The Capital Asset Pricing Model Chapter 9

Our focus in this chapter CAPM theory versus what we actually observe in the data.

Review: What is the CAPM The CAPM is: Where: E(Ri) is the expected return for stock i rf is the risk free rate of interest Bi is the beta for security i E(Rm) is the expected return for the market In practice, we use historical data to estimate the risk free rate, a firm’s beta, and the expected market return.

How is the CAPM used? 1) Portfolio managers use it to get an idea of whether or not a stock is overpriced or underpriced. A firm’s alpha 2) Corporations use it to perform capital budgeting. Obtain cost of capital from the CAPM If IRR > k then accept project Where IRR = the rate at which all discounted future cash flows from a project equals zero.

Capital Asset Pricing Model (CAPM) It is the 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.

Assumptions Individual investors are price takers. Single-period investment horizon. Investments are limited to traded financial assets. No taxes and transaction costs.

Assumptions (cont’d) Information is costless and available to all investors. Investors are rational mean-variance optimizers (only mean and standard deviation matter, that is, the return distribution is normal) All investors have homogeneous expectations. Basic derivation: Max wealth subject to budget constraints. Obtain “first order conditions.” Solve for E(Ri)

Resulting Equilibrium Conditions All investors will hold the same portfolio of risky assets – the VW market portfolio. Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value.

Resulting Equilibrium Conditions (cont’d) Risk premium on the market depends on the average risk aversion of all market participants. As average risk aversion goes up, the market risk premium goes up.

Resulting Equilibrium Conditions (cont’d) Risk premium on an individual security is a function of its covariance with the market. That is, a stock’s risk premium is a function of its beta. Betai = [Cov (ri,rm)] / sm2

Capital Market Line E(r) CML M E(rM) rf  m

Slope and Market Risk Premium M = Market portfolio rf = Risk free rate E(rM) - rf = Market risk premium E(rM) - rf = Slope of the CML  M

Return and Risk For Individual Securities The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio. Wi[COV(ri,rm)] An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio. [E(rm) - rf]

Security Market Line E(r) SML E(rM) rf b bM = 1.0

SML Relationships = [COV(ri,rm)] / m2 Slope SML = E(rm) - rf = market risk premium SML = rf + [E(rm) - rf] Betam = [Cov (ri,rm)] / sm2 if ri =rm, then Cov(rm, rm)= psmsm = 1* sm2 = sm2 / sm2 = 1

Sample Calculations for SML E(rm) - rf = .08 rf = .03 x = 1.25 E(rx) = .03 + 1.25(.08) = .13 or 13% y = .6 E(ry) = .03 + .6(.08) = .078 or 7.8%

Graph of Sample Calculations E(r) SML Rx=13% .08 Rm=11% Ry=7.8% 3% b .6 by 1.0 1.25 bx

Example of estimating beta See class web page, Excel handouts, chapter 9, “How to calculate CAPM beta for IBM.” Regress excess monthly returns of IBM on the market excess returns and an intercept. Coefficient on the market excess returns is beta.

Beta In practice, most providers of betas use 3 to 5 years of monthly data to estimate betas. General Electric beta: http://finance.yahoo.com/q/ks?s=GE It’s “party time” with high beta stocks… http://www.nasdaq.com/article/4-high-beta-stocks-for-todays-surging-market-cm649487 Of course, what happens when the market goes down? High beta stock index: http://etfdb.com/index/sp-500-high-beta-index/

A CAPM example

Disequilibrium Example Suppose a security with a  of 1.25 is offering expected return of 15%. According to SML, it should be 13%. Under-priced: offering too high of a rate of return for its level of risk. Price should go up if expected return moves from 15% to 13%.

Disequilibrium Example Analyst consensus forecast E(r) SML 15% Positive alpha Rm=11% rf=3% b 1.0 1.25

Example

Extensions of the CAPM: Liquidity Illiquidity Premium Research supports a premium for illiquidity. Amihud and Mendelson

CAPM with a Liquidity Premium f (ci) = liquidity premium for security i f (ci) increases at a decreasing rate

Liquidity and Average Returns Average monthly spread between high and low liquidity stocks is about 70 basis points Average monthly return(%) Bid-ask spread (%)

Basic statistical tests to determine if a variable (like CAPM beta) is priced in the cross-section of expected returns Example

Study: Fama-French study on the CAPM Beta, Size, and Book-to-Market See class web page, handouts, chapter 9, “Fama-French study on the CAPM Beta, Size, and Book-to-Market ” Motivation Is average return positively related to CAPM beta (as theory says it should be)? Method Cross-sectional regressions and annual sorts on lagged betas. Individual firm betas are estimated by regressing monthly excess returns on the excess returns of the market.

Study:Fama-French study on the CAPM Beta, Size, and Book-to-Market

Study:Fama-French study on the CAPM Beta, Size, and Book-to-Market Conclusions CAPM beta does not appear to be positively related to the cross-section of future returns! Thus, the CAPM appears to not work. However, firm size (ME) and firm book-to-market (BE/ME) appear to do a good job predicting the cross-section of returns. Asset pricing models should instead focus on multi-factor models (chapter 10).

NEWS FLASH

Oh no! One more study to go for this chapter!

Study: Mutual Funds and Beta's Death See class web page, handouts, chapter 9, “Mutual Funds and Beta's Death” Motivation Could the behavior of investors (especially mutual fund managers) in-part cause beta to not be priced?

Study: Mutual Funds and Beta's Death The idea Investor inflows to mutual funds tend to be determined by fund performance. In general, the best performing funds get the biggest inflows, especially in up markets. If high beta stocks outperform low beta stocks in up markets, then fund managers may tilt their portfolios towards high beta stocks. If stock prices are elastic, then this buying of high beta stocks will push up prices and lower expected returns for the high beta stocks. Thus, the presence of actively managed mutual funds may cause beta risk to be priced to a lesser degree than predicted by the CAPM.

Study: Mutual Funds and Beta's Death The results According to previous research, beta’s relationship with the cross-section of future returns became much weaker in the 1980’s relative to previous periods. This coincided with the spectacular growth of mutual funds in the 1980’s. Data also shows that funds tend to overweight high beta stocks. This does not prove the hypothesis, but is consistent with it.

Some concluding thoughts on the CAPM Even thought the CAPM may not work empirically, it is still a good way to think about risk. The CAPM may not work empirically because our proxy for the market portfolio is imperfect – it is not the true VW portfolio of all wealth in the economy (which is what theory says it should be). The CAPM is still widely used in cost of capital estimations for capital budgeting, although the 3-factor model of Fama and French, and other 4 and even 5 factor models are gaining in popularity.