modern portfolio theory the factor models and the arbitrage pricing theory chapter 8 by ding...

Modern Portfolio TheoryThe Factor Models and

The Arbitrage Pricing Theory

Chapter 8

By Ding zhaoyong

Return-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 be greatly simplified by introducing this process.

– The market model is a kind of this process, and there are many others.


• Factor models– These models assume that the return on a s

ecurity is sensitive to the move-ments of various factors or indices.

– In attempting to accurately estimate expected returns, variances, and covariances for securities, multiple-factor models are potentially more useful than the market model.


– Implicit in the construction of a factor model is the assumption that the returns on two securities will be correlated only through common reactions to one or more of the specified in the model. Any aspect of a security’s return unexplained by the factor model is uncorrelated with the unique elements of returns on other securities.


– A factor model is a powerful tool for portfolio management.« It can supply the information needed to ca

lculate 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 portfolio’s sensitivity to movement in the factors.


• 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, abstraction is an essential step in identifying the Markowitz set.


• 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 respond 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 factors 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 11.1

GDPfor factor zero the

growth GDP predicted Widget to ofy sensitivit

tperiod inWidget on return specific oe unique the

tperiod in GDP in return of rate predicted the

tperiod inWidget on return the

:where

a

b

e

GDP

r

ebGDPar

t

t

t

ttt

One-Factor Models

• Generalizing the example

– Assumptions« The random error term and the factor are

uncorrelated. (Why?)« The random error terms of any two

securities are uncorrelated. (Why?)

ittiiit eFbar

One-Factor Models

– Expected return

– Variance

– Covariance

Fbar iii

2222eiFii b

2F j i ijb b

One-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 identifying 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+2

One-Factor Models

– The feature of diversification is true of any one-factor model.« Factor risk:« Nonfactor risk:« Diversification leads to an averaging of fa

ctor risk« Diversification reduces nonfactor risk

)( 22Fib

2ei

One-Factor Models

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Multiple-Factor Models

• The health of the economy effects most firms, but the economy is not a simple, monolithic entity. Several 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 price


• Two-Factor Models

– Assume that the return-generating process contains two factors.

ittitiiit eFbFbar 2211

tttt eINFbGDPbar 21


« The second equation provides a two-factor model of a company’s stock, whose returns are affected by expectations concerning both the growth rate in GDP and the rate of inflation.

« Page 301: Figure 11.2« To this scatter of points is fit a two-

dimensional plane by using the statistical technique of multiple-regression analysis.


– 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 the expected value of each factor and the variance of each factor. Finally, the covariance between factors.


– Expected return

– Variance

– Covariance

2211 FbFbar iiii

22121

22

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2 ),(2 eiiiFiFii FFCOVbbbb

),()( 2112212

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– The tangency portfolio« The investor can proceed to use an optimizer to d

erive the curve efficient set.

– Diversification« Diversification leads to an averaging of factor ris

k.« Diversification can substantially reduce nonfacto

r risk.« For a well-diversified portfolio, nonfactor risk wi

ll be insignificant.


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• 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 prospects for that sector.

– To create a sector-factor model, each security must be assigned to a sector.


• 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 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.


– Differing from the two-factor model, with two-sector-factor model, F1 and F2 now denote sector-factors 1 and 2, respectively. Any particular security belongs to either sector-factor 1 or sector-factor 2 but not both.

jjjj

iiii

eFbar

eFbar

22

11


– 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-sector-factor model. (ai1,ei, and eitherbi1 or bi2 ).

• Multiple-factor models

itktiktitiiit eFbFbFbar 2211

Estimating 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 approaches

Factor 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 asset’s return.

)( fMiMfi

iii

rrrr

Fbar


– 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 stock that needs to be estimated to determine the stock’s expected return, aii.


« As 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 returns 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 a

nd bi of the one-factor model and the single parameter i of the CAPM.

– If the expected 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.

)( fMiMfi

iii

rrrr

Fbar

Arbitrage Pricing Theory

• APT is a theory which describes how a security is priced just like CAPM.

– Moving away from construction 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 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 differential 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 activity is a critical element of 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.


• 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.


– To construct a zero-investment 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.


• An example:

– Four 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.


• The critical property of an arbitrage portfolio is that any investor, 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 opportunities 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 324

Factor Models and Principle of Arbitrage

• A factor model implies that securities or portfolios with equal-factor 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 arbitrage portfolio must satisfy:

– A net market value of zero

– No sensitivity to any factor

– A positive expected return

0321 XXX

0 3 3 2 2 1 1 X b X b X b

0332211 rXrXrX

Arbitrage Portfolios

The arbitrage portfolio is attractive to any investor who desires a higher return and is not concerned with nonfactor risk.

It requires 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 approximately linear relationship between expected returns and sensitivities of the following sort:

ii br 10


– The equation is the asset pricing equation of the APT when returns are generated by one factor« The linear equation means that in equili-briu

m there will be a linear relationship between expected returns and sensitivities.

« The expected return on any security is, in equilibrium, a linear function of the security’s sensitivity to the factor, bi


– Any security that has a factor sensitivity and expected return such that it lies off the line will be mispriced according to the APT and will present investors with the opportunity of forming arbitrage portfolios.

– Page 327: Figure 12.1


• Interpreting the APT pricing equation

– Riskfree asset, rf

– Pure factor portfolio, p*

ififi brrrr 10

11 fpfp rrrr

iffip brrrr 11 :let

Two-Factor Model And APT

• The two-factor model

• Arbitrage portfolios

– A net market value of zero

– No sensitivity to any factor

– A positive expected return

iiiii eFbFbar 2211

22110 iii bbr


• Pricing effects

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)()(

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1 is the expected return on the portfolio 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 risk« This portfolio is a well-diversification portfolio th

at has unit sensitivity to the first factor and zero sensitivity to the second factor.


– 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 industrial production and unit sensitivity to predicted inflation would have an expected return of 6%.

• Multiple-factor model

• The APT pricing equation

Multiple-Factor Model And APT

ikikiiii eFbFbFbar 2211

ikkiii bbbr 22110

ikfkiffi brbrrr )()( 11

The APT And The CAPM

• Common point

– Both require equilibrium

– Both have almost similar equation

• Distinctions

– Different equilibrium mechanism« Many investors v.s. Few investors

– Different Portfolio« Market 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 equation

Assignments 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

modern portfolio theory the factor models and the arbitrage pricing theory chapter 8 by ding...

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nonfactor risk slide

ding zhaoyong slide