back start next[start] [1] [2] [3] [4] [5] [6] [ 7 ] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] 7 15. Show that the ratio of the accumulated value of 1 invested at rate ; for n periods, to the accumulated value of 1 invested at rate J for n periods, i > J, is equal to the accumulated value of 1 invested for n periods at rate r. Find an expression for as a function of ; and /.16. At a certain rate of compound interest, 1 will increase to 2 in a years, 2 will increase to 3 in i years, and 3 will increase to 15 in years. If 6 will increase to 10 in n years, express n as a function of a, b, and c.17. An amount of money is invested for one year at a rate of interest of 3 % per quarter. Let D(k) be the difference between the amount of interest earned On a compound interest basis and on a simple interest basis for quarter k, where k= 1, 2, 3, 4. Find the ratio of I>(4) to D(3).1.6 Present value18. Find an expression for the discount factor during the nth period from the date of investment, i.e. (1 + („)" in terms of the amount function.19. The sum of the present value of 1 paid at the end of n periods and 1 paid at the end of 2 periods is 1. Find (1 + 0"-20. Show that the current value of a payment of 1 made n periods ago and a payment of 1 to be made n periods in the future is greater than 2, if / > 0.21. It is known that an invesunent of \$500 will increase to \$4000 at the end of 30 years. Find the sum of the present values of three payments of \$10,000 each which will occur at the end of 20, 40, and 60 years.1.7 The effective rate of discounti 72. The amount of interest earned on A for one year is \$336, while the equivalent amount of discount is \$300. Find A.a) Find «/5 if the rate of simple interest is 10%.b) Find «/5 if the rate of simple discount is 10%.a) Assuming compound discount, show that d„ is constant for all n.b) Assuming simple discount, show that \ - dt if t > \.This exercise verifies the relative magnitudes Of present values at simple and compound discount over various periods of time.2. a) Prove that A(n) - (0) = /, + + • • • + /„. b) Verbally interpret the result obtained in (a).3. Find the amount of interest earned between time t and time n, where t < n, if:a) I, = r.b) /, =24. It is known that a(0 is of the form ar + A. If \$100 invested at time 0 accumulates to \$172 at time 3, find the accumulated value at time 10 of \$100 invested at time 5.1.3 The effective rate of interest5. Assume that/4(0 = lOO + 5/.a) Find /5.b) Find / -6. Assume that A{f) = 100(1.1).a) Find /5.b) Find 1,0-7. Show that A(n) = (1 + i„)A(n - 1).8. If/4(4) = 1000 and i„ = .Oln, find /4(7).1.4 Simple interest9. a) At what rate of simple interest will \$500 accumulate to \$615 in 2 1/2 years?b) In how many years will \$500 accumulate to \$630 at 7.8% simple interest?10. If j. is the rate of simple interest for period k, where = 1, 2, . . . , n, show that a{n) - a(0) = jj + + • • • + („.11. At a certain rate of simple interest \$1000 will accumulate to \$1110 after a certain period of time. Find the accumulated value of \$500 at a rate of simple interest three fourths as great over twice as long a period of time.12. Simple interest of j = 4% is being credited to a fund. In which period is this equivalent to an effective rate of 2 1/2%?1.5 Compound interest13. Assuming thatO < i < 1, show that: a) (1 + 0 < 1 + ft if 0 < f < 1. *) (1 + 0 = 1 + it if t = 1.c) (1 + 0 > 1 + /f if f > 1.This exercise verifies the relative magnitudes of accumulated values at simple and compound interest over various periods of time.14. It is known that \$600 invested for two years will earn \$264 in interest. Find the accumulated value of \$2000 invested at the same rate of compound interest for three years.27. Show thatd (i-df1.8 Nominal rates of interest and discount28. a) Express d as a function of i-K b) Express as a function of29. On occasion, interest is convertible less frequently than once a year. Define j™ and d"" to be nominal annual rates of interest and discount convertible once every m years. Find a formula analogous to formula (1.22a) for this situation.30. Find the accumulated value of \$100 at the end of two years:a) If the nominal annual rate of interest is 6% convertible quarterly.b) If the nominal annual rate of discount is 6% convertible once every four years.31. Derive formula (1.23).32. a) Show that i"" = d"\i + """-b) Verbally interpret the result obtained in (a).33. Given that J" = .1844144 and rf™ = .1802608, find m.34. It is known that,•(4)Find n.;(4)35. If r =-, express v in terms of r.rf(4)1.9 Forces of interest and discount36. Derive formula (1.37).37. Use formula (1.23) to give a third proof of the result that 5 = 5.38. Rank i, i-"\d,d"\ and 6 in increasing order of magnitude, assuming m> I.39. Show that 4«( = "dt A(t) J ,40. a) Obtain an expression for 5, if A(t) = Kab.b) Is formula (1.24) or (1.25) more convenient in this case?41. Show that:a) J5, 0 and b > 0. (2) Find the accumulation factor during the nth period from the date ofinvestment, i.e. 1 + <„. b) (1) Derive an expression for a{t) assuming 5 is exponential and positive,i.e. 6 = ab, where a > 0 and > 0. (2) Find the accumulation factor during the nth period from the date ofinvesunent, i.e. 1 + /„.60. Stoodleys formula for the force of interest is5,=p +Show thata-\t) =1 + r1 +re.v. +1 + rwhere Vj = e~* and V2 = eP.Solution of problems in interest2.1 INTRODUCTIONThe basic principles in the theory of interest are relatively few. In Chapter 1, various quantitative measures of interest were analyzed. Chapter 2 discusses general principles to be followed in the solution of problems in interest. The 05 of this chapter is to develop a systematic approach by which the basic principles from Chapter 1 can be applied to more complex financial transactions.With a thorough understanding of the first two chapters it is possible to solve most problems in interest. Successive chapters have two main 05 :1. To familiarize the reader with more complex types of financial transactions, including definitions of terms, which occur in practice.2. To provide a systematic analysis of these financial transactions, which will often lead to a more efficient handling of the problem than resorting to basic principles.As a result of the second above, on occasion, simplifying formulas will be derived. Fortunately, the number of formulas requiring memorization is small. Even so, a common source of difficulty for some is blind reliance on formulas without an understanding of the basic principles upon which the formulas are based. It is important to realize that problems in interest can generally be solved from basic principles and that in many cases resorting to basic principles is not as inefficient as it may first appear to be.2.2 OBTAINING NUMERICAL RESULTSIn Chapter 1 answers to many of the examples and exercises involving accumulated values and present values and involving equivalent rates of interest[start] [1] [2] [3] [4] [5] [6] [ 7 ] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]