Chapter 4/The Time Value of Money 43
4-31. Your brother has offered to give you $100, starting next year, and after that growing at 3% for
the next 20 years. You would like to calculate the value of this offer by calculating how much
money you would need to deposit in the local bank so that the account will generate the same
cash flows as he is offering you. Your local bank will guarantee a 6% annual interest rate so long
as you have money in the account.
a. How much money will you need to deposit into the account today?
b. Using an Excel spreadsheet, show explicitly that you can deposit this amount of money into
the account, and every year withdraw what your brother has promised, leaving the account
with nothing after the last withdrawal.
a. The amount to be deposited in the account is $1456.15.
Year Cash flows of Brother’s deal PV of Brother’s deal with 6% discount factor
0 – –
1 100.00$ 94.34$
2 103.00$ 91.67$
3 106.09$ 89.08$
4 109.27$ 86.55$
5 112.55$ 84.10$
6 115.93$ 81.72$
7 119.41$ 79.41$
8 122.99$ 77.16$
9 126.68$ 74.98$
10 130.48$ 72.86$
11 134.39$ 70.80$
12 138.42$ 68.79$
13 142.58$ 66.85$
14 146.85$ 64.95$
15 151.26$ 63.12$
16 155.80$ 61.33$
17 160.47$ 59.59$
18 165.28$ 57.91$
19 170.24$ 56.27$
20 175.35$ 54.68$
1,456.15$
Sum of cash flows with 6% discount factor ->
44 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
b.
Year Payout Remaining Balance
0 –$ 1,456.15$
1 100.00$ 1,443.52$
2 103.00$ 1,427.13$
3 106.09$ 1,406.67$
4 109.27$ 1,381.80$
5 112.55$ 1,352.16$
6 115.93$ 1,317.36$
7 119.41$ 1,276.99$
8 122.99$ 1,230.63$
9 126.68$ 1,177.79$
10 130.48$ 1,117.98$
11 134.39$ 1,050.66$
12 138.42$ 975.28$
13 142.58$ 891.22$
14 146.85$ 797.84$
15 151.26$ 694.45$
16 155.80$ 580.32$
17 160.47$ 454.67$
18 165.28$ 316.67$
19 170.24$ 165.43$
20 175.35$ (0.00)$
4-32. Suppose you currently have $4800 in your savings account, and your bank pays interest at a rate
of 0.5% per month. If you make no further deposits or withdrawals, how much will you have in
the account in four years?
4-33. Your firm spends $4,700 every month on printing and mailing costs, sending statements to
customers. If the interest rate is 0.48% per month, what is the present value of eliminating this
cost by sending the statements electronically?
4-34. You have just entered an MBA program and have decided to pay for your living expenses using
a credit card that has no minimum monthly payment. You intend to charge $1090 per month on
the card for the next 21 months. The card carries a monthly interest rate of 1.09%. How much
money will you owe on the card 22 months from now, when you receive your first statement post–
graduation?
Chapter 4/The Time Value of Money 45
©2017 Pearson Education, Ltd.
Our charges correspond to a 21-month annuity. Therefore, using the PV of an annuity formula, the
present value is:
21
1,090 1
PV 1 $20,360.62
0.0109 1.0109
==
−
Of course, we are not quite done. When we receive our statement in the 22nd month, there will be one
more month’s worth of interest charged. Therefore, we need to calculate the future value at time 22,
i.e.
FV = 20,360.62(1.0109)22 = $25,844.69
4-35. Your credit card charges an interest rate of 1.98% per month. You have a current balance of
$1120, and want to pay it off. Suppose you can afford to pay off $85 per month. What will your
balance be at the end of one year?
We want to compute the future value of our account balance. Let’s begin with the timeline over the
next 12 months:
1
2
12
1120
–85
–85
–85
From the timeline we can see that we need to combine the FV of our current balance with the FV of
our annuity payments of $100 per month:
FVbalance = 1,120(1.0198)12 = $1,417.09
FVpayments =
12
12
1 $1,138.7
85 1 (1.0198)
0.0198 1.0198 5
−
=−
−
Future account balance = $278.35
4-36. You have decided to buy a perpetuity. The bond makes one payment at the end of every year
forever and has an interest rate of 4%. If you initially put $5000 into the bond, what is the
payment every year?
r
4-37. You are thinking of purchasing a house. The house costs $250,000. You have $36,000 in cash that
you can use as a down payment on the house, but you need to borrow the rest of the purchase
price. The bank is offering a 30-year mortgage that requires annual payments and has an
interest rate of 5% per year. What will your annual payment be if you sign up for this mortgage?
Timeline: (From the perspective of the bank)
0
1
2
3
30
–214,000
C
C
C
C
0
1
2
3
46 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
©2017 Pearson Education, Ltd.
30
250, 000 36, 000
C $13, 921.01
11
1
0.05 1.05
−
==
−
4-38. You would like to buy the house and take the mortgage described in Problem 37. You can afford
to pay only $23,500 per year. The bank agrees to allow you to pay this amount each year, yet still
borrow $300,000. At the end of the mortgage (in 30 years), you must make a balloon payment;
that is, you must repay the remaining balance on the mortgage. How much will this balloon
payment be?
Timeline: (where X is the balloon payment.)
0
1
2
3
30
–300,000
23,500
23,500
23,500
23,500 + X
The present value of the loan payments must be equal to the amount borrowed:
( )
30
30
23,500 1 X
300, 000 1
0.07 1.07 1.07
.= − +
Solving for X:
( )30
30
23, 500 1
X 300, 000 1 1.07 $63, 848
0.07 1.07
= − − =
4-39. You have just made an offer on a new home and are seeking a mortgage. You need to borrow
$600,000.
a. The bank offers a 30-year mortgage with fixed monthly payments and an interest rate of
0.5% per month. What is the amount of your monthly payment if you take this loan?
b. Alternatively, you can get a 15-year mortgage with fixed monthly payments and an interest
rate of 0.4% per month. How much would your monthly payments be if you take this loan
instead?
( ) ( )
0.005
1 1.005
$3597.30
rr
+
=
Or, using the annuity calculator:
Chapter 4/The Time Value of Money 47
©2017 Pearson Education, Ltd.
b. Note that a 15-year loan has 15 × 12 = 180 monthly payments. Here is the timeline:
1
2
180
P = 600,000
–C
–C
–C
We can solve for the loan payment using the formula
( ) ( )
360
600,000
1 1 1 1
11
0.004
1 1.004
$4682.49
N
P
C
rr
==
−−
+
=
Or, using the annuity calculator:
4-40. Suppose you take the 30-year mortgage described in Problem 38 part (a). How much will you
still owe on the mortgage after 15 years?
Note that a 30-year loan has 30 × 12 = 360 monthly payments. Here is the timeline:
1
2
360
P = 600,000
–C
–C
–C
We can solve for the loan payment using the formula:
( ) ( )
360
600,000
1 1 1 1
11
0.005
1 1.005
$3597.30
N
P
C
rr
==
−−
+
=
Or, using the annuity calculator:
Now, after 15 years only 15 × 12 = 180 monthly payments have been made, and 180 remain. Here is
the timeline of the remaining payments:
1
2
180
–3597.30
–3597.30
–3597.30
We can solve for the remaining loan amount by calculating the present value of these payments at the
loan rate:
48 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
©2017 Pearson Education, Ltd.
( ) ( )
180
1 1 1 1
11
0
$3597.30 .005
1 1.005
$426,293
N
PCrr
= − = − −
+
=
Or, using the annuity calculator:
Note that over 70% of the original balance remains!
4-41. You are thinking about buying a piece of art that costs $30,000. The art dealer is proposing the
following deal: He will lend you the money, and you will repay the loan by making the same
payment every two years for the next 30 years (i.e., a total of 15 payments). If the interest rate is
9% per year, how much will you have to pay every two years?
15
11
1
0.1881 1.1881
−
4-42. You are saving for retirement. To live comfortably, you decide you will need to save $1 million
by the time you are 65. Today is your 23rd birthday, and you decide, starting today and
continuing on every birthday up to and including your 65th birthday, that you will put the same
amount into a savings account. If the interest rate is 5%, how much must you set aside each year
to make sure that you will have $1 million in the account on your 65th birthday?
50 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
©2017 Pearson Education, Ltd.
First, we calculate the PV of the annuity (at age 39):
7500 1
PV 1 $74, 467.29
26
0.09 1.09
= − =
In FV at age 65, this is equal to 74,467.29(1.09)26 = $699,929.83
Then, we calculate the value of the payment that we can cash out for an annuity that will pay 20 times,
i.e. from the day we turn 66 to the day we turn 85:
20
699, 929.83
C $76, 674.85
11
1
0.09 1.09
==
−
4-45. You have just turned 30 years old, have just received your MBA, and have accepted your first
job. Now you must decide how much money to put into your retirement plan. The plan works as
follows: Every dollar in the plan earns 7% per year. You cannot make withdrawals until you
retire on your 70th birthday. After that point, you can make withdrawals as you see fit. You
decide that you will plan to live to 100 and work until you turn 70. You estimate that to live
comfortably in retirement, you will need $90,000 per year starting at the end of the first year of
retirement and ending on your 100th birthday. You will contribute the same amount to the plan
at the end of every year that you work. How much do you need to contribute each year to fund
your retirement?
Timeline:
30
31
32
70
71
72
100
0
1
2
40
41
42
70
–C
–C
–C
90
90
90
The present value of the costs must equal the PV of the benefits. So begin by dividing the problem into
two parts, the costs and the benefits.
Costs: The costs are the contributions, a 40-year annuity with the first payment in one year:
costs 40
C1
PV 1
0.07 1.07
=−
Benefits: The benefits are the payouts after retirement, a 30-year annuity paying $90,000 per year with
the first payment 41 years from today. The value of this annuity at age 70 is:
70
90,000 1
PV 1 $1.117 million
30
0.07 1.07
= − =
The value today is just this value discounted 40 years:
benefits 40
1,116, 813.71
PV $74, 581.24
1.07
==
Since the PV of the costs must equal the PV of the benefits (or equivalently the NPV of the cash flow
must be zero):
40
C1
74, 581.24 1
0.07 1.07
=−
Chapter 4/The Time Value of Money 51
Thus,
40
74,581.24
C $5,594.27
11
1
0.07 1.07
==
−
4-46. Problem 45 is not very realistic because most retirement plans do not allow you to specify a fixed
amount to contribute every year. Instead, you are required to specify a fixed percentage of your
salary that you want to contribute. Assume that your starting salary is $75,000 per year and it
will grow 2% per year until you retire. Assuming everything else stays the same as in Problem
45, what percentage of your income do you need to contribute to the plan every year to fund the
same retirement income?
Chapter 4/The Time Value of Money 53
Timeline:
0
1
2
3
4
–61,200
18,000
18,000
18,000
18,000
The PV of the car payments is a 4-year annuity:
( )
4
18, 000 1
PV 1
r1r
=−
+
Setting the NPV of the cash flow stream equal to zero and solving for r gives the IRR:
( )
4
18, 000 1
NPV 0 61, 200 1
IRR 1 IRR
= = − + − +
To find the IRR we either need to guess or use the annuity calculator. You can check and see that IRR
= 6.833% solves this equation.
4-50. A local bank is running the following advertisement in the newspaper: “For just $4000 we will
pay you $280 forever!” The fine print in the ad says that for a $4000 deposit, the bank will pay
$280 every year in perpetuity, starting one year after the deposit is made. What interest rate is
the bank advertising (what is the IRR of this investment)?
Timeline:
0
1
2
3
–4,000
280
280
280
The payments are a perpetuity, so:
280
PV r
=
Setting the NPV of the cash flow stream equal to zero and solving for r gives the IRR:
280
NPV 0 4,000 IRR 7%
IRR
= = − + → =
4-51. You are considering purchasing a warehouse. The cost to purchase the warehouse is $492,000.
Renting the equivalent space costs $19,700 per year. If the annual interest rate is 5.6%, at what
rate must rental cost increase each year to make the cost of renting comparable to purchasing?
We can think of the rental cost as a growing perpetuity. Using the formula for the present value of
growing perpetuity, we want to find the growth rate g so that:
19,700 19,700
4-52. The Tillamook County Creamery Association manufactures Tillamook Cheddar Cheese. It
markets this cheese in four varieties: aged 2 months, 9 months, 15 months, and 2 years. At the
shop in the dairy, it sells 2 pounds of each variety for the following prices: $7.50, $9.50, $11.50,
and $12.50, respectively. Consider the cheese maker’s decision whether to continue to age a
particular 2-pound block of cheese. At 2 months, he can either sell the cheese immediately or let
54 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
it age further. If he sells it now, he will receive $7.50 immediately. If he ages the cheese, he must
give up the $7.50 today to receive a higher amount in the future. What is the IRR (expressed in
percent per month) of the investment of giving up $75.00 today by choosing to store 20 pounds of
cheese that is currently 2 months old and instead selling 10 pounds of this cheese when it has
aged 9 months, 6 pounds when it has aged 15 months, and the remaining 4 pounds when it has
aged 2 years?
Timeline:
2
3
9
10
15
16
24
0
1
7
8
13
14
22
–(20)(7.50/2)
(10)(9.50/2)
(6)(11.50/2)
(4)(12.50/2)
The PV of the cash flows generated by storing the cheese is:
( ) ( ) ( )
7 13 22
47.45 34.5 25
NPV 75
1 r 1 r 1 r
= − + + +
+ + +
The IRR is the r that sets the NPV equal to zero:
( ) ( ) ( )
7 13 22
47.5 34.5 25
NPV 75 0
1 IRR 1 IRR 1 IRR
= − + + + =
+ + +
By iteration or by using a spreadsheet, the r that solves this equation is IRR = 3.019% per month.
4-A.1. Your grandmother bought an annuity from Rock Solid Life Insurance Company for $200,000
when she retired. In exchange for the $200,000, Rock Solid will pay her $25,000 per year until
she dies. The interest rate is 5%. How long must she live after the day she retired to come out
ahead (that is, to get more in value than what she paid in)?
Timeline:
0
1
2
3
N
–200,000
25,000
25,000
25,000
25,000
She breaks even when the NPV of the cash flows is zero. The value of N that solves this is:
( )
( )
( ) ( )
( ) ( )
( )
( )
N
N
N
N
25, 000 1
NPV 200, 000 1 0
0.05 1.05
1 200, 000 0.05
1 0.4
25, 000
1.05
11
0.6 1.05 0.6
1.05
N log 1.05 log 0.6
log 0.6
Nlog 1.05
10.5.
= − + − =
− = =
= =
=−
−
=
=
Chapter 4/The Time Value of Money 55
©2017 Pearson Education, Ltd.
So if she lives 10.5 or more years, she comes out ahead.
4-A.2. You are thinking of making an investment in a new plant. The plant will generate revenues of $1
million per year for as long as you maintain it. You expect that the maintenance cost will start at
$50,000 per year and will increase 5% per year thereafter. Assume that all revenue and
maintenance costs occur at the end of the year. You intend to run the plant as long as it continues
to make a positive cash flow (as long as the cash generated by the plant exceeds the maintenance
costs). The plant can be built and become operational immediately. If the plant costs $10 million
to build, and the interest rate is 6% per year, should you invest in the plant?
Timeline:
0
1
2
N
-10,000,000
1,000,000 –
1,000,000 –
1,000,000
50,000
50,000(1.05)
50,000(1.05)N – 1
The plant will shut down when:
( )
( )
( ) ( ) ( )
( )
( )
N1
N1
1, 000, 000 50, 000 1.05 0
1, 000, 000
1.05 20
50, 000
N 1 log 1.05 log 20
log 20
N > 1 62.4.
log 1.05
−
−
−
=
−
+=
So the last year of production will be in year 62.
The cash flows consist of two pieces, the 62-year annuity of the $1,000,000 and the growing annuity.
The PV of the annuity is:
( )
A62
1, 000, 000 1
PV 1 16, 217, 006.
0.06 1.06
= − =
The PV of the growing annuity is:
62
GA
50, 000 1.05
PV 1 2, 221, 932.
0.06 0.05 1.06
−
= − = −
−
So the PV of all the cash flows is:
PV 16, 217, 006 2, 221,932 $13,995, 074.= − =
So
the NPV 13, 995, 07 10, 000,000 $3,995, 074, and you should build it.= − =