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 value

18. 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 discount

i 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 <f„ is increasing for increasing n if Q < n - i < .

Assuming that Q < d < 1, show that:

a) 0- - d) < \ - dtifO < t < 1.

b) {\ - d) = i - dt if t = i.

c) {\ - d)* > \ - 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) /, =2

4. 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 interest

5. 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 interest

9. 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 interest

13. 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 that

d (i-df

1.8 Nominal rates of interest and discount

28. a) Express d as a function of i-K b) Express as a function of

29. 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 discount

36. 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,<U = -logv".

b) V05, = /i +/2+ • • • +/„.

42. Fund A accumulates at a simple interest rate of 10%. Fund accumulates at a simple discount rate of 5 %. Find the point in time at which the forces of interest on the two funds are equal.

43. An investment is made for one year in a fund whose accumulation function is a second degree polynomial. The nominal rate of interest earned during the first half of the year is 5 % convertible semiannually. The effective rate of interest earned for the entire year is 7%. Find 65.

44. Find an expression for the fraction of a period at which the excess of accumulated values computed at simple interest over compound interest is a maximum.

1.10 Varying interest

45. If bf = .01/, 0 < f < 2, find the equivalent annual effective rate of interest over the interval 0 < f < 2.

46. Find the accumulated value of 1 at the end of 19 years if 5, = .04(1 +0".

47. Find the level effective rate of interest over a three-year period which is equivalent to an effective rate of discount of 8% the first year, 7% the second year, and 6% the third year.

48. a) Find the accumulated value of 1 at the end of n periods where the effective rate

of interest for the hh period, 1 < < n,is defined by

= (1 + r)\\ + 0 - 1. b) Show that the answer to (a) can be written in the form (1 + f)". Find J.

49. The force of interest at time t is f/lOO. Find " ).

50. In Fund X money accumulates at a force of interest

5, = .01/ + .1 for 0 < / < 20.

In Fund Y money accumulates at an annual effective interest rate /. An amount of 1 is invested in each fund for 20 years. The value of Fund X at the end of 20 years is equal to the value of Fund Y at the end of 20 years. Calculate the value of Fund Y at the end of 1.5 years.

51. You are given 5, = --- for 2 < t < 10. For any one year interval between n and n + 1, with 2 < n < 9, calculate the equivalent

52. If the effective rate of discount in year is equal to .Oik + .06 for Jt = 1,2, 3,fmd the equivalent rate of simple interest over the three-year period.

26. If / and d are equivalent rates of simple interest and simple discount over t periods, show that

i-d = idt.

34 The theory of interest Miscellaneous problems

53. a) Show that S =---

, where = denotes approximate equality.

b) Assuming m = 1, find an exact expression for the error in (o) expressed as a series expansion in 5.

54. Show that

,m-l j-m

= 5.

55. Find the following derivatives:

a) p di

b) ±b di

56. Find the following in the form of series expansions:

d) i as a function of d. d) /"as a function of /.

b) d&s & function of /. e) as a function of d.

c) V as a function of 5.

57. Show that

58. Show that

2 4 6

d"

(v"-5) = -(1 + /)(«- 1)!

dv"

59. a) (1) Derive an expression for a{t) assuming 5 is linear and positive,

i.e. 5 = a + br, where a > 0 and b > 0. (2) Find the accumulation factor during the nth period from the date of

investment, 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 of

invesunent, i.e. 1 + /„.

60. Stoodleys formula for the force of interest is

5,=p +

Show that

a-\t) =

1 + r

1 +re

.v. +

1 + r

where Vj = e~* and V2 = eP.

Solution of problems in interest

2.1 INTRODUCTION

The 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 RESULTS

In Chapter 1 answers to many of the examples and exercises involving accumulated values and present values and involving equivalent rates of interest