX B LOGARITHMS OF NUMBERS^
No.
0
I
2
3
4
5
6
7
8
9
lOO
00 000
00 043
00 087
00 130
00 173
00 217
00 260
00 303
00 346
00 389
lOI
00 432
00475
00 518
00 561
00 604
00 647
00 689
00 732
00 775
00 817
102
00 860
00903
00 945
00 988
01 030
01 072
01 IIS
01 IS7
01 199
01 242
103
01 284
01 326
01 368
01 410
01 452
or 494
01 536
01 578
01 620
01 662
104
01 703
01 745
01 787
01 828
01 870
01 912
01 953
01 995
02 036
02 078
105
02 119
02 160
02 202
02 243
02 284
02 32s
02 366
02 407
02 449
02 490
106
02 S3 I
02 572
02 612
02 653
02 694
02 735
02 776
02 816
02 857
02 898
107
02 938
02 979
03 019
03 060
03 100
03 141
03 181
03 222
03 262
03 302
108
03 342
03 383
03 423
03 463
03 5 03
03 543
03 583
03 623
03 663
03 703
109
03 743
03 782
03 822
03 862
03 902
03 941
03 981
04 021
04 060
04 100
IIO
04 139
04 179
04 218
04 258
04 297
04 336
04 376
04 4IS
04 454
04 493
III
04 S32
04 S7I
04 610
04 650
04 689
04 727
04 766
04 805
04 844
04 883
112
04 922
04 961
04 999
05 038
OS 077
OS 115
OS 154
05 192
05 231
OS 269
113
OS 308
OS 346
05 385
05 423
OS 461
OS 500
OS 538
OS 576
OS 614
OS 652
114
05 690
05 729
OS 767
05 80s
OS 843
OS 881
05 918
OS 956
OS 994
06 032
IIS
06 070
06 108
06 14s
06 183
06 221
06 258
06 296
06 333
06 371
06 408
116
06 446
06 483
06 521
06 SS8
06 595
06 633
06 670
06 707
06 744
06 781
117
06 819
06 856
06 893
06 930
06 967
07 004
07 041
07 078
07 lis
07 isi
118
07 188
07 225
07 262
07 298
07 335
07 372
07 408
07 445
07 482
07 S18
119
07 555
07 591
07 628
07 664
07 700
07 737
07 773
07 809
07 846
07 882
120
07 918
07 954
07 990
08 027
08 063
08 099
08 US
08 171
08 207
08 243
121
08 279
08 314
08 350
08 386
08 422
08 458
08 493
08 529
08 56s
08 600
122
08 636
08 672
08 707
08 743
08 778
08 814
08 849
08 884
08 920
08 9SS
123
08 991
09 026
09 061
09 096
09 132
09 167
09 202
09 237
09 272
09 307
124
09 342
09 377
09 412
09 447
09 482
09 517
09 552
09 587
09 621
09 656
125
09 691
09 726
09 760
09 795
09 830
09 864
09 899
09 934
09 968
10 003
126
10 037
10 072
10 106
10 140
10 175
10 209
10 243
10 278
10 312
10 346
127
10 380
10 41S
10 449
10 483
10 S17
10 551
10 585
10 619
10 653
10 687
128
10 721
10 755
10 789
10 823
10 857
10 890
10 924
10 958
10 992
II 02s
129
II 059
II 093
II 126
II 160
II 193
II 227
II 261
II 294
II 327
II 361
130
ir 394
II 428
II 461
II 494
II 528
II 561
II 594
II 628
II 661
II 694
131
II 727
1 1 760
II 793
II 826
II 860
II 893
1 1 926
II 959
II 992
12 024
132
12 057
12 090
12 123
12 156
12 189
12 222
12 254
12 287
12 320
12 352
133
12 385
12 418
12 450
12 483
12 S16
12 548
12 S8l
12 613
12 646
12 678
134
12 710
12 743
12 775
12 808
12 840
12 872
12 905
12 937
12 969
13 001
' E. H Barker, Computing Tables and Formulas. 1913, pages 22-39.
233
234
APPENDIX
No.
0
I
2
3
4
5
6
7
8
9
135
13 033
13 066
13 098
13 130
13 162
13 194
13 226
13 258
13 290
13 322
136
13 354
13 386
13 418
13 450
13 481
13 SI3
13 S4S
13 577
13 609
13 640
137
13 672
13 704
13 73S
13 767
13 799
13 830
13 862
13 893
13 92s
13 956
138
13 988
14 019
14 051
14 082
14 114
14 14s
14 176
14 208
14 239
14 270
139
14 301
14 333
14 364
14 395
14 426
14 457
14 489
14 520
14 551
14 582
140
14 613
14 644
14 675
14 706
14 737
14 768
14 799
14 829
14 860
14 891
141
14 922
14 953
14 983
IS 014
15 045
IS 076
IS 106
IS 137
IS 168
IS 198
142
15 229
IS 259
IS 290
IS 320
IS 3SI
IS 381
IS 412
IS 442
IS 473
IS S03
143
IS 534
IS 564
IS 594
IS 62s
15 6s5
IS 685
IS 715
IS 746
IS 776
IS 806
144
IS 836
IS 866
IS 897
IS 927
15 957
IS 987
16 017
16 047
16 077
16 107
14s
16 137
16 167
16 197
16 227
16 256
16 286
16 316
16 346
16 376
16 406
146
16 435
16 46s
16 495
16 524
16 554
16 584
16 613
16 643
16 673
16 702
147
16 732
16 761
16 791
16 820
16 850
16 879
16 909
16 938
16 967
16 997
148
17 026
17 056
17 08s
17 114
17 143
17 173
17 202
17 231
17 260
17 289
149
17 319
17 348
17 377
17 406
17 435
17 464
17 493
17 522
17 551
17 580
ISO
17 609
17 638
17 667
17 696
17 72s
17 754
17 782
17 811
17 840
17 869
151
17 898
17 926
17 955
17 984
18 013
18 041
18 070
18 099
18 127
18 156
152
18 184
18 213
18 241
18 270
18 298
18 327
18 355
18 384
18 412
18 441
153
18 469
18 498
18 526
18 554
18 583
18 611
18 639
18 667
18 696
18 724
154
18 752
18 780
18 808
18 837
18 865
18 893
18 921
18 949
18 977
19 005
155
19 033
19 061
19 089
19 1 17
19 14s
19 173
19 201
19 229
19 257
19 285
156
19 312
19 340
19 368
19 396
19 424
19 451
19 479
19 S07
19 535
19 562
157
19 590
19 618
19 645
19 673
19 700
19 728
19 756
19 783
19 811
19 838
158
19 866
19 893
19 921
19 948
19 976
20 003
20 030
20 058
20 085
20 112
159
20 140
20 167
20 194
20 222
20 249
20 276
20 303
20330
20 3S8
2038s
160
20 412
20 439
20 466
20 493
20 520
20 548
20 575
20 602
20 629
20 656
161
20 683
20 710
20 737
20 763
20 790
20 817
20 844
20 871
20 898
2092s
162
20952
20 978
21 005
21 032
21 059
21 085
21 112
21 139
21 165
21 192
163
21 219
21 24s
21 272
21 299
21 32s
21 352
21 378
21 405
21 431
21 4S8
164
21 484
21 511
21 537
21 564
21 590
21 617
21 643
21 669
2 1 696
21 722
165
21 748
21 775
21 801
21 827
21 854
21 880
21 906
21 932
21 9S8
21 985
166
22 on
22 037
22 063
22 089
22 IIS
22 141
22 167
22 194
22 220
22 246
167
22 272
22 298
22 324
22 350
22 376
22 401
22 427
22 453
22 479
22 SOS
168
22 531
22 557
22 583
22 608
22 634
22 660
22 686
22 712
22 737
22 763
169
22 789
22 814
22 840
22 866
22 891
22 917
22 943
22 968
22 994
23 019
170
23 045
23 070
23 096
23 121
23 147
23 172
23 198
23 223
23 249
23 274
171
23 300
23 32s
23 350
23 376
23 401
23 426
23 452
23 477
23 S02
23 528
172
23 553
23 578
23 603
23 629
23 654
23 679
23 704
23 729
23 754
23 779
173
23 80s
23 830
23 855
23 880
23 90s
23 930
23 9S5
23 980
24 005
24030
174
24 055
24 080
24 105
24 130
24 155
24 180
24 204
24 229
24 254
24 279
175
24 304
24 329
24 353
24 378
24 403
24 428
24 452
24477
24 502
24 527
176
24 551
24 S76
24 601
24 62s
24 650
24 674
24699
24 724
24 748
24 773
177
24 797
24 822
24 846
24 871
24 895
24 920
24 944
24 969
24 993
25 018
178
25 042
2S 066
25 091
25 IIS
25 139
25 164
25 188
25 212
25 237
25 261
179
25 28s
f
25 310
25 334
25 358
25 382
25 406
2S 431
25 455
25 479
25 S03
LOGARITHMS OF NUMBERS
235
No.
0
I
2
3
4
5
6
7
8
9
180
25 527
25 551
25 575
25 600
25624
25 648
25 672
25 696
25 720
25 744
181
25 768
25 792
25 816
25 840
25 864
25 888
25 912
25 935
25 959
25 983
182
26 007
26 031
26 05s
26 079
26 102
26 126
26 150
26 174
26 198
26 221
183
26 24s
26 269
26 293
26 316
26 340
26 364
26 387
26 411
26 435
26 458
184
26 482
26 50s
26 529
26 553
26 576
26 600
26 623
26 647
26 670
26 694
I8S
26 717
26 741
26 764
26 788
26 811
26 834
26 858
26 881
26 905
26 928
186
26 951
26 975
26 998
27 021
27 04s
27 068
27 091
27 114
27 138
27 161
187
27 184
27 207
27 231
27 254
27 277
27 300
27 323
27 346
27 370
27 393
188
27 416
27 439
27 462
27 48s
27 SO8
27 531
27 554
27 577
27 600
27623
189
27 646
27 669
27 692
27 715
27 738
27 761
27 784
27 807
27 830
27 852
190
27 87s
27 898
27 921
27 944
27 967
27 989
28 012
28 035
28 058
28 081
191
28 103
28 126
28 149
28 171
28 194
28 217
28 240
28 262
28 285
28 307
192
28 330
28 353
2837s
28 398
28 421
28 443
28 466
28 488
28 SIX
28 533
193
28 5S6
28 578
28 601
28 623
28 646
28 668
28 691
28 713
28 735
28 758
194
28 780
28 803
28 82s
28 847
28 870
28 892
28 914
28 937
28 959
28 981
195
29 003
29 026
29 048
29 070
29 092
29 lis
29 137
29 159
29 i8r
29 203
196
29 226
29 248
29 270
29 292
29 314
29 336
29358
29 380
29403
29 425
197
29447
29 469
29491
29 513
29 535
29 557
29 579
29 601
29 623
29 645
198
29 667
29 688
29 710
29 732
29 754
29 776
29 798
29 820
29 842
29 863
199
29 88s
29 907
29 929
29 951
29 973
29 994
30 016
30 038
30 060
30 081
200
30 103
30 125
30 146
30 168
30 190
30 211
30 233
30 255
30 276
30 298
201
30 320
30341
30363
30384
30 406
30 428
30 449
30 471
30492
30 S14
202
30 535
30 557
30578
30 600
30 621
30 643
30 664
30 685
30 707
30 728
203
30 750
30 771
30 792
30 814
30 835
30 856
30 878
30 899
30 920
30 942
204
30 963
30 984
31 006
31 027
31 048
31 069
31 09X
31 112
31 133
31 154
20s
31 175
31 197
31 218
31 239
31 260
31 281
31 302
31 323
31 345
31366
206
31 387
31 408
31 429
31 450
31 471
31 492
31 S13
31 534
31 555
31 576
207
31 597
31 618
31 639
31 660
31 681
31 702
31 723
31 744
31 765
31 78s
208
31 806
31 827
31 848
31 869
31 890
31 911
31 931
31 952
31 973
31 994
209
32 015
32 035
32 056
32 077
32 098
32 118
32 139
32 160
32 181
32 201
210
32 222
32 243
32 263
32 284
32 305
32 325
32 346
32 366
32 387
32 408
211
32 428
32 449
32 469
32 490
32 510
32 531
32 552
32 572
32 593
32 613
212
32 634
32 654
i2 67s
32 695
32 715
32 736
32 756
32 777
32 797
32 818
213
32 838
32 858
32 879
32 899
32 919
32 940
32 960
32 980
33 001
33 021
214
a 041
a 062
3i 082
33 102
33 122
33 143
33 163
33 183
33 203
33 224
215
a 244
a 264
33 284
33 304
33 325
33 345
33 36s
33 385
33 405
33 42s
216
a 445
3i 465
33 486
33 S06
33 526
33 546
33 566
33 586
33 606
33 626
217
3i 646
a 666
33 686
33 706
33 726
33 746
33 766
33 786
33 806
33 826
218
a 846
a 866
33 88s
33 905
33 9-25
33 945
33 965
33 985
34 005
34 025
219
34 044
34 064
34 084
34 104
34 124
34 143
34 163
34 1S3
34 203
34 223
220
34 242
34 262
34 282
34 301
34 321
34 341
34 361
34 380
34 400
34 420
221
34439
34 459
34 479
34 498
34518
34 537
34 557
34 577
34 596
34616
222
34 63s
34 655
34674
34694
34 713
34 733
34 753
34 772
34 792
34 811
223
34 830
34 850
34 869
34 889
34 908
34 928
34 947
34 967
34986
35 OOS
224
35 025
35 044
35 064
35 083
35 102
35 122
35 141
35 i6o
35 180
35 199
236
APPENDIX
No.
0
I
2
3
4
5
6
7
8
9
225
3S 218
35 238
35 257
35 276
35 295
35 31S
35 334
35 353
35 372
35 392
226
35 411
35 430
35 449
35 468
35 488
35 507
35 526
35 545
35 564
35 583
227
35 603
35 622
35 641
35 660
35 679
35 698
35 717
35 736
35 755
35 774
228
35 793
35 813
35 832
35 8SI
35 870
35 889
35 908
35 927
35 946
35 96s
229
35 984
36 003
36 021
36 040
36 059
36 078
36 097
36 1x6
36 135
36 154
330
36 173
36 192
36 211
36 229
36 248
36 267
36 286
36 305
36 324
36 342
231
36 361
36 380
36 399
36 418
36 436
36 455
36 474
36 493
36 Sii
36 530
232
36 549
36 568
36 586
36 605
36 624
36 642
36 661
36 680
36 698
36 717
233
36 736
36 754
36 773
36 791
36 810
36 829
36 847
36 866
36 884
36 903
234
36 922
36 940
36959
36 977
36996
37 014
37 033
37 051
37 070
37 088
235
37 107
37 125
37 144
37 162
37 181
37 199
37 218
37 236
37 254
37 273
236
37 291
37 310
37 328
37 346
37 365
37 383
37 401
37 420
37 438
37 457
237
37 475
37 493
37 511
37 530
37 548
37 S66
37 585
37 603
37 621
37 639
238
37 658
37 676
37 694
37 712
37 731
37 749
37 767
37 785
37 803
37 822
239
37 840
37 858
37 876
37 894
37 912
37 931
37 949
37 967
37 98s
38 003
240
38 021
38 039
38 057
38 07S
38 093
38 112
38 130
38 148
38 166
38 184
241
38 202
38 220
38 238
38 256
38 274
38 292
38310
38328
38346
38364
242
38 382
38 399
38 417
38 435
38 453
38 471
38489
38 S07
38 525
38 543
243
38 S6i
38 578
38 596
38614
38 632
38 650
38 668
38686
38 703
38 721
244
38 739
38 757
38 775
38 792
38 810
38 828
38 846
38 863
38 881
38 899
24s
38 917
38 934
38 952
38 970
38 987
39 005
39 023
39 041
39 058
39 076
246
39 094
39 III
39 129
39 146
39 164
39 182
39 199
39 217
39 235
39 252
247
39 270
39 287
39 305
39 322
39 340
39 358
39 375
39 393
39 410
39428
248
39 445
39 463
39 480
39498
39 SIS
39 533
39 SSO
39 568
39 585
39602
249
39 620
39637
39 655
39672
39 690
39 707
39 724
39 742
39 759
39 777
250
39 794
39 811
39 829
39 846
39 863
39 881
39 898
39915
39 933
39 950
2SI
39 967
39 985
40 002
40 019
40 037
40 054
40 071
40 088
40 106
40 123
2S2
40 140
40 157
40 175
40 192
40 209
40 226
40 243
40 261
40 278
40 295
253
40 312
40 329
40 346
40 364
40 381
40 398
40 41S
40 432
40 449
40 466
254
40 483
40 500
40 S18
40 535
40 552
40 569
40 586
40 603
40 620
40 637
255
40 654
40 671
40 688
40 70s
40 722
40 739
40 756
40 773
40 790
40 807
256
40 824
40 841
40 858
40 875
40 892
40 909
40 926
40943
40 960
40976
257
40 993
41 010
41 027
41 044
41 061
41 078
41 095
41 II I
41 128
41 145
2S8
41 162
41 179
41 196
41 212
41 229
41 246
41 263
41 280
41 296
41 313
259
41 330
41 347
41 363
41 380
41 397
41 414
41 430
41 447
41 464
41 481
260
41 497
41 514
41 531
41 547
41 564
41 S8l
41 597
41 614
41 631
41 647
261
41 664
41 681
41 697
41 714
41 731
41 747
41 764
41 780
41 797
41 814
262
41 830
41 847
41 863
41 880
41 896
41 913
41 929
41 946
41 963
41 979
263
41 996
42 012
42 029
42 045
42 062
42 078
42 095
42 III
42 127
42 144
264
42 160
42 177
42 193
42 210
42 226
42 243
42 259
42 275
42 292
42 308
265
42 325
42 341
42 357
42 374
42 390
42 406
42 423
42 439
42 455
42 472
266
42 488
42 SO4
42 521
42 537
42 553
42 570
42 586
42 602
42 619
42 635
267
42 651
42 667
42 684
42 700
42 716
42 732
42 749
42 765
42 781
42 797
268
42 813
42 830
42 846
42 862
42 878
42 894
42 911
42 927
42 943
42 959
269
42 97S
42 991
43 008
43 024
43 040
43 056
43 072
43 088
43 104
43 120
LOGARITHMS OF NUMBERS
237
No.
0
I
2
3
4
5
6
7
8
9
270
43 136
43 152
43 169
43 i8s
43 201
43 217
43 233
43 249
43 26s
43 281
271
43 297
43 313
43 329
43 345
43 361
43 377
43 393
43 409
43 425
43 441
272
43 457
43 473
43 489
43 505
43 521
43 537
43 553
43 569
43 584
43 600
273
43 616
43 632
43648
43 664
43 680
43 696
43 712
43 727
43 743
43 759
274
43 775
43 791
43 807
43 823
43 838
43 854
43 870
43 886
43 902
43 917
27s
43 933
43 949
43 965
43 981
43 996
44 012
44 028
44 044
44 059
44 075
276
44 091
44 107
44 122
44 138
44 154
44 170
44 i8s
44 201
44 217
44 232
277
44 248
44 264
44 279
44 295
44 311
44 326
44 342
44 358
44 373
44389
278
44 404
44 420
44 436
44 451
44 467
44 483
44 498
44 514
44 529
44 545
279
44 560
44 576
44 592
44 607
44623
44638
44654
44 669
4468s
44 700
380
44 716
44 731
44 747
44 762
44 778
44 793
44 809
44 824
44 840
44 855
281
44871
44 886
44 902
44 917
44 932
44 948
44 963
44 979
44 994
45 010
282
45 025
45 040
45 056
45 071
45 086
45 102
45 117
45 133
45 148
45 163
283
45 179
45 194
45 209
45 22s
45 240
45 255
45 271
45 286
45 301
45 317
284
45 332
45 347
45 362
45 378
45 393
45 408
45 423
45 439
45 454
45 469
28S
45 484
45 500
45 515
45 530
45 545
45 561
45 576
45 591
45 606
45 621
286
45 637
45 652
45 667
45 682
45 697
45 712
45 728
45 743
45 758
45 773
287
45 788
45 803
45 818
45 834
45 849
45 864
45 879
45 894
45 909
45 924
288
45 939
45 954
45 969
45 984
46 000
46 015
46 030
46 045
46 060
46 075
289
46 090
46 105
46 120
46 135
46 150
46 165
46 180
46 195
46 210
46 22s
290
46 240
46 255
46 270
46 28s
46 300
46 315
46 330
46 345
46 359
46374
291
46 389
46 404
46 419
46 434
46 449
46 464
46 479
46 494
46 S09
46 523
292
46 538
46 553
46 568
46583
46 598
46 613
46 627
46 642
46 657
46 672
293
46 687
46 702
46 716
46 731
46 746
46 761
46 776
46 790
46 805
46 820
294
46 835
46 850
46 864
46 879
46 894
46 909
46 923
46 938
46 953
46 967
295
46 982
46 997
47 012
47 026
47 041
47 056
47 070
47 085
47 100
47 114
296
47 129
47 144
47 159
47 173
47 188
47 202
47 217
47 232
47 246
47 261
297
47 276
47 290
47 305
47 319
47 334
47 349
47363
47 378
47 392
47 407
298
47 422
47 436
47 451
47 465
47 480
47 494
47 509
47 524
47 538
47 553
299
47 567
47 582
47 596
47 611
47 62s
47 640
47 6S4
47 669
47 683
47 698
300
47 712
47 727
47 741
47 7S6
47 770
47 784
47 799
47 813
47 828
47 842
301
47 857
47 871
47 88s
47 900
47 914
47 929
47 943
47 958
47 972
47 986
302
48 001
48 015
48 029
48 044
48 058
48 073
48 087
48 lOI
48 116
48 130
303
48 144
48 159
48 173
48 187
48 202
48 216
48 230
48 244
48 259
48 273
304
48 287
48 302
48316
48 330
48 344
48 359
48 373
48 387
48 401
48 416
30s
48 430
48 444
48 4S8
48 473
48487
48 SOI
48 S15