Financial Economics: Capital Asset Pricing Model

Transcription

1 Financial Economics: Capital Asset Pricing Model Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY April, / 66

2 Outline Outline MPT and the CAPM Deriving the CAPM Application of CAPM Strengths and Shortcomings of the CAPM 2 / 66

3 MPT and the CAPM MPT and the CAPM The Capital Asset Pricing Model builds directly on Modern Portfolio Theory. It was developed in the mid-1960s by William Sharpe (US, b.1934, Nobel Prize 1990), John Lintner (US, ), and Jan Mossin (Norway, ). William Sharpe, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance Vol.19 (September 1964): pp John Lintner, The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, Review of Economics and Statistics Vol.47 (February 1965): pp Jan Mossin, Equilibrium in a Capital Asset Market, Econometrica Vol.34 (October 1966): pp / 66

4 MPT and the CAPM MPT and the CAPM Assumptions of the Capital Asset Pricing Model Investors are risk-averse individuals who maximize the expected utility of their end-of-period wealth Investors are price-takers and have homogeneous expectations about asset returns that have a joint normal distribution There exists a risk-free asset such that investors may borrow or lend unlimited amounts at the risk-free rate. The quantities of assets are fixed. All assets are marketable and perfectly divisible. Asset markets are frictionless and information is costless and simultaneously available to all investors. There are no market imperfections such as taxes, regulations, or restrictions on short selling. 4 / 66

5 MPT and the CAPM MPT and the CAPM But whereas Modern Portfolio Theory is a theory describing the demand for financial assets, the Capital Asset Pricing Model is a theory describing equilibrium in financial markets. By making an additional assumption namely, that supply equals demand in financial markets the CAPM yields additional implications about the pricing of financial assets and risky cash flows. 5 / 66

6 MPT and the CAPM MPT and the CAPM Like MPT, the CAPM assumes that investors have mean-variance utility and hence that either investors have quadratic Bernoulli utility functions or that the random returns on risky assets are normally distributed. Thus, some of the same caveats that apply to MPT also apply to the CAPM. For example, one might hesitate before applying the CAPM to price options. The traditional CAPM also assumes that there is a risk free asset as well as a potentially large collection of risky assets. Under these circumstances, as we ve seen, all investors will hold some combination of the riskless asset and the tangency portfolio: the efficient portfolio of risky assets with the highest Sharpe ratio. 6 / 66

7 MPT and the CAPM MPT and the CAPM But the CAPM goes further than the MPT by imposing an equilibrium condition. Because there is no demand for risky financial assets except to the extent that they comprise the tangency portfolio, and because, in equilibrium, the supply of financial assets must equal demand, the market portfolio consisting of all existing financial assets must coincide with the tangency portfolio. In equilibrium, that is, everyone must own the market. 7 / 66

8 CML, price of risk Deriving the CAPM Capital Market Line In the CAPM, equilibrium in financial markets requires the demand for risky assets the tangency portfolio to coincide with the supply of financial assets the market portfolio. The CAPM s first implication is immediate: the market portfolio is efficient. 8 / 66

9 CML, price of risk Deriving the CAPM Capital Market Line The line originating at (0, r f ) and running through (σ M, E( r M )) is called the capital market line (CML). 9 / 66

10 CML, price of risk Deriving the CAPM Capital Market Line Endowed with point A, we always have two choices available when there is a capital market: moving along the MVF or moving along CML by borrowing and lending. First move to B where MRS=MRT of the MVF, and U 1 increases to U 2 ; Then we can better off by moving to M and borrowing to reach C, utility increases to U / 66

11 Deriving the CAPM Capital Market Line CML, price of risk Nearly everyone is better off in a world with capital markets. Two-fund theorem obtains. In equilibrium, the MRS is the same for all individuals, regardless of their subjective attitude to risk. 11 / 66

12 CML, price of risk Deriving the CAPM Capital Market Line Hence, it also follows that all individually optimal portfolios are located along the CML and are formed as combinations of the risk free asset and the market portfolio. 12 / 66

13 CML, price of risk Deriving the CAPM Capital Market Line Recall that the trade-off between the standard deviation and expected return of any portfolio combining the riskless asset and the tangency portfolio is described by the linear relationship E( r P ) = r f + [ E( r T ) r f ]σ P Since the CAPM implies that the tangency and market portfolios coincide, the formula for the Capital Market Line is likewise σ T E( r P ) = r f + [ E( r M) r f ]σ P σ M 13 / 66

14 CML, price of risk Deriving the CAPM Capital Market Line And since all individually optimal portfolios are located along the CML, the equation E( r P ) = r f + [ E( r M) r f ]σ P σ M implies that the market portfolio s Sharpe ratio E( r M ) r f σ M measures the equilibrium price of risk: the expected return that each investor gives up when he or she adjusts his or her total portfolio to reduce risk, i.e., it shows how much must be given up in expected portfolio rate of return in order to reduce standard deviation by one unit. 14 / 66

15 Deriving the CAPM Motivation of CAPM Motivation Capital Asset Pricing Model shows what determines prices of individual assets. First motivation: Covariance is important. Comparing alternative portfolios, when only one of them can be chosen, have assumed variances of rates of return are the relevant measure of risk. But for individual assets, which can be combined in portfolios, the relevant measure turns out to be a covariance with other rates of return. 15 / 66

16 Motivation Deriving the CAPM Motivation of CAPM Consider making an equally weighted portfolio of n assets, i.e., with all w j = 1/n. Assume that among the rates of return, one has the maximum variance, σ 2 max. Then lim n σ2 p = σ ij the average covariance between rates of return, and σp 2 lim n w i = 2 σ ij 16 / 66

17 Proof Deriving the CAPM Motivation of CAPM Observe that An equally weighted portfolio has σ 2 p = Σ n i=1σ n j=1w i w j σ ij σ 2 p = 1 n 2 Σn i=1σ n j=1σ ij = 1 n 2 Σn i=1σ 2 i + 1 n 2 Σn i=1σ j i σ ij Observe that the first term satisfies 1 n 2 Σn i=1σ 2 i The second term satisfies < 1 n 2 n σ2 max 0 n 1 n 2 Σn i=1σ j i σ ij = n2 n n 2 which proves the first result. σ ij σ ij n 17 / 66

18 Proof Deriving the CAPM Motivation of CAPM Observe next that for any portfolio σ 2 p w i = 2w i σ 2 i + 2Σ j i w j σ ij Evaluated where all w i = 1/n, this becomes 2 σ2 i n + 2n 1 σ ij 2 σ ij n n 18 / 66

19 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Consider an equilibrium, everyone holds combination of risk free asset and market portfolio. Next, let s consider an arbitrary asset asset j with random return r j, expected return E( r j ), and standard deviation σ j.(possible, even though M already contains j.)) MPT would take E( r j ) and σ j as data that is, as given. The CAPM again goes further and asks: if asset j is to be demanded by investors with mean-variance utility, what restrictions must E( r j ) and σ j satisfy? 19 / 66

20 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula To answer this question, consider an investor who takes the portion of his or her initial wealth that he or she allocates to risky assets and divides it further: using the fraction w to purchase asset j and the remaining fraction 1 w to buy the market portfolio. Note that since the market portfolio already includes some of asset j, choosing w > 0 really means that the investor overweights asset j in his or her own portfolio. Conversely, choosing w < 0 means that the investor underweights asset j in his or her own portfolio. 20 / 66

21 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Based on our previous analysis, we know that this investor s portfolio of risky assets now has random return expected return and variance r P = w r j + (1 w) r M E( r P ) = we( r j ) + (1 w)e( r M ) σp 2 = w 2 σj 2 + (1 w) 2 σm 2 + 2w(1 w)σ jm where σ jm is the covariance between r j and r M. We can use these formulas to trace out how σ P and E( r P ) vary as w changes. 21 / 66

22 Deriving the CAPM Derivation of CAPM formula Deriving the CAPM The red curve traces out how σ P and E( r P ) vary as w changes, that is, as asset j gets underweighted or overweighted relative to the market portfolio. The red curve passes through M, since when w = 0 the new portfolio coincides with the market portfolio. For all other values of w, however, the red curve must lie below the CML. 22 / 66

23 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Otherwise, a portfolio along the CML would be dominated in mean-variance by the new portfolio. Financial markets would no longer be in equilibrium, since some investors would no longer be willing to hold the market portfolio. 23 / 66

24 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Together, these observations imply that the red curve must be tangent to the CML at M. 24 / 66

25 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Tangent means equal in slope. We already know that the slope of the Capital Market Line is E( r M ) r f σ M But what is the slope of the red curve? 25 / 66

26 Deriving the CAPM Derivation of CAPM formula Deriving the CAPM Let f (σ P ) be the function defined by E( r P ) = f (σ P ) and therefore describing the red curve. 26 / 66

27 Deriving the CAPM Derivation of CAPM formula Deriving the CAPM Next, define the functions g(w) and h(w) by so that g(w) = we( r j ) + (1 w)e( r M ) h(w) = [w 2 σ 2 j + (1 w) 2 σ 2 M + 2w(1 w)σ jm ] 1/2 E( r P ) = g(w) σ P = h(w) 27 / 66

28 Deriving the CAPM Derivation of CAPM formula Deriving the CAPM Substitute E( r P ) = g(w) σ P = h(w) into then E( r P ) = f (σ P ) g(w) = f (h(w)) and use the chain rule to compute g (w) = f (h(w))h (w) = f (σ P )h (w) 28 / 66

29 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula Let f (σ P ) be the function defined by E( r P ) = f (σ P ) and therefore describing the red curve. Then f (σ P ) is the slope of the curve. Hence, to compute f (σ P ), we can rearrange and get g (w) = f (h(w))h (w) = f (σ P )h (w) f (σ P ) = g (w) h (w) and compute g (w) and h (w) from the formulas we know g(w) = we( r j ) + (1 w)e( r M ) implies g (w) = E( r j ) E( r M ) 29 / 66

30 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula implies h(w) = [w 2 σ 2 j + (1 w) 2 σ 2 M + 2w(1 w)σ jm ] 1/2 so h (w) = 1 2 ( 2wσ 2 j 2(1 w)σ 2 M + 2(1 2w)σ jm [w 2 σ 2 j + (1 w) 2 σ 2 M + 2w(1 w)σ jm] 1/2 ) = wσ 2 j (1 w)σ 2 M + (1 2w)σ jm [w 2 σ 2 j + (1 w) 2 σ 2 M + 2w(1 w)σ jm] 1/2 f (σ P ) = g (w) h (w) = [E( r j ) E( r M )] [w 2 σ 2 j + (1 w) 2 σ 2 M + 2w(1 w)σ jm] 1/2 wσ 2 j (1 w)σ 2 M + (1 2w)σ jm 30 / 66

31 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula The red curve is tangent to the CML at M. Hence f (σ P ) equals the slope of the CML when w = 0 31 / 66

32 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula When w = 0 f (σ P ) = [E( r j ) E( r M )] [w 2 σj 2 + (1 w) 2 σm 2 + 2w(1 w)σ jm] 1/2 wσj 2 (1 w)σm 2 + (1 2w)σ jm implies f (σ P ) = [E( r j) E( r M )]σ M σ jm σm 2 Meanwhile, we know that the slope of the CML is E( r M ) r f σ M 32 / 66

33 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula The tangency of the red curve with the CML at M therefore requires [E( r j ) E( r M )]σ M σ jm σm 2 = E( r M) r f σ M E( r j ) E( r M ) = [E( r M) r f ][σ jm σm 2 ] σm 2 E( r j ) E( r M ) = σ jm σ 2 M E( r j ) = r f + σ jm [E( r σm 2 M ) r f ] [E( r M ) r f ] [E( r M ) r f ] 33 / 66

34 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula E( r j ) = r f + σ jm [E( r σm 2 M ) r f ] Let β j = σ jm σm 2 so that this key equation of the CAPM can be written as E( r j ) = r f + β j [E( r M ) r f ] where β j, the beta for asset j, depends on the covariance between the returns on asset j and the market portfolio. This equation summarizes a very strong restriction. It implies that if we rank individual stocks or portfolios of stocks according to their betas, their expected returns should all lie along a single security market line with slope E( r M ) r f 34 / 66

35 Deriving the CAPM Deriving the CAPM Derivation of CAPM formula According to the CAPM, all assets and portfolios of assets lie along a single security market line. Those with higher betas have higher expected returns. 35 / 66

36 Deriving the CAPM Derivation of CAPM formula Security market line Observe β M = 1 Observe β j = ρ jm σ j /σ M May have σ j > σ M, and ρ jm close to 1;Thus possible to have β j > 1 for some assets May also have σ jm < 0, β j < 0, not very common in practice; Such assets contribute to reducing σ M (when included in M). 36 / 66

37 Security market line Deriving the CAPM Derivation of CAPM formula Any portfolio of m securities locate on the SML. Show for m = 2 µ p = wµ i + (1 w)µ j = w[r f + β i (µ M r f )] + (1 w)[r f + β j (µ M r f )]) = r f + [wβ i + (1 w)β j ](µ M r f ) = r f + [w cov( r i, r M ) + (1 w) cov( r j, r M ) ](µ M r f ) σ 2 M σ 2 M = r f + cov(w r i + (1 w) r j, r M ) (µ M r f ) σ 2 M = r f + cov( r p, r M ) (µ M r f ) σ 2 M = r f + β p (µ M r f ) β p is a value-weighted average of β i and β j. 37 / 66

38 CML V.S. SML Deriving the CAPM Derivation of CAPM formula Capital Market Line Efficient set given 1 risk free and n risky assets. Relevant for choice between alternative portfolios. Drawn in (σ p, µ p ) diagram. A ray starting at (0, r f ) in that diagram. Security market line Location of all traded assets in equilibrium. Also location of any portfolio of these assets. Not relevant for choice between assets which are already traded, so that equilibrium prices are observable. But relevant if equilibrium price at t = 0 is unknown. Drawn in (β j, µ j ) diagram. A line through (0, r f ) in that diagram 38 / 66

39 Deriving the CAPM Interpretation Interpretation E( r j ) = r f + σ jm [E( r σm 2 M ) r f ] The expected rate of return on any asset depends on only one characteristic of that asset, namely its rate of return s covariance with the rate of return on the market portfolio. The expected rate of return is equal to the risk free interest rate plus a term which depends on a measure of risk. (Higher risk means higher expected rate of return.) The relevant measure of risk is the asset s beta. This is multiplied with the expected excess rate of return on the market portfolio. 39 / 66

40 Interpretation Deriving the CAPM Interpretation E( r j ) = r f + σ jm [E( r σm 2 M ) r f ] Risk measure depends on covariance because the covariance determines how much that asset will contribute to the risk of the agent s portfolio.this is true for any agent, since all hold the same risky portfolio. 40 / 66

41 Deriving the CAPM Interpretation Interpretation There are two complementary ways of interpreting this result. Both bring us back to the theme of diversification emphasized by MPT. Both take us a step further, by emphasizing as well the idea of aggregate risk, which cannot be diversified away, and idiosyncratic risk, which can be diversified away. 41 / 66

42 Deriving the CAPM first interpretation Aggregate risk & idiosyncratic risk The first approach uses the CAPM equation in its original form E( r j ) = r f + σ jm [E( r σm 2 M ) r f ] together with the definition of correlation, which implies Then the CAPM relationship ρ jm = σ jm σ j σ M E( r j ) = r f + [ E( r M) r f ]ρ jm σ j σ M 42 / 66

43 Deriving the CAPM first interpretation Aggregate risk & idiosyncratic risk E( r j ) = r f + [ E( r M) r f ]ρ jm σ j σ M The term inside brackets is the equilibrium price of risk. And since the correlation lies between 1 and 1, the term ρ jm σ j, satisfying ρ jm σ j σ j represents the portion of the total risk σ j in asset j that is correlated with the market return. 43 / 66

44 Deriving the CAPM first interpretation Aggregate risk & idiosyncratic risk E( r j ) = r f + [ E( r M) r f ]ρ jm σ j σ M The idiosyncratic risk in asset j, that is, the portion that is uncorrelated with the market return, can be diversified away by holding the market portfolio. Since this risk can be freely shed through diversification, it is not priced. Hence, according to the CAPM, risk in asset j is priced only to the extent that it takes the form of aggregate risk that, because it is correlated with the market portfolio, cannot be diversified away. 44 / 66

45 Deriving the CAPM first interpretation Aggregate risk & idiosyncratic risk Thus, according to the CAPM: E( r j ) = r f + [ E( r M) r f ]ρ jm σ j σ M Only assets with random returns that are positively correlated with the market return earn expected returns above the risk free rate. They must, in order to induce investors to take on more aggregate risk. Assets with returns that are uncorrelated with the market return have expected returns equal to the risk free rate, since their risk can be completely diversified away. 45 / 66

46 Deriving the CAPM first interpretation Aggregate risk & idiosyncratic risk Thus, according to the CAPM: E( r j ) = r f + [ E( r M) r f ]ρ jm σ j σ M Assets with negative betas that is, with random returns that are negatively correlated with the market return have expected returns below the risk free rate! For these assets, E( r j ) r f < 0 is like an insurance premium that investors will pay in order to insulate themselves from aggregate risk. 46 / 66

47 Deriving the CAPM Statistical interpretation Second interpretation The second approach to interpreting the CAPM uses together with the definition of E( r j ) = r f + β j [E( r M ) r f ] β j = σ jm σ 2 M 47 / 66

48 Deriving the CAPM Statistical interpretation Second interpretation Consider a statistical regression of the random return r j on asset j on a constant and the market return r M r j = α + β j r M + ε j This regression breaks the variance of r j down into two orthogonal (uncorrelated) components: The component β j r M that is systematically related to variation in the market return. The component ε j that is not. Do you remember the formula for β j, the slope coefficient in a linear regression? It is β j = σ jm σm 2 the same beta as in the CAPM! 48 / 66

49 Deriving the CAPM Statistical interpretation Second interpretation r j = α + β j r M + ε j But this is not an accident: to the contrary, it restates the conclusion that, according to the CAPM, risk in an individual asset is priced and thereby reflected in a higher expected return only to the extent that it is correlated with the market return. 49 / 66

50 Deriving the CAPM Second interpretation Statistical interpretation r j = α + β j r M + ε j E( r j ) = r f + β j [E( r M ) r f ] the CAPM equation implies E(ε j ) = 0 and cov(ε j, r M ) = cov( r j α β j r M, r M ) = cov( r j, r M ) β j var( r M ) = σ jm β j σ 2 M = 0 50 / 66

51 Deriving the CAPM Statistical interpretation Second interpretation This allows us to split σ 2 j in two parts: var( r j ) σ 2 j = β 2 j σ 2 M + σ 2 ε j = β j σ jm + σ 2 ε j First term is the aggregate risk. This is reflected in the market valuation. Second term is the unsystematic risk or idiosyncratic risk. As we have seen, it is not reflected in market valuation. The sum of the two called total risk or variance risk. This is relevant for portfolios, evaluated for being the total wealth of someone, but not for individual securities, to be combined with other securities in portfolios. 51 / 66

52 Deriving the CAPM Statistical interpretation Second interpretation r j = α + β j r M + ε j For a portfolio: Variance is the relevant risk measure. For each security: Covariance with r M is relevant. Because covariance measures contribution to portfolio variance. Generally: Covariance with each agent s marginal utility; but with mean-variance preferences: Covariance with agent s wealth. In equilibrium, all have same risky portfolio (composition); Thus covariance with r M is relevant for everyone. 52 / 66

53 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows We can also use the CAPM to value risky cash flows. Let C t+1 denote a random payoff to be received at time t + 1 ( one period from now ) and let Pt C denote its price at time t ( today. ) If C t+1 was known in advance, that is, if the payoff were riskless, we could find its value by discounting it at the risk free rate: P C t = C t r f 53 / 66

54 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows But when C t+1 is truly random, we need to find its expected value E( C t+1 ) and then penalize it for its riskiness either by discounting at a higher rate P C t = E( C t+1 ) 1 + r f + ψ or by reducing its value more directly P C t = E( C t+1 ) Ψ 1 + r f The CAPM can help us identify the appropriate risk premium ψ or Ψ. 54 / 66

55 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows Our previous analysis suggests that, broadly speaking, the risk premium implied by the CAPM will somehow depend on the extent to which the random payoff C t+1 is correlated with the return on the market portfolio. To apply the CAPM to this valuation problem, we can start by observing that with price Pt C today and random payoff C t+1 one period from now, the return on this asset or investment project is defined by or 1 + r C = C t+1 Pt C r C = C t+1 Pt C where the notation r C emphasizes that this return, like the future cash flow itself, is risky. 55 / 66 P C t

56 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows Now the CAPM implies that the expected return E( r C ) must satisfy E( r C ) = r f + β C [E( r M ) r f ] where the project s beta depends on the covariance of its return with the market return: β C = σ CM σ 2 M This is what takes skill: with an existing asset, one can use data on the past correlation between its return and the market return to estimate beta. With a totally new project that is just being planned, a combination of experience, creativity, and hard work is often needed to choose the right value for β C. 56 / 66

57 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows But once a value for β C is determined, we can use E( r C ) = r f + β C [E( r M ) r f ] together with the definition of the return itself to write which implies ( 1 P C t E( C t+1 P C t r C = C t+1 1 Pt C 1) = r f + β C [E( r M ) r f ] )E( C t+1 ) = 1 + r f + β C [E( r M ) r f ] P C t = E( C t+1 ) 1 + r f + β C [E( r M ) r f ] 57 / 66

58 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows P C t = E( C t+1 ) 1 + r f + β C [E( r M ) r f ] Right-hand side is expected present value, with Risk-adjusted discount rate(radr). Risk-adjustment again depends on (E( r M ) r f )/σm 2 and covariance. E( r C ) = r f + β C [E( r M ) r f ] the CAPM implies a risk premium of ψ = β C [E( r M ) r f ] depends critically on the covariance between the return on the risky project and the return on the market portfolio. 58 / 66

59 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows Alternatively, ( 1 P C t can be rewritten as )E( C t+1 ) = 1 + r f + β C [E( r M ) r f ] P C t = E( C t+1 ) Pt C β C [E( r M ) r f ] 1 + r f indicating that the CAPM also implies Ψ = Pt C β C [E( r M ) r f ] which, again as expected, depends critically on the covariance between the return on the risky project and the return on the market portfolio. 59 / 66

60 Application of CAPM Valuing Risky Cash Flows Valuing Risky Cash Flows Note that P C t = E( C t+1 ) Pt C β C [E( r M ) r f ] 1 + r f E( C t+1 ) Pt C β C [E( r M ) r f ] is called the certainty equivalent (in the CAPM sense) of C t+1. Price today is present value of expression, just as if that would be received with certainty one period into the future. Not what is called certainty equivalent in expected utility theory. 60 / 66

61 Application of CAPM Valuation of Firms Valuation of firms, value additivity P C 0 = E( C 1 ) Pt C β C [E( r M ) r f ] V ( 1 + r C 1 ) f defines valuation function V (), given some r f, r M. C 1 is value of share at t = 1, including dividends.. Consider firm is financed 100% by equity (no debt). Firm s net cash flow at t = 1 goes to shareholders, thus plug in the net cash flow instead of C 1, Then formula gives value of all shares in firm. 61 / 66

62 Application of CAPM Valuation of Firms Valuation of firms, value additivity What if C 1 = a C i1 + b C j1? (a > 0, b > 0) Linearity of E() and cov() implies P C 0 = ap i 0 + bp j 0 P0 C = E( C 1 ) cov( C 1, r M )[E( r M ) r f ]/σm r f = E(a C i1 + b C j1 ) λcov( C 1, r M ) 1 + r f = E(a C i1 + b C j1 ) λcov(a C i1 + b C j1, r M ) 1 + r f = ae( C i1 ) + be( C j1 ) λ[acov( C i1, r M ) + bcov( C j1, r M )] 1 + r f = ap0 i + bp j 0 where λ = (µ M r f )/σm 2 Diversification is no justification for mergers; Diversification can be done by shareholders. 62 / 66

63 Pros and Cons of CAPM Pros and Cons of CAPM An enormous literature is devoted to empirically testing the CAPM s implications. Although results are mixed, studies have shown that when individual portfolios are ranked according to their betas, expected returns tend to line up as suggested by the theory. A famous article that presents results along these lines is by Eugene Fama (Nobel Prize 2013) and James MacBeth, Risk, Return, and Equilibrium, Journal of Political Economy Vol.81 (May-June 1973), pp Early work on the MPT, the CAPM, and econometric tests of the efficient markets hypothesis and the CAPM is discussed extensively in Eugene Fama s 1976 textbook, Foundations of Finance. 63 / 66

64 Pros and Cons of CAPM Pros and Cons of CAPM More recent evidence against the CAPM s implications is presented by Eugene Fama and Kenneth French, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics Vol.33 (February 1993): pp This paper shows that equity shares in small firms and in firms with high book (accounting) to market value have expected returns that differ strongly from what is predicted by the CAPM alone. Quite a bit of recent research has been directed towards understanding the source of these anomalies.. 64 / 66

65 Pros and Cons of CAPM Pros and Cons of CAPM Despite some empirical shortcomings, however, the CAPM quite usefully deepens our understanding of the gains from diversification. Related, the CAPM alerts us to the important distinction between idiosyncratic risk, which can be diversified away, and aggregate risk, which cannot. Like MPT, the CAPM must rely on one of the two strong assumptions either quadratic utility or normally-distributed returns that justify mean-variance utility. And while the CAPM is an equilibrium theory of asset pricing, it stops short of linking asset returns to underlying economic fundamentals. 65 / 66

66 Pros and Cons of CAPM Pros and Cons of CAPM These last two points motivate our interest in other asset pricing theories, which are less restrictive in their assumptions and/or draw closer connections between asset prices and the economy as a whole. Arbitrage Pricing Theory, to which we will turn our attention next, yields many of the same implications as the CAPM, but requires less restrictive assumptions about preferences and the distribution of asset returns. The equilibrium version of Arrow-Debreu theory draws links between asset prices and the economy that are only implicit in the CAPM. 66 / 66