CAPM
1Finance - Pedro Barroso
Finance - Pedro Barroso
CAPM Assumptions• Frictionless markets
– No trading costs– No taxes
• Unlimited borrowing and lending– No restrictions on short sales
• Lending and borrowing rates are the same• Investors care only about means and variances• All investors are fully rational and have the same
information (homogeneous expectations)
2
Finance - Pedro Barroso
Market Equilibrium• Since
– Everyone have same efficient frontier– Everyone holds the same tangency portfolio (of
stocks)
• Then THE TANGENCY PORTFOLIO IS
THE MARKET PORTFOLIO
3
Market Portfolio
Finance - Pedro Barroso 4
Market Portfolio
Finance - Pedro Barroso 5
Finance - Pedro Barroso
Market Equilibrium• Every investor solves mean-variance problem and
holds a combination of risk-free asset and portfolio of risky assets (tangency)
• The sum of all investors’ risky portfolios will have the same weights as tangency
• In equilibrium the sum of all investors’ desired portfolios must equal the supply of assets
• Aggregate supply and demand of assets is the market portfolio
• Market portfolio is the tangency portfolio
6
Capital Market Line
• Investors choose a point along the line – Capital Market Line (CML)• Efficient portfolios are combination of the risk-free asset and the
market portfolio M
P
rf
E(re
turn
)CML
Marketportfolio M
PM
fMfP
rrErrE
][][
7Finance - Pedro Barroso
Capital Market Line
Where the investor chooses along the CML depends on his risk aversion, but all investors have the same CML
rf
M
CML
P
E(re
turn
)
8Finance - Pedro Barroso
Security Market Line• CML gives risk-return trade off for efficient portfolios• How is the relation for individual stocks (inneficient)?
– Individual stocks are below CML– Relation is named Security Market Line (SML)– Individual stock risk is measured by its covariance with
market portfolio because it is the marginal variance – How does a small increment to the weight of a stock
change the variance of the portfolio?– Likewise in economics, it is the marginal cost of goods that
determines their prices, not their total or average cost
9Finance - Pedro Barroso
Finance - Pedro Barroso
Security Market Line• Suppose you hold a portfolio M and are considering
adding a little more of asset i (weight w):
iMMiP
MiP
wwww
rEwrwErE
)1(2])1[()(
)()1()()(222
rf
M
CML
P
E(re
turn
)
i
10
Finance - Pedro Barroso
Security Market Line• Now evaluate change in risk and return
• In equilibrium (M) excess demand for stock i is zero; now evaluate change in risk and return for w = 0
iMMiP
MiP
wwwwσ
rrwr
)21(2)1(22
)(E)(E)E(
222
M
iMM
wP
PP
w
P
Miw
P
σσ
wσ
wσ
rrwr
222
)(E)(E)E(
2
02
2
0
0
11
Finance - Pedro Barroso
Security Market Line• Risk-return trade off in equilibrium (M)
• Risk-return trade off in equilibrium (M) given by CML is the same:
M
MiM
Mi
M
iMM
Mi
P
P
σ
rr
σ
rrwσwr
22
)(E)(E
222
)(E)(E)E(
fMM
iMfi
M
MiM
Mi
M
fM rrσ
rr
σ
rrσ
rr
)(E)(E)(E)(E)(E
22
12
Finance - Pedro Barroso
Beta• Security Market Line:
where
• Beta measures the responsiveness of a stock to movements in the market portfolio (i.e., systematic risk)
])([)E( fMifi rrErr
)()( ,
M
Mii rVar
rrCov
13
Expected Return on a Stock
])([β)( fMifi rrErrE
fr
1.0
)( MrE
14Finance - Pedro Barroso
)( irE
i
Expected Return on a Stock
• Assume i = 0, then• Assume i = 1, then• Assume i < 1, then• Assume i > 1, then
)()( Mi rErE
Expected return on
a stock= Risk-free
rate + Beta of stock × Market risk
premium
15Finance - Pedro Barroso
])([β)( fMifi rrErrE
fi rrE )(
)()( Mi rErE )()( Mi rErE
Expected Return on a Stock
i1.5
%5.13
%5.13%)3%10(5.1%3)( irE
16Finance - Pedro Barroso
%3
%rE%r Mfi 10)( 3 5.1
)( irE
Finance - Pedro Barroso
Beta of a Portfolio• Beta of a portfolio is a portfolio-weighted average of
individual assets
• Thus, we can use SML for any portfolio:
N
iiiP w
1
17
])([β)( fMPfP rrErrE
Finance - Pedro Barroso
Intuition behind the CAPM• High beta stocks are risky, and must therefore offer a
higher return on average to compensate for the risk• Why are high beta stocks risky?
– Because they pay up just when you need the money least, when the overall market is doing well
– And they loose money when you really need it – when the overall market is doing poorly
– If anyone is to hold this security, it must offer a high expected return
18
Risk or insurance?• Consider house insurance.
• Say it costs 0.5 to buy fire insurance. Then the expected return of the “investment” will be -80%. The standard deviation will be 632%. Beta is -333.33 (check).
Prob. WealthInsurance
Payoff
No fire 0,999 200 0
Fire 0,001 80 100
Finance - Pedro Barroso
CML vs. SML• CML plots the relation between expected returns and
standard deviation for efficient portfolios• SML is the relation between expected returns and • All portfolios, whether efficient or not, must lie on
the SML but only efficient portfolios are on the CML– with the same mean return can have different standard
deviations, but must have the same – in other words, the only relevant measure of risk for
pricing securities is (a measure of covariance or marginal variance)
20
Finance - Pedro Barroso
Market Proxy and Risk-free Rate• What market proxy?
– CAPM says it should be all the assets in the world– Typically people use broad, value-weighted stock
market index (e.g. S&P 500)
• What risk-free rate?– CAPM says it should be riskless and match the
horizon of the investment– People use short-term T-bill
21
Finance - Pedro Barroso
Market Premium• This is the hardest input to measure in the CAPM
equation• From January 1926 to December 2005, the market
risk premium has been 8.5% – Depending on the sample and on whether we use the
arithmetic or geometric mean, we can come up with numbers between 5% and 8%
• Can we trust this historical average?– Standard error of the estimate is 2.2%
22
Finance - Pedro Barroso
Estimation of Beta• i usually estimated using a time-series regression
• Typical R2=25%• Estimation issues
– Betas may change over time– Don’t use data from too long ago– Five years of weekly or monthly data is reasonable– Use Data Analysis / Regression or Linest in Excel
titftMiitfti rrrr ,,,,, )(
23
Finance - Pedro Barroso
Estimation of Risk• Standard deviation of stock returns can be break-
down into systematic risk and idiosyncratic risk
222
,,2
,
,,,2
,,
,,,,,
)()()(
)()()(
)(
iMii
titMiti
titftMitfti
titftMiitfti
VarrVarrVar
VarrrVarrrVar
rrrr
24
Finance - Pedro Barroso
MSFT Beta: Example
tMSFTtftMtftMSFT rrrr ,,,,, )(993.0002.0
Microsoft Office Excel 97-2003 Worksheet
25
Finance - Pedro Barroso
MSFT Example• Assume risk-free rate of 3% and equity premium of
6% then expected return (annual) on MSFT: 3% + 0.993 x 6%= 8.96%
• S&P 500 standard deviation is 3.71% and standard deviation of residuals is 5.54% then standard deviation on MSFT (annual):
(0.9932 x 0.03712 + 0.05542)1/2 x (12) 1/2 = 6.65% x (12) 1/2 = 23%
26
Finance - Pedro Barroso
MSFT Beta - Bloomberg
27
Finance - Pedro Barroso
Jensen’s Alpha• Excess return over that predicted by CAPM
• If alpha is positive– Security has earned a higher return on average than is required for its
level of risk (It has been attractive historically)– Could say that it was mispriced, but be careful drawing conclusions for
the futureOR– The security might not be mispriced, but rather the CAPM is wrong!
• Measure of portfolio performance
ff rrrr miii
28
Finance - Pedro Barroso
Testing the CAPM
• Take a large number of stocks or portfolios• Over some long time period, e.g. 1950-2000,
estimate alpha and beta for each of them by running a regression
• Then look at the alphas, are they statistically different from zero?
29
Finance - Pedro Barroso
Testing the CAPM• Fama and French (1992) finds even weaker results
– There does not seem to be any relation between and average returns, once you control for
• Size (market capitalization) • Ratio of book value of equity to market value (book-
to-market)
– Three-factor Fama-French model• Market return (rM)
• Small minus big (SMB) – small cap premium• High minus low (HML) – value premium
30
Finance - Pedro Barroso
Bottom Line
• Assumption of CAPM are restrictive• Gives a simple and elegant relation for
expected returns and a nice measure of risk• Research shows that it is not very accurate• But, widely used in corporate finance and
investments
31