Minimum-variance weights and the efficient frontier

EXERCISE 01

Asset A has expected return 6 and variance 1.5; asset B has expected return 3.5 and variance 4. Covariance is −0.35. Returns use a common unit, and variances use its square. Portfolio weights sum to one.

A.Find minimum-variance weights and portfolio risk.

A.

\sigma_p^2=x^2\sigma_A^2+(1-x)^2\sigma_B^2+2x(1-x)\sigma_{AB}

\sigma_p^2=1.5x^2+4(1-x)^2-0.70x(1-x)=6.2x^2-8.7x+4

\frac{d\sigma_p^2}{dx}=12.4x-8.7=0

x_{min}=\frac{\sigma_B^2-\sigma_{AB}}{\sigma_A^2+\sigma_B^2-2\sigma_{AB}}

x_A=\frac{4+0.35}{1.5+4+0.70}

70.161290% in A.

x_B=1-x_A

29.838710% in B.

\sigma_{min}^2=6.2x_A^2-8.7x_A+4

Variance = 0.947983871; standard deviation = 0.973644633. Expected return = 5.254032258.

Minimum-variance weights and the efficient frontier

EXERCISE 02

Asset A takes returns −1, 1 and 3 with probability 1/3 each. Asset B takes returns −1, 1 and 2 with probabilities 1/4, 1/4 and 1/2. Weights sum to one. First compare their moments; then consider the stipulated correlation −1 and check whether it is consistent with these marginal distributions.

A.Apply mean–variance dominance.
B.Calculate the formal minimum-variance weights for correlation −1, and assess feasibility.

A.

E[R]=\sum_j p_jR_j,\quad Var(R)=\sum_jp_jR_j^2-E[R]^2

\mu_A=\frac{-1+1+3}{3}=1

\sigma_A^2=\frac{1+1+9}{3}-1=\frac83

\mu_B=-\frac14+\frac14+\frac22=1

\sigma_B^2=\frac14+\frac14+\frac42-1=\frac32

B dominates A under the mean–variance criterion: identical expected return and lower variance.

B.

\sigma_p^2=x^2\sigma_A^2+(1-x)^2\sigma_B^2+2x(1-x)\sigma_{AB}

\rho=-1\quad\Rightarrow\quad \sigma_p^2=[x\sigma_A-(1-x)\sigma_B]^2

x_A=\frac{\sqrt{3/2}}{\sqrt{8/3}+\sqrt{3/2}}=\frac37

A: 42.857143%; B: 57.142857%.

\mu_p=1,\qquad \sigma_p^2=0\quad(\rho=-1\text{ assumed})

This zero-risk result is conditional. Perfect negative correlation requires B to be an affine decreasing function of A almost surely.

B=1-\frac{\sigma_B}{\sigma_A}(A-1)=1-\frac34(A-1)

A values −1, 1 and 3 would require B values 2.5, 1 and −0.5, each with probability 1/3. Those are not the supplied B distribution. Therefore no joint distribution with the stated marginals can have correlation −1; the formal portfolio is not feasible for those exact distributions.

Minimum-variance weights and the efficient frontier

EXERCISE 03

A has expected return 0.09 and variance 0.14; B has expected return 0.13 and variance 0.18. Covariance is 0.10. Weights sum to one and portfolio variance must not exceed 0.15. Short selling is permitted; also report whether the optimum is long-only.

A.Maximise expected return subject to the variance cap.

A.

\sigma_p^2=x^2\sigma_A^2+(1-x)^2\sigma_B^2+2x(1-x)\sigma_{AB}

\mu_p=0.09x+0.13(1-x)=0.13-0.04x

\sigma_p^2=0.14x^2+0.18(1-x)^2+0.20x(1-x)

\sigma_p^2\le0.15\quad\Longleftrightarrow\quad0.12x^2-0.16x+0.03\le0

x_{\pm}=\frac{0.16\pm\sqrt{0.16^2-4(0.12)(0.03)}}{0.24}

Feasible A weights lie between 0.225708115 and 1.107625219. Expected return decreases with x, so choose the lower endpoint.

x_A=x_-

22.570811%.

x_B=1-x_-

77.429189%.

\mu_{max}=0.13-0.04x_-

12.097168%.

Variance equals 0.15 and standard deviation equals 0.387298335. Both weights are positive, so the same optimum is feasible without short selling.

Minimum-variance weights and the efficient frontier

EXERCISE 04

A has expected return 1 and variance 4; B has expected return 2 and variance 9. Correlation is −1/2 and weights sum to one. Use the stated common return unit and squared-unit variances. Short selling is permitted; the reported optima also have non-negative weights.

A.Find the minimum-variance portfolio and quantify diversification.
B.Derive the efficient frontier.
C.Find the maximum expected return when variance is at most 2.

A.

\sigma_p^2=x^2\sigma_A^2+(1-x)^2\sigma_B^2+2x(1-x)\sigma_{AB}

\sigma_{AB}=\rho\sigma_A\sigma_B=-\frac12(2)(3)=-3

\sigma_p^2=4x^2+9(1-x)^2-6x(1-x)=19x^2-24x+9

x_{min}=\frac{\sigma_B^2-\sigma_{AB}}{\sigma_A^2+\sigma_B^2-2\sigma_{AB}}

x_A=\frac{9+3}{4+9+6}=\frac{12}{19},\qquad x_B=\frac7{19}

\sigma_{min}^2=\frac{27}{19},\quad\sigma_{min}=\sqrt{\frac{27}{19}},\quad\mu_{min}=\frac{26}{19}

A: 63.157895%; B: 36.842105%. Variance 1.421052632; standard deviation 1.192079121. Compared with the lower standalone standard deviation of 2, diversification reduces risk by 0.807920879 return units.

B.

\mu_p=x+2(1-x)=2-x\quad\Rightarrow\quad x=2-\mu_p

\sigma_p^2=19(2-\mu_p)^2-24(2-\mu_p)+9

\sigma_p^2=19\mu_p^2-52\mu_p+37

\mu_{efficient}(\sigma)=\frac{52+\sqrt{76\sigma^2-108}}{38}

Choose the upper-return branch, with variance at least 27/19 and expected return at least 26/19. The lower-return branch is inefficient. With long-only weights, also restrict expected return to at most 2.

C.

19\mu^2-52\mu+35=0

\mu_{max}=\frac{52+\sqrt{52^2-4(19)(35)}}{38}

1.542980252 return units.

x_A=2-\mu_{max}\approx0.457019748

x_B=\mu_{max}-1\approx0.542980252