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1、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 programmi
2、ng for large portfolio CAPM is a single, static factor model,3,Single-Index Models,A Mini Case,4,Regression Model,Macro or systematic factor,Firms or unsystematic factor,Exogenous,5,Covariance,Systematic risk,Unsystematic risk,6, Market Model,CAPM is a special case of Single-Index Models taking as t
3、he 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,The hyperbola through A and M cannot be tangent to the efficient frontier The point A cannot be located on the efficient frontier,8,Multi-Index Mo
4、dels,The Mini Case,Growth of GDP,Inflation,Firms or unsystematic factor,9,Covariance,10, More About Arbitrage,A riskless arbitrage opportunity exists if and only if either:,Two portfolios can be created that have identical payoffs in every state but have different costs; or Two portfolios can be cre
5、ated 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 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.,1
6、1,A Mini Case,12,13,The Portfolio,Comparing an equally weighted portfolio of the stocks A, B and C with the stock D,D,14,Expected return and standard deviation and correlation between the portfolio and the stock D,0.94,The Portfolio,D,Is there a reskless arbitrage opportunity,?,15,Making arbitrage p
7、ositions,Investing in A,Investing in B,Investing in C,Short sell D,Net position,0,16,Arbitrage Pricing Theory (APT), Single-Factor APT,Macro-economy factor: the deviation from the expectation,Pure unsystematic risk,Sensitivity of the security is return to the unexpected change of the macro-economy f
8、actor,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 p
9、ortfolio 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,There is an arbitrage opportunity between portfolios D and C,!,Security Market Line of APT,21,APT for individual
10、 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 charact
11、eristics:,Factor portfolio 1:,Factor portfolio 2:,23,Factor portfolios (cont.),For factor portfolio 1:,For factor 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
12、 of the diversified portfolio P,Expected return of P,Expected 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 factors,Example:,26, Multi-F
13、actor 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 portfolios change is limited the aggregation creates a large volume of buying and selling to restore equilibri
14、um,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 equil
15、ibrium is violated,many investors make portfolio changes each portfolios 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 b
16、ring the price pressures to restore equilibrium,CAPM,APT,Stronger,28,Summary of Chapter Four,Index Models Strict Separation of Systematic and Unsystematic Risks CAPM A Special Case of Single-Index Model. Whats the Difference? How to Beat the Markets? The Key of APT Factor Portfolios No Arbitrage Equilibrium vs. Risk/Return Dominance Arguments APT vs. CAPM,APT,
