McGraw-Hill/Irwin

CHAPTER 7

Capital Asset Pricing and Arbitrage Pricing Theory

© 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.

The Markowitz Model The Previous Chapter Outlined a Version of the Markowitz Model The next group of slides summarizes that approach 7-2

Markowitz Portfolio Selection Model First: identify the efficient frontier, the boundary of the portfolio choice set Second: add the risk free asset, and find the “best” CAL, the one maximizing the Sharpe ratio. This gives the optimal risky portfolio.

Third, pick the point on CAL the determines the mix of risk-free asset and our optimal risky portfolio, using indifference curves.

7-3

Figure 7.10 The Minimum-Variance Frontier of Risky Assets 7-4

Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL 7-5

Figure 7.12 The Efficient Portfolio Set 7-6

Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set 7-7

The Separation Property The portfolio choice problem can be separated into two independent tasks 1. Determine the optimal risky portfolio 2. Determine the complete portfolio, the allocation between the optimal risky portfolio and the risk free asset 7-8

More on Diversification and Covariance Covariance is like an aggregate risk element.

Consider diversification by holding equally weighted portfolio of n securities in two situations, one with uncorrelated securities, one with positively correlated securities With correlation = 0, adding securities always reduces portfolio risk With positive correlation, adding securities eventually does not reduce risk. There is a minimum risk, and it is not zero.

7-9

Portfolio Risk and Covariance  2

p



E

{[( 1 2

R

1  1 2 (

R

2 )] 2 }   1 2  2  1 2 cov 1 4 [

E

(

R

1 2 ) 

E

(

R

2 2 )  2

E

(

R

1

R

2 )]  2

p



E

{[( 1 3

R

1  1 3

R

2  1 3

R

3 )] 2 }  1 9 [ 3

E

(

R i

2 )  6

E

(

R

1

R

2 )]  1 3  2  2 3 cov  2

p



E

{[( 1 4

R

1  1 4

R

2  1 4  2  3 4 cov  1 4

R

3  1 4

R

4 )] 2 }  1 16 [ 4

E

(

R i

2 )  12

E

(

R

1

R

2 )] 7-10

Portfolio Risk and Covariance General Rule given below Note the behavior as n -> infinity  2

p

 1 

n

2  (

n

 1 ) cov

n

7-11

Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes 7-12

7.1 THE CAPITAL ASSET PRICING MODEL

7-13

Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern financial theory Derived using principles of diversification with simplified assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development 7-14

Assumptions Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, no transaction costs 7-15

Assumptions (cont.) Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations i.e. all have same expectations and agree on expected returns and risk of returns 7-16

Resulting Equilibrium Conditions All investors will hold the same portfolio for risky assets – the market portfolio The market portfolio contains all securities, and the proportion of each security is its market value as a percentage of total market value 7-17

Resulting Equilibrium Conditions (cont.) 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 7-18

Figure 7.1 The Efficient Frontier and the Capital Market Line 7-19

The Risk Premium of the Market Portfolio M r f E(r M ) - r f = = = Market portfolio Risk free rate Market risk premium E(r M ) - r f  M = Market price of risk = Slope of the CAPM 7-20

Expected Returns On Individual Securities The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio 7-21

Expected Returns On Individual Securities: an Example Using the Dell example: Rearranging gives us the CAPM’s expected return-beta relationship 7-22

Figure 7.2 The Security Market Line and Positive Alpha Stock 7-23

SML Relationships b = [COV(r i ,r m )] /  m 2 E(r m ) – r f = market risk premium SML = r f + b [E(r m ) - r f ] 7-24

Sample Calculations for SML E(r m ) - r f = .08; r f = .03

b x = 1.25

E(r x ) = .03 + 1.25(.08) = .13 or 13% b y = .6

E(r y ) = .03 + .6(.08) = .078 or 7.8% 7-25

Graph of Sample Calculations

E(r) R x =13% R m =11% R y =7.8% 3% SML .08

ß .6

ß y 1.0

ß m 1.25

ß x

7-26

7.2 THE CAPM AND INDEX MODELS

7-27

Estimating the Index Model Using historical data on T-bills, S&P 500 and individual securities Regress risk premiums for individual stocks against the risk premiums for the S&P 500 Slope is the beta for the individual stock 7-28

Table 7.1 Monthly Return Statistics for T-bills, S&P 500 and General Motors 7-29

Figure 7.3 Cumulative Returns for T-bills, S&P 500 and GM Stock 7-30

Figure 7.4 Characteristic Line for GM 7-31

Table 7.2 Security Characteristic Line for GM: Summary Output 7-32

GM Regression: What We Can Learn GM is a cyclical stock Required Return: Next compute betas of other firms in the industry 7-33

Predicting Betas The beta from the regression equation is an estimate based on past history Betas exhibit a statistical property – Regression toward the mean 7-34

THE CAPM AND THE REAL WORLD

7-35

CAPM and the Real World The CAPM was first published by Sharpe in the

Journal of Finance

in 1964 Many tests of the theory have since followed including Roll’s critique in 1977 and the Fama and French study in 1992 7-36

7.4 MULTIFACTOR MODELS AND THE CAPM

7-37

Multifactor Models Limitations for CAPM Market Portfolio is not directly observable Research shows that other factors affect returns 7-38

Fama French Three-Factor Model Returns are related to factors other than market returns Size Book value relative to market value Three factor model better describes returns 7-39

Table 7.3 Summary Statistics for Rates of Return Series 7-40

Table 7.4 Regression Statistics for the Single-index and FF Three-factor Model 7-41

7.5 FACTOR MODELS AND THE ARBITRAGE PRICING THEORY

7-42

Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero beta investment portfolio with a return greater than the risk-free rate If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high-priced portfolio In efficient markets, profitable arbitrage opportunities will quickly disappear 7-43

Figure 7.5 Security Line Characteristics 7-44

APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models 7-45