1. Comm 324 --- W. Suo Slide 1

2. Comm 324 --- W. Suo Slide 2 Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit  Since no investment is required, an investor can create large positions to secure large levels of profit  In efficient markets, profitable arbitrage opportunities will quickly disappear

3. Comm 324 --- W. Suo Slide 3 Example: Returns with Equal Probability StocksState 1State 2State 3State 4 A-20%20%40%60% B0%70%30%-20% C90%-20%-10%70% D15%23%15%36%

4. Comm 324 --- W. Suo Slide 4 Arbitrage Example StockCurrent Price ($) Expected Return (%) Standard Deviation (%) A1025.029.58 B1020.033.91 C1032.548.15 D1022.58.58

5. Comm 324 --- W. Suo Slide 5 Arbitrage Action and Returns Action: Short 3 shares of D and buy 1 of A, B & C to form portfolio P Returns: You earn a higher rate on the investment than you pay on the short sale E(r) s P D 25.83 22.25 6.40 8.58

6. Comm 324 --- W. Suo Slide 6 Payoffs (short 300 shares of D. buy 100 shares of A. B & C each) Stock$ InvState 1State 2State 3State 4 A1,000-200200400600 B1,0000700300-200 C1,000900-200-100700 D-3,000-450-690-450-1,080 Portfolio$0$250$10$150$20

7. Comm 324 --- W. Suo Slide 7 Example - continued  What will happen in the market?

8. Comm 324 --- W. Suo Slide 8 APT & Well-Diversified Portfolios  F is some macroeconomic factor  For a well-diversified portfolio e P approaches zero  The result is similar to CAPM

9. Comm 324 --- W. Suo Slide 9 Arbitrage Pricing Theory Line E(r i ) = Expected Return Slope = = risk premium Risk class of assets O and U Overpriced asset O U Underpriced asset 0 RFR Factor beta

10. Comm 324 --- W. Suo Slide 10 F E(r)(%) Portfolio F E(r)(%) Individual Security Portfolio & Individual Security Comparison

11. Comm 324 --- W. Suo Slide 11 E(r)% Beta for F 10 7 6 Risk Free = 4 A D C.51.0 Disequilibrium Example

12. Comm 324 --- W. Suo Slide 12 Disequilibrium Example  Short Portfolio C  Use funds to construct an equivalent risk higher return Portfolio D D is comprised of A & Risk-Free Asset  Arbitrage profit of 1%

13. Comm 324 --- W. Suo Slide 13 M Beta (Market Index) Risk Free 1.0 [E(r M ) - r f ] Market Risk Premium E(r) APT with Market Index Portfolio

14. Comm 324 --- W. Suo Slide 14  APT applies to well diversified portfolios and not necessarily to individual stocks  With APT it is possible for some individual stocks to be mispriced - not lie on the SML  APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio  APT can be extended to multifactor models APT and CAPM Compared

15. Comm 324 --- W. Suo Slide 15 A Two-Factor APT Model  The single factor APT can be extended to include more independent risk factors that work together to determine market prices  APT is more flexible than SML  However, APT offers no clues as to what factors are relevant Research must be done to determine best explanatory factor

16. Comm 324 --- W. Suo Slide 16 Three Highly Diversified Portfolios  Three risk-averse investors form portfolios B, C and D (each contains N assets) with two risk factors These are arbitrage portfolios, requiring no cash investment When N is large, unsystematic residual risk is diversified away Portfolio Expected Return Risk Factor b p1 Risk Factor b p2 B16%1.00.7 C14%0.61.0 D11%0.50.4

17. Comm 324 --- W. Suo Slide 17 Three Highly Diversified Portfolios  The general form of the APT model with two factors is:  The specific APT model for the three portfolios on the previous slide is:

18. Comm 324 --- W. Suo Slide 18 The Arbitrage Portfolio  Consider a mispriced asset PortfolioE(r p )B i1 B i2 Definition S13.66%0.7 3B + 3C + 3D U15.66%0.7 Underpriced  Portfolios S and U have the same risk but different expected returns  Portfolio U is underpriced Smart investors would buy portfolio U

19. Comm 324 --- W. Suo Slide 19 The Arbitrage Portfolio  It is possible to set up a perfect hedge with portfolios S and U to create a riskless profit opportunity Portfolio Initial Cash flow Ending Cash flowB i1 B i2 S = short position+$100-$113.66-0.7 U = underpriced (long position)-$100$115.660.7 A = arbitrage (perfectly hedged)0+$2.0000

20. Comm 324 --- W. Suo Slide 20 The k-Dimensional APT Hyperplane  A more elaborate model with k risk factors is:  Salomon Smith Barney uses a multi-factor arbitrage pricing model including factors such as:  The market’s trend or drift  Economic growth  Credit quality  Interest rates  Inflation shock  Small-cap premiums