Chapter 5
Interest Rates
5-1. Your bank is offering you an account that will pay 20% interest in total for a two-year deposit.
Determine the equivalent discount rate for a period length of:
a. Six months.
b. One year.
c. One month.
5-2. Which do you prefer: a bank account that pays 5% per year (EAR) for three years or:
a. An account that pays 2
12%
every six months for three years?
b. An account that pays 7
12%
every 18 months for three years?
c. An account that pays
12%
per month for three years?
5-3. Many academic institutions offer a sabbatical policy. Every seven years a professor is given a
year free of teaching and other administrative responsibilities at full pay. For a professor
earning $70,000 per year who works for a total of 42 years, what is the present value of the
amount she will earn while on sabbatical if the interest rate is 6% (EAR)?
Timeline:
5-4. You have found three investment choices for a one-year deposit: 10% APR compounded
monthly, 10% APR compounded annually, and 9% APR compounded daily. Compute the EAR
for each investment choice. (Assume that there are 365 days in the year.)
For a $1 invested in an account with 10% APR with monthly compounding you will have:
5-5. You are considering moving your money to new bank offering a one-year CD that pays an 8%
APR with monthly compounding. Your current bank’s manager offers to match the rate you
have been offered. The account at your current bank would pay interest every six months. How
much interest will you need to earn every six months to match the CD?
With 8% APR, we can calculate the EAR as follows:
5-6. Your bank account pays interest with an EAR of 5%. What is the APR quote for this account
based on semiannual compounding? What is the APR with monthly compounding?
Using the formula for converting from an EAR to an APR quote:
5-7. Suppose the interest rate is 8% APR with monthly compounding. What is the present value of an
annuity that pays $100 every six months for five years?
5-8. You can earn $50 in interest on a $1000 deposit for eight months. If the EAR is the same
regardless of the length of the investment, determine how much interest you will earn on a $1000
deposit for:
a. 6 months.
b. 1 year.
c. 1
12
years.
12/8
1.05 1 7.593%EAR = − =
5-9. Suppose you invest $100 in a bank account, and five years later it has grown to $134.39.
a. What APR did you receive, if the interest was compounded semiannually?
b. What APR did you receive if the interest was compounded monthly?
The EAR can be calculated as follows:
5-10. Your son has been accepted into college. This college guarantees that your son’s tuition will not
increase for the four years he attends college. The first $10,000 tuition payment is due in six
months. After that, the same payment is due every six months until you have made a total of
eight payments. The college offers a bank account that allows you to withdraw money every six
months and has a fixed APR of 4% (semiannual) guaranteed to remain the same over the next
four years. How much money must you deposit today if you intend to make no further deposits
and would like to make all the tuition payments from this account, leaving the account empty
when the last payment is made?
Timeline:
1
2
8
5-11. You make monthly payments on your mortgage. It has a quoted APR of 5% (monthly
compounding). What percentage of the outstanding principal do you pay in interest each month?
5-12. Capital One is advertising a 60-month, 5.99% APR motorcycle loan. If you need to borrow $8000
to purchase your dream Harley Davidson, what will your monthly payment be?
Timeline:
0
1
2
3
4
60
C
C
C
C
C
62 Berk/DeMarzo, Corporate Finance, Fourth Edition
5-13. Oppenheimer Bank is offering a 30-year mortgage with an EAR of 5
38%.
If you plan to borrow
$150,000, what will your monthly payment be?
Timeline:
0
1
2
3
4
360
C
C
C
C
C
5-14. You have decided to refinance your mortgage. You plan to borrow whatever is outstanding on
your current mortgage. The current monthly payment is $2356 and you have made every
payment on time. The original term of the mortgage was 30 years, and the mortgage is exactly
four years and eight months old. You have just made your monthly payment. The mortgage
interest rate is 63⁄8% (APR). How much do you owe on the mortgage today?
Timeline:
56
57
58
360
0
1
2
304
5-15. You have just sold your house for $1,000,000 in cash. Your mortgage was originally a 30–year
mortgage with monthly payments and an initial balance of $800,000. The mortgage is currently
exactly 181⁄2 years old, and you have just made a payment. If the interest rate on the mortgage is
5.25% (APR), how much cash will you have from the sale once you pay off the mortgage?
First we need to compute the original loan payment
Timeline #1:
0
1
2
3
360
Chapter 5/Interest Rates 63
5-16. You have just purchased a home and taken out a $500,000 mortgage. The mortgage has a 30–
year term with monthly payments and an APR of 6%.
a. How much will you pay in interest, and how much will you pay in principal, during the first
year?
b. How much will you pay in interest, and how much will you pay in principal, during the 20th
year (i.e., between 19 and 20 years from now)?
5-17. Your mortgage has 25 years left, and has an APR of 7.625% with monthly payments of $1449.
a. What is the outstanding balance?
b. Suppose you cannot make the mortgage payment and you are in danger of losing your house
to foreclosure. The bank has offered to renegotiate your loan. The bank expects to get
$150,000 for the house if it forecloses. They will lower your payment as long as they will
receive at least this amount (in present value terms). If current 25-year mortgage interest
rates have dropped to 5% (APR), what is the lowest monthly payment you could make for
the remaining life of your loan that would be attractive to the bank?
a. The monthly discount rate is
5-18. You have an outstanding student loan with required payments of $500 per month for the next
four years. The interest rate on the loan is 9% APR (monthly). You are considering making an
extra payment of $100 today (that is, you will pay an extra $100 that you are not required to
pay). If you are required to continue to make payments of $500 per month until the loan is paid
off, what is the amount of your final payment? What effective rate of return (expressed as an
APR with monthly compounding) have you earned on the $100?
We begin with the timeline of our required payments
Using the annuity spreadsheet to compute the present value, we get the same number:
Chapter 5/Interest Rates 65
you will pay off the loan faster; that is, it will reduce the payments you need to make at the very end of
the loan. How much smaller will the final payment be? With the extra payment, the timeline changes:
5-19. Consider again the setting of Problem 18. Now that you realize your best investment is to prepay
your student loan, you decide to prepay as much as you can each month. Looking at your budget,
you can afford to pay an extra $250 per month in addition to your required monthly payments of
$500, or $750 in total each month. How long will it take you to pay off the loan?
66 Berk/DeMarzo, Corporate Finance, Fourth Edition
We can also use the annuity spreadsheet to solve for N.
N
I
PV
PMT
FV
30.02
0.75 %
20,092.39
0
N
I
PV
PMT
0.75 %
20,092.39
FV
5-20. Oppenheimer Bank is offering a 30-year mortgage with an APR of 5.25% based on monthly
compounding. With this mortgage your monthly payments would be $2,000 per month. In
addition, Oppenheimer Bank offers you the following deal: Instead of making the monthly
payment of $2,000 every month, you can make half the payment every two weeks (so that you
will make 52 ⁄ 2 = 26 payments per year). With this plan, how long will it take to pay off the
mortgage if the EAR of the loan is unchanged?
To compute the number of payments N, we set the PV of the loan payments equal to the original
balance
5-21. Your friend tells you he has a very simple trick for shortening the time it takes to repay your
mortgage by one-third: Use your holiday bonus to make an extra payment on January 1 of each
year (that is, pay your monthly payment due on that day twice). Assume that the mortgage has
an original term of 30 years and an APR of 12%.
a. If you take out your mortgage on January 1 (so that your first payment is due on February 1),
and you make your first extra payment at the end of the first year, in what year will you finish
repaying your mortgage?
b. If you take out your mortgage on July 1 (so the first payment is on August 1), and you make the
extra payment each January, in how many months will you pay off your mortgage?
c. How will the amount of time it takes to pay off the loan given this strategy vary with the interest
rate on the loan?
Let’s consider the payments you will make over the first year, including the extra January
payment. The timeline is:
68 Berk/DeMarzo, Corporate Finance, Fourth Edition
We can solve for N using an annuity calculator, with Excel (NPER function), by trial and error, or
directly as follows:
b. Following the same process in a, but with the new timeline: Given the loan APR, the discount rate
Let’s consider the payments you will make over the first year, including the extra January
payment. The timeline is:
Chapter 5/Interest Rates 69
We can solve for N using an annuity calculator, with Excel (NPER function), by trial and error, or
directly as follows:
5-22. You need a new car and the dealer has offered you a price of $20,000, with the following
payment options: (a) pay cash and receive a $2000 rebate, or (b) pay a $5000 down payment and
finance the rest with a 0% APR loan over 30 months. But having just quit your job and started
an MBA program, you are in debt and you expect to be in debt for at least the next 2
12
years.
You plan to use credit cards to pay your expenses; luckily you have one with a low (fixed) rate of
15% APR (monthly). Which payment option is best for you?
5-23. The mortgage on your house is five years old. It required monthly payments of $1402, had an
original term of 30 years, and had an interest rate of 10% (APR). In the intervening five years,
interest rates have fallen and so you have decided to refinance—that is, you will roll over the
outstanding balance into a new mortgage. The new mortgage has a 30-year term, requires
monthly payments, and has an interest rate of 6 5⁄8% (APR).
a. What monthly repayments will be required with the new loan?
b. If you still want to pay off the mortgage in 25 years, what monthly payment should you
make after you refinance?
c. Suppose you are willing to continue making monthly payments of $1402. How long will it
take you to pay off the mortgage after refinancing?
d. Suppose you are willing to continue making monthly payments of $1402, and want to pay off
the mortgage in 25 years. How much additional cash can you borrow today as part of the
refinancing?
70 Berk/DeMarzo, Corporate Finance, Fourth Edition
a. First, we calculate the outstanding balance of the mortgage. There are 25 × 12 = 300 months
remaining on the loan, so the timeline is as follows.
Timeline #1:
0
1
2
300
1,402
1,402
1,402
To determine the outstanding balance we discount at the original rate, i.e.,
Next we calculate the loan payment on the new mortgage.
Timeline #2:
0
Using the formula for the loan payment:
5-24. You have credit card debt of $25,000 that has an APR (monthly compounding) of 15%. Each
month you pay the minimum monthly payment only. You are required to pay only the
outstanding interest. You have received an offer in the mail for an otherwise identical credit card
with an APR of 12%. After considering all your alternatives, you decide to switch cards, roll
over the outstanding balance on the old card into the new card, and borrow additional money as
well. How much can you borrow today on the new card without changing the minimum monthly
payment you will be required to pay?
Chapter 5/Interest Rates 71
The discount rate on the original card is:
Timeline:
5-25. In 1975, interest rates were 7.85% and the rate of inflation was 12.3% in the United States. What
was the real interest rate in 1975? How would the purchasing power of your savings have
changed over the year?
5-26. If the rate of inflation is 5%, what nominal interest rate is necessary for you to earn a 3% real
interest rate on your investment?
5-27. Can the nominal interest rate available to an investor be significantly negative? (Hint: Consider
the interest rate earned from saving cash “under the mattress.”) Can the real interest rate be
negative? Explain.
5-28. Consider a project that requires an initial investment of $100,000 and will produce a single cash
flow of $150,000 in five years.
a. What is the NPV of this project if the five-year interest rate is 5% (EAR)?
b. What is the NPV of this project if the five-year interest rate is 10% (EAR)?
72 Berk/DeMarzo, Corporate Finance, Fourth Edition
c. What is the highest five-year interest rate such that this project is still profitable?
5-29. Suppose the term structure of risk-free interest rates is as shown below:
a. Calculate the present value of an investment that pays $1000 in two years and $2000 in five
years for certain.
b. Calculate the present value of receiving $500 per year, with certainty, at the end of the next
five years. To find the rates for the missing years in the table, linearly interpolate between
the years for which you do know the rates. (For example, the rate in year 4 would be the
average of the rate in year 3 and year 5.)
c. Calculate the present value of receiving $2300 per year, with certainty, for the next 20 years.
Infer rates for the missing years using linear interpolation. (Hint: Use a spreadsheet.)
a. Timeline:
0
1
2
3
4
5
b. Timeline:
0
1
2
3
4
5
c. Timeline:
Chapter 5/Interest Rates 73
Since the opportunity cot of capital is different for investments of different maturities, we must
use the cost of capital associated with each cash flow as the discount rate for that cash flow.
Unfortunately, we do not have a rate for a number of years, so we linearly interpolate.
13
r 4.37
=
5-30. Using the term structure in Problem 29, what is the present value of an investment that pays
$100 at the end of each of years 1, 2, and 3? If you wanted to value this investment correctly
using the annuity formula, which discount rate should you use?
5-31. What is the shape of the yield curve given the term structure in Problem 29? What expectations
are investors likely to have about future interest rates?
5-32. Suppose the current one-year interest rate is 6%. One year from now, you believe the economy
will start to slow and the one-year interest rate will fall to 5%. In two years, you expect the
economy to be in the midst of a recession, causing the Federal Reserve to cut interest rates
drastically and the one-year interest rate to fall to 2%. The one-year interest rate will then rise to
3% the following year, and continue to rise by 1% per year until it returns to 6%, where it will
remain from then on.
a. If you were certain regarding these future interest rate changes, what two–year interest rate
would be consistent with these expectations?
b. What current term structure of interest rates, for terms of 1 to 10 years, would be consistent
with these expectations?
c. Plot the yield curve in this case. How does the one-year interest rate compare to the 10-year
interest rate?
5-33. Figure 5.4 shows that Johnson and Johnson’s five-year borrowing rate is 1.9% and Xerox’s is
4.0%. Which would you prefer? $500 from Johnson and Johnson paid today or a promise that
the firm will pay you $575 in five years? Which would you choose if Xerox offered you the same
alternative?
5-34. Your best taxable investment opportunity has an EAR of 4%. You best tax-free investment
opportunity has an EAR of 3%. If your tax rate is 30%, which opportunity provides the higher
after-tax interest rate?
5-35. Your uncle Fred just purchased a new boat. He brags to you about the low 7% interest rate
(APR, monthly compounding) he obtained from the dealer. The rate is even lower than the rate
he could have obtained on his home equity loan (8% APR, monthly compounding). If his tax rate
is 25% and the interest on the home equity loan is tax deductible, which loan is truly cheaper?
5-36. You are enrolling in an MBA program. To pay your tuition, you can either take out a standard
student loan (so the interest payments are not tax deductible) with an EAR of 5
1
2
%
or you can
use a tax-deductible home equity loan with an APR (monthly) of 6%. You anticipate being in a
very low tax bracket, so your tax rate will be only 15%. Which loan should you use?
Using the formula to convert an APR to an EAR:
5-37. Your best friend consults you for investment advice. You learn that his tax rate is 35%, and he
has the following current investments and debts:
■ A car loan with an outstanding balance of $5000 and a 4.8% APR (monthly compounding)
■ Credit cards with an outstanding balance of $10,000 and a 14.9% APR (monthly
compounding)
■ A regular savings account with a $30,000 balance, paying a 5.50% EAR
■ A money market savings account with a $100,000 balance, paying a 5.25% APR (daily
compounding)
■ A tax-deductible home equity loan with an outstanding balance of $25,000 and a 5.0% APR
(monthly compounding)
a. Which savings account pays a higher after-tax interest rate?
b. Should your friend use his savings to pay off any of his outstanding debts? Explain.
5-38. Suppose you have outstanding debt with an 8% interest rate that can be repaid anytime, and the
interest rate on U.S. Treasuries is only 5%. You plan to repay your debt using any cash that you
don’t invest elsewhere. Until your debt is repaid, what cost of capital should you use when
evaluating a new risk-free investment opportunity? Why?
5-39. In the summer of 2008, at Heathrow Airport in London, Bestofthebest (BB), a private company,
offered a lottery to win a Ferrari or 90,000 British pounds, equivalent at the time to about
$180,000. Both the Ferrari and the money, in 100-pound notes, were on display. If the U.K.
interest rate was 5% per year, and the dollar interest rate was 2% per year (EARs), how much
did it cost the company in dollars each month to keep the cash on display? That is, what was the
opportunity cost of keeping it on display rather than in a bank account? (Ignore taxes.)
5-40. You firm is considering the purchase of a new office phone system. You can either pay $32,000
now, or $1000 per month for 36 months.
a. Suppose your firm currently borrows at a rate of 6% per year (APR with monthly
compounding). Which payment plan is more attractive?
b. Suppose your firm currently borrows at a rate of 18% per year (APR with monthly
compounding). Which payment plan would be more attractive in this case?
5-41. After reading the Novy-Marz and Rauh paper you decide to compute the total obligation of the
state that you live in. After some research you determine that your state’s promised pension
payments amount to $1 billion dollars annually, and you expect this obligation to grow at 2% per
year. You determine that the riskiness of this obligation is the same as the riskiness of the state’s
debt. Based on the pricing of that debt you determine that the correct discount rate for the
fund’s liabilities is 3% per annum. Currently, based on actuarial calculations using 8% as the
discount rate, the plan is neither over nor underfunded—the value of the liabilities exactly
matches the value of the assets. What is the extent of the true unfunded liability?