15.

401

15.401 Finance Theory

MIT Sloan MBA Program

Andrew W. Lo
Harris & Harris Group Professor, MIT Sloan School

Lectures 15–17: The CAPM and APT

Critical Concepts 15.401

 Review of Portfolio Theory

 The Capital Asset Pricing Model
 The Arbitrage Pricing Theory
 Implementing the CAPM
 Does It Work?
 Recent Research
 Key Points

Reading
 Brealey and Myers, Chapter 8.2 – 8.3

Lectures 15–17: The CAPM and APT Slide 2

Review of Portfolio Theory 15.401

Risk/Return Trade-Off
 Portfolio risk depends primarily on covariances
– Not stocks’ individual volatilities
 Diversification reduces risk
– But risk common to all firms cannot be diversified away
 Hold the tangency portfolio M
– The tangency portfolio has the highest expected return for a given
level of risk (i.e., the highest Sharpe ratio)
 Suppose all investors hold the same portfolio M; what must M be?
– M is the market portfolio
 Proxies for the market portfolio: S&P 500, Russell 2000, MSCI, etc.
– Value-weighted portfolio of broad cross-section of stocks

Lectures 15–17: The CAPM and APT Slide 3

Review of Portfolio Theory 15.401

2.4%

1.8% Motorola
Tangency
portfolio M
IBM
1.2%
Expected

GM
Return

0.6%

T-Bill

0.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Standard Deviation of
Return

Lectures 15–17: The CAPM and APT Slide 4

The Capital Asset Pricing Model 15.401

Implications of M as the Market Portfolio

 Efficient portfolios are combinations of the market portfolio and T-Bills
 Expected returns of efficient portfolios satisfy:

 This yields the required rate of return or cost of capital for efficient
portfolios!
 Trade-off between risk and expected return
 Multiplier is the ratio of portfolio risk to market risk
 What about other (non-efficient) portfolios?

Lectures 15–17: The CAPM and APT Slide 5

The Capital Asset Pricing Model 15.401

Implications of M as the Market Portfolio

 For any asset, define its market beta as:

 Then the Sharpe-Lintner CAPM implies that:

 Risk/reward relation is linear!

 Beta is the correct measure of risk, not sigma (except for efficient
portfolios); measures sensitivity of stock to market movements

Lectures 15–17: The CAPM and APT Slide 6

The Capital Asset Pricing Model 15.401

The Security Market Line

 Implications:

Lectures 15–17: The CAPM and APT Slide 7

The Capital Asset Pricing Model 15.401

What About Arbitrary Portfolios of Stocks?

 Therefore, for any arbitrary portfolio of stocks:

Lectures 15–17: The CAPM and APT Slide 8

The Capital Asset Pricing Model 15.401

We Now Have An Expression for the:

 Required rate of return
 Opportunity cost of capital
 Risk-adjusted discount rate

 Risk adjustment involves the product of beta and market risk premium
 Where does E[Rm] and Rf come from?

Lectures 15–17: The CAPM and APT Slide 9

The Capital Asset Pricing Model 15.401

Example:
Using monthly returns from 1990 – 2001, you estimate that Microsoft’s
beta is 1.49 (std err = 0.18) and Gillette’s beta is 0.81 (std err = 0.14).
If these estimates are a reliable guide going forward, what expected
rate of return should you require for holding each stock?

Lectures 15–17: The CAPM and APT Slide 10

The Capital Asset Pricing Model 15.401

Security Market Line

25%

20%

15%
Expected
Return

10% β = 1, Market Portfolio

5%

0%
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Beta

Lectures 15–17: The CAPM and APT Slide 11

The Capital Asset Pricing Model 15.401

The Security Market Line Can Be Used To Measure Performance:

 Suppose three mutual funds have the same average return of 15%
 Suppose all three funds have the same volatility of 20%
 Are all three managers equally talented?
 Are all three funds equally attractive?
25%

20%

A B
Expected Return

15%
C
10% β = 1, Market Portfolio

5%

0%
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Beta

Lectures 15–17: The CAPM and APT Slide 12

The Capital Asset Pricing Model 15.401

Example:
Hedge fund XYZ had an average annualized return of 12.54% and a
return standard deviation of 5.50% from January 1985 to December
2002, and its estimated beta during this period was −0.028. Did the
manager exhibit positive performance ability according to the CAPM?
If so, what was the manager’s alpha?

Lectures 15–17: The CAPM and APT Slide 13

The Capital Asset Pricing Model 15.401

Example (cont):
Cumulative Return of XYZ and S&P
500
January 1985 to December 2002
16

14

12

10
Cumulative Return

Month

XYZ S&P 500

Lectures 15–17: The CAPM and APT Slide 14

The Arbitrage Pricing Theory 15.401

What If There Are Multiple Sources of Systematic Risk?

 Let returns following a multi-factor linear model:

 Then the APT implies the following relation:

 Cost of capital depends on K sources of systematic risk

Lectures 15–17: The CAPM and APT Slide 15

The Arbitrage Pricing Theory 15.401

Strengths of the APT

 Derivation does not require market equilibrium (only no-arbitrage)
 Allows for multiple sources of systematic risk, which makes sense

Weaknesses of the APT

 No theory for what the factors should be
 Assumption of linearity is quite restrictive

Lectures 15–17: The CAPM and APT Slide 16

Implementing the CAPM 15.401

Parameter Estimation:
 Security market line must be estimated
 One unknown parameter: β
 Given return history, β can be estimated by linear regression:

Lectures 15–17: The CAPM and APT Slide 17

Implementing the CAPM 15.401

Lectures 15–17: The CAPM and APT Slide 18

Does It Work? 15.401

Biogen vs. VWRETD

40%

30%
y = 1.4242x -
0.0016
R 2 = 0.3336 20%

10%

0%
-20.0% -15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%
-10%

-20%

-30%

-40%

Lectures 15–17: The CAPM and APT Slide 19

Does It Work? 15.401

NASDAQ vs. VWRETD

20%

15%

10%

5%

0%
-20% -15% -10% -5% 0% 5% 10% 15% 20%
-5%

-10%

-15%

-20%

Lectures 15–17: The CAPM and APT Slide 20

Does It Work? 15.401

Market-Cap Portfolios:
Over the past 40 years, the smallest firms (1st decile) had an average
monthly return of 1.33% and a beta of 1.40. The largest firms (10th
decile) had an average return of 0.90% and a beta of 0.94. During the
same time period, the Tbill rate averaged 0.47% and the market risk
premium was 0.49%. Are the returns consistent with the CAPM?

Lectures 15–17: The CAPM and APT Slide 21

Does It Work? 15.401

Size-Sorted Portfolios, 1960 – 2001

1.40

1.30

1.20

1.10
Average Monthly

1.00

0.90
Returns

0.80

0.70

0.60
0.70 0.90 1.10 1.30 1.50 1.70
Beta

Lectures 15–17: The CAPM and APT Slide 22

Does It Work? 15.401

Beta-Sorted Portfolios, 1960 – 2001

18%

16%

14%

12%
Average Annual

10%
Returns

8%

6%

4%
0.50 0.70 0.90 1.10 1.30 1.50 1.70
Beta

Lectures 15–17: The CAPM and APT Slide 23

Does It Work? 15.401

Beta-Sorted Portfolios, 1926 – 2004

16.0

14.0

12.0

10.0

8.0

6.0

4.0
Low 2 3 4 5 6 7 8 9 High
Firms sorted by ESTIMATED BETA

Lectures 15–17: The CAPM and APT Slide 24

Does It Work? 15.401

Volatility-Sorted Portfolios, 1926 – 2004

16.0

14.0

12.0

10.0

8.0

6.0

4.0
Low 2 3 4 5 6 7 8 9 High
Firms sorted by ESTIMATED VOLATILITY

Lectures 15–17: The CAPM and APT Slide 25

Recent Research 15.401

Other Factors Seem To Matter

 Book/Market (Fama and French, 1992)
 Liquidity (Chordia, Roll, and Subrahmanyam, 2000)
 Trading Volume (Lo and Wang, 2006)

But CAPM Still Provides Useful Framework For Applications

 Graham and Harvey (2000): 74% of firms use the CAPM to estimate
the cost of capital
 Asset management industry uses CAPM for performance attribution
 Pension plan sponsors use CAPM for risk-budgeting and asset
allocation

Lectures 15–17: The CAPM and APT Slide 26

Key Points 15.401

 Tangency portfolio is the market portfolio

 This yields the capital market line (efficient portfolios)

 The CAPM generalizes this relationship for any security or portfolio:

 The security market line yields a measure of risk: beta

 This provides a method for estimating a firm’s cost of capital
 The CAPM also provides a method for evaluating portfolio managers
– Alpha is the correct measure of performance, not total return
– Alpha takes into account the differences in risk among managers
 Empirical research is mixed, but the framework is very useful

Lectures 15–17: The CAPM and APT Slide 27