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# Time Value Money

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1. Future Values. You deposit \$1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest?

2. Calculating Interest Rate. Find the annual interest rate.
Present Value Future Value Time Period
100 115.76 3 years
200 262.16 4 years
100 110.41 5 years

3. Present Values. What is the present value of the following cash-flow stream if the interest rate is 5 percent?
Year Cash Flow
1 \$200
2 \$400
3 \$300

4. Calculating Interest Rate. Lenny Loanshark charges "one point" per week (that is, 1 percent per week) on his loans. What APR must he report to consumers? Assume exactly 52 weeks in a year. What is the effective annual rate?

5. 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 rate on the loan?

6. 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?

7. 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?

8. Amortizing Loan. You take out a 30-year \$100,000 mortgage loan with an APR of 8 percent and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?

9. Perpetuities. A local bank will pay you \$100 a year for your lifetime if you deposit \$2,500 in the bank today. If you plan to live forever, what interest rate is the bank paying?

10. Applying Time Value. You invest \$1,000 today and expect to sell your investment for \$2,000 in 10 years.

a. Is this a good deal if the discount rate is 5 percent?
b. What if the discount rate is 10 percent?

11. Calculating Interest Rate. A store will give you a 3 percent discount on the cost of your purchase if you pay cash today. Otherwise, you will be billed the full price with payment due in 1 month. What is the implicit borrowing rate being paid by customers who choose to defer payment for the month?

12. Real versus Nominal Dollars. Your consulting firm will produce cash flows of \$100,000 this year, and you expect cash flow to keep pace with any increase in the general level of prices. The interest rate currently is 8 percent, and you anticipate inflation of about 2 percent.

a. What is the present value of your firm's cash flows for Years 1 through 5?
b. How would your answer to (a) change if you anticipated no growth in cash flow?

13. Rule of 72. Using the Rule of 72, if the interest rate is 8 percent per year, how long will it take for your money to quadruple in value?

#### Solution Preview

Time Value Money
1. Future Values. You deposit \$1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest?

(1,000 x 0.04 x 10) + 1,000 = 1,400
FV = 1,000(1.04)10
FV = 1,480.24
Interest on interest = 1,480.24 - 1,400 = 80.24

2. Calculating Interest Rate. Find the annual interest rate.

Present Value Future Value Time Period
100 115.76 3 years
200 262.16 4 years
100 110.41 5 years

115.76 = 100(1 + r)3
r = 5%
262.16 = 200(1 + r)4
r = 7%
110.41 = 100(1 + r)5
r = 2%

3. Present Values. What is the present value of the following cash-flow stream if the interest rate is 5 percent?

Year Cash Flow
1 \$200
2 \$400
3 \$300

Present value = 200/1.05 + 400/(1.05)2 + 300/1.05)3
= 190.48 + 362.81 + 259.15
= 812.44

4. Calculating Interest Rate. Lenny Loanshark charges "one point" per week (that is, 1 percent per week) on his loans. What APR must he report to consumers? Assume exactly 52 weeks in a year. What is the effective annual rate?

Effective annual interest rate = (1 + R / P)P - 1 ,
where R is the nominal rate and P is the number of compounding periods.
R = 1% x 52 = 52%
EAR = (1 + 0.52/52)52 - 1 = 67.77%

5. Loan Payments. If you take out an \$8,000 ...

#### Solution Summary

This solution is comprised of a detailed explanation to answer how much will you accumulate in your account after 10 years.

\$2.19
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## Calculations of the Time Value of Money

1) You invest \$20,000 today, at a rate of 10% compound quarterly. What will the investment be worth at the end of year twenty?
2) You are offered an annuity that will pay you \$9,000 at the end of each of the next 10 years. What is the maximum amount you would be willing to pay today for this annuity? (Assume you require a 15% rate of return on an investment of this nature.)
3) You have \$15,000 to put down on a new house that cost \$200,000, and you have been quoted the following finance terms by your local banker: 6% Annual Percentage Rate, for 30 years. If you decide to purchase this home, what will your monthly payment be? Additionally, over the life of the loan what would your total interest expense be?
4) You want to start saving for your child's education. You project that your child will need \$170,000 to attend school 15 years from now. If you can earn a rate of return of 10% compounded semi-annually on a given investment, what dollar amount will you need to invest today to ensure your child can attend college?
5) Steaks Galore has \$190,000 in excess cash that it wishes to invest. Bank One offers a certificate of deposit that is paying 10%, compounded monthly. Bank Two offers a certificate of deposit paying 9.5%, compounded daily. In which Bank should the firm opt to invest its' surplus cash? (You must use the EAR formula to solve this problem. In addition, you must show all of your work.) Additionally, what is the nominal and period rate of interest offered by Bank One?
6) You plan on depositing \$3,000 in an account at the end of each of the next 5 years. If the account is paying interest at an annual rate of 10% per year, what will the total value of your investment be at the end of the 10th year?
7) Your Life Insurance Agent is trying to sell you an investment that will pay you \$5,000 a year forever. If your required rate of return is 11% on an investment of this nature, what would you be willing to pay your agent today for this investment opportunity?
8) It is forecasted that you will receive the following cash inflows at the end of the next four years: Year 1 \$1,000, Year 2 \$2,000, Year 3 \$4,000 and Year 4 \$1,000. If upon receipt of these cash inflows, you can re-invest the amounts received at a rate of 10%, what will the total future value of this investment be?
9) While Bob Jones was a student at Tiffin University, he borrowed \$43,063 in student loans at an annual rate of 7 percent. If Bob repays \$500 per month, how long, to the nearest year, will it take him to repay the loan?
10) Company XYZ plans to invest \$5 million to clear a tract of land and to set out some young trees. These trees will mature in 12 years, at which time XYZ plans to sell all the trees at an expected price of \$10 million. What is XYZ's expected rate of return? In addition, given this rate of return, would you recommend that XYZ proceed with the plan? Why or why not.

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