978-1259285295 Chapter 4 Solution Manual Part 1

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Chapter 04 – Future Value, Present Value, and Interest Rates

Chapter 4

Future Value, Present Value, and Interest Rates

Conceptual and Analytical Problems

1. Compute the future value of $100 at an 8 percent interest rate 5, 10, and 15 years into the

future. What would the future value be over these time horizons if the interest rate were 5

percent? (LO2)

Answer:

Future value in 5 years = $100*(1.08)5 = $146.93

Future value in 10 years = $100*(1.08)10 = $215.89

2. Compute the present value of a $100 investment made 6 months, 5 years, and 10 years from

now at 4 percent interest. (LO2)

Answer:

6 months: Present Value = 100/(1.04)0.5 = $98.06

Remember, you are calculating the present value of an investment to be made in the future.

3. Assuming that the current interest rate is 3 percent, compute the present value of a five-year,

5 percent coupon bond with a face value of $1,000. What happens when the interest rate goes

to 4 percent? What happens when the interest rate goes to 2 percent? (LO2)

Answer:

Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=3%) =

Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=4%) =

The present value falls when the interest rate rises to 4 percent.

Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=2%) =

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Chapter 04 – Future Value, Present Value, and Interest Rates

The present value rises when the interest rate falls to 2 percent.

4. *Given a choice of two investments, would you choose one that pays a total return of 30

percent over five years or one that pays 0.5 percent per month for five years? (LO1)

Answer: To compare the investments, you need to measure their returns in the same units.

One option would be to convert both these returns to annual rates. The first investment gives

Alternatively, you could convert the first investment to a monthly return:

A third option would be to convert the monthly rate on the second investment into a 5-year

5. A financial institution offers you a one-year certificate of deposit with an interest rate of 5

percent. You expect the inflation rate to be 3 percent. What is the real return on your

deposit? (LO3)

6. Consider two scenarios. In the first, the nominal interest rate is 6 percent and the expected

rate of inflation is 4 percent. In the second, the nominal interest rate is 5 percent and the

expected rate of inflation is 2 percent. In which situation would you rather be a lender? In

which would you rather be a borrower? (LO3)

Answer: In the first scenario the real interest rate is 2 percent (the difference between the

nominal interest rate and the expected inflation rate) and in the second the real interest rate is

7. You decide you would like to retire at age 65, and expect to live until you are 85 (assume

there is no chance you will die younger or live longer). You figure that you can live nicely

on $50,000 per year. (LO3)

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Chapter 04 – Future Value, Present Value, and Interest Rates

a. Describe the calculation you need to make to determine how much you must save to

purchase an annuity paying $50,000 per year for the rest your life. Assume the

interest rate is 7 percent.

b. If you want to keep your purchasing power constant, how would your calculation

change if you expected inflation to average 2 percent for the rest of your life?

Answer:

b. If you want to have $50,000 in purchasing power for each year of your retirement,

you would need to calculate:

8. Most businesses replace their computers every two to three years. Assume that a computer

costs $2,000 and that it fully depreciates in 3 years, at which point it has no resale value and

is thrown away. (LO1)

a. If the interest rate for financing the equipment is equal to i, show how to compute the

minimum annual cash flow that a computer must generate to be worth the purchase.

Your answer will depend on i.

b. Suppose the computer did not fully depreciate but still had a $250 value at the time it

was replaced. Show how you would adjust the calculation given in your answer to

part (a).

c. What if financing can only be had at a 10 percent interest rate? Calculate the

minimum cash flow the computer must generate to be worth the purchase using your

answer to part (a).

Answer:

a. If x = minimum annual cash flow:

9. Some friends of yours have just had a child. Thinking ahead, and realizing the power of

compound interest, they are considering investing for their child’s college education, which

will begin in 18 years. Assume that the cost of a college education today is $125,000. Also

assume there is no inflation and no tax on interest income used to pay college tuition and

expenses. (LO2)

a. If the interest rate is 5 percent, how much money will your friends need to put into

their savings account today to have $125,000 in 18 years?

b. What if the interest rate were 10 percent?

c. The chance that the price of a college education will be the same 18 years from now

as it is today seems remote. Assuming that the price will rise 3 percent per year, and

that today’s interest rate is 8 percent, what will your friend’s investment need to be?

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Chapter 04 – Future Value, Present Value, and Interest Rates

d. Return to part (a), the case with a 5 percent interest rate and no inflation. Assume that

your friends don’t have enough financial resources to make the entire investment at

the beginning. Instead, they think they will be able to split their investment into two

equal parts, one invested immediately and the second invested in five years. Describe

how you would compute the required size of the two equal investments, made five

years apart.

Answer:

a. PV = $125,000/(1.05)18 = $51,940.08

c. If the price rises 3 percent per year, the cost of a college education in 18 years will be:

d. If x is the size of each investment:

10. You are considering buying a new house, and have found that a $100,000, 30-year fixed-rate

mortgage is available with an interest rate of 7 percent. This mortgage requires 360 monthly

payments of approximately $651 each. If the interest rate rises to 8 percent, what will

happen to your monthly payment? Compare the percentage change in the monthly payment

with the percentage change in the interest rate. (LO1)

Answer: If the annual interest rate is 8 percent, then the monthly rate is (1.08)1/12 – 1 =

0.006434

Using the equation from Appendix 4A:

11. *Use the Fisher equation to explain in detail what a borrower is compensating a lender for

when he pays her a nominal rate of interest. (LO3)

Answer: The Fisher equation illustrates that the nominal interest rate (i) can be broken down

into two components where i = r + пe. Taking i to be an annual interest rate, the borrower is

12. If the current interest rate increases, what would you expect to happen to bond prices?

Explain. (LO2)

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Chapter 04 – Future Value, Present Value, and Interest Rates

Answer: Interest rates and bond prices are inversely related so bond prices will fall when

13. Which would be most affected in the event of an interest rate increase– the price of a

five-year coupon bond that paid coupons only in years 3, 4, and 5 or the price of a five-year

coupon bond that paid coupons only in years 1, 2, and 3, everything else being equal?

Explain. (LO2)

Answer: The price of the bond with the later payments will fall by relatively more. The

payments are made further into the future, so the change in the interest rate has a greater

14. Under what circumstances might you be willing to pay more than $1,000 for a coupon bond

that matures in three years, has a coupon rate of 10 percent, and a face value of $1,000?

(LO2)

Answer: If the interest rate in the market were less than 10 percent, the present value of the

payment flows associated with the bond would be higher than $1000. You can use the

15. *Approximately how long would it take for an investment of $100 to reach $800 if you

earned 5 percent? What if the interest rate were 10 percent? How long would it take an

investment of $200 to reach $800 at an interest rate of 5 percent? Why is there a difference

between doubling the interest rate and doubling the initial investment? (LO1)

Answer: Using the rule of 72, we know that if the interest rate is 5 percent, it will take

(You can check that your calculations are approximately correct using the future value

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Chapter 04 – Future Value, Present Value, and Interest Rates

16. Rather than spending $100 today on paint today, you decide to save the money until next

year, at which point you will use it to paint your room. If a can of paint costs $10 today, how

many cans will you be able to buy next year if the nominal interest rate is 21 percent and the

expected inflation rate is 10 percent? (LO3)

Answer: Saving $100 today means forgoing 10 cans of paint. Since the funds grow in

nominal terms by 21 percent, $121 will be available in one year. Expected inflation is 10

percent, so the anticipation is that can of paint in one year will cost $11. Thus, the number of

17. Recently, some lucky person won the lottery. The lottery winnings were reported to be $85.5

million. In reality, the winner got a choice of $2.85 million per year for 30 years or $46

million today. (LO2)

a. Explain briefly why winning $2.85 million per year for 30 years is not equivalent to

winning $85.5 million.

b. The evening news interviewed a group of people the day after the winner was

announced. When asked, most of them responded that, if they were the lucky winner,

they would take the $46 million up-front payment. Suppose (just for a moment) that

you were that lucky winner. How would you decide between the annual installments

or the up-front payment?

Answer:

a. $2.85 million per year is not equivalent to winning $85.5 million because of the time

b. I would calculate which payment option gave me the highest present value. I would

look at the market to determine the appropriate interest rate to use and calculate the

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