Portfolio A Define alpha as α = f – β fB. Let hA be the characteristic portfolio for alpha, the minimum-risk portfolio with alpha of 100 percent. (Portfolio A will involve significant leverage.) According to Eq. (2A.5), we can express hA in terms of hB and hq. From Eq. (2A.4), we see that the relationship between alpha and beta is
Page 34 . However, αB = 0 by construction, and so portfolios A and B are uncorrelated, and βA = 0. In many cases, we will find it convenient to assume that there is a fully invested portfolio that explains expected excess returns. That will be the case if the expected excess return on portfolio C is positive. This is not an unreasonable assumption, and we will use it throughout the book. The next proposition details some of its consequences. Proposition 3 Assume that fC > 0. 1. Portfolio q is net long: Let portfolio Q be the characteristic portfolio of eqf. PortfolioQ is fully invested, with holdings hQ = hq/eq. In addition, SRQ = SRq, and for any portfolio P with a correlation ρP,Q with portfolio Q, we have Note that Eq. (2A.36) specifies exactly how portfolio Q "explains" expected returns. 4. If the benchmark is fully invested, eB = 1, then Proof For part 1, note that and fC > 0 imply eq > 0. From part 5 of proposition 1,
Page 35 The holdings in portfolio Q are a positive multiple of the holdings in q, and so their Sharpe ratios and correlations with other portfolios are the same. For item 2, start with and use . This yields . If we multiply this by hQ, we get . For item 3, premultiply Eq. (2A.27) by hB. This yields which gives 3. For item 4, note that eB = 1 and imply that . When this is combined with , we get 4. Partial List of Characteristic Portfolios Characteristic Portfolio f hq eqf hQ (if fC > 0) β hB e hC α = f – βfB hA We have built portfolios capturing the important characteristics for portfolio management. These portfolios will play significant roles as we further develop the theory. For example, if we want to build a portfolio based on our alphas, but with a beta of 1, full investment, and conforming to our preferences for risk and return, we will build a linear combination of portfolios A, B, and C.
Page 36 The Efficient Frontier Now focus on two characteristic fully invested portfolios: portfolioC and portfolio Q. At this point we would like to introduce a set of distinctive portfolios called the efficient frontier. Portfolio C and portfolio Q are both elements of this set. In fact, we will see that all efficient-frontier portfolios are weighted combinations of portfolioC and portfolio Q, so each element of the efficient-frontier is a characteristic portfolio. The return and risk characteristics of efficient frontier portfolios are simply parameterized in terms of the return and risk characteristics of portfolio C and portfolio Q. A fully invested portfolio is efficient if it has minimum risk among all portfolios with the same expected return. Efficient frontier portfolios solve the minimization problem subject to the full investment and expected excess return constraints (but not a long-only constraint): We can solve this minimization problem to find: where we have used the definitions of hC and hQ and have assumed that f e. So efficient frontier portfolios are weighted combinations of portfolio C and portfolio Q. Remember that the correspondence between characteristics and portfolios is one-to-one. We can therefore solve for the characteristicaP that underlies each efficient portfolio, using Eq. (2A.45) and (2A.5). In each case, the characteristic is a linear combination of e and eqf, the characteristics underlying portfolios C and Q, respectively:
Page 37 Figure 2A.1 The efficient frontier. We can now use Eq. (2A.45) to solve for the variance of the efficient-frontier portfolios. We find We depict this relationship in Fig. 2A.1. In this figure, portfolio Q has a volatility of 20 percent and an expected excess return of 7 percent. Portfolio C has a volatility of 12 percent and, therefore, an expected excess return of 2.52 percent. The risk-free asset appears at the origin. The Capital Asset Pricing Model We establish the CAPM in two steps. We have already accomplished step 1, showing in Eq. (2A.36) that the vector of asset expected excess returns is proportional to the vector of asset betas with respect to portfolio Q. In step 2, we show that certain assumptions
Page 38 lead us to the conclusion that portfolio Q is the market portfolio M, i.e., that the market portfolio M is indeed the portfolio with the highest ratio of expected excess return to risk among all fully invested portfolios. Theorem If • All investors have mean/variance preferences. • All assets are included in the analysis. • All investors know the expected excess returns. • All investors agree on asset variances and covariances. • There are no transactions costs or taxes. then portfolio Q is equal to portfolio M, and Proof If all investors are free of transactions costs, have the same information, and choose portfolios in a mean/variance-efficient way, then each investor will choose a portfolio that is a mixture of Q and the risk-free portfolio F. That would place each investor somewhere along the line FQF' in Fig. 2A.1. Portfolios from F to Q combine the risk-free portfolio (lending) and portfolio Q. Portfolios from Q to F' represent a levered position (borrowing) in portfolio Q. When we aggregate (add up, weighted by value invested) the portfolios of all investors, they must equal the market portfolio M, since the net supply of borrowing and lending must equal zero. The only way that the portfolios along FQF' can aggregate to a fully invested portfolio is to have that aggregate equal Q. The aggregate must equal M, and the aggregate must equal Q. ThereforeM = Q. Exercises 1. Show that . Since portfolio C is the minimum-variance portfolio, this relationship implies that βC ≤ 1, with βC = 1 only if the market is the minimum-variance portfolio. 2. Show that .
Page 39 3. What is the ''characteristic" associated with the MMI portfolio? How would you find it? 4. Prove that the fully invested portfolio that maximizes has expected excess return f* = fC + 1/(2λκ). 5. Prove that portfolio Q is the optimal solution in Exercise 4 if . 6. Suppose portfolio T is on the fully invested efficient frontier. Prove Eq. (2A.45), i.e., that there exists a wT such that hT = wThC + (1 – wT)hQ. 7. If T is fully invested and efficient and T C, prove that there exists a fully invested efficient portfolio T* such that Cov{rT,rT*} = 0. 8. For any T C on the efficient frontier and any fully invested portfolio P, show that we can write where T* is the fully invested efficient portfolio that is uncorrelated with T. 9. If P is any fully invested portfolio, and T is the efficient fully invested efficient portfolio with the same expected returns as P, µP = µT, we can always write the returns to P as rP = rC + {rT – rC} + {rP – rT}. Prove that these three components of return are uncorrelated. We can interpret the risks associated with these three components as the cost of full investment, Var{rC}; the cost of the extra expected return µP – µC, Var{rT – rC}; and the diversifiable cost, Var{rP – rT}. Applications Exercises10 For ease of calculation, focus on just MMI assets when considering these application exercises. The MMI is a share-weighted 20-stock 10Applications exercises will appear on occasion throughout the book. These are exercises that require access to applications tools, e.g., a risk model and an optimizer. Applications exercises often aim to demonstrate results in the book, not through mathematical proof, but through software "experiments."
Page 40 index (you can consider it a portfolio with 100 shares of each stock). Also define the market as the capitalization-weighted MMI, or CAPMMI for short. 1. Restricting attention to MMI stocks, build the minimum-variance fully invested portfolio (portfolio C). What are the betas of the constituent stocks with respect to this portfolio? Verify Eq. (2A.16). 2. Build an efficient, fully invested portfolio with CAPM expected returns (proportional to betas with respect to the CAPMMI, which has an assumed expected excess return of 6 percent). Use a risk aversion of where is the risk of the CAPMMI. a. What are the beta and expected return to the portfolio? b. Compare this portfolio to the linear combination of portfolios C and B described in Eq. (2A.45). In this case, portfolio B is the CAPMMI.
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