MVS with N risky assets and a risk–free asset The weight of the risk–free asset is xf = 1 − 10N x, and the expected portfolio return is µp = xf rf + x0µ = rf + x0(µ − 1N rf ). Thus, the MVS–problem with a risk–free asset becomes 1 min x0Σx x 2 subject to µp = rf + x0(µ − 1N rf ) Note that we can drop the restriction 10N x = 1 on the weights of the risky assets, because the risk–free asset makes up the residual. 1 MVS with N risky assets and a risk–free asset The Lagrange function is 1 0 L = x Σx + λ [µp − rf − x0(µ − 1N rf )] . 2 The solution to this problem is xM V S (µp) µp − r f −1 (µ − 1N rf ) = Σ 0 −1 (µ − 1N rf ) Σ (µ − 1N rf ) " # µp − r r −1 = (µ − 1N rf ). Σ 2 a − 2brf + crf 2 MVS with N risky assets and a risk–free asset Thus, provided rf 6= b/c, minimum variance portfolios are combinations of the risk–free asset and a portfolio of risky assets M with weight vector xM = 1 Σ−1(µ − 1N rf ), b − rf c (1) the “tangency portfolio”. To see that (i) M ∈ M V S, and (ii) that M is indeed the tangency portfolio, left–multiply (1) by x0M Σ to get 2 σM x0M (µ − 1N rf ) µM − rf = = . b − rf c b − rf c (2) 3 MVS with N risky assets and a risk–free asset The mean of M is obtained by left–multiplying (1) by µ0, i.e., µM = a − brf bµM − a ⇔ rf = . b − crf cµM − b (3) Substituting (the equality on the right–hand side of) (3) into (2), we observe 2 σM = M −a µM − bµ cµ −b M M −a b − c bµ cµ −b M = (cµ2M − 2bµM + a)/(cµM − b) (bcµM − b2 − bcµM + ac)/(cµM − b) = cµ2M − 2bµM + a , d which is the equation of the MVS, so M ∈ M V S. 4 MVS with N risky assets and a risk–free asset For (ii), note that the right–hand side equality in (3) implies that rf is the mean of the zero–beta portfolio with respect to M . Thus, a tangency to the hyperbola at M intersects the return axis at rf , or, portfolio M is at the tangency point of the hyperbola and a line passing through rf . That is, M is the tangency portfolio. 5 MVS with N risky assets and a risk–free asset The tangency portfolio is on the upper (lower) branch of the hyperbola if rf < (>)b/c = µGM V P , i.e., we either have µM > µGM V P > rf or µM < µGM V P < rf . To prove this claim, just evaluate the product = (µM − µGM V P ) (µGM V P − rf ) a − brf b b − crf − b − crf c c = (ac − cbrf − b2 + bcrf )(b − crf ) c2(b − crf ) = d > 0. c2 6 MVS with N risky assets and a risk–free asset In the economically more relevant case, where rf < b/c, efficient portfolios are combinations of a long position in portfolio M and lending or borrowing at the risk–free rate. In the case where rf > b/c, efficient portfolios are generated by short (or zero) positions in the tangency portfolio (which is not efficient) and risk–free lending. The efficient set is above the hyperbola. The first analysis of these situations appeared in Robert C. Merton (1972): An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis, 7: 1851–1872. 7 Example Consider a two–asset example with µ= 2 5 , Σ= 4 2 2 6 . The MVS–constants are given by a = 4.2, b = 0.9, c = 0.3, and d = ac − b2 = 0.45. Thus, µGM V P = b/c = 0.9/0.3 = 3. 8 Example Assume rf = 1 < 3 = µGM V P . Then xM = Σ −1 (µ − rf 12) = b − crf −0.1667 1.1667 , with µM = 5.5 and σM = 2.7386, and θM µM − r f = = 1.6432. σM (4) Thus, the MVS is µp(M V S) = 1 ± σp(M V S)1.6432. 9 Example case 1: 1 = rf < µGMVP = b/c = 3 10 8 6 4 tangency portfolio M 2 0 −2 −4 −6 −8 0 1 2 3 4 5 6 10 Example Assume rf = 4 > 3 = µGM V P . Then xM = Σ −1 (µ − rf 12) = b − crf 2.3333 −1.3333 , with µM = −2 and σM = 4.4721, and θM µM − r f = = −1.3416. σM (5) Thus, the MVS is µp(M V S) = 4 ± σp(M V S)(−1.3416). The efficient set is above the hyperbola. 11 Example case 2: 4 = rf > µGMVP = b/c = 3 12 10 8 6 4 2 tangency portfolio M 0 −2 −4 0 1 2 3 4 5 6 12 Example Now consider the case rf = 3 = µGM V P . Then, 10N Σ−1(µ − 1N rf ) = b − rf c = 0, so that the portfolio of risky assets is a zero–investment portfolio, i.e., a portfolio with zero net value, created by buying and shorting equal amounts of securities. 13 Example Holding this portfolio on a scale γ > 0 results in a portfolio mean µQ = γx0M µ = (µ − rf 12)0Σ−1µ b = γ(a − brf ) = γ a − b c = (γ/c)(ac − b2) = γd/c > 0, so efficient portfolios are combinations of a full investment in the risk–free asset and a long position in the zero–investment portfolio. 14 Example To figure out the MVS, we compute the variance, 2 σQ = γ 2(µ − rf 12)0Σ−1ΣΣ−1(µ − rf 12) = γ 2(µ − rf 12)0Σ−1(µ − rf 12) = γ 2(a − 2brf + crf2 ) 2 2 2b b = γ2 a − + c c = γ2 d, c or σQ = γ r d ⇔ γ(σQ) = σQ c r c . d 15 Example Substituting γ(σQ) into µQ = γd/c, we find µQ = r cd σQ = dc r d σQ , c and combining a full investment in the risky asset with the zero–investment portfolio Q produces the MVS µp = r f ± r d σQ = µGM V P ± c r d σp , c so that the MVS is given by the asymptotes of the hyperbola which describes the risky assets–only MVS. There is no tangency portfolio. 16 Example In our example, xQ = 4 2 2 6 −1 2−3 5−3 = −0.5 0.5 . The MVS is µp = r f ± r d σp = 3 ± c r 0.45 σp = 3 ± 1.225σp. 0.3 17 Example case 3: 3 = r = µ f GMVP = b/c = 3 20 15 10 5 0 −5 −10 0 2 4 6 8 10 12 18