MATH346

Fall 2020

Classwork 2

Due: 11/08/2020

1. A loan of amount L > 0 is repaid with 72 end of month payments of 500.

The annual e¤ective interest rate is i > 0.

(a) Write an expression for L that is entirely in terms of interest functions

at an annual e¤ective interest rate i.

(b) Suppose i = 7%. Use your formula you determined in (a) to compute

the loan amount L.

(c) Repeat (b) with values i = 8%; 9%; 10%; 12%. What do you think is

happening to L as i increases?

2. A perpetuity has level payments of 100 at the end of years 5, 10, 15, 20,

25, 30, …. The price of the perpetuity is calculated based on an annual e¤ective

interest rate of i > 0.

(a) Write an expression for the present value of the perpetuity that is entirely

in terms of interest functions at an annual e¤ective interest rate i.

(b) Suppose i = 9%. What is the present value of the perpetuity?

(c) Suppose the present value is 1000. What is i?

3. At time t = 0, Karl buys three perpetuities which pay according to the

following.

(i.) Perpetuity X pays 1 at times 2, 4, 6, 8, 10, 12, ….

(ii.) Perpetuity Y pays 2 at times 3, 6, 9, 12, 15, ….

(iii.) Perpetuity Z pays 3 at times 6, 12, 18, 24, 30, ….

Suppose the annual e¤ective interest rate for all three perpetuities is i > 0.

(a) Write an expression for the total present value of the three perpetuities

that is entirely in terms of interest functions at an annual e¤ective interest rate

i.

(b) Compute the present value for i = 5%; 10%; 15%; 20%; 25%.

4. At time t = 0, Karl buys three perpetuities which pay according to the

following.

(i.) Perpetuity X pays 1 at times 2, 4, 6, 8, 10, 12, ….

(ii.) Perpetuity Y pays 2 at times 3, 6, 9, 12, 15, ….

(iii.) Perpetuity Z pays 3 at times 6, 12, 18, 24, 30, ….

Additionally suppose Perpetuity X has an annual e¤ective interest rate of

i > 0, Perpetuity Y has an annual e¤ective interest rate of 2i and Perpetuity Z

has an annual e¤ective interest rate of 3i.

1

(a) What proportion of the total present value is accounted for by Perpetuity

X?

(b) What proportion of the total present value is accounted for by Perpetuity

Y ?

(c) What proportion of the total present value is accounted for by Perpetuity

Z?

(d) Verify that (a)-(c) sum to one.

5. Phil plans to retire in 25 years. Beginning on the day he retires, the

fund will provide monthly payments of 600, 650, 700, 750, 800, 850, ….. and

continuing in this pattern forever. To make this go according to plan, Phil makes

300 monthly deposits of X with the …rst occurring one month from today. The

annual e¤ective interest rate is i > 0.

(a) Find an expression for X that is entirely in terms of interest functions

with annual e¤ective interest rate i.

(b) Evaluate your expression in (a) for i = 1%; 2%; 5%; 10%; 12%.

6. Susan deposits 2000 on January 1 1990. On June 30 1990, Susan withdrawals 500 and then on December 31 1990, Susan withdrawals 500. Susan

repeats the same activity for each of the years 1991 through 2000, inclusive.

Suppose the nominal rate of interest convertible semi-annually is 6%. What is

the balance in Susan’s fund just after her last withdrawal on December 31 2000?

7. Loan of 100,000 is to be repaid by 40 equal end of quarter payments

totaling 120,000. The borrower has the alternative of making 10 equal annual payments with the …rst payment being one year from today. For i =

1%; 2%; 5%; 10%; 12%, …nd the total of the 10 payments under this alternate

repayment plan.

8. Gwen continuously deposits a total of 1000 per year in her savings account

for a total of 15 years. Starting 16 years from now, Gwen begins to make annual

withdrawals of 5000 per year for as long as possible and then make a possibly

smaller …nal withdrawal. The savings account credits interest at an annual

e¤ective rate of 8%.

(a) How many annual withdrawals can Gwen make?

(b) What is the amount of her …nal withdrawal?

2

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