Question

3. Consider the following portfolio of two risky assets: the asset 1 with return r1 and the asset 2 with return r2. We invest x dollars in the asset 1 and (1-x) dollars in the asset 2, where 0<=x<=1.

a. Calculate the expected value of the portfolio E[rp]

b. Calculate the variance of the portfolio, Var(rp)

c. Based on your findings on the part b. what kind of assets you should choose when constructing the portfolio.

d. CAPM assets that all investors will hold the optimal portfolio. What is the optimal portfolio in this context?

Answer 1

Part (a)

E[r_p] = weighted average return of individual securities = xr₁ + (1 - x)r₂

Part (b)

Variance of the portfolio = σ²_p =  x²σ²₁ +  (1 - x)²σ²₂ + 2ρ₁₂x(1 - x)σ₁σ₂

Where σ₁ and σ₂ are the standard deviation of return of asset 1 and 2 respectively and ρ₁₂ is the coefficient of correlation between the returns of asset 1 and 2.

Part (c)

We should construct the portfolio in a manner such that variance of the portfolio σ_P is minimized. In order to reduce this, we should combine stocks which are highly negatively correlated or have very low level of correlation.

Part (d)

As per CAPM, the portfolio to hold will be the minimum variance portfolio.This can be done by differentiating the variance σ²_p with respect to x and equating the same to zero.

Such a portfolio will have weight of x in asset 1 given by x = (σ₂²-ρ₁₂σ₁σ₂) / (σ₁² + σ₂² – 2ρ₁₂σ₁σ₂) and (1 - x) invested in asset 2.