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5 1 + / = (l.20a) since each side of the equation gives the accumulated value of 1 invested for one measurement period. Rearranging, we have i(m)  1, (1.20b) = m[(l +0""  1]. (120C) Figure 1.4 illustrates accumulation at a nominal rate of interest for one measurement period. The diagonal arrows to the right can be inte reted as plus signs and downward arrows as equal signs. Time: 0 Interest; Balance: 1 (.4 ( Figure 1.4 Illustration of nominal rates of interest The symbol for a nominal rate of discount payable m times per period is By a nominal rate of discount d"\ we mean a rate payable mthly, i.e. the effective rate of discount is rf"V/n for each mth of a period. \  d = (1.21a) since each side of the equation gives the present value of 1 to be paid at the end of one measurement period. Rearranging, we have = 1  dC") = m[l (1 d)"] =m[l v"] . (1.21b) (1.21c) Figure 1.5 illustrates discounting at a nominal rate of discount for one measurement period. The diagonal arrows to the left can be inte reted as minus signs and downward arrows as equal signs. Time: Discount; I = 1 Figure 1.5 niustration of nominal rates of discount There is a close relationship between nominal rates of interest and nominal rates of discount. The following relationship holds, since both sides of the equation are equal to 1 + /, Um = p, then formula (1.22a) becomes (1.22a) (1.22b) cautioned not to rely on such connotations when encountering these terms, but to ascertain exactly how the interest is computed and paid. This section defines nominal rates of interest and discount and develops a systematic method of finding effective and nominal rates of interest and discount which are equivalent. The definition of "equivalency" was given in Section 1.7. The symbol for a nominal rate of interest payable m times per period is i\ where m is a positive integer > 1. By a nominal rate of interest /"\ we mean a rate payable mthly, i.e. the rate of interest is i"Vm for each mth of a period and not i\ For example, a nominal rate of 8% convertible quarterly does not mean an interest rate of 8% per quarter but rather an interest rate of 2% per quarter. In reality, we can say that a nominal rate of interest of / per period is identical to an effective rate of interest of / "V/n per mth of a period. Thus, from the definition of equivalency we have The nominal rate of discount rf" is a measure of interest paid at the beginning of mths of a period in much the same manner as d is a measure of interest paid at the beginning of the period. By an argument similar to the one used in developing the relationship between /" and /, it is possible to develop a formula relating rf" and d such that they are equivalent. From the definition of equivalency we have
/(«) (n) i(m) (m) The verbal inte retation of this result is similar to the verbal inte retation of formula (1.17). The derivation is left as an exercise. In developing the above formulas involving rates of interest and discount which are equivalent, the definition of equivalency given in Section 1.7 was used with a comparison over one measurement period. Although the use of one measurement period was arbitrary, the reader should verify that under compound interest and discount the rates that are equivalent do not depend on the period of time chosen for the comparison. However, for other patterns of interest development, such as simple interest and simple discount, the rates that are equivalent will depend on the period of time chosen for the comparison. a(/) = (l+/) S4 Figure 1.6 Accumulation function illustrating nominal rates of interest and discount It is instructive to relate nominal rates of interest and discount to the accumulation function a(t). An example is given in Figure 1.6 for m = 2. The reader is encouraged to construct other examples. The following relationships hold graphically: AB = d A2B2 d B2C2 = 2 ,1/2 = 1  ;(2) = (1 + i) = 1 + measurement of interest 21 (2) B4C4, = l+i ,•(2) It is interesting to note that nominal rates of interest and discount are not relevant under simple interest and simple discount. Since the amount of interest or discount is directly proportional to the time involved, a rate of interest of discount payable mthly is no different than one payable once per measurement period. As with the word "discount," unfortunately the word "nominal" also has multiple meanings. In Section 7.3 "nominal" is used in another sense in connection with yields on bonds, while in Section 9.4 "nominal" takes on a different meaning in connection with the reflection of inflation in rates of interest. Example 1.7 Find the accumulated value of $500 invested for five years at 8% per annum convertible quarterly. The answer is = 500(1.02) It should be noted that this situation is equivalent to one in which $500 is invested at a rate of interest of 2% for 20 years. Example 1.8 Find the present value of $1000 to be paid at the end of six years at 6% per annum payable in advance and convertible semitmnuaUy. The answer is 1000 = 1000(.97) It should be noted that this situation is equivalent to one in which the present value of $1000 to be paid at the end of 12 years is calculated at a rate of discount of 3%. Example 1.9 Find the nominal rate of interest convertible quarterly which is equivalent to a nominal rate of discount of 6% per annum convertible monthly. Using formula (1.22e) ,.(4) = (.995) ,(4) = 4[(.995) 1]. It m = 1, then " = /, the effective rate of interest; and if /? = 1, then d = d, the effective rate of discount. Thus formula (1.22a) can be used in general to find equivalent rates of interest or discount, either effective or nominal, convertible with any desired frequency. Another relationship between /" and d" which is analogous to formula (1.17) is ,(/71) j(m] :m> Jim) (1.23)
1.9 FORCES OF INTEREST AND DISCOUNT The measures of imerest defined in the preceding sections are useful for measuring interest over specified intervals of time. Effective rates of interest and discount measure interestover one measurement period, while nominal rates of interest and discount measure interest over /nths of a measurement period. It is important in many cases to be able to measure the intensity with which interest is operating at each moment of time, i.e. over infinitesimally small intervals of time. This measure of interest at individual moments of time is called the force of interest. Consider the investment of a fund such that the amount in the fund at time t is given by the amount function A{t). Recall that the only factor operating on the fund is the growth of the iund through interest, i.e. no principal is added or withdrawn. The intensity with which interest is operating at time t is measured by the rate of change or the slope of the A{t) curve at time t. From elementary calculus, the slope of the A{t) curve at time t is given by the derivative at that point. However, as a measure of interest, A{t)\s unsatisfactory, since it depends on the amount invested. For example, if $200 and $100 are invested under identical conditions, the rate of change of the $200 fund will be twice as great as the rate of change of the $100 fund. However, interest is not operating with twice the intensity on the $200 fund; in fact, we would say that it is operating with the same intensity on both funds. We can compensate for this by dividing A{t)by the amount in the fund at time namely A(f). This gives a measure of the intensity with which interest is operating at time t expressed as a rate independent of the amount in the fimd, i.e. as a rate per dollar in the fund. Thus, the force of interest at time t, denoted by 6,, is defined as 5 = £(0 = Ait) ait) (1.24) The following properties of 5, should be kept in mind: 1. 6, is a measure of the intensity of interest at exact time t. 2. 6, expresses this measurement as a rate per measurement period. It is possible to write an expression for the value of Ait) and ait) in terms of the function 5,. It will be seen from formula (1.24) that an alternate expression for 6, is 6, = 1 log. Ait) = 1 log, ait). (1.25) Replacing thy r and integrating both sides between the limits 0 and / ± log. Air)dr .0 dr bdr = = log, Air)] = log and hence we have /1(0) Ait) ait) AiQ) a(0) ait). (1.26) Another formula can be obtained from formula (1.24) written as Ait)bf  Aif). Integrating between the limits 0 and n, we obtain Ait)5,dt = Ait)dt = AOJo = («) (0). (1.27) Formula (1.27) has a rather interesting verbal 1 1 1 11 . The term Ain)  /4(0) is the amount of interest earned over n measurement periods. The differential expression Ait)bfdt may be inte reted as the amount of interest earned on amount Ait) at exact time t because of the force of interest 6,. When this expression is integrated between limits 0 and n, it gives the total amount of interest earned over the n periods. Further insight into the nature of the force of interest may be gained by analyzing formula (1.24) in terms of the definition of the derivative. The derivative of Ait) may be expressed as Ait) = LJl 0 h and bj from formula (1.24) may then be written 6 = = Ait+ h)Ait) Ait) hAit) (1.28) Now the expression  may be regarded as the rate of interest based upon the interest earned during the interval from time t to time t + h. For example, if = 1, we have +1) it) Lich is one periods mcrement in the fund divided by the amount in the fund at the beginning of the period. If = 1/2, we have 2 • +1/2)/1(0 . period s increment in the fund divided by the amount in the fund at the beginning of the period. As h approaches 0, the limit of this expression, the force of interest, may be described as the nominal rate of interest based upon the intensity of interest at time t.
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