1、.0An Alternative View of Risk and Return: The Arbitrage Pricing TheoryChapter 1212-1.1Key Concepts and SkillsoDiscuss the relative importance of systematic and unsystematic risk in determining a portfolios returnoCompare and contrast the CAPM and Arbitrage Pricing Theory12-2.2Chapter Outline12.1 Int

2、roduction12.2 Systematic Risk and Betas12.3 Portfolios and Factor Models12.4 Betas and Expected Returns12.5 The Capital Asset Pricing Model and the Arbitrage Pricing Theory12.6 Empirical Approaches to Asset Pricing12-3.312.1 IntroductionArbitrage Pricing TheoryArbitrage arises if an investor can con

3、struct a zero investment portfolio with a sure profit.nSince no investment is required, an investor can create large positions to secure large levels of profit.nIn efficient markets, profitable arbitrage opportunities will quickly disappear.12-4.4Total RiskoTotal risk = systematic risk + unsystemati

4、c riskoThe standard deviation of returns is a measure of total risk.oFor well-diversified portfolios, unsystematic risk is very small.oConsequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.12-5.5Risk: Systematic and UnsystematicSystematic Risk: m N

5、onsystematic Risk: n 2Total risk We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk:risk icunsystemat theis risk systematic theis wherebecomesmmRRURR 12-6.612.2 Systematic Risk and BetasoThe beta coefficient, b, tells us the response of the

6、 stocks return to a systematic risk.oIn the CAPM, b measures the responsiveness of a securitys return to a specific risk factor, the return on the market portfolio.)()(2,MMiiRRRCovb We shall now consider other types of systematic risk.12-7.7Systematic Risk and BetasoFor example, suppose we have iden

7、tified three systematic risks: inflation, GNP growth, and the dollar-euro spot exchange rate, S($,).oOur model is:risk icunsystemat theis beta rate exchangespot theis beta GNP theis betainflation theis FFFRRmRRSGNPISSGNPGNPII12-8.8Systematic Risk and Betas: ExampleoSuppose we have made the following

8、 estimates:nbI = -2.30nbGNP = 1.50nbS = 0.501.Finally, the firm was able to attract a “superstar” CEO, and this unanticipated development contributes 1% to the return.FFFRRSSGNPGNPII%1%150. 050. 130. 2SGNPIFFFRR12-9.9Systematic Risk and Betas: ExampleWe must decide what surprises took place in the s

9、ystematic factors. If it were the case that the inflation rate was expected to be 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. 2SGNPIFFFRR%150. 050. 1%530. 2SGNPFFRR12-10.10Systematic Risk and Betas: ExampleIf it

10、 were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then: FGNP = Surprise in the rate of GNP growth = actual expected = 1% 4% = 3%150. 050. 1%530. 2SGNPFFRR%150. 0%)3(50. 1%530. 2SFRR12-11.11Systematic Risk and Betas: ExampleIf it were the case that the dollar-euro

11、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%530. 2SFRR%1%)10(50. 0%)3(50. 1%530. 2 RR12-12.12Systematic Risk and Betas: ExampleFinally, if it we

12、re the case that the expected return on the stock was 8%, then:%1%)10(50. 0%)3(50. 1%530. 2 RR%12%1%)10(50. 0%)3(50. 1%530. 2%8RR%8R12-13.1312.3 Portfolios and Factor ModelsoNow let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model.oWe will create po

13、rtfolios from a list of N stocks and will capture the systematic risk with a 1-factor model.oThe ith stock in the list has return:iiiiFRR12-14.14Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FiiiiiFRRIf we assume that there is no unsyst

14、ematic risk, then i = 0.12-15.15Relationship Between the Return on the Common Factor & Excess ReturnExcess returnThe return on the factor FIf we assume that there is no unsystematic risk, then i = 0.FRRiii12-16.16Relationship Between the Return on the Common Factor & Excess ReturnExcess retu

15、rnThe return on the factor FDifferent securities will have different betas.0 . 1B50. 0C5 . 1A12-17.17Portfolios and DiversificationoWe know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:NNiiPRXRXRXRXR2211)()()(22221111NNNNPFRXFRXFRXRNNNNNN

16、PXFXRXXFXRXXFXRXR222222111111iiiiFRR12-18.18Portfolios 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 unsystematic risk is diversified away.NNPRXRXRXR2211The weighted average of expected

17、returns.FXXXNN)(2211The weighted average of the betas times the factor.NNXXX2211The weighted average of the unsystematic risks.12-19.19Portfolios and DiversificationSo the return on a diversified portfolio is determined by two sets of parameters:nThe weighted average of expected returns.1.The weight

18、ed average of the betas times the factor F.FXXXRXRXRXRNNNNP)(22112211In a large portfolio, the only source of uncertainty is the portfolios sensitivity to the factor.12-20.2012.4 Betas and Expected ReturnsThe return on a diversified portfolio is the sum of the expected return plus the sensitivity of

19、 the portfolio to the factor.FXXRXRXRNNNNP)(1111FRRPPPNNPRXRXR11 that RecallNNPXX11 andPRP12-21.21Relationship Between b & Expected ReturnoIf shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.FRRPPP12-22.22Relationship Between

20、b & Expected ReturnExpected returnb bFRABCDSML)(FPFRRRR12-23.2312.5 The Capital Asset Pricing Model and the Arbitrage Pricing TheoryoAPT applies to well diversified portfolios and not necessarily to individual stocks.oWith APT it is possible for some individual stocks to be mispriced - not lie o

21、n the SML.oAPT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio.oAPT can be extended to multifactor models.12-24.2412.6 Empirical Approaches to Asset PricingoBoth the CAPM and APT are risk-based models. oEmpirical methods are

22、based less on theory and more on looking for some regularities in the historical record.oBe aware that correlation does not imply causality.oRelated to empirical methods is the practice of classifying portfolios by style, e.g.,nValue portfolionGrowth portfolio12-25.25Quick QuizoDifferentiate systema

23、tic risk from unsystematic risk. Which type is essentially eliminated with well diversified portfolios?oDefine arbitrage.oExplain how the CAPM be considered a special case of Arbitrage Pricing Theory?12-26.261.Suppose a factor model is appropriate to describe the returns on a stock. Information abou

24、t those factors is presented in the following chart.Beta ofExpectedActualFactorFactorValue (%)Value (%)Growth in GNP2.043.5%4.8%Interest rates-1.9014.0%15.2%Stock return10.0%a. What is the systematic risk of the stock return?b. The firm announced that its market share had unexpectedly increased from

25、 23 percent to 27 percent. Investors know from their past experience that the stock returns will increase by .36 percent per an increase of 1percent in its market share. What is the unsystematic risk of the stock?c. What is the total return of this stock?12-27.27Answer:11.81%1.440.3710.0ReturnTotalc.1.44%=23-270.36=Return icUnsystematb.0.372%=14.0%15.2%1.903.5%-4.8%2.04=Risk Systematica.12-28.282.The following three stocks are available in the market. Assume the market model is valid.a.write the market-model equati