# On the sensitivity of the Black capital asset pricing model to the market portfolio

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1 Risk and ecision Analysis 4 (03) OI 0.333/RA IOS Press On the sensitivity of the Black capital asset pricing model to the market portfolio Winston S. Buckley a, Oneil Harris b and Sandun Perera c, a Institute of Mathematics, Actuarial and omputer Sciences, Jamaica b epartment of Finance, Eastern arolina University, N, USA c Jindal School of Management, The University of Texas at allas, Richardson, TX, USA Abstract. We show that Black apital Asset Pricing Model (Black APM) is extremely sensitive to the choice of the market portfolio and becomes unstable as market portfolios approach the Global Minimum-Variance portfolio. When market portfolios approach the minimum-variance portfolio, the expected return on the zero beta asset approaches negative infinity and its variance increases rapidly. Moreover, expected return on a fixed portfolio becomes indefinite (i.e., takes infinitely many values), and betas of all portfolios approach one. Unlike the Sharpe Lintner APM, the market risk premium in the Black APM always has a positive minimum, while beta may have a negative minimum value, dependent on the underlying covariance matrix. Keywords: Asset pricing, Black APM, global minimum variance asset, zero-beta asset. Introduction The capital asset pricing model (APM) seeks to explain the relation between risk and return in a rational market. The Sharpe [7] and Lintner [4] version of the APM is an extension of the Markowitz [5] mean variance efficient portfolio. Markowitz argues that investors are risk averse, so they choose portfolios that maximize expected return at a given level of risk, or that minimize the variance of the portfolio s return. The Sharpe Lintner APM builds on Markowitz [5,6] by also assuming that investors not only agree on the joint probability distribution of asset returns, but that they can unrestrictedly borrow and lend at a given riskless rate of interest. The Sharpe Lintner extension implies that investors combine the riskless asset with the best attainable mean variance efficient portfolio, MVEP (i.e., tangency portfolio) depending on their risk return preferences [8]. Under the Sharpe Lintner APM, the expected return ER i on any asset i, can be expressed as: ER i = R f + β i E[R m R f ], * orresponding author: Sandun Perera, Jindal School of Management, The University of Texas at allas, Richardson, TX 75080, USA. Tel.: ; where R f represents the riskless interest rate, β i symbolizes asset i s market beta and E[R m R f ] denotes the market risk premium (per unit of beta risk). This market risk premium is sometimes negative. Asset i s market beta represents the market risk-adjusted covariance of the return on the asset and the return of the mean variance efficient market portfolio, and is explicitly given as β i = ov(r i, R m ) Var R m. Since the APM provides an intuitive mechanism to price risk, it is widely used by analysts, corporations and investors to estimate the cost of capital and to evaluate performance. The 00 0 sovereign debt crisis has provoked debate concerning the applications of the APM, as hardly any asset actually appears to be risk free. ue to the nature of sovereign debt, practitioners usually consider short-term government bonds to be riskless. However, in wake of rising national debt levels, a crisis of confidence has emerged in sovereign bonds markets around the world. Rating agencies have cast doubt on the credit rating of the United States, the United Kingdom, Austria, France, and other countries, suggesting that riskless rates are illusions /3/\$ IOS Press and the authors. All rights reserved

2 78 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio Market participants in global finance have long struggled to apply the APM, as there is no appropriate risk free rate. Seeing that many sovereign bonds no longer appear to be riskless owing to recent downgrades, practitioners in the domestic setting have to now question their reference risk free rate. Yet, the APM assumes that a riskless asset can be specified. Because financial decisions are more complicated when there is uncertainty, we examine whether Black s approach to the APM is an appropriate alternative. Black [] relaxes the assumption that investors can borrow and lend at the riskless rate and develops a version of the APM where all assets are risky, provided that there is a zero-beta asset (a portfolio constructed to have zero systematic risk). The Black APM gives the expected return on an asset or portfolio i, with return R i, in terms of the expected return on the zerobeta asset zp, with return R zp, which is uniquely determined from any mean variance efficient market portfolio p, with return R p. onsequently, the expected return on any portfolio i, is given by the simple relationship: E[R i R zp ] = β ip E[R p R zp ], () β ip = ov(r i, R p ), Var(R p ) () where β ip is the market risk-adjusted covariance of the return on the asset, R i, with the return of the mean variance efficient portfolio or asset, R p. efine the expected return of asset a byμ a = ER a, where a {i, g, p, z, pz, zp}. Then μ i = ( β ip )μ zp + β ip μ p. (3) Hence, Black [] implies that the APM does not strictly require the existence of a riskless asset. In this paper, we show that Eq. (3) holds only if μ zp is not very large in absolute value. That is, if p, the efficient market portfolio, is not close to the global minimum variance portfolio, g. We demonstrate that the Black APM becomes unstable as portfolios approach the global minimum variance portfolio on the mean variance efficient frontier. The results indicate that as the market portfolio moves towards the global minimum variance portfolio, the expected return on the zero-beta asset approaches negative infinity, while its variance approaches positive infinity. Therefore, unlike the Sharpe Lintner APM, with a risk free (zero variance) asset, the unique zero-beta asset of the Black APM has a variance that may become infinite. In addition, expected returns on a fixed portfolio becomes indefinite (that is, takes infinitely many values), and betas of all portfolios approach. We also show that, unlike the Sharpe Lintner APM, the market risk premium in the Black APM always has a positive minimum value which varies with the underlying covariance matrix. onsequently, betas of portfolios are sometimes negative, a situation that does not generally occur in the Sharpe Lintner APM. We also show that given the mean variance efficient frontier (MVEF), the zero-beta asset uniquely determines the market portfolio on the MVEF. We then give a simplified Black APM that has a mean zero-beta return of zero, and a mean market return of B A, where A and B are extracted from the underlying covariance matrix. The betas of assets can become negative, and have a minimum value of A B. The Security Market Line has zero intercept, and all portfolios have cost of capital dependent only on the MVEP with mean B A.. The Black mean variance frontier In this section, we briefly review the construction and properties of the Black [] mean variance frontier. Recall that there is no risk-free asset in the Black framework. Rather, Black posits that the return on the zero-beta asset is uncorrelated with the return of the efficient market portfolio. onsider n risky assets, with random returns R, R,..., R n and return vector R = (R, R,..., R n ), and mean return vector μ = ER = (ER, ER,..., ER n ). Black [] assumes that no linear combination of these risky assets have zero variance. That is, the covariance matrix of returns, denoted Ω = ov(r, R ), is non-singular. Let R p be the return on a portfolio built from these n assets, with weight vector w p = (w, w,..., w n ). Since there are no risk-free assets, the portfolio p is fully invested in the assets,,..., n, with R p = wr, μ p = wμ, wi = and I is a column of ones. The mean variance frontier is the locus of (σ p, μ p ), where σp = min w w Ωw, subject to μ p = wμ, wi =. Thus, for a given mean return μ p, the minimum variance of the return on the portfolio is σ p. One can easily

3 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 79 show that the weight vector w p is a linear function of the expected return μ p of the portfolio (see []). That is, w p = g + h μ p, (4) where the constant vectors g and h are built from the (inverse) covariance matrix Ω, the mean return vector μ and I as follows: g = [B(Ω I) A(Ω μ)], (5) h = [(Ω μ) A(Ω I)]. (6) The parameters A, B, and are constants constructed from the inverse covariance matrix of the underlying base portfolio from which all portfolios are constructed. Explicitly, A = μ T Ω I, B = μ T Ω μ, = I T Ω I, = B A. (7) We now give an explicit formula for the minimum variance conditioned on the expected mean return on the portfolio. The reader is directed to [] for proofs of the following proposition and its corollary. Proposition. The mean variance frontier is the locus of (σ p, μ p ), where σ p = Var R p = ( μ p A ) +. Moreover, if R p and R q are mean variance portfolios, then ov(r p, R q ) = ( μ p A )( μ q A ) +. (8).. The global minimum mean variance portfolio (GMVP) It follows immediately from Proposition that the variance, σp, has a global minimum of when μ p = A. The unique portfolio with this mean and variance is called the Global Minimum Mean Variance Portfolio (GMVP). We reserve the subscript g to reference GMVP. Figure plots the mean variance frontier with the corresponding mean return for each zero-beta portfolio. The graph emphasizes that as the market portfolio approaches the global minimum variance portfolio, the mean return of the zero-beta portfolio approaches negative infinity. Fig.. μ p versus σp is the mean variance frontier with the corresponding mean return for each zero-beta portfolios. The graph emphasizes that as the market portfolio approaches the global minimum variance portfolio, the mean of the zero-beta portfolio approaches negative infinity. The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP).

4 80 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio orollary. The Global Minimum Portfolio g, with return R g, has mean, variance and weight vector given by μ g = A, σ g = Var R g =, w g = Ω I = Ω I σ g. Moreover, the covariance between R g and the return on any other mean variance portfolio (MVP) R p, is equal to the global minimum variance,. 3. The zero-beta portfolio From Proposition, one can obtain the mean return of a given MVP in terms of the mean return of any other MVP provided that the covariance between them is known. We offer these results in this section. Theorem. Let R q and R p be returns of MVPs, with expected returns, μ q and μ p, respectively and covariance γ pq. Then provided μ p A, μ q = A + (/)(γ pq /). (9) μ p A/ Moreover, γ pq = if and only if q is the GMVP. Proof. Rewriting Eq. (8) in terms of μ q yields, γ pq = (μ p A )(μ q A ), whence μ q A = (/)(γ pq /), which gives the result. μ p A/ If γ pq =, then μ q = A, which is the mean return of the GMVP. onversely, if q is the GMVP, then from orollary, μ q = A, whence γ pq = 0. Equation (9) indicates that, given the mean return of an MVP, we can find the mean return of any other MVP provided we know how both covary. Since covariance is a proxy for correlation, we can find the mean of any other MVP, given the mean and correlation coefficient, ρ, of some fixed MVP. We explicitly give this relationship in the following corollary. orollary. Let R q and R p be returns of MVPs, with expected returns and volatilities, μ q, μ p and σ p, σ q, respectively, and correlation coefficient ρ pq. Then, provided μ p A, μ q = A + (/)(ρ pqσ p σ q /). (0) μ p A/ Moreover, ρ pq = σpσq if and only if q is the GMVP. This representation is a non-linear equation in μ q since σ q can be written as a quadratic function thereof. It is fairly complicated since it requires more inputs that just correlation; e.g., the variances of both portfolios are also required, in addition to the standard input μ p, the mean market return. We now provide another simple consequence of Proposition. orollary 3. Let R q and R p be returns of MVPs, with expected returns and volatilities μ q, μ p and σ p, σ q, respectively, and correlation coefficient ρ pq. Then provided μ p A, the portfolios are always positively correlated, with ρ pq σ p σ q = σ g. () σ p σ q There is a lower bound on, given by σ pσ q. Proof. Equation () follows from (0) by setting γ pq = ρ pq σ p σ q. Since ρ pq, and the portfolios are MVPs, then μ p > A. Thus ρ pqσ p σ q > 0, whence ρ pq σ pσ q. The other representation follows from the fact that σg =. Since ρ pq, then σ pσ q, from which the result follows. Alternatively, the minimum variance portfolio has variance = σ g = σ g σ g σ p σ q, for any MVPs, p and q. Thus, correlation between MVPs is bounded below by the inverse of the product of and the volatilities, σ p and σ q. We now introduce the zero-beta portfolio zp, for each mean variance market portfolio p. This is the MVP with return R zp, which has minimum variance and zero covariance with the MVP portfolio p having return R p. Thus γ zp,p = 0. Unlike the risk-free asset, the zero-beta asset has mean and variance that are tied to the mean and variance of the market return R p.this portfolio is unique. We give these measures in the following theorem.

5 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 8 Theorem. Let R p be the return of a MVP, with mean μ p and volatility σ p. Provided p is not the global minimum variance portfolio GMVP, there exists a unique zero-beta portfolio zp, with return R zp, that is also MVP, with mean μ zp and volatility σ zp given by μ zp = A / μ p A/, () σzp = σp / +. Remark. (i) The mean of the zero-beta asset is a constant minus a positive component that is inversely related to the excess of the mean return of p and the global minimum variance portfolio, g. (ii) The variance of the return on zp is a constant plus a positive component that is inversely related to the excess variance of the market portfolio over the GMVP. Proof. Equation () comes directly from setting γ zp,p = 0 in Theorem. Therefore μ zp A = μ zp μ g = / μ = / p A/. Since R zp is MVP, then from Proposition, it follows that Var R zp = σzp = ov(r zp, R zp ) = (μ zp A ) + = (μ zp μ g ) + = + 3 (). However σp = (μ p μ g ) +. Thus σ p = σ p σg = (μ p μ g ). Therefore, σzp = + 3 () = σp σ g σp σ g. σp σ g + = σ4 g σ p σ g σ g + σg = σg( + ) = σp σ g Observe from the proof that the excess volatility of the zero-beta portfolio over the global minimum variance portfolio is inversely proportional to the excess volatility of the market portfolio over the global minimum variance portfolio. That is, excess variances and hence, volatilities covary negatively. σ zp σ g = (σ p σ g). The mean returns and variances of the GMVP and zero-beta portfolio are linked as follows. orollary 4. Let μ g = A be the mean and Var R g = be the variance of the GMVP. Then provided μ p μ g, the mean and variance of the unique zero-beta portfolio zp, of the market portfolio p, with return R p, are given by: μ zp = μ g / μ p μ g, Var R p = (μ p μ g ) + Var R g, Var R zp = Var R p Var R g + Var R g = σ pσg σp σg. (3) Proof. The results follow when we set μ g = A, Var R g = = σ g and Var R p = σ p in Theorem and Proposition. In addition, σzp = σp σg + σg = σ4 g σp σg + σg ( σ = σg ) g σp σg + = σ p σg σp σg. Figure shows that the mean zero-beta return μ zp, is a strictly increasing function of the mean market return μ p, with μ zp being strictly negative if the mean market return is less than B A = In addition, the mean zero-beta return μ zp, is a strictly decreasing function of μ g, which is the mean global minimum variance portfolio as shown in Fig. 3. It approaches negative infinity as μ g approaches the vertical asymptote, B A = Furthermore, as shown in Fig. 4, the volatility of the zero-beta portfolio, σ zp, is a strictly decreasing function of σ p, the market return volatility. It becomes infinitely large when σ p approaches the volatility σ g, of the global minimum variance portfolio, and is bounded below by it. 4. Minimum risk premia This section shows that the expected excess return of the market portfolio p, over the return of the zerobeta asset zp, has a positive minimum of.thisis a surprising result, since in the Sharpe Lintner APM, this can be negative. We formalize this result in the following theorem. Theorem 3. Let p be the market portfolio and zp its zero-beta portfolio, with mean returns μ p and μ zp, respectively. Let g be the GMVP, with mean return μ g.

6 8 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio Fig.. The mean zero-beta return μ zp is a strictly increasing function of the mean market return μp, with μzp being strictly negative if the mean market return is less than B A = The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). Fig. 3. The mean zero-beta return μ zp is a strictly decreasing function of μg, the mean global minimum variance portfolio. It becomes infinitely negatively large as μ g approach the vertical asymptote, B A = The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). Then the excess expected return of the market over the zero-beta portfolio is at least ; that is μ p μ zp. In particular, the minimum (equality) is achieved when μ p = μ g + = A +, where A, and are defined by (7).

7 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 83 Fig. 4. The volatility of the zero-beta portfolio σ zp is a strictly decreasing function of σp, the market return volatility. It becomes infinitely large when σ p approaches the volatility σg, of the global minimum variance portfolio, and is bounded below by it. The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). Proof. From orollary 4, μ zp = μ g /, whence y μ p μ zp = μ p μ g + = x + / x f(x), / μ p μ g where x μ p μ g. The market risk premium y has a turning point at f (x) = / = 0. Thus x = x. At this value of x, f (x) = / > 0, which x 3 confirms a minimum value of y. Thus μ p μ zp f at x = μ p μ g = μ g + ( ) = + / / =,. Since μ g = A, then μ p = = A+. An immediate consequence of Theorem 3 is that the expected return on any arbitrary portfolio i, also has a guaranteed minimum that is a function of its beta. We formalize this in the following corollary. orollary 5. Let p be the market portfolio and zp its zero-beta portfolio, with mean returns μ p and μ zp, respectively. Let g be the GMVP, with mean return μ g and let i be any arbitrary portfolio with mean μ i and beta β ip. (i) If β ip 0, then the excess expected return of the portfolio over the zero-beta portfolio is at least β ip ; that is μ i μ zp β ip. In particular, the minimum expected return on portfolio i, is min μ i = μ zp + β ip = A + (β ip ). (ii) If β ip < 0, then the excess expected return of the portfolio over the zero-beta portfolio is at most β ip ; that is μ i μ zp β ip.

8 84 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio In particular, the maximum expected return on portfolio i, is max μ i = μ zp + β ip = A + (β ip ). For both cases, optimality is achieved when μ zp = μ g = A, or equivalently, when μ p = μ g + = A+, where A, and are positive numbers defined by (7). Proof. By the Black APM formula (), and Theorem 3, min(μ i μ zp ) = min E[R i R zp ] = min β ip E[R p R zp ] = β ip min E[R p R zp ] = β ip, which is achieved when μ p = μ g + = A+. But from orollary 4, μ zp = μ g / = A = A, which gives the result. Since μ i, then its minimum value must be greater than. Therefore, A + (β ip ). Rearranging yields, β ip A. Part (ii) follows similarly using the fact that for any function f(x), max f(x) = min( f(x)). Note that the minimum return on a portfolio i, is positive only if the mean zero-beta return is greater than the negative of the product of its beta and the minimum market risk premium. Likewise, the maximum return is negative if the betas are less than or equal to A, which is a function of the underlying covariance matrix. Let us now examine the trade-off between the expected return on the zero-beta asset, and the expected returns on the GMVP g, and the market portfolio, p. 4.. Sensitivity of μ zp to μ g ifferentiating μ zp = μ g / partially with respect to μ g, yields μ zp μ g = (μ p μ g ) = (μ p μ g /)(μ p μ g + /) (μ p μ g ). learly, μzp μ g 0 if only if μ p μ g.itis zero at μ p μ g =, which, from Theorem 3, is the point where the minimum excess expected return of the market over the zero-beta portfolio is attained. Thus, for fixed μ p, the mean zero-beta return increases with the mean global minimum return when μ g <μ p and decreases with the mean global minimum return when μ g >μ p. 4.. Sensitivity of μ zp to μ p Fixing μ zp and differentiating μ zp = μ g / partially with respect to μ p, yield μ zp μ p = (μ p μ g ) > 0, which confirms that the mean zero-beta return increases monotonically with the mean market return. We now give the limiting behaviour of the mean and variance of the zero-beta asset zp, when the underlying portfolio p, gets close to the global minimum asset, g. Theorem 4. Let g be the GMVP. For any choice of the market portfolio p on the MV efficient frontier, μ zp is monotonic increasing in μ p. In particular: (i) μ zp <μ g and σ zp σ g; (ii) μ zp μ g and σ zp σ g as μ p ; (iii) μ zp and σ zp as μ p μ g that is as p q. Proof. (i) On the mean variance efficient frontier, μ p μ g, whence from Eq. (3) of orollary 4, μ zp = μ g / <μ g, since is always positive. (ii) Since μzp μ p is always positive then μ zp is an strictly increasing function of μ p. Thus μ zp increases monotonically to μ g. Similarly, from orollary 4, if μ p explodes then μ zp converges to the GMVP, whence σzp σg. (iii) If μ p μ g, then by Eq. (3) of orollary 4, /, whence μ zp = μ g /. Since σp σg then by the variance formula σzp = σg( σg + ). σp σ g

9 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 85 Fig. 5. The excess mean market return over the zero-beta portfolio is a strictly positive function of the excess mean market return over the global minimum variance portfolio. It decreases strictly from positive infinity to a positive minimum of = when the mean excess market return over the GMVP is = It then increases strictly at a positive rate to infinity. The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). Figure 5 shows that the excess mean market return over the return of the zero-beta portfolio is a strictly positive function of the excess mean market return over the global minimum variance portfolio. It decreases strictly from positive infinity to a positive minimum of = when the mean excess market return over the GMVP is = It then increases strictly at a positive rate to infinity. In addition, Fig. 6 shows that μ zp is a strictly increasing function of μ p, which increases at a decreasing rate as μ p, the mean market return, increases. μ zp is also a strictly decreasing function of μ g, the mean global minimum variance portfolio, but does so at an increasing rate as μ g increases. 5. Instability of Black APM near the GMVP In this section, we prove that the Black APM fails to give stable cost of capital whenever the underlying efficient market portfolio is close to the global minimum portfolio. Assume that μ zp 0 and the returns on market p and an arbitrary asset i, are finite and fixed. From Eq. (), the beta of the APM equation is given by β ip = ov(r i, R p ) Var(R p ) = E[R i R zp ] E[R p R zp ] = μ i μ zp μ p μ zp = μ i/μ zp μ p /μ zp. (4) Now let the market portfolio p converge to the global minimum portfolio g. Then from Theorem 4, μ zp which forces the ratios μ i μ zp and μp μ zp to zero, and hence, by Eq. (4), the asset s beta β ip goes to one! In this situation, an infinite number of expected returns satisfy Eq. (4); that is, μ i {x: x< } and the return on the asset becomes arbitrary! We subsequently show that given the mean variance (MV) frontier, a zero-beta asset z of finite expected return, μ z, uniquely determines the expected market portfolio return, μ pz, and hence the market portfolio, pz. Usinga similar argument, we see that if z, which lives on the lower half of the MV frontier, gets close to the global minimum portfolio g, then μ pz and the ratios μ i μ pz

10 86 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio Fig. 6. μ zp is a strictly increasing function of μp, which increases at a decreasing rate as μp, the mean market return increases. It is also a strictly decreasing function of μ g, the mean global minimum variance portfolio, but does so at an increasing rate as μg increases. The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). and μz μ pz go to zero, whence from Eq. (4), β i,pz = μ i/μ pz μ z /μ pz μ z /μ pz 0. In this case, asset i degenerates to an asset that has mean return equal to that of the GMVP. That is, μ i = μ g for any arbitrary asset i. Thus, the model now considers all asset to be the GMVP, a clearly unreasonable and impractical state of affair. That is, the model can no longer distinguish between assets! In fact, this situation implies that even the MV portfolios have the same mean, and hence the same variance, thereby forcing the MV frontier to be a single point at the GMVP. We encapsulate these findings in the following theorem. Theorem 5. (a) Let g the GMVP, and p any MV portfolio. Let i be any fixed portfolio. If p g, then β ip and μ i is any value in the set {x : x< }. (b) Let i be any fixed arbitrary portfolio. Given the mean variance (MV) frontier, if z is any zero-beta portfolio, with finite mean μ z, and matching MV efficient portfolio pz, then β i,pz 0 and μ i = μ g as z g. That is, all arbitrary portfolios have the same expected mean return as the mean of the zerobeta portfolio approaches the mean of the GMVP. Remark. In the sequel we simply refer to pz as p and z as zp, since they are obviously the same objects. 6. A zero-beta asset determines a unique market portfolio Given the MV frontier and a zero-beta asset z,offinite expected return μ z, we now show that this portfolio or rather mean, uniquely determines the (mean of) MV efficient portfolio pz. Proposition. Let z be any arbitrary zero-beta asset with finite mean μ z. Then there exists an MV efficient portfolio pz, with mean μ pz, given by μ pz = μ g / μ z μ g. (5) Proof. Let μ z be the given zero-beta portfolio. onstruct or choose a portfolio p, with mean return μ p on the MV efficient frontier, such that its unique zerobeta portfolio has mean μ zp = μ z. Then from Theorem, μ z = μ g / μ p μ g. Solving for μ p yields μ p = μ g / μ z μ g. Taking μ pz = μ p yields the desired result. Replacing pz by p and z by zp, we can now rewrite Eq. (5) in the usual notation. μ p = μ g / μ zp μ g, (6)

11 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 87 with μ p as μ zp. Equation (6) is striking! We can now increased our degree of freedom in the use of Black APM, in the sense that we can now choose returns of mean variance efficient portfolios (MVEP), given any zero-beta expected return that is less than the mean return of the GMVP. That is, we can start with a zero-beta mean return and work backwards to find the mean market return via Eq. (6). This leads to considerable simplification of the Black APM, in that we may do away with the intercept term by choosing a mean zero-beta return that is actually zero! We explore this concept with the help of the following proposition. Proposition 3. Let μ zp be the mean of the zero-beta asset corresponding to the MVEP p, with mean μ p. Then μ p = B A if μ zp = 0, with variance σp = B, and A μ p > B A iff μ zp > 0. Proof. Recall that If μ zp = 0 then μ p = μ g / μ g = A / A = A A Eq. (6) μ p μ g = / = B A = B A.Ifμ p > B A then from > B A A = B A A = μ zp μ g /. Thus μ zp μ g > A, whence (μ zp μ g ) < A, A which yields, μ zp > 0. The converse follows similarly! From Proposition, if μ p = B A, then σ p = ( B A A ) + A + = +A A = ( B A A ) + = = B A = B A. + A = Remark. It follows immediately from the last result and Theorem 3, that A, B,, must satisfy the condition, B A. This can also be proven directly by algebraic manipulation of (B A ) 0. Since we are now free to choose the mean return of the zero-beta asset, this leads to a zero-intercept Black APM. Theorem 6. Let R i be the return on any asset with finite mean μ i. There exists β i such that μ i = β i,a,b B A, where β i,a,b = A ov(r i,r A,B ) B is greater than or equal to A B and R A,B is the zero-market MVEP with mean return B A and weight vector w A,B = h + g B A. Proof. hoosing μ zp = 0, yields μ p = B A. Substituting these values in Black APM yields, μ i = ( β ip )μ zp + β ip μ p = β ip μ p = β B i,a,b A. Since μ i then β B ip A. Therefore, β i,a,b A B. The weight vector for the zero-market MVEP R A,B follows from Eqs (4) (7). Thus, the expected return on any asset is positive or negative if it covaries positively or negatively with the zero-market MVEP R A,B, which has mean return B A. Moreover, unlike the non-negative betas in the Sharpe Lintner APM model, the betas in the Black APM model can be negative, with a lower bound of A B. 7. Example We use the six Fama-French Size and Book-to- Market portfolios to generate the mean variance frontier. We collect data for 60 months from July 00 to June 007. We obtain A = B = , = , = The mean variance frontier is generated by the equation: σ = (μ μ g) + = (μ ) In Fig. 7, we plot a scaled version of the mean variance frontier of the 6 Fama-French size and book-to-market portfolios. The zero-beta portfolio has mean return μ zp = 0, with variance σzp = The zero-market MVEP has mean return μ p = B A = , variance σp = and volatility σ p = The betas for the Black APM model must satisfy the condition β i A B = 6.5. The GMVP has mean return μ g = and variance σg = = In Fig. 8, we plot the Security Market Line of the Black APM model. This is given as μ i = β i, where β i 6.5.

12 88 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio Fig. 7. This is a scaled version of the mean variance frontier of the 6 Fama-French size and Book-to-Market portfolios. The data are obtained from the enter for Research in Security Prices (RSP). We collect data for the 60 months from July 00 to June 007. Fig. 8. The security Market Line μ = β for β in the range 6 to 00. The graph is based on the 6 Fama-French Size and Book-to-Market portfolios. We collect data for 60 months from July 00 to June 007 from the enter for Research in Security Prices (RSP). Thus, the expected return of assets increases by basis points for each positive unit change in beta. In addition, betas can take negative values greater than 6.5. The Black APM model breaks down in the general case if μ p The mean-zero-beta return for each corresponding expected return of a MVEP is μ zp = μ g μ p μ g = E 05 μ p ,

13 W.S. Buckley et al. / On the sensitivity of the Black capital asset pricing model to the market portfolio 89 with variance σzp = σ pσg σp σg = σ p σ p The minimum market risk premium is min [μ p μ zp ] = p MV EP = onclusion We show that the Black APM [] is extremely sensitive to the choice of the market portfolio and becomes unstable as portfolios approach the Global Minimum (GMVP) portfolio on the mean variance efficient frontier. Unlike the Sharpe [7] and Lintner [4] versions of APM with a risk free (zero variance) asset, the unique zero beta asset of Black APM has variance that may become extremely large or infinite. As the market portfolio approaches the GMVP, the expected return on the zero beta asset approaches negative infinity while its variance approaches positive infinity. In addition, expected returns on portfolios become indefinite (that is, have infinitely many values), and betas of portfolios approach one. We also show that unlike the Sharpe Lintner APM, the market risk premium in the Black APM always has a positive minimum value dependent on the underlying covariance matrix. We show that given the mean variance frontier, the zero-beta asset uniquely determines the market portfolio on the MVEF. We then give a simplified Black APM model that has a mean zero-beta return of zero and a mean zero-market return of B A, where A, B are constants extracted from the underlying covariance matrix. Unlike the Sharpe Lintner APM, the beta of a portfolio can be negative, and takes a minimum value of A B, in accordance to how it covaries with the zero-market efficient portfolio. The Security Market Line has zero intercept, and all portfolios have cost of capital dependent only on the zeromarket MVEP. We believe that our findings may encourage investors to empirically test the Black APM to see whether it generates portfolios near the GMVP before they use it for their investment decisions. In addition, when the market portfolios degenerates to the GMVP, the instability in the model could possibly lead to numerical instability and large rounding errors when practically applying the Black APM. It is also argued in [3] that equity investors should invest in the GMVP. However, our results show that it might not be optimal to all the investors to invest in the GMVP. Our main results can also persuade scholars to further investigate other applications of the relative pricing concept in the finance literature. In particular, we would like to bring the following analogous application of the relative pricing concept used in the complete markets. First, we recall that APM extends the Markowitz model by introducing a risk free asset and hence uses a relative pricing concept. The Black APM replaces the risk-free asset (or the numeraire) with zero-beta asset which is independent of the market portfolio. Arrow s and ebreu s state preference theory provides the fundamental framework for defining a relative pricing concept which uses an appropriate martingale measure. However, the martingale approach in the complete market can possibly lead to a martingale which is different from the risk neutral one. Hence, the relative pricing concept used in the complete market could employ a numeraire which is different from the risk free numeraire which may be analogous to the zero-beta asset in the Black APM. Acknowledgements The authors thank harles Tapiero, o-editor-in- hief, and anonymous reviewers for the prompt review process and their helpful comments. References [] F. Black, apital market equilibrium with restricted borrowing, J. Business 45 (97), [] J. ampbell, A. Lo and. MacKinley, The Econometrics of Financial Markets, Princeton Univ. Press, Princeton, NJ, 997. [3] R. Jagannathan and T. Ma, Risk reduction in large portfolios: why imposing the wrong constraints helps, J. Finance 58 (003), [4] J. Lintner, The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets, Rev. Econ. Statist. 47 (965), [5] H. Markowitz, Portfolio selection, J. Finance 7 (95), [6] H. Markowitz, Portfolio Selection: Efficient iversification of Investments, Wiley, New York, 959. [7] W. Sharpe, apital asset prices: a theory of market equilibrium under conditions of risk, J. Finance 9 (964), [8] J. Tobin, Liquidity preferences as behavior towards risk, Rev. Econ. Stud. 5 (958),

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