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22

120 The theory of interest

A simplified expression can be obtained by performing an integration by parts. We have

)

tvdt

logv

fv logv

nv"

0 logv

0 (log.v)

1 - v" nv"

- nv"

(4.44)

It should be noted that formula (4.44) can also be derived from formula (4.42)

( ) = lim (/(")a)J> = lim L J

« - nv" nv"

In general, if the amount of the payment being made at exact moment t is f{t)dt, then an expression for the present value of an n-period continuous varying annuity would be

"AOvdt. (4.45)

An even more general continuous varying aimuity would not only be one in which the payments are being made continuously and the variations are occurring continuously, but also one in which the force of interest is varying continuously. In this case, a generalized version of formula (4.45) would be

n - ls,dr

me jo dt.

(4.46)

Example 4.18 Find an expression for the present value of a continuously increasing annuity with a term of n years if the force of interest is and if the rate of payment at time t is t per annum.

The answer is obtained by performing an integration by parts

More general annuities 121

te-dt =

+ -0 5 Jo

"te-dt

e-dt

= - Ule" - e"*" - -e~" + -

4.9 SUMMARY OF RESULTS

Table 4.1 summarizes the expressions for present values of level annuities for various payment periods and interest conversion periods. The annuity illustrated pays 60 per annum for 10 years at 12% per annum.

Table 4.1 Summary of Relationships for Level Annuities in Chapter 4

Interest conversion period

Payment period

Annual

Quarterly

Monthly

Continuous

Annual

Quarterly

60 «401 .03 *4l .03

.03

15401

Monthly

j5 «120 .01

5«120 .01

5«T20l

31.01

Continuous

15

60» -

e-

. » - 1



122 The theory of interest EXERCISES

4.2 Annuities payable at a different frequency than interest is convertible

1. Find the accumulated value 18 years after the first payment is made of an annuity on which there are 8 payments of $2000 each made at two-year intervals. The nominal rate of interest convertible semiannually is 7%. Answer to the nearest dollar.

2. Find the present value of a ten-year annuity which pays $400 at the beginning of each quarter for the first 5 years, increasing to $600 per quarter thereafter. The annual effective rate of interest is 12%. Answer to the nearest dollar.

3. A sum of $100 is placed into a fiind at the beginning of every other year for eight years. If the fund balance at the end of eight years is $520, find the rate of simple interest earned by the fund.

4.3 Further analysis of annuities payable less frequently than interest Is convertible

4. Rework Exercise 1 using the approach developed in Section 4.3.

5. Give an expression in terms of functions assuming a rate of interest per month for the present value, 3 years before the first payment is made, of an annuity on which there are payments of $200 every 4 months for 12 years:

a) Expressed as an annuity-immediate.

b) Expressed as an annuity-due.

6. Show that the present value at time 0 of 1 payable at times 7, 11, 15, 19, 23, and 27 is

7. A perpetuity of $750 payable at the end of every year and a perpetuity of $750 payable at the end of every 20 years are to be replaced by an annuity of R payable at the end of every year for 30 years. If = .04, show that

R = 37,500

where all functions are evaluated at 2% interest.

8. Find an expression for the present value of an annuity-due of $600 per annum payable semiannually for 10 years if d = .09.

9. The present value of a peipetuity paying 1 at the end of every three years is 125/91. Find /.

..(»>)

1 +,(").

16. Derive the following formulas analogous to formulas (3.13) and (3.14):

a) d- = a-\l+if-.

b) if = 4>(i.o-.

17. Derive the following formulas analogous to formulas (3.15) and (3.16): a)

(m) ,, , (m)

a- = \lm

..(12) .

- l/m.

18. Express d in terms of with an adjustment factor.

19. a) Show that = """"

b) Verbally 1 1 the result obtained in (a).

20. A sum of $10,000 is used to buy a deferred peipetuity-due paying $5(X) every six months forever. Find an expression for the deferred period expressed as a function of d.

10. Find an expression for the present value of an annuity on which payments are $100 per quarter for five years, just before the first payment is made, if 5 = .08.

11. A peipetuity paying 1 at the beginning of each year has a present value of 20. If this perpetuity is exchanged for another perpetuity paying R at the beginning of every two years, find R so that the values of the two peipetuities are equal.

12. Find an expression for the present value of an annuity on which payments are 1 at the beginning of each 4-month period for 12 years, assuming a rate of interest per 3-month period.

4.4 Further analysis of annuities payable more frequently than Interest is convertible

13. Rework Exercise 2 using the approach developed in Section 4.4.

14. Derive formula (4.11).

15. Derive the following formulas analogous to formulas (3.6) and (3.12):



21- If 3ag = 20=454-. findi.

22. Find an expression for die present value of an annuity which pays 1 at the beginning of each 3-nionth period for 12 years, assuming a rate of interest per 4-month period.

4.5 Continuous annuities

23. Find die value of r, 0 < ; < 1, such that 1 paid at time t is equivalent to 1 paid continuously between time 0 and 1.

24. Show algebraically and verbally that - < < < d- < - where m > 1. /

25. Find an expression for - if 5, = j-i-.

26. There is $40,000 in a fiind which is accumulating at 4% per annum convertible continuously. If money is withdrawn continuously at the rate of $2400 per annum, how long will die fimd last?

27. If = 4 and 1 = 12, find 5.

28. Show diat « ! =""/1, .

an

4.6 Basic varying annuities

29. Verbally inte ret the result obtained in Example 4.13.

30. Simphfy e( + 5)v.

31. Show algebraically, and by means of a time diagram, the following relationship between (1 ) and ( )- :

{ ) =( +\) -{1 ) .

32. The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of die sixth year, decreasing by 1 each year until nothing is paid. Show that the present value is

lO-a, +fl(l -lOQ /

33. Find die present value of a perpetuity under which a payment of 1 is made at die end of the first year, 2 at the end of the second year, increasing until a payment of n is made at the end of the nth year, and thereafter payments are level at n per year forever.

34. A perpetuity-immediate has annual payments of 1, 3, 5, 7. . . If die present value of the sixth and seventh payments are equal, find the present value of die perpetuity.

35. If is the present value of a pe etuity of 1 per year with the first payment at the end of the second year and 20X is the present value of a series of annual payments 1, 2, 3,. . . widi die first payment at die end of die tiiird year, fmd d.

36. An annuity-immediate has semiannual payments of 800, 750, 700, . . . , 350, at P-) = .16. If 08 = f" present value of die annuity in terms of .

37. Annual deposits are made into a fiind at die beginning of each year for 10 years. The first 5 deposits are $1000 each and deposits increase by 5% per year dierafter. If die fund earns 8% effective, find die accumulated value at die end of 10 years. Answer to die nearest dollar.

38. Find die present value of a 20-year annuity widi annual payments which pays $600 immediately and each subsequent payment is 5 % greater dian die preceding payment. The annual effective rate of interest is 10.25%. Answer to die nearest doUar.

4.7 More general varying annuities

39. Derive formula (4.42).

40. a) Find die sum of die payments in (/a)-

b) Find die sum of die payments in {I-af.

41. Show diat (!"W = .

42. Show diat die present value of a perpetuity on which payments are 1 at die end of die 5di and 6tfi years, 2 at die end of die 7di and 8 1 years, 3 at die end of die 9di and 10th years, . . . , is 4

i-vd

43. A perpetuity has payments at die end of each four-year period. The first payment at die end of four years is 1. Each subsequent payment is 5 more dian die previous payment. It is known diat v"* = 0.75. Calculate die present value of diis perpetuity.

44. A 10-year annuity has die following schedule of payments:

On each January 1 ............... 100

On each April 1 ................. 200

On each July 1.................. 300

On each October 1 ............... 400

Show diat die present value of diis annuity on January 1 just before die first payment is made is

45. A perpetuity provides payments every six mondis starting today. The fu-st payment is 1 and each payment is 3% greater dian die immediately preceding payment. Find die present value of die 1 11 if die effective rate of interest is 8% per annum.



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