91Ó°ÊÓ

Annuity Values. a. What is the present value of a 3 -year annuity of \(\$ 100\) if the discount rate is 8 percent? b. What is the present value of the annuity in (a) if you have to wait 2 years instead of 1 year for the payment stream to start?

Annuities and Interest Rates. Professor's Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of \(\$ 80,000\) at age \(65,\) the firm will pay the retiring professor \(\$ 600\) a month until death. a. If the professor's remaining life expectancy is 20 years, what is the monthly rate on this annuity? What is the effective annual rate? b. If the monthly interest rate is .5 percent, what monthly annuity payment can the firm offer to the retiring professor?

Annuity Values. You want to buy a new car, but you can make an initial payment of only \(\$ 2,000\) and can afford monthly payments of at most \(\$ 400\) a. If the APR on auto loans is 12 percent and you finance the purchase over 48 months, what is the maximum price you can pay for the car? b. How much can you afford if you finance the purchase over 60 months?

Calculating Interest Rate. In a discount interest loan, you pay the interest payment up front. For example, if a 1 -year loan is stated as \(\$ 10,000\) and the interest rate is 10 percent the borrower "pays" \(.10 \times \$ 10,000=\$ 1,000\) immediately, thereby receiving net funds of \(\$ 9,000\) and repaying \(\$ 10,000\) in a year a. What is the effective interest rate on this loan? b. If you call the discount \(d\) (for cxample, \(d=10 \%\) using our numbers), express the effective annual rate on the loan as a function of \(d\) c. Why is the effective annual rate always greater than the stated rate \(d\) ?

Annuity Value. Your landscaping company can lease a truck for \(\$ 8,000\) a year (paid at yearend) for 6 years. It can instead buy the truck for \(\$ 40,000\). The truck will be valueless after 6 years. If the interest rate your company can earn on its funds is 7 percent, is it cheaper to buy or lease?

Annuity Valuc. The \(\$ 40\) million lottery payment that you just won actually pays \(\$ 2\) million per year for 20 years. If the discount rate is 10 percent, and the first payment comes in 1 year, what is the present value of the winnings? What if the first payment comes immediately?

Valuing Delayed Annuities. Suppose that you will receive annual payments of \(\$ 10,000\) for a period of 10 years. The first payment will be made 4 years from now. If the interest rate is 6 percent, what is the present valuc of this stream of payments?

Amortizing Loan. Consider a 4-year amortizing loan. You borrow \(\$ 1,000\) initially, and repay it in four equal annual year-end payments. a. If the interest rate is 10 percent, show that the annual payment is \(\$ 315.47\) b. Fill in the following table, which shows how much of each payment is comprised of interest versus principal repayment (that is, amortization), and the outstanding balance on the loan at cach date.

EAR versus APR. You invest \(\$ 1,000\) at a 6 percent annual interest rate, stated as an APR. Interest is compounded monthly. How much will you have in 1 year? In 1.5 years?

Continuous Compounding. How much will \(\$ 100\) grow to if invested at a continuously compounded interest rate of 10 percent for 6 years? What if it is invested for 10 years at 6 percent?