Two assets have volatilities 20% and 30% with correlation 0.2. Find the weight on the first asset in the minimum variance portfolio when the weights must sum to 1, and give the resulting portfolio volatility.

Two assets have volatilities 20% and 30% with correlation 0.2. Find the weight on the first asset in the minimum variance portfolio when the weights must sum to 1, and give the resulting portfolio volatility.

Approach: Set up the variance as a function of one weight after eliminating the other with the budget constraint, differentiate, and solve the resulting linear equation.

0.7358. The covariance is 0.2 * 0.2 * 0.3 = 0.012, with variances 0.04 and 0.09. Writing w_2 = 1 - w_1 from the budget constraint, the variance is 0.04 w_1^2 + 0.09 (1-w_1)^2 + 2 * 0.012 w_1 (1-w_1). Setting the derivative to zero gives the standard minimum variance portfolio weight w_1 = (var_2 - cov)/(var_1 + var_2 - 2 cov) = (0.09 - 0.012)/(0.04 + 0.09 - 0.024) = 0.078/0.106 = 0.7358, so w_2 = 0.2642. The same result comes from the lagrange multiplier condition Sigma w = lambda * 1, which says every asset must have equal covariance with the portfolio. The portfolio variance is 0.7358^2 * 0.04 + 0.2642^2 * 0.09 + 2 * 0.7358 * 0.2642 * 0.012 = 0.03261, so the volatility is 18.06%, below the 20% of the safer asset on its own.

Follow-up: At what correlation does the minimum variance portfolio start shorting the second asset, and what does that say about using estimated correlations near 1?

Key concepts: minimum variance portfolio, lagrange multiplier, covariance, budget constraint.