指数模型和套利定价理论课件

1、CHAPTER FOUR: Index Models and APT1Problems 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 f

2、or large portfolio CAPM is a single, static factor model2 5.7% 1.1% 14.3%6.4 4.419.27.94.423.47.04.615.65.16.1 9.22.93.113.0Single-Index Models123456YearGrowth of GDP（ ）Inflation（ ）Difference of the realized return of Stock i and the risk-free rate（ ） A Mini Case 3 Regression ModelMacro or systemati

3、c factorFirms or unsystematic factorExogenous 4 CovarianceSystematic riskUnsystematic risk5 Market ModelCAPM is a special case of Single-Index Models taking as the factor.CAPM:The market is at equilibrium 6 Can you beat the market?If you can find a portfolio manager with a positive you can beat the

4、market!CML0The hyperbola through A and M cannot be tangent to the efficient frontierThe point A cannot be located on the efficient frontier7Multi-Index Models The Mini Case Growth of GDPInflationFirms or unsystematic factor8 Covariance9 More About ArbitrageA riskless arbitrage opportunity exists if

5、and only if either:Two portfolios can be created that have identical payoffs in every state but have different costs; orTwo 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 stat

6、e; orA 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.10 A Mini Case ABCD-20 20 40 60 0 70 30-20 90-20-10 70 15 23 15 3625%25%25%25% High real rates Low real rates Probability of the statesHigh inflationLow infla

7、tionHigh inflationLow inflationStockReturn（%）11ABCD$10$10$10$102520 32.5 22.2529.58 33.91 48.15 8.58DABC 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 1.00Correlation MatrixPriceExpected Return（%）StockStandard Deviation (%)12The Portfolio Comparing an equally weighted po

8、rtfolio of the stocks A, B and C with the stock DD High real rates Low real rates High inflationLow inflationHigh inflationLow inflationStock or PortfolioReturn（%）23.33 23.33 20.00 36.67 15.00 23.00 15.0036.0013 Expected return and standard deviation and correlation between the portfolio and the sto

9、ck DExpected returnStandard deviationCorrelationStock or Portfolio25.83% 6.40% 0.94The PortfolioD22.25% 8.58% Is there a reskless arbitrage opportunity?14 Making arbitrage positions High real rates Low real rates Cash FlowHigh inflationLow inflationHigh inflationLow inflationPositionInvesting in AIn

10、vesting in BInvesting in CShort sell DNet 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 m015Arbitrage Pricing Theory (APT) Single-Factor APTMacro-e

11、conomy factor: the deviation from the expectation Pure unsystematic riskSensitivity of the security is return to the unexpected change of the macro-economy factor16 Well-diversified portfolios and the APTA well-diversified portfolio consisting of securities: Variance of macro-economy factor017 Well-

12、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.Arbitrage18Proposition!If two well-diversified portfolios have same value, they would have same expected return in the market.19 Risk pr

13、emium must be proportional to value 760.5101.0risk premium0Expected return of portfolioThere is an arbitrage opportunity between portfolios D and C!Security Market Line of APT20 APT for individual securitiesFor two diversified portfolios and :It holds almost for all individual securities i and jFor

14、any diversified portfolio, is the same. 21 Multi-Factor APTMacro-economy factors are the deviations from their expectations Factor portfoliosDiversified portfolios with the following characteristics:Factor portfolio 1:Factor portfolio 2:22 Factor portfolios (cont.)For factor portfolio 1:For factor p

15、ortfolio 2:For a diversified portfolio P:Replicating portfolio Q: weightRisk-free security:For the replicating portfolio Q:23 The replicating portfolio Q is the arbitrage portfolio of the diversified portfolio PExpected return of PExpected return of QIf Arbitrage opportunity:Long position of QShort

16、position of PNet profit:24Proposition : The risk premium for a diversified portfolio is the sum of the contributions from all the macro-economy factorsExample:25 Multi-Factor APT ModelsFor a portfolio P:For a security i:The extension of Security Market LineIt holds almost for all securities in the m

17、arkets!26 many investors make portfolio changes each portfolios 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

18、pressures to restore equilibriumDifference Between APT and CAPM Risk free arbitrage vs. risk/return dominantsSupport of equilibrium price relationshipWhen equilibrium 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 bring the price pressures to restore equilibriumCAPMAPTStrong