MVS with N risky assets and a risk free asset

Transcription

1 MVS with N risky assets and a risk free asset The weight of the risk free asset is x f = 1 1 Nx, and the expected portfolio return is µ p = x f r f + x µ = r f + x (µ 1 N r f ). Thus, the MVS problem with a risk free asset becomes 1 min x 2 x Σx subject to µ p = r f + x (µ 1 N r f ) Note that we can drop the restriction 1 N x = 1 on the weights of the risky assets, because the risk free asset makes up the residual. 1

2 MVS with N risky assets and a risk free asset The Lagrange function is L = 1 2 x Σx + λ [µ p r f x (µ 1 N r f )]. The solution to this problem is x MV S (µ p ) [ ] µ p r f = (µ 1 N r f ) Σ 1 Σ 1 (µ 1 N r f ) (µ 1 N r f ) [ ] µ p r r = a 2br f + crf 2 Σ 1 (µ 1 N r f ). 2

3 MVS with N risky assets and a risk free asset Thus, provided r f b/c, minimum variance portfolios are combinations of the risk free asset and a portfolio of risky assets M with weight vector x M = 1 b r f c Σ 1 (µ 1 N r f ), (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 x MΣ to get σ 2 M = x M (µ 1 Nr f ) b r f c = µ M r f b r f c. (2) 3

4 MVS with N risky assets and a risk free asset The mean of M is obtained by left multiplying (1) by µ, i.e., µ M = a br f b cr f r f = bµ M a cµ M b. (3) Substituting (the equality on the right hand side of) (3) into (2), we observe σ 2 M = µ M bµ M a cµ M b b c bµ M a cµ M b = (cµ 2 M 2bµ M + a)/(cµ M b) (bcµ M b 2 bcµ M + ac)/(cµ M b) = cµ2 M 2bµ M + a, d which is the equation of the MVS, so M MV S. 4

5 MVS with N risky assets and a risk free asset For (ii), note that the right hand side equality in (3) implies that r f 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 r f, or, portfolio M is at the tangency point of the hyperbola and a line passing through r f. That is, M is the tangency portfolio. 5

6 MVS with N risky assets and a risk free asset The tangency portfolio is on the upper (lower) branch of the hyperbola if r f < (>)b/c = µ GMV P, i.e., we either have µ M > µ GMV P > r f or µ M < µ GMV P < r f. To prove this claim, just evaluate the product (µ M µ GMV P ) (µ GMV P r f ) = = = ( a brf b ) b crf b cr f c c (ac cbr f b 2 + bcr f )(b cr f ) c 2 (b cr f ) d c 2 > 0. 6

7 MVS with N risky assets and a risk free asset In the economically more relevant case, where r f < 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 r f > 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:

8 Consider a two asset example with µ = [ 2 5 ], Σ = [ ]. The MVS constants are given by a = 4.2, b = 0.9, c = 0.3, and d = ac b 2 = Thus, µ GMV P = b/c = 0.9/0.3 = 3. 8

9 Assume r f = 1 < 3 = µ GMV P. Then x M = Σ 1 (µ r f 1 2 ) b cr f = [ ], with µ M = 5.5 and σ M = , and θ M = µ M r f σ M = (4) Thus, the MVS is µ p (MV S) = 1 ± σ p (MV S)

10 10 case 1: 1 = r f < µ GMVP = b/c = tangency portfolio M

11 Assume r f = 4 > 3 = µ GMV P. Then x M = Σ 1 (µ r f 1 2 ) b cr f = [ ], with µ M = 2 and σ M = , and θ M = µ M r f σ M = (5) Thus, the MVS is µ p (MV S) = 4 ± σ p (MV S)( ). The efficient set is above the hyperbola. 11

12 12 case 2: 4 = r f > µ GMVP = b/c = tangency portfolio M

13 Now consider the case r f = 3 = µ GMV P. Then, 1 NΣ 1 (µ 1 N r f ) = b r f 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

14 Holding this portfolio on a scale γ > 0 results in a portfolio mean µ Q = γx Mµ = (µ r f 1 2 ) Σ 1 µ = γ(a br f ) = γ (a bc ) b = (γ/c)(ac b 2 ) = 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

15 To figure out the MVS, we compute the variance, σ 2 Q = γ 2 (µ r f 1 2 ) Σ 1 ΣΣ 1 (µ r f 1 2 ) = γ 2 (µ r f 1 2 ) Σ 1 (µ r f 1 2 ) = γ 2 (a 2br f + crf) 2 ( ) = γ 2 a 2b2 c + b2 c = γ2 c d, or σ Q = γ d c c γ(σ Q) = σ Q d. 15

16 Substituting γ(σ Q ) into µ Q = γd/c, we find µ Q = c d d c σ Q = d c σ Q, and combining a full investment in the risky asset with the zero investment portfolio Q produces the MVS µ p = r f ± d c σ Q = µ GMV P ± d c σ p, 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

17 In our example, x Q = [ ] 1 [ ] = [ ]. The MVS is µ p = r f ± d c σ p = 3 ± σ p = 3 ± 1.225σ p. 17

18 20 case 3: 3 = r f = µ GMVP = b/c =