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CHAPTER 6
DISCOUNTED CASH FLOW VALUATION
Answers to Concepts Review and Critical Thinking Questions
1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and the
4. It’s deceptive, but very common. The basic concept of time value of money is that a dollar today is
5. If the total money is fixed, you want as much as possible as soon as possible. The team (or, more
7. Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they are
easier to compute, but with modern computing equipment, that advantage is not very important.
8. A freshman does. The reason is that the freshman gets to use the money for much longer before
9. The problem is that the subsidy makes it easier to repay the loan, not obtain it. However, ability to
repay the loan depends on future employment, not current need. For example, consider a student
CHAPTER 27 - 2
10. In general, viatical settlements are ethical. In the case of a viatical settlement, it is simply an
exchange of cash today for payment in the future, although the payment depends on the death of the
seller. The purchaser of the life insurance policy is bearing the risk that the insured individual will
live longer than expected. Although viatical settlements are ethical, they may not be the best choice
11. This is a trick question. The future value of a perpetuity is undefined since the payments are
12. The ethical issues surrounding payday loans are more complex than they might first appear. On the
one hand, the interest rates are astronomical, and the people paying those rates are typically among
the worst off financially to begin with. On the other hand, and unfortunately, payday lenders are
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. The time line is:
01234
50
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
CHAPTER 27 - 3
PV@10% = $680 / 1.10 + $810 / 1.102 + $940 / 1.103 + $1,150 / 1.104 = $2,779.30
2. The times lines are:
012345678
012345
To find the PVA, we use the equation:
At a 5 percent interest rate:
And at a 15 percent interest rate:
Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 15
percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the
total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a
higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate, these
bigger cash flows early are more important since the cost of waiting (the interest rate) is so much
greater.
3. The time line is:
01234
$1,2
25
$1,3
45
$1,4
60
$1,5
90
To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a
lump sum, we use:
FV = PV(1 + r)t
CHAPTER 27 - 4
FV@8% = $1,225(1.08)3 + $1,345(1.08)2 + $1,460(1.08) + $1,590 = $6,278.76
Notice, since we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of
the other cash flows. In other words, we do not need to compound this cash flow.
4. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r) t] } / r )
0 1
…
15
0 1
…
40
0 1
…
75
To find the PV of a perpetuity, we use the equation:
PV = C / r
0 1
…
∞
Notice that as the length of the annuity payments increases, the present value of the annuity
approaches the present value of the perpetuity. The present value of the 75-year annuity and the
present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years
is only $1,159.50.
5. The time line is:
CHAPTER 27 - 5
0 1
…
15
$38,000 C C C C C C C C C
Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the PVA equation:
We can now solve this equation for the annuity payment. Doing so, we get:
6. The time line is:
01234567
$57,00
0
CCCCCCC
To find the PVA, we use the equation:
PVA = C({1 – [1 / (1 + r) t] } / r )
7. Here we need to find the FVA. The equation to find the FVA is:
FVA = C{[(1 + r)t – 1] / r}
0 1
…
20
0 1
…
40
Notice that because of exponential growth, doubling the number of periods does not merely double
the FVA.
8. The time line is:
0 1 …12
CHAPTER 27 - 6
$50,000
C C C C C C C C C
Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the FVA equation:
FVA = C{[(1 + r)t – 1] / r}
We can now solve this equation for the annuity payment. Doing so, we get:
9. The time line is:
012345
0
Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the PVA equation:
PVA = C({1 – [1/(1 + r)t] } / r)
We can now solve this equation for the annuity payment. Doing so, we get:
10. The time line is:
0 1 …∞
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C / r
11. The time line is:
0 1 …∞
–
0
$40,0
$40,0
$40,0
$40,0
$40,0
$40,0
$40,0
$40,0
$40,00
CHAPTER 27 - 7
Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C / r
We can now solve for the interest rate as follows:
CHAPTER 27 - 8
12. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m – 1
To find the EAR with continuous compounding, we use the equation:
EAR = eq – 1
13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding:
EAR = [1 + (APR / m)]m – 1
We can now solve for the APR. Doing so, we get:
APR = m[(1 + EAR)1/m – 1]
EAR = .1240 = [1 + (APR / 2)]2 – 1 APR = 2[(1.1240)1/2 – 1] = .1204, or 12.04%
Solving the continuous compounding EAR equation:
EAR = eq – 1
We get:
APR = ln(1 + EAR)
14. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m – 1
So, for each bank, the EAR is:
Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding
periods within a year will also affect the EAR.
CHAPTER 27 - 9
15. The reported rate is the APR, so we need to convert the EAR to an APR as follows:
EAR = [1 + (APR / m)]m – 1
APR = m[(1 + EAR)1/m – 1]
16. The time line is:
0 1
…
34
$2,400 FV
For this problem, we need to find the FV of a lump sum using the equation:
FV = PV(1 + r)t
It is important to note that compounding occurs semiannually. To account for this, we will divide the
17. For this problem, we need to find the FV of a lump sum using the equation:
FV = PV(1 + r)t
It is important to note that compounding occurs daily. To account for this, we will divide the interest
rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by
365. Doing so, we get:
0 1
…
5(365)
$7,000 FV
0 1
…
10(365
)
$7,000 FV
0 1
…
20(365
)
$7,000 FV
CHAPTER 27 - 10
18. The time line is:
0 1
…
10(365
)
PV $65,000
For this problem, we need to find the PV of a lump sum using the equation:
PV = FV / (1 + r)t
It is important to note that compounding occurs daily. To account for this, we will divide the interest
19. The APR is simply the interest rate per period times the number of periods in a year. In this case, the
interest rate is 32 percent per month, and there are 12 months in a year, so we get:
To find the EAR, we use the EAR formula:
Notice that we didn’t need to divide the APR by the number of compounding periods per year. We do
this division to get the interest rate per period, but in this problem we are already given the interest
rate per period.
20. The time line is:
0 1
…
60
We first need to find the annuity payment. We have the PVA, the length of the annuity, and the
interest rate. Using the PVA equation:
PVA = C({1 – [1 / (1 + r)t] } / r)
Solving for the payment, we get:
To find the EAR, we use the EAR equation:
EAR = [1 + (APR / m)]m – 1
CHAPTER 27 - 11
21. The time line is:
0 1
…
?
$18,000
Here we need to find the length of an annuity. We know the interest rate, the PV, and the payments.
Using the PVA equation:
PVA = C({1 – [1 / (1 + r)t] } / r)
Now we solve for t:
22. The time line is:
0 1
–$3 $4
Here we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
FV = PV(1 + r)
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
And using the equation to find the EAR:
EAR = [1 + (APR / m)]m – 1
