Modern Portfolio Theory

Modern Portfolio Theory (MPT), developed by Harry Markowitz in 1952, revolutionized investment management by providing a mathematical framework for constructing portfolios that optimize the trade-off between expected return and risk. This framework remains the foundation of institutional portfolio management.

Portfolio Return and Risk

Portfolio Return

For a portfolio of $n$ assets with weights $w_i$ and expected returns $\mu_i$: $E[R_p] = \sum_{i=1}^{n} w_i \mu_i = \mathbf{w}^T \boldsymbol{\mu}$

where weights must sum to 1: $\sum_{i=1}^{n} w_i = 1$

Portfolio Variance

The portfolio variance is not simply the weighted average of individual variances. It depends on covariances: $\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij} = \mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}$

where $\boldsymbol{\Sigma}$ is the covariance matrix.

For two assets: $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2$

The Power of Diversification

Diversification reduces risk when assets are not perfectly correlated. Consider $n$ assets with equal weights ($w_i = 1/n$), identical variances ($\sigma^2$), and identical pairwise correlations ($\rho$):

$\sigma_p^2 = \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2$

As $n \to \infty$: $\sigma_p^2 \to \rho\sigma^2$

The first term (diversifiable/idiosyncratic risk) vanishes with many assets. The second term (systematic/market risk) cannot be eliminated through diversification.

The Efficient Frontier

Mean-Variance Optimization

MPT frames portfolio construction as an optimization problem:

Minimum variance for target return: $\min_{\mathbf{w}} \mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}$ subject to: $\mathbf{w}^T \boldsymbol{\mu} = \mu_{target}$ and $\mathbf{w}^T \mathbf{1} = 1$

Maximum return for target risk: $\max_{\mathbf{w}} \mathbf{w}^T \boldsymbol{\mu}$ subject to: $\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w} = \sigma_{target}^2$ and $\mathbf{w}^T \mathbf{1} = 1$

The Efficient Frontier Curve

The set of optimal portfolios forms the efficient frontier - a hyperbola in mean-standard deviation space. Portfolios on the frontier offer:

• Maximum return for a given risk level
• Minimum risk for a given return level

The Global Minimum Variance Portfolio (GMVP) sits at the leftmost point of the frontier.

Capital Market Line

Risk-Free Asset

Introducing a risk-free asset (rate $R_f$) transforms the problem. Any portfolio combining the risk-free asset with a risky portfolio lies on a straight line.

The Capital Market Line (CML) is the line from $R_f$ tangent to the efficient frontier: $E[R_p] = R_f + \frac{E[R_M] - R_f}{\sigma_M} \sigma_p$

The slope $\frac{E[R_M] - R_f}{\sigma_M}$ is the Sharpe ratio of the market portfolio.

Tangency Portfolio

The tangency portfolio (market portfolio in equilibrium) maximizes the Sharpe ratio: $\max_{\mathbf{w}} \frac{\mathbf{w}^T \boldsymbol{\mu} - R_f}{\sqrt{\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}}}$

Capital Asset Pricing Model (CAPM)

The CAPM, derived from MPT, describes the relationship between systematic risk and expected return: $E[R_i] = R_f + \beta_i (E[R_M] - R_f)$

where beta measures systematic risk: $\beta_i = \frac{Cov(R_i, R_M)}{Var(R_M)} = \frac{\sigma_{iM}}{\sigma_M^2}$

Interpretation:

• $\beta = 1$: Moves with the market
• $\beta > 1$: More volatile than market
• $\beta < 1$: Less volatile than market
• $\beta < 0$: Moves opposite to market

Alpha ($\alpha$) is the return unexplained by CAPM: $\alpha_i = R_i - [R_f + \beta_i(R_M - R_f)]$

Positive alpha suggests outperformance; generating alpha is the goal of active management.

Performance Metrics

Sharpe Ratio

$SR = \frac{E[R_p] - R_f}{\sigma_p}$ Risk-adjusted return per unit of total risk.

Treynor Ratio

$TR = \frac{E[R_p] - R_f}{\beta_p}$ Risk-adjusted return per unit of systematic risk.

Information Ratio

$IR = \frac{\alpha_p}{\sigma(\alpha_p)}$ Active return per unit of tracking error.

Sortino Ratio

$Sortino = \frac{E[R_p] - R_f}{\sigma_{downside}}$ Uses only downside deviation, addressing the criticism that upside volatility isn't risk.

Practical Considerations

Estimation Error

MPT requires estimates of expected returns and covariances. These estimates are subject to error, and optimization amplifies these errors (the "error maximization" problem).

Solutions:

• Shrinkage estimators (e.g., Ledoit-Wolf)
• Black-Litterman model
• Robust optimization
• Resampling methods

Constraints

Real portfolios face constraints:

• No short selling: $w_i \geq 0$
• Position limits: $w_i \leq w_{max}$
• Sector constraints
• Turnover limits
• Transaction costs

Implementation

Portfolio optimization requires:

• Quadratic programming solvers
• Efficient covariance matrix estimation
• Handling of large-scale problems ($n$ > 1000 assets)
• Regular rebalancing algorithms