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Practical applications


Chapter 8 contains several practical applications of the theory of interest not previously discussed. Sections 8.2 through 8.4 cover the area of consumer credit, including a discussion of "truth in lending" requirements.

Section 8.5 and 8.6 deal with topics involved in the financial analysis of investments in fixed assets which have not previously been discussed. Section 8.5 introduces the reader to various depreciation methods which are in common usage. Section 8.6 discusses a type of financial analysis known as capitalized cost, a concept related to capital budgeting which was discussed in Section 5.8.

Section 8.7 analyzes the calculation of yield rates on short sales, a type of financial transaction not previously considered. Some unusual difficulties arise in connection with such calculations.

Finally, Section 8.8 provides an overview of the wide variety of modem j financial instruments which are in existence today. This material is of j considerable value in introducing the reader to a number of investment vehicles which may be unfamiliar. It also provides the foundation for further analytical development in Chapters 9 and 10.


In 1968 the United States Congress enacted the Consumer Credit Protection Act. Title I of this law is widely known as the "Truth in Lending Act" (15 use 1601).

The primary purpose of this law is to require lenders to provide fair and accurate disclosure of the terms of consumer loans to borrowers. The law does not attempt to control the amount which lenders may charge on loans, but only

) require proper disclosure. Also, the law applies only to consumer loans, and not business loans.

The law requires the disclosure of two key financial values; namely, the nee charge and the annual percentage rate (often called the "APR"). The former is designed to express the amount of interest to be charged over the term of the loan in dollars, while the latter expresses the interest to be paid as an nual rate. In addition to these two financial values, a number of additional Jisclosures in narrative form are required.

At the time a loan is taken out, the borrower may incur certain charges in onnection with obtaining the loan. Examples of such charges would be points (to be defined in Section 8.3), other loan fees, service charges, credit report s, and premiums for credit insurance.

The finance charge, as defined in the Truth in Lending Act, includes all ynterest charges. Moreover, it also includes some, but not necessarily all, of the dditional charges cited in the preceding paragraph. Detailed regulations have en released by the Federal Reserve Board, the federal agency responsible for iministering the Truth in Lending Act, which define items that must be eluded in the finance charge and those that do not have to be included. 1 It is specified that the annual percentage rate is to be computed by the ctuarial method. This method is defined on a basis compatible with compound interest theory. Using the actuarial method leads to a subdivision of payments jbetween principal and interest which is consistent with the amortization schedule discussed in Section 6.3.

One interesting aspect of the annual percentage rate is that it is quoted as i a nominal rate convertible at the frequency with which payments are being made, rather than as an effective rate. For example, two loans might both quote "APR = 12%," but if one is repaid with monthly installments and the other is j repaid with quarterly installments, the two rates are not equivalent. Thus, armual percentage rates on various loans cannot validly be compared directly \ unless they are all at the same frequency. The length of the payment interval ; is called the imit period in the law.

Truth in lending requirements make a distinction between open end credit and closed end credit. Examples of open erul credit are a revolving charge account or a credit card. For open end credit the finance charge need only be disclosed each period as it is computed. The annual percentage rate is merely the nominal annual rate being charged on the outstanding loan balances. For example, if a credit card company charges interest each month at 1.75% on the outstanding balance, it must quote "APR = 21%."

The above brief discussion is intended to give the reader a general appreciation of the Truth in Lending Act and its application under typical conditions. However, the discussion is far from complete. Readers interested in learning more details about truth in lending are referred to "Regulation Z Truth in Lending" published by the Federal Reserve Board (12 CFR 226). An "Official Staff Commentary on Regulation Z Truth in Lending" has also been published to elaborate upon the formal regulation.

Some historical perspective on the computation of interest rates on loans is instructive. Until the early 1800s the most common method was the Merchants Rule, which is essentially equivalent to simple interest. The Merchants Rule worked satisfactorily for short-term loans, but produced illogical results for longer-term loans.

The issue reached the courts in Virginia in 1795. The landmark Ross v. Pleasants case involved a dramatic result. The roles of the debtor and creditor I were reversed when the Merchants Rule was applied over an extended period! of time. Needless to say, the original creditor who became a debtor was nonej too pleased at this turn of events!

The issue was finally resolved by the United States Supreme Court in the! decision on Story v. Livingston (38 U.S. 359) 1839. The court decided that! payments by the borrower should first be applied to pay any accrued interest, I with any excess being applied to reduce the outstanding loan balance. This I decision is consistent with the amortization scheduled developed in Section 6.3 and with compound interest theory. In recognition of this Supreme Court i decision, the above procedure for dividing installment payments between interest \ and principal is often called the United States Rule, a term used in contrast to] the Merchants Rule.

The following is an extract from the decision in the Story v. Livingston case] written by Mr. Justice Wayne:

"The correct rule in general is that the creditor shall calculate interest whenever a payment is made. To this interest the payment is first to be applied; and if it exceed the interest due, die balance is to be applied to diminish the principal. If die payment fall short of the interest, the balance of interest is not to be added to the principal, so as to produce interest. Thus rule is equally applicable whether the debt be one which expressly draws interest, or one in which interest is given in the name of damages."

Two aspects of this decision are worthy of comment. First, the United States Rule will result in a compounding of interest whenever an installment payment is made. If installment payments are made at a regular frequency, dien

Interest will be convertible at the same regular frequency as payments are made, i However, if installment payments are made at an irregular frequency, then the United States Rule produces the rather odd result that the rate of interest quoted is convertible at the same irregular frequency as payments are made. This result is illustrated in Example 8.3.

Second, it is often said that the United States Rule and the actuarial method are equivalent. Under most circumstances, they are equivalent. However, the ynited States Rule is not consistent with the actuarial method when a payment ris made that is insufficient to cover the accrued interest. Under the United i iStates Rule any such deficiency is not added to the outstanding loan balance to Iaccrue additional interest. However, under the actuarial method, which is rionsistent with compound interest theory, any such deficiency must be pitalized, i.e. added to the outstanding loan balance to accrue additional Ifaiterest.

Example 8.1 Find the annual percentage rate (APR) on a consumer loan bf$1000 which is repaid with monthly installments of $90 at the end of each 4th for one year. The finance charge is equal to

= (12)(90) - 1000 = 80.

The mondily rate j is determined from an application of formula (8.2)

90 ,- = 1000

a = 11.1111.

1 The Federal Reserve Board tables promulgated under Regulation Z give the Jmswer 12J = 14.50% for this example. The tables contain answers accurate only to nearest .25%, which is die minimum level of accuracy required under trudi in ng-

A more accurate answer can be obtained using iteration for an unknown rate of as discussed in Section 3.8. Such an iteration produces j = .012043, so that APR = 12J = .1445, or 14.45%.

Prior to the passage of the Trudi in Lending Act, this type of loan arrangement was iuendy called an "8% add-on," since the finance charge for one year was 8% of the ( amount. This misleading statement implied a much lower rate of interest than the Imie rate given by the APR. It was this type of widespread misrepresentation that led to enactment of truth in lending.

Example 8.2 A purchaser of a new automobile needs to finance $10,000 pfthe purchase price. The dealer offers two options for financing the loan vith monthly payments over four years. Under Option A the APR is 9%. \ Under Option the dealer offers a "cash back" of $600 and an APR of 12%.

We define L, K, R, n, i, and./ as in Section 8.2 and set m = 12. We theJ maice the following additional definitions:

Q = expenses at settlement that must be reflected in the APR

L* = amount financed for truth in lending purposes, i.e. reflecting Q ,,

f = monthly rate of interest on the loan

/ = quoted annual rate of interest on the loan

We then have the following relationships. The quoted annual rate of! interest on the loan is

= 12/ (8.4)j

The monthly payment on the loan is


The amount financed for truth in lending purposes is the amount of the loan 1 the expenses at settlement that must be reflected, i.e.

L* = L-Q. (8.

The finance charge is equal to the difference between die total payments to made and the amount financed for truth in lending purposes, i.e.

K = nR-L*. (8.7)1

To obtain die rate of interest per month for truth in lending we solve following equation for j

Finally, the annual percentage rate for truth in lending is

= 127. (8.5

The loan amortization schedules considered in Chapter 6 involved loans witl relatively short terms. The reader will find it instructive to examine illustrative 30-year amortization schedule for a real estate mortgage containe in Appendix VII. The early payments are almost entirely interest, while ending payments are almost entirely principal. Many people find it frustratir to buy a home, make payments for several years, and then find that outstanding loan balance on their real estate mortgage has declined very litde!

In recent years a new type of mortgage loan has appeared, the adjustabU rate mortgage. This term is in contrast to the traditional mortgage, which often called fixed rate mortgage. Under the adjustable rate mortgage the rati of interest charged on the loan can be periodically adjusted upward downward by the lender under certain conditions and subject to cer restrictions.

Ttje m.otivating factor for the development of the adjustable rate mortgage ; the fact that lenders are willing to provide such mortgages at lower initial t(gs of interest than on traditional fixed rate mortgages. This reaction by ders is understandable, since lenders are making firm commitments on the ! of interest to be charged on fixed rate mortgages over periods ranging from to 30 years. In periods of rising interest rates lenders will be locked into I rate portfolios with yield rates well below prevailing market rates. On the er hand, in periods of falling interest rates borrowers will be able to cfinance their loans at the lower rates.

f- From the borrowers perspective the adjustable rate mortgage may be active, since the initial rate of interest and the initial monthly payment are than with a fixed rate mortgage. Moreover, if interest rates fall, the ndily payment will be adjusted even lower. However, there is a significant risk with an adjustable rate mortgage that i of interest and monthly payments will increase and exceed those that would ve existed had a fixed rate mortgage been chosen originally. Thus, with a [ rate mortgage the risk associated with interest rate fluctuations is borne by i lender, whereas with adjustable rate mortgages much of that risk is shifted i;the borrower.

The interval at which the lender can adjust the interest rate on an adjustable i mortgage is called the adjustment period. This interval is typically every e, three, or five years.

The interest rates on an adjustable rate mortgage are usually tied by formula >an index rate. This prevents the potential abuse of the lender making [bitrary upward adjustments in the interest rate even if prevailing market rates aid not justify such increases. One example of a possible index rate is the i on Treasury securities of a specified maturity. A second example would Ian index rate based on the interest rates being paid to depositors in order to ure funds to lend by a broad cross section of institutional lenders. The erest rate on an adjustable rate mortgage is computed as the index rate plus 5xed margin, where the margin is defined by the original mortgage ement.

Most adjustable rate mortgages have limitations on the amount of increase he interest rate and/or payments which are called caps. One type of cap is interest rate cap. This type of cap places a limit on the amount of any nodic increase in the interest rate, or a limit on the total increase over die life I the loan, or both. A second type of cap is the payment cap. This type of places a percentage limit on the dollar increase in monthly payment that can cur at any one adjustment date.

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