Mean-Variance Portfolio Optimization

Mathematical foundations

The mean-variance optimization framework is built on several key mathematical components:

Expected return

For a portfolio of n assets, the expected return is calculated as:

$E(R_p) = \sum_{i=1}^n w_i E(R_i)$

Where:

• $E(R_p)$ is the expected portfolio return
• $w_i$ is the weight of asset i
• $E(R_i)$ is the expected return of asset i

Portfolio variance

The portfolio variance, which measures risk, is given by:

$\sigma_p^2 = \sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_{ij}$

Where:

• $\sigma_p^2$ is the portfolio variance
• $\sigma_{ij}$ is the covariance between assets i and j

The optimization problem

The mean-variance optimization problem can be formulated in two equivalent ways:

1. Maximize expected return subject to a risk constraint: $\max_{w} \sum_{i=1}^n w_i E(R_i)$ subject to: $\sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_{ij} \leq \sigma_{target}^2$ $\sum_{i=1}^n w_i = 1$

2. Minimize risk subject to a return constraint: $\min_{w} \sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_{ij}$ subject to: $\sum_{i=1}^n w_i E(R_i) \geq R_{target}$ $\sum_{i=1}^n w_i = 1$

The efficient frontier

The efficient frontier represents the set of optimal portfolios that offer the highest expected return for a given level of risk. It can be visualized as a curve in risk-return space:

Implementation considerations

Data requirements

• Historical price data for all assets
• Expected returns estimates
• Covariance matrix estimation
• Investment constraints

Practical challenges

1. Parameter uncertainty: Return and risk estimates are subject to estimation error
2. Non-stationarity: Market relationships change over time
3. Transaction costs: Portfolio rebalancing incurs costs
4. Concentration risk: Solutions may suggest extreme weights

Extensions and variations

Black-Litterman model

The Black-Scholes Model for Option Pricing inspired approach combines market equilibrium with investor views to improve portfolio optimization.

Risk parity

Risk Parity Portfolio Construction allocates assets based on their risk contribution rather than capital allocation.

Robust optimization

Accounts for parameter uncertainty by considering worst-case scenarios within a confidence interval.

Multi-period optimization

Extends the framework to consider multiple investment periods and dynamic rebalancing.

Applications in modern finance

Systematic trading

Algorithmic trading systems use mean-variance optimization for portfolio construction and rebalancing.

Risk management

Financial Risk Modeling incorporates mean-variance analysis for portfolio risk assessment.

Performance attribution

Sharpe Ratio Calculation and other risk-adjusted metrics build on mean-variance foundations.

Regulatory considerations

Mean-variance optimization plays a role in various regulatory frameworks:

1. Basel capital requirements
2. Investment fund risk monitoring
3. Portfolio disclosure requirements
4. Risk limit compliance

Technology and implementation

Modern portfolio optimization relies on:

1. High-performance computing
2. Real-time market data processing
3. Advanced optimization algorithms
4. Risk management systems integration

The implementation often requires Real-Time Portfolio Optimization capabilities for institutional investors.