30
Chapter 4
The Time Value of Money
4-1. You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs
$5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments
of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would
the timeline differ if you created it from the bank’s perspective?
4-2. You currently have a four-year-old mortgage outstanding on your house. You make monthly
payments of $1500. You have just made a payment. The mortgage has 26 years to go (i.e., it had
an original term of 30 years). Show the timeline from your perspective. How would the timeline
differ if you created it from the bank’s perspective?
0
1
2
3
4
312
4-3. Calculate the future value of $2000 in:
a. Five years at an interest rate of 5% per year.
b. Ten years at an interest rate of 5% per year.
c. Five years at an interest rate of 10% per year.
d. Why is the amount of interest earned in part (a) less than half the amount of interest earned
in part (b)?
a. Timeline:
0
1
2
5
Chapter 4/The Time Value of Money 31
b. Timeline:
0
1
2
10
0
1
2
4-4. What is the present value of $10,000 received:
a. Twelve years from today when the interest rate is 4% per year?
b. Twenty years from today when the interest rate is 8% per year?
c. Six years from today when the interest rate is 2% per year?
a. Timeline:
0
1
2
3
12
10,000
b. Timeline:
0
1
2
3
20
10,000
c. Timeline:
0
1
2
3
4
5
6
PV = ?
10,000
32 Berk/DeMarzo, Corporate Finance, Fourth Edition
4-5. Your brother has offered to give you either $5000 today or $10,000 in 10 years. If the interest
rate is 7% per year, which option is preferable?
Timeline:
0
1
2
3
4
10
PV = ?
10,000
4-6. Consider the following alternatives:
i. $100 received in one year
ii. $200 received in five years
iii. $300 received in ten years
a. Rank the alternatives from most valuable to least valuable if the interest rate is 10% per
year.
b. What is your ranking if the interest rate is only 5% per year?
c. What is your ranking if the interest rate is 20% per year?
a. Option ii > Option iii > Option i
4-7. Suppose you invest $1000 in an account paying 8% interest per year.
a. What is the balance in the account after 3 years? How much of this balance corresponds to
“interest on interest”?
b. What is the balance in the account after 25 years? How much of this balance corresponds to
interest on interest?
4-8. Your daughter is currently eight years old. You anticipate that she will be going to college in 10
years. You would like to have $100,000 in a savings account to fund her education at that time. If
the account promises to pay a fixed interest rate of 3% per year, how much money do you need
to put into the account today to ensure that you will have $100,000 in 10 years?
Timeline:
0
1
2
3
10
PV = ?
100,000
4-9. You are thinking of retiring. Your retirement plan will pay you either $250,000 immediately on
retirement or $350,000 five years after the date of your retirement. Which alternative should you
choose if the interest rate is:
a. 0% per year?
b. 8% per year?
c. 20% per year?
Timeline: Same for all parts
0
1
2
3
4
5
PV = ?
350,000
4-10. Your grandfather put some money in an account for you on the day you were born. You are now
18 years old and are allowed to withdraw the money for the first time. The account currently has
$3996 in it and pays an 8% interest rate.
a. How much money would be in the account if you left the money there until your 25th
birthday?
b. What if you left the money until your 65th birthday?
c. How much money did your grandfather originally put in the account?
a. Timeline:
18
19
20
21
25
0
1
2
3
7
3,996
b. Timeline:
18
19
20
21
65
0
1
2
3
47
c. Timeline:
0
1
2
3
4
18
PV=?
4-11. Suppose you receive $100 at the end of each year for the next three years.
a. If the interest rate is 8%, what is the present value of these cash flows?
b. What is the future value in three years of the present value you computed in (a)?
c. Suppose you deposit the cash flows in a bank account that pays 8% interest per year. What
is the balance in the account at the end of each of the next three years (after your deposit is
made)? How does the final bank balance compare with your answer in (b)?
rate 8%
4-12. You have just received a windfall from an investment you made in a friend’s business. He will be
paying you $10,000 at the end of this year, $20,000 at the end of the following year, and $30,000
at the end of the year after that (three years from today). The interest rate is 3.5% per year.
a. What is the present value of your windfall?
b. What is the future value of your windfall in three years (on the date of the last payment)?
a. Timeline:
0
1
2
3
0
1
2
3
4-13. You have a loan outstanding. It requires making three annual payments at the end of the next
three years of $1000 each. Your bank has offered to allow you to skip making the next two
payments in lieu of making one large payment at the end of the loan’s term in three years. If the
interest rate on the loan is 5%, what final payment will the bank require you to make so that it is
indifferent between the two forms of payment?
Timeline:
0
1
2
3
1,000
1,000
4-14. You have been offered a unique investment opportunity. If you invest $10,000 today, you will
receive $500 one year from now, $1500 two years from now, and $10,000 ten years from now.
a. What is the NPV of the opportunity if the interest rate is 6% per year? Should you take the
opportunity?
b. What is the NPV of the opportunity if the interest rate is 2% per year? Should you take it
now?
Timeline:
0
1
2
3
10
-10,000
1,500
4-15. Marian Plunket owns her own business and is considering an investment. If she undertakes the
investment, it will pay $4000 at the end of each of the next three years. The opportunity requires
an initial investment of $1000 plus an additional investment at the end of the second year of
$5000. What is the NPV of this opportunity if the interest rate is 2% per year? Should Marian
take it?
4-16. Your buddy in mechanical engineering has invented a money machine. The main drawback of
the machine is that it is slow. It takes one year to manufacture $100. However, once built, the
machine will last forever and will require no maintenance. The machine can be built
immediately, but it will cost $1000 to build. Your buddy wants to know if he should invest the
money to construct it. If the interest rate is 9.5% per year, what should your buddy do?
Timeline:
4-17. How would your answer to Problem 16 change if the machine takes one year to build?
38 Berk/DeMarzo, Corporate Finance, Fourth Edition
4-18. The British government has a consol bond outstanding paying £100 per year forever. Assume the
current interest rate is 4% per year.
a. What is the value of the bond immediately after a payment is made?
b. What is the value of the bond immediately before a payment is made?
4-19. What is the present value of $1000 paid at the end of each of the next 100 years if the interest
rate is 7% per year?
Timeline:
4-20. You are head of the Schwartz Family Endowment for the Arts. You have decided to fund an arts
school in the San Francisco Bay area in perpetuity. Every five years, you will give the school $1
million. The first payment will occur five years from today. If the interest rate is 8% per year,
what is the present value of your gift?
Chapter 4/The Time Value of Money 39
4-21. When you purchased your house, you took out a 30-year annual-payment mortgage with an
interest rate of 6% per year. The annual payment on the mortgage is $12,000. You have just
made a payment and have now decided to pay the mortgage off by repaying the outstanding
balance. What is the payoff amount if:
a. You have lived in the house for 12 years (so there are 18 years left on the mortgage)?
b. You have lived in the house for 20 years (so there are 10 years left on the mortgage)?
c. You have lived in the house for 12 years (so there are 18 years left on the mortgage) and you
decide to pay off the mortgage immediately before the twelfth payment is due?
a. Timeline:
12
13
14
15
30
0
1
2
3
18
b. Timeline:
21
22
23
24
30
0
1
2
3
10
c. Timeline:
12
13
14
15
30
4-22. You are 25 years old and decide to start saving for your retirement. You plan to save $5000 at
the end of each year (so the first deposit will be one year from now), and will make the last
deposit when you retire at age 65. Suppose you earn 8% per year on your retirement savings.
a. How much will you have saved for retirement?
b. How much will you have saved if you wait until age 35 to start saving (again, with your first
deposit at the end of the year)?
4-23. Your grandmother has been putting $1000 into a savings account on every birthday since your
first (that is, when you turned 1). The account pays an interest rate of 3%. How much money will
be in the account on your 18th birthday immediately after your grandmother makes the deposit
on that birthday?
Timeline:
0
1
2
3
18
4-24. A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year
and will be $1000. Each year after that, you will receive a payment on the anniversary of the last
payment that is 8% larger than the last payment. This pattern of payments will go on forever. If
the interest rate is 12% per year,
a. What is today’s value of the bequest?
b. What is the value of the bequest immediately after the first payment is made?
a. Timeline:
Chapter 4/The Time Value of Money 41
2
1
4-25. You are thinking of building a new machine that will save you $1000 in the first year. The
machine will then begin to wear out so that the savings decline at a rate of 2% per year forever.
What is the present value of the savings if the interest rate is 5% per year?
Timeline:
0
1
2
3
4-26. You work for a pharmaceutical company that has developed a new drug. The patent on the drug
will last 17 years. You expect that the drug’s profits will be $2 million in its first year and that
this amount will grow at a rate of 5% per year for the next 17 years. Once the patent expires,
other pharmaceutical companies will be able to produce the same drug and competition will
likely drive profits to zero. What is the present value of the new drug if the interest rate is 10%
per year?
Timeline:
0
1
2
3
17
2
4-27. Your oldest daughter is about to start kindergarten at a private school. Tuition is $10,000 per
year, payable at the beginning of the school year. You expect to keep your daughter in private
school through high school. You expect tuition to increase at a rate of 5% per year over the 13
years of her schooling. What is the present value of the tuition payments if the interest rate is 5%
per year? How much would you need to have in the bank now to fund all 13 years of tuition?
42 Berk/DeMarzo, Corporate Finance, Fourth Edition
Timeline:
0
1
2
3
12
13
10,000
10,000(1.05)
0
4-28. A rich aunt has promised you $5000 one year from today. In addition, each year after that, she
has promised you a payment (on the anniversary of the last payment) that is 5% larger than the
last payment. She will continue to show this generosity for 20 years, giving a total of 20
payments. If the interest rate is 5%, what is her promise worth today?
Timeline:
0
1
2
3
20
5,000
5000(1.05)
4-29. You are running a hot Internet company. Analysts predict that its earnings will grow at 30% per
year for the next five years. After that, as competition increases, earnings growth is expected to
slow to 2% per year and continue at that level forever. Your company has just announced
earnings of $1,000,000. What is the present value of all future earnings if the interest rate is 8%?
(Assume all cash flows occur at the end of the year.)
Chapter 4/The Time Value of Money 43
Timeline:
0
1
2
3
4
5
6
7
4-30. Ten years ago Diana Torres wrote what has become the leading Tort textbook. She has been
receiving royalties based on revenues reported by the publisher. These revenues started at $1
million in the first year, and grew steadily by 5% per year. Her royalty rate is 15% of revenue.
Recently, she hired an auditor who discovered that the publisher had been under reporting
revenues. The book had actually earned 10% more in revenues than had been reported on her
royalty statements.
a. Assuming the publisher pays an interest rate of 4% on missed payments, how much money
does the publisher owe Diana?
b. The publisher is short of cash, so instead of paying Diana what is owed, the publisher is
offering to increase her royalty rate on future book sales. Assume the book will generate
revenues for an additional 20 years and that the current revenue growth will continue. If
Diana would otherwise put the money into a bank account paying interest of 3%, what
royalty rate would make her indifferent between accepting an increase in the future royalty
rate and receiving the cash owed today.