Portfolio Theory & Asset Pricing

Module 5 in the curriculum. Starting point: a vague MBA memory of Markowitz and the efficient frontier, plus practical diversification intuitions from active personal investing. The goal is to rebuild this from the ground up --- the optimization math, the equilibrium models, the empirical failures, and modern factor-based alternatives.

Why this module matters

Portfolio theory answers the most fundamental question an investor faces: how should you allocate capital across risky assets? The answer turns out to depend on deep questions about market efficiency, risk measurement, and whether historical patterns persist. This module connects the math of constrained optimization (a comfort zone) to the economics of asset pricing.

Learning roadmap

Core ideas to internalize

Mean-variance optimization

Given $n$ assets with expected return vector $\mathbf{\mu }$ and covariance matrix $\Sigma$, the investor solves:

${\min }_{\mathbf{w}}{\mathbf{w}}^{\top }\Sigma \mathbf{w}\text{subject to}{\mathbf{w}}^{\top }\mathbf{\mu }={\mu }_{\text{target}},{\mathbf{w}}^{\top }1=1$

The set of solutions traces the efficient frontier. With a risk-free asset at rate $r_f$, the optimal risky portfolio is the tangency portfolio --- the point where the Capital Allocation Line is tangent to the frontier. This is pure constrained optimization, and the math is clean.

The practical problem: $\Sigma$ must be estimated, and estimation error in a $500\times 500$ covariance matrix dominates the optimization. This is why shrinkage estimators (Ledoit-Wolf), Black-Litterman, and risk-parity approaches exist.

CAPM and its discontents

If all investors hold mean-variance optimal portfolios, the market portfolio $M$ must be on the efficient frontier. This yields CAPM:

$\mathbb{E}[R_i]-r_f={\beta }_i(\mathbb{E}[R_M]-r_f),{\beta }_i=\frac{\text{Cov}(R_i,R_M)}{\text{Var}(R_M)}$

Empirical problems:

• Roll’s critique: the true market portfolio is unobservable, so CAPM is untestable
• Low-beta anomaly: low-beta stocks earn higher risk-adjusted returns than CAPM predicts
• Size and value effects: small-cap and value stocks earn excess returns unexplained by $\beta$

Factor models: from CAPM to Fama-French

The Fama-French 5-factor model explains returns via:

$R_i-r_f={\alpha }_i+{\beta }_i^{MKT}MKT+{\beta }_i^{SMB}SMB+{\beta }_i^{HML}HML+{\beta }_i^{RMW}RMW+{\beta }_i^{CMA}CMA+{ϵ}_i$

where SMB (size), HML (value), RMW (profitability), CMA (investment) capture anomalies that CAPM misses. The APT provides the theoretical justification: if returns are driven by $k$ factors, arbitrage forces risk premia to be linear in factor exposures.

Risk measures: VaR and beyond

Value at Risk at confidence level $\alpha$: the loss $L$ such that $P(L>{\text{VaR}}_{\alpha })=1-\alpha$. Three approaches:

• Parametric: assume normality, ${\text{VaR}}_{\alpha }=\mu +z_{\alpha }\sigma$
• Historical simulation: use empirical return distribution
• Monte Carlo: simulate paths from a fitted model

VaR has a fatal flaw: it is not subadditive (diversification can appear to increase risk). CVaR / Expected Shortfall fixes this:

${\text{CVaR}}_{\alpha }=\mathbb{E}[L\mid L>{\text{VaR}}_{\alpha }]$

Taleb’s critique goes deeper: all these measures assume we can estimate tail probabilities, which is precisely where our data is thinnest.

Prerequisites

• Probability Theory --- distributions, expectation, covariance
• Statistics --- estimation, regression, hypothesis testing
• Linear algebra --- eigenvalues of $\Sigma$, quadratic forms

Key references

• Cochrane --- Asset Pricing (the theoretical backbone)
• Ang --- Asset Management: A Systematic Approach to Factor Investing
• Markowitz --- Portfolio Selection (1952 paper)
• Fama & French --- key papers (1993, 2015)
• MIT OCW 15.401 --- Finance Theory I (portfolio theory sessions)