FM250 - Finance
The CAPM
Dong Lou
London School of Economics
LSE Summer School
Topics Covered
• The CAPM
• Measuring betas
• Use of the CAPM
• Empirical evidence on the CAPM
• Multifactor models
How does the investor choose?
Increasing
Return utility for
risk averse
investor
Low Risk High Risk
High Return High Return
Low Risk High Risk
Low Return Low Return
Risk
Efficient Frontier
Return
Portfolio Return rT
T
.
Efficient Portfolio
Risk Free
rf
Return
Risk
The Capital Asset Pricing Model
• The CAPM accomplishes two goals:
– To give an economic meaning/interpretation to
the tangency portfolio.
– To determine the expected returns of assets and
portfolios of assets by using some nice features
of the tangency portfolio.
The Capital Asset Pricing Model
• Assumptions:
– There are N stocks and one riskless asset in the
economy (i.e., lending and borrowing at a single
riskless rate);
– Each investor holds a mean-variance efficient
portfolio, i.e., a portfolio with the highest expected
return given volatility;
– Investors have a one-period horizon;
– Investors have the same beliefs;
– Market clears, i.e., demand = supply.
The Capital Asset Pricing Model
• As we discussed, all investors hold a combination of the
risk free asset and the tangency portfolio (with different
weights).
• If we add up all investors, and consider them as a single
group, this group also invests in the risk free asset and the
tangency portfolio (with some average weights).
• Hence, the total demand for risky assets is represented by
the tangency portfolio.
The Capital Asset Pricing Model
• We now characterize the supply of stocks.
• Suppose, there are N stocks in the economy and stock i has
market capitalization (number of shares times share price) Vi.
• The proportion of wealth allocated to asset i is given by:
Vi
M
w =
i .
V1 + ... + VN
M
w
• The market portfolio has portfolio weight i in stock i.
• We can approximate the market portfolio by a value-
weighted index of all stocks traded in the market.
The Capital Asset Pricing Model
• The supply of all risky assets is the market portfolio.
• Matching demand and supply, we see that the market
portfolio is the tangency portfolio!
• Said differently, market weights are determined by stocks’
expected returns, standard deviations, and return
correlations.
Capital Market Line
Return
Market Return rM
M
.
Efficient Portfolio
Risk Free
rf
Return
Risk
CAPM: derivation by intuition
• Recall that investors are maximizing the
expected returns while minimizing the risk,
in a mean-variance analysis framework.
• Now let’s derive CAPM by perturbation.
CAPM: derivation by intuition
• If you increase the weight of asset j in your portfolio (a
combination of the market portfolio and the risk free
asset) by Δ and borrow at the risk free rate:
– Then expected returns increase by:
∆ ( E ( rj ) − rf )
– Then the variance of the portfolio increases by:
2∆Cov ( rj , rM )
– Hence, the return/risk gain is:
∆ ( E ( rj ) − rf ) ( E ( rj ) − rf )
=
2∆Cov ( rj , rM ) 2Cov ( rj , rM )
– This must be the same for all assets. Why?
CAPM: derivation by intuition
• Suppose that for two assets A and B:
E (rA ) − rf E (rB ) − rf
>
Cov(rA , rM ) Cov(rB , rM )
– Asset A offers a better return/risk ratio than asset B
• Buy A, sell B
• What if everybody does this?
– Hence, in equilibrium, all return/risk ratios must be equal for all
assets
E (rA ) − rf E (rB ) − rf
=
Cov(rA , rM ) Cov(rB , rM )
CAPM: derivation by intuition
• If the risk-return trade-off is the same for all assets, then
the equality also holds for the market:
E (rA ) − rf E (rB ) − rf E (rM ) − rf
= =
Cov(rA , rM ) Cov(rB , rM ) Var (rM )
This gives the relationship between risk and expected
return for individual stocks and portfolios.
Cov( rA , rM )
E ( rA ) = rf +
Var ( rM )
( E (rM ) − rf ) = rf + β A ( E (rM ) − rf )
Cov( rA , rM )
βA =
Var ( rM )
This is called the Security Market Line.
Security Market Line
Expected
Return
Expected
Market
Return
Expected
market risk
premium
Risk free
rate
0 0.5 1.0 Beta
Expected Risk free Beta Expected market
= + x
return rate factor risk premium
Implications of CAPM
• If β = 0 then E[rj ] = rf as the asset does not contribute to the
riskiness of an efficient portfolio.
– Risks associated with this asset can be diversified away, and
hence investors do not require a risk premium for holding this
asset.
• If β > 0 then E[rj ] > rf as the asset increases the riskiness of an
efficient portfolio.
– Investors require a risk premium for holding the asset.
• If β < 0 then E[rj ] < rf as the asset decreases the riskiness an
efficient portfolio.
– The asset is valuable, and investors are willing to buy it even if
its expected return is lower than the risk free rate.
Measuring Betas
Market Portfolio – the portfolio of all assets in the
economy. In practice a broad stock market index,
such as the S&P Composite, is used to represent the
market.
Beta – How do we estimate beta? A linear regression.
Beta measures the sensitivity of a stock’s return to the
return on the market portfolio.
Measuring Betas
Rate of Return
on Stock A Slope = Beta
x x
x
x x
x x x
x
x
x x Rate of Return
on the Market
x x
x
Jan 1995
Measuring Betas
Dell Computer
Dell return (%)
Price data: May 91- Nov 97
R2 = .10
B = 1.87
Slope determined from plotting the Market return (%)
line of best fit.
Measuring Betas
Dell Computer
Dell return (%)
Price data: Dec 97 - Apr 04
R2 = .27
B = 1.61
Slope determined from plotting the Market return (%)
line of best fit.
Measuring Betas
General Motors
GM return (%)
Price data: May 91- Nov 97
R2 = .07
B = 0.72
Market return (%)
Slope determined from plotting the
line of best fit.
Measuring Betas
General Motors
GM return (%)
Price data: Dec 97 - Apr 04
R2 = .29
B = 1.21
Slope determined from plotting the Market return (%)
line of best fit.
Betas of selected common stocks
Stock Beta Stock Beta
AT&T 0.96 Ford Motor 1.03
Boston Ed. 0.49 Home Depot 1.34
BM Squibb 0.92 McDonalds 1.06
Delta Airlines 1.31 Microsoft 1.20
Digital Equip. 1.23 Nynex 0.77
Dow Chem. 1.05 Polaroid 0.96
Exxon 0.46 Tandem 1.73
Merck 1.11 UAL 1.84
Betas based on 5 years of monthly returns.
Uses of the CAPM
• Estimate expected returns for:
– Project evaluation, project selection
– Benchmark to evaluate investment strategies
(risk adjustment)
• It tells us what the required return on a stock should
be, accounting for risk.
– Performance evaluation and attribution of
money managers.
Uses of the CAPM
• The CAPM gives us a way to estimate the
expected (or required) rate of return on equity.
( ) [
E rj = r f + β j E( rM ) − r f ]
• We need estimates of three things:
– Risk-free interest rate, rf.
– Market risk premium, [E(rM)-rf].
– Beta for the stock, βj.
Uses of the CAPM
• A company’s cost of capital can be estimated
by the CAPM required return
SML
Required
return
13
Company Cost
of Capital
5.5
0
Project Beta
1.26
Testing the CAPM
• Let’s sort all stocks by beta into ten portfolios:
– Portfolio 1 contains 10% of all stocks with the lowest beta;
– Portfolio 2 contains 10% of stocks with the next-lowest beta ;…;
– Portfolio 10 contains 10% of stocks with the highest beta.
• We now look at portfolios formed in 1931 and held for 34
years and portfolios formed in 1966 and held for 39 years.
• We see that the relationship for CAPM is weaker since the
mid-1960s, especially for high-beta portfolios.
Testing the CAPM
Beta vs. Average Risk Premium
Avg Risk Premium
1931-2002
30
SML
Empirical SML
20
10
Market
Portfolio
0
Portfolio Beta
1.0
Testing the CAPM
Beta vs. Average Risk Premium
Avg Risk Premium
1931-65 SML
30
Empirical SML
20
10 Market
Portfolio
0
Portfolio Beta
1.0
Testing the CAPM
Beta vs. Average Risk Premium
Avg Risk Premium
1966-2002
30
20 Empirical SML SML
10
Market
0 Portfolio
Portfolio Beta
1.0
Empirical evidence on the CAPM
• Some evidence that expected returns increase with risk
• But at the same time:
– high beta stocks have lower returns than predicted by
the CAPM
– low beta stocks have higher returns than predicted by
the CAPM
Empirical evidence on the CAPM
• Is CAPM dead?
– Theory says that true betas are related to average returns.
– We regress average returns on estimated betas.
• Estimation error in betas may explain part of this problem
– We use a proxy for the market return (the “Roll critique”)
• Market should include bonds, real estate, foreign assets, human
capital, …
• The real market portfolio is unobservable, thus the CAPM cannot be
tested!
Empirical evidence on the CAPM
• CAPM is imperfect, but still useful in the estimation of the
costs of capital.
• It has been found in the data that other sources of risk which are
not captured by the market beta also matter.
• We can try to improve CAPM by:
– Accounting for market frictions that limit borrowing,
lending, and short selling;
– Accounting for consumption (Consumption CAPM);
– Accounting for multiple periods (Intertemporal CAPM);
– Accounting for additional sources of risk (APT).
Multifactor Pricing Models
• Alternative to the CAPM.
Include more factors in the regression, beyond the market
factor. For example:
FAt = RM,t – rf
FBt = Rsmall,t – Rbig,t.
FCt = Rhigh,t – Rlow,t.
• As in the CAPM, the betas (sensitivities) are the
coefficients of the regression
Rit = αi + βAi FAt + βBi FBt + βCi FCt + εit
Multifactor Pricing Models
Dollars
(log scale)100
High-minus low book-to-market
10
Small minus big
1
1926
1936
1946
1956
1966
1976
1986
1996
2003
0.1
Multifactor Pricing Models
How to use a multi-factor model:
• Construct a series of factors Fkt through time
• Examples of factors:
(1) Macroeconomic and financial variables: Chen,
Roll, and Ross (1986)
(2) Return spreads between index portfolios formed
according to cross-sectional sorts based on size and
book-to-market: Fama and French (1993)
• Estimate the sensitivities bik for each asset i by
running a time-series regression of Rit on the Fkt.
Multifactor Pricing Models
Estimated risk premia for macroeconomic risk factors
(1978-1990)
Factor Estimated Risk Premium
Term spread 5.10%
Market 6.36%
Exchange rate -0.59%
Real GNP growth 0.49%
Inflation -0.83%
Multifactor Pricing Models
• An example: you are considering making investment in the
pharmaceutical industry. Your project is expected to have the
same risk exposures as an average firm in this industry.
– Fama-French betas for this industry are given by:
βmarket=0.68, βSMB=-0.62, βHML=-0.43;
– Factor premiums can be estimated from historical data
(1926-2006):
rM − rf = 7%, rSMB = 3.7%, rHML = 5.2%;
– What is the expected return (cost of capital) of your
project?
Multifactor Pricing Models
• Based on the multifactor pricing model:
rnew = rf + βmarket × 7% + βSMB × 3.7% + βHML × 5.2%
= 5% + 0.68 × 7% − 0.62 × 3.7% − 0.43 × 5.2%
= 5.23%.
