Differential Calculus and Optimization Basics › Key Parameters
Michael BrenndoerferOctober 24, 20251 min read
Quantitative FinanceAll chapters Part 1 of 15Mathematical Foundations
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The key parameters for portfolio optimization are:
- μ (mu): Expected returns vector. These estimates affect the return constraint or objective, but the direction of an asset's optimal-weight response also depends on covariance estimates, the other assets, and the active constraints.
- Σ (Sigma): Covariance matrix capturing asset variances and correlations. Holding weights and marginal volatilities fixed, lower pairwise correlation reduces the contribution for same-sign nonzero positions, increases it for opposite-sign positions, and has no effect for a zero exposure.
- Target return: A feasible exact target selects a point on the minimum-variance locus. In this example, targets at or above the global-minimum-variance return lie on the upper efficient branch; lower targets lie on a dominated branch.
- Bounds: Constraints on individual weights (e.g., no short selling requires w ≥ 0).
- Learning rate (α): For gradient descent, controls step size. A sufficiently large rate can cause divergence; a small rate slows convergence.