Two-Asset Sharpe Ratio Maximization

This quant interview question is about building the optimal combination of two risky assets when you care about risk-adjusted return, not just raw expected return. It forces you to translate expected excess returns and a covariance structure into a single performance metric that reflects both reward and variability, under a basic capital constraint that mimics real portfolio construction.

It trains your ability to handle fractional optimization in a continuous setting, where a linear numerator is divided by a quadratic form in the denominator. You need comfort with covariance matrices, portfolio variance, and how weights interact through correlation. It also tests your understanding of scale invariance and how constraints shape the location of an optimum.

This matters in quant interviews because real quant roles focus on maximizing risk-adjusted performance, not just returns. Strong quant prep must cover Sharpe-based optimization, tangency portfolios, and interpreting covariance input into portfolio choice. You are expected to connect mathematical optimization with practical portfolio design, and this type of question crystallizes that link.

What it tests

When optimizing the Sharpe ratio for a portfolio of risky assets, the key structure is that the Sharpe ratio is a fractional function: the expected excess return (a linear function of weights) divided by the portfolio volatility (the square root of a quadratic form in the weights). The maximum Sharpe ratio is achieved at the point where the marginal increase in expected return per unit increase in risk is exactly offset by the marginal increase in risk per unit increase in expected return. This is fundamentally a problem of maximizing a ratio of a linear to a quadratic function, which leads to a first-order condition involving the derivative of the ratio. The solution often lies outside the feasible region (e.g., weights exceeding 1 or being negative), reflecting that unconstrained optimization of the Sharpe ratio can suggest leveraging or shorting assets. The underlying reason is that the Sharpe ratio is scale-invariant: only the direction of the weight vector matters, not its magnitude, so the optimum is determined by the tangency point with the efficient frontier.