Markowitz mean-variance methods

All four methods trade expected return off against risk using the same two inputs: an expected-return estimate and a covariance matrix, both measured over the lookback window, and all four honour the same constraints. They differ only in what they hold fixed and what they optimise. For where they sit among the six methods, see Portfolio optimisation.

Risk Level maximises expected return for a chosen level of risk, set by a single slider. At 0% it is the minimum-volatility portfolio; at 100% it is the maximum-Sharpe portfolio; in between it sits on the efficient frontier between the two. For a Performance Portfolio, 80% to 100% is the usual range; for a Hedge Portfolio, 0% to 10%. It is the general-purpose default.

Efficient Risk maximises expected return while holding portfolio volatility at a target you set. Use it when a specific volatility is the objective. The target is annualised by default.

Efficient Return minimises volatility while holding expected return at a target you set. Use it when a specific return is the objective. A target higher than the best attainable return makes the optimiser fail.

Max Quadratic Utility maximises expected return minus a risk-aversion penalty on variance (expected return − risk aversion × variance). A low coefficient, around 0.5, is aggressive and close to maximum return; a medium one, 1 to 2, is balanced; a high one, above 3, is defensive and close to minimum volatility.

Two estimators feed every Markowitz method, both measured over the lookback.

• Expected-return estimator. Mean Return weights every day equally; Exponentially Weighted Mean Return gives recent days more weight. The estimate of expected return often matters more than the choice of method.
• Risk (covariance) estimator. Sample Covariance is the traditional choice; Ledoit-Wolf, a shrinkage estimator, is more robust and is the recommendation. Semicovariance and Exponentially Weighted Covariance are also available.