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  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 Investments 11 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[r P ] M E[r M ] r f



M

Investments 11

CML



P

5

CAPM – a Single Factor Model

 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

Investments 11 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  is a function of the covariance of returns with the assets that make up the market portfolio  is equal to one for market portfolio Investments 11 7

Security Market Line (SML)

 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

Investments 11 b

M

= 1.0

b

i

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 :

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 %  13 % )  3 %  0 .

6  8 %  7 .

8 % Investments 11 9

Security Market Line (SML)

 Graph of Samples

E(r) SML r x =13% r M =11% r y =7.8% r f =3% Market risk premium: 8%

Investments 11 b

y =0.6

b

M =1.0

b

x =1.25

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

R i

 a

i

b

i R M



e i

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

r_i (GM) r_M (Mkt) r_f (Tbill) r_i - r_f 6.06% 7.89% 0.65% 5.41% r_M - r_f 7.24% -2.86% -8.18% -7.36% 7.76% 0.52% -1.74% -3.00% 1.51% 0.23% -0.29% 5.58% 1.73% -0.21% -0.36% 0.58% 0.62% 0.72% 0.66% 0.55% 0.62% 0.55% -3.44% -8.80% -8.08% 7.10% -0.03% -2.36% -3.55% 0.93% -0.39% -1.01% 4.92% 1.18% -0.83% -0.91% -0.56% -0.37% 6.93% 3.08% 0.02% 4.97% -3.58% 4.62% 6.85% 4.55% 2.38% 3.33% 0.60% 0.65% 0.61% 0.65% 0.62% 0.05% -1.16% -1.02% 6.32% 2.43% -0.60% 4.97% -4.18% 3.97% 6.24% 3.90% 1.76% 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

Investments 11 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:

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

 Risk free two bucks? I’ll take it anytime!

Investments 11 16

Application

 Graph of disequilibrium

E[r i ] 15%

a

= 2% r m =11% SML r f =3%

Investments 11

1.0

1.25

b 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