# THE CAPITAL ASSET PRICING MODEL

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1 THE CAPITAL ASSET PRICING MODEL Szabolcs Sebestyén Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) CAPM Investments 1 / 30

2 Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 2 / 30

3 Introduction Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 3 / 30

4 Introduction MPT versus CAPM MPT chooses a portfolio, given the expected returns and covariances It does not examine whether there is equilibrium Equilibrium: a situation where all investors are satisfied with their holdings and do not want to change anything MPT needs estimates of expected returns, which are hard to obtain The CAPM describes the relation between expected returns and covariances of all assets It provides what returns should be in the equilibrium Sebestyén (ISCTE-IUL) CAPM Investments 4 / 30

5 Introduction CAPM as an Equilibrium Theory The Capital Asset Pricing Model (CAPM) is an equilibrium theory relying on MPT, but with a somewhat peculiar structure 1 The CAPM is a theory of financial equilibrium only, it does not link returns with the real economy 2 It assumes that the observed asset prices are equilibrium prices, i.e., the supply equals demand, without calculating asset supply or demand explicitly If investor i invests a fraction ω ij of her initial wealth w 0i in asset j, the value of her asset j holding is ω ij w 0i = her demand for asset j Aggregate demand = Total value of all holdings of asset j, and in equilibrium we have I ω ij w 0i = p j Q j i=1 3 The CAPM expresses equilibrium in terms of relationships between the return distributions of individual assets and those of the portfolio of all assets Sebestyén (ISCTE-IUL) CAPM Investments 5 / 30

6 The Traditional Approach to the CAPM Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 6 / 30

7 The Traditional Approach to the CAPM Basic Assumptions and Consequences The CAPM assumes that 1 all agents have the same beliefs about future returns (i.e., homogenous expectations) 2 assets are all tradable and infinitely divisible 3 there is perfect competition 4 there are no transaction costs and taxes Hence, the mean-variance frontier is the same for every investor Moreover, all investors optimal portfolios have an identical structure: a fraction of initial wealth is invested in the risk-free asset, the rest in the tangency portfolio T (two-fund separation) All risky assets must belong to T by the definition of equilibrium The share of any asset j in T must correspond to the ratio of the market value of that asset and that of all assets Thus, the tangency portfolio T is essentially the market portfolio M, the portfolio of all existing assets Sebestyén (ISCTE-IUL) CAPM Investments 7 / 30

8 The Traditional Approach to the CAPM Some Results The market portfolio M is efficient because it is on the efficient frontier All individual optimal portfolios are located on the half-line originating at point ( 0, r f ) and going through (σm, r M ), this is the Capital Market Line (CML) The slope of the CML is ( r M r f ) /σm, i.e., an investor considering a marginally riskier efficient portfolio would obtain an increase in expected return of ( r M r f ) /σm This is the price of risk, and, for efficient portfolios, r p = r f + r M r f σ M σ p Sebestyén (ISCTE-IUL) CAPM Investments 8 / 30

9 The Traditional Approach to the CAPM Figure: The Capital Market Line Source: Danthine and Donaldson (2005), Figure 7.1 Sebestyén (ISCTE-IUL) CAPM Investments 9 / 30

10 The Traditional Approach to the CAPM The CML and Non-Efficient Portfolios The CML applies only to efficient portfolios Consider a portfolio with a fraction 1 α of wealth invested in an arbitrary security j, and a fraction α in the market portfolio Then we have µ p = αr M + (1 α) r j σ 2 p = α 2 σ 2 M + (1 α) 2 σ 2 j + 2α (1 α) σ jm As α varies, we trace a locus that passes through M (and through j) it is tangent to the CML at M It can be shown that r j = r f + ( r M r f ) σ jm σ 2 M Sebestyén (ISCTE-IUL) CAPM Investments 10 / 30

11 The Traditional Approach to the CAPM Systematic Risk Let s define β j σ jm /σ 2 M Rewrite the last equation as r j = r f + r M r f σ M β j σ M = r f + r M r f σ M ρ jm σ j Only a portion of the total risk of an asset j, σ j, is remunerated by the market The risk premium on a given asset is the market price of risk, ( rm r f ) /σm, multiplied by the relevant measure of risk for that asset For an inefficient asset j, this risk differs from σ j Systematic risk (market risk or undiversifiable risk): the portion of total risk that is priced, i.e., β j σ M = ρ jm σ j σ j Sebestyén (ISCTE-IUL) CAPM Investments 11 / 30

12 The Traditional Approach to the CAPM Intuition Behind Systematic Risk Since every investor holds the market portfolio (T = M), the relevant risk is the variance of the market portfolio Thus, what is important to the investor is the contribution of asset j to the risk of the market portfolio This is measured by ρ jm σ j or β j σ M Investors must be compensated to persuade them to hold an asset with high covariance with the market, and this compensation is in the form of higher expected return An efficient portfolio is one for which all diversifiable risks are eliminated For an efficient portfolio, total risk and systematic risk are the same Sebestyén (ISCTE-IUL) CAPM Investments 12 / 30

13 The Traditional Approach to the CAPM The Security Market Line Write the return on asset j as an affine function of the market return with a random error term, independent of the market return, r j = α + β j r M + ε j The variance of return on asset j is σ 2 j = β 2 j σ2 M + σ 2 ε j The OLS estimator of the market return coefficient is β j = σ jm σ 2 M We can write the equation of systematic risk as r j r f = ( r M r f ) βj Security Market Line (SML): the expected excess return (risk premium) of an asset j is proportional to its β j Sebestyén (ISCTE-IUL) CAPM Investments 13 / 30

14 The Traditional Approach to the CAPM Figure: The Security Market Line Source: Danthine and Donaldson (2005), Figure 7.2 Sebestyén (ISCTE-IUL) CAPM Investments 14 / 30

15 The Traditional Approach to the CAPM CML versus SML Both CML and SML are half-lines, connecting the risk-free rate and the market portfolio M Both represent the relation between risk and expected return; however, the CML is plotted in the σ (total risk)-µ space, while the SML in the β (systematic risk)-µ space Efficient portfolios lie on the CML and inefficient ones lie below it All portfolios lie on the SML in the CAPM world In reality, assets can lie above (underpriced) or below (overpriced) the SML Sebestyén (ISCTE-IUL) CAPM Investments 15 / 30

16 Valuing Risky Cash Flows with the CAPM Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 16 / 30

17 Valuing Risky Cash Flows with the CAPM Valuation of Risky Cash Flows One can use the CAPM not only to price assets, but also to value non-traded risky cash flows The CAPM helps determine the appropriate discount factor Although it is a one-period model, in practice it is used in multiperiod settings Consider a project j that costs at time t p j,t and yields an uncertain cash flow at time t + 1 of CF j,t+1 The return on the project is Hence we have r j,t+1 = CF j,t+1 p j,t p j,t 1 + E ( r j,t+1 ) = E ( CF j,t+1 p j,t ) = 1 p j,t E ( CF j,t+1 ) Sebestyén (ISCTE-IUL) CAPM Investments 17 / 30

18 Valuing Risky Cash Flows with the CAPM CAPM Provides the Right Discount Rate By the CAPM, we know that E ( r j ) = rf + β j [ E ( rm ) rf ] Thus the equation for project return can be rewritten as E ( CF j,t+1 ) p j,t = 1 + r f + β j [ E ( rm ) rf ] Or, equivalently, p j,t = E ( CF j,t+1 ) 1 + r f + β j [ E ( rm ) rf ] The project is then priced as the PV of its expected cash flows discounted at the risk-adjusted rate appropriate to its risk class (β j ) Sebestyén (ISCTE-IUL) CAPM Investments 18 / 30

19 Valuing Risky Cash Flows with the CAPM Example: Project Valuation Example A media company is considering going into the cell phone business. By investing \$100 million today, it is expecting to receive \$20 million in 1 year, \$30 million in 2 years, and \$90 million in 3 years. Telecom companies have an average beta of 0.7. The risk-free rate is 3% and the average market risk premium is 6%. Should the company expand its operations into the cell phone business? Solution The relevant discount rate is r = 1 + r f + β j [ E ( rm ) rf ] = = The NPV of the project is NPV = = > 0 Sebestyén (ISCTE-IUL) CAPM Investments 19 / 30

20 Valuing Risky Cash Flows with the CAPM Modify the Numerator of the Pricing Equation An alternative is to modify the numerator of the pricing equations so that we can discount at the risk-free rate Hence, replace the expected future cash flow by its CE Now we need a market CE rather than an individual one as we are interested in equilibrium valuation The CAPM helps find the appropriate market risk premium Sebestyén (ISCTE-IUL) CAPM Investments 20 / 30

21 Valuing Risky Cash Flows with the CAPM Derivation The CAPM implies that E ( r ) [ ) ] j = rf + β j E ( rm rf = ( ) CF Cov j,t+1 p j,t p j,t, r M [ ) ] = r f + E ( rm rf = Solving for p j,t yields σ 2 M = r f + 1 Cov ( CF ) )E ( rm rf j,t+1, r M p j,t σ 2 M E ( CF ( ) ) E r M ) r f j,t+1 Cov ( CF j,t+1, r M σm p j,t = 2 = 1 + r f = E( CF ) [ ) ] j,t+1 pj,t β j E ( rm rf 1 + r f Sebestyén (ISCTE-IUL) CAPM Investments 21 / 30

22 The Standard CAPM Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 22 / 30

23 The Standard CAPM Assumptions We want to identify a portfolio on the efficient frontier without specifying a priori r and V The CAPM tells us that this portfolio is the market portfolio (M), under one of the two assumptions: agents maximise increasing and strictly concave expected utility, and asset returns are multivariate Gaussian; or every agents maximises a mean-variance concave utility function u ( µ, σ 2) with u 1 > 0 and u 2 < 0 Also assume that all agents have a common time horizon and homogeneous beliefs about r and V Under these assumptions, investors will only hold efficient frontier portfolios This implies that, in equilibrium, the market portfolio is also on the efficient frontier The CAPM does not describe how the equilibrium is achieved Sebestyén (ISCTE-IUL) CAPM Investments 23 / 30

24 The Standard CAPM Derivation of the Standard CAPM (1) It is easy to show that the covariance between any two portfolios is Cov ( r q, r p ) = ω q Vω p Substitution yields Cov ( r [ ) q, r p = ω µp r f q V H V 1( r r f 1 ) ] ( ) ( ) µp r f µq r f = H = µ p r f H ω q ( r rf 1 ) = Solving for µ q r f and using that σ 2 p = ( µ p r f ) 2 /H, we have that µ q r f = H Cov( r ) q, r p = Cov ) ( rq, r p ( ) µp r µ p f r f σ 2 p Sebestyén (ISCTE-IUL) CAPM Investments 24 / 30

25 The Standard CAPM Derivation of the Standard CAPM (2) Since T is a frontier portfolio, we can choose p = T In equilibrium T = M, so µ q r f = Cov( r ) q, r M ( ) µm r f σ 2 M Or, equivalently, µ q = r f + β q,m ( µm r f ) This is the standard CAPM Sebestyén (ISCTE-IUL) CAPM Investments 25 / 30

26 The Standard CAPM Economic Interpretation of β Assume that asset a is riskier than asset b, i.e., β a > β b Why does this imply that E ( r a ) > E ( rb )? All investors hold the market portfolio, and they like when markets go up and dislike when they go down Due to decreasing MU, investors prefer additional pay-offs in bad times to those in good times Thus, agents like assets with low covariance with the market, i.e., low beta Asset b offers higher pay-off in bad times, so investors are willing to hold it with a lower expected return, i.e., they are willing to pay a higher price for it Sebestyén (ISCTE-IUL) CAPM Investments 26 / 30

27 The CAPM and Empirical Evidence Outline 1 Introduction 2 The Traditional Approach to the CAPM 3 Valuing Risky Cash Flows with the CAPM 4 The Standard CAPM 5 The CAPM and Empirical Evidence Sebestyén (ISCTE-IUL) CAPM Investments 27 / 30

28 The CAPM and Empirical Evidence Some Remarks on the CAPM The model is static, i.e., only one period of returns is studied It is a model of partial equilibrium, since the real economy is ignored Expectations are homogeneous in that all investors share the same information It is a cross-sectional model Whether the CAPM holds in practice has generated much debate over the decades Even if the CAPM performs poorly in the data, it is worth studying it: Its failure means that we can beat the market portfolio The CAPM provides a framework to think about risk and return Sebestyén (ISCTE-IUL) CAPM Investments 28 / 30

29 The CAPM and Empirical Evidence Testing the CAPM If the CAPM is true, then all assets should lie on the SML, E ( r i ) = rf + β i [ E ( rm ) rf ] Implications: The relation between the asset s expected return and β i is linear Only β i explains differences between returns on different assets An asset with β i = 0 has an expected return of r f An asset with β i = 1 has an expected return of E ( r M ) Several tests have been proposed to test the CAPM in the data Tests typically reject the CAPM Sebestyén (ISCTE-IUL) CAPM Investments 29 / 30

30 The CAPM and Empirical Evidence Possible Reasons for Rejection Measurement error in β i This can be tackled by forming portfolios Measurement error in expected returns What is really the market portfolio? Roll s critique (Roll, 1977) Only 1/3 of non-governmental assets are owned by the corporate sector Only 1/3 of corporate assets are financed by equity Intangible assets, such as human capital, are hard to price Roll shows that even if two potential proxies for M are correlated greater than 0.9, their estimated betas can be very different More and more evidence suggests that firm characteristics beyond beta may provide explanatory power for returns (We ll study this issue later) Sebestyén (ISCTE-IUL) CAPM Investments 30 / 30

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