Capital Asset Pricing Model Lecture 5 FINC3017 INVESTMENTS AND PORTFOLIO MANAGEMENT DR ANDREW AINSWORTH

Learning Objectives Where does the CAPM come from? Utility maximising investors + mean-variance portfolios + assumptions Why is the CAPM important? Equilibrium expected returns on risky assets are calculated by the covariance of the stock with the market What lessons can we learn from the CAPM? What happens if we relax the CAPM assumptions? If there is no risk-free rate we can obtain the Zero-beta CAPM Reading BKM Ch. 9 Fama and French (2004) “The Capital Asset Pricing Model: Theory and Evidence”, Journal of Economic Perspectives, Vol. 18, No. 3, pp. 25-46.

Assets and risk factors Assets are bundles of risk factors Each different factor defines a different set of bad times Investors exposed to losses during bad times are compensated by risk premiums in good times “Factors are to assets what nutrients are to food” Factors matter, not assets Understand the factor content of your assets Different investors need different risk factors If we endure the bad times we are compensated with risk premiums How does the CAPM define bad times? How would you define bad times?

Derivation of CAPM

CAPM assumptions There are many investors, each with an initial level of wealth which is small compared with the total level of wealth in the market (perfect competition) All investors plan for a single holding period of the same length Investments are restricted to a universe of publicly available assets (stocks, bonds) and may borrow and invest at a fixed risk-free rate There are no taxes or transactions costs All investors are mean-variance optimisers. This means they all use the Markowitz model to select their portfolios. Investors have quadratic utility functions or returns have a multivariate normal distribution All investors create their opinions in the same way. They thus have identical estimates for the probability distribution of assets returns and thus provide the exact same inputs into the Markowitz model to select their portfolios (homogenous expectations)

CAPM assumptions If all investors have the same information and form their beliefs in the same way then all investors find the exact same efficient frontier All investors have the same utility function with different risk aversion (A) With a risk-free asset the efficient frontier is the CML The tangency portfolio is the market portfolio (M) and contains all assets If all investors hold this portfolio it must be the market portfolio The market portfolio is the optimal risky portfolio Prices will adjust in equilibrium such that every stock is included in the market portfolio in proportion to its market weight

CAPM Derivation Please refer to the Lecture 5.pdf notes for details Assume there are N risky assets whose Expected returns are given by the vector e Finite variances are given by the positive definite matrix V Solve individual investor’s portfolio problem Choose portfolio weights (x) to maximise expected return (x’e) subject to a given portfolio variance (x’Vx = c) Gives the holding of mean variance efficient portfolios

CAPM Derivation Solving the optimisation problem yields the following first order conditions for a given efficient portfolio x:

CAPM Derivation The market return is an efficient portfolio, which implies: Where the market portfolio weights are represented by xm From here, we can find the expected return on the market portfolio by pre-multiplying by : Or

CAPM Derivation Since we know that return vector is a function of the risk free rate and standard deviation of the market we can calculate the expected return of any individual asset i We know the following is true as the market portfolio is efficient: Pre-multiplying by , where xi has a weight of one on asset i and zero on all other assets Or

CAPM Derivation From the equation for the expected return on the market: We can solve for lambda and substitute into the equation for the return on an individual asset: Substitute this expression for lambda into the equation for security i expected return:

CAPM Derivation Which can be rearranged: Or

Lessons from the CAPM Don’t hold individual assets, hold the factor Diversify away idiosyncratic risk as it is not priced Hold the market as it is efficiently diversified (it has the max Sharpe ratio) Same expectations implies same portfolio of risky assets just in different quantities depending on individual investor risk aversion The market is the factor because everyone holds this mean-variance efficient portfolio In equilibrium, prices changes so supply = demand Each investor has their own optimal exposure to factor risk Investor’s portfolio betas will differ The average investor holds the market portfolio

Lessons from the CAPM 4. Factor risk premium has an economic story As var(Rm) ↑ then E(Rm) ↑ so asset prices ↓ 5. Risk is factor exposure (i.e. beta) High beta means lower diversification benefit: Investors accept a lower return for low beta securities because they provide higher diversification benefits A lower expected return implies that investors are willing to pay a higher price for these assets In bad times, low beta assets pay off well and high beta assets pay off poorly

Lessons from the CAPM 6. The risk premium in the CAPM is a reward for investing in an asset that pays off in bad times Consider a high beta asset: High Rm → High Ri Low Rm → Low Ri But, investors are risk averse so they require a risk premium

The Security Market Line (SML) The equation for the CAPM plots a linear relationship between a stock’s, or a portfolio’s, beta and its expected return CAPM says all stocks lie on SML In reality, some do not The difference between the SML and a stock’s actual return is called alpha, a Stocks above the SML are positive a stocks Are these under- or over-priced? Stocks below the SML are negative a stocks Is the underpricing/overpricing retrospective or prospective?

The Security Market Line (SML) E(r) SML A rf B CAPM tells us that returns are linear in risk as measured by Beta. b

Extensions to CAPM Assumptions underlying CAPM can be relaxed: Cover taxes, differing borrowing and lending rates, no risk free asset, skewness, international markets, intertemporal effects, heterogenous expections etc. Basic structure of model survives Linearity Covariance of returns with risk factors, ie. systematic risk is what matters Sometimes a multi-factor model emerges What happens if there is no risk-free rate? Black, Fischer (1972) “Capital Market Equilibrium with Restricted Borrowing”, Journal of Business, vol. 45, pp.444-455. Investors choose different points on the efficient frontier

Derivation of Zero-Beta CAPM Assume investors want to hold mean-variance efficient portfolios Solve individual investor’s portfolio problem Choose portfolio weights (x) to maximise expected return (x’e) subject to a given portfolio variance (x’Vx = c) Solving the optimisation problem yields the following first order conditions for a given efficient portfolio x: Gives the holding of mean variance efficient portfolios

Derivation of Zero-Beta CAPM The market is an efficient portfolio, which implies: Where the market portfolio weights are represented by xm From here, we can find the expected return on the market portfolio by pre-multiplying by : Or

Derivation of Zero-Beta CAPM Since we know that return vector is a function of g and standard deviation of the market we can calculate the expected return of any individual asset i We know the following is true as the market portfolio is efficient: Pre-multiplying by , where xi has a weight of one on asset i and zero on all other assets Or

Derivation of Zero-Beta CAPM Using the equation for the expected return on the market we can solve for lambda and substitute into the equation for the return on an individual asset to yield: If we define xz to be a portfolio uncorrelated with the market portfolio and premultiply the following equation by : This provides us with the Black (1972) CAPM

Zero-Beta CAPM Combinations of efficient portfolios are also efficient portfolios Every efficient portfolio has a companion portfolio that lies on the bottom half of the mean-variance frontier with which it is uncorrelated For the market portfolio, this companion portfolio is called the zero-beta portfolio. To find this portfolio, we compute the tangent to the market portfolio and follow this to the expected return axis Moving horizontally along to the frontier provides the appropriate portfolio

Zero-Beta CAPM E(R) Q P Z(Q) Z(P) s

Zero-Beta CAPM The expected return can be expressed as the function of the expected returns of any two frontier portfolios The two fund theorem Select portfolios, M and ZB, to replace P and Q cov(RM,RZB) = 0

Zero-Beta CAPM E(R) S M T ZB sR

Testable implications of CAPM Return and risk (beta) are linearly related The slope of the Security Market Line is positive Higher beta gives higher return Beta is the only variable that explains expected return Idiosyncratic risk is not priced Alphas are zero CAPM should completely explain return The market risk premium should be positive

Can we test whether the CAPM is true? The CAPM is a model of equilibrium expected returns But we only observe actual returns Beta estimation Stability: historical stock betas do not predict future betas well Thin trading problems Standard CAPM is a one period model But applications and tests use multiperiod data Econometric issues

Can we test whether the CAPM is true? Roll’s Critique The CAPM is about all capital assets, but observability usually dictates tests based on a market portfolio of shares By definition if the proxy for the market portfolio is on the efficient set there will be a perfect linear relation between returns and beta measured relative to that portfolio Therefore the only test of the CAPM is whether the market portfolio is efficient Unfortunately, we can’t observe the market portfolio for all capital assets (i.e. human capital)

Can we test whether the CAPM is true? The CAPM fails because the assumptions are unreasonable Investors have only financial wealth Investors have mean-variance utility Single period investment Investors have homogenous expectations No taxes and transaction costs Investors are price takers Information is costless and available to all When assumptions are removed we get a different model Some models are similar to CAPM as they only have one factor Some models results in multiple factors

Conclusion Fama and French (1992) 50 year analysis in the US After controlling for size, there is actually a negative relationship between beta and realised return Next week, we will discuss alternative asset pricing models that can overcome the shortcomings of the CAPM What other risk factors are priced in addition to beta? The answer gets us closer to explaining how security returns are generated