Hedge ratio for minimal portfolio variance

This question introduces a simple two-asset hedging problem: you hold one risky stock and choose how large a short position to take in another stock to reduce overall risk. The setup is framed in terms of return variances and correlation, so the candidate must think in terms of portfolio risk rather than prices or trade execution. It is a canonical example of a static hedge ratio calculation, similar in spirit to equity relative-value and index-component hedging seen in sell-side trading and buy-side risk management roles.

The solution relies on writing the hedged portfolio's return as a linear combination of the two asset returns and then expressing the portfolio variance in terms of their variances and correlation. The key step is recognizing that the variance is a quadratic function of the hedge weight and then using calculus to find the minimizing value. An interviewer looks for comfort with variance and covariance, correct handling of the correlation term, and a clean optimization argument, rather than memorized formulas. They also watch whether the candidate interprets the sign and scaling of the resulting hedge ratio sensibly.

What it tests

The core structure here is variance minimization in linear portfolios, which is a general principle in risk management. When combining two assets, the variance of the portfolio depends not only on the variances of the individual assets but also on their covariance (or correlation). The minimum-variance hedge ratio is derived by differentiating the portfolio variance with respect to the hedge size and solving for the point where incremental changes in the hedge no longer reduce risk. This works because variance is a quadratic function in the hedge size, so its minimum can be found analytically by calculus. The reason this approach works is that the cross-term (the covariance) allows you to offset risk in one asset by taking the right-sized position in another, and the optimal size is proportional to both the correlation and the ratio of standard deviations.