Formulate a Single-Period Stat-Arb Objective

# Formulate a Single-Period Stat-Arb Objective Formulate the objective for a single-period statistical-arbitrage portfolio that chooses target holdings to maximize expected return minus a covariance-based risk penalty. Ignore transaction costs for this part. Define every term, explain the role and units of the risk-aversion parameter, and state when the optimization is convex. ### Constraints & Assumptions - `h` denotes target holdings or portfolio weights. - `alpha` contains expected asset returns in units compatible with `h`. - `Sigma` is a symmetric positive-semidefinite return covariance matrix. - Feasibility constraints are handled separately. ### Clarifying Questions to Ask - Are holdings expressed in dollars, shares, or fractions of capital? - Is `alpha` a return forecast or expected profit per holding unit? - What horizon do `alpha` and `Sigma` represent? - Should risk be written with a one-half convention? ### What a Strong Answer Covers - A consistent linear expected-return term and quadratic variance penalty - Equivalent maximization and minimization forms - Convexity or concavity conditions - Dimensional consistency and the effect of risk aversion ### Follow-up Questions 1. How do current holdings matter once transaction costs are introduced? 2. What happens if the covariance matrix is not positive semidefinite? 3. How does multiplying all alphas by a constant affect the chosen holdings? ```hint Keep forecast and covariance horizons aligned The linear reward and quadratic penalty must use compatible holding units and time scales before their trade-off parameter has meaning. ```