Portfolio Optimization

Why Portfolio Optimization?

Portfolio optimization is the process of selecting asset weights to maximize expected return for a given level of risk, or minimize risk for a given expected return. This mathematical framework, pioneered by Harry Markowitz in 1952, revolutionized investment management by formalizing the risk-return tradeoff and the benefits of diversification.

Modern portfolio theory provides the foundation for asset allocation, risk budgeting, and performance evaluation across all areas of finance — from individual retirement planning to institutional asset management and hedge fund strategies.

Mean-Variance Optimization

Markowitz Framework

Objective: Minimize portfolio variance subject to expected return constraint.

Setup:

Asset returns: $r_i$ with expected return $\mu_i$ and covariance matrix $\Sigma$
Portfolio weights: $w = (w_1, \ldots, w_n)^T$ with $\sum_{i=1}^n w_i = 1$
Portfolio return: $r_p = w^T r$
Portfolio expected return: $\mu_p = w^T \mu$
Portfolio variance: $\sigma_p^2 = w^T \Sigma w$

Optimization Problem

Minimum Variance Portfolio:

\min_{w} w^T \Sigma w \quad \text{subject to} \quad \mathbf{1}^T w = 1

Solution:

w^{MV} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}}

Target Return Portfolio:

\min_{w} w^T \Sigma w \quad \text{subject to} \quad \mathbf{1}^T w = 1, \quad \mu^T w = \mu_p

Solution:

w = g + h\mu_p

where:

g = \frac{\Sigma^{-1}(\mathbf{1} - \mu C)}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}}, \quad h = \frac{\Sigma^{-1}(\mu B - \mathbf{1} A)}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}}

with:

A = \mathbf{1}^T \Sigma^{-1} \mathbf{1}, \quad B = \mu^T \Sigma^{-1} \mathbf{1}, \quad C = \mu^T \Sigma^{-1} \mu

Efficient Frontier

The efficient frontier is the locus of all mean-variance efficient portfolios:

\sigma_p^2(\mu_p) = \frac{C - 2B\mu_p + A\mu_p^2}{AC - B^2}

Key Properties:

Hyperbolic shape in mean-variance space
Two-fund separation: Any efficient portfolio is a combination of any two efficient portfolios
Minimum variance portfolio lies at the vertex

Capital Asset Pricing Model (CAPM)

When a risk-free asset with return $r_f$ is available:

Capital Allocation Line:

\mu_p = r_f + \frac{\mu_T - r_f}{\sigma_T} \sigma_p

where the tangent portfolio has weights:

w^T = \frac{\Sigma^{-1}(\mu - r_f \mathbf{1})}{\mathbf{1}^T \Sigma^{-1}(\mu - r_f \mathbf{1})}

Black-Litterman Model

Motivation

Traditional mean-variance optimization suffers from:

Estimation error: Small changes in inputs cause large weight changes
Extreme positions: Optimizers concentrate in few assets
Counterintuitive results: Negative weights in "good" assets

Framework

Prior: Market capitalization weights

w_m

with equilibrium returns:

\Pi = \delta \Sigma w_m

where $\delta$ is the risk aversion coefficient.

Views: Investor's views on specific returns:

P\mu = Q + \varepsilon

where:

$P$ : Picking matrix (which assets the views concern)
$Q$ : Vector of view returns
$\varepsilon \sim \mathcal{N}(0, \Omega)$ : View uncertainty

Bayesian Update:

\bar{\mu} = [(\tau\Sigma)^{-1} + P^T\Omega^{-1}P]^{-1}[(\tau\Sigma)^{-1}\Pi + P^T\Omega^{-1}Q]

\bar{\Sigma} = [(\tau\Sigma)^{-1} + P^T\Omega^{-1}P]^{-1}

Optimal Weights:

w = \frac{\bar{\Sigma}^{-1}\bar{\mu}}{\mathbf{1}^T\bar{\Sigma}^{-1}\bar{\mu}}

Risk Parity

Equal Risk Contribution

Objective: Each asset contributes equally to portfolio risk.

Risk Contribution:

RC_i = w_i \frac{\partial \sigma_p}{\partial w_i} = w_i \frac{(\Sigma w)_i}{\sigma_p}

Equal Risk Constraint:

RC_i = \frac{\sigma_p^2}{n} \quad \forall i

Maximum Diversification

Objective: Maximize the ratio of weighted average volatility to portfolio volatility:

MD = \frac{w^T \sigma}{\sqrt{w^T \Sigma w}}

where $\sigma = (\sigma_1, \ldots, \sigma_n)^T$ are individual asset volatilities.

Minimum Variance

Risk parity often approximates the minimum variance portfolio when correlations are moderate.

Factor Models in Optimization

Single-Factor Model

r_i = \alpha_i + \beta_i f + \varepsilon_i

Covariance Matrix:

\Sigma = \beta\beta^T \sigma_f^2 + D

where $D = \text{diag}(\sigma_{\varepsilon_1}^2, \ldots, \sigma_{\varepsilon_n}^2)$ .

Multi-Factor Model

r_i = \alpha_i + \sum_{k=1}^K \beta_{i,k} f_k + \varepsilon_i

Benefits:

Dimension reduction: $K \ll n$
Structural interpretation: Economic factors
Estimation efficiency: Fewer parameters

Fama-French Factors

Three-Factor Model:

r_{i,t} - r_{f,t} = \alpha_i + \beta_{i,M}(r_{M,t} - r_{f,t}) + \beta_{i,SMB}SMB_t + \beta_{i,HML}HML_t + \varepsilon_{i,t}

Factors:

Market: Excess market return
SMB: Small minus big (size factor)
HML: High minus low (value factor)

Robust Optimization

Uncertainty Sets

Box Uncertainty:

\mathcal{U} = \{\mu : \mu_i^L \leq \mu_i \leq \mu_i^U\}

Ellipsoidal Uncertainty:

\mathcal{U} = \{\mu : (\mu - \hat{\mu})^T \Sigma_\mu^{-1} (\mu - \hat{\mu}) \leq \kappa^2\}

Robust Formulation

Max-Min Problem:

\max_{w} \min_{\mu \in \mathcal{U}} \mu^T w - \frac{\gamma}{2} w^T \Sigma w

Solution (ellipsoidal uncertainty):

w^* = \frac{1}{\gamma} \Sigma^{-1}(\hat{\mu} - \kappa\sqrt{\frac{w^T \Sigma^{-1} \Sigma_\mu \Sigma^{-1} w}{w^T \Sigma^{-1} w}}\Sigma^{-1} w)

Dynamic Portfolio Optimization

Merton's Problem

Continuous-time setup:

\max_{c_t, w_t} \mathbb{E}\left[\int_0^T U(c_t) dt + B(X_T)\right]

subject to:

dX_t = (r + w_t^T(\mu - r\mathbf{1}) - c_t)X_t dt + w_t^T \sigma X_t dW_t

Solution (power utility):

w_t = \frac{1}{\gamma} \Sigma^{-1}(\mu - r\mathbf{1})

Multi-Period Discrete Model

Dynamic Programming:

V_t(x) = \max_{w_t} \mathbb{E}[V_{t+1}(x_{t+1}) | x_t = x]

Challenges:

Curse of dimensionality: State space grows exponentially
Parameter uncertainty: Must update beliefs
Transaction costs: Rebalancing costs

Alternative Risk Measures

Value at Risk (VaR) Optimization

Objective: Minimize portfolio VaR:

\min_{w} \text{VaR}_\alpha(w^T r) \quad \text{subject to} \quad \mu^T w \geq \mu_{\min}

Linear approximation (normal returns):

\text{VaR}_\alpha \approx \mu^T w - \Phi^{-1}(\alpha) \sqrt{w^T \Sigma w}

Conditional Value at Risk (CVaR)

\text{CVaR}_\alpha = \mathbb{E}[L | L \geq \text{VaR}_\alpha]

Advantages:

Coherent risk measure: Satisfies desirable properties
Convex optimization: Easier to solve
Tail risk focus: Captures extreme scenarios

Optimization with CVaR

\min_{w,\zeta} \zeta + \frac{1}{1-\alpha} \mathbb{E}[\max(0, -w^T r - \zeta)]

Constraints and Practical Considerations

Common Constraints

Long-only:

w_i \geq 0 \quad \forall i

Sector limits:

\sum_{i \in \text{sector } s} w_i \leq u_s

Turnover constraints:

\sum_{i=1}^n |w_i - w_{i,\text{prev}}| \leq T

Tracking error:

\sqrt{(w - w_b)^T \Sigma (w - w_b)} \leq TE

Transaction Costs

Linear costs:

\text{Cost} = \sum_{i=1}^n c_i |w_i - w_{i,\text{prev}}|

Market impact:

\text{Cost} = \sum_{i=1}^n \alpha_i (w_i - w_{i,\text{prev}})^2

Machine Learning in Portfolio Optimization

Feature Engineering

Technical indicators: Moving averages, momentum, volatility Fundamental ratios: P/E, P/B, ROE, debt-to-equity Macroeconomic variables: GDP growth, inflation, term structure Alternative data: Sentiment, satellite data, credit card spending

Regularization

Ridge regression:

\min_w ||y - X\beta||^2 + \lambda ||\beta||^2

Lasso regression:

\min_w ||y - X\beta||^2 + \lambda ||\beta||_1

Elastic net: Combines ridge and lasso penalties

Neural Networks

Deep learning for:

Return prediction: Non-linear factor models
Risk modeling: Time-varying covariance
Regime detection: Hidden market states

Performance Evaluation

Risk-Adjusted Returns

Sharpe Ratio:

SR = \frac{\mu_p - r_f}{\sigma_p}

Information Ratio:

IR = \frac{\mu_p - \mu_b}{TE}

Sortino Ratio:

\text{Sortino} = \frac{\mu_p - r_f}{\text{Downside deviation}}

Alpha Decomposition

Jensen's Alpha:

\alpha = \mu_p - r_f - \beta(\mu_m - r_f)

Multi-factor alpha:

\alpha = \mu_p - r_f - \sum_{k=1}^K \beta_k (\mu_{f_k} - r_f)

ESG and Sustainable Investing

ESG Integration

ESG scores as additional constraints or tilts:

\min_{w} w^T \Sigma w \quad \text{subject to} \quad w^T s_{ESG} \geq s_{\min}

Exclusionary screening: Remove assets below ESG threshold Best-in-class: Select top ESG performers within sectors

Impact Measurement

Carbon footprint: Portfolio-weighted carbon intensity UN SDGs: Alignment with Sustainable Development Goals Engagement metrics: Proxy voting, shareholder resolutions

Algorithmic Implementation

Quadratic Programming

Standard mean-variance problems reduce to:

\min_w \frac{1}{2} w^T Q w + c^T w \quad \text{s.t.} \quad Aw = b, \quad Gw \leq h

Interior Point Methods

Efficient for large-scale problems with many constraints.

Heuristic Approaches

Genetic algorithms: Global optimization for complex objectives Simulated annealing: Escape local minima Particle swarm: Population-based optimization

Connection to Other Topics

Portfolio optimization integrates many quantitative concepts:

Built on probability theory and random variables
Uses normal distribution assumptions extensively
Connects to linear regression for factor models
Applies optimization algorithms for solution
Foundation for risk management and asset allocation
Links to option pricing via risk-neutral measures
Enables sophisticated quantitative strategies