Application to Portfolio Theory and the Capital Asset Pricing Model

Transcription

1 Appendix C Application to Portfolio Theory and the Capital Asset Pricing Model Exercise Solutions C.1 The random variables X and Y are net returns with the following bivariate distribution. y x f Y (y) 0 0 3/8 3/8 0 6/8 1 1/ /8 2/8 f X (x) 1/8 3/8 3/8 1/8 Collecting results from Appendix B, these marginal distributions can be used to obtain the following means and variances. E(X) = 3 2 E(Y ) = 1 4 V (X) = 3 4 V (Y ) = 3 16 As well, we saw in Appendix B that although these random variables are not statistically independent they nevertheless have zero covariance: cov(x, Y ) = 0. A portfolio of these returns is of the form R p = (1 a)x + ay. 209

2 210 INSTRUCTOR S MANUAL (a) The expected return on the portfolio is: E(R p ) = (1 a)e(x) + ae(y ) = (1 a) a1 4 = a. (b) The variance of the portfolio is: V (R p ) = (1 a) 2 V (X) + a 2 V (Y ) = (1 a) a (c) The derivative of this variance with respect to a is dσ 2 p da = 2(1 a)( 1) a 3 16 = a. Setting this derivative equal to zero yields a first order condition that may be solved for the risk minimizing portfolio weight: a = 4 5. The value of this minimum variance is σp 2 = (1 a) ( ) 2 ( ) 2 1 a2 16 = = Note that this is less than either of the return variances individually. (d) The minimum-variance expected portfolio return is E(R p ) = a = = 1 2. Note that this is between the extremes of E(X) = 3 2 and E(Y ) = 1 4, as it must be. C.2 Let the umbrella manufacturer s return be denoted R 1 and the resort owner s return R 2. The bivariate distribution of these random variables is as follows. R 1 25% 50% R 2 f 2 (R 2 ) 25% 0 1/2 1/2 50% 1/2 0 1/2 f 1 (R 1 ) 1/2 1/2 (a) Using the marginal distribution for R 1, the mean of the umbrella manufacturer s return is and the variance is E(R 1 ) = ( 0.25) (0.50)1 2 = 1 8 V (R 1 ) = E(R 2 1) (ER 1 ) 2 = ( 0.25) (0.50)2 1 2 ( 1 8 ) 2 = 9 64.

3 EXERCISE SOLUTIONS 211 (b) Since R 1 and R 2 have the same marginal distributions they have the same expected return and variance. (c) Using the bivariate distribution, E(R 1 R 2 ) = ( 0.25)( 0.25)0 + ( 0.25)(0.50) 1 2 The covariance is therefore + (0.50)( 0.25) (0.50)(0.50)0 = 1 8. cov(r 1, R 2 ) = E(R 1 R 2 ) E(R 1 )E(R 2 ) = 1 8 ( 1 8 and the correlation is ρ = cov(r 1, R 2 ) V (R1 ) V (R 2 ) = 9/64 = 1. 9/64 9/64 ) 2 = 9 64 That is, these two returns are exactly offsetting. (d) A diversified investor with half his wealth in each asset earns expected return E(R p ) = 1 2 E(R 1) E(R 2) = = 1 8. This is, of course, the common expected return of the two assets individually. The variance is ( ) 2 ( ) 2 ( ) V (R p ) = V (R 1 ) + V (R 2 ) + 2 σ 1 σ 2 ρ ( ) 2 ( ) 2 ( ) = ( 1) = 0. That is, investing 50/50 in the two assets makes it possible to earn a certain return of R p = 1 8. C.3 The assets R 1 and R 3 have the following means and standard deviations. E(R 1 ) = 0.10 σ 1 = 0.04 E(R 3 ) = 0.05 σ 3 = 0.02 (a) The expected return on a portfolio is E(R p ) = (1 a)e(r 1 )+ae(r 3 ) = (1 a)(0.10)+a(0.05) = a. (b) From Example 3, the standard deviation of the portfolio is σ p = (1 a)σ 1 + aσ 3 = (1 a)(0.04) + a(0.02) = a.

4 212 INSTRUCTOR S MANUAL (c) Solving this for a and substituting into the expression for E(R p ) yields E(R p ) = (0.04 σ p) 0.02 = 5 2 σ p. (d) The relationship between E(R p ) and σ p takes the form of a straight line between the points R 1 and R 3. All points on the line are efficient. C.4 The assets R 1 and R 2 have the following means and standard deviations. E(R 1 ) = 0.10 σ 1 = 0.04 E(R 2 ) = 0.05 σ 2 = 0.04 (a) The expected return on a portfolio is E(R p ) = (1 a)e(r 1 )+ae(r 2 ) = (1 a)(0.10)+a(0.05) = a. (b) The variance of the portfolio is σ 2 p = (1 a) 2 σ a 2 σ (1 a)aσ 1 σ 2 ρ = (1 a) 2 (0.016) + a 2 (0.016) + 2(1 a)a(0.04)(0.04) 1 2 = [(1 a) 2 + a 2 + (1 a)a](0.016). Thus the standard deviation is σ 2 p = 0.04 (1 a) 2 + a 2 + (1 a)a. (c) Numerical values for the mean and standard deviation of the portfolio return are as follows. a σ p E(R p ) (d) Graphed, this relationship is a curve joining R 1 and R 2 with minimum risk at a = 0.5. C.5 The assets R 2 and R 3 have the following means and standard deviations. E(R 2 ) = 0.05 σ 2 = 0.04 E(R 3 ) = 0.05 σ 3 = 0.02

5 EXERCISE SOLUTIONS 213 (a) The expected return on a portfolio is E(R p ) = (1 a)e(r 2 ) + ae(r 3 ) = (1 a)(0.05) + a(0.05) = (b) From Example 4, the variance of the portfolio is σ 2 p = (1 a) 2 σ a 2 σ 2 3 = (1 a) 2 (0.04) 2 + a 2 (0.02) 2. (c) The derivative of this variance with respect to a is dσ 2 p da = 2(1 a)( 1)(0.0016) + 2a(0.0004) = a. Setting this derivative equal to zero yields a first order condition that may be solved for the risk minimizing portfolio weight: a = 4 5. The value of this minimum variance is σp 2 = (1 a) 2 (0.04) 2 + a 2 (0.02) 2 ( ) 2 ( ) = (0.0016) + (0.0004) = = Note that this is less than either of the return variances individually. (d) Numerical values for the mean and standard deviation of the portfolio return are as follows. a σ p E(R p ) (e) Graphed, this relationship is a straight line segment, horizontal at E(R p ) = It terminates on the right at R 2 and on the left at the minimum risk of σ p = 2/ The efficient set consists only of this minimum-risk point. C.6 Let the returns on asset i and the market be scaled by a factor v, and so are vr i and vr m. Prior to scaling the return on a portfolio is R p = (1 a)r i + ar m. (a) After scaling, the return on a portfolio is (1 a)vr i + avr m = v[(1 a)r i + ar m ] = vr p. Thus the return on a portfolio is also scaled by v.

6 214 INSTRUCTOR S MANUAL (b) The expected return on each asset is scaled by v. E(vR i ) = ve(r i ) E(vR m ) = ve(vr m ) (c) The expected return on a portfolio is scaled by v. E(vR p ) = ve(r p ) (d) The variance of the individual returns is scaled by v 2. V (vr i ) = v 2 V (R i ) V (vr m ) = v 2 V (R m ) The standard deviations are scaled by v. V (vri ) = vσ i V (vrm ) = vσ m ) (e) The covariance is scaled by v 2. cov(vr i, vr m ) = v v cov(r i, R m ) = v 2 σ im The correlation is unaffected. cor(vr i, vr m ) = cov(vr i, vr m ) vσ i vσ m (f) The variance of a portfolio is scaled by v 2. V (vr p ) = v 2 V (R p ) The standard deviation is scaled by v. V (vr p ) = vσ p ) (g) Using the expression β im = ρ im σ i σ m, = v2 σ im v 2 σ i σ m = σ im σ i σ m = ρ im we have seen that a scaling affects these quantities as follows: Thus beta is invariant to scaling. ρ im vσ i vσ m = ρ im vσ i vσ m = β im. (h) The measure of total systematic risk is β im σ m. A scaling leaves β im unchanged but affects σ m : β im vσ m = vβ im σ m. Thus this measure of total risk is scaled by the factor v.

7 EXERCISE SOLUTIONS 215 C.7 (a) i. The return on the portfolio is R p = (1 a)r 1 + ar 2. ii. The expected return on the portfolio is E(R p ) = (1 a)e(r 1 ) + ae(r 2 ). iii. The variance of the return on the portfolio is V (R p ) = (1 a) 2 σ a 2 σ (1 a)aσ 12. (b) i. The relationship between σ pm = cov(r p, R m ) and the individual asset covariances σ 1m and σ 2m is linear. σ pm = cov(r p, R m ) = E[(R p E(R p ))(R m E(R m ))] = E{[(1 a)r 1 + ar 2 ((1 a)e(r 1 ) + ae(r 2 ))](R m ER m )} = E{[(1 a)(r 1 ER 1 ) + a(r 2 ER 2 )](R m ER m )} = E{(1 a)(r 1 ER 1 )(R m ER m ) + a(r 2 ER 2 )(R m ER m )} = (1 a)e[(r 1 ER 1 )(R m ER m )] + ae[(r 2 ER 2 )(R m ER m )] = (1 a)σ 1m + aσ 2m The coefficients in this linear combination are the portfolio weights 1 a and a. ii. The beta of the portfolio with the market is β p = σ pm σ 2 m = (1 a)σ 1m + aσ 2m σ 2 m = (1 a) σ 1m σm 2 +a σ 2m σm 2 = (1 a)β 1 +aβ 2. Thus the beta of the portfolio is a linear combination of the betas of the individual assets, where the coefficients in the linea combination are the portfolio weights. C.8 Inclusion of a risk-free asset as an (n + 1)st asset with portfolio weight a n+1. (a) The appropriate expression for the return on the portfolio is R p = a i R i + a n+1 r f. (b) The appropriate expression for the expected return on the portfolio is E(R p ) = a i E(R i ) + a n+1 r f. (c) The appropriate expression for the restriction on the portfolio weights is n+1 a i = 1.

8 216 INSTRUCTOR S MANUAL (d) The variance of the portfolio return is ( n ) ( n ) V (R p ) = V a i R i + a n+1 r f = V a i R i = j=1 a i a j σ ij. That is, because the term a n+1 r f is nonrandom it does not affect the expression for the portfolio variance. (e) The Lagrangean for this optimization problem is ( n+1 ) a i E(R i ) + a n+1 r f λ 1 a i λ 2 a i a j σ ij σp 2 j=1 (f) No; because the new terms are additive, the derivatives of the Lagrangean with respect to the a i (i = 1,..., n) are unchanged. (g) The derivative of the Lagrangean with respect to a n+1 is Setting this equal to zero yields L a n+1 = r f λ 1. λ 1 = r f. The Lagrange multiplier λ 1 has the interpretation as the risk-free rate of return. C.9 The empirical version of the CAPM is R it r ft = β im (R mt r ft ) + ε t. Taking the variance of both sides, and using the fact that r f is nonrandom, V (R it ) = V (β im (R mt ) + ε t ). The right hand side may be evaluated according to Law B.4 which, since the idiosyncratic component ε t is uncorrelated with the market return R m, simplifies to Corollary B.4.1: V (R it ) = β 2 imv (R mt ) + V (ε t ) = (β im σ m ) 2 + V (ε). Thus the empirical CAPM implies the same decomposition into systematic and idiosyncratic risk as followed from the diversification effect.