Minimum-variance weights for two stocks

Question

Two stocks have the same expected return. Their volatilities are $20\%$ and $30\%$ , and the correlation between them is $0.5$ . How should you split your money between them to make the portfolio's variance as small as possible?

Short answer

$6/7\approx85.7\%$ in the $20\%$ -vol stock and $1/7\approx14.3\%$ in the other. The minimum-variance weight on the first stock is $\tfrac{\sigma_2^2-\rho\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}=\tfrac{0.09-0.03}{0.04+0.09-0.06}=\tfrac67$ . With equal expected returns, minimising risk is the whole objective.

Adapted from Crack, Heard on the Street, Q3.19, p.36; question and answer rewritten by SelfMadeQuant.com.

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