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

2. Arbitrage Opportunity and Profit
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
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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.
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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
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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!
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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
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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”
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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
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9. HML =1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth)
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
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10. Are All Risk Factors Covered Now?
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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!
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