Modern Portfolio Theory and Mean-Variance Optimization › Key Parameters

The key parameters for Mean-Variance Optimization are:

• $\boldsymbol{\mu}$ (Expected Returns): The vector of forecast returns. Errors can strongly affect allocations; their impact relative to covariance errors depends on estimation precision, constraints, and conditioning.
• $\Sigma$ (Covariance Matrix): Captures the risk of individual assets (diagonal elements) and their co-movements (off-diagonal elements). At fixed long-only weights, increasing a pairwise correlation cannot reduce portfolio variance, within the valid covariance domain; short positions can reverse that response.
• $r_f$ (Risk-Free Rate): The return on a risk-free asset, used to calculate Sharpe ratios and the Capital Market Line.
• $\gamma$ (Risk Aversion): Determines the trade-off between risk and return in the utility maximization formulation ($U = \mathbf{w}^\top \boldsymbol{\mu} - \frac{\gamma}{2}\mathbf{w}^\top \Sigma \mathbf{w}$).
• $\mathbf{w}$ (Weights): The decision variables representing the fraction of capital allocated to each asset. Constraints typically require $\sum w_i = 1$ and $w_i \ge 0$ (for long-only portfolios).