Understanding Arbitrage Pricing Theory and Multifactor Models

1. Arbitrage Pricing Theoryand Multifactor Models Arbitrage Opportunity and Profit Diversification and APT APT and CAPM Comparison Multifactor Models

2. Arbitrage Opportunity and Profit • Arbitrage • The opportunity of making riskless profit by exploiting relative mispricing of securities • E.g., IBM: $96 on NYSE and $96.15 on NASDAQ creates an arbitrage opportunity • Zero-Investment Portfolio • A portfolio of zero value by long and short the same amount of securities • E.g., Buy $10,000 of stock A and short $10,000 of stock B creates a zero-investment portfolio

3. Arbitrage Opportunity and Profit • Example: Two stocks A, B and a bond C. • If it rains tomorrow, A pays $1.3 and B pays $0.2 • if it does not rain, A pays $0.3 and B pays $1.5 • C pays $2 regardless. • Price today: PA = PB = $1, PC= $2 • Find the arbitrage opportunity and profit from it • Solution • Short 1 share of A and B each to get $2 • Use the proceeds to buy bond C • Total initial investment = $0 • P/L = $0.5 if it rains, and P/L = $0.2 if it does not.

4. Diversification and APT • Well-diversified Portfolio • A portfolio sufficiently diversified such that non-systematic risk is negligible • Arbitrage Pricing Theory (APT) • A theory of risk-return relationships derived from no-arbitrage conditions in large capital market • Individual stock: • Well-diversified portfolio: RF is the factor return • No-arbitrage means: αP = 0

5. Diversification and APT • How does it work? • Factor portfolio: • If portfolio C has αP = 2%, βP = 0.5 • Show me the money • Short $100 of the factor portfolio • Long $200 of portfolio C • Net payoff • Risk-free four bucks? I’ll take it anytime!

6. APT and CAPM Comparison • APT applies to well-diversified portfolios and not necessarily to individual stocks • It is possible for some individual stocks not to lie on the SML • APT is more general in that its factor does not have to be the market portfolio • Both models can be extended to multifactor setup

7. Multifactor Models • Possible to consider more than one benchmark factor! • Consider a two-factor model: • Ri: excess return = ri – rf • RMi: factor portfolios excess return = rMi – rf • : return sensitivity to systematic factors - also called as “factor loadings” “factor betas”

8. Multifactor Models • Where do the factors come from? • Variables that reflect macroeconomic picture • E.g. industrial production, inflation, bond spreads • Variables that serve as proxies for exposure to systematic risk • E.g. Fama-French (1993) model approach

9. Fama-French (1993) Model • Three-factor model: • Ri: stock excess return = ri – rf • RM: market excess return = rM – rf • SMB: “Small Minus Big” factor return SMB =1/3 (Small Value + Small Neutral + Small Growth)- 1/3 (Big Value + Big Neutral + Big Growth) • HML: “High Minus Low” factor return HML =1/2 (Small Value + Big Value)- 1/2 (Small Growth + Big Growth) • : return sensitivity to factors

10. Are All Risk Factors Covered Now?

11. Wrap-up • What is arbitrage and how to do it? • What are the major differences between APT and CAPM? • Multifactor models – the way to go!