CAPM
Security Market Line
CAPM and Market Efficiency
Alpha ( a
) vs. Beta ( b
)


Capital Asset Pricing Model

An equilibrium model underlying modern finance theory

Based on diversification principle and simplified assumptions

Who developed it?

Markowitz: Nobel Prize

Sharpe: Nobel Prize

Treynor, Lintner and Mossin
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
Assumptions

Individual investors are price takers

Individual’s action inconsequential to stock prices

Single-period investment horizon

Investors maximize expected utility

Homogeneous expectations

Investors do not know the actual outcome

Investors agree on the likelihood of each outcome

Investors risk aversion may be different

Market is frictionless

No taxes, and transaction costs
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
Resulting Equilibrium Outcome

All investors will hold the same portfolio for risky assets – the market portfolio

Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Risk premium on the market depends on the average risk aversion of all market participants

Risk premium on an individual security is a function of its covariance with the market
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
Capital Market Line
E[r
P
]
M
E[r
M
] r f

M
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CML

P
5


CAPM is just a single factor model!
E [ r i
]
 r f
 b i
( E [ r
M
]
 r f
)
M : Market portfolio r f
: Risk
E [ r
M
]
 r f
E [ r i
]
 r f
free rate
:
: Market
Risk risk premium premium of security i
E [ r
M

]

M r f
: Market price of risk
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
Expected return on individual security

The risk premium on individual securities
 is equal to its expected return above the risk free rate of return
 depends on its contribution to the risk of the market portfolio
 depends on its level of systematic risk

The systematic risk
 is a function of the covariance of returns with the assets that make up the market portfolio
 is equal to one for market portfolio
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
Math and Graphical Representation
E(r i
)
E [ r i
]
 r f
 b i
( E [ r
M
]
 r f
) b i

Cov [
 r i
2
M
, r
M
]
SML
E(r
M
) r f b i
8 Investments 11 b
M
= 1.0


Sample calculations

Market risk premium is 8%, risk free rate is 3%, security x and y have beta of 1.25 and 0.6, what is the expected return of each based on CAPM?

Solution: risk free rate : r f

3 % market risk premium : E [ r
M
]
 r f

8 %


Security x:
E [ r x
]
 r f

Security y:
E [ r y
]
 r f
 b x
( E [ r
M b y
( E [ r
M
]
 r f
]
 r f
)
)

3 %

1 .
25

8 %

3 %

0 .
6

8 %


13 %
7 .
8 %
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
Graph of Samples
E(r)
SML r x
=13% r
M
=11% r y
=7.8% r f
=3%
Market risk premium: 8% b
10 Investments 11 b y
=0.6
b
M
=1.0
b x
=1.25


How to find beta?

Find the return data of individual stocks

Find the market return data

Find the T-bill data

Calculate the excess return of

Individual stocks

Market

Run the regression
R i
 a i
 b i
R
M
 e i
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
GM Example (is it such a good stock?)
Jun
Jul
Aug
Spet
Oct
Nov
Month
Jan
Feb
Mar
Apr
May
Dec
Mean
Std Dev.
alpha beta r_i (GM) r_M (Mkt) r_f (Tbill) r_i - r_f r_M - r_f
6.06% 7.89% 0.65% 5.41% 7.24%
-2.86% 1.51% 0.58% -3.44% 0.93%
-8.18% 0.23% 0.62% -8.80% -0.39%
-7.36% -0.29% 0.72% -8.08% -1.01%
7.76% 5.58% 0.66% 7.10% 4.92%
0.52% 1.73% 0.55% -0.03% 1.18%
-1.74% -0.21% 0.62% -2.36% -0.83%
-3.00% -0.36% 0.55% -3.55% -0.91%
-0.56% -3.58% 0.60% -1.16% -4.18%
-0.37% 4.62% 0.65% -1.02% 3.97%
6.93% 6.85% 0.61% 6.32% 6.24%
3.08%
0.02%
4.97%
4.55% 0.65% 2.43% 3.90%
2.38% 0.62% -0.60% 1.76%
3.33% 0.05% 4.97% 3.32%
-5.00%
Coeff
-0.03
1.14
Stan Err
0.01
0.31
t Stat
-2.24
3.68
P-value
0.05
0.00
8.00%
6.00%
4.00%
2.00%
0.00%
-4.00%
-6.00%
-8.00%
-10.00%
5.00% 10.00%
Regression Statistics
Multiple R
R Square
0.76
0.57
Adj R Square
Standard Error
Observations
0.53
0.04
12.00
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
If markets are perfectly efficient, there would be no non-zero alphas!

Did this stop people in search for alpha?
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Where can we see Alphas (and how to tell them from Betas)?
1) Traditional sources: Active Managers
Alpha production on top of benchmark

Alpha is integrated into the product, but is easily identifiable b
Benchmark Return
2) “Pure Alpha” sources: Hedge Funds

The product is the alpha, with or without some residual market beta a Alpha of strategy
Residual beta of strategy
(may be zero)
3) “Embedded Alpha” Sources: Private
Investments

The alpha is inseparable from the beta, but dispersion of returns among managers suggests that alpha exists and can be large
Alpha and Beta are integrated in strategy
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

Investments – Active vs. Passive

Alpha ( a
) vs. Beta ( b
)
Beta is easy – it is the market

Beta should be free

Hedge Funds manage to charge for b (and not just a token)…

Alpha is hard, but does it require frequent trading?


Not necessarily – it is about taking right long-term positions, and identifying underpriced factors
Good old “ Buy Low – Sell High ” always works!!!

Not having too many constraints helps
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
Suppose a security with b
= 1.25 is offering expected return of 15%, what’s your decision?

Solution:


According to SML (CAPM), it should offer 13% a
= 15% – 13%=2%


Under-priced: offering too high a rate of return for its level of risk, what to do?
What is then over-priced? – It is the market index!!!

Long a portfolio C of similar stocks and short a market portfolio!
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
How does it work?


Market portfolio: α
M
If portfolio C has α
C
= 0, and β
M
= 2%, β
C
=
= 1
1.25

Show me the money

Long $100 of portfolio C

Short $125 of the market portfolio

Net payoff
100

R
C

125

R
M

100

( a
C
 b
C
R
M
)

125

R
M

2

Riskfree two bucks? I’ll take it anytime!
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
Graph of disequilibrium
E[r i
]
15% a
= 2% r m
=11%
SML r f
=3% b
18 Investments 11
1.0
1.25


What is CAPM?

Market risk premium
 beta

What does CAPM tell us?

How to capture the excess risk adjusted return (non-zero a
)?
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