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Loan Payments

21. Loan Payments. If you take out an $8,000 car loan that calls for 48 monthly payments at an APR of 10 percent, what is your monthly payment? What is the effective annual interest rateon the loan?

22. 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?

23. 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?

24. 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?

25. 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 $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 example, 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?

26. Annuity Due. Recall that an annuity due is like an ordinary annuity except that the first payment is made immediately instead of at the end of the first period.

a. Why is the present value of an annuity due equal to (1 + r) times the present value of an ordinary annuity?

b. Why is the future value of an annuity due equal to (1 + r) times the future value of an ordinary annuity?

27. Rate on a Loan. If you take out an $8,000 car loan that calls for 48 monthly payments of $225 each, what is the APR of the loan? What is the effective annual interest rate on the loan?

28. Loan Payments. Reconsider the car loan in the previous question. What if the payments are made in four annual year-end installments? What annual payment would have the same present value as the monthly payment you calculated? Use the same effective annual interest rate

as in the previous question. Why is your answer not simply 12 times the monthly payment?

29. 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?

30. Annuity Due Value. Reconsider the previous problem. What if the lease payments are an annuity due, so that the first payment comes immediately? Is it cheaper to buy or lease?

31. Annuity Due. A store offers two payment plans. Under the installment plan, you pay 25 percent down and 25 percent of the purchase price in each of the next 3 years. If you pay the entire bill immediately, you can take a 10 percent discount from the purchase price. Which is a better deal if you can borrow or lend funds at a 6 percent interest rate?

32. Annuity Value. Reconsider the previous question. How will your answer change if the payments on the 4-year installment plan do not start for a full year?

33. Annuity and Annuity Due Payments.

a. If you borrow $1,000 and agree to repay the loan in five equal annual payments at an interest rate of 12 percent, what will your payment be?

b. What if you make the first payment on the loan immediately instead of at the end of the first year?

34. 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 value of this stream of payments?