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17

22. 23.

At an annual effective interest rate i it is known that:

(/) The present value of 2 at the end of each year for 2n years, plus an

additional 1 at die end of each of die first n years, is 36. (h) The present value of an n-year deferred annuity-immediate paying 2 per year for n years is 6.

Find i.

It is known diat

« 1 +

Find X, y, and z.

Simplify aj (1 -t- v -(- v°) to one symbol.

Find the present value to the nearest dollar on January 1 of an annuity which pays $2000 every six months for five years. The first payment is due on the next April 1 and die rate of interest is 9% convertible semiannually.

3.5 Perpetuities

24. A sum P is used to buy a deferred perpetuity-due of 1 payable annually. The annual effective rate of interest is > 0. Find an expression for die deferred period.

25. Deposits of $1000 are placed into a fund at the beginning of each year for the next 20 years. After 30 years annual payments commence and continue forever, widi die first payment at the end of the 30th year. Find an expression for the amount of each payment.

26. A benefactor leaves an inheritance to four charities. A, B, C, and D. The total inheritance is a series of level payments at the end of each year forever. During the first n years A, B, and share each payment equally. All payments after n years revert to D. If die present values of die shares of A, B, C, and D are all equal, fmd (1 -t- 0".

27. A level perpetuity-immediate is to be shared by A, B, C, and D. A receives the first n payments, the second payments, the third n payments, and D all payments thereafter. It is known that the ratio of the present value of Cs share to As share is .49. Find the ratio of die present value of Bs share to Ds share.

3.6 Nonstandard terms and interest rates

28. Find an expression for the error involved in approximating

29. Compute aj- if i = 5% using the following definitions:

a) Formula (3.22).

b) A payment of .25 at time 5.25.

c) A payment of .25 at time 6.

(1 + iy-l

by k.

a) Formula (3.22).

b) A payment of .25 at time 5.25.

c) A payment of .25 at time 6.

30. In the text annuity symbols were defined only for positive values of n. Find expressions consistent with formulas (3.2) and (3.4) for:

a) a. b)

31. Derive the following formulas for annuity values computed at a negative rate of interest:

3.7 Unknown time

32. A loan of $1000 is to be repaid by annual payments of $100 to commence at die end of the fifdi year and to continue thereafter for as long as necessary. Find die time and amount of the final payment, if the final payment is to be larger than the regular payments. Assume i = 4 1/2%.

33. A fimd of $2000 is to be accumulated by n annual payments of $50, followed by n annual payments of $100, plus a smaller final payment made one year after the last regular payment. If die effective rate of interest is 4 1/2%, find n and die amount of the final irregular payment.

34. One annuity pays 4 at the end of each year for 36 years. Another annuity pays 5 at the end of each year for 18 years. The present values of both annuities are equal at effective rate of interest j. If an amount of money invested at the same rate i will double in n years, find n.

35. A fiind earning 8% effective is being accumulated widi payments of $500 at die beginning of each year for 20 years. Find the maximum number of withdrawals of $1000 which can be made at the ends of years under the condition that once withdrawals start they must continue through the end of the 20-year period.

36. A borrower has the following two options for repaying a loan:

(0 Sixty monflily payments of $100 at the end of each month.

( ) A single payment of $6000 at the end of months. Interest is at the nominal annual rate of 12% convertible monthly. The two options have the same present value. Find K.

3.8 Unknown rate of interest

37. Derive flie following:



+ 2 -

fiJs+\)-fiJs)

widi die two starting values obtained in Example 3.8, i.e. let Jq = .0200 and yl = .0250.

41. A fund of $17,000 is to be accumulated at the end of five years widi payments at die end of each half-year. The first five payments are $1000 each, while the second five payments are $2000 each. Find die nominal rate of interest convertible semiannually earned on die fund.

42. A beneficiary receives a $10,000 life insurance benefit. If die beneficiary uses die proceeds to buy a 10-year annuity-immediate, die annual payout will be $1538. If a 20-year annuity-immediate is purchased, die annual payout will be $1072. Both calculations are based on an annual effective interest rate of i. Find (.

3.9 Varying interest

43. a) Find the present value of an annuity-immediate which pays 1 at the end of each

half-year for five years, if the rate of interest is 8% convertible semiannually for die first three years and 1% convertible semiannually for the last two years.

b) Find the present value of an annuity-immediate which pays 1 at the end of each half-year for five years, if the payments for the first three years are discounted at 8% convertible semiannually and the payments for the last two years are discounted at 1% convertible semiannually.

c) Justify from general reasoning that the answer to {b) is larger than the answer to {a).

44. A loan of P is to be repaid by 10 annual payments beginning 6 months from the date of the loan. The first payment is to be half as large as the others. For the first 4 1/2 years interest is i effective; for the remainder of die term interest is j effective. Find an expression for die first payment.

45. You are given:

(0 X is the current value at time 2 of a 20-year annuity-due of 1 per annum.

(«) The annual effective interest rate for year t is

8 + r

Find X.

Basic annuities 93 (<() The annual effective interest rate for year t is -i- Find X.

3,10 Annuities not involving compound interest

46. Find an expression for - assuming each payment is valued at simple discount rate

47 If ait) =-?-, find an expression for a-, by direcfly taking

log2(/+ 2)-logger+1)

the present value of the payments.

48. Given diat 5, = 2qT7 0. fi"** To]-

49. For time / > 0, die discount function is defined by

a-\t) =

1 +

A five-year annuity has payments of 1 at times r=l,2,3,4,5. A calculates the present value of this annuity at time 0 direcdy. However, first accumulates the payments according to the accumulation function

ait) = 1 + .Qlt.

then multipUes die result by a"(5). By how much do the answers of A and differ?

Miscellaneous problems

50. a) Show diat 5

b) Show diat

51. A loan of $1000 is to be repaid widi annual payments at the end of each year for the next 20 years. For die first 5 years the payments are per year; the second 5 years, 2k per year; the tfiird 5 years, per year; and the fourth 5 years, 4t per year. Find an expression for it.

52. The present value of an annuity-immediate which pays $200 every 6 mondis during die next 10 years and $100 every 6 months during the following 10 years is $4000. The present value of a 10-year deferred annuity-immediate which pays $250 every 6 months for 10 years is $2500. Find the present value of an annuity-immediate which pays $2 every 6 months during the next 10 years and $300 every 6 months during the following 10 years. (Hint: Payments made during the first 10 years are discounted at a different rate dian payments made during the second 10 years.)

38. If Oj =1.75, find an exact expression for i.

39. Rework Example 3.8 using die ad hoc iteration technique illustrated in Example 2.10.

40. Rework Example 3.8 using die iteration formula



= 15

55. Show diat sd

> if ( > 0 and n > 1.

More general annuities

4.1 INTRODUCTION

In Chapter 3 we discussed annuities for which the payment period and the interest conversion period are equal and coincide, and for which the payments are of level amount. In Chapter 4 annuities for which payments are made more or less frequently than interest is convertible and annuities with varying payments will be considered.

4.2 ANNUITIES PAYABLE AT A DIFFERENT FREQUENCY THAN INTEREST IS CONVERTIBLE

We first address annuities for which the payment period and the interest conversion period differ and for which the payments are level. There are two distinct approaches that can be followed in handling such annuities.

The first approach is applicable if the only objective is to compute the numerical value of an annuity and a calculator with exponential and logarithmic fiinctions is available. In this case a two-step procedure can be followed:

1. Find the rate of interest, convertible at the same frequency as payments are made, which is equivalent to the given rate of interest.

2. Using this new rate of interest, find the value of the annuity using the techniques discussed in Chapter 3.

This approach is general and can be used for annuities payable more or less frequently than interest is convertible. Morever, unknown time and unknown rate of interest problems can be handled in this fashion. This direct approach will be illustrated in the examples.

53. A depositor puts $10,000 into a bank account that pays an annual effective interest rate of 4% for 10 years. If a withdrawal is made during die first 5 1/2 years, a penalty of 5 % of die wididrawal amount is made. The depositor wididraws at die end of each of years 4, 5, 6, and 7. The balance in die account at die end of year 10 is $10,000. Find K.

Simplify 52



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