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2 The measurement of interest 1,1 INTRODUCTION Interest may be defined as the compensation that a borrower of capital pays to a lender of capital for its use. Thus, interest can be viewed as a form of rent that the borrower pays to the lender to compensate for the loss of use of the capital by the lender while it is loaned to the borrower. In theory, capital and interest need not be expressed in terms of the same commodity. For example. Farmer A may lend a tractor to Farmer for use in harvesting Bs wheat crop in return for a percentage of the wheat harvested. In this example, the tractor is capital and the portion of wheat that gives to A is interest. However, for almost all applications, both capital and interest are expressed in terms of money. In Chapter 1 the various quantitative measures of interest are analyzed. This chapter includes most of the basic principles involved in the measurement of interest. Chapters 2 through 8 elaborate and extend these basic principles to more complex financial transactions. These chapters explore the various methods by which interest is calculated and by which capital and interest are repaid by the borrower to the lender. Chapters 18 in essence are concerned with the mathematical theory of interest on a deterministic basis. Chapter 9 introduces the student to the economic and financial theory of interest and related topics. Finally, Chapter 10 approaches the subject of interest from a stochastic rather than deterministic approach. 1.2 THE ACCUMULATION AND AMOUNT FUNCTIONS A common financial transaction is the investment of an amount of money at interest. For example, a person may invest in a savings account at a bank. 10.4 An asset pricing model, 349 10.5 An option pricing model, 355 10.6 A random walk model, 361 10.7 Scenario testing, 365 10.8 A review of the literature, 370 Appendix I Table of compound interest functions.......... 376 Appendix Table numbering the days of the year ......... 393 Appendix III Basic mathematical review ................ 394 Appendix IV Statistical background ................... 396 Appendix V Iteration methods...................... 399 Appendix VI Further analysis of varying annuities.......... 401 Appendix VII Illustrative mortgage loan amortization schedule . . . 402 Appendix VIII Full immunization ..................... 406 Appendix IX Derivation of the variance of an annuity........ 408 Appendix X Derivation of the BlackScholes formula........ 410 Bibliography.................................... 412 Answers to the exercises ............................ 415 Glossary of notation............................... 431 Index ........................................ 439
(c) (d) Figure 1.1 Four illustrative amount functions Actually, the accumulation function is a special case of the amount fiinction for which A: = 1. However, the accumulation function will be significant enough in the rest of this chapter to warrant a separate definition. In many cases, the accumulation fiinction and the amount function can be used interchangeably. Figure 1.1 shows four examples of amount functions. Figure (a) is a linear amount function. Figure {b) is nonlinear, in this case an exponential curve. Figure (c) is an amount function which is horizontal, i.e. the slope is zero. This figure represents an amount function in which the principal is accruing no interest. Figure (d) is an amount function in which interest is not accruing We will denote the amount of interest earned during the mh period from the date of investment by /„. Then /„ = A{n)A{n\) for integral > 1. (1.2) It should be noted that /„ involves the effect of interest over an interval of time, whereas A{n) is an amount at a specific point in time. The initial amount of money (capital) invested is called the principal and the total amount received after a period of time is called the accumulated value. The difference between the accumulated value and the principal is the amount of interest, or just interest, earned during the period of investment. For the moment, assume that given the original principal invested, the accumulated value at any point in time can be determined. We will assume that no principal is added or withdrawn during the period of investment, i.e. that any change in the fund is due strictly to the effect of interest. Later we will relax this assumption and allow for contributions and withdrawals during the period of investment. Let t measure time from the date of investment. In theory, time may be measured in many different units, e.g., days, months, decades, etc. The unit in which time is measured is called the measurement period, or just period. The most common measurement period is one year, and this will be assumed unless stated otherwise. Consider the investment of one unit of principal. We can define an accumulation Junction a(t) which gives the accumulated value at time r > 0 of an original investment of 1. What properties does this function possess? 1. It is clear that a(0) = 1. 2. a(t) is generally an increasing function. A decrease in the functional values for increasing t would imply negative interest. Although negative interest is possible mathematically, it is not relevant to most situations encountered in practice. However, there are situations in which negative interest does appear, e.g. an investment fund which loses money over a certain period of time. Constant fiinctional values would imply zero interest, a situation occurring occasionally. 3. If interest accrues continuously, as is usually the case, the function will be continuous. However, there are situations in which interest does not accrue continuously between interest payment dates, in which case a(t) possesses discontinuities. In general, the original principal invested will not be one unit but will be some amount > 0. We now define an amount Junction A(t) which gives the accumulated value at time r > 0 of an original investment of k. Then we have A(t) = k a{t) (1.1) A(0) = k. The second and third properties of a{t) listed above clearly also hold for A(f).
i a(0) A{0) ) Thus, an alternate definition is: The effective rate of interest i is the ratio of the amount of interest earned during the period to the amount of principal invested at the beginning of the period. The same four observations made above also apply to this alternate definition. Effective rates of interest can be calculated over any measurement period. Let i„ be the effective rate of interest during the nth period from the date of investment. Then we have . A{n)A{n\) for integral > 1. ) A{n\) 1) Within this framework, the "i" in formula (1.4a) might more properly be labeled jj. Although formula (1.4b) allows the various effective rates of interest J„ to vary for different n, it will be demonstrated in Section 1.5 that for one very important accumulation function, the effective rate of interest is constant over successive measurement periods, i.e. for all integral n > 1. 1.4 SIMPLE INTEREST It was shown in the preceding sections that o(0) = 1 and a(l) = 1 + j. There are an infinite number of accumulation functions that pass through these two points. Two of these are most significant in practice. The first, simple interest, will be discussed in this section; and the second, compound interest, will be discussed in Section 1.5. Consider the investment of one unit such that the amount of interest earned during each period is constant. The accumulated value of 1 ai the end of the first period is 1 + i, at the end of the second period it is 1 + 2i, etc. Thus, in general, we have a linear accumulation function a{t) = I + it for integral t > 0. (1.5) The accruing of interest according to this pattern is called simple interest. It can be shown that a constant rate of simple interest does not imply a constant effective rate of interest. Let i be the rate of simple interest and let be the effective rate of interest for the nth period, as defined in Section 1.3. Then we have . ajn)  a{n  1) [1 +in]  [1 +/(nl)] i " a(n  1) 1 + i(.n  1) 1 + i(n  1) (1.6) continuously but is accruing in discrete segments with no interest accruing between interest payment dates. In the following sections, various measures of interest will be developed from the accumulation function. In practice, two particular accumulation functions will handle most situations which arise. However, the reader should understand the properties of a general accumulation function as defined in this section and be able to work with it. 1.3 THE EFFECTIVE RATE OF INTEREST The first measure of interest is called the effective rate of interest and is denoted by i. A precise definition is: The effective rate of interest i is the amount of money that one unit invested at the beginning of a period will earn during the period, where interest is paid at the end of the period. Note that in terms of the accumulation function, this definition is equivalent to saying that i = a{V)  aiO) a{\) = 1 + r. (1.3) Several observations about this definition are important: 1. The use of the word "effective" is not intuitively clear. This term is used for rates of interest in which interest is paid once per measurement period. This will be contrasted with "nominal" rates of interest, in which interest is paid more frequently than once per measurement period, to be considered in Section 1.8. 2. The effective rate of interest is often expressed as a percentage, e.g. J =8%. The concept of the effective rate of interest as a percentage is not inconsistent with the definition above, which states that it is an amount of money, since 8% can be looked upon as .08 per unit of principal. 3. The amount of principal remains constant throughout the period, i.e. no new principal is contributed and no principal is withdrawn during the period. 4. The effective rate of interest is a measure in which interest is paid at the end of the period. The significance of this statement is not immediately clear, but it will become evident in Section 1.7, where a situation is described in which interest is paid at the beginning of the period. The effective rate of interest can be defined in terms of the amount function as follows:
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