Question

6) Consider a simple stock market where there exist two risky securities, s stock in a two-stock portfolio be w, and w, the projected return on each stock be denoted by E(R )and E(R,), the variance of each stock be σ' and σ| respectively, and the covariance of the two stocks be σ12 . ecurity 1 and security 2. Let the weights attached to each The expected return of the portfolio is given by the expression E(R,)-w,E(R, ) + w2E(R2) and the variance of the portfolio by , = w,σ' + wİo; + 2w,w20n. A risk averse investor with risk aversion coefficient Φ will choose weights w, and w, to maximize the function subject to the constraint w, w2-1. (a) Using the information above derive expressions for the optimal weights of the two-stock portfolio w, and w (b) Use your answer in (a) to calculate the optimal weights if: E(R): 0.12 E(R)-0.20 120.10 Φ=1.00 o, = 0.40

Answer 1

Part (a)

Since lots of symbols and notations are involved, I am providing a hand written solution below. Please see the picture below:

Double derivative of U can be easily seen to be negative and hence the above points correspond to a maxima.

2 2.

Part (b)

Now that we have the expressions we have to just input the values to get optimal weights as:

w₁ = [0.12 - 0.20 + 1 x (0.40² - (-0.10))] / [1 x (0.20² + 0.40² - 2 x (-0.1)] = 0.45 = 45%

w = 1 - w₁ = 1 - 0.45 = 0.55 = 55%