CHAPTER LII. Some Applications of Interest and Proportion

SOME APPLICATIONS OF INTEREST AND PROPORTION

=The Nature of Interest.=—Interest may be defined as the charge made for the use of money. Sprague defines it as the increase in principal due to the lapse of time. The ethics of the practice of charging interest was questioned by the ancient world and not fully conceded as right until modern times. Various economic theories have been evolved to explain the true character of interest. Whatever they may be, interest as a commercial phenomenon is thoroughly established and countenanced by the law, although in many states an exorbitant interest charge is declared to be usury.

=Commercial Interest.=—Commercial interest, so-called, usually contains an element in addition to the time-charge for the use of money. That element may be: (1) in the nature of a premium for insurance against the risk of losing the money loaned; or (2) where capital in some fixed form is loaned, in the nature of an allowance or additional charge to cover the shrinkage in the asset loaned due to wear and tear.

=Simple and Compound Interest.=—As to its method of calculation, interest may be simple or compound. Simple or single interest is figured on the single base known as the principal, the only other element being the length of time. Compound interest periodically adds the unpaid interest to the previous principal, and so secures interest not only on the original principal but on all unpaid interest as well.

In accounting both kinds of interest are recorded under the common title Interest. Some applications of the interest principle to certain special accounts will be discussed.

=Equation of Payments.=—The practice of averaging accounts is occasionally met with at the present time in American business. In proceedings in bankruptcy, all claims against the bankrupt on open or running account comprising several items, when filed with the trustee, must show the average due date of the items if interest is to be secured on the overdue amounts.

The problem involved may best be shown by an example. The following account appears on A’s books, showing charges against B:

B ============================================================= Jan. 5 Mdse. 2/10, n/30. 100.00 | Feb. 1 Mdse. 2/10, n/60. 350.00 | Apr. 10 Mdse. net 200.00 | June 2 Mdse. 2/10, n/60. 2,000.00 |

If B does not settle the various amounts as they come due, A is deprived of the use of his money longer than contemplated in the sale contract. In justice to him, interest on the overdue amounts ought to be allowed. If B should pay any of the amounts earlier than the terms of sale require, he should be allowed a discount, i.e., a rebate equal to the interest for the time of prepayment. Further, shortly after the last purchase on June 2 at 60 days, amounting to $2,000, B may desire to settle his entire account, taking his discount for prepayment on the $2,000 and allowing A interest on the overdue amounts. If the date of settlement is fixed, the amount necessary for an equitable settlement may be determined by the method used for the account current in the previous chapter.

=Average Due Date.=—But B may want to know the date on which he can settle equitably by paying the exact amount of the account without either paying interest on the overdue items or taking discount on the $2,000. The problem involved is that of averaging or equating accounts. The equated date, due date, or average date of payment are the terms variously applied to the date of equitable settlement. If the account has only debits or only credits, the equation is called a simple or single equation or average; if it has both debits and credits, the equation is called a compound or double equation.

In order to determine the equated date, an arbitrary one, called the focal date, is taken for the purpose of computing the interest charges and credits, and from that date the days of interest are counted backwards or forwards according to the result arrived at through use of the arbitrary date. Interest is calculated at an arbitrary rate, usually 6% (100% per day is used by another method of calculation), and in the case of compound equation the same rate must be used on both debits and credits.

To illustrate the method of calculation for the simple equation and the interest principle involved, the account above cited will be equated. In order that the expired time between the focal date and each date of value may be easily computed, the last day of the previous year is taken as the focal date. Interest is at 6%.

Int. on Total Int. on Each Date of Date of Expired Amount for Amount for Entry Value Time Amount 1 Day Expired Time

1/5 2/4 35 da. $ 100 $ .58 2/1 4/2 92 ” 350 5.37 4/10 4/10 100 ” 200 3.33 6/2 8/1 213 ” 2,000 71.00 ------ ------- $2,650 .44⅙ )$80.28 ====== ------ 182 da.

This calculation shows that theoretically, had the various transactions been under contemplation on December 31, the focal date, payment of the total $2,650 could equitably have been made with a discount of $80.28. The interest (or discount) on $2,650 for 1 day is 44⅙ cents. A discount amounting to $80.28 can therefore be demanded on $2,650 only as the result of an offer to prepay 182 days (80.28 ÷ .44⅙ = 182) before the payments are equitably due. Hence, payment of $2,650 without discount would settle the account equitably 182 days after the focal date, or on July 1. That this is true can easily be proved by using July 1 as the settlement date and figuring as for a current account. It will be found that interest on the overdue items on that date amounts to $10.42, while the discount on the item not yet due amounts to $10.33; the difference .09 not being a large enough fraction (9 ÷ 44⅙) to justify payment one full day earlier.

=The 100% Method.=—A short method of calculation may be used, employing the 100% per day method. Any date may be taken as a focal date, and very frequently the date of the first or last transaction is used. In the illustration below, November 30 of the previous year is taken as the focal date so that the expired time on each item is immediately indicated by the number of the month and the day in the “date of value” column. The use of the 100% per day method makes the calculation of interest on each item a simple matter of multiplication by time and amount, i.e., it reduces each amount to a “day-dollars” figure, and on that basis one day’s interest on the account total is equal to that total, and therefore the divisor in the division made to determine the focal date is the amount of the account. This greatly simplifies all the operations. Sometimes the expired time is calculated by calendar months and days, converting fractions of a month on a 30-day basis. The method is used in the illustration below, where the problem shown above by the accurate interest method is solved by the 100% method.

The “month-dollars” column divided by the “amount” column gives 6, shown in the “equated date, months” column, with a remainder of 2,500. This is reduced, by multiplication by 30, to day-dollars and carried to that column, whose total, 80,100, is divided by 2,650, giving 30 as shown in the “equated date, days” column. The equated date is therefore June 30 (6/30). The one day’s difference between this and the other method is accounted for because each calendar month is counted as 30 days.

Time Month- Day- Equated Date Months Days Amount Dollars Dollars Months Days

2 4 $ 100 $ 200 $ 400 4 2 350 1,400 700 4 10 200 800 2,000 8 1 2,000 16,000 2,000 ----- -------- ------- $2,650 )$18,400 $ 5,100 6 30 ------ 15,900 -------- $ 2,500 30 75,000 6 30 ------- ------- $80,100 79,500 ------- $ 600

=Compound Equation.=—Where the account has both debits and credits, the estimate is made similarly. Calculation of the month- and day-dollars is made for each side separately. At this point the totals on both sides are combined to find the balance of the account and the balance of the discounts, and these two balances are used to find the equated date. If the balance of the account is on the same side as the balance of the discount, the equated date is forward from the focal date because, if settlement were made on that date, the man who owes the balance is entitled to the theoretical discount also. If the balance of the account and the balance of interest are on different sides, the count is backward from the focal date. The following account and solution will illustrate:

S. L. DAVIS ================================================================== 19— | 19— Mar. 8 Mdse. net 1,000.00 | Apr. 30 Note, 30 da., 6% 500.00 June 20 ” n/30 1,500.00 | Aug. 30 Cash 1,500.00 Sept. 5 ” n/60 2,000.00 | Sept. 10 Note, 60 da., no | interest 2,000.00

Debits: Expired Time Interest Months Days Amount Month-Dollars Day-Dollars 3 8 $1,000 $ 3,000 $ 8,000 7 20 1,500 10,500 30,000 11 4 2,000 22,000 8,000 ------ ------- ------- Totals $4,500 $35,500 $46,000

Credits: 4 30 $ 500 $ 2,000 $15,000 8 30 1,500 12,000 45,000 11 9 2,000 22,000 18,000 ------- ------- ------- Totals $4,000 $36,000 $78,000 ------ ------- ------- Balances: Amount Dr. $500 Interest Cr. $500 Cr. $32,000

Dividing we get 1 month, 64 days, i.e., 3 months, 4 days. The balances being on opposite sides, the equated date is 3 months, 4 days, backward from November 30 (11/30), i.e., 11/30-3/4 = 8/26 or August 26. Equitable settlement could therefore be made by interest-bearing note for $500, dated August 26, or by cash payment of $500 plus interest on $500 from August 26 until date of actual settlement, as would be the case had the account been handled as an account current with adjustment as of August 26.

=The Cash Balance.=—When an account has been equated, to determine the cash sum which will be required for equitable settlement on a given date subsequent to the equated date, the balance of the account plus interest on that balance from the equated date to the date of settlement will be the correct amount. This amount is technically called the cash balance of the account. It is exactly the same as the adjusted balance of an account current, and may be determined by such adjustment of the account instead of by the method of equation of payments just described.

=Interest on Partial Payments.=—Under the heads of accounts current and equation of payments, the question of partial payments on open account has been treated. There remains to be discussed a statement of the practices governing partial payments on notes. Two methods of calculating are in use, the legal or United States method and the so-called merchants’ method. The merchants’ method is used for short-time notes and on any other kind by agreement. The method is exactly similar to that of adjustment of current accounts. Interest is charged on the face of the note from its date of issue till its due date, and allowed on each partial payment from its date of payment till the due date of the note. The difference between the sum of the face of the note plus its interest and the partial payments plus their interest accruals is, of course, the balance due.

=United States Rule.=—The United States Supreme Court has ordered the application of the partial payments somewhat differently. The first partial payment must first be applied to the payment of the accrued interest on the principal up to the date of the first payment. Any excess shall be applied to a reduction of the principal. Each succeeding payment is similarly applied first to cancellation of accrued interest on each new principal and then to a reduction of the principal. In case any payment is insufficient to meet the accrued interest, the payment is held in reserve, the principal remaining unchanged until a payment or payments are made which added to the previously reserved payment or payments are sufficient to cancel all interest accrued to that date. Any excess is used to retire the principal.

=Interest on Daily and Savings Bank Balances.=—In the handling of balances between banks, interest on daily balances is usually figured in the settlement. Calculation is on a 365-day basis in the larger banks. Banks frequently allow large depositors a low rate of interest on daily balances maintained above a certain fixed minimum. Take the following account:

X. Z. & CO. ==================================================== Date Dr. Cr. Balance Interest Base Interest Jan. 2 1,500 500 3 300 500 1,700 700 4 800 100 1,000 5 400 600 1,200 200 6 500 400 1,100 100 7 700 1,100 1,500 500 .11 ----- 2,000

In the above account interest is allowed on all amounts above $1,000 at the rate of 2%. If settlement is periodical, interest may be calculated on the total of the “interest base” column for one day. Usually, however, if a monthly settlement basis is used, the total minimum balance for the month is subtracted from the total of the “balance” column, and the remainder is the interest base. In savings banks no interest is allowed on amounts which have been withdrawn during the interest period, regardless of how long the sum may have been on deposit previous to date of withdrawal. There is no uniform practice as to when deposits shall begin to draw interest, in some banks at the beginning of the month after deposit, unless deposit is made on the first day of the current month; in others, not until the beginning of the next interest period. Great care must be exercised, therefore, in handling deposit and withdrawal dates.

=Bank and True Discount.=—Bank discount has been defined as the prepaid or collected interest on a discounted note, calculation being on the basis of the amount to be collected on the note at its maturity.

True discount is the difference between the face of a debt and its present worth, meaning by present worth that sum of money which placed at interest now will equal or be worth the face of the debt at maturity.

=Proportion, Simple and Weighted.=—Proportion is defined as an equality of ratios. Thus, if the ratio of a to b is the same as the ratio of c to d, this relation may be expressed as follows:

a c ——— = ——— or a : b = c : d b d

the fractional form being preferred. In accounting it is often required to divide amounts in certain ratios, as when profits must be apportioned among partners, when insurance, taxes, and other expense charges must be distributed over departments, etc. This is usually entitled “apportioning.” It is not necessary to illustrate simple proportion, but a problem in “weighted proportion” will be given here.

Apportion an insurance charge of $1,000 over departments A, B, and C according to the property values in those departments, taking account of the fact that the rate on A is double that on B and C. The property values are: A $10,000; B $15,000; C $35,000.

The proper basis for distribution cannot be found, as in simple proportion, by an addition of the values in the departments. Before addition, the value in A must be weighted by 2, i.e., doubled. This gives a basis of $70,000 ($20,000 for A + $15,000 for B + $35,000 for C = $70,000). Of the $1,000 insurance cost, department A will have to bear ²⁰/₇₀; B ¹⁵/₇₀; and C ³⁵/₇₀. The charges will be therefore:

A, ²⁰/₇₀ of $1,000 or $ 285.71 B, ¹⁵/₇₀ ” 1,000 ” 214.29 C, ³⁵/₇₀ ” 1,000 ” 500.00 --------- $1,000.00

=Apportioning Freight Charges.=—In-freight and cartage are treated as additions to the cost of goods bought; consequently, at inventory time it is necessary to add the correct amount of in-freight to the cost of goods on hand. However, it is seldom possible to apply the freight costs directly to each unit of product on hand and yet theoretically this should be done. Usually the ratio of freight costs to the total amount of purchases during a given period is taken as the basis for adding freight to the inventory. Thus, if that ratio has been 5% for the period, a commodity costing $100 would be valued at $105 for the inventory. Thus the freight expense is deferred.

This usually is deemed sufficiently accurate for most purposes. Where departmental records are kept, or where accurate factory costs are required, a closer apportioning is sometimes necessary. The freight classifications are such that the rates are not proportionate to the values of the goods; but other factors such as weight, kind of goods, method of crating, etc., all enter into the freight rate. Of these, the only factor which is easily obtainable is the weight. A distribution of freight on the basis of value and weight has been suggested—a weighted proportion method of apportionment. This, of course, requires an involved calculation which is usually “shied” at by bookkeepers and is not necessary except where very accurate and detailed costs are required.