Module 4

Portfolio Choice

Overview

The previous module identified systematic factors that drive differences in average returns across stocks. This module asks the complementary question: given a set of assets with different expected returns and risks, how should an investor allocate across them?

The module starts with the geometry of diversification: how combining two risky assets traces out an investment opportunity set in mean–standard deviation space, and why correlation determines the magnitude of the diversification benefit. It then introduces the numerical tools needed to extend this analysis to any number of assets, and applies those tools to two classical portfolio problems — maximizing the Sharpe ratio and minimizing variance.

Topics

Diversification and the Investment Opportunity Set

  • The mean–standard deviation frontier for two risky assets and how correlation determines its curvature
  • Lower correlation produces greater diversification benefit and a larger feasible region in \((\mu, \sigma)\) space
  • Illustrating the opportunity set using pairs of sector ETFs

Numerical Optimization in Python

  • scipy.optimize.minimize: objective function, initial guess, constraints, and bounds
  • Simple minimization and maximization examples to build intuition for the financial applications that follow
  • Recovering OLS beta numerically by minimizing the sum of squared residuals

Mean-Variance Optimization with Multiple Assets

  • Finding the maximum-Sharpe-ratio portfolio across seven sector ETFs, with and without a short-selling constraint
  • A twenty-year training window to reduce noise in expected return estimates
  • Out-of-sample performance evaluation; the long-only constraint as a regularizer against estimation error

The Minimum Variance Portfolio

  • Minimizing portfolio variance without any expected return inputs, bypassing the need for return forecasts
  • A ten-year training window; why covariance estimates are more reliable than mean estimates over short horizons
  • Out-of-sample comparison against an equal-weight portfolio and SPY as a passive benchmark