1 CHAPTER FOUR: Index Models and APT

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

3 5.7% 1.1% 14.3% Single-Index Models Year Growth of GDP （ ） Inflation （ ） Difference of the realized return of Stock i and the risk-free rate （ ） A Mini Case

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

5 Covariance Systematic risk Unsystematic risk

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

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

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

9 Covariance

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.

11 A Mini Case A B C D % High real rates Low real rates Probability of the states High inflation Low inflation High inflation Low inflation Stock Return （ % ）

12 A B C D $ DABC Correlation Matrix Price Expected Return （ % ） Stock Standard Deviation (%)

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 （ % ）

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?

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

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

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

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

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

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

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.

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:

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:

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:

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

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 !

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

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