likelihood is the same as minimizing the sum of squared errors. Therefore, in situations where we make the assumptions as discussed, then the sum ofsquares multiplied by a negative sign may be used as a proxy for the maximum likelihood.
CHAPTER
3
Factor Models
INTRODUCTIONFactor models are models that are used to explain the risk/return characteristics of assets. It is actually a rather loose term that serves to describe a widevariety of models. However, all the models share the common characteristicthat they may be viewed as extensions to the CAPM model. The premise ofthe CAPM model is that the returns of assets are explicable almost completely by the behavior of the overall market. Each asset is sensitive to themarket in its own characteristic way, and this sensitivity is termed beta.37
38
BACKGROUND MATERIAL
Thus in the CAPM model there is a single explanatory factor and exposurevalue; namely, the market return and beta. A natural extension to this ideawould then be to have multiple explanatory factors and exposure/sensitivityvalues. For instance, it is possible to construe that the return on a stock depends on the sector of the economy in which it operates, the market capitalization, and a good number of other explanatory factors that can bedrawn from the available repertoire of market variables. In this context ofmultiple explanatory factors, the return of a stock would then be an aggregate of the return contributions of the factors scaled according to the sensitivity/factor exposure. Thus, the return of a stock in a factor model isexplained by the return contributions of the various factors.Depending on the type of the factors used, factor models may be looselycategorized into three main groups: statistical factor models, macroeconomic factor models, and fundamental factor models. The factors in astatistical factor model are what we shall call eigen portfolios. They are aset of building-block portfolios with the property that their returns are uncorrelated with each other. Also, the return on any portfolio can be expressed as a linear combination of the returns on the eigen portfolios.However, the eigen portfolios are actually statistical artifacts deduced fromdata, and interpreting the results is a task that is easier said than done. So,when looking to answer questions from a valuation or a risk control standpoint, one would have to examine the returns closely to answer the question: What is the predominant theme or themes that characterize the eigenportfolio? It is this problem of interpretation that makes the statistical factor models more of a black box and hard to use. Not surprisingly, the preference for practitioners has been models that allow them to specify thefactors (macroeconomic or fundamental) allowing for a more intuitive explanation for the factor returns. These models are different from the statistical factor model in that the role of the eigen portfolios is actually assumedby some macroeconomic or fundamental variable that can be observeddirectly.The macroeconomic factor models are constructed using historical stockreturns and observable macroeconomic variables. An example of proprietary macroeconomic factor models is the Burmeister, Ibbotson, Roll, andRoss (BIRR) model. The factors or attributes in such models typically include short-term bond yield changes, long-term bond yield changes, dollarvalue versus other currencies, investor confidence, and changes in long-runeconomic growth. In contrast to the macroeconomic model, the fundamental factor model uses company and industry attributes and market data asraw descriptors to explain the returns. Examples of commercially availablemodels of this type are the BARRA and Wilshire Atlas models. The inputsto these models are typically industry factors comprising the industries inwhich the firms operate, and other fundamental factors like price/earnings
Factor Models
39
ratio, the price/book ratio, attributes relating to the capital structure of thefirm like debt/equity ratios, and the like.Even though there exists a wide variety of models, it may not be necessary to discuss each of the models on an individual basis. The theoretical underpinning for the models is provided by arbitrage pricing theory (APT).Thus, by treating the factors used as inputs in an abstract way and discussing arbitrage pricing theory, we can cover a lot of ground on the behavior and use of these different models.
ARBITRAGE PRICING THEORYArbitrage pricing theory was originally proposed by Stephen A. Ross in 1976.Unlike the preceding introduction, in which APT was presented as an extension of CAPM, the original proposal by Ross is actually embedded in an arbitrage argument and is appropriately reflected in the name of the theory.In this chapter, however, we will avoid an elaborate discussion on the foundations of APT. For that, we direct the reader to the material listed in the references. Instead, we will provide simple definitions and focus on a fewapplications to familiarize the reader with the concepts and their application.In the multifactor framework, an asset is fully characterized by its factor exposure/sensitivity profile. The contribution to the overall asset returndue to each factor is commensurate with the exposure/sensitivity of the assetto the different factors. The total return is the aggregate of the contributions.Therefore, if APT was to be summed up in one sentence, it would probablybe something like this: “Give me the risk factor profile of a security, and Iwill tell you all about its risk and return characteristics.” Let us now describesome terminology and notation surrounding APT.We will first start with risk factor exposures. Keeping with the idea ofAPT being an extension of the CAPM model, let us denote the factor exposures as β1 , β 2 , β 3 , … , β k . If r1 , r2 , r3 , … , rk denote the return contributions of each factor, then the return on the stock is given as
(
)
(
)
r = b1r1 + b2r2 + b3r3 +...+ bkrk + re
(3.1)
where re is the idiosyncratic return or specific return on the stock that is notexplicable by the factors in the model. One of the key assumptions of APTis that the specific return for a given stock is uncorrelated with both the factor returns and the specific returns of any other stock.Let us now focus on the evaluation of risk. The risk in a stock is measured as the variance of the return. The variance of return may in some waysbe likened to the range of possible values that the return can assume. A smallvariance is indicative of a narrow range and therefore lower risk, whereas alarge variance or wide range is indicative of higher uncertainty in the returns
BACKGROUND MATERIAL
40
and therefore greater risk. This approach to measuring risk as the secondmoment of the return distributions was originally proposed by Markowitz,in the context of portfolio optimization. The Markowitz approach to portfolio design is also sometimes referred to as mean-variance optimizationand was awarded the Nobel Prize in economics. Today it has become common practice to use the variance of the return as a measure of risk. We willalso keep with this practice and illustrate how risk/variance of return is calculated in the APT framework. We do this by way of an example. Let usconsider an APT model with two factors.The returns on the stock in the two factor model case is given asr = b1r1 + b2r2 + re
(3.2)
The risk is then measured as the variance of this return. To evaluate it, let usfirst expand the squared return of the stock using the algebraic identity
(a + b + c) = a + b + 2ab + 2ac + 2bc + c2
2
2
2
(3.3)
We then haver 2 = β12 r12 + β 22 r22 + 2β1β 2 r1r2 + 2β1r1re + 2β 2 r2 re + re2
(3.4)
Applying expectations on both sides and using the formulas in the appendixin the first chapter, we have
()
( )
( )
(
)
( )
var r = β12 var r1 + β 22 var r2 + 2β1β 2 cov r1 , r2 + var re
(3.5)
Note that since re is uncorrelated with both r1 and r2, the terms with theirproducts do not feature on the value for the variance. Also, Equation 3.5 canbe written in matrix form as follows:
( )( )
var r1var r = β1 β 2 cov r1 , r2
() [
]
( )( )
cov r1 , r2  β1    + var revar r2  β 2 
( )
(3.6)
Notice the structure of the equation. We have the factor exposure profile and its transpose on either side of a square matrix. This square matrix isstructured such that it has the variance of the factor returns on its diagonaland the covariance as the off-diagonal elements. It is also commonly referredto as the covariance matrix of factor returns and plays a central role in thecalculation of the risk of the security. We can simplify the notation for riskeven further:
Factor Models
41
var(r) = eVeT + var(re)
(3.7)
where V is the covariance matrix and e is the factor exposure vector. Alsonote from Equation 3.7 that the variance of the return is expressed as a simple sum of two terms. The first term is the variance due to the common factors, and the second term is the idiosyncratic/specific variance. Also, giventhat the standard deviation is the square root of variance, Equation 3.7 mayalso be written as22σ ret= σ cf2 + σ specific
(3.8)
One can easily remember the formula by drawing parallels between this andthe Pythagorean theorem from high school geometry. The standard deviations may be represented as the sides of a right-angled triangle as shown inFigure 3.1. In practice, it turns out that the specific variance is the smallercomponent of the total variance, and a significant portion of the total variance is explained by the common factor variance. Note that key to the evaluation of the common factor variance is the knowledge of the covariancematrix of factor returns.So, how is the covariance matrix calculated in practice? If we have asample of past historic factor returns, then it is a simple matter of using theformulas in the appendix of the first chapter to evaluate each of the entriesof the covariance matrix. The question therefore now becomes, how do weget a sample of past historic factor returns? To do this, we first write outthe linear equations for the return of each stock with known stock returns,treating the factor returns as unknown variables. Next, we solve this systemof equations to obtain an estimate of the factor and specific returns. Wenow have the past factor returns that may be used to estimate the covariancematrix.In other words, the covariance matrix can be deduced from the factorreturns. The converse of this statement is also true. Knowledge of the
σtotal
σspecific
σcommon factor
FIGURE 3.1
The Risk Diagram.
BACKGROUND MATERIAL
42
covariance matrix with the factor exposures and specific variances is sufficient for us to deduce the vector of expected factor returns. The reader is referred to the book by Grinold and Kahn on how that is done. Consequently,knowledge of the factor covariance matrix and the specific variances is sufficient in order to specify an APT model completely. With that said, let usformally list the parameters that are typically provided in the specification ofa factor model. They are as follows:Factor Exposure Matrix. This is the matrix of exposure/sensitivity factors. If there are N stocks in our universe and k factors in the model, wecan construct a N × k matrix with the exposures for each stock in a row.Let us denote this matrix as X.Factor Covariance Matrix. This is denoted as V.Specific Variance Matrix. This is the specific variance for each of the Nstocks assembled in an N × N matrix with the specific variances on thediagonal. Because the specific variances are assumed to be uncorrelated,the nondiagonal elements are zero. This matrix is denoted as ∆.Of the three parameters, the factor covariance matrix is by far the most interesting. We will therefore discuss some of the properties of the covariancematrix in its own section.
THE COVARIANCE MATRIXThe factor covariance matrix plays a key role in the determination of therisk. It is in fact a square matrix. In a model with k factors, the dimensionsof the covariance matrix is k × k. The diagonal elements form the varianceof the individual factors, and the nondiagonal elements are the covariancesand may have nonzero values. A nonzero covariance implies that the returnsof two explanatory factors share some correlation. For example, considerthe situation where market capitalization and the leverage of the firm areused as explanatory variables. It is not uncommon within an industry to findthat the small cap names have a high amount of leverage. If we assume thatthe small cap names outperformed the overall market, then we can expect tosee a nonzero correlation between the returns attributed to the leverage andcapitalization factors. Hence, it is possible to have nonzero entries in the offdiagonal elements of the covariance matrix.The covariance matrix is also symmetric. This is self-evident becausethe (i,j)th element and the (j,i)th element contain the entry for the covariance between the ith factor and the jth factor and are therefore the same.Additionally, the covariance matrix is also positive definite. This means
Factor Models
43
that the matrix has a square root; that is, V = B2 for some B, where V, B arematrices.Consider the situation where we are required to evaluate the covariancebetween the returns of securities A and B. Let eA and eB be the factor exposure vectors for the two stocks. Adapting the formula for variance previouslydiscussed, we have the covariance ascov(rA , rB ) = e AVeBT
(3.9)
We can therefore calculate the covariance between all the securities in ouruniverse and make them entries in a covariance matrix. This matrix wouldcome in handy to evaluate correlations between securities. Note that if thetotal universe of securities is about 5000 stocks, then the covariance matrixfor the list is a square matrix with 25 million entries. Calculating the variance and covariance of each stock pair individually by sampling past datacan be a tedious endeavor. Armed with the factor exposure vectors and thefactor covariance matrix, the covariance and correlations between securitiesmay be calculated readily. Thus, the use of the factor covariance matrix reduces the complexity of evaluating the correlations between securities in adramatic way. Even so, the full and complete covariance matrix for all thestocks in the universe is given byCovMatrix = XVXT
(3.10)
It is also prudent to be aware of certain potential issues when working withthe factor covariance matrix. For example, it is not uncommon in investment circles to hear someone say, “But you don’t want to be mining the covariance matrix!” Let us examine what they mean by that. Mining hererefers to data mining, albeit with a negative connotation. The word is usedsynonymously with bias, indicating that since the covariance matrix is deduced from historical data, the values are a reflection of the past and maynot hold going forward. While the empirical observation has been that thecovariance matrices are relatively stable, it is still subject to the fact that thevalues used in the covariance matrix may not be exact, and it may be usefulto do some sensitivity analysis on applications where we use the covariancematrix. It is probably worthwhile to also bear in mind that along with thecovariance matrix, the specific variance is also backward looking and issubject to the “mining syndrome.”Another issue that is commonly cited with regard to the covariance matrix and its use is the underlying assumption of the Gaussian distribution forthe computed variance. The so-called fat tails that are ubiquitous in the variance of returns in financial time series are not accounted for by the model.
BACKGROUND MATERIAL
44
Nevertheless, in the absence of any information whatsoever, the covariance matrix serves as a critical piece of information to assess correlations between securities and for use in factor models. Careful use of the covariancematrix can help keep the reconciliation process between the risk and returnof a large universe of securities tractable.
ExampleFactor exposure for stock A in two-factor model = (0.5, 0.75)The specific variance on stock A = .0123
.0625 .0225The factor covariance matrix for the two-factor model is = .0225 .1024The variance of return for the stock.0625 .0225  0.5 0.75  + .0123.0225 .1024 0.75= .0901 + .0123
[
A = 0.5
]
= .1024The square root of the variance is the standard deviation = .32, or 32%.Thus, stock A has a volatility value of 32%.Factor exposures for stock B in the two-factor model = (0.75, 0.5)Covariance between stocks
[
A and B = 0.5
.0625 .0225 0.750.75  = 0.08010.225 .1024  0.5 
]
APPLICATION: CALCULATING THE RISK ONA PORTFOLIOIn the earlier sections we discussed how the APT model may be used to calculate the risk on a particular asset. Now we will focus on assessing the riskof an entire portfolio. Just as with a single security, the risk can be expressed as a sum of two components; namely, common factor risk and specific risk. We will adopt the approach in which we reason out the formulasfor each of these components, leading in turn to the expression for the riskin a portfolio.Let us start with the common factor variance for a security, which canbe computed if we know the factor exposures for the security and the factorcovariance matrix. The logic to evaluate the common factor variance for the
Factor Models
45
TALKING POINTSBear in mind that the APT model is a linear model; that is, the returnis a linear combination of factor returns. Is that a limitation of themodel? Some would argue that linearity in returns is probably validonly in a fixed range of values for the factor exposures. As the valuesstretch more and more to the extremes, linearity leads to progressivelypoor predictions of asset returns. This is akin to quite a few phenomena encountered in the physical sciences where linear relationships arevalid only in a certain operating range.With that said, if you happen to find yourself in a finance conference, in the middle of a discussion on APT at cocktail hour, here issomething to try out. Look to a distance, put a finger to your cheekand proclaim wistfully, “I wonder what the consequences are of neglecting the potentially inherent nonlinearities in the modelingprocess. . . .”In all likelihood, this will cause a whole lot of other people to alsowonder with you. If, however, someone were to probe further and askyou to elaborate on your musings, you could always pretend to recognize someone at a distance, smile politely, and excuse yourself, sayingthat you need to mingle.
portfolio is also along the same lines. If we are able to evaluate the net factor exposure for the portfolio, then we can treat it as a single security andevaluate the common factor variance using the formula discussed in the previous section. Let us therefore concentrate our efforts on the evaluation ofthe factor exposure for the portfolio. The factor exposure of the portfolio issimply the weighted sum of the factor exposures of all the securities in it.The weight to be used for the exposure of each security is determined by theweight of the security in the portfolio.Let us consider a portfolio composed of two securities, A and B, withexposure vectors given by eA and eB. Let the weights of the two securities inthe portfolio be hA and hB, respectively. The exposure vector of the portfolio is given asep = hAeA + hBeB
(3.11)
Assuming a two-factor model and writing out the formula in matrix form,we have
BACKGROUND MATERIAL
46
β A,1e p = hA hB β B,1
[
]
β A, 2 β B, 2 
(3.12)
orep = hX
(3.13)
That is, the exposure of a portfolio is given by the matrix product of theholdings vector and the exposure matrix. Using the value in Equation 3.13for the factor exposure of the portfolio, the common factor variance is givenby substituting the calculated exposure value:
σ cf2 = e pVe Tp
(3.14)
Performing the substitutions and applying the matrix identity on the transpose of a matrix product, we have
σ cf2 = hXVX T hT
(3.15)
Let us now tackle the specific risk component. The specific risk is calculatedas the variance of the specific return. Now, the specific return on the portfolio is the sum of the specific returns of the assets in the portfolio weightedby the share of the security in the portfolio holdings. Also, note that bymodel assumptions the specific returns of the securities are uncorrelatedwith each other. If the returns are uncorrelated with each other, then theoverall variance is a weighted sum of the individual specific variances. In thepreceding two-security example, the specific variance is given as
( )
( )
(3.16)
 h  1 var re , B  h2 
(3.17)
2σ specific= h12 var re , A + h22 var re , B
Writing this out in matrix form, we have
( )
var re, A2= h1 h2 σ specific0
[
]
0
( )
or2σ specific= h∆hT
(3.18)
Adding it all together we are now able to put down an expression for thetotal risk in a portfolio based on the portfolio holdings and the parametersof the APT model. The risk/portfolio variance is therefore given as
Factor Models
47
22σ portfolio= σ cf2 + σ specific
(3.19)
= hXVX T hT + h∆hTThe formula to evaluate the variance may be easily adapted to evaluate thecovariance between two portfolios. We will leave it to the reader to reasonout that the covariance between two portfolios Y and Z is given by
(
)
cov rY , rZ = hY XVX T hZT + hY ∆hZT
(3.20)
In any case, the formula for variance in Equation 3.20 can be used to calculate the risk in a portfolio.Assuming normality, a two-standard-deviation movement on either sideof the daily mark to market should be able to catch the price movement forthe next day, 95 percent of the time. Yet in practice that may not turn outto be the case. Let us therefore examine the points of failure of the model.We start by listing the inputs to the risk calculation and then examine thedifferent scenarios. The key inputs are the factor exposures and the covariance matrix.When a big news event occurs with respect to a particular stock, the factor exposures that we use in the model for that stock are no longer valid. Themarket now trades on the expectation that the factor exposures after the newsevent are likely to be a lot different. The covariance structure between the factors is still intact. In any case, since the factor exposure input to the risk modelis no longer valid, the calculation breaks down for that particular stock.The second scenario is the occurrence of a scenario-altering, huge macroeconomic event, for example, relating to interest rates. The big event typically manifests itself in the form of a liquidity crisis. In these situations, thecovariance structure breaks down, leading to the breakdown of the model.Another explanation for observing price moves of more than two standard deviations of that expected by the Gaussian assumption is attributed tothe nonnormal fat-tailed distribution of asset returns observed in practice.This can be somewhat addressed by calibrating the number of standard deviations to use in our assessment of the range of price movement. It is therefore important that users of the multifactor technology also be aware of thepotential points of failure in the model.
APPLICATION: CALCULATION OFPORTFOLIO BETAThe topic of this section is more of a misnomer, and it likely to strike thereader as an anomaly. We had earlier stated that the APT is a more
BACKGROUND MATERIAL
48
advanced version of CAPM. Then, if we have access to the APT model, whywould we want to calculate the parameters of a simpler CAPM model? Isthat not going back full circle? That would indeed be true. However, if weare looking to hedge our portfolio with the market portfolio, then the hedgeratio that provides the best possible hedge is given by the beta of the portfolio. Therefore, it does make sense to calculate beta. While the title may aswell read “Determination of Hedge Ratio,” having the word beta in the titlehelps make the association between beta and the hedge ratio explicit.To see the relationship between beta and the hedge portfolio, considerthe linear combination of the two portfolios in the ratio 1:l. The return ofthe linear combination is given by rp – lrm, where rp is the return on the portfolio and rm is the return on the market. The value of l that results in the leastreturn variance of the combined portfolio is indeed the best hedge ratio. Toevaluate the variance of the return, we will apply the algebraic identity
(a + b) = a + b + 2ab2
2
2
(3.21)
to the combined portfolio.
(r − λr ) = r + λ r − 2λr r2
p
2p
m
2 2m
(3.22)
p m
Applying expectations on both sides and using the formulas in the appendixof Chapter 1, we have
(
)
( )
( )
( )
var rp − λrm = var rp + λ2 var rm − 2λ cov rp rm
(3.23)
To find the value for l that minimizes the variance, we differentiate withrespect to l and equate the differential to zero. It is easy to then see that
λ =
(
cov rp , rm
( )
)
var rm
(3.24)
Equation 3.24 also happens to be the definition of beta, and we thereforeconclude that beta is the best hedge ratio. Equation 3.24 is composed of thecovariance and the variance terms, which we know how to evaluate in theAPT framework. Applying the substitutions, we have
λ =
TTe pVem+ hp ∆hmTTemVem+ hm ∆hm
(3.25)
Factor Models
49
where ep and em are the factor exposures of the portfolio and the market,and hp and hm are their respective holdings vectors. We are thus able to evaluate the optimal hedge ratio.
APPLICATION: TRACKING BASKET DESIGNA tracking basket is a basket of stocks that tracks an index. If the basket isthe same as the index, then the prices of both will be the same at all times.However, if the tracking basket is composed of fewer stocks than the index,then there is likely to be tracking error; that is, the returns for the trackingbasket are not exactly the same as the returns for the index. The discrepancyin the returns is expressed in terms of tracking error.Tracking error may be defined as the standard deviation of the difference in the return between the tracking basket and the index. In the definition of tracking error, the mean value of the difference is assumed to be zero.To see this more clearly, consider a long–short portfolio where we are longthe index and short the tracking basket. The expectation is that the return onthe index and tracking basket is the same. Therefore, a profit made on oneleg of the portfolio is likely to be neutralized by an equal loss on theother leg. The expected value of total return on the portfolio is thereforezero. However, while the expected return is zero, it is possible for the actualreturn value to be a nonzero value. The extent of this variation from zero iscaptured by the standard deviation of returns of the long–short portfolioand forms a measure of the tracking error.A natural deduction from the preceding discussion is that the design ofa tracking basket involves designing a portfolio such that it minimizes thetracking error. Writing out the equations for the variance of the error in returns, we have
(
Min: E rm − rp
) = var(r ) + var(r ) − 2 cov(r r )2
m
p
m p
(3.26)
where rp is the return on the tracking basket and rm is the return on the market. Expanding the terms using APT constructs, we haveTTMin: hm XVX T hm+ hm ∆hm+ hp XVX T hpT + hp hpT −
[
− 2 hm XVX T hpT + hm ∆hp
(3.27)
]
Bear in mind that there are some constraints on the values of hp; that is,some of them are forced to have zero values even though they are part of theindex.
50
BACKGROUND MATERIAL
Note that in Equation 3.27, the variance of the market portfolio is notat all affected by changing the composition of the tracking basket. Theonly two terms that are affected when we change the tracking basket composition is the tracking basket variance and the covariance term. Therefore,minimizing the tracking error is equivalent to minimizing the sum of thetwo terms.The error variance may also be viewed as a sum of two components;namely, a common factor component and a specific component. Now, if wewere to design the tracking basket such that the factor exposures of the basket match the factor exposures of the index exactly (even though their contents may not be identical), then the common factor component goes tozero. We are now left only with the specific components of the variance. Recall from our earlier discussion that the contributions to the total variancefrom the specific components are a lot less than they would be from the common factor component. Furthermore, if the two portfolios are highly diversified, the expected value of the specific returns on the portfolios is zero.Hence, the tracking error contribution in this case is solely due to the different specific returns in the portfolios. It is now easy to appreciate that agood starting point to the design of tracking baskets will ensure that the factor exposures match as closely as possible. Hence, APT constructs may beused to design tracking baskets.
SENSITIVITY ANALYSISIn our discussion on the covariance matrix we talked about the mining syndrome, the idea being that the estimation of the covariance matrix is biasedto the past and may not hold going forward into the future. This apprehension may be objectively examined by studying the stability of the covariancematrix.One approach would be to estimate a sequence of covariance matricesand study the variations between two consecutive ones. The extent to whichthe variations affect the particular situation—say, the evaluation of beta, themeasurement of risk, or the design of tracking baskets—may be gauged byperturbing the current covariance matrix with the sequence of observedchanges and running the calculations with the perturbed matrix.This results in a set of values for the estimated parameter. We can thentreat the set of values as realizations from a probability distribution and getan idea of the error in our estimates using the current covariance matrix tohelp us quantify the extent of uncertainty due to the mining syndrome.
Factor Models
51
SUMMARYFactor models are models that are used to explain the risk return characteristics of assets and come in many flavors.Even though the details may vary, factor models are firmly based on theprinciples of arbitrage pricing theory (APT).A factor model is considered fully specified by the factor exposures, thefactor covariance matrix, and the specific variance matrix.A factor model may be used as a framework to estimate many commonplace parameters that may be needed in the course of the investmentprocess.Examples of such computations include the estimation of risk on a portfolio, the evaluation of portfolio beta, and computing the contents of atracking basket.The factor covariance matrix is a crucial piece of information that thefactor model provides. However, it must be used with care.
FURTHER READING MATERIALKing, Benjamin F. “Market and Industry Factors in Stock Price Behavior.” Journalof Business 3