Problem 1: Let the effective annual rate be 5 percent (i.e., r = .05) for all maturities.

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a. Calculate the present value of a perpetuity that makes annual payments of $1,000,000 every year forever, with the next payment being made exactly one year from now.

b. Calculate the present value of a perpetuity that makes annual payments of $1,000,000 every year forever, with the next payment being made exactly ten years from now.

c. Calculate the present value of an annuity that makes annual payments of $1,000,000 every year for 9 years, with the next payment being made exactly one year from now.

d. Calculate the present value of a perpetuity that makes a payment of $1,000,000 every 6 months, with the next payment being made in exactly 6 months from now. Hint: Use the standard perpetuity formula but let the “period” be six months instead of a year and use the effective 6-month rate implied by the annual effective rate of 5 percent.

e. Calculate the present value of an annuity that makes a payment of $1,000,000 every other year for 10 payments with the first payment being made exactly two years from now. Hint: Use the standard annuity formula, but let the “period” be two years (rather than just one year) and use the effective two-year rate implied by the annual effective rate of 5 percent.

f. Calculate the present value of an annuity that makes a payment of $1,000,000 every other year for 10 payments with the first payment being made exactly one year from now. Hint: How is the present value of this annuity related to the annuity value in e above?

Problem 2: Let the interest rate be 10 percent (r = .10) at all maturities.

a. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 7 percent (g = .07).

b. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9 percent (g = .09).

c. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9.5 percent (g = .095).

d. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9.9 percent (g = .099).

e. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 10.5 percent (g = .105). Hint: Consider the trend in the present values as the growth rate increases by comparing your answers to a through d. If you use the standard growing perpetuity formula when g > r, you get a silly answer. Using your intuition from your answers to a through d, what is the real answer?

Problem 3: You want to buy a new car for $30,000 and want to finance $20,000 of the purchase price (you will put $10,000 down out of your own pocket). You want to get an idea of the difference between the monthly payments on a 4-year (i.e., 48-month) versus a 5-year (60-month) loan. The stated rate on the 4-year loan is 3 percent (.03) while the stated rate on the 5-year loan is 3.5 percent (.035); both are compounded monthly. What are the monthly payments of each of these loans? Both loans require the first payment to be made exactly one month from now.

Some loans allow you to make payments every half month rather than every month. Consider the 5-year loan above. If instead of making 60 monthly payments (with the first payment exactly a month from now) you make 120 half-monthly payments (with the first payment exactly one half month from now), what is the difference in the total monthly payment between these two cases?

Problem 4: The effective annual discount rate is 5% (r = .05) at all maturities. What is the present value of a stream of payments that starts at $100 at t = 1 (time in years) and grows at 3 percent for 5 years (to t = 6), then growing at 10 percent for 10 years (to t = 16), then growing thereafter at 1 percent?

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