Capital Asset Pricing Model
CAPM
Security Market Line
CAPM and Market Efficiency
Alpha (a) vs. Beta (b)
CAPM

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
Investments 11
2
CAPM

Assumptions

Individual investors are price takers


Single-period investment horizon


Investors maximize expected utility
Homogeneous expectations




Individual’s action inconsequential to stock prices
Investors do not know the actual outcome
Investors agree on the likelihood of each outcome
Investors risk aversion may be different
Market is frictionless

Investments 11
No taxes, and transaction costs
3
CAPM

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
Investments 11
4
CAPM

Capital Market Line
E[rP]
CML
E[rM]
M
rf
M
Investments 11
P
5
CAPM – a Single Factor Model

CAPM is just a single factor model!
E[ri ]  r f  b i ( E[rM ]  r f )
M : Market portfolio
r f : Risk - free rate
E[rM ]  r f : Market risk premium
E[ri ]  r f : Risk premium of security i
E[rM ]  r f
M
Investments 11
: Market price of risk
6
CAPM

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


Investments 11
is a function of the covariance of returns with
the assets that make up the market portfolio
is equal to one for market portfolio
7
Security Market Line (SML)

Math and Graphical Representation
E
[
r
]

r

b
(
E
[
r
]

r
)
i
f
i
M
f
E(r )
i
bi 
Cov[ri , rM ]
SML
 M2
E(rM)
rf
bM = 1.0
Investments 11
bi
8
Security Market Line (SML)

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 : rf  3%
market risk premium : E[rM ]  rf  8%

Security x:
E[rx ]  rf  b x ( E[rM ]  rf )  3%  1.25  8%  13%

Security y:
E[ry ]  rf  b y ( E[rM ]  rf )  3%  0.6  8%  7.8%
Investments 11
9
Security Market Line (SML)

Graph of Samples
E(r)
SML
rx=13%
rM=11%
ry=7.8%
Market risk premium: 8%
rf=3%
by=0.6 bM=1.0 bx=1.25
Investments 11
b
10
CAPM Estimation

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
Ri  ai  bi RM  ei
Investments 11
11
CAPM Estimation

GM Example (is it such a good stock?)
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Spet
Oct
Nov
Dec
Mean
Std Dev.
alpha
beta
Investments 11
r_i (GM) r_M (Mkt) r_f (Tbill) r_i - r_f r_M - r_f
8.00%
6.06%
7.89% 0.65% 5.41% 7.24%
6.00%
-2.86%
1.51% 0.58% -3.44% 0.93%
4.00%
-8.18%
0.23% 0.62% -8.80% -0.39%
-7.36%
-0.29% 0.72% -8.08% -1.01%
2.00%
7.76%
5.58% 0.66% 7.10% 4.92%
0.00%
0.52%
1.73% 0.55% -0.03% 1.18% -5.00%
0.00%
5.00%
10.00%
-2.00%
-1.74%
-0.21% 0.62% -2.36% -0.83%
-4.00%
-3.00%
-0.36% 0.55% -3.55% -0.91%
-6.00%
-0.56%
-3.58% 0.60% -1.16% -4.18%
-0.37%
4.62% 0.65% -1.02% 3.97%
-8.00%
6.93%
6.85% 0.61% 6.32% 6.24%
-10.00%
3.08%
4.55% 0.65% 2.43% 3.90%
0.02%
2.38% 0.62% -0.60% 1.76%
Regression Statistics
4.97%
3.33% 0.05% 4.97% 3.32%
Multiple R
0.76
R Square
0.57
Coeff Stan Err t Stat P-value
Adj R Square
0.53
-0.03
0.01
-2.24
0.05
Standard Error
0.04
1.14
0.31
3.68
0.00
Observations
12.00
12
CAPM and Market Efficiency

If markets are
perfectly
efficient, there
would be no
non-zero
alphas!

Did this stop
people in
search for
alpha?
Investments 11
13
Investments - It Is All about Alpha!


Investments – Active vs. Passive
 Alpha (a) vs. Beta (b)
Beta is easy – it is the market


Beta should be free!
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
Investments 11
14
Application - Disequilibrium Example


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!
Investments 11
15
Arbitrage – How to Get It Done

How does it work?


Market portfolio: αM = 0, and βM = 1
If portfolio C has αC = 2%, βC = 1.25




Show me the money
Long $100 of portfolio C
Short $125 of the market portfolio
Net payoff
100 RC  125 RM  100 (aC  b C RM )  125 RM  2

Investments 11
Risk-free two bucks? I’ll take it anytime!
16
Application

Graph of disequilibrium
SML
E[ri]
15%
a = 2%
rm=11%
rf=3%
b
1.0
Investments 11
1.25
17
Wrap-up

What is CAPM?




Market risk premium
beta
What does CAPM tell us?
How to capture the excess risk adjusted
return (non-zero a)?
Investments 11
18