2. 1 CHAPTER FOUR: Index Models and APT

3. 2 Problems of Markowitz Portfolio Selection There are some problems for Markowitz portfolio selection: Huge number of estimates of covariance between all pairs of available securities Vast computing capacity required to resolve an optimization quadratic programming for large portfolio CAPM is a single, static factor model

4. 3 5.7% 1.1% 14.3% 6.44.419.2 7.94.423.4 7.04.615.6 5.16.1 9.2 2.93.113.0 Single-Index Models 1 2 3 4 5 6 Year Growth of GDP （ ） Inflation （ ） Difference of the realized return of Stock i and the risk-free rate （ ） A Mini Case

5. 4 Regression Model Macro or systematic factor Firm’s or unsystematic factor Exogenous

6. 5 Covariance Systematic risk Unsystematic risk

7. 6 — Market Model CAPM is a special case of Single-Index Models taking as the factor. CAPM: The market is at equilibrium

8. 7 Can you beat the market? If you can find a portfolio manager with a positive you can beat the market! CML 0 1.The hyperbola through A and M cannot be tangent to the efficient frontier 2.The point A cannot be located on the efficient frontier

9. 8 Multi-Index Models The Mini Case Growth of GDP Inflation Firm’s or unsystematic factor

10. 9 Covariance

11. 10 — More About Arbitrage A riskless arbitrage opportunity exists if and only if either: 1.Two portfolios can be created that have identical payoffs in every state but have different costs; or 2.Two portfolios can be created with equal costs, but where the first portfolio has at least the same payoff as the second in all states, but has a higher payoff in at least one state; or 3.A portfolio can be created with zero cost, but which has a non-negative payoff in all states and a positive payoff in at least one state.

12. 11 A Mini Case A B C D -20 20 40 60 0 70 30-20 90-20-10 70 15 23 15 36 25% High real rates Low real rates Probability of the states High inflation Low inflation High inflation Low inflation Stock Return （ % ）

13. 12 A B C D $10 25 20 32.5 22.2529.58 33.91 48.15 8.58 DABC 0.68 -0.38 0.22 1.00 -0.15 -0.29 0.68 -0.15 1.00 -0.87 -0.38 -0.29 -0.87 1.00 0.22 1.00 Correlation Matrix Price Expected Return （ % ） Stock Standard Deviation (%)

14. 13 The Portfolio Comparing an equally weighted portfolio of the stocks A, B and C with the stock DComparing D High real rates Low real rates High inflation Low inflation High inflation Low inflation Stock or Portfolio Return （ % ） 23.33 20.00 36.67 15.00 23.00 15.0036.00

15. 14 Expected return and standard deviation and correlation between the portfolio and the stock Dportfolio Expected return Standard deviation Correlation Stock or Portfolio 25.83% 6.40% 0.94 The Portfolio D 22.25% 8.58% Is there a reskless arbitrage opportunity?

16. 15 Making arbitrage positions High real rates Low real rates Cash Flow High inflation Low inflation High inflation Low inflation Position Investing in A Investing in B Investing in C Short sell D Net position $ 0.25 m$ 0.01 m $ 0.15 m $ 0.02 m -$ 0.2 m $ 0.2 m $ 0.4 m $ 0.6 m - $ 1 m 0 $ 0.7 m $ 0.3 m-$ 0.2 m-$ 1 m $ 0.9 m-$ 0.2 m-$ 0.1 m $ 0.7 m -$ 1 m - $ 0.45 m -$ 0.69 m -$ 0.45 m -$ 1.08 m $ 3 m 0

17. 16 Arbitrage Pricing Theory (APT) — Single-Factor APT Macro-economy factor: the deviation from the expectation Pure unsystematic risk Sensitivity of the security i’s return to the unexpected change of the macro-economy factor

18. 17 Well-diversified portfolios and the APT A well-diversified portfolio consisting of securities: Variance of macro- economy factor 0

19. 18 Well-diversified portfolios and the APT (Cont.) Two diversified portfolio A and B, A Mini Case: Short selling $ 1 million portfolio B Investing the amount in portfolio A. Arbitrage

20. 19 Proposition! If two well-diversified portfolios have same value, they would have same expected return in the market.

21. 20 Risk premium must be proportional to value 7 6 0.5 10 1.0 risk premium 0 Expected return of portfolio There is an arbitrage opportunity between portfolios D and C ! Security Market Line of APT

22. 21 APT for individual securities For two diversified portfolios and : It holds almost for all individual securities i and j For any diversified portfolio, is the same.

23. 22 — Multi-Factor APT Macro-economy factors are the deviations from their expectations Factor portfolios Diversified portfolios with the following characteristics: Factor portfolio 1:Factor portfolio 2:

24. 23 Factor portfolios (cont.) For factor portfolio 1: portfolio 1 For factor portfolio 2: portfolio 2 For a diversified portfolio P: Replicating portfolio Q: weight Risk-free security: For the replicating portfolio Q:

25. 24 The replicating portfolio Q is the arbitrage portfolio of the diversified portfolio P Expected return of PExpected return of Q If Arbitrage opportunity: Long position of Q Short position of P Net profit:

26. 25 Proposition : The risk premium for a diversified portfolio is the sum of the contributions from all the macro-economy factorscontributions Example:

27. 26 — Multi-Factor APT Models For a portfolio P: For a security i: The extension of Security Market Line It holds almost for all securities in the markets !

28. 27 many investors make portfolio changes each portfolio’s change is limited the aggregation creates a large volume of buying and selling to restore equilibrium implying arbitrage opportunity exists each arbitrageur wants to take as large position as possible a few arbitrageurs bring the price pressures to restore equilibrium Difference Between APT and CAPM Risk free arbitrage vs. risk/return dominants Support of equilibrium price relationship When equilibrium is violated many investors make portfolio changes each portfolio’s change is limited the aggregation creates a large volume of buying and selling to restore equilibrium implying there exists arbitrage opportunity each arbitrageur wants to take as large position as possible a few arbitrageurs bring the price pressures to restore equilibrium CAPM APT Stronger

29. 28 Summary of Chapter Four 1.Index Models  Strict Separation of Systematic and Unsystematic Risks 2.CAPM  A Special Case of Single-Index Model. What’s the Difference? 3.How to Beat the Markets? 4.The Key of APT — Factor Portfolios 5.No Arbitrage Equilibrium vs. Risk/Return Dominance Arguments  APT vs. CAPM