The Markowitz Portfolio Model

CPSC 406 – Computational Optimization

Rate of Return and Risk

• Rate of Return for stock \(i\) at time \(t\):
  • \(r_t^i = \frac{p_t^i - p_1^i}{p_1^i}\) where \(p_t^i\) is the price at time \(t\)
• Expected Return for stock \(i\):
  • \(r_i = \frac{1}{m} \sum_{t=1}^m r_t^i\) (mean of returns over time)
• Risk measured by variance/standard deviation:
  • \(\sigma_i^2 = \frac{1}{m} \sum_{t=1}^m (r_t^i - r_i)^2\)
  • Higher variance = Higher risk
• Covariance between stocks \(i\) and \(j\):
  • \(\Sigma_{ij} = \frac{1}{m} \sum_{t=1}^m (r_t^i - r_i)(r_t^j - r_j)\)
  • Measures how returns move together

Portfolio Return and Risk

• Portfolio weights: \(x = (x_1, x_2, \ldots, x_n)\) where:
  • \(x_i\) = fraction of investment in stock \(i\)
  • \(x_i \geq 0\) (no short selling)
  • \(\sum_{i=1}^n x_i = 1\) (full allocation)
• Portfolio expected return:
  • \(\mu_p = r^T x = \sum_{i=1}^n r_i x_i\)
• Portfolio risk (variance):
  • \(\sigma_p^2 = x^T \Sigma x = \sum_{i=1}^n \sum_{j=1}^n x_i x_j \Sigma_{ij}\)
• Key insight: Portfolio risk can be lower than individual stock risks through diversification

The Markowitz Optimization Model

• Objective: Minimize risk for a given level of return

• Mathematical formulation: \[\min_{x} \frac{1}{2} x^T \Sigma x\]

• Subject to constraints:

  • Target return: \(r^T x \geq \mu\) (achieve desired return)
  • No short selling: \(x \geq 0\) (non-negative weights)
  • Full allocation: \(\sum_{i=1}^n x_i = 1\) (weights sum to 1)

• Solution approaches:

  • Projected gradient methods (as in HW7)
  • Convex optimization solvers
  • The solution forms the “efficient frontier”