Optimal Portfolio Risk Allocation

This question is a focused exercise in basic portfolio theory using a simple two-asset setup. The candidate is told that two risky assets have the same expected return but differ in volatility and are positively correlated, and is asked to find the weight in each that minimizes overall portfolio risk. It is a canonical mean-variance diversification problem, the kind that appears in buy-side quant research or risk-focused roles, where the goal is not to optimize return but to construct the least risky combination given a set of inputs.

To answer it, the candidate must work with the analytical expression for the variance of a two-asset portfolio, recognize its quadratic form in the allocation weight, and differentiate to find the minimizer. The interviewer is checking comfort with variance-covariance algebra, careful handling of correlation versus covariance, and the ability to reason about interior versus boundary solutions. They are also looking for qualitative intuition: how the relative volatilities and correlation shape the optimal mix, and why the least volatile asset does not always get 100% weight once imperfect correlation is introduced.

What it tests

Whenever you combine two risky assets, the total portfolio risk depends not just on the individual risks but also on how the assets move together, captured by their correlation. The key structure is that the portfolio variance is a quadratic function of the weights, with cross-terms reflecting the covariance. Minimizing this quadratic function with respect to the weights always yields a unique solution unless the assets are perfectly correlated, in which case the minimum may be at a boundary. The reason this pattern holds is that diversification benefits arise when assets are not perfectly positively correlated, allowing the combined risk to be less than the weighted sum of individual risks. The optimal allocation is always a function of the variances and the covariance, reflecting the trade-off between reducing individual asset risk and exploiting imperfect correlation.