Minimizing Portfolio Risk with Two Stocks

This question focuses on constructing a two-asset portfolio where both securities have the same expected return, but different levels of volatility and a specified degree of correlation. The candidate is asked to determine how to split capital between the two stocks so that the overall portfolio risk is as low as possible, given those characteristics. It is a classic minimum-variance portfolio setup that appears frequently in interviews for buy-side and sell-side quantitative roles, including risk, asset allocation, and portfolio construction positions.

To answer it well, a candidate must translate the verbal description into the variance formula for a two-asset portfolio and recognize that this is an unconstrained (except for full-investment) convex optimization problem. The solution uses differential calculus or known closed-form expressions for optimal weights with two risky assets. Interviewers look for comfort with variance-covariance representations, correct handling of correlation versus covariance, clear algebra, and an understanding of why diversification reduces risk even when expected returns are equal. They also watch for interpretation: checking the plausibility of the resulting allocation and recognizing its dependence on relative volatilities and correlation.

What it tests

When constructing a portfolio from multiple assets, the key mathematical structure is the interplay between individual asset risks (variances) and how those risks interact (covariances or correlations). The minimum-variance portfolio exploits the fact that imperfectly correlated assets can combine to reduce overall risk below that of any single asset, even if their expected returns are identical. This happens because the covariance term can offset some of the portfolio's total variance, depending on the weights and the correlation. The optimal allocation is found by minimizing the quadratic form of the portfolio variance, subject to the constraint that all weights sum to one. The principle holds because diversification works best when assets are not perfectly correlated: the less they move together, the more one asset's fluctuations can cancel out another's, reducing total risk.