Two-Asset Portfolio Math

This question asks you to compare two stylized two-asset portfolio setups and explain how changing the relative volatilities and the correlation structure alters the minimum-variance allocation story. Instead of crunching numbers, you are asked to reason qualitatively about what happens when one asset is much riskier than the other and the correlation is positive but not extreme. The focus is on how that configuration differs from a previous scenario, for example in terms of whether diversification can push risk below the safer asset's variance or whether the optimal solution hugs a corner.

To answer it well, you need comfort with the two-asset variance formula, the geometry of the mean–variance frontier, and how correlation and volatility ratios shape the feasible set. The interviewer is looking for recognition of when diversification remains beneficial, how the critical correlation thresholds shift with a large volatility gap, and whether the minimum-variance portfolio stays interior. Clear verbal reasoning, not just formulas, matters here: you are expected to articulate how the comparative statics change the optimal weight pattern and diversification gains.

What it tests

The core structure of two-asset portfolio variance minimization problems is governed by the interplay between the assets' volatilities and their correlation. The minimum-variance portfolio weight formula emerges from minimizing a quadratic function, and the qualitative behavior of the solution changes at critical thresholds: when the correlation equals the ratio of volatilities, or when it reaches its bounds of -1 or 1. These thresholds mark transitions between interior solutions (weights strictly between 0 and 1), corner solutions (all in one asset), or the possibility of constructing a riskless portfolio. The reason these thresholds matter is that the risk-reducing benefit of diversification depends on how much the assets' returns move together relative to their individual risks; when correlation is low enough, combining assets can reduce variance below that of either asset alone, but as correlation increases, this benefit disappears and the optimal allocation shifts to the less risky asset. The mathematical structure is thus a competition between the risk reduction from diversification and the drag from correlation, with the ratios of volatilities and correlation acting as the key levers.