Banking Chapter 5 Homework Interest Rates Your Bank Offering You Account That Will Pay Interest

Chapter 5

Interest Rates

5-1. Your bank is offering you an account that will pay 13% 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.

61

1

5-2. Which do you prefer: a bank account that pays 6% per year (EAR) for three years or:

a. An account that pays 3% every six months for three years?

b. An account that pays 9% every 18 months for three years?

c. An account that pays 0.6% 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 $100,000 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)?

Chapter 5/Interest Rates 57

Timeline:

0

7

14

42

100,000

Because

( )

7

1.06 1.50363=

, the equivalent discount rate for a 7-year period is 50.363%.

Using the annuity formula:

( )

6

100,000 1

PV 1 181,377.62

0.50363 1.50363

= − =







5-4. You have found three investment choices for a one-year deposit: 9% APR compounded monthly,

10% APR compounded annually, and 8% APR compounded daily. Compute the EAR for each

investment choice. (Assume that there are 365 days in the year.)

For an account with 9% APR with monthly compounding you will have:

12

0.09

EAR 1 1 9.38%

12



= + − =





For an account with 10% APR with annual compounding you will have:

( )

1

EAR 1 0.1 1 10%= + − =

For an account with 8% APR with daily compounding you will have:

365

0.08

EAR 1 1 8.33%

365



= + − =





5-5. You are considering moving your money to new bank offering a one-year CD that pays an APR

of 2% 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 2% APR, we can calculate the EAR as follows:

12

0.02



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:

k

APR

1 1.05

k

+=







Solving for the APR:

( )

1

k

APR 1.05 1 k=−

With annual payments k = 1, so APR = 5%

58 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

With semiannual payments k = 2, so APR = 4.939%

With monthly payments k = 12, so APR = 4.889%

5-7. Suppose the interest rate is 6.8% APR with monthly compounding. What is the present value of

an annuity that pays $108 every six months for six years?

12

0.0345 1.0345





5-8. You can earn $38 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. 9 months.

b. 1 year.

c. 1.6 years.

12/8

1000 38

EAR 1 5.75%

1000

+



= − =





a)

9/12

1000(1.0575 1) 42.85−=

b)

1

1000(1.0575 1) 57.54−=

c)

1.6

1000(1.0575 1) 93.64−=

5-9. Suppose you invest $101 in a bank account, and five years later it has grown to $136.4.

a. What APR did you receive, if the interest was compounded semiannually?

b. What APR did you receive if the interest was compounded monthly?

1/5

134.39

EAR 1 6.03%

100



= − =





a) Using the formula for EAR, we can calculate the APR for semiannual compounding.

( )

12

APR 2 1 0.0603 1 5.94%=  + − =

b) Similarly we can calculate the APR for monthly compounding:

( )

112

APR 12 1 0.0603 1 5.87%=  + − =

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 $8500 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 8% (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

Chapter 5/Interest Rates 59

and would like to make all the tuition payments from this account, leaving the account empty

when the last payment is made?

Timeline:

0

1

2

1

4

0

1

2

8

8,500

8% APR (semiannual) implies a semiannual discount rate of

8% 4%

2=

So,

8

8,500 1

PV 1 $57,228.33

0.04 1.02



= − =





5-11. You make monthly payments on your mortgage. It has a quoted APR of 10% (monthly

compounding). What percentage of the outstanding principal do you pay in interest each month?

5-12. Capital One is advertising a 60-month, 6.61% APR motorcycle loan. If you need to borrow

$11,000 to purchase your dream Harley Davidson, what will your monthly payment be?

0

1

2

3

4

C

60

11

1

0.0055 1.0055





−





5-13. Oppenheimer Bank is offering a 30-year mortgage with an EAR of 5.5%. If you plan to borrow

$170,000, what will your monthly payment be?

Timeline:

0

1

2

3

4

360

–170,000

C

5.5% EAR implies a monthly interest rate of 1.0551/12 – 1 = 0.4472%

Using the formula for computing a loan payment

60 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

30 12

170,000

C $951.00

11

1

0.0045 1.0045 





==





−





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 $1991 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 5.537% (APR). How much do you owe on the mortgage today?

Timeline:

56

57

58

360

0

1

2

304

1,991

To find out what is owed, compute the PV of the remaining payments using the loan interest rate to

compute the discount rate:

5.537% APR implies a monthly interest rate of 5.537% / 12 = 0.4614%

304

1991 1

PV 1 $325,036

0.0046 1.0046



= − =





5-15. You have just sold your house for $1,100,000 in cash. Your mortgage was originally a 30-year

mortgage with monthly payments and an initial balance of $750,000. The mortgage is currently

exactly 18.5 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

0

1

2

3

360

Using the formula for a loan payment

360

750,000

C $4,465.78

11

1

0.0044 1.0044





==





−





Now we can compute the PV of continuing to make these payments

Chapter 5/Interest Rates 61

The timeline is

Timeline #2:

222

223

224

225

360

0

1

2

3

138

4,465.78

Using the formula for the PV of an annuity

138

4,465.63 1

PV 1 $461,911.57

0.0044 1.0044



= − =





So, you would keep $1,100,000 – $461,911.57 = $638,088.43.

5-16. You have just purchased a home and taken out a $460,000 mortgage. The mortgage has a 30–

year term with monthly payments and an APR of 6.08%.

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 26 years left, and has an APR of 7.449% 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

$149,638 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 26-year mortgage interest

62 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

rates have dropped to 4.842% (APR), what is the lowest monthly payment you could make

for the remaining life of your loan that would be attractive to the bank?

312

11

1

0.0040 1.0040





−





5-18. You have an outstanding student loan with required payments of $600 per month for the next

four years. The interest rate on the loan is 10% APR (monthly). You are considering making an

extra payment of $175 today (that is, you will pay an extra $175 that you are not required to

pay). If you are required to continue to make payments of $600 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 $175?

We begin with the timeline of our required payments

(1) Let’s compute our remaining balance on the student loan. As we pointed out earlier, the remaining

47

–600

0

2

–600

1

–600

48

–600

64 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

( )

1000 1

362,185 1

0.0020169 1.0020169 N

=−







and solve for N using an annuity calculator, Excel NPER function Excel Goal Seek function, trial and

error, or directly as follows:

1 362,185 0.0020169

N



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?

Timeline #1:

0

1

2

360

100,000

–C

Using the formula for the loan payment,

360

100,000 0.01

C $1,028.61.

1

11.01



==

−







Let’s consider the payments you will make over the first year, including the extra January

Chapter 5/Interest Rates 65

Let’s find the equivalent one time annual payment to these cash flows (as though we only made a

single payment each February). The future value of the above cash flows is the future value of the

1yr =1,028.61

0.01 1.0112 –1

So, the new payment plan is equivalent to paying $14,073.99 at the end of every year. At that

0.12683 1–1

1.12683

è

ø

è

ç

ø

÷

We can solve for N using an annuity calculator, with Excel (NPER function), by trial and error, or

directly as follows:

1

1.12683

æ

è

ø

N

14,073.99 =0.098834

Therefore,

1

2

100,000

360

1

11.01

−







Let’s consider the payments you will make over the first year, including the extra January

payment. The timeline is:

0

1=Aug

6=Jan

7=Feb

12=July

–1028.61

-1028.61

Let’s find the equivalent one time annual payment to these cash flows (as though we only made a

single payment each July). The future value of the above cash flows is the future value of the

monthly annuity plus the future value of the extra January payment:

0.12683 1.12683



We can solve for N using an annuity calculator, with Excel (NPER function), by trial and error, or

directly as follows:

1 $100,000 0.12683

N



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 21⁄2 years.

You plan to use credit cards to pay your expenses; luckily you have one with a low (fixed) rate of

14.51% APR. Which payment option is best for you?

You can use any money that you don’t spend on the car to pay down your credit card debt. Paying

5-23. The mortgage on your house is five years old. It required monthly payments of $1390, 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 5.625% (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 $1390. How long will it

take you to pay off the mortgage after refinancing?

Chapter 5/Interest Rates 67

d. Suppose you are willing to continue making monthly payments of $1390, and want to pay off

the mortgage in 25 years. How much additional cash can you borrow today as part of the

refinancing?

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

1,390

To determine the outstanding balance we discount at the original rate, i.e.,

10 0.8333%.

12 =

300

1,390 1

PV 1 152,965.65

0.0083 1.0083



= − =





Next we calculate the loan payment on the new mortgage.

Timeline #2:

0

1

2

360

152,965.65

–C

The discount rate on the new loan is the new loan rate:

5.625% 4.6875%.

12 =

Using the formula for the loan payment:

152,965.65





152,965.65





5-24. You have credit card debt of $30,000 that has an APR (monthly compounding) of 16%. 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 10%. 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

68 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

well. How much can you borrow today on the new card without changing the minimum monthly

payment you will be required to pay?

Assuming that your current monthly payment is the interest that accrues, it equals $30,000 × 1.33% =

$400

Timeline:

0

1

2

400

5-25. In 1975, interest rates were 7.98% and the rate of inflation was 12.26% 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?

7.98% 12.26% 3.96%

1 1.0798

i

ri



−−

= = = −

+

The purchasing power of your savings declined by 3.96% over the year.

5-26. If the rate of inflation is 5.1%, what nominal interest rate is necessary for you to earn a 2.2%

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 $102,000 and will produce a single cash

flow of $153,000 in five years.

a. What is the NPV of this project if the five-year interest rate is 4.9% (EAR)?

b. What is the NPV of this project if the five-year interest rate is 9.8% (EAR)?

c. What is the highest five-year interest rate such that this project is still profitable?

70 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

r1 = 1.97%

r5 = 3.21%

r6 = 3.47%

r17 = 4.71%

r18 = 4.79%

r19 = 4.87%

( )

1

1i

i

=

+

5-30. Using the term structure in Problem 29, what is the present value of an investment that pays

$110 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 5.7%. One year from now, you believe the economy

will start to slow and the one-year interest rate will fall to 4.7%. 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 1.7%. The one-year interest rate will then rise

74 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition

Because the value of the assets equals the value of the liabilities at this discount rate, this is also the

value of the assets.