15.401

15.40115.401 FinanceFinance TheoryTheory MIT Sloan MBA Program

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

Lectures 15–17: The CAPM and APT

© 2007–2008 by Andrew W. Lo Critical Concepts 15.401 ƒ Review of Portfolio Theory ƒ The Capital Asset Pricing Model ƒ The Arbitrage Pricing Theory ƒ Implementing the CAPM ƒ Does It Work? ƒ Recent Research ƒ Key Points

Reading ƒ Brealey and Myers, Chapter 8.2 – 8.3

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 2 Review of Portfolio Theory 15.401 Risk/Return Trade-Off ƒ Portfolio risk depends primarily on covariances – Not stocks’ individual volatilities ƒ Diversification reduces risk – But risk common to all firms cannot be diversified away ƒ 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) ƒ Suppose all investors hold the same portfolio M; what must M be? – M is the market portfolio ƒ Proxies for the market portfolio: S&P 500, Russell 2000, MSCI, etc. – Value-weighted portfolio of broad cross-section of stocks

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 3 Review of Portfolio Theory 15.401

2.4%

1.8% Motorola Tangency rn u

t portfolio M e IBM R

d 1.2% e

t GM c e p

Ex 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

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 4 The Capital Asset Pricing Model 15.401 Implications of M as the Market Portfolio ƒ Efficient portfolios are combinations of the market portfolio and T-Bills ƒ Expected returns of efficient portfolios satisfy:

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

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 5 The Capital Asset Pricing Model 15.401 Implications of M as the Market Portfolio ƒ For any asset, define its market beta as:

ƒ Then the Sharpe-Lintner CAPM implies that:

ƒ Risk/reward relation is linear! ƒ Beta is the correct measure of risk, not sigma (except for efficient portfolios); measures sensitivity of stock to market movements

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 6 The Capital Asset Pricing Model 15.401 The Security Market Line

ƒ Implications:

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 7 The Capital Asset Pricing Model 15.401 What About Arbitrary Portfolios of Stocks?

ƒ Therefore, for any arbitrary portfolio of stocks:

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 8 The Capital Asset Pricing Model 15.401 We Now Have An Expression for the: ƒ Required rate of return ƒ Opportunity cost of capital ƒ Risk-adjusted discount rate

ƒ Risk adjustment involves the product of beta and market risk premium

ƒ Where does E[Rm] and Rf come from?

© 2007–2008 by Andrew W. Lo 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?

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 10 The Capital Asset Pricing Model 15.401

Security Market Line

25%

20% n r u t 15% e R

ed β 10% = 1, Market Portfolio ect p x E 5%

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

© 2007–2008 by Andrew W. Lo 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: ƒ Suppose three mutual funds have the same average return of 15% ƒ Suppose all three funds have the same volatility of 20% ƒ Are all three managers equally talented? ƒ Are all three funds equally attractive? 25%

20% n

r AB u t 15% e R

d C e t

c 10% β = 1, Market Portfolio e p Ex 5%

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

© 2007–2008 by Andrew W. Lo 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?

© 2007–2008 by Andrew W. Lo 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

n 10 r u t Re e

v 8 i t a l u

m 6 Cu

4

2

0

5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 0 0 0 n- n- an- an- an- an- an- an- an- an- an- an- an- an- an- an- an- an- J J Ja J J J J J J J J J J J Ja J J J Month

XYZ S&P 500

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 14 The Arbitrage Pricing Theory 15.401 What If There Are Multiple Sources of Systematic Risk? ƒ Let returns following a multi-factor linear model:

ƒ Then the APT implies the following relation:

ƒ Cost of capital depends on K sources of systematic risk

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 15 The Arbitrage Pricing Theory 15.401 Strengths of the APT ƒ Derivation does not require market equilibrium (only no-arbitrage) ƒ Allows for multiple sources of systematic risk, which makes sense

Weaknesses of the APT ƒ No theory for what the factors should be ƒ Assumption of linearity is quite restrictive

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 16 Implementing the CAPM 15.401 Parameter Estimation: ƒ Security market line must be estimated ƒ One unknown parameter: β ƒ Given return history, β can be estimated by linear regression:

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 17 Implementing the CAPM 15.401

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 18 Does It Work? 15.401

Biogen vs. VWRETD

40%

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

© 2007–2008 by Andrew W. Lo 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%

© 2007–2008 by Andrew W. Lo 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?

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 21 Does It Work? 15.401

Size-Sorted Portfolios, 1960 – 2001

1.40

1.30 s n

r 1.20 tu e 1.10 y R l h

t 1.00 n o 0.90 e M

ag 0.80 ver

A 0.70

0.60 0.70 0.90 1.10 1.30 1.50 1.70 Beta

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 22 Does It Work? 15.401

Beta-Sorted Portfolios, 1960 – 2001

18%

16% s n r 14% tu e

R 12% al u n

n 10% e A 8% erag v

A 6%

4% 0.50 0.70 0.90 1.10 1.30 1.50 1.70 Beta

© 2007–2008 by Andrew W. Lo 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 Low23456789High Firms sorted by ESTIMATED BETA

© 2007–2008 by Andrew W. Lo 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 Low23456789High Firms sorted by ESTIMATED VOLATILITY

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 25 Recent Research 15.401 Other Factors Seem To Matter ƒ Book/Market (Fama and French, 1992) ƒ Liquidity (Chordia, Roll, and Subrahmanyam, 2000) ƒ Trading Volume (Lo and Wang, 2006)

But CAPM Still Provides Useful Framework For Applications ƒ Graham and Harvey (2000): 74% of firms use the CAPM to estimate the cost of capital ƒ Asset management industry uses CAPM for performance attribution ƒ Pension plan sponsors use CAPM for risk-budgeting and asset allocation

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 26 Key Points 15.401 ƒ Tangency portfolio is the market portfolio ƒ This yields the capital market line (efficient portfolios)

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

ƒ The security market line yields a measure of risk: beta ƒ This provides a method for estimating a firm’s cost of capital ƒ 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 ƒ Empirical research is mixed, but the framework is very useful

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 27 Additional References 15.401

ƒ Bernstein, 1992, Capital Ideas. New York: Free Press. ƒ Bodie, Z., Kane, A. and A. Marcus, 2005, Investments, 6th edition. New York: McGraw-Hill. ƒ Brennan, T., Lo, A. and T. Nguyen, 2007, Portfolio Theory: A Review, to appear in Foundations of Finance. ƒ Campbell, J., Lo, A. and C. MacKinlay, 1997, The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. ƒ Chordia, T., Roll, R. and A. Subrahmanyam, 2000, “Commonality in Liquidity”, Journal of Financial Economics 56, 3–28. ƒ Fama, Eugene F., and Kenneth French, 1992, The cross section of expected stock returns", Journal of Finance 47, 427–465. ƒ Grinold, R. and R. Kahn, 2000, Active Portfolio Management. New York: McGraw-Hill. ƒ Lo, A. and J. Wang, 2006, “Trading Volume: Implications of an Intertemporal Capital Asset Pricing Model”, Journal of Finance 61, 2805–2840.

© 2007–2008 by Andrew W. Lo Lectures 15–17: The CAPM and APT Slide 28 MIT OpenCourseWare http://ocw.mit.edu

15.401 Finance Theory I Fall 2008

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