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Solution Key for Chapter 4 Problems:
5.Plan: You are being offered a choice between $5000 today and $10,000 in 10 years. One way
to evaluate this decision is determine how much the $10,000 in 10 years is worth today.
In this way we can compare the $5000 today against the present value of the $10,000 in
10 years.
Execute:
10
PV
PV ? 10,
Evaluate: The 10,000 in 10 years is worth $5083.49 today. It is preferable to the $
payment today because it is worth more.
8. a.
7
FV 3996(1.08)
You would have $6858.44 at age 25, which is seven years from today.
3996 FV ?
b.
47
FV 3996(1.08)
3996 FV?
You would have $148,799 at age 65, which is 47 years from today.
c.
18
PV
PV ? 3996
10.Plan: First, create a timeline to understand when the cash flows are occurring.
Second, calculate the present value of the cash flows:
Once you know the present value of the cash flows, compute the future value (of this
present value) at date 3.
Execute:
2 3
PV
3
3
FV 2723 1.
Evaluate: Because of the bank’s offer, you now have two choices as to how you will
repay this loan. Either you will pay the bank $1000 per year for the next three years as
originally promised. Or you can decide to skip the three annual payments of $1000 and
pay $3152 in year three.
You now have the information to make you decision.
11. Plan: Draw a timeline to show the relevant cash flows of attending college and its
subsequent financial benefits. Estimate the net present value of going to college.
To pay off the loan you must repay the remaining balance. The remaining balance is
equal
to the present value of the remaining payments. The remaining payments are a four-
year annuity, so:
b.
Execute:
a.
4
PV 1
b.
PV
Evaluate: To pay off the loan after owning the vehicle for one year will require
To pay off the loan after owning the vehicle for four years will require $4716.98.
20. a.
C C C C
First, we need to calculate the PV of $160,000 in 18 years.
18
PV
In order for the parents to have $160,000 in your college account by your 18th
birthday, the 18-year annuity must have a PV of $40,039.84. Solving for the annuity
payments:
18
C
which must be saved each year to reach the goal.
C
Evaluate: The bequest is worth $25,000 today and will be worth $27,000 in one years
time.
*22. Plan: The machine will produce a series of savings that are growing at a constant rate.
The rate of growth is negative, but the constant growth model can still be used.
Execute: The timeline for the saving would look as follows.
2
We must value a growing perpetuity with a negative growth rate of –0.02:
PV
Evaluate: The value of the savings produced by the machine is worth $14,285.
today.
25.Plan: Your company’s earnings are expected to grow in the future. The earnings consist of
two parts. For the first five years you have an annuity growing at 30%. After five years
you
have an annuity growing at 2%. You must value each of the parts in today’s dollars and
add them together to get the value of all earnings.
Execute: Draw a timeline indicating when the cash flows will occur.
2
(1.3)
3
(1.3)
4
(1.3)
5
(1.3)
5
(1.02) (1.3)
5
(1.02)
2
This problem consists of two parts:
a. A growing annuity (at 30%) for 5 years
b. A growing perpetuity (at 2%) after 5 years
First we find the PV of (a)
5
GA
PV 1
$9.02 million
Now we calculate the PV of (b). The value at date 5 of the growing perpetuity is
5
5 0 5
PV $63.12 million PV $42.96 million
Adding the present value of (a) and (b) together gives the PV value of future earnings:
$9.02 $42.96 $51.98 million
Evaluate: The Internet company is worth $51.98 million dollars.
27. a.
The amount in the retirement account in 43 years would be:
43
43
FV ((1.10) 1)
b. To solve for the lump sum amount today, find the PV of the $2,962,003.
43
PV
29.Plan: Draw a timeline to determine when the cash flows occur. Solve the problem to
determine the annual payments.
Timeline (from the perspective of the bank):
Execu te:
–300,000 C C C C
30
C
which is the annual payment.
Evaluate: You will have to pay the bank $24,176 per year for 30 years in mortgage
payments.
31.Plan: Draw a timeline to determine when the cash flows occur. Timeline (where X is the
balloon payment):
–300,000 23,500 23,500 23,500 23,500 X
Note that the PV of the loan payments must be equal to the amount borrowed.
Execute:
30 30
X
Solving for X :
30
30
X
Evaluate: At the end of 30 years you would have to make a $63,848 single (balloon)
payment to the bank.
32. Plan: Draw a timeline to demonstrate when the cash flows occur. We know that you
intend to fund your retirement with a series of annuity payments and the future value of
that annuity is
$2 million.
C C C C C
Setting these equal gives:
43
C
C
Solving for C
43
C
Evaluate: You would have to put aside $3612.94 in year 1 to reach your retirement
goal.
38.Plan: Draw a timeline and solve the problem for the breakeven number of time periods.
Execute:
0 1 2 3 N
She breaks even when the NPV of the cash flows is zero. The value of N that solves
this is:
( )
NPV 200,000 1 0
log(1.05) log
log(1.05) log(0.6)
log(0.6)
log 1.
N N N N N N N
Evaluate: So if she lives 10.5 or more years she comes out ahead.
40.Plan: Draw a timeline to show when the cash flows will occur. Then determine how much
you will have to put into the retirement plan annual to meet your goal.
Execute:
– C – C – C 100 100 100
The PV of the costs must equal the PV of the benefits, so begin by dividing the
problem into two parts: the costs and the benefits.
