# Mean-Variance Portfolio Theory

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Assignment 7: Mean-Variance Portfolio Theory

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1 .
Consider, as in Lecture 7.1, a portfolio of two risky assets, with expected returns rˉ1,rˉ2, variances σ21,σ22 and covariance σ1,2. No other assets are available. You have to allocate \$1 mln of investment in the portfolio of the two assets in order to minimize total portfolio variance. What is the optimal amount of investment in asset 1 (in mln dollars)? Assume expected returns are positive. radio button to select σ_22rˉ_2−σ_12rˉ_1σ_12+σ_22−2σ_1,2 as your response

σ22rˉ2−σ21rˉ1σ21+σ22−2σ1,2
radio button to select σ_12rˉ_1−σ_1,2rˉ_2σ_12+σ_22−2σ_1,2 as your response

σ21rˉ1−σ1,2rˉ2σ21+σ22−2σ1,2
radio button to select σ_22−σ_1,2σ_12+σ_22−2σ_1,2 as your response

σ22−σ1,2σ21+σ22−2σ1,2
radio button to select σ_22−σ_1,2σ_12+σ_22−σ_1,2 as your response

σ22−σ1,2σ21+σ22−σ1,2
Correct Answer
σ22−σ1,2σ21+σ22−2σ1,2

Instructor Explanation
Call a the proportion of total portfolio value invested in first stock. Then we have to minimize the total variance f(a)=a2σ21+(1−a)2σ22+2a(1−a)σ1,2
First-order conditions imply
0=∂f∂a=2aσ21−2(1−a)σ22+(2−4a)σ1,2
Solving this for a yields a=σ22−σ1,2σ21+σ22−2σ1,2
Second order conditions also hold, which is trivial to check.

2 .
Similarly to in-class problem, consider 3 assets with expected returns 1, 2 and 3, respectively. Assume all variances to be 1, σ1,2=0.5,σ1,3=σ2,3=0.5. Solve the Markowitz problem for this setup, and required portfolio return rˉ=2. What are the resulting portfolio weights?

radio button to select -3/2, 1, 3/2 as your response

-3/2, 1, 3/2
radio button to select -1/4, 1/2, 3/4 as your response

-1/4, 1/2, 3/4
radio button to select 3/8, 1/4, 3/8 as your response

3/8, 1/4, 3/8
radio button to select -5/8, 3/4, 9/8 as your response

-5/8, 3/4, 9/8
Correct Answer
3/8, 1/4, 3/8

Instructor Explanation
After writing down the Langrangian for the problem...