## Solutions to More Time Value of Money Practice Problems

Prepared by Pamela Peterson Drake
1. How much must I deposit in an account today so that I can withdraw \$100 per year for four years, beginning two years from now, if my deposits earn 5% interest, compounded annually?
• PV of ordinary annuity one year from today is \$100 (3.5460) = \$354.60.
• Value today of the \$354.60 one year from today is \$354.60 (0.9524) = \$337.71.

2. How much must I deposit in an account today so that I can withdraw \$1,000 each year for four years, my deposits earn 4% interest (compounded annually), and my first withdrawal is ten years from today?
• PV of ordinary annuity nine years from today is \$1,000 (3.6299) = \$3,629.90
• Value today of the \$3,629.90 nin years from today is \$3,629.90 (0.7026) = \$2,550.32

3. How much must I deposit in an account each year starting today so that I can withdraw \$1,000 each year for four years, my deposits earn 4% interest (compounded annually), my first withdrawal is ten years from today, and my last deposit is nine years from today?
• PV of ordinary annuity of \$1,000 for four periods is \$3,629.90
• To solve for the payment, consider the \$3,629.90 to be the future value (what you want in the account at the end of the ninth period and solve for cash flows (deposits) that produce this future value.
• FV = \$3,629.90; T = 10 (year 0 through 9); r = 4%
• Annual deposit = \$302.34

4. Suppose Charlie borrows \$100,000 today and must make monthly payments of \$3,874.81 at the end of each month for thirty months. What is the annual percentage rate (APR) on Charlie's loan? What is the effective annual rate (EAR) on Charlie's loan?
• PV = \$100,000
• CF = \$3,874.81
• T = 30
• Solve for the monthly interest rate, r
• r = 1% per month
• APR = 1% x 12 = 12%
• EAR = (1 + 0.01)12 - 1 = 12.68%

5. Calculate the effective annual rate (EAR) on a savings account with an annual percentage rate (APR) of 10% for the following compounding frequencies:
• Semi-annual: r = .05 EAR = 10.25%
• Quarterly: r = .025 EAR = 10.3813%
• Monthly: r = .008333 EAR = 10.4713%
• Daily: r = .000274 EAR = 10.5156%
• Continuous: e0.12 EAR = 10.5171%