The CAPM and Risky Arbitrage
Professor David McLean
Alberta School of
Business
What We Will Learn
This lecture will cover the following:
The CAPM assumptions
The resulting equilibrium conditions
Capital Market Line (CML)
Security Market Line (SML)
Hedging and Risky Arbitrage
T
he CAPM
The CAPM is the lynchpin of modern finance theory
It
r
elies on mean-variance efficiency, which we covered in the last lecture
The CAPM has many applications, such as estimating discount rates and evaluating mutual fund performance
In this lecture we will learn a little bit about the assumptions that underlie the CAPM, but focus more on its applications
The CAPM Assumptions
Individual investors are price takers; trading does not impact prices
Investors
are
rational; they desire mean
-variance
efficiency
Investments
are limited to financial assets, like stocks and bonds
Ignore
taxes and transaction costs, although we can include those and the effects are the same
Investors
have the same expectations of risk and returns
CAPM Equilibrium
Conditions
All investors
hold
the same portfolio of risky
assets; the market portfolio
The market
portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
The market portfolio is on the efficient frontier and, moreover, it is
the optimal risk portfolio
Optimal Risky Portfolio
Recall that in class we learned that all mean-variance optimizing investors should hold some combination of the optimal risky (tangency) portfolio and the risk free asset
How
to calculate the optimal P:
Find
w
i
that results in the highest the slope of the CAL (that is to maximize the reward – to – variability ratio)
CAL becomes the CML
In the CAPM, the market portfolio (M) is the optimal risky portfolio
So now the Capital Allocation Line (CAL) is the Capital Market Line (CML)
In the CAPM every investor holds some combination of M and the risk free asset
These combinations are plotted on the CML
The Capital
Market Line
CML Slope
and Market Risk Premium
M = The market portfolio
r
f
= Risk free rate
E(
r
M
) -
r
f
= Market risk premium
= Slope of the CML
The CAPM and Individual Securities
The risk
premium
on
a individual security
is a function of the individual security’s contribution to the risk of the market portfolio
Put differently, an individual security’s risk premium is a function of its covariance with the market portfolio
The
intuition for this can be seen by looking back at our lecture on portfolio theory
Diversified Portfolio’s Variance
In a portfolio
that
is well diversified
(many assets; each
asset has
only a
small weight) covariances are more important than variances
A
stock’s covariance with other stocks determines its contribution to the portfolio’s overall variance
Hence, investors only care about
the risk that is common to many
stocks
R
isks
that are unique to each stock
are diversified away (each asset’s weight is so small)
Diversified Portfolio’s Variance
Consider the variance of an
n
asset
portfolio
Portfolio risk comes from 2 places: firm-specific risk and covariances
Diversified Portfolio’s Variance
The portfolio variance equation contains
n
2
terms
n
terms are
variances
n
2
–
n
terms are covariances
Covariances are far more important!
Diversified Portfolio’s Variance
Consider an equally weighted portfolio
For portfolios with large n, the variance is essentially the average covariance
CAPM and Individual Security Risk
In the CAPM every investor holds the market portfolio, which is a large diversified portfolio
As we just showed, idiosyncratic risk in this setting is irrelevant
A risky asset in the CAPM is therefore one that has high covariance with other risky assets
We refer to this average covariance risk as
i
=
Cov
(
r
i
,r
m
) /
m
2
Expected Returns
of
a Risky Asset
CAPM investors are risk averse, so riskier assets need to yield higher returns
Riskiness is measured as
β
, so expected returns are a function of
β
E(
r
i
)
=
r
f
+
i
[E(
r
m
) -
r
f
]
If we plot this relation, we refer to the line as the Security Market Line (SML)
Security Market Line
SML Relationships
i
=
Cov
(
r
i
,r
m
) /
m
2
E(r)
i
=
r
f
+
i
[
E(
r
m
) -
r
f
]
M
=
Cov
(
r
M
,r
M
) /
s
M
2
=
s
M
2
/
s
M
2
=
1
CAPM Alpha
In the CAPM, expected returns are
:
E(
r
i
)
=
r
f
+
i
[E(
r
m
) -
r
f
]
If an asset does not adhere to this equation, we refer to the difference in expected return as “α”
E(
r
i
)
=
α
i
+
r
f
+
i
[E(
r
m
) -
r
f
]
A stock with an alpha does not plot on the SML
An Alpha Example
A financial analyst believes that a security with a β of will have a return of 17%
r
f
is 6%, and
r
m
is 14%
In the CAPM, the expected return is %
Can you show this?
So the alpha is 17% - % = %
Is the stock overpriced or underpriced?
The
SML and
Alpha
The CAPM’s Applications
Estimating mutual funds’ alphas
Discount rates for company valuations
Discount rates for project NPVs
The SML and Actual Stock
R
eturns
Empirically,
does not predict stock returns
The CAPM is perhaps a better
normative
theory than a
positive
theory
The CAPM does provide us with a “fair” rate of return given an asset’s risk
This is why it is used so widely in practice
Applying the CAPM…
Risky Arbitrage
Risky Arbitrage
Let’s assume that you think a stock is mispriced
How do you use this information optimally?
Do you put all of your wealth into this stock?
How does y* change? Or does it?
A Primer on Short Sales
What if a stock has a negative alpha?
Y
ou could sell it “short”.
In a short sale you…
Borrow and sell a security
Buy it back later, hopefully at a lower price
Return the security to the lender
Example: Borrow a stock for $100; sell it short; buy it back for $90
Ignore fees and other issues for now
What is your HPR? What if the stock paid a dividend of $5?
Realized Returns vs. Expected Returns
Expected
Returns: E
(
r
i
)
=
r
f
+
i
[E(
r
m
)
–
r
f
]
Realized
Returns: (
r)
i
=
r
f
+
i
[(
r
m
) -
r
f
] +
e
i
E
(
e
i
)
=
0
The variance of
e
i
is
not
zero
Why?
Decomposing Risk
Recall that an asset’s variance can be decomposed into two types of
variances…
Systematic risk and Idiosyncratic risk
Total Risk = Systematic Risk + Idiosyncratic Risk
Decomposing Risk
Systematic risk can be hedged by taking an offsetting position in an asset with the same beta
Idiosyncratic risk cannot be hedged
Hence, we need to consider it when deciding how much to invest in a risky asset
Idiosyncratic risk is the variance in returns that is
not
explained by beta
Hedging Systematic Risk
Consider a stock with β= and α=
r
f
is 6%, and
r
m
is 14
%
We can use the market portfolio and the risk free asset to create a portfolio with
β=
How?
Getting rid of Systematic Risk
If we buy the asset and short the hedging portfolio in equal amounts, what is the β and expected return of the total portfolio?
Answer: β=0 and
α+
r
f
= 8%
The proceeds
from the short sale
are
invested in a
safe
asset, so we assume the investor
also gets
r
f
There is no systematic risk
(
β=0
)
B
ut
there is idiosyncratic risk!
Lets build on y*
The investor has
r
m
and
r
f
, as before
The investor also has
n
mispriced securities that she can invest in
Assume all systematic risk is hedged
Expected returns are therefore
α
i
+
r
f
Variance is therefore only idiosyncratic,
σ
2
i,e
Recall from Portfolio Theory…
The Arbitrager’s Portfolio Problem
The Arbitrageur's Solution
The Arbitrageur's Solution
The arbitrageur therefore invests in the risk-free asset, the market, and mispriced assets
The arbitrage portfolio weights trade-off the benefit of alpha with the cost of idiosyncratic risk
Utility is maximized by investing some amount in each mispriced asset
Investing in mispriced assets increases utility; results in
a steeper CAL than the
CML
A CAL with Arbitrage
M (market or Passive portfolio)
A (Active
portfolio)
(Optimal risky Portfolio) P
E(r)
CML
CAL
R
f
Idiosyncratic
R
isk and Arbitrage
Idiosyncratic
risk is in the
denominator of the portfolio weight; investor’s dislike risk
Most
of a stock’s variance is
idiosyncratic
High
idiosyncratic risk stocks therefore get fewer arbitrage resources
Across
stocks, those with higher idiosyncratic risk should maintain higher
mispricing
What You Should Know
What the CML and SML are
Idiosyncratic risk vs. systematic risk in a CAPM world
Estimating β and α
How to use the CAPM to estimate a discount rate
The equivalence of discount rates and expected returns
How
to invest
optimally
in mispriced assets
