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Formulate a Single-Period Stat-Arb Objective

Point72Quantitative Researcher · Online Assessment
MediumQuantitative ResearcherOnline AssessmentStatistics & Math

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Formulate a convex single-period statistical-arbitrage objective balancing expected return against covariance risk, with consistent units.

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?

Hints

Hint 1

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.

Follow-up questions

  • How do current holdings matter once transaction costs are introduced?
  • What happens if the covariance matrix is not positive semidefinite?
  • How does multiplying all alphas by a constant affect the chosen holdings?