第十一章 套利定价理论

1、McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-0Chapter Outline11.1 Factor Models: Announcements, Surprises, and Expected Returns11.2 Risk: Systematic and Unsystematic11.3 Systematic Risk and Betas11.4 Portfolios and Factor Models11.5 Betas and Expected Ret

2、urns11.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory11.7 Parametric Approaches to Asset Pricing11.8 Summary and ConclusionsMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-1Arbitrage Pricing TheoryArbitrage - arises if an investor can con

3、struct 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.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companie

4、s, Inc. All rights reserved.11-211.1 Factor Models: Announcements, Surprises, and Expected Returns The return on any security consists of two parts. First the expected returns Second is the unexpected or risky returns. A way to write the return on a stock in the coming month is:return theofpart unex

5、pected theis return theofpart expected theis whereURURRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-311.1 Factor Models: Announcements, Surprises, and Expected Returns Any announcement can be broken down into two parts, the anticipated or expected part an

6、d the surprise or innovation: Announcement = Expected part + Surprise. The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock.The surprise is the news that influences the unanticipated return on the stock, U. McGraw-Hill

7、/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-411.2 Risk: Systematic and Unsystematic A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or s

8、mall group of assets. Unsystematic risk can be diversified away. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation. On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examp

9、les of unsystematic risk.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-511.2 Risk: Systematic and UnsystematicSystematic Risk; m Nonsystematic Risk; n Total risk; UWe can break down the risk, U, of holding a stock into two components: systematic risk and u

10、nsystematic risk:risk icunsystemat theis risk systematic theis wherebecomesmmRRURR McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-611.3 Systematic Risk and Betas The beta coefficient, b, tells us the response of the stocks return to a systematic risk. In th

11、e CAPM, b measured the responsiveness of a securitys return to a specific risk factor, the return on the market portfolio.)()(2,MMiiRRRCovb We shall now consider many types of systematic risk.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-711.3 Systematic R

12、isk and BetasFor example, suppose we have identified three systematic risks on which we want to focus:1.Inflation2. GDP growth3.The dollar-euro spot exchange rate, S($,)Our model is:risk icunsystemat theis beta rate exchangespot theis beta GDP theis betainflation theis FFFRRmRRSGDPISSGDPGDPIIMcGraw-

13、Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-8Systematic Risk and Betas: ExampleSuppose we have made the following estimates:1.bI = -2.302.bGDP = 1.503.bS = 0.50.Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contribut

14、es 1% to the return.FFFRRSSGDPGDPII%1%150. 050. 130. 2SGDPIFFFRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-9Systematic Risk and Betas: ExampleWe must decide what surprises took place in the systematic factors. If it was the case that the inflation rate

15、was expected to be by 3%, but in fact was 8% during the time period, then FI = Surprise in the inflation rate= actual expected= 8% - 3%= 5%150. 050. 130. 2SGDPIFFFRR%150. 050. 1%530. 2SGDPFFRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-10Systematic Risk

16、and Betas: ExampleIf it was the case that the rate of GDP growth was expected to be 4%, but in fact was 1%, then FGDP = Surprise in the rate of GDP growth = actual expected= 1% - 4%= -3%150. 050. 1%530. 2SGDPFFRR%150. 0%)3(50. 1%530. 2SFRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies,

17、 Inc. All rights reserved.11-11Systematic Risk and Betas: ExampleIf it was the case that dollar-euro spot exchange rate, S($,), was expected to increase by 10%, but in fact remained stable during the time period, then FS = Surprise in the exchange rate= actual expected= 0% - 10%= -10%150. 0%)3(50. 1

18、%530. 2SFRR%1%)10(50. 0%)3(50. 1%530. 2 RRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-12Systematic Risk and Betas: ExampleFinally, if it was the case that the expected return on the stock was 8%, then%150. 0%)3(50. 1%530. 2SFRR%12%1%)10(50. 0%)3(50. 1%53

19、0. 2%8RR%8RMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-1311.4 Portfolios and Factor Models Now let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model. We will create portfolios from a list of N stocks and

20、will capture the systematic risk with a 1-factor model. The ith stock in the list have returns:iiiiFRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-14Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the fac

21、tor FiiiiiFRRIf we assume that there is no unsystematic risk, then i = 0McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-15Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FIf we assume that there i

22、s no unsystematic risk, then i = 0FRRiiiMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-16Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FDifferent securities will have different betas0 . 1B50. 0

23、C5 . 1AMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-17Portfolios and Diversification We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:NNiiPRXRXRXRXR2211)()()(22221111NNNNPFRXFRXFRXRNNNNNNPX

24、FXRXXFXRXXFXRXR222222111111iiiiFRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-18Portfolios and DiversificationThe return on any portfolio is determined by three sets of parameters:In a large portfolio, the third row of this equation disappears as the uns

25、ystematic risk is diversified away.NNPRXRXRXR22111. The weighed average of expected returns.FXXXNN)(22112. The weighted average of the betas times the factor.NNXXX22113. The weighted average of the unsystematic risks.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reser

26、ved.11-19Portfolios and DiversificationSo the return on a diversified portfolio is determined by two sets of parameters:1.The weighed average of expected returns.2.The weighted average of the betas times the factor F.FXXXRXRXRXRNNNNP)(22112211In a large portfolio, the only source of uncertainty is t

27、he portfolios sensitivity to the factor.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-2011.5 Betas and Expected ReturnsThe return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.FXXRXRXRNNNNP

28、)(1111FRRPPPNNPRXRXR11 that RecallNNPXX11 andPRPMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-21Relationship Between b & Expected Return If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected

29、return.FRRPPPMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-22Relationship Between b & Expected ReturnExpected returnb bFRABCDSML)(FPFRRRRMcGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved.11-2311.6 The Capital Asset

30、Pricing Model and the Arbitrage Pricing Theory 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.McGraw-Hill/IrwinCopyright 2002 by The McGraw-Hill Companies