Minimum-variance split when variances are 10 and 40 percent

Question

Two stocks again have the same expected return. This time their return variances are $0.10$ and $0.40$ , and their correlation is $0.5$ . What portfolio of the two has minimum variance?

Short answer

Put everything in the first stock. The covariance is $0.5\sqrt{0.1\times0.4}=0.1$ , which equals the first stock's variance. The minimum-variance weight on stock 1 is $\tfrac{0.4-0.1}{0.1+0.4-0.2}=1$ , so the second stock adds nothing: any small amount of it raises variance, and the minimum is $0.10$ .

Adapted from Crack, Heard on the Street, Q3.20, p.36; question and answer rewritten by SelfMadeQuant.com.

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