**Chapter 9**

**Time Value of Money**

**Problems**

1. **Future value (LO2)** You invest $2,500 a year for three years at 8 percent.

*a*. What is the value of your investment after one year? Multiply $2,500 × 1.08.

*b*. What is the value of your investment after two years? Multiply your answer to part *a* by 1.08.

*c*. What is the value of your investment after three years? Multiply your answer to part *b* by 1.08. This gives your final answer.

*d*. Confirm that your final answer is correct by going to Appendix A (future value of $1), and looking up the future value for n = 3, and i = 8 percent. Multiply this tabular value by $2,500 and compare your answer to the answer in part *c*. There may be a slight difference due to rounding.

**9-1. Solution:**

a. $2,500 × 1.08 = $2,700

b. $2,700 × 1.08 = $2,916

c. $2,916 × 1.08 = $3,149.28

d. Appendix A (8%, 3 periods)

FV = PV × FVIF

$2,500 × 1.260 = $3,150

2. **Present value (LO3)** What is the present value of:

*a*. $8,000 in 10 years at 6 percent?

*b*. $16,000 in 5 years at 12 percent?

*c*. $25,000 in 15 years at 8 percent?

**Solution:**

3. **Present Value (LO3)**

*a.* What is the present value of $100,000 to be received after 40 years with an 18 percent discount rate?

*b.* Would the present value of the funds in part *a *be enough to buy a $125 concert ticket?

**Solution:**

Appendix B

PV = FV × PVIF (18%, 40 periods)

a. $100,000 × .001 = $100

b. NO. You only have $100 in present value.

4. **Present Value (LO4)** You will receive $4,000 three years from now. The discount rate is 10 percent.

*a*. What is the value of your investment two years from now? Multiply $4,000 × .909 (one year’s discount rate at 10 percent).

*b*. What is the value of your investment one year from now? Multiply your answer to part *a* by .909 (one year’s discount rate at 10 percent).

*c*. What is the value of your investment today? Multiply your answer to part *b* by .909 (one year’s discount rate at 10 percent).

*d*. Confirm that your answer to part *c* is correct by going to Appendix B (present value of $1) for n = 3 and i = 10%. Multiply this tabular value by $4,000 and compare your answer to part *c*. There may be a slight difference due to rounding.

**9-4. Solution:**

5. **Future value (LO2)** If you invest $12,000 today, how much will you have:

*a*. In 6 years at 7 percent?

*b*. In 15 years at 12 percent?

*c*. In 25 years at 10 percent?

*d*. In 25 years at 10 percent (compounded semiannually)?

**9-5. Solution:**

Appendix A

FV = PV × FVIF

a. $12,000 × 1.501 = $ 18,012

b. $12,000 × 5.474 = $ 65,688

c. $12,000 × 10.835 = $130,020

d. $12,000 × 11.467 = $137,604 (5%, 50 periods)

6. **Present value (LO3)** Your aunt offers you a choice of $20,000 in 50 years or $45 today. If money is discounted at 13 percent, which should you choose?

**9-6. Solution:**

7. **Present Value (LO3)** Your uncle offers you a choice of $100,000 in 10 years or $45,000 today. If money is discounted at 8 percent, which should you choose?

**9-7. Solution:**

Appendix B

PV = FV × PVIF (8%, 10 periods)

PV = $100,000 × .463 = $46,300

Choose $100,000 after 10 years.

8. **Present Value (LO3)** In Problem 7, if you had to wait until 12 years to get the $100,000, would your answer change? All other factors remain the same.

**9-8. Solution:**

9. **Present Value (LO3)** You are going to receive $200,000 in 50 years. What is the difference in present value between using a discount rate of 15 percent versus 5 percent?

**9-9. Solution:**

Appendix B

EMBED Equation.DSMT4

The difference is $17,200

EMBED Equation.DSMT4

10. **Present Value (LO3)** How much would you have to invest today to receive:

*a*. $12,000 in 6 years at 12 percent?

*b*. $15,000 in 15 years at 8 percent?

*c*. $5,000 each year for 10 years at 8 percent?

*d*. $40,000 each year for 40 years at 5 percent?

**9-10. Solution:**

11. **Future value (LO2)** If you invest $8,000 per period for the following number of periods, how much would you have?

*a*. 7 years at 9 percent.

*b*. 40 years at 11 percent.

**9-11. Solution:**

Appendix C

FVA = A × FV IFA

a. $8,000 × 9.20 = $ 73,600

b. $8,000 × 581.83 = $ 4,654,640

12. **Future value (LO2)** You invest a single amount of $12,000 for 5 years at 10 percent. At the end of 5 years you take the proceeds and invest them for 12 years at 15 percent. How much will you have after 17 years?

**9-12. Solution:**

13. **Present value (LO3)** Mrs. Crawford will receive $6,500 a year for the next 14 years from her trust. If a 8 percent interest rate is applied, what is the current value of the future payments?

**9-13. Solution:**

Appendix D

PVA = A × PVIFA (8%, 14 periods)

= $6,500 × 8.244 = $53,586

14. **Present value (LO3)** John Longwaite will receive $100,000 in 50 years. His friends are very jealous of him. If the funds are discounted back at a rate of 14 percent, what is the present value of his future “pot of gold”?

**9-14. Solution:**

15. **Present Value (LO3)** Sherwin Williams will receive $18,000 a year for the next 25 years as a result of a picture he has painted. If a discount rate of 10 percent is applied, should he be willing to sell out his future rights now for $160,000?

**9-15. Solution:**

Appendix D

PVA = A × PVIFA (10%, 25 periods)

PVA = $18,000 × 9.077 = $163,386

No, the present value of the annuity is worth more than $160,000.

16. **Present value (LO3)** General Mills will receive $27,500 per year for the next 10 years as a payment for a weapon he invented. If a 12 percent rate is applied, should he be willing to sell out his future rights now for $160,000?

**9-16. Solution:**

17. **Present value (LO3)** The Western Sweepstakes has just informed you that you have won $1 million. The amount is to be paid out at the rate of $50,000 a year for the next 20 years. With a discount rate of 12 percent, what is the present value of your winnings?

**9-17. Solution:**

Appendix D

PVA = A × PVIFA (12%, 20 periods)

PVA = $50,000 × 7.469 = $373,450

18. **Present value (LO3)** Rita Gonzales won the $60 million lottery. She is to receive $1 million a year for the next 50 years plus an additional lump sum payment of $10 million after 50 years. The discount rate is 10 percent. What is the current value of her winnings?

**9-18. Solution:**

19. **Future value (LO2)** Bruce Sutter invests $2,000 in a mint condition Nolan Ryan baseball card. He expects the card to increase in value 20 percent a year for the next five years. After that, he anticipates a 15 percent annual increase for the next three years. What is the projected value of the card after eight years?

**9-19. Solution:**

Appendix A

FV = PV × FVIF (20%, 5 periods)

= $2,000 × 2.488 = $4,976

FV = PV × FVIF (15%, 3 periods)

= $4,976 × 1.521 = $7,568.50

20. **Future value (LO2)** Christy Reed has been depositing $1,500 in her savings account every December since 2001. Her account earns 6 percent compounded annually. How much will she have in December 2010? (Assume that a deposit is made in December of 2010. Make sure to count the years carefully.)

**9-20. Solution:**

21. **Future value (LO2)** At a growth (interest) rate of 8 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer.

**9-21. Solution:**

Appendix A

If the sum is doubling, then the tabular value must equal 2.

In Appendix A, looking down the 8% column, we find the factor closest to 2 (1.999) on the 9-year row. The factor closest to 3 (2.937) is on the 14-year row.

22. **Present value (LO3)** If you owe $30,000 payable at the end of five years, what amount should your creditor accept in payment immediately if she could earn 11 percent on her money?

**9-22. Solution:**

23. **Present value (LO3)** Barney Smith invests in a stock that will pay dividends of $3.00 at the end of the first year; $3.30 at the end of the second year; and $3.60 at the end of the third year. Also, he believes that at the end of the third year he will be able to sell the stock for $50. What is the present value of all future benefits if a discount rate of 11 percent is applied? (Round all values to two places to the right of the decimal point.)

**9-23. Solution:**

Appendix B

PV = FV × PVIF

Discount rate = 11%

$ 3.00 × .901 = $ 2.70

3.30 × .812 = 2.68

3.60 × .731 = 2.63

50.00 × .731 = 36.55

$44.56

24. **Present value (LO3)** Mr. Flint retired as president of Color Title Company but is currently on a consulting contract for $45,000 per year for the next 10 years.

*a*. If Mr. Flint’s opportunity cost (potential return) is 10 percent, what is the present value of his consulting contract?

*b*. Assuming that Mr. Flint will not retire for two more years and will not start to receive his 10 payments until the end of the third year, what would be the value of his deferred annuity?

**Solution:**

**Using a Two Step Procedure**

a.

**Alternative Solution**

b.

Chapter 09: Time Value of Money

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