# Capital Asset Pricing and Arbitrage Pricing Theory II

```Arbitrage Pricing Theory
and Multifactor Models
Arbitrage Opportunity and Profit
Diversification and APT
APT and CAPM Comparison
Multifactor Models
Arbitrage Opportunity and Profit
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Arbitrage
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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
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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
Investments 12
2
Arbitrage Opportunity and Profit
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Example: Two stocks A, B and a bond C.
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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
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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.
Investments 12
3
Diversification and APT
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Well-diversified Portfolio
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A portfolio sufficiently diversified such that nonsystematic risk is negligible
Arbitrage Pricing Theory (APT)
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A theory of risk-return relationships derived from
no-arbitrage conditions in large capital market
Individual stock:
Ri  i  i RF  ei
Well-diversified portfolio: RF is the factor return
Rp   p   p RF
No-arbitrage means: αP = 0
Investments 12
4
Diversification and APT
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How does it work?
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Factor portfolio: Rp  RF   F  0, and  F  1
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
200 Rp 100 RF  200 ( p   p RF ) 100 RF  4
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Investments 12
Risk-free four bucks? I’ll take it anytime!
5
APT and CAPM Comparison
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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
Investments 12
6
Multifactor Models
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Possible to consider more than one
benchmark factor!
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Consider a two-factor model:
Ri   i   i1RM1   i 2 RM 2  ei
Ri: excess return = ri – rf
 RMi: factor portfolios excess return = rMi – rf
  ij : return sensitivity to systematic factors
“factor betas”
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Investments 12
7
Multifactor Models
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Where do the factors come from?
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Variables that reflect macroeconomic
picture
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E.g. industrial production, inflation, bond
Variables that serve as proxies for
exposure to systematic risk
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Investments 12
E.g. Fama-French (1993) model approach
8
Fama-French (1993) Model
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Three-factor model:
Ri  i   iM RM   iSMB SMB   iHML HML ei
 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)
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HML: “High Minus Low” factor return
HML =1/2 (Small Value + Big Value)
- 1/2 (Small Growth + Big Growth)
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i : return sensitivity to factors
Investments 12
9
Are All Risk Factors Covered Now?
Investments 12
10
Wrap-up
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What is arbitrage and how to do it?
What are the major differences between
APT and CAPM?
Multifactor models – the way to go!
Investments 12
11
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