CHAPTER 4
DISCOUNTED CASH FLOW VALUATION
Answers to Concept Questions
1. Assuming positive cash flows and interest rates, the future value increases and the present value
2. Assuming positive cash flows and interest rates, the present value will fall and the future value will
rise.
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 interest
starts to accrue.
6. It’s a reflection of the time value of money. TMCC gets to use the $24,099 immediately. If TMCC
7. Oddly enough, it actually makes it more desirable since TMCC only has the right to pay the full
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 receipt
of the $100,000 grows shorter, and the present value rises. In 2019, the price will probably be higher
Solutions to Questions and Problems
0
10
$4,800
FV
The simple interest per year is:
$4,800 × .07 = $336
So, after 10 years, you will have:
2. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
The time line for the cash flows is:
0
10
$3,550
FV
b. The time line for the cash flows is:
0
10
$3,550
FV
FV = $3,550(1.08)10 = $7,664.18
c. The time line for the cash flows is:
0
20
$3,550
FV
FV = $3,550(1.06)20 = $11,385.33
d. Because interest compounds on the interest already earned, the interest earned in part c is more
than twice the interest earned in part a. With compound interest, future values grow
exponentially.
3. To find the PV of a lump sum, we use:
PV = FV/(1 + r)t
0
9
PV
$15,451
PV
$51,557
0
PV
$886,073
PV = $886,073/(1.14)16 = $108,890.97
0
24
PV
$550,164
PV = $550,164/(1.11)24 = $44,951.14
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:
FV = PV(1 + r)t
0
3
$293
$1,053
$162,181
19
–$54,382
$483,500
FV = $483,500 = $54,382(1 + r)19; r = ($483,500/$54,382)1/19 – 1 = .1219, or 12.19%
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
$1,284
–$18,400
$234,162
–$21,500
$215,000
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
The length of time to double your money is:
0
?
–$1
$2
FV = $2 = $1(1.0575)t
t = ln 2/ln 1.0575 = 12.40 years
The length of time to quadruple your money is:
0
?
–$1
$4
12. The times lines are:
0
1
2
3
4
5
6
7
8
9
PV
$4,850
$4,850
$4,850
$4,850
$4,850
$4,850
$4,850
$4,850
$4,850
0
1
2
3
4
5
PV
$6,775
$6,775
$6,775
$6,775
$6,775
X@21%: PVA = $4,850{[1 – (1/1.21)9]/.21} = $18,941.36
Y@21%: PVA = $6,775{[1 – (1/1.21)5]/.21} = $19,823.54
Notice that the PV of Cash Flow X has a greater PV at an interest rate of 5 percent, but a lower PV at
an interest rate of 21 percent. The reason is that X has greater total cash flows. At a lower interest rate,
greater.
13. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)t] }/r )
0
1
…
15
PV
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
PVA@15 yrs: PVA = $5,500{[1 – (1/1.075)15]/.075} = $48,549.16
0
1
…
40
PV
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
PVA@40 yrs: PVA = $5,500{[1 – (1/1.075)40]/.075} = $69,269.25
0
1
…
75
PV
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
PVA@75 yrs: PVA = $5,500{[1 – (1/1.075)75]/.075} = $73,009.99
To find the PV of a perpetuity, we use the equation:
PV = C/r
0
1
…
∞
PV
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
$5,500
PV = $5,500/.075
PV = $73,333.33
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 $323.35.