The Pricing Of Risk
Understanding the Systematic Risk Return Relation
Systematic Risk
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Last time we discussed the dramatic impact that diversification
has on the risk of a portfolio as compared to the risk of an
individual asset.
We also saw, in several different forms, that it was the
correlation or covariance between the returns on the
individual assets in a portfolio that tells us how much of an
impact diversification can have.
Holding the volatility of the individual assets constant:
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A portfolio of assets whose returns are weakly correlated will have
very little risk.
A portfolio of assets whose returns are highly correlated will have
lots of risk.
Covariance (or correlation) between the returns of assets in a
portfolio determines the amount of measure of systematic risk
of the portfolio.
Systematic Risk
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Suppose we have a portfolio of 3 stocks A, B, and C.
When we think about stock A’s contribution to the
volatility of the portfolio it is determined by Cov(RA, RB),
Cov(RA, RC), and Cov(RA, RA).
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The last term equals the variance of the return on stock A. Its
influence declines as more stocks are added to the portfolio.
The contribution stock A makes to portfolio variance is
given by the weighted sum:
wA wB Cov( RA , RB )  wA wC Cov ( RA , RC )  wA wACov ( RA .RA )
 wACov( RA , wB RB )  wACov( RA , wC RC )  wACov ( RA , wA RA )
 wACov( RA , wB RB  wC RC  wA RA )
 wACov( RA , RP )
Systematic Risk
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With our portfolio of 3 stocks we see that the way stock
A contributes to the variance of the return on a portfolio
is determined by how much we add of stock A and the
either (1) all the paired covariances of the return on
stock A with the returns on all the stocks in the portfolio
or (2) the covariance of the return on stock A and the
return on the portfolio.
We keep the “weight” for stock A separate since we want
a measure that will be useful for all possible weights.
Then we see that the contribution stock A makes to the
variance of the return on a portfolio is determined by the
covariance of the return on stock A and the return on
the entire portfolio.
Systematic Risk
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Recall, however, that covariance itself is not a particularly
informative statistic.
“Only the sign is informative” someone once said.
The size is not.
How did we solve that problem?
We standardized by the product of the standard deviations of
the random variables being compared and found correlation; a
more useful statistic.
However, that is not a useful way to standardize in this
context.
Recall, we want to put all measures of systematic risk in terms
that are relative to our measure of “one unit” of risk.
Measuring Systematic Risk
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How can we measure systematic risk or estimate the amount
or proportion of an asset's risk that non-diversifiable?
The Beta Coefficient is the slope coefficient in an OLS regression
of stock returns on market returns:
Cov(Ri , R M )
i 
Var(R M )
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Beta is a measure of sensitivity: it describes how strongly a
stock’s excess return moves with the market excess return.
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What is the expected percent change in the excess return of security i
for a 1% change in the excess return of the market portfolio?
It is standardized covariance, standardized by our measure of
“one unit” of risk.
Why the Market Portfolio?
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Intuitively we focused on the market portfolio of all risky
assets because it represented the total amount of risk
that the market had to bribe/compensate investors to
hold.
The market portfolio in this sense is a well diversified
portfolio by definition.
It therefore became our definition of one unit of risk.
The efficient portfolio analysis developed by Harry
Markowitz and James Tobin discussed in the text provides
the theoretical underpinnings for the intuition.
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It is “the” efficient portfolio and all assets are most usefully
considered in relation to it.
Determining Beta
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Estimating Beta from Historical Returns
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Recall, beta is the expected percent change in the excess
return of the security for a 1% change in the excess return
of the market portfolio.
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Consider Lockheed Martin stock and how it changes with the
market portfolio.
Scatterplot of Monthly Excess Returns for
Lockheed Martin Versus the S&P 500, 2001–2010
Scatter plot
Lockheed excess returns
1
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
0.5
S&P excess returns
1
1.5
Determining Beta
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Estimating Beta from Historical Returns
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As the scatterplot on the previous slide shows, Lockheed tends
to be up when the market is up, and vice versa.
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We can see that a 10% change in the market’s return
corresponds to about a 10% change in Lockheed’s return.
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What does this imply about Lockheed’s beta?
Thus, Lockheed’s return moves about two for one with the overall
market, so Lockheed’ss beta is about 1.
Beta corresponds to the slope of the best-fitting line in the plot of
the security’s excess returns versus the market excess return.
Using Linear Regression
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Linear Regression
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The statistical technique that identifies the best-fitting line
through a set of points.
(Ri  rf )  i  i (RMkt  rf )  i
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αi is the intercept term of the regression.
βi(RMkt – rf) represents the sensitivity of the stock to market risk.
εi is the error term and represents the deviation from the best-fitting
line and is zero on average.
Using Linear Regression
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Linear Regression
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Since E[εi] = 0:
E[Ri ]  rf  i ( E[RMkt ]  rf ) 
Expected return for i from the SML
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i
Distance above / below the SML
αi represents a risk-adjusted performance measure for the historical
returns.
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If αi is positive, the stock has performed better than predicted by the CAPM.
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If αi is negative, the stock’s historical return plots below the SML.
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A regression for Lockheed using the monthly returns for 2001 – 2010
indicates the estimated beta is 0.96 (standard error is 0.017).
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The estimate of Lockheed’s alpha from the regression is 0.0087 (the
standard error is .006 so this alpha is insignificantly different from zero).
Estimates of Expected Return
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With a beta estimate of 0.96 and a standard error on the
beta estimate of .017 we have a 95% confidence interval
on beta for Lockheed of 0.93 – 1.00.
If the risk free rate is 2% and the market risk premium is
5% the confidence interval for beta translates to a
confidence interval for expected return of 6.65% – 7.0%.
If we simply use average return for Lockheed stock over
the last ten years we find an average of 11.3%.
With a standard error on this estimate of mean of 0.068
this translates to a confidence interval of -2.04% to 24.7%
for expected return.
The CAPM Intuition
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re,i = E[Ri] = RF (risk free rate) + Risk Premium
= Appropriate Discount Rate
Risk free assets earn the risk-free rate (think of this as a
rental rate on capital).
If the asset is risky, we need to add a risk premium.
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The size of the risk premium for a given asset depends on the
amount of systematic risk for that asset (stock, bond, or investment
project) and the price per unit risk.
Aside: could a risk premium ever be negative?
The CAPM Intuition Formalized
Cov(R i , R M )
E[R i ]  R F 
[E[R M ]  R F ]
Var(R M )
or,
E[Ri ]  R F   i [E[RM ]  R F ]
Number of units of
systematic risk ()
Market Risk Premium
or the price per unit risk
• The expression above is referred to as the “Security
Market Line” (SML) or commonly just the CAPM.
Betas and Portfolios
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The beta of a portfolio is the weighted average of the
component assets’ betas.
Example: You have 30% of your money in asset X, which
has X = 1.4 and 70% of your money in asset Y, which has
Y = 0.8.
Your portfolio beta is:
P = .30(1.4) + .70(0.8) = 0.98.
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What if your portfolio has 50 assets, not 2?
Why do we care about this feature of betas?
 It further demonstrates that an asset’s beta measures the
contribution that asset makes to the systematic risk of a
portfolio!
 Note that this is a linear relation just like expected return.
Risk and the Cost of Capital
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Three inputs are required:
(i) An estimate of the risk free interest rate.
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The current yield on short term treasury bills is one proxy.
Practitioners tend to favor the current yield on longer-term treasury bonds
but this may be a fix for a problem we don’t fully understand.
Must adjust the market risk premium accordingly.
(ii) An estimate of the market risk premium, E(Rm - Rf).
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Expectations are not observable.
Use a historically estimated value. Use the average spread between the risk
free rate and the market return.
(iii) An estimate of beta. Is the asset or a close substitute for the asset
traded in financial markets? If so, gather data and run an OLS regression
or look it up from a variety of sources. If not, it gets fuzzy.
The Market Risk Premium
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The market is defined as a portfolio of all wealth including real
estate, human capital, etc.
In practice, a broad based stock index, such as the S&P 500 or
the portfolio of all NYSE stocks, is generally used.
We want the expected return on the market portfolio above the
risk free rate.
Again, we use the average of this difference over time.
Historically, the average market risk premium has been about
8% - 9% above the return on treasury bills.
The average market risk premium has been about 6% - 7%
above the return on treasury bonds.
More recent averages are considerably lower.
Cost of Equity Capital Estimates
Cost of Equity = 2% + Risk Premium
= 2% + Beta*5.0%
Company
Microsoft
Cisco Systems
UAL (United Airlines)
Home Depot
Lockheed Martin
SBC Communications
Homestake Mining
Southern Company
Beta
1.63
1.39
1.20
1.16
0.96
0.87
0.62
0.10
Risk Premium
8.15
6.95
6.00
5.80
4.80
4.35
3.10
0.50
Cost of Equity
10.15
8.95
8.00
7.80
6.80
6.35
5.10
2.50
Example
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Suppose that in the coming year you expect Microsoft
stock to have a volatility of 23% and a beta of 1.28.
You expect McDonalds to have a volatility of 37% and a
beta of 0.99.
Which stock carries more total risk?
Which has more systematic risk?
If the risk free rate is 4% and the market’s expected
return is 10%, estimate the cost of capital of McDonalds
and for Microsoft. Which has a higher cost of capital?
Example - Solution
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Total risk is measured by volatility:
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Systematic risk is measured by beta:
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E(RMSFT) = rf + 1.28 × (10% - 4%) = 4% + 7.7% = 11.7%
McDonalds stock has a lower beta (0.99) and so has a risk
premium slightly below the market risk premium:
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Microsoft has a higher beta so more systematic risk.
Microsoft’s beta is 1.28 so it has a risk premium that is 1.28
times the market risk premium:
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McDonalds stock has more total risk.
E(RMcD) = rf + 0.99 × 6% = 4% + 5.9% = 9.9%
Because systematic risk cannot be diversified, it is systematic
risk that determines the cost of capital. Therefore, Microsoft
has a higher cost of capital than McDonalds, even though it has
less volatility.
The Cost of Debt Capital
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Firms use debt as well as equity capital. How do we
measure the cost of debt capital?
We could use the same approach we developed for
equity.
Table 12.3 Average Debt Betas by Rating and Maturity
By Rating
A & above
BBB
BB
B
CCC
Avg. Beta
< 0.05
0.10
0.17
0.26
0.31
By Maturity
BBB & above
1 – 5 Year
5 – 10 Year
10 – 15 Year
> 15 Year
0.01
0.06
0.07
0.14
Avg. Beta
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For investment grade bonds betas are near zero. Then
there is the general lack of data and measurement error.
The Cost of Debt Capital
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A commonly used way to estimate the cost of debt
capital is to use the yield to maturity on a firm’s existing
debt (or other firms’ existing debt with the same rating as
the debt to be issued).
As was discussed before, the YTM of a bond is only the
expected return if there is no possibility of default: only
then is promised return equal to expected return.
The Cost of Debt Capital
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Consider a one-year bond with a promised payment of $(1
+ y) for each $1 invested. If the bond has a probability of
default of p over the year and has an expected loss of L
per $1 invested in the event of default.
Then expected return is written:
rd = E(r) = prob of no default × promised return
+ prob of default × return in default
= (1 – p)y + p(y – L) = y – pL
= YTM – prob of default × expected loss rate
So if p (or L) is near zero the YTM is a reasonable estimate.
Average loss rate (L) for unsecured debt is 60%.
Default Rates
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When not in a recession, in the US investment grade
bonds have trivial default rates making YTM a reasonable
approximation.
In recessions this is far from true.
Table 12.2 Annual Default Rates by Debt Rating (1983 – 2008)
Rating
AAA
AA
A
BBB
BB
B
CCC
CC-C
Avg.
0.0%
0.0%
0.2%
0.4%
2.1%
5.2%
9.9%
12.9%
In recessions
0.0%
1.0%
3.0%
3.0%
8.0%
16.0% 43.0% 79.0%
Default Rate
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In average times a BBB bond sees expected return of
0.004x0.60 = 0.24% below quoted yield.
In recession, a CCC rated bond sees 0.43x0.60 = 25.8%.
Cost of Capital for a Project
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Suppose you are considering an investment project and
want to know an appropriate cost of capital with which
to evaluate the project.
Let’s first assume the project will be financed only with
equity so we can avoid some complications.
What is the appropriate cost of unlevered equity capital –
also the cost of capital of the assets?
How do we find the cost of capital of a project that has
not yet been established?
We turn to comparable firms.
Cost of Capital for a Project
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Comparable Firms:
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Firms that share characteristics related to risk with the target
project.
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Industry
Size
Fixed versus Variable costs (operating leverage)
Financing
Geographical Location
Find a set of firms that match the project on these
dimensions as closely as possible. Why a set?
If the comparable firms are all equity financed then it is
easy: use the average of their costs of (unlevered) equity
capital as an estimate of the project’s cost of capital.
Cost of Capital for a Project
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When the comparable firms we identify are levered firms
the problem becomes a little more complicated.
Find a cost of (levered) equity capital and a cost of debt
capital for the comparable firm(s). Now think of the
securities for each comparable firm as a portfolio.
Recall, the costs of capital are the expected return on the
securities an investor demands for holding them given
their level of risk.
What is the expected return an investor demands for
holding all the debt and all the equity of a levered firm?
Now think about the balance sheet of the firm.
Cost of Capital for a Project
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As with an unlevered comparable firm we would still like to
identify the cost of unlevered equity capital – aka the cost of
capital of the assets.
Letting E be the market value of equity and D be the market
value of net debt we have:
E
D
rU 
rE 
rD
ED
ED
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Similarly:
E
D
U 
E 
D
ED
ED
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The unlevered cost of capital is the expected return investors
can expect to receive holding the securities of the firm.
Example 12.7 from the Text
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You have decided to open a department store and want
an estimate of an asset beta for such an endeavor.
Company
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Equity Beta
D/V
Debt Rating
Dillard’s
2.38
0.59
B
J.C. Penney Co.
1.60
0.17
BB
Kohl’s
1.37
0.08
BBB
Macy’s
2.16
0.62
BB
Nordstrom
1.94
0.35
BBB
Saks
1.85
0.50
CCC
Sears Holdings
1.36
0.23
BB
Note the large range of equity betas (1.36 – 2.38).
Example 12.7 from the Text
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You need to find the asset beta for each comparable, this
tells us about the risk of operating a department store.
Company
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Equity Beta
D/V
Debt Beta
Asset Beta
Dillard’s
2.38
0.59
0.26
1.13
J.C. Penney Co.
1.60
0.17
0.17
1.36
Kohl’s
1.37
0.08
0.10
1.27
Macy’s
2.16
0.62
.017
0.93
Nordstrom
1.94
0.35
0.10
1.30
Saks
1.85
0.50
0.31
1.08
Sears Holdings
1.36
0.23
0.17
1.09
Average
Median
1.16
1.13
Asset betas are much more homogeneous.
The Weighted Average Cost of Capital
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Looking ahead we will see that if a firm undertakes a
project and uses debt financing for the project and pays
taxes at the rate , the firm gains a valuable tax benefit
from the project. To include the value of the tax savings
we calculate NPV using the WACC:
E
D
rWACC 
rE 
rD (1   C )
ED
ED
E
D
D

rE 
rD 
rD C
ED
ED
ED
D
 rU 
rD C
ED
WACC is the effective after-tax cost of capital to the firm
Problem
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Buffet.com equity currently sells for $21 per share and
has 2 million shares outstanding. The firm also has $30
million worth of debt outstanding. The firm’s debt has a
BBB rating and it’s equity beta is estimated as 1.9. The
current risk free rate is 3%, the market risk premium is
5% and the corporate tax rate is 35%.
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What is Buffet’s cost of capital of assets?
What is Buffet’s weighted average cost of capital?
Solution
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The firm has $42 million in equity value and $30 million in
debt value for a total market value of $72 million. This
means that the equity percent of value is 58.3% and the
debt percent of value is 41.7%.
The security market line tells us that the cost of equity
capital is: rE = 3% + 1.9 × 5% = 12.5%.
The security market line tells us that the cost of debt
capital is: rD = 3% + 0.1 × 5% = 3.5%.
Solution
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Using these inputs we can find the cost of capital of the
assets or the cost of unlevered equity capital as:
E
D
rU 
rE 
rD  0.583 12.5%  0.417  3.5%  8.747%
ED
ED
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This is the expected return to an investor holding the
debt and equity of the firm in a value weighted portfolio.
It is also the cost of capital to apply to an unlevered
project with a systematic risk level equal to the firm’s.
Solution
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The firm’s weighted average cost of capital is:
E
D
rE 
rD (1   C )  0.583 12.5%  0.417  3.5%(1  .35)  8.236%
ED
ED
D
 rU 
rD C  8.747%  0.417  3.5%  .35  8.236%
ED
rWACC 
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This is the cost of capital to apply to a project with a
systematic risk level equal to the firm’s that is financed
with 41.7% debt capital and 58.3% equity capital.
Market Efficiency
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A large debate in the finance profession centers around
the “efficiency” of market prices. Do prices reflect value?
Your book makes a convincing case that the question is
really one of how liquid and competitive is the relevant
market.
Consider the market for US publicly traded equities.
How many investors are watching? How easy/costless is it
for them to trade?
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If a firm announces information relevant to the valuation of its
securities, how quickly do you think the price should respond?
The implications of an efficient market are far reaching.