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1、Modern Portfolio TheoryThe Factor Models and The Arbitrage Pricing Theory Chapter 8 By Ding zhaoyongReturn-generating Processand Factor Models Return-generating process Is a statistical model that describe how return on a security is produced. The task of identifying the Markowitz efficient set can
2、be greatly simplified by introducing this process. The market model is a kind of this process, and there are many others.Return-generating Processand Factor Models Factor models These models assume that the return on a security is sensitive to the move-ments of various factors or indices. In attempt
3、ing to accurately estimate expected returns, variances, and covariances for securities, multiple-factor models are potentially more useful than the market model.Return-generating Processand Factor Models Implicit in the construction of a factor model is the assumption that the returns on two securit
4、ies will be correlated only through common reactions to one or more of the specified in the model. Any aspect of a securitys return unexplained by the factor model is uncorrelated with the unique elements of returns on other securities.Return-generating Processand Factor Models A factor model is a p
5、owerful tool for portfolio management.It can supply the information needed to calculate expected returns, variances, and covariances for every security, which are the necessary conditions for determining the curved Markowitz efficient set.It can also be used to characterize a portfolios sensitivity
6、to movement in the factors.Return-generating Processand Factor Models Factor models supply the necessary level of abstraction in calculating covariances. The problem of calculating covariances among securities rises exponentially as the number of securities analyzed increase. Practically, abstractio
7、n is an essential step in identifying the Markowitz set. Return-generating Processand Factor Models Factor models provide investment managers with a framework to identify important factors in the economy and the marketplace and to assess the extent to which different securities and portfolios will r
8、espond to changes in these factors. A primary goal of security analysis is to determine these factors and the sensitivities of security return to movements in these factors.One-Factor Models The one-factor models refer to the return-generating process for securities involves a single factor. These f
9、actors may be one of the followings: The predicted growth rate in GDP The expected return on market index The growth rate of industrial produc-tion, etc. One-Factor Models An example Page 295: Figure 11.1GDPfor factor zero thegrowth GDP predicted Widget to ofy sensitivit tperiod inWidget on return s
10、pecific oe unique the tperiod in GDP in return of rate predicted the tperiod inWidget on return the:whereabeGDPrebGDParttttttOne-Factor Models Generalizing the example AssumptionsThe random error term and the factor are uncorrelated. (Why?)The random error terms of any two securities are uncorrelate
11、d. (Why?)ittiiiteFbarOne-Factor Models Expected return Variance CovarianceFbariii2222eiFiib2FjiijbbOne-Factor Models Two important features of one-factor model The tangency portfolio is easy to get.The returns on all securities respond to a single common factor greater simplifies the task of identif
12、ying the tangency portfolio.The common responsiveness of securities to the factor eliminates the need to estimate directly the covariances between the securities.The number of estimates: 3N+2One-Factor Models The feature of diversification is true of any one-factor model.Factor risk:Nonfactor risk:D
13、iversification leads to an averaging of factor riskDiversification reduces nonfactor risk)(22Fib2eiOne-Factor ModelsNNNXbXbbeNeeNieiepNieiiepNiiipepFpp22221122212221222211:whereMultiple-Factor Models The health of the economy effects most firms, but the economy is not a simple, monolithic entity. Se
14、veral common influences with pervasive effects might be identified The growth rate of GDP The level of interest rate The inflation rate The level of oil priceMultiple-Factor Models Two-Factor Models Assume that the return-generating process contains two factors.ittitiiiteFbFbar2211tttteINFbGDPbar21M
15、ultiple-Factor ModelsThe second equation provides a two-factor model of a companys stock, whose returns are affected by expectations concerning both the growth rate in GDP and the rate of inflation.Page 301: Figure 11.2To this scatter of points is fit a two-dimensional plane by using the statistical
16、 technique of multiple-regression analysis.Multiple-Factor Models Four parameters need to be estimated for each security with the two-factor model: ai, bi1, bi2, and the standard deviation of the random error term. For each of the factors, two parameters need to be estimated. These parameters are th
17、e expected value of each factor and the variance of each factor. Finally, the covariance between factors.Multiple-Factor Models Expected return Variance Covariance2211FbFbariiii22121222221212),(2eiiiFiFiiFFCOVbbbb),()(21122122222111FFCOVbbbbbbbbjijiFjiFjiijMultiple-Factor Models The tangency portfol
18、ioThe investor can proceed to use an optimizer to derive the curve efficient set. DiversificationDiversification leads to an averaging of factor risk.Diversification can substantially reduce nonfactor risk.For a well-diversified portfolio, nonfactor risk will be insignificant.Multiple-Factor Modelsp
19、ttptppNiititNiiitNiiiNiiiNiittitiiiNiitipteFbFbaeXFbXFbXaXeFbFbaXrXr221112121111122111)(Multiple-Factor Models Sector-Factor Models Sector-factor models are based on the acknowledge that the prices of securities in the same industry or economic sector often move together in response to changes in pr
20、ospects for that sector. To create a sector-factor model, each security must be assigned to a sector.Multiple-Factor Models A two-sector-factor model There are two sectors and each security must be assigned to one of them. Both the number of sectors and what each sector consists of is an open matter
21、 that is left to the investor to decide. The return-generating process for securities is of the same general form as the two-factor model. Multiple-Factor Models Differing from the two-factor model, with two-sector-factor model, F1 and F2 now denote sector-factors 1 and 2, respectively. Any particul
22、ar security belongs to either sector-factor 1 or sector-factor 2 but not both.jjjjiiiieFbareFbar2211Multiple-Factor Models In general, whereas four parameters need to be estimated for each security with a two-factor model (ai1,bi1,bi2 , ei,), only three parameters need to be estimated with a two-sec
23、tor-factor model. (ai1, ei, and eitherbi1 or bi2 ). Multiple-factor modelsitktiktitiiiteFbFbFbar2211Estimating Factor Models There are many methods of estimating factor models. There methods can be grouped into three major approaches: Time-series approaches Cross-sectional approaches Factor-analytic
24、 approachesFactor Models and Equilibrium A factor model is not an equilibrium model of asset pricing.Both equation show that the expected return on the stock is related to a characteristic of the stock, bi or i. The larger the size of the characteristic, the larger the assets return.)(fMiMfiiiirrrrF
25、barFactor Models and Equilibrium The key difference is ai and rf.The only characteristic of the stock that determine its expected return according to the CAPM is ii, as rff denotes the risk-free rate and is the same for all securities.With the factor model, there is a second characteristic of the st
26、ock that needs to be estimated to determine the stocks expected return, aii.Factor Models and EquilibriumAs the size of ai differs from one stock to another, it presents the factor model from being an equilibrium model.Two stocks with the same value of bi can have dramatically different expected ret
27、urns according to a factor model.Two stocks with the same value of i will have the same expected return according to the equilibrium-based CAPM.Factor Models and Equilibrium The relationship between the parameters ai and bi of the one-factor model and the single parameter i of the CAPM. If the expec
28、ted returns are determined according to the CAPM and actual returns are generated by the one-factor market model, then the above equations must be true. )(fMiMfiiiirrrrFbarArbitrage Pricing Theory APT is a theory which describes how a security is priced just like CAPM. Moving away from construction
29、of mean-variance efficient portfolio, APT instead calculates relations among expected rates of return that would rule out riskless profits by any investor in well-functioning capital markets.Arbitrage Pricing Theory APT makes few assumptions. One primary assumption is that each investor, when given
30、the opportunity to increase the return of his or her portfolio without increasing its risk, will proceed to do so.There exists an arbitrage opportunity and the investor can use an arbitrage portfolios.Arbitrage Opportunities Arbitrage is the earning of riskless profit by taking advantage of differen
31、tial pricing for the same physical asset or security. It typically entails the sale of a security at a relatively high price and the simultaneous purchase of the same security (or its functional equivalent) at a relatively low price.Arbitrage Opportunities Arbitrage activity is a critical element of
32、 modern, efficient security markets. It takes relatively few of this active investors to exploit arbitrage situations and, by their buying and selling actions, eliminate these profit opportunities. Some investors have greater resources and inclination to engage I arbitrage than others.Arbitrage Oppo
33、rtunities Zero-investment portfolio A portfolio of zero net value, established by buying and shorting component securities . A riskless arbitrage opportunity arises when an investor can construct a zero-investment portfolio that will yield a sure profit.Arbitrage Opportunities To construct a zero-in
34、vestment portfolio, one has to be able to sell short at least one asset and use the proceeds to purchase on or more assets. Even a small investor, using borrowed money in this case, can take a large position in such a portfolio. There are many arbitrage tactics.Arbitrage Opportunities An example: Fo
35、ur stocks and four possible scenarios the rate of return in four scenarios Page 180-181 in the textbook The expected returns, standard deviations and correlations do not reveal any abnormality to the naked eye.Arbitrage Opportunities The critical property of an arbitrage portfolio is that any invest
36、or, regardless of risk aversion or wealth, will want to take an infinite position in it so that profits will be driven to an infinite level. These large positions will force some prices up and down until arbitrage opportunities vanishes. Factor Models and Principle of Arbitrage Almost arbitrage oppo
37、rtunities can involve similar securities or portfolios. That similarity can be defined in many ways. One way is the exposure to pervasive factors that affect security prices. An example Page 324Factor Models and Principle of Arbitrage A factor model implies that securities or portfolios with equal-f
38、actor sensitivities will behave in the same way except for nonfactor risk. APT starts out by making the assumption that security returns are related to an unknown number of unknown factors. Securities with the same factor sensitivities should offer the same expected returns.Arbitrage Portfolios An a
39、rbitrage portfolio must satisfy: A net market value of zero No sensitivity to any factor A positive expected return0321XXX0332211XbXbXb0332211rXrXrXArbitrage PortfoliosThe arbitrage portfolio is attractive to any investor who desires a higher return and is not concerned with nonfactor risk. It requi
40、res no additional dollar investment, it has no factor risk, and it has a positive expected return.One-Factor Model and APT Pricing effects on arbitrage portfolio The buying-and-selling activity will continue until all arbitrage possibilities are significant reduced or eliminated There will exist an
41、approximately linear relationship between expected returns and sensitivities of the following sort:iibr10One-Factor Model and APT The equation is the asset pricing equation of the APT when returns are generated by one factorThe linear equation means that in equili-brium there will be a linear relati
42、onship between expected returns and sensitivities.The expected return on any security is, in equilibrium, a linear function of the securitys sensitivity to the factor, biOne-Factor Model and APT Any security that has a factor sensitivity and expected return such that it lies off the line will be mis
43、priced according to the APT and will present investors with the opportunity of forming arbitrage portfolios. Page 327: Figure 12.1One-Factor Model and APT Interpreting the APT pricing equation Riskfree asset, rf Pure factor portfolio, p*ififibrrrr1011fpfprrrriffipbrrrr11 :letTwo-Factor Model And APT
44、 The two-factor model Arbitrage portfolios A net market value of zero No sensitivity to any factor A positive expected returniiiiieFbFbar221122110iiibbrTwo-Factor Model And APT Pricing effects221102211rate riskfree)()(fffififfirrrbrbrrrTwo-Factor Model And APT 1 is the expected return on the portfol
45、io which is known as a pure factor portfolio or pure factor play, because it has:Unit sensitivity to one factor (F1, b1=1)No sensitivity to any other factor (F2, b2=0)Zero nonfactor riskThis portfolio is a well-diversification portfolio that has unit sensitivity to the first factor and zero sensitiv
46、ity to the second factor.Two-Factor Model And APT It is the same with 2 . It is the well-diversification portfolio that has zero sensitivity to the first factor and unit sensitivity to the second factor, meaning that it has b1=0 and b2=1. Such as a portfolio that has zero sensitivity to predicted in
47、dustrial production and unit sensitivity to predicted inflation would have an expected return of 6%. Multiple-factor model The APT pricing equationMultiple-Factor Model And APTikikiiiieFbFbFbar2211ikkiiibbbr22110ikfkiffibrbrrr)()(11The APT And The CAPM Common point Both require equilibrium Both have
48、 almost similar equation Distinctions Different equilibrium mechanismMany investors v.s. Few investors Different PortfolioMarket portfolio v.s. Well-diversifyed P.Summary The Factor Models One-factor models Multi-factor models Factor models and equilibrium Arbitrage opportunity and portfolio The arbitrage pricing equation One-factor equation Multi-factor equationAssignments For chapter 8 Readings Page 282 through 301 Page 308 through 321 Exercises Page 304: 14,15; Page 323: 4, 13 Q/A: Page 302: 3 Page 324: 8