CHAPTER 4
DISCOUNTED CASH FLOW VALUATION
Answers to Concepts Review and Critical Thinking Questions
1. Assuming positive cash flows and interest rates, the future value increases and the present value
decreases.
3. The better deal is the one with equal installments.
4. Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they are
5. A freshman does. The reason is that the freshman gets to use the money for much longer before
6. It’s a reflection of the time value of money. TMCC gets to use the $24,099 immediately. If TMCC
7. This will probably make the security less desirable. TMCC will only repurchase the security prior to
maturity if it is to its advantage, i.e. interest rates decline. Given the drop in interest rates needed to
8. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to
9. The Treasury security would have a somewhat higher price because the Treasury is the strongest of
all borrowers.
10. The price would be higher because, as time passes, the price of the security will tend to rise toward
$100,000. This rise is just a reflection of the time value of money. As time passes, the time until
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 for the cash flows is:
0 10
$7,0
00
FV
The simple interest per year is:
2. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
a.
0 10
$1,0
00
FV
b.
0 10
$1,0
00
FV
c.
0 20
$1,0
00
FV
d. Because interest compounds on the interest already earned, the interest earned in part c is more
3. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
0 8
PV $13,8
27
0 13
PV $43,8
52
0 17
PV $725,3
80
0 26
PV $590,7
10
4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
Solving for r, we get:
0 4
– $345
$242
0 8
–
$410
$927
0 16
–
$51,70
0
$152,1
84
0 27
–
$18,75
0
$538,6
00
5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV / PV) / ln(1 + r)
0 ?
–
$625
$1,2
84
0 ?
–
$810
$4,3
41
0 ?
–
$16,50
0
$402,6
62
0 ?
–
$21,50
0
$147,3
50
6. To find the length of time for money to double, triple, etc., the present value and future value are
irrelevant as long as the future value is twice the present value for doubling, three times as large for
tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the
same answer since they are the inverse of each other. We will use the FV formula, that is:
The length of time to double your money is:
0 ?
–$1 $2
0 ?
–$1 $4
Notice that the length of time to quadruple your money is twice as long as the time needed to double
7. The time line is:
0 20
PV –$550,000,000
8. The time line is:
0 4
–
$1,680,000
$1,100,000
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
Notice that the interest rate is negative. This occurs when the FV is less than the PV.
9. The time line is:
0 1
…
∞
PV $125 $125 $125 $125 $125 $125 $125 $125 $125
A consol is a perpetuity. To find the PV of a perpetuity, we use the equation:
10. To find the future value with continuous compounding, we use the equation:
FV = PVert
a.
0 9
$1,9
00
FV
b.
0 5
$1,9
00
FV
c.
0 17
$1,9
00
FV
d.
0 10
$1,9
00
FV
11. The time line is:
01234
PV $675 $880 $985 $1,5
30
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
12. The times lines are:
0123456789
PV $3,9
00
$3,9
00
$3,9
00
$3,9
00
$3,9
00
$3,9
00
$3,9
00
$3,9
00
$3,9
00
012345
PV $6,1
00
$6,1
00
$6,1
00
$6,1
00
$6,1
00
To find the PVA, we use the equation:
At an interest rate of 5 percent rate:
And at an interest rate of 22 percent:
Notice that the PV of Cash flow X has a greater PV than Cash flow Y at an interest rate of 5 percent,
but a lower PV at an interest rate of 22 percent. 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
13. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)]t } / r )
0 1
…
15
PV $5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,650
0 1
…
40
PV $5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,650
0 1
…
75
PV $5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,650
To find the PV of a perpetuity, we use the equation:
PV = C / r
0 1
…
∞
PV $5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,65
0
$5,650
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
14. The time line is:
0 1 …∞
PV $12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,00
0
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
To find the interest rate that equates the perpetuity cash flows with the PV of the cash flows, we can
use the PV of a perpetuity equation:
PV = C / r
0 1
…
∞
–
$275,00
0
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,0
00
$12,00
0
15. 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 = er – 1
16. 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]
Solving the continuous compounding EAR equation:
EAR = er – 1
We get:
17. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m – 1
So, for each bank, the EAR is:
18. The cost of a case of wine is 10 percent less than the cost of 12 individual bottles, so the cost of a
case will be:
Now, we need to find the interest rate. The cash flows are an annuity due, so:
0 1 …12
–$108 $10 $10 $10 $10 $10 $10 $10 $10 $10
$10
Solving for the interest rate, we get:
r = .0198, or 1.98% per week
So, the APR of this investment is:
The analysis appears to be correct. He really can earn about 177 percent buying wine by the case.
The only question left is this: Can you really find a fine bottle of Bordeaux for $10?