Portfolio Risk and Performance Measurement

Portfolio Theory Recap

Two-Asset Portfolio Risk

σ_P² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂

Diversification benefit: When ρ₁₂ < 1, portfolio risk is less than the weighted average of individual risks.

Efficient Frontier

The set of portfolios that offer the highest expected return for each level of risk. All rational investors should hold portfolios on the efficient frontier.

Capital Market Line (CML)

When a risk-free asset is available: E(R_P) = R_f + [(E(R_M) − R_f) / σ_M] × σ_P

The slope is the Sharpe ratio of the market portfolio — the "price of risk."

---

Factor Models

CAPM (Single Factor)

E(R_i) = R_f + β_i × (E(R_M) − R_f)

Beta measures systematic risk. Only systematic risk is rewarded — diversifiable (idiosyncratic) risk earns no premium.

Fama-French Three-Factor Model

E(R_i) = R_f + β₁(R_M − R_f) + β₂(SMB) + β₃(HML)

Where:

• SMB = Small Minus Big (size premium)
• HML = High Minus Low (value premium)

Carhart Four-Factor Model

Adds UMD (Up Minus Down) — the momentum factor.

Arbitrage Pricing Theory (APT)

Multi-factor generalization: E(R_i) = R_f + β₁λ₁ + β₂λ₂ + ... + βₖλₖ

Where λⱼ are risk

...