Assets A and B have annual volatilities of 20% and 30% and a correlation of 0.2. Shorting is allowed. What weight in A minimises the variance of a two asset portfolio?

Assets A and B have annual volatilities of 20% and 30% and a correlation of 0.2. Shorting is allowed. What weight in A minimises the variance of a two asset portfolio?

Approach: Differentiate the two asset variance with respect to the weight and set it to zero. Keep the covariance term as rho*sigma_A*sigma_B.

39/53. The minimum variance weight is w_A = (sigma_B^2 - rho*sigma_A*sigma_B)/(sigma_A^2 + sigma_B^2 - 2*rho*sigma_A*sigma_B). Here sigma_A^2 = 0.04, sigma_B^2 = 0.09 and the covariance rho*sigma_A*sigma_B = 0.2*0.2*0.3 = 0.012, so w_A = (0.09 - 0.012)/(0.04 + 0.09 - 0.024) = 0.078/0.106 = 39/53 = 0.7358. The resulting variance is sigma_A^2*sigma_B^2*(1 - rho^2)/0.106 = 0.0036*0.96/0.106 = 0.032604, so the portfolio volatility is 18.06%, below the 20% of A alone. That 1.94 point saving is the diversification gain, and it shrinks to zero as the correlation rises to sigma_A/sigma_B = 2/3, where the minimum variance portfolio is A alone and B earns no weight.

Follow-up: At what correlation does the minimum variance portfolio start shorting B?

Key concepts: minimum variance portfolio, covariance, correlation, diversification.