AlmostSurely
Problem 24.7 Easy Free 3 variations

Minimum-Variance Hedge Ratio

You hold exactly 11 share of stock AA, whose return has variance 44. You will also buy xx shares of stock BB, whose return has variance 99; the returns of AA and BB have correlation −1/2-1/2. Treating xx as a real number (short sales allowed), find the value of xx that minimises the variance of the combined portfolio’s return.

Work it out, then compare with the answer.

SolutionShow the worked solution

Let RA,RBR_A,R_B be the two returns, with σA=2\sigma_A=2, σB=3\sigma_B=3 and ρ=−12\rho=-\tfrac12. Then

Cov⁡(RA,RB)=ρ σAσB=−12⋅2⋅3=−3.\operatorname{Cov}(R_A,R_B)=\rho\,\sigma_A\sigma_B=-\tfrac12\cdot 2\cdot 3=-3 .

The portfolio return is RA+xRBR_A+xR_B, so

Var⁡(RA+xRB)=σA2+2xCov⁡(RA,RB)+x2σB2=9x2−6x+4.\operatorname{Var}(R_A+xR_B)=\sigma_A^2+2x\operatorname{Cov}(R_A,R_B)+x^2\sigma_B^2=9x^2-6x+4 .

This is a convex quadratic in xx. It is minimised where its derivative 18x−618x-6 vanishes, so x=13x=\tfrac13. The minimum variance is 1−2+4=31-2+4=3.

In general the minimiser is x∗=−Cov⁡(RA,RB)/σB2=−ρ σA/σBx^*=-\operatorname{Cov}(R_A,R_B)/\sigma_B^2=-\rho\,\sigma_A/\sigma_B. The minimum variance is then σA2(1−ρ2)\sigma_A^2(1-\rho^2). Here ρ<0\rho<0, so the hedge is a long position in BB.

x=13\boxed{x=\tfrac13}

Key idea. The minimum-variance hedge ratio is the regression coefficient of −RA-R_A on RBR_B.

Variations

The same idea with a twist. Each solution gives only the changed step.

Variation 24.7a

Let X1X_1 and X2X_2 be independent random variables with variances 2525 and 100100 respectively, and let Y=cX1+(1−c)X2Y = cX_1 + (1-c)X_2 for some 0≤c≤10 \le c \le 1. Find the value of cc that minimises Var⁡(Y)\operatorname{Var}(Y).

Work it out, then compare with the answer.

Solution

By independence the covariance term vanishes, so

Var⁡(Y)=25c2+100(1−c)2.\operatorname{Var}(Y)=25c^2+100(1-c)^2 .

Setting the derivative to zero gives 50c−200(1−c)=050c-200(1-c)=0, so c=45c=\tfrac45. This lies in [0,1][0,1]. The weights are inverse-variance weights: c=1/251/25+1/100c=\frac{1/25}{1/25+1/100}. The minimum variance is 16+4=2016+4=20.

c=45\boxed{c=\tfrac45}
Variation 24.7b

Asset 11 has expected return 4%4\% and return variance 15%15\%; asset 22 has expected return 2%2\% and return variance 4%4\%. Their returns have correlation ρ=−1\rho = -1. You form a portfolio with weights w1w_1 and w2=1−w1w_2 = 1-w_1 in assets 11 and 22, chosen to make the portfolio’s return risk-free (zero variance). Find w1w_1, to two decimal places.

Work it out, then compare with the answer.

Solution

Here σ1=0.15=15/10\sigma_1=\sqrt{0.15}=\sqrt{15}/10 and σ2=0.2\sigma_2=0.2. With ρ=−1\rho=-1 the portfolio variance is a perfect square:

w12σ12−2w1(1−w1)σ1σ2+(1−w1)2σ22=(w1σ1−(1−w1)σ2)2.w_1^2\sigma_1^2-2w_1(1-w_1)\sigma_1\sigma_2+(1-w_1)^2\sigma_2^2=\bigl(w_1\sigma_1-(1-w_1)\sigma_2\bigr)^2 .

It vanishes exactly when

w1=σ2σ1+σ2=22+15=215−411≈0.3405.w_1=\frac{\sigma_2}{\sigma_1+\sigma_2}=\frac{2}{2+\sqrt{15}}=\frac{2\sqrt{15}-4}{11}\approx 0.3405 .

The expected returns play no role.

w1≈0.34\boxed{w_1\approx 0.34}
Variation 24.7c

Asset 11 has expected return 4%4\% and return variance 15%15\%; asset 22 has expected return 2%2\% and return variance 4%4\%. Their returns have correlation ρ=−1\rho = -1. You form a portfolio with weights w1w_1 and w2=1−w1w_2 = 1-w_1 in assets 11 and 22, chosen to make the portfolio’s return risk-free (zero variance). Find the expected return of this portfolio, to three significant figures.

Work it out, then compare with the answer.

Solution

Take w1=215−411w_1=\frac{2\sqrt{15}-4}{11} from the previous part. The expected return is

μ=0.04 w1+0.02(1−w1)=0.02(1+w1)=0.02⋅7+21511=7+215550≈0.02681.\mu=0.04\,w_1+0.02(1-w_1)=0.02(1+w_1)=0.02\cdot\frac{7+2\sqrt{15}}{11}=\frac{7+2\sqrt{15}}{550}\approx 0.02681 .
μ≈2.68%\boxed{\mu\approx 2.68\%}
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