Homework 7
October 25, 2022
1. (Problem 26.1) Smith receives $400 in 1 year, $800 in 2 years, $1,200 in 3 years and so on until the final payment of $4,000. Using an effective interest rate of 6%, determine the present value of these payments at time 0.
2. (Problem 26.21) A perpetuity costs 77.1 and makes annual payments at the end of the year. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, . . . , n at the end of year (n+1): After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%. Calculate n.
3. (Problem 26.27) The present value of a 25-year annuity-immediate with a first payment of 2500 and decreasing by 100 each year thereafter is X. Assuming an annual effective interest rate of 10%, calculate X.
4. (Problem 37.4) Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. What is the outstanding balance at the end of 10 years immediately after the 120th payment?
5. (Problem 37.9) A loan is being repaid with 20 payments of 1,000. The total interest paid during the life of the loan is 5,000. Calculate the amount of the loan.
6. (Problem 37.15) A 20-year loan of 1,000 is repaid with payments at the end of each year. Each of the first ten payments equals 150% of the amount of interest due. Each of the last ten payments is X. The lender charges interest at an annual effective rate of 10%. Calculate X.
Hint for #6: It’s a bit annoying to calculate the amounts of the first 10 payments. Instead, you can reason in this way–at each time, the payment is 150% of the interest due. Then 50% of the payment is used to reduce the principal. If the loan balance at time t is Bt, then the interest payment is 0.1Bt and the loan payment is then 0.05Bt. Therefore, the principal is reduced by 5% with each of the first 10 payments. You can then calculate the loan balance at t = 10.
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