Chapter 09 - Risk and the Cost of Capital
CHAPTER 9
Risk and the Cost of Capital
Answers to Problem Sets
1. Overestimate. High-risk projects will have a higher cost of capital than the
company’s existing securities. Therefore, the value of a high-risk project would be
overestimated if the company cost of capital is used.
Est. Time: 01 - 05
2.
Company cost of capital = 10 x .4 + (10 + .5 x 8) x .6 = 12.4%.
After-tax WACC = (1 - .35) x 10 x .4 + (10 + .5 x 8) x .6 = 11.0%.
Est. Time: 01 - 05
3.
R-squared measures the proportion of the total variance in the stock’s returns
that can be explained by market movements. Citi’s R squared shows that 0.49, or
49% of variation was due to market movements; the remainder, (1-.49) = 0.51, or
51%, of the variation was diversifiable. Diversifiable risk shows up in the scatter
about the fitted line. The standard error of the estimated beta was 0.34. If you
said that the true beta was 2 x 0.34 = 0.68 for either side of your estimate, you
would have a 95% chance of being right.
Est. Time: 01 - 05
4.
a.
The expected return on debt. If the debt has very low default risk, this is
close to its yield to maturity.
b.
The expected return on equity
c.
A weighted average of the cost of equity and the after-tax cost of debt,
where the weights are the relative market values of the firm’s debt and
equity
d.
The change in the return of the stock for each additional 1% change in the
market return
e.
The change in the return on a portfolio of all the firm’s securities (debt and
equity) for each additional 1% change in the market return
f.
A company specializing in one activity that is similar to that of a division of
a more diversified company
g.
A certain cash flow occurring at time t with the same present value as an
uncertain cash flow at time t
9-1
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
Est. Time: 06 -10
5.
Beta of assets = .5 × .15 + .5 × 1.25 = .7.
Est. Time: 01 - 05
6.
A diversifiable risk has no affect on the risk of a well-diversified portfolio and
therefore no affect on the project’s beta. If a risk is diversifiable it does not
change the cost of capital for the project. However, any possibility of bad
outcomes does need to be factored in when calculating expected cash flows.
7.
Suppose that the expected cash flow in Year 1 is 100, but the project proposer
provides an estimate of 100 × 115/108 = 106.5. Discounting this figure at 15%
gives the same result as discounting the true expected cash flow at 8%.
Adjusting the discount rate, therefore, works for the first cash flow but it does not
do so for later cash flows (e.g., discounting a two-year cash flow of 106.5 by 15%
is not equivalent to discounting a two-year flow of 100 by 8%).
Est. Time: 01 - 05
8.
a.
A (a project with a higher fixed cost generally has higher operating
leverage, which leads to a higher beta)
b.
C (more cyclical revenues)
Est. Time: 01 - 05
9.
a.
False. The company cost of capital is the correct discount rate for new
projects only if the new projects have the same risk level as the existing business. If
a new project is riskier, a higher cost of capital should be used. If the new project is
less risky, a lower cost of capital should be used.
b.
False. In order to account for the riskiness inherent in distant cash flows, it
is necessary to account for several possible outcomes in cash flows and calculate
the probability-weighted cash flow for each scenario. The discount rates should not
be adjusted based on uncertainty in cash flows.
c.
True
Est. Time: 01 - 05
9-2
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
10.
a.
PV =
=
110
121

1  r f   (rm  r f ) 1  r f   (rm  r f ) 2


110
121

 $200
1.10 1.10 2
b.
To solve for the certainty equivalent, we set CEQ1/1.05 = 110/1.10.
Therefore, CEQ1 = (110 x 1.05) / 1.10.
CEQ1 = $105.
For Year 2, CEQ2 /1.052 = 121/1.102.
Therefore, CEQ2 = (121 x 1.052) / 1.102 = $110.25.
c.
Ratio1 = 105/110 = .95; Ratio2 = 110.25/121 = .91.
Est. Time: 06 -10
11.
a.
requity = rf +   (rm – rf) = 0.04 + (1.5  0.06) = 0.13 = 13%.
b.
rassets 
D
E
 $4 million
  $6 million

rdebt  requity  
 0.04   
 0.13  .
V
V
 $10 million
  $10 million

rassets = 0.094 = 9.4%.
c.
The cost of capital depends on the risk of the project being evaluated.
If the risk of the project is similar to the risk of the other assets of the
company, then the appropriate rate of return is the company cost of
capital. Here, the appropriate discount rate is 9.4%.
d.
requity = rf +   (rm – rf) = 0.04 + (1.2  0.06) = 0.112 = 11.2%.
rassets 
D
E
 $4 million
  $6 million

rdebt  requity  
 0.04   
 0.112  .
V
V
 $10 million
  $10 million

rassets = 0.0832 = 8.32%.
Est. Time: 06 -10
12.
a.
D 
P 
C

β assets   β debt     β pref erred     β common   
V 
V 
V

$100millio n  
$40million  
$299millio n 

0 
   0.20 
  1.20 
  0.836
$439millio n  
$439millio n  
$439millio n 

b.
r = rf +   (rm – rf) = 0.05 + (0.836 0.06) = 0.10016 = 10.016%.
9-3
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
Est. Time: 01 - 05
13.
a.
The R2 value for Toronto Dominion was 0.66, which means that 66% of
total risk comes from movements in the market (i.e., market risk).
Therefore, (1-.66)= .34, or 34% of total risk is unique risk.
The R2 value for Research in Motion was .08, which means that 8% of
total risk comes from movements in the market (i.e., market risk).
Therefore, (1-.08) = .92, or 92% of total risk is unique risk.
b.
The variance of Toronto Dominion is: (25)2 = 625.
Market risk for Toronto Dominion: 0.25 × 625 = 156.25.
c.
The t-statistic for RIM is: .82/.25 = 3.28.
This is significant at the 1% level, so that the confidence level is 99%.
d.
rTD = rf + TD  (rm – rf) = 0.05 + [1.26  (0.12 – 0.05)] = 0.1382 = 13.82%.
e.
rTD = rf + TD  (rm – rf) = 0.05 + [1.26  (0 – 0.05)] = -.013, or -1.3%.
Est. Time: 06 -10
14.
The total market value of outstanding debt is $300,000. The cost of debt
capital is 8%. For the common stock, the outstanding market value is:
$50  10,000 = $500,000. The cost of equity capital is 15%. Thus, Golden
Fleece’s company cost of capital is:




300,000
500,000
  0.08  
  0.15
rassets  
300,000

500,000
300,000

500,000




rassets = 0.124 = 12.4%
Est. Time: 06 -10
15.
a.
rNS = rf + NS  (rm – rf) = 0.02 + (1.42  0.07) = 0.1194 = 11.94%.
rIND = rf + IND  (rm – rf) = 0.02 + (1.34  0.07) = 0.1138 = 11.38%.
b.
No, we cannot be confident that Norfolk Southern’s true beta is not the
industry average. The difference between NS and IND (0.08) is less than
two times the standard error (2  0.09 = 0.18), so we cannot reject the
hypothesis that NS = IND with 95% confidence.
9-4
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
c.
Norfolk Southern’s beta might be different from the industry beta for a
variety of reasons. For example, Norfolk Southern’s business might be
more cyclical than is the case for the typical firm in the industry. Or
Norfolk Southern might have more fixed operating costs so that operating
leverage is higher. Another possibility is that Norfolk Southern has more
debt than is typical for the industry so that it has higher financial leverage.
Est. Time: 06 -10
16.
Financial analysts or investors working with portfolios of firms may use industry
betas. To calculate an industry beta we would construct a series of industry
portfolio investments and evaluate how the returns generated by this portfolio
relate to historical market movements.
Est. Time: 01 - 05
17.
We should use the market value of the stock, not the book value shown on the
annual report. This gives us an equity value of 500,000 shares times $18 = $9
million. Therefore the total value of the company is $9 million + $5 million = $14
million. So Binomial Tree Farm has a debt/value ratio of 5/14 = 0.36 and an
equity /value ratio of 9/14 = 0.64.
Est. Time: 06 -10
18.
a.
If you agree to the fixed price contract, operating leverage increases.
Changes in revenue result in greater than proportionate changes in profit.
If all costs are variable, then changes in revenue result in proportionate
changes in profit. Business risk, measured by assets, also increases as a
result of the fixed price contract. If fixed costs equal zero, then:
assets = revenue. However, as PV(fixed cost) increases, assets increases.
b.
With the fixed price contract:
PV(assets) = PV(revenue) – PV(fixed cost) – PV(variable cost)
$20million
$10million
PV(assets) 
 ($10million  annuity factor 6%,10years) 
0.09
(0.09)  (1.09)10
PV(assets) = $101,686,818
Without the fixed price contract:
PV(assets) = PV(revenue) – PV(variable cost)
PV(assets) 
$20 million $10 million

0.09
0.09
PV(assets) = $111,111,111
Est. Time: 06 -10
9-5
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
19.
a.
The threat of a coup d’état means that the expected cash flow is less
than $250,000. The threat could also increase the discount rate, but
only if it increases market risk.
b.
The expected cash flow can be found by finding an unbiased forecast. The
probability of having a $0 cash flow is 25%, and the probability of having a
$250,000 cash flow is therefore (1-.25) = .75, or 75%.Therefore, the
unbiased forecast is: (0.25  0) + (0.75  250,000) = $187,500.
Assuming that the cash flow is about as risky as the rest of the
company’s business:
PV = $187,500/1.12 = $167,411
Est. Time: 01 - 05
20.
a.
Expected daily production =
(0.2  0) + 0.8  [(0.4 x 1,000) + (0.6 x 5,000)] = 2,720 barrels
Expected annual cash revenues = 2,720 x 365 x $100 = $99,280,000.
b.
The possibility of a dry hole is a diversifiable risk and should not affect
the discount rate. This possibility should affect forecasted cash flows,
however. See Part a.
Est. Time: 06 -10
21.
a.
Using the Security Market Line, we find the cost of capital:
r = 0.07 + [1.5  (0.16 – 0.07)] = 0.205 = 20.5%
Therefore:
PV   100 
b.
40
60
50


 3.09
2
1.205 1.205
1.205 3
CEQ1 = 40(1.07/1.205) = 35.52.
CEQ2 = 60(1.07/1.205)2 = 47.31.
CEQ3 = 50(1.07/1.205)3 = 35.01.
c.
a1 = 35.52/40 = 0.8880.
a2 = 47.31/60 = 0.7885.
a3 = 35.01/50 = 0.7002.
9-6
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
d.
Using a constant risk-adjusted discount rate is equivalent to assuming
that at decreases at a constant compounded rate.
Est. Time: 11 -15
22.
At t = 2, there are two possible values for the project’s NPV:
NPV2 ( if test is not successful )  0
NPV2 ( if test is successful )   5,000,000 
Therefore, at t = 0:
NPV0   500,000 
700,000
 $833,333
0.12
(0.40  0)  (0.60  833,333)
  $152,778
1.202
Est. Time: 06 -10
23.
It is correct that, for a high beta project, you should discount all cash flows at a
high rate. Thus, the higher the risk of the cash outflows, the less you should
worry about them because the higher the discount rate, the closer the present
value of these cash flows is to zero. This result does make sense. It is better to
have a series of payments that are high when the market is booming and low
when it is slumping (i.e., a high beta) than the reverse.
The beta of an investment is independent of the sign of the cash flows. If an
investment has a high beta for anyone paying out the cash flows, it must have a
high beta for anyone receiving them. If the sign of the cash flows affected the
discount rate, each asset would have one value for the buyer and one for the
seller, which is clearly an impossible situation.
Est. Time: 06 -10
24.
a.
Since the risk of a dry hole is unlikely to be market related, we can use
the same discount rate as for producing wells. Thus, using the Security
Market Line:
rnominal = 0.06 + (0.9  0.08) = 0.132 = 13.2%
We know that:
(1 + rnominal) = (1 + rreal)  (1 + rinflation)
Therefore:
rreal 
1.132
 1  0.0885  8.85%
1.04
10
b.
NPV1   10 million  
t 1
3 million
  10 million  [ (3 million)  (3.1914)]
1.2885 t
9-7
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Risk and the Cost of Capital
NPV1   $425,800
15
NPV2   10 million  
t 1
2 million
  10 million  [ (2 million)  (3.3888)]
1.2885 t
NPV2   $3,222,300
c.
Expected income from Well 1: [(0.2  0) + (0.8  3 million)] = $2.4 million.
Expected income from Well 2: [(0.2  0) + (0.8  2 million)] = $1.6 million.
Discounting at 8.85% gives:
10
NPV1   10 million  
t 1
2.4 million
  10 million  [ (2.4 million)  (6.4602)]
1.0885 t
NPV1  $5,504,600
15
NPV2   10 million  
t 1
1.6 million
  10 million  [ (1.6 million)  (8.1326)]
1.0885 t
NPV2  $3,012,100
d.
For Well 1, one can certainly find a discount rate (and hence a “fudge
factor”) that, when applied to cash flows of $3 million per year for 10
years, will yield the correct NPV of $5,504,600. Similarly, for Well 2, one
can find the appropriate discount rate. However, these two “fudge factors”
will be different. Specifically, Well 2 will have a smaller “fudge factor”
because its cash flows are more distant. With more distant cash flows, a
smaller addition to the discount rate has a larger impact on present value.
Est. Time: 16 -20
9-8
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.