Question

Consider two assets with means and standard deviations of returns given by arbitrary μ1 and σ1 > 0 for asset 1, and μ2 and σ2 > 0 for asset 2. Suppose that correlation of returns is different from −1, that is ρ12 > −1. Show that there is no portfolio of these two assets that has risk-free return. Consider only portfolios with positive weights, i.e., without short sales.

Answer 1

Consider a portfolio consist of “asset 1” and “asset 2”. So, the average return of the portfolio is given below.

=> μP = w1*μ1 + w2*μ2, where “wi” be the share of budget being spend on the “ith” asset.

Now, the “variance of P” is given by.

=> V(P) = w1^2*σ1^2 + w2^2*σ2^2 + 2*w1*w2*ρ12* σ1* σ2, where w1, w2, σ1, σ2 all are positive.

Now, we can see that for “ρ12 ≥ 0”, the “V(P)” will be always positive, => for this value of “correlation coefficient”, the portfolio will not be risk free. Now, if “w1=w2” and “σ1= σ 2”, the corresponding value of “correlation coefficient” is “ρ12=-1”, which is not possible because we have given that “ρ12 > -1, => V(P) > 0”. Now, if “w1≠w2” and “σ1≠σ 2”, the corresponding value of “correlation coefficient” is “ρ12 <-1”, which is not possible because we have given that “ρ12 > -1” , => V(P) > 0.

So, the portfolio consist of “asset 1” and “asset2” will not be risk free because the standard deviation of the portfolio is always positive.