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35

d) How much of the $1500 should be considered as principal?

e) What is the sinking ftind balance at die end of die 1 Idi year?

21. A loan of $1000 is being repaid with level annual payments of $120 plus a smaller final payment made one year after the last regular payment. The effective rate of interest is 8 %. Show algebraically and verbally that the outstanding loan balance after the fifth payment has been made is: a) 1000(1.08)-12055.

b) 1000 - 405j.

22. Show diat a

n\i&j

1 + is

23. On a loan of $10,000 interest at 9% effective must be paid at the end of each year. The borrower also deposits $X at die beginning of each year into a sinking fiind earning 7% effective. At the end of 10 years the sinking fiind is exacdy sufficient to pay off die loan. Calculate X.

24. A borrower is repaying a loan with 10 annual payments of $1000. Half of the loan is repaid by the amortization mediod at 5% effective. The odier half of the loan is repaid by the sinking fimd method in which the lender receives 5 % effective on die investment and the sinking fund accumulates at 4% effective. Find the amount of the loan. Answer to the nearest dollar.

25. A borrows $12,000 for 10 years and agrees to make semiannual payments of $1000. The lender receives 12% convertible semiannually on the investment each year for the first 5 years and 10% convertible semiannually for the second 5 years. The balance of each payment is invested in a sinking fiind earning 8% convertible semiannually. Find the amount by which the sinking fiind is short of repaying the loan at the end of the 10 years. Answer to the nearest dollar.

26. a) A borrower takes out a loan of $3000 for 10 years at 8% convertible

semiaimually. The borrower replaces one third of the principal in a sinking fimd earning 5% convertible semiannually and the other two thirds in a sinking fimd earning 7% convertible semiaimually. Find the total semiannual payment.

b) Rework (a) if the borrower each year puts one third of the total sinking fiind deposit into die 5% sinking fiind and die odier two dtirds into die 7% sinking fimd.

c) Justify from general reasoning the relative magrutude of die answers to (a) and (b).

27. A payment of $36,000 is made at the end of each year for 31 years to repay a loan of $400,000. If the borrower replaces the capital by means of a sinking fiind earning 3% effective, find the effective rate paid to the lender on die loan.

28. A 20-year annuity-immediate has a present value of $10,000 where interest is 8% effective for the first 10 years and 7% effective for the second 10 years. An investor buys diis annuity at a price which over die entire period yields 9% on dis purchase price and fiirther allows the replacement of capital by means of a sinking

fund earning 6% for die first 10 years and 5% for die second 10 years. Find an expression for die amount diat is placed in die sinking fiind each year.

29. A loan of 1 yields die lender rate i per period for n periods, while die borrower replaces die capital in a sinking ftmd earning ratey per period. Find expressions for the following if I t < n:

a) Periodic interest paid to die lender.

b) Periodic sinking fund deposit.

c) Interest earned on sinking fiind during period /.

d) Amount in sinking fiind at end of / periods.

e) Net amount of loan at the end of / periods. fi Net interest paid in period t.

g) Principal repaid in period /.

6.5 Differing payment periods and interest conversion periods

30. An investor buys an annuity with payments of principal and interest of $500 per quarter for 10 years. Interest is at die effective rate of 8% per annum. How much interest does die investor receive in total over die lO-year period? Answer to die nearest dollar.

31. A borrows $10,000 for five years at 12% convertible semiannually. A replaces die principal by means of deposits at die end of every year for five years into a sinking fiind which earns 8 % effective. Find die total dollar amount which A must pay over the five-year period to completely repay the loan. Answer to die nearest dollar.

32. A borrower is repaying a loan widi payments of $3000 at die end of every year over an unknown period of time. If the amount of interest in the third installment is $2000, find die amount of principal in die sixdi instalhnent. Assume diat interest is 10% convertible quarterly.

; 33. A borrows $5000 for 10 years at 10% convertible quarterly. A does not pay interest currendy and will pay all accrued interest at the end of 10 years together with the principal. Find the annual sinking fiind deposit necessary to liquidate the loan at the end of 10 years if die sinking fiind earns 7% convertible semiannually.

! 34. A loan of $3000 is being amortized by 20 quarterly payments. Payments 11 and 12 are not made. At die designated time of die 12di payment, die loan is renegotiated so diat the 13di payment is $Aand payments 14, 16, 18, and 20 are each $40 more I*; dian die preceding payment. If die rate of interest is 8% convertible semiannually, i find die value of N which would provide diat die loan be completely repaid in die original time period. Answer to the nearest dollar.

6.6 Varying series of payments

r35. A loan is being repaid widi 10 annual payments. The first payment is equal to die interest due only, die second paym.ent is twice die first, die diird payment is diree times the first, and so forth. Prove that at the rate of interest on die loan



36. A loan is being repaid widi 10 payments. The first payment is 10, die second 9, and so fordi widi die tendi payment being 1. Show diat die amount of interest in the sixth payment is

5-5, .

37. A loan is repaid with payments which start at $200 die first year and increase by $50 per year until a payment of $1000 is made, at which time payments cease. If interest is 4% effective, find die amount of principal in die fourth payment.

38. A borrower is repaying a $1000 loan widi 10 equal payments of principal. Interest at 6% convertible semiannually is paid on die outstanding balance each year. Find die price to yield an investor 10% convertible semiannually.

39. A mortgage of $8000 is repayable in 20 years by semiannual instaUments of $200 each plus interest on die unpaid balance at 5%. Just after the 15th payment die lender sells die mortgage at a price which yields die new lender 6 % and allows the accumulation of a sinking fund to replace die capital at 4%. Assume diat all interest rates are convertible semiannually.

a) Show that die price assuming a level net rehirn every six mondis is

75 s

551 02

+ 6250

= $4412.

1 + ms 02

b) Show that the price assuming a level sinking fund deposit every six months is («551 .03 + 125)555, .02 54453

03 «551 .03 (1 + -03*551 .02)

c) Justify from general reasoning die relative magnimde of die answers to (a) and (b).

40. A borrows $2000 at an effective rate of interest of 10% per annum and agrees to repay the loan with payments at the end of each year. The first payment is to be $400 and each payment diereafter is to be 4% greater dian die preceding payment, with a smaller final payment made one year after the last regular payment.

d) Find the outstanding loan balance at the end of three years. b) Find the principal repaid in the third payment.

41. Two loans for equal amounts are amortized at 4% effective. Loan L is to be repaid by 30 equal annual payments. Loan M is to be repaid by 30 annual payments, each containing equal principal amounts widi die interest portion of each payment based upon the unpaid balance. The payment for loan L first exceeds the payment for loan M at the end of year k. Find k.

42. A has money invested at effective rate 1. At die end of die first year A wididraws 162 1/2% of die interest eamed, at die end of die second year A wididraws 325%

of the interest earned, and so forth with the withdrawal factor increasing in aridimetic progression. At the end of 16 years die fund is exhausted. Find i.

6.7 Amortization with continuous payments

43. A loan of a 251 being repaid widi continuous payments at die annual rate of 1 per annum for 25 years. If i = .05, find die total amount of interest paid during die sixdi dirough die tendi years inclusive.

44. «) Show diat

«;n

b) Verbally interpret die result obtained in (a).

j 45. A loan is being repaid over n periods widi continuous payments at die rate of / per period at time t. Find expressions for die outstanding loan balance at time k,0<k<n:

a) Prospectively.

b) Retrospectively.

; 46. A loan of 1 is being amortized over a 10-year period widi continuous payments which vary in such a fashion that die outstanding loan balance is Unear. The force of interest is 10%. Find:

a) The principal repaid over die first 5 years.

b) The interest paid over die first 5 years.

47. Express formulas (6.14), (6.15), and (6.16) in more general form in which die force of interest varies continuously.

148. It is known diat die remaining undiscounted payout on an insurance claim t periods after the claim was incurred is given by ae

a) If the instantaneous rate of claim payment is P(t), find an expression for P(t).

b) Find die undiscounted total payout on die claim at time 0.

c) Find the present value of the total payout on die claim at time 0, if the force of interest is 5.

d) Find the present value of the remaining payout on the claim at time t, if the force of interest is 5.

6.8 Step-rate amounts of principal

[49. In order to pay off a $2000 loan, payments of $P are made at die end of each quarter. Interest on the first $500 of the unpaid balance is at rate i* = 16%, while interest on die excess is at i = 14 %. If die outstanding loan balance is $1000 at the end of die first year, find P. Answer to the nearest dollar.

50. A loan of $1000 is to be amortized widi quarterly installments of $1( for as long as necessary plus a smaller final payment one quarter after the last regular payment.



P, + k

= 1 for / = 1, 2, . . . ,a - 1,

(1) (2)

Find k. Find J.

51. Consider a loan of $3000 which is being repaid with level monthly payments over 12 months. Interest is computed at 1 1/2% per month on the first $1000 of outstanding loan balance, at 1 1/4% per month on the next $1000, and at 1% per month on any excess over $1000. Find the level payment which will exacdy amortize this loan. (Hint: Assume that the two "crossover" points are / = 5 and r = 9. These can be confirmed as correct from the resulting amortization schedule.)

Miscellaneous problems

52. a) Show diat

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

53. a) Show that if a loan is amortized widi n level payments of R

B, = R{a - v".;,). b) Verbally interpret the result obtained in (a).

54. The original amount of an inheritance was just sufficient at 3 1/2% effective to pay $10,000 at the end of each year for 10 years. The payments were made as scheduled for the first five years even though the fimd achially earned 5 % effective. How much excess interest was in the fiind at the end of the fifth year? Answer to the nearest dollar.

55. An investor is making level payments at the beginning of each year for 10 years to accumulate $10,000 at die end of die 10 years in a bank which is paying 5% effective. At die end of five years die bank drops its interest rate to 4% effective.

a) Find the annual deposit for the first five years.

b) Find the annual deposit for the second five years.

56. A family was making annual payments of on a 10% 30-year mortgage. After making 15 payments they renegotiate to pay off the debt in 5 more years with the lender being satisfied widi 9% effective over die entire period. Find an expression for the revised annual payment.

57. Nine years ago a family incurred a 20-year $80,000 mortgage at 8% effective on which diey were making annual payments. They desire now to make a lump-sum payment of $5000 and to pay off die mortgage in nine more years. Find an expression for die revised annual payment:

a) If die lender is satisfied widi an 8 % yield for the past nine years but insists on a 9% yield for die next nine years.

b) If die lender insists on a 9 % yield during die entire hfe of die mortgage.

58. A loan of $1000 is to be repaid widi equal annual payments at 5% effective over a 10-year period. The borrower may accelerate the amortization of the loan. However, diere is a prepayment penalty of 2% of die excess of any payment over die originally scheduled payment. If die borrower makes a $300 payment at die end of die first year and a $250 payment at die end of die second year, find die outstanding loan balance just prior to die payment at die end of die third year. Answer to nearest dollar.

Interest is computed at 12% convertible quarterly on the first $500 of outstanding loan balance and at 8% convertible quarterly on any excess.

a) Find the principal repaid in the fourth installment.

b) Show that prior to the "crossover" point, the successive principal repayments plus a constant form a geometric progression, i.e.



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