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w2reading_2 - “AF-{ER 15 THE Afifili’MGE PRICING...

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Unformatted text preview: “AF-{ER 15 THE Afifili’MGE PRICING MOQEL APTM NEW APPROACH TO EXPINNING ASSET PRJCES '- , An assumption of homogeneous expectations is necessary. The assumption of eswfg utilizing a mean variance framework is repiaced by an assumption of the process gng security returns. AP’I‘ requires that the returns on any stock be linearly rotated dexcs as shown in Equation {16.} }.‘ '6 n: m ii. ncrat ‘ .3 set of In The Arbitrage Pricing Model APT—MA New Approach to Explaining Asset Prices Rizai+biill+bi2f2 l"“+bijlr‘+ei 06'” a; e the expected ievel of return for stock 1' if a1} indices have a value of zero If. e the value of the jth index that impacts the return on stock i b1}. : the sensitivity of stock i's returnto the jth index a random error term with mean cquai to zero and variance equal to 0‘3,- . . For the model to fully describe the process'generating security returns:2 19(ng = o is an r and j where i a j Elam — .7 = 0 for ail stocks and indexes If you are 1oeginning to get the feeling that you have seen all this before, you are right. This representation is nothing more or less than the description of the multi—index modei presented in Chapter 8. Ai’l‘ is the description of the expected returns that can be derived when returns are generated by a single— or multi—index model meeting the conditions defined before. The contribution of APT is in demonstrating how (and under what conch“ tions) one can go from a multi—index modei to a description of equilibrium. _ In the following pages we will demonstrate the derivation of an APT equilibrium in two different ways. The first proof stresses the economic rationale behind APT, whereas the second proof is mathematically more rigorous. All of the equilibrium models discussed in Chapters 13, i4 and 15 have it -‘ ' mean—variance analysis. All require that it is optima} for the investor to cho 1w ham on the hasts of expected return and variance. However, definitions of retusfa mvwme‘n means E‘li’ld variances are caiculated differ between modcis. For exam le inullns for W " the'caprtal asset pricing model (CAPM) involving taxes, investors Iexam' m versmn variances of aftentax returns. As a second example, Bitch and Gruber ( 198gl ehmeans and: that the aiternativc version of CAPM under conditions of uncertain infiation can :1? Show“. majaor cigigaiéeslgzt/‘csting any of these equilibrium theories oss , } has proposed a new and di ' ' ' ‘ ' of assets. Ross had developed a mechanism that,f 2:231:31???1mm returns, derives asset prices from arbitrage arguments analogous to (hit m3 as security than) those used in the beginning of Chapter 13 to derive CAPMS In this ch or: omplex': present the mechanism of arbitrage pricing theory (APT). This is the dc ' a? e! We firsgi librium conditions given any prespecified return-generating process manon 0f equ I following this, we diseuss implementation of the APT. APT theo. rov’d ' ' ' insight into the nature of equilibrium. However, the theory is far frorripe 1 es’mleresmg' Empirical research is stili in the early stages in this area Furthe asy m mlpleme'm '- approaches have been advocated for implementing the theory: After dig-23m: a mmatm : gigglaltemauvef, we PLBSSHE an examination of whether evidence suppolrtinsgsglig’lfoiglsgf' 1 y‘inconsm ent Wit t e standard form or an alte ' I of equrlihrium. We close with a discussion of bofh aplilliactaii‘EJJtizrxg iglizgtigg :1? :gdel I I A Simple Proof of APT We will demonstrate the expected returns that must arise from the AP? with a two-index model. Suppose that the following two-index model describes retorns: R- = Q‘- +anl +b£212 +8; I I Furthermore, assume that Emmi) m i). If an investor holds a well-diversified portfoiio, rcsiduai risic will tend to go to zero and oniy systematic risk. wiii matter. The only terms in the preceding equation that affect the systematic risk in a portfoiio are big and bfg. Since the investor is assumed to be concerned with expected return and risk. he or she need be concerned only with three attributes of any portfolio (p): 17p, by“, and 171,2. Let us hypothesize the existence of the three widely diversified portfolios shown in the following table. APT—WHAT IS 1T? :rbitrage' pricing theory is a new and different approach to determining asset rices it is ' ased on the law of one price: two items that are the same can’t seIi at differentprices‘ T he I gézontghasztgnrp‘tjions‘made about utility theory in deriving the CAPM are not neiessary In I , e escnption of equilibrium is more general than ' ' I I ‘ . . that rovrded b - type model in that prictng can be affected by influences beyorfd simply 11:32:21); lThe iiriearity assumption is not as restrictive as it might at first appear. Any of the indexes can be a noniinear function of a variable. It couid be a variable squared. the tog of a variahie. or any other noniinear transformation that seems appropriate. 2it is convenient, though unnecessary. to assume the indexes are uncorrelated with each other. We show in Chapter 8 that a set of corrciated indexes can always be converted to a set of uncorrelated indexes. 'i‘hc results remain the same with uncoireialed indexes but the mathematics is more complex. 362 PART 3 MODELS OF EOUIIJBREUM W THE Wi‘i‘AL Portfolio Expected Rem . A 15 1.0 B 14 .5 C i0 3 We know from the concepts of geometry that three points determine a potnts determine a line. The equation of th three portfolios is3 I _ plane just as {w 3 Plan‘3 In Rpa 13,91, and bpg space defined by i; as R,- a 7.75 +51%l + 3.7512“; .The expected return and risk measures of an given by y portfolio of these three portfolios N RF 3 E Xipr. i=1 N bl)! = z Xrbu What happens if we consider a new portfolio not on this piano? For exam portfolio E extsts With an expected return or” 15 ‘ ‘ %, a b” of 0.6, and a b of 0.6. I compare th1s W1th a portfoiiio {call it D) constructed by placing 3L oilthe funds in pa ' folio A, "3'111 portfolio B, and 5 in portfoiio C. The lam-s on this portfolio are ple. assume a _ 1 1 1 by; ~ 'i‘ + = i 1 t b :— 0. ._ __ ._ p2 3{ 6)+3(l.{})+3{0.2)_,5 The risk for portfolio D is identical to the risk on portfol pmzmlio D R to E. The expected retain on: ét15)+%(14)+%(10}=33 3The reader interested in verifying this can recle that the hzbn. By substituting in the values of R}, b.- three unknowns: he. AI, and k2. tion in the text. equation of a plane can be written as R,- “ lo, + lob” ’ i. and 1.1,: for portfolios A, B, and C, We obtain three equations wit Solving the three equations gives the values of Ag. Al, and A2 shown in the £2an "The reader is encouraged to form a portfolio of can then see that this portfolio lies on the plane portfolio D anaiyzed shortly in the text. portfolios_A, B, and C with any set of X, summing to one. 0" given by R,- m 7.75 ~l- 5 b“ + 3.75 {42.0112 example of thisl '- cl _ :33 this quite easily. Assume the investor selis $160 worth of portfolio D short and buys g ER [6 THE ARBTTRAGE PilfClNG MOEDEL RAW-MA NEW N’i’iiOACi-i TO EXPLAINiNG A555? PRICES iiernaiively, since portfolio I) must lie on the plane described above, we could have by,in its expected return from the equation of the plane: ,‘ Te. a 7.75 + 5(0.6) + 3.75(0.6) e13 By the law of one price, two portfolios that have the same risk cannot sell at a different ‘ acted return. in this situation it would pay arbitrageurs to step in and buy portfolio E X '13 white selling an equal amount of portfolio D short. Buying portfolio E and financing it by ring 1) Short would guarantee a riskless profit with no investment and no risk. We can we. worth of portfolio E. The results are shown in the following table. 1 Initiai Cash ‘ " End of Period Flow 1 l _. Cash Flow in; be isostatic D +s100 ‘ ~s113n ~e.s was Portfolio E “M ' silk-0 .. [1.6 9.6. Arbitrage portfolio 0 2.0 i) 0 WW 3 The arbitrage portfolio involves zero investment, has no systematic risk (11,-; and 19,2), and - cams $2. Arbitrage would continue until portfolio E lies on the same plane as portfolio A, B, and C. We have established that alt investments and portfolios must be on a plane in expected mum, b“, 1952 space. If an investment were to iic above or beiow the plane, an opportunity '- would exist for rislcless arbitrage. The arbitrage would continue until all investments con- ' verged to a piano. The general equation of a plane in expected return, on, ba space is R;- Wlo +7»;be +K2bf2 This is the equilibrium modei produced by the APT when returns are generated by a two- index model. Notice that it; is the increase in expected return for a one-unit increase in b“. Thus in and h? are returns for hearing the risks associated with I. and 13, respectively. More insight can be gained into the meaning of the its by using Equation (16.3) to exam- ine a particular set of portfolios. Examine a portfolio with bi] and by; both equal to zero. The expected return on this portfolio equais AD. This is a Zero 195 portfolio, and we denote its return by RF. If the riskless asset is not available, RF is replaced with R3 the return on a zero Beta portfoiio. Most researchers in this area assume that the intercept is in fact RF. Substituting KP for to, and examining a portfolio with a h; of zero and a b” of one, we see that R1 2 R1 '"" RF where E1 is the return on a portfolio having a by] of one and a big of zero. In generai, lo 5 R; - R1: or it) is the expected excess return on a portfoiio only subject to risk of index j and having a unit measure of this risk. The analysis in this section can be generalized to the J index case Rf =ai+buli +b52I2 “l”""i'buIJ+€,' ' By analogous arguments it can be shown that ail securities and portfolios have expected returns described by the J—dimensionai hyperpiane 175; 2 Re "t kii-ln +1251? +"‘+ Mbu (16.4) _' with x0 = R; and i] m E,- " RE PART 3 MODELS OF EOUILIBRiUM N THE CN’iTAL MARKETS 366 A More Rigorous Proof of APT Once again we will derive APT assuming a two—index return-generating process. This def; ivation is sufficientiy rich to allow generalization to any arbitrary number of indices. Tim twouindex model we use is that presented in Equation (16.2). Taking the expected vaiue of Equation (16.2) and subtracting it from Equation (16.2), we have 2,. medium, —f])+bf,[12—f2)+e, (165) _ Now a sufficient condition for an APT proof to hold is that there are enough securities in the market so that a portfolio with the following characteristics can be formed: .v 2X, = 0 Ni=1 2&th = 0 int N E ion, a o ir—“l N EXf-efr—O 6:: The last condition is a requirement that residual risk be approximately zero.5 The first of these four equations states that this portfolio involves zero investment. The remaining equations imply that this portfoiio has no risk. This portfolio involves no investment and no risk; therefore, it must produce an expected return of zero. In other words, the titres equations plus the condition on residual risk just discussed imply that N 2X..an Now there is another more mathematical interpretation of these equations. The equation N 2X:in "—7 0 iml means that the vector of security proportions is orthogonal to the vector of bus. Similarly, the first equation means that the vector of security proportions is orthogonal to a vector of ones. We have just. shown, in the previous paragraph, that if the vector of portfolio proportions is orthogonal to 5'i‘he assumption of zero residual risk might seem bothersome. Original proofs of AE’T assumed an infinite nutri- hcr of securities and well—diversified arbitrage portfolios. Because with uncorrelated residuais each residual vari: ance enters with a weight equal to the square of the fraction of money placed in that security, for well-diversified portfoiios selected from an infinite or, in fact. a very large population of securities, residual risk will he very slow lo zero. A series of papers by Dybvig (i983), Grinblatt and 'iitrnan (1983, 1985). and ingersoll (1984) investigate how closely the Airl‘ holds for finite economies and economies where residual risks are not uncorrelated, A!"r continues to hold, although it does not necessarily hold exactly the same for all securities (there can be very small errors for many securities and there can be large pricing errors fora few securities). cHAPTER l6 THE nRBiTRfiGE l’RiClNG MODEL AWWA NEW APPROACH TO EXi’lNNlNG ASSET FRECES 367 a Vector of ones. a vector of bits, and a vector of has, this implies that the vector of secu- rity proportions is orthogonal to the vector of expected returns. But there is a well—known - moorem in iinear algebra that states that if the fact that a vector is orthogonal to N ~—~ i vec» tors implies it is orthOgnnai to the Nth vector, then the Nth vector can he expressed as a linear combination of the N w l vectors. In this case, the vector of expected returns can be expressed as a iinear combination of a vector of ones, a vector of 12515, and a vector of bias. Thus we can write the expected value for any security as a constant times 1, plus a second constant tithes b“, plus a third constant times £752 or E m 7m + 7L119:1 +7L213l2 This equation must hoid for all securities and ail portfolios. The As can be evaluated by following the procedure used in the previoussection of this chapter, namely, forming three - portfolios with the following characteristics . .l ‘ l. by] = O hand bpg = 0 2. lap, “we 1 u and lap m 3. by); = and bpz = 1 we find that w wTer‘rbullél "RF)+bi2(§2 ""er or for the general case Ki ” RF +bil(§i *RF)+"‘+bu(EJ "‘er Defining no as RF and n}- as R} - RF, we can write this equation as is; 7" 7L0 Hi'lbil + k219:2 i“ “‘"l‘ Ari-in The principal strength of the APT approach is that it is based on the no arbitrage condi— tion. Because the no arbitrage conditions should hold for any subset of securities, it is not necessary to identify all risky assets or a “market portfolio” to test the APT. It is reason- able to test it over a class of assets such as common stocks or even a smaller set such as the stocks making up the Standard 8r. Poor’s (385?) index or all stocks on the New York Stock Exchange. One has to be somewhat careful in that the correct APT model for a iarger class of securities can be different from {contain more influences than) an APT model appropriate for a smalier set of securities. Faiiure to find a model for a small set (type) of securities does not mean that a model does not exist across different types of securities However, it is appropriate to use the APT to describe relative prices for a set of securities of interest to the investigator rather than deal with the whole population of risky assets. In fact, it has been argued that many tests of the CAPM were realiy tests of a single or - muitiple—factor APT model. An important characteriStic of the APT theory is that it is extremely general. This gen- . erality is both a strength and a weakness. Although it allows us to describe equiiibrium in ' terms of any multi~index model, it gives us no evidence as to what might be an appropri- _ ate muiti—index model. Furthermore, APT teils us nothing about the size or the signs of the _ his. This makes interpretation of tests difficult. We’ll have more to say about this shortly. " ESTlMATING AND resume APT The proof of any economic theory is how weli it describes reaiity. Tests oi APT are par- ticularly difficult to formulate because all the theory specifies is a structure for asset 363 mm 3 MODELS (BF €GUIUERlUM as THE CAPll'AL mmth pricing: the economic or firm characteristics that should affect expected return are no specified. Let us review the structure of APT that will enter any test procedure. We can write the multifactor return- generating process as J . R,- = (2‘.- + 8) i=1 The APT modei that arises from this return—generating process can be written as J' I it, a RF + Esp,- (16.7) in"! it’s worth spending a little time discussing the meaning of the variables [7,311, and hi. Notice from Equation (166) that each security i has a unique sensitivity to each I}- but that any [Jr has a value that is the same for aii securities. Any [j affects more than one secu. rity (if it did not, it would have been compounded in the residuai term ei). These {is have generally been given the name factors in the APT literature. They are identicai to the influ. ences we called indexes in earlier chapters. The factors affect the returns on more than one security and are the sources of covariance between securities. The bps are unique to each security and represent an attribute of the security. This attribute may be simply the sensi- tivity of the security to a particuiar factor, or it can be a characteristic of the security such I as dividend yield. Finally, from Equation (16.7) we see that hj is the extra expected return required because of a security’s sensitivity to the jth attribute of the security. At this point the reader might note that Equation 86.6) looks suspiciously like the type of relationship we used in first- .' pass regression tests of the CAPM in Chapter 15, whereas Equation (16.?) beats a close resemblance to the type of equation used in second-pass tests. This intuition is correct. The problem is that, whereas for the CAPM the correct 5- is defined {e.g., the excess return on the market portfolio for the simpie CAPM), for the multifactor model and the APT, the set of he is not defined by the theory. In order to test the APT one must rest Equation (16.7), which means that one must have estimates of the bps. Most tests of APT use Equation {16.6} to estimate the bus. However, to estimate the has We must have definitions of the rel- evant [is The most general approach to this problem is to estimate simultaneously factors (1)5} and firm attributes (has) for Equation U63). Most of the early tests of the APT employed this methodology. It still continues to be widely used in the finance literature and in pram tice. We examine this type of simultaneous estimation technique shortiy. Before we do so, however, let us point out two alternative methods. One alternative method is to specify a set of attributes {firm characteristics) that might affect expected return. When using this method the has are directly specified. The bijs might inciude such characteristics as divided yield and the firm’s Beta with the market. -' Once the bus are specified, Equation (16.7) is used to estimate the M and thus the APT modei. The second alternative method is to specify the factors 13s in Equation 86.6) and then to estimate the security attributes bps and market prices of risk his. Two approaches have been used to specify the factors. One approach is to first hypothesize (we hope on the basis of economic theory) a set of macroeconomic influences that might affect return and then to use Equation (i6.6) to estimate the bus. These infiuences might include variables such as the rate of infiation and the rate of interest.6 eRecently BIRR has offered a commercial version of this research. A detailed description of their model can be found in Burmeister, Roli, and Ross (i994). cHAflER 16 THE ARBETMGE PiiiClNG MODEL APT—A NEW APPROACH TO EXi’ifilNiNG ASSET PRICES 369 A second approach is to specify a set of portfolios as factors that the researcher believes captures the relevant influences affecting security returns. As in the previous case, Equation (16.6) is mod to estimate the has With the return on the hypothesized portfolios used as the [is and Fags estimated via regression analysis. For either approach, Equation - {16.7} is then estimated to obtain the his and the associated APT model. If any method other than factor analysis is used to obtain the bps for testing APT, one is really conducting a joint test of the APT and the relevancy of the factors or characteristics that have been hypothesized as determining ecguiiibrium. Each of these generai approaches will now be discussed in more detail. simultaneous Determination of, Factors and Characteristics A complete specification of Equation (i6,6) would call for all factors Hi) and attributes (bi) to be defined, so that the covariance hetween any residual return (the e,-s not expiaincd by the equation) was zero. While it is not'possible to produce this exact resuit, there IS a body of statisricai methodology that is very wet] suited to approximating this resuit. These techniques are called factor analysis. We present a simpie example of a factor anaiytic soiution in Appendix A to provide the reader who has not worked with this technique some feel for what it accomplishes. Factor anaiysis determines a specific set of [Is and bps such that the covariance of resid- uai returns (retuérrns after the influence of these indexes has been removed) is as small as possible.7 In the terminology of factor analysis, the fjs are called factors and the lag-s are called factor loadings. A specific factor anaiysis is performed for a specific number of hypothesized factors. By repeating this process for alternative hypotheses about the num— ber of factors, a soiution for two factors, three factors, . . . , and j factors is obtained. One can stop when the probability that the next factor expiains a statistically significant portion of the covariance matrix drops beiow some loves—for example, 50%.8 Using this tech— nique, it is not possible to be sure that one has captured all reievant factors. At best, state— ments such as the foilowing can be made: “There is less than a 50% probability that another factor is needed." Whether one chooses to step extracting factors when there is a 50% chance that no more are needed, or a 10% chance, or some other levei is a matter of taste rather than mathematical rigor. Without a theory of how many factors should he pres- ent, the decision as to how many to extract from the data has to be made subiectively. Factor analysis produces estimates of the factor loadings (by) and the factors (1;). Recali that the factor loadings by are sensitivity measures and are like the Bis of the simple CAPM. At this point, a set of tests analogous to the first—pass regression tests discussed in Chapter 15 has been performed. The major difference is that one has not oniy identified the has but one has estimated how many factors (indices) there should be and has deter— mined the definition of each 1;. Each 1} is an index consisting of a (different) weighted aver- age ot‘ the securities on which the factor analysis is performed. 7Principal component anaiysis is somewhat analogous to factor analysis. Recali from Chapter 8 that principal component analysis extracts from the data a set of indexes that best explains the variance of the data. indexes are extracted in order of importance and as many indexes are extracted as the sunailer of the number of stocks or the number of observations. Factor analysis is coovariance rather than variance driven. For a specified number of indexes it finds the set of that many indexes that best explains the covariance in the original data. There are taken native ways of performing factor analysis. Most empiricai work in this area ates maximum likeiihood factor analysis, and the techniques developed by loreskog (i963, 1967, 197"!) are often used. ESet: lawlcy and Maxwell (1963) for a discussion of the test procedure described. The reader shouid be aware that these tests are based on the assumption of multivariate normality. This is the procedure appiicd by Roll and Ross (1980). 370 PART 3 MODELS OF EOUiLlBRiUM N THE CAPlTAL WRKETS. The next stop in testing the APT is to form a set of tests directly analogous to the sccon pass tests performed by Fama and MacBeth (1973) on the simple CAPM.9 By running a cross—sectional test, estimates of he can be computed for each time period and the average value of each h] and its variance over time computed. Roll and Ross (1980) were the first to perform this type of test. The mathematics of factor analysis allows this to be done mom easily than with regression techniques, but the results are analogous to these that would be obtained by using the generalized least squares regression procedure. However, there are some problems with the use of factor analysis of which the reader should be aware. First, we have the same error in variables problem that we had when testing the standard CAPM. The factor loading bps, like the Betas from the first-pass regression, are estimated with error. This means that significance tests of his are only asymptotically correct. There are three additional problems that are unique to factor anaiysis. First, there is no meaning [a the signs of the factors produced by factor analysis, so the signs on the has and on the his could be reversed. Second, the scaling of the has and the his is arbitrary. For example, all has could be doubled and the resultant his halved. Third, there is no guarantee that factors are produced in a particular order, so when analysis is performed on separate samples, the ' first factor from one sample may be the third from another sample. The procedure discussed above is that used by R013 and Ross (1980) in their classic _ study of APT. They applied factor analysis to 42 groups of 30 stocks using daily data for the time period Suly 3, 1962, to December 1972. The resuits of their first~pass test are rather striking. These tests show that, in over 38% of the groups, there was less than a 10% chance that a sixth factor had expianatory power and in over three-fourths of the groups there was a 50% chance that five factors were sufficient. While Roll and Ross try several ' different second—pass tests, their major results are that at least three factors are significant 3 in explaining equilibrium prices but that it is uniiitely that four are significant. On the sur— face it would appear that they find more factors significant than one would expect to find under the standard CAPM model or the zero Beta version of the CAPM. It is logical to question whether there is any way these results could be consistent with the CAPM or whether there seem to be additional factors at work in the market. Aithough we cannot answer definitely, the analysis of Cbo, Elton, and Gruber {1984) wouid seem to indicate that there are additional influences at work. They repeat the Roi] and Ross methodology for a later period and find more factors to be significant than do Roll and Ross. They then simniate a set of data using the zero Beta form of CAPM while enforcing the same means and variances on the returns for each stock that were present in the origi- nal data. In doing so, they allow the rate on the zero Beta portfolio and the Beta on each asset to change over time. When the R011 and Ross methodology is appiied to these data, ' the number of factors that are found to be significant is consistent with the zero Beta form of the CAPM. The fact that many more factors were found to be significant when actual returns were analyzed lends support to R01: and Ross‘s argument that additional factors beyond those embodied by the zero Beta form of the CAPM determine equiiibrium prices. Whiie this analysis would seem to suggest that more than one or two factors are important in determining both returns and equilibrium returns, there still remain some questions about the implementation of APT through the use of factor anaiysis.10 The usefulness of an APT model cannot be differentiated from the methodology used to estimate it. The theory may well be correct, but if it cannot be implemented or estimated °Altcrnate tests such as those advocated by Gibbons (1982) described in the previous chapter or those advocated by Bunneistcr ct al. (1988) described Eater in this chapter can be used instead of the second~pass test. min the last section of this chapter we discuss an alternative way in which the zero Beta form of the CAPM could be consistent with the Rail and Ross results. Gwyn-1R 16 THE ARBlTRAGE PRICENG MODELAPTM NEW APPROACH TO EXPINNNG ASSET PRiCES 37 'I if, a meaningful sense, then, while it remains useful as a way of thinking about the world, “cannot be used as part of the investment process. A test of the APT is a joint test of the “gory and the methodology used to implement the theory. Factor analysis is the principal methodology used to estimate simultaneously the factors mat affect equilibrium return and the sensitivity of firms to these factors. One problem with employing this methodology to estimate factors is that the mathematics of factors analysts is so complex that oniy a limited number of securities can be analyzed at one tune. A set of fac~ [m5 and factor loadings are extracted that can best describe the behavror of a small sample of risky assets rather than all risky assets. Roll and Ross used groups of 30 assets. The reader may well ask, “So what? If the arbitrage pricing theory is correct, why don’t we obtain the true fac- tors whether we use 30 securities or 2600 securities?” Dhrymes, Friend, and Gnltekin {1984) present evidence that the number of factors that appear significant is an increasing function of the size of the group analyzed. In their samples, the number of significant factors increased from 3 for groups of 15 securities to 7 for groups of 60 securities, the largest groups studied. The authors suggest that dividing the sampleinto subgroups may ignore important sources of covariance between the securities in different groups and, further, that the factors identified within any subgroup may not be the same as the factor identified in a second subgroup. While the necessity of estimating the APT for small groups provides some major prob lems with respect to the applicability of the result, it does provide a unique opportunity for testing the theory and methodology jointly. According to the theory, 4" J R, = R2 ~2— Zea-ti i=1 Now if the theory is correct and if through factor anaiysis we have identified the correct fac- tors in the return—generating process, and thus the lags, then the value of the market price of all factor his and the intercept should be the same for each group. Testing this is not as easy as it may seem at first. Remember that the sign of the has and )Ljs are not uniquely deter— mined, nor is the order in which factors appear in different groups uniquely determined. Methodology does exist for evaiuating whether the intercept is constant across groups and whether the factor prices estimated are the same across groups. The methodology and test results have been described very weli in an article by Brown and Weinstein (1983). They are able to test {1) whether the intercept term is the same for all groups, (2) whether the factor prices are the same for ail groups given that the intercept is constrained to be the same, and (3) whether both the intercept and factor prices are the same for all groups, a joint hypothesis. Unfortunately their results are ambiguous. Although they cast some doubt on the use of the maximum iilteiihood factor analysis to explain equilibrium return suc- cessfully, as Brown and Weinstein recognize, their “results cannot be viewed as com- pelling evidence against the APT.” Dhryrnes, Friend, and Gultekin (1984) in another set of tests find that, depending on the method of grouping stocks employed, the intercept term may be significantly different or not significantly different across groups. ' Other tests of the AFT have failed to demonstrate its clear superiority pvcr other mod« els although results are mixed. 33hrymes, Friend, and Guiteltin find that a multifactor model of the APT has better explanatory power than a one—factor model. This tends to support "Chen (1981) has described a procedure that allows AFT to be estimated and tested across large numbers of Securities. However, his procedure, which involves forming a smail number of portfolios of securities based on an initiai factor solution for use in further tests, has been questioned by Din-ymcs, Friend. and Gultekin (i984). The resolution of the adequacy of this procedure. and in particular the value of estimates for some securities ver~ sus the loss of information involved in his portfolio aggregation technique, will have to await further study. PART 3 MODELS OF EOUliJElRiUM 3N THE WJTAL mm five risk factors employed by Roii and Ross are significantly different from zero. Other tests of the Roll and Ross type APT have produced equally ambiguons results. i3 ' example, one test of the APT that would give us great confidence is if a stock‘s resiou risk was not priced when added as another factor in the equilibrium pricing aquatic Recall that the bus are supposed to capture the impact of all systematic components of its Any other attribute of a security, and in particular its residual risk, should be unique to each; security and therefore diversifiable. Roll and Ross test for the impact of residual risk and find almost no evidence that it is priced. Dhrymes, Friend, and Gultekln also test and find' that both a stock’s own standard deviation and schness generally yield insignificant coef-. ficients. However, they find that these two influences are significant at least as frequenti as the factors suggested by Roll and Ross.12 The literature has contained a number of tests of empiricaily estimated factor models and APT models. Although a detailed descriptiOn of the empirical methodology goes beyond the level of analysis, we wish to present here two articles that are Well worth mentioning. Lehmann and Modest (1938) implemented the idea of forming portfolios of assets that mimic factor realizations {returns}. By forming a portfolio that has minimum residual risk for each factor, they can then use this set of portfolios as independent variables to estimate the sensitivities of each of a large number of securities to each influence (factor). Each portfolio is identified by finding a set of weights summing to one across stocks, so that the = portfolio has minimum residual risk and a sensitivity of zero to all factors except the one under study. Lehmann and Modest are able to explain certain phenomena not explained by the standard CA‘PM. In Chapter 17 on efficient markets we show that the standard CAPM ' does not satisfactorily account for extra returns associated with high dividends. the stock‘s '_ own variance, smali size (low capitalization), and the January effect. Lehmann and Modest - show that a multi~index APT can explain away discrepancies due to dividend yield and " own variance, but that the extra return on small firms and in January are only partialiy accounted for by the model. Nevertheless, the ability of the model to account for some influences not cxpiained by the CAPM is in fact support for the modei as an alternative to the simple CAPM. Lest we get too excited, recall that an after—tax CAPM tested by - Litzenberger and Ramaswamy {1979) was also successful in accounting for returns vary- " ing with dividend yield. Connor and Korajczyk (1986) provide a test of APT using the asymmetric principle components technique proposed iay Chamberlain and Rothschild {1981). They find that with five factors, they can explain the extra return on small firms and in January better than the CAPM based on a vaiue—weighted index. The ability of an APT model employing a small number of factors to account for return patterns unexpiained by the CAPM strongly suggests that the APT is a useful model for .' explaining relative prices. All of the tests just described are joint tests of the APT and a particular statisticai ‘ methodoiogy used to identify both the bus and [Is of the factor rnodel. The results of this l2One other type of test has been performed on Roll and Ross type of multifactor modeis, In applying the slow dnrd CAPM and zero Beta CAPM, certain anomalies have been noted. For exampie. small (low capitalization) firms tend consistently to produce returns in excess of those we would expect based on CAPM. This anomaly is either due to a market inelliciency or a deficiency in the CAPM as a model of cquiilbrlum returns If a Roll and _ Ross type of multifactor of equilibrium better explained anomalies. such as the smallwfirm effect, one would haw added faith in Such modeis. Reinganum (198i) has investigated this issue and finds that a Roll and Ross mulliv factor model couid not explain the size anomaly any better than the standard CAidvl. : {£55 [a if: THE ARBITRAGE FRlClNG MODEL APT—A NEW APPROACH TO HPWNING ASSET PRECES § 3 m arch are inconclusive. There is fairly strong indication that more than two factors affect returns and that more than two factors are priced. Statistical methodology has been devel- ped and continues to be developed that allows us to better‘definc the factors and to better form portfolios that mimic them. However, research is just beginning to expiore the sta- ' bility of the factor structure over time. In Japan, APT has been tested and shows a clear superiority over the CAPM in select« - mg securities as weil as in explaining past returns. For example, Elton and Gruber (1982, 1933) find that a five~factor APT modei does a better joh of explaining and predicting expected returns than does a single—factor or CAPM mode}. In particular, in the Japanese ' stock market the CAPM model appears to break down. In Japan, unlike other markets, smali stocks have smailer Betas than‘large stocks. This shouid imply a lower expected return given the CAPM and yet smali stocks have significantiy higher excess returns. This happens when smail is defined as anything but the largest 160 stocks on the Tokyo Stock Exchange. These problems are not nearly as great when a inultifactor model is used. . Furthermore, a maitifactor model does a much better job of allowing mimicking portfolios : to be constructed (both as index funds and hedge portfolios for futures and option trading) than does a single-index model. The APT model is almost universally used by industry as - a replacement for the CAPM model in Japan. An Alternative Approach to Testing the APT '. g If we could specify a priori either the factors that affected stocit returns or the character- istics of stocks that affected returns, we would then have a much easier estimation prob~ lent to soive. A debate exists among academics and practitioners about whether part of the modei should he prespecitied on the basis of theory or whether all of the parameters should be determined einpiricaliy. This type of debate has gone on since the dawn of modern science. The issue is discussed by Roli and Ross (1980). They state that “we do consider the basic underlying causes of the generating process of returns to be a poten— ' tially important area of research but we think it is an area that can be investigated sepa» rately from testing asset pricing theories." The probiem is that, without a theory, the empirical tools one uses are a lot weaker and the results of tests harder to interpret. For exampie, in the APT we have no idea of what the size or even the sign of factor prices should be. All we can say is that we expect some of them to be statistically different from zero. On the other hand, in the Sharpe-whatnot CA?M the price of Beta Was supposed to be R”, M RF, a quantity that we expected to be positive and about which we have some rough idea of magnitude. The controversy we are discussing would be easy to resolve if we had a theory of the appropriate factors or characteristics that determine security returns. Someday we hope to have one. in the absence of such a theory ail we can do is examine three attempts to pre- specify one set of variabies in the multifactor model. One attempt hypothesized a set of firm characteristics, another hypothesized a set of macroeconomic indexes, and the third specifies a set of portfolios as the indexes. Specifying Attributes of Securities in the preceding section of this chapter we examined the use of maximum likelihood fac- tor analysis to determine simultaneousiy the characteristics that affect return and the extra return required because of a security‘s sensitivity to these characteristics. if a set of characteristics that affects return couid be specified a priori, then the market price of these ' CharacteristiCs over any period of time could be measured fairly easily. 374 PART 3 moons or EQUFUBRiUM uv nae want MARK The estimating equation would be the form E = K0 4'1le] +1.21%? +"'+?\.Jbu for the case of J characteristics. In this equation the bus would be the value each Chara teristic took on, and the his the average extra return required because of these characten’ ties. The values of the his would be estimated via regression analysis. This procedure is directly analogous to a second-pass test of the CAPM. in fact, we haVe already examine : two models that could be viewed as this type of test. The first was the model tested by Fama and MacBeth (1973) and reviewed in Chapter i5; although they viewed the mode[ as a test of the CAPM, it could be viewed as a test of APT. The model they tested was ‘ El 3 7to + M3: "l" 125i? “Laser The firm characteristics examined were the Beta for each firm, the Beta for each stock squared, and the residual risk of each stock. These tests clearly show that, at least with: respect to the hypothesized set of characteristics, a multifactor model did not outperform the zero Beta form of the CAPM. None of the added characteristics were price I Remember that tests of this type are ajoint test of the A91“ in general and the specific char acteristics that were hypothesized as explaining equilibrium returns. Fania and MacBeth tested characteristics that on the basis of economic theory should not explain equilibrium returns and concluded that they did not. We examined a second model in Chapter 15 that hypothesized an additionai firm char - acteristic as affecting equilibrium return. Recall that Litzenberger and Ramaswamy (1979) included dividend yield as an added variable and found its impact was statistically signi icant. This should encourage the pursuit of models containing more characteristics. One such model has been constructed and tested by Sharpe (E982). He starts with the: hypothesis that equilibrium returns should be affected by the following characteristics: a stock’s Beta with the S&P index, its dividend yield, the size of the firm (market value of equity), its Beta with long—term bonds, its past value of Alpha (the intercept of the regre sion of past excess return against excess returns on the S&P index), and eight~sector mem~ bership variables. Sharpe does not attempt an eiaborate economic rationale for these variables but rather states that he has selected them more or less “ex cathedra." We would expect both Beta and dividend yield to be related positively to expected returns based on. the theory discussed in Chapters 13, 14, and 15. Size may well be, at least in part, a proxy for liquidity. If so, size should enter the model with a negative sign. If sensitivity to inter-- est rates is an important variable, we would expect bond Beta to play a role in determin- ing equilibrium returns. If the past value of Alpha proves significant, it would be evidence of autocorrelation of the residuals from the CAPM. This might indicate that there are some added variables explaining cross—sectional returns that were not captured in the model. The use of sector membership as an additional set of variables implies that membership in a particular sector of the economy has an important effect on equilibrium return. The results of appiying this model to 219? stocks on a monthly basis for all months. between 2932 and i979 are summarized in Table 16.1, which reports the average coeffi: cients (on an annualized basis) over the entire period and the percent of months in which the coefficients were significantly different from zero at the 5% level. Note that for those variables where we had clear expectations about the sign of the relationship and return, our expectations are borne out. Furthermore, note that while on the basis of chance we would expect any firm characteristic to be significant about 5% of the time, each characteristic: was significant at much higher percentage of the time. Another way to judge the importance of including more than one characteristic in th __ description of equilibrium is by examining the explanatory power (coefficient of determination) . “Am-Era 16 THE ARBITRAGE PRJCING MODEL APTWA NEW APPROACH TO EXPWNiNG ASSET PRECES able 16.1 ss~sectional Data on Sharpe’s Moi 'f Percent of Months in Which 7 Associated It Was Significantly Annualized Value minute of Associated it Different from Zero Beta 5.36 58.3 - yield 0.24 39.5 Bond Beta —0. 12 28.2 - alpha *2.00 43.5 'Sector Membership I , Basic industries ‘ 1.65 32.5 Capital goods 0.36 18.7 Construction 4.; .59 15.3 Consumer goods #0,.”18 39.3 Energy 6.28 36.9 Finance “4.48 16.3 Transportation ~05? 43,9 Utilities m2.62 35 .0 W an . . . . . of the model as more characteristics are employed. The average coefficient of determination for _' monthly data when Beta is used as the only characteristic to explain cross—sectional returns is 0.037. This might seem low relative to the results reported in Chapter 15, but recall that monthly data are being used and portfolio grouping is not being done. This is, in fact, consistent with .- elite: studies employing similar research designs. When the security characteristics of yield, _ ize, bond Beta, and Alpha are added, the coefficient of determination adjusting for added vali— . ables more than doubles to 0.079. When all the characteristics in Tabie 16.1 are used, it goes up to 0.104. The use of firm characteristics in addition to Beta has increased the explanatory _ power of the model. In addition, these factors seem to be significant a considerably higher per» centage of the time than chance alone would explain. Sharpe seems to have identified some additional characteristics, beyond a stock‘s Beta with a proxy for the market portfolio, that are useful for explaining cross-sectional returns _ over time. He recognizes that his model is rather ad hoc in nature, but it is an indication _' that increased reseflch into significant economic characteristics of a stock should allow us - to build better models of equilibrium. A second model that is widely used in industry and that specifies a set of firm charac— tenstics is that employed by Banal.” This model uses nine firm characteristics in place of 'the five characteristics used by Sharpe. These are volatility, momentum, size, liquidity, 'growtii, value, earnings volatility, financial leverage, and industry membership. __Specifying the Influences Affecting the Return-Generating Process Another alternative to the joint determination of factor loadings and factors diScussed in Ihc earlier section of this chapter is the specification {one hopes on the basis of economic :Lhfiory) of the set of influences or indexos (II-s) that should enter the return-generating process. n . . . . . Sec Gnnold and Kahn (2994) for a description of this model. As explained in this article, the model actually Wilkes use of a combination of firm specific characteristics and macroeconomic variables. 37$ PART 3 MODELS C‘F EGUJUEREUM W THE WWAL Mini u WEI? “5 THE ARBH‘RAGE PRICiNG MGDEL APTwaA NEW APPROACH TO EXPLAINiNG ASSET PRICES 377 Chen, Roll, and R055 (1986) have hypothesized and tested a set of economic variabl" They reason that return on stocks should be affected by any influence that affects eith future cash flows from holding a security or the veins of these cash flows to the invest (cg, changes in the appropriate discount rate on future cash flows). Chen, Roll, and R0 construct sets of alternative measures of unanticipated changes in the following jnfi ences: grime premium as measured by the return on tongmterm government bonds minus the onemonth Treasury bill rate one month ahead. ,1 Deflation as measured by expected inflation at the beginning of the month minus actual inflation during the month. [4 5 Change in expected sales. a The market return not captured by the first four variables. 1. Inflation. Inflation impacts both die level of the discount rate and the size of the rum cash flows. Tfie fifth variable is a proxy for any unobserved general influences. As explained in the appendix, it is estimated by taking the residuals from a regression of a diversified portfo- r e authors use the S&P composite index) against the first four observable variables described earlier. The regression the authors found was 2. The term structure of interest rates. Differences betWeen the rate on bonds with a gag maturity and a short maturity affect the value of payments far in the future relative in near-term payments. ' ‘6 C3 :1- 3. Risk premia. Differences betWeen the return on safe bonds (AAA) and more risky , RM — to 2 0.00224 "1.330111.h+'0.55812 + 2.28613 a 0.93514 bonds (BAA) are used to measure the market’s reaction to risk. - . . . {0.6‘t9) (4.94) (4.96) {1.997} (—2.27) 4. Industrial production. Changes in industrial production affect the opportunities facing ‘ " investors and the real value of cash flows. - The first four factors account for about 25% (R2 = .24) of the variation in the return on the 885? composite index and each of the four coefficients is significant. .- When the sensitivities (by) are estimated for each firm, more than twouthirds of the sen- sitivities are statistically different from zero at the 5% level, and the five variables typically 3 account for 30% to 50% of the variation of returns of individual firms. In general, 19,-] appears with a significant negative coefficient, whereas by; and bg appear with significant positive coefficients. The remaining two variables have a more ambiguous impact on stock . returns. The prices (Ag) of each of the five sensitivities implied by the model are all positive and - all statistically significantly different from zero. The average value of the its using monthly _ returns is contained in the following table: Chen, Roll, and Ross then examined these measures or indexes 1. To see if they were correlated with the set of indexes extracted by the factor analysis used by Roll and Ross as described in a previous section of this chapter. 2. To see if they explained equilibrium returns. When they examine the relationship between the macroeconomic variables and the fac- tors (indexes) over the period to which the factors were formed (fit), they find a strong rela- tionship. Furthermore, when the relationship is tested over a hold—out period (a period following the lit period). the relationship continues to be strong. There appears to be a Sig- nificant relationship between the hypothesized macroeconomic variables and the statisti cally identified systematic factors in stock market returns. it Statlsttc Mean A Value The second set of tests involves investigating whether returns are related to the sensi tivity of a stock to their macroeconomic variables. The procedure is analogous to the two it! 04:; 427 step procedure used by Fama and MacBeth (and discussed in the previous chapter) to it; 1.00 4.76 investigate the CA?M. In the first stage, timeuseries regressions are run for each of a set-1e A3 0.04 1.83 of portfolios to estimate each portfolio’s sensitivity to each macroeconomic variable [th is 0-15 2-21 has of Equation {16.6)}. Then the market price of risk [the his of Equation (16.7)} is esti is 0'51 3-21 mated by running a cross-Sectional regression each month and looking at the average 0 the market price in each month. Chen, Roll, and Ross find that the macrovariables are sig . nil’icant explanatory influences on pricing. Furthermore, when the Beta of each portfoli with the market was introdoced as an additional variable along with the sensitivity of sec portfolio to the macroeconomic variables, it did not show up as significant in the second . stage (crossoectional) regression. Chen, Roll, and Ross recognize that they cannot claim to have found the (correct) state vari ables for asset pricing. However. they certainly have matte an important start in that direction Their work is continued in a series of papers by Burmeister and Mcfilroy. Burrneiste and McElroy have integrated tests of the {actor models, CAPM, and APT. It is worthwhil reviewing two of their tests. The first test is constructed using the mold-index mode described in Chapter 8. More specifically, returns are assumed to be generated by th lowing five indexes {see Chapter 8). Additional interesting questions can be addressed with the author’s methodology. The -'first is whether the APT form of the return equation [Equation (A-fi) in the appendix] explains returns significantly better than the return-generating process (five‘index model). 3 The difference between these two models is that in estimating the APT form, the expected return on each stock is constrained to take on a value that does not allow arbitrage oppor— tunities between securities. This difference is analogous to earlier tests of the CAPM - where the intercept was constrained to be R p(1 — 8;), the rte-arbitrage condition from the market model. Imposing this constraint cannot increase the explanatory power of the model; it can only decrease it. If the APT is correct, however, the decrease should be small. The fact that the decrease is not statistically significant shows that we can’t reject the APT version of the return-generating process and is, at least, Weak evidence in support of it. Another ted: the authors perform is to restrict the coefficients to see if the market index ' alone explains a statistically different amount than the five-index model. The additional 3 explanation of the four variables is statistically significant even when the APT form of the return-generating process is used. (9 Ht 9.. I; = Default risk as measured by the return on long-term government bonds minus the' return on iong~term corporate bonds plus onewhalf of 1 percent. 378 PART 3 MODELS OF EOUiUBRlUM 3N THE CAPITAL MKES 91mm 16 me assurance FRiCtNG MODEL APT-A new APPROACH r0 axrwwwo ASSET PRICES 379 In a later paper, Burmeister and McElroy (1988) continue their attempt to differentiate between three models: the return-generating model {the factor model), APT, and CAP This study differs from their previous work in two ways.’1‘hey modify their definition 0 the observable factors, but more important, they assume there are three unobservable f3 " tors rather than one. They use three portfolios to represent these unobservable factors: the return on the S&P 500 stock index, the return on 20uyear corporate bonds, and the return on 20—year government bonds. - Burmeister and McEiroy conclude that at the 1% significance levei the CAPM mode can be rejected in favor of their APT model, Furthermore, the APT restrictions cannot be rejected at any reasonable significance levei in favor of the more general factor modei. This work represents the strongest evidence so far in favor of the APT model as a useful expia nation of expected return. ' whether one describes these as measures of macroeconomic variables or portfoiios is largely a matter of taste. The unique aspect of this model is in the formulation of the apayable-s representing size and book to market ratios. In Sharpe’s modei (described badger), size enters as a firm characteristic or a bi}. Size is measured in doliars (actuaily [the natural logarithm of doilars) and a A is associated with it via cross—sectional regres“ sion. What Fame and French have done is to convert the size component from a direct measure to a return concept by constructing a portfolio to capture this ini‘luonce. The by emaciated with size is not the iog of size for any company i, but rather is the sensitivity of that company to the return on the size portfolio. Because sizr: is measured by the return on a portfolio, it now enters the return-generating proaess as weil as the pricing Equation. This allows Fama and French to investigate both the time—series and cross-sec- tional properties of size.‘6 ’ ' Fama and French test the model described above in a number of time-series tests. The .' cross-sectionai implications are tested by examining whether the intercepts of the time series of excess returns indeed equals zero'as APT would suggest.” They find that, in fact, the intercepts are zero and that this portfolio model is successful in expiaining expected _ stock returns. More specifically, they conclude that “at a minimum, our results show that five factors do a good job explaining a) common variations in bond and stock returns and - b) the cross«section of average returns." Specifying a Set of Portfolios Affecting the Return-Generating Process Another alternative is to specify a set of portfolios (5-5) (which may or may not include the market portfolio) that a priori are thought to capture the influences affecting security returns. These portfolios are seiected on the basis of a belief about the types of securities - and/or economic influences that affect security returns.” . 7 An example of this type of approach is that used by Patna and French (1993) to con struct a modci to expiain returns and expected returns on both stocics and bonds. In addition to using the returns on a market portfolio of stocks, they use the returns on ' other portfolios to represent the Ijs in the retumwgenerating process. These portfolio " HIE -; APT AND chem - Before continuing our examination of APT models, we should discuss the fact that the APT model and, in fact, the existence of a multifactor model, including one where more than one factor is priced, is not necessariiy inconsistent with the Sharpe—Lintner—Mossin form or one of the other forms of the CAPM. The simpiest case in which an APT model is consistent with the simple form of the CAPM is the case where the mum—generating function is of the form RI! xiii +fiiR I" 1. The difference in return on a portfolio of small stocks and a portfolio of large stocks 1 (smali minus iarge). 2. The difference in return between a portfolio of high book to market stocks and a port— _ folio of low book to market stocks (high minus low). + 9: If returns are generated by a singledndex model, the single index is the return on the market portfoiio, and a riskless rate exists, then the methodology at the beginning of the chapter can be used to show that 3. The difference between the monthly long—term government bond return and the one month Treasury hill return. 4. The difference in the monthly return on a portfolio of iong»terrn corporate bonds and ' a portfolio of longwterm government bonds. 2 RF Milli", —RF) Note that all variables are either the return on portfoiios of assets or the difference in the return of two portfolios of assets.ls The latter can be considered a portfolio with a _5 set of stocks sold short. Cleariy this model has elements in common with the models that ._ have been presented eariier in this chapter. We saw that Chen, Roll, and Ross and . Burmeister and McElroy use bond return variables simiiar to those used in this model - if the return—generating function is more complex than this, does it imply that the sins» pie CAPM cannot hold? The answer is no. Recall that the simple CAPM does not assume that the market is the oniy source of covariance between returns. Let us assume that the return-generating function is of the muitbindex type R,- = a,- + tit-111+ 19,212 +e,- (E63) The indexes can be industry indexes, sector indexes, or indexes of broad economic influ— ences such as the rate of inflation. All we assume is that the set of indexes used captures all the sources of covariance between securities {e.g., E(e,eJ-} m 0]. “We shouid point out that this is fundamentaliy different from the approach of factor replicating portfolios that _ has been discussed by ithann and Modest (1988) and Huhcrman, Kandel, and Stambaugh (198‘?) among oth era. In these approaches, either factor analysis is used to extract factors or macroeconomic variables are hypoth- ' esizecl as important and then a mathematical programming probiem is solved to find portfolios that mimic the underlying factors. “Biron. Gmber. Das, and Hlavka (1993). Blake, Eilon, and Gruber (1993), and Elton, Gruber, and Blake (1994i _ investigate alternative return—generating processes. In the latter paper, they develop an APT mode] where 5011“? or ail of the indexes represent portfoiios of assets. This work is distrusted more fully in Chapter 25, “Evaluation of Portfolio Performance." I("flinch of these approaches (measuring the by; directly from firm size or estimating it from a regression on a portfolio) is better awaits further empirical investigation. "This is basically a multl-variabie form of the Black, Jensen. and Scholcs (1972) procedure discussed in Chapter {5. 330 PART 3 MODELS OF" EGUILIBRJUM iN THE CAP“. THE MEFRAGE flilCiNC-i MODEL/tan NEW APPROACH TO EXFEAlNiNG ASSET PRICES 38'! on on the uses of multi—index models and APT folloWS. Although there are many of adding this section, most of which are discussed later, perhaps the key reason .3; teaching APT, so many of our students have remarked that it seems more com a}, the CAPM and asked why bother with it. "i he Aiyi“ equilibrium model for this moitifactor returnugenerating process w; less asset is fit 7“ RF ‘l’ bail "3‘ bola Index Models, APT, and Portfolio Management of multi—indcx modeis and multi~index equiiibriuin models (AFT models) in the in of securities and the management and evaluation of portfoiios is growing rapidiy. bwkel-age firms, financial institutions, and financial consulting firms have devei— gr awn amid-index inodeis to aid indhe investment process. These models have 'mcseasingly popular because they'allow risk to he more tightly controiled and they investor to protect against specific types of risit to which he or she is particoiariy this or to make specific bets on certain'ty‘pes of risk. 13 section we wili discuss the use (if APT and rnuiti—indcx models to aid in pas— nagement, active management, and portfolio evaluation. Before we do so, we View mold—index models and APT briefly and present a simple example of an del that we will use to illustrate some of the phenomena we diSCuss in this each A} is given by the CAPM or 7H = fin (Em ‘ RF) 7&2 3 i512 (Em "" RF) Sobstituting into Equation (16.9) yields 1% m RF +bflBMU-5m - RF) 'i" brziz’mifim ’ RF) R; a RF +(bni5t1 'i' braBazXEm “‘ RF) Defining B,- as (bush; + bats”) results in the expected return of R:- bcing price CAPM. 35,, a RF + syn”, m RF) w'or Muiti~lndex Modeis and APT The A9? solution with multiple factors appropriately priced is fully consistent, Shamchintner—Mossin form of the CAPM. We wish to stress this point. EmpEOying the Roll and Ross procedure and find more than one A; is significantly different from zero is not sofficient proof to CAPM. If the his are not significantly different from Skim," ~ R;), the empiri could be fully consistent with the Sharpe—Lintneerossin form of the CAPM. I fectly pas ' ' ' ' ' but that the CAPM holds. While we have demonstrated this with the simple CAPM, it shooid be appar reader that other values of he can exist that are fully consistent with the root nonstandard forms of the CAPM reviewed in Chapter 14. specth return on security i. ‘ that Equation (16.1a) leads to a description of expected returns given by 111 this chapter we presented a rcturn~gcncrating process that expressed the return security as a linear fonction of a series of indexes: R,- =a,.+b,.,rl +b,212+b,313+---+e, (16.121) 'radjusted to have a mean equal to zero. Since the indexes and residuals have a zero, taking the expected value of both sides of Equation (16.1) resoits in m H E at S no ,_. {D H (a n E- O :3 G) s CL (5 M ("D X "D _. o) :— :5 m r, m“ (9 n 0 g m :3 O m (T m ,_,. 5 rs r: '.:l U: a n a: :1 R,- =a,- RECAPITULATEON Rf - RF = be: + labs + has + - -- (16.19) The APT theory remains the newest and most promising explanation of relative The theory promises to supply us with a more complete description of returns CAPM. Recent work, some of which employs a set of macro variables an which employs a set of portfolios, is qnite encouraging. The fact that a numb ics have found a set of macro variables and portfoiios that impact average rot are not oniy priced but are priced differently than the CAPM wouid imply practical and theoretical significance. One word of caution is in order. It is pos__ these additional influences are priced not because the APT is the corrcc expected returns, but because we have not correctiy identified the market in? ing our model. The residual market plus the other variable employed in the in: together simply serve as a proxy for the (true but unobserved) market in the marl gested in Chapter 15. Even if this is correct, the use of these mum—index mod! I a practical level, a better explanation for returns than any of the market proxifi *3 been proposed to date. represent the sensitivity of a security’s retorn to index j and is a measure of the in the security uncier study its represent the reward for hearing these risks r15 . hitting Equations (16.1) and (16.20) by recognizing that a,- m R", R! =RF+ltbu+bii2+7tsbia+“ (16‘11) +5311!l «thigh +bt-3I3 +~-+e,- . Severai ways of identifying the Is in Equation {16.i) and the bps and his in 116.61 i). However, for iliustrating the use of these types of modeis, it helps to deal 89.1fm model. slime that we have identified foot infiuences in the totem-generating model 6.1) and that ,pccted change in inflation, denoted by I; 332 PART 3 Meiosis OF EOUIUBREUM in: THE 019nm im- 12 == unexpected change in aggregate sales, denoted by [5 13 = unexpected change in oil prices, denoted by 19 14 = the return in the S&P index constructed to be orthogonai to the other influen denoted by 1M Furthermore, assume that oil risk is not priced (kg m 0}. Equation (i6.t0} becomes Er “ RF 3 115:1 +A'Sbi5 + yLMitre while Equation (16.11) becomes R: ’ RF = Rib” +13% + lerM + bill! "‘"brsls “9:010 “l” biMIM + er Recali that all Is have an expected value of zero.18 The set of its on these factors consistent with the results reported by Burmeister, R911 and Ross are K xm W432 is m 1.49 While the sensitivities (1)) values for the S&P index were bsaPr m m 0-37 ems = 1.71 hmpo = 6.00 bs&}= M = 1.00 The parameterization of the model allows us to recognize the importance of any fact in determining the expected excess return on the S&P index. To do so, simpiy multiply th 17 associated with a factor times the associated price of risk 0t}. Contribution to 8&1? Factor I) it Expected Excess Return (9’0) Inflation ~03? w“4.32 1.59 Sales growth L71 1.49 2.54 Oil prices 0.00 0.00 0.00 Market $.00 3.96 3.96 Expected excess return for S&P index 8.09 This table shows that the expected excess return (return above the riskless rate} fort 88;? index is 8.09%. Sales growth contributes 2.54% to the expected return for the S& in other words, sensitivity to sales growth accounts for 2.54 + 8.09 or 31.4% of the total expected excess return. J The same type of analysis can he used to examine the importance of the sources 0 risk for the expected excess return on any security or portfolio. For example, for a po “Who model we describe here and the values We present represent a simplified version of the model and par eters described in Burmeistcr. Roll, and Ross (1994). Their mode: contains additional influences to those cit- and does not contain an oil index. We wanted 10 include an unpriccd index to show the role of unpriced indexes in portfoiio management. The Salomon Brothers risk index model. discussed in Chapter 8 on Multidndet' Models, also does not include an oil index for us stocks though they find this index is an important influeflfé in Japan, the United Kingdom, Germany, and France. TER :5 THE ARBl'TRAGE PRICING MODEL APT—A NEW/APPROACH TO EXFWNFNG ASSET FRiCES .110 Of growth stocks the bs, As, and contribution to expected excess return are shown :er:19 ’ Portfolio Expected Excess m, b A Return (9%) inflation MSG W432 2gp, files growth 0'1} prices m l .00 0.00 6,00 Market 1.30 3.96 5.15 Expected excess return for growth stock portfolio. 11.41 W Notice that the expected excess return for the growth stool: portfolio (1}..41) is higher than it was for the 88:? index (8.09). Thisli's not surpdsing because the growth stock port- folio has more risk, with respect to each index, than the S&P portfolio?“ individual influences (indexes) have a different absoiute and relative contribution to the expected excess return on a growth stock portfolio than they have on the 8&1J index. For example, the contribution of sales growth to expected excess return is now 4.10%. Sales growth accounts for 35.9% of the excess return on the growth stock portfolio. It is not sur— prising that growth stocks are more sensitive to sales growth than the typical stocit. What might be surprising, though it is generaliy true, is that growth stocks are more sensitive to aii important indexes. So although the increase in sensitivity to sales growth causes the largest increaSe in expected excess returns, changes in all influences lead to greater excess return. Let’s now turn to the use of this model for investment and portfolio management. Portfolio managers can be divided into passive and active managers. Passive managers beiieve that mispriced securities can’t be identified and thus try to hold a portfolio that mimics some set of stocks. The most common way passive management is practiced is to hold a portfolio of stocks that closely tracks a selected index. Active management involves making bets about some securities or set of securities in the sense of designing a portfolio hosed on a belief that one or more securities are mispriced. Passive Management The maid-index model can play a major role in improving passive management. it can be used to do a better 30b of tracking an index or to design a passive portfolio that is appro— priate for a particular client. . : The simplest use of a mold-index model is to create a portfolio of stocks that closely tracks an index. An obvious way to construct an index fund is to hold stocks in the same proportion they represent of the index. However, many index funds do not simply hold each stock in an index in the propeltion the stock represents of the index, but rather attempt ' i9.“illhough estimating the cost of equity capital falls beyond the scope of this book, the preceding analysis leads . naturally to estimates of cost of capital. For example, the cost of capital of any stock or portfolio can be found by adding the riskless rate to the estimate of excess return from the APT model. For growth stocks this would be RF + l1.4l. For a detailed explanation of using APT to determine cost of equity capital, see Elton. Gusher, and Mei (1994). 2“Note that all D values. except for sensitivity to inflation, are larger for the growth stock portfolio than for the 3&1) portfolio. Though the b value for inflation is smaller for the growth stock portfolio, this portfolio is still toss desirable with respect to inflation sensitivity because (unlike other As) the price of inflation sensilivity (it!) is negative. 3 84 mar 3 MODELS OF EOUILiBRlUM in THE CAPITAL mm EWER n5 l‘HE assumes PRlClNG noon. new/r new APPROACH To sxpwmmo ASSET mines 335 to replicate the index with a smaller number of stocks. The more issues in an index, a smaller the companies represented in an index, and the less liquid the stocks in an indé the more costly it is to match the index by purchasing stocks in the same proportion th- represent in the index. Clearly, once one becomes concerned with tracking an index 1h represents a very large segment of a market, exact matching of proportions becomes 1 and less appropriate. An index fund can be created using the single-index model by f ing the portfolio that has a Beta of one with the desired index and that has minimum resi uai risks for a given portfolio size (minimum variance of the as in a single index form Equation £16.13). Empioying a maid-index model rather than a one—index model allows the creation of index fund that more closely matches the desired index.” The reason for this is clear, A' properly constructed multivindex model ensures that the index has been matched in terms' of all important sources of return movements (risk). On the other hand, inst matching market risk can leave the portfolio and the index with different sensitivities to the cornde factors affecting both, such as sensitivity to inflation. Let’s consider a simple example of this. Reviewing the sensitivity coefficients associated with the market from Table 8.1, hr Chapter 8, we see that both oil stocks and cyclical stocks have a sensitivity with the 8&1} index of 1.14, Thus, in a single-index model, except for resideai risk, one would be ind ferent to holding oil stocks or cyclical stocks in matching the S&P index. However, oil stocks and cyclical stocks have very different sensitivities (lbs) to sales growth. Thus, a portfolio that was matched to an index on sensitivity with the S&P but was not matched on the I: value with sales growth might not track the index very well in periods when unex~ pected changes in sales growth were large. . In general, the fewer stocks in an index»matching portfolio, the less likely that the port folio will be matched on the common factors affecting the portfolio and the index and the greater the superiority of multi-index models over singiewindex models.22 This is true because unexpected changes in the missing indices will differentially impact the residue rislt in future periods if sensitivity to these missing indices is not held constant. Portfolio are often formed to serve as arbitrage portfolios in the trading of options or futures on a index. Firms typically attempt to form a small basket of stocks (25 or 50) that they ca actively trade as they change their futures or options position. The number must be kep " small. becaUse the basket of stocks will be bought and sold frequently. The use of multi index models becomes critical in these instances. _ inflation goes up. The investor who wanted zero sensitivity to inflation would expect to have Another problem frequently encountered in passive management is the desire to match 3 3("4.32) X 0-37 3 "1.636hangc (decreaSc) in expected return to obtain the preferred posi» an index with a portfolio that excludes certain types of stocks. Social goals or management 1 _ ' lion. Like most of economics, this is not a free lunch. Instead, it is a method of allowing the preferences frequently restrict the set of stocks that can be used to match an index. In the ';_ investor to make specified trade-offs between types of risir and expected returns.25 . last 10 years. for example, it was not Uncommon for a pension fund to declare that it would . There is one variable in our model that allows the investor to take an action that is very not own tobacco stocks or gambling stocks. It is likely that a sector of the market such as _- close to a free lunch. Let’s reexamine our model. One of the factors, oil price changes, had tobacco Stocks has sensitivity to inflation or interest rates that is different from the aVer .. a zero A (was given a zero price by the market). While oil prices affect returns on some age stock. If an index fund is formed from a set of stocks that precludes tobacco stocks 5 Stocks, changes in oil prices are not a pervasive enough influence to be priced by the mar" using the single-index model, then the sensitivity to the single—index will be matched hut ket. At first glance, one might think that the sensitivity on a portfolio to oil should be set the sensitivity to other important influences will probably he different. Use of a multi— to zero. After all, why take on a risk (increased variability in returns) With n0 Commensu- index model improves tracking an index_23 rate increase in expected returns? For the average investor this is correct. However, think Multi-index models also help improve performance under a set of conditions that are directly opposite to those just described. An investor may decide to match an index with a ortfolio that must contain certain stocks. This is very common in Japan where stocks are often held for reasons that have their foundations in the‘business relationship betWecn 3 firms, In the United States, an investor may want to maintain (or add} certain holdings in ' Eportfolio for business reasons or because the investor does not want to recognize certain accumulated but unrealized capital losses or gains for either tax purposes or reporting pur— oses?“ The problem then is to find an overail portfolio matching as closely as possible an - index but including a defined set of stocks. Because these stocks may have sensitivities to - important influences that are different from the index being matched, it is important to explicitly match on each of the key risk factors. . There is one type of passive management which can be performed with a multivindex ' model, which is fundamentally differentfr'orn what can be done with a single-index model. The mold-index model allows one to closely match an index while purposely taking posi— . [ions with respect to certain types of risk.» different from the positions contained in the ' ndex. For example, consider a pension'fund that has cash outflows affected by inflation ': (CGLA or costuof—living adjustments). The payments for such a pension fund increase with inflation. Thus the overaccrs want a portfolio that will perform especially well when the rate of inflation increases. This can be illustrated more fully by returning to the data presented for the S&P index earlier in this chapter. The b value (sensitivity) for the.S&P '. index with inflation was —0.37, which implies (other things held constant) that an investment in the S&P index will tend to go down by 0.37% if the rate of inflation goes up by 1%. lfatpension fund is particularly sensitive to inflation risk (because its liabil- ity payments go up with inflation), it might wish to hold a portfolio that has a zero son- I Isitivity to inflation (or even a positive sensitivity). it could form a portfolio that had the ' same response to all factors affecting the 5&3? (except for the inflation factor) by solv- _ ing a quadratic programming problem to form a portfolio that matched all S&P bs except for the b on inflation, had a zero or positive I) with inflation, and had minimum residual risic. The applications we have just discussed can be done using a multi-index model; however, assuming an APP adds additional insight into the process. It tells the investor the expected cost of changing the exposure to inflation. Observing the It with inflation, we see that the market will accept a lower return of 4.32 for every one unit increase in sensitivity to infla» tion. This is because the aggregate of investors prefer stocks that offer higher return when E 2'See Elton and Gntber (£988) for a demonstration of the improvement in index tracking that results from USEHE '. . RAH example of the latter occurs in insurance companies where the realization of gains or losses impacts the sur— e four~index as opposed to a one-index model. '_ Plus account and thus the ability of the firm to write new business. 728% Elton and Gruher (1988) for empirical evidence on this issue. 23See Elton and Gruber “988) for empirical evidence. Deviatmg from market .55 to better match liabilities is different from deviating in order to latte active bets on :_ ll“! change in one or more underlying influences. This active use of factor bets will be discussed shortly. 336 rarer 3 moons or courtrooms/i w ms wont mama 16 THE assumes PRECING MODELAW—A new APPROACH TO exuwmwo ASSET PRECES 337 '_ Another application of APT is to use APT to determine stocks that are under— or over- x.agued. In this procedure an analyst produces a forecast of the return on a stock. The APT is {hen used together with estimates of the sensitivity of the stock to the factors to calcum We a required return for the stock (using an equation such as 16.10). If the estimated return is above what’s required given the stock‘s sensitivity and the its, the stock is purchased. - This is a generalization of the analysis that is used when the CAPM rather than the APT ' is used as an equilibrium model. Recail as shown in Chapter 14, that the CAPM is a Straight line in expected return Beta space (see Figure 14.2). If a firm’s expected return and _ Beta are such that it plots above the CAPM line. it offers a higher return (given its Beta) . than is required in equiiibrium and is a buy. Similarly, if it plots below the iine, its expected return is less than required in equiiibrium it should be sold. The analysis with APT has the same logic. Consider a two-factor APT model. In this case, the APT piots as a plane in ' a three-dimensional space where the axes are sensitivities to the two factors and expected rctum. Firms that plot above the plane offerza higher expected return than is required given the sensitivities and its and should be purchased.27 Why the APT rather than the CAPM? If the APT is the appropriate equilibrium model and the CAPM is used, then stocks with different sensitivities to the factors but the same market Beta wili be incorrectly classified as equally risky. The CAPM model incorrectly implies that they have the same expected return. 'i‘o better understand this, let’s return to the example we have been discussing in this chapter. Note that: the iambda on growth is positive. This implies that investors require a higher expected return for stocks that have higher sensitivity to unexpected changes in growth. A stock with a high sensitivity to growth wili tend (because growth has a positive - price or iambda) to have a higher expected return than a stock with a lower sensitivity to growth. But this is ignored (except for the pan captured in the market Beta) by the stan— dard CAPM modcis. Thus the extra return investors require (as reflected in the market price of risk or lambda associated with high sensitivity to growth) will be interpreted as underpricing by the standard CAPM modei. Stocks that are very sensitive to unexpected changes in growth will tend to lie above the security market iine. Stocks that are sensitive to other priced influences not included in the singie-index model are likely to show up as Systematically tinderpriccd or overpriced by the CAPM and to lie above or below the secu- rity market line. One of the most common uses of the APT model is to form a portfolio of stocks that while closely tracking a target will also produce a return in excess of that index. One way to implement this type of procedure is simply to employ the index—matching procedure described earlier in this chapter, but only allow selection from among a set of stocks that analysts have earmarked as superior performers. Other techniques use either numeric dis- crete ranking of stocks or expected return on stocks in an attempt to produce an excess return above an index whiie using the mold—index model to track an index as closely as possible.28 Portfolios designed this way have become known as research titled index funds. Although some additional risk is invoived {the index can’t be matched as closeiy when selecting from a restricted set of stocks), investors who use this technique feel that an excess return can be earned with oniy a siight loss in the ability to track the index. The advantage of the muiti—index model over the simpledndex model is that the target index of an investor whose cash outflow increases with increases in oil prices. Such an llivesm would want to hold a portfolio of securities that has a positive sensitivity with oil pric': Furthermore, because oil sensitivity is not priced by the market, increasing the sensim. ity to oil prices does not change expected return. Of course, if everybody wanted to ho' portfolios that exhibited increased return with increases in oii prices. then the it assoc‘ ated with oil prices would be positive. The fact that an investor desires, with respect to 0 sensitivity, a position different from the aggregate aliows the investor to improve his a her portfolio with no decrease in expected return, although there will be some increasej total risk. _ Keep in mind that matching an index while making quantitative judgments on the among} of a particular type of risk to take can only be done if indexes representing these risks 3; contained in the mold-index mode]. Furthermore, the expected return (or expected cost) 0 these nonaverage risk positions can only be determined from an APT model. Active Management Most uses of multi-index models for active management parallei their use in passive man agement. it’s easier to discuss them in reverse order to that presented previousiy. What multi—index model does that cannot be done with a single—index model is allow the user to . make factor bets. if you believe that unexpected inflation wiil accelerate at a rate abov that anticipated by the market (I, > 0), then you may want to piace a hot by increasing yon exposure (5:: value} with inflation. This can be done holding a portfolio with a sensitivit the more active bets you can make. For example, in the Salomon model described earlier-- in this chapter, you can take active bets on economic growth. the stage of the business __ cycle, iong-term interest rates, shortuterm interest rates, inflation rates, the value of the. U.S. doliar, or the state of the stock market. Return to the simple model we have been discussing, assume that the S&P index is the. appropriate benchmark and that an analyst believed that sales were going to increase by -. 1% more than the market expected. The analyst might increase the 1: value with respeth . sales on the portfoiio from the 1.71 value found for the S&P index to 2.21. Under the APT modei and recognizing the A for sales is 1.49, the increase in saies sensitivity of 6.5? would lead to a 0.5(1 .49) = 0.745% increase in expected return, which is just sufficientt reward the investor for the additional risk. However, the additionai 1% increase in sates wouid lead to an additional 2.21% increase in the return on the portfoiio should it mated alize. Of this 2.21% increase, 0.5% arose from increasing the sensitivity to sates whiie 1.71 would have arisen had the 17 been left at the level of the S&P index. The 0.5% increase i often called the excess risk adjusted return that arises from an ability to forecast factors, better than the market. _ Multidndex models and APT modeis can be used just as the single—index model and. CAPM modcis are used to form optimal portfolios building upon estimates of the per formance of individual securities. The simplest approach is that discussed back in Chapter - 8, where a multi-index model is used to generate the convariance between securities whil expected returns and variances are supplied by some combination of analysts’ forecasts and historical data. 1"lilac marl-tat prices of risk (As) for the CAPM or APT models can be specified by theory or estimated using ana- IYSIS‘ forecasts of expected return and sensitivities fora set of stocks. See Chapter 19 for a discussion of how firm forecasts are used. 2‘SWe assume the S&P index is the mievant benchmark in this section. Actually the analysis holds with the sea- sitivities of any benchmark (growth stocks or the New York Stock Exchange index) substituted for sensitivitifii to the S&P index. .' 28llriany firms have their anaiysts place stocks into groups (often five) with group i being the best purchases and 5 the stocks that should be sold. 383 mm 3 MODELS OF EOU'UWUM ‘N THE Wim MARKET (“were 1 e no announce swerve MODEL MM NEW APPROACH ro sxrwmmo ASSET FRtCES 339 can be tracked more closely because the different sources of risk are explicitly taken in consideration. The more the target being tracked differs from a diversified market portfolio, the m0 important it is to use a mold-index model. The extreme case and one that has received a1 of attention is the long~short investment strategy or risk neutral strategy. If one has supen' ability to identify stocks that will perform above average on an APT risk—adjusted basis and Stocks that will perform below average on an APT riskwadjusted basis, then using the APT» index, one can form portfolios that offer an excess return and have no risk {zero [9 risk) wt respect to any factor {c.g., no risk due to change in the market level, inflation, or interest rate" movements). Obviously there is also no expected return due to any factor because the Beta on each factor is set to zero. What one gets is a pure payoff from security selection with all factors inciuding the market neutralized. We can examine this by returning to Equation: (16.i 1). If we believe that an analyst can predict the extra return from any security over a' period of time (return from security selection} Equation (1631} can be written as nfluences that enter the return» generating process and APT are ignored in doing perform ance evaluation, not oniy can't the analyst's performance lac-attributed to the type of man- ag‘ément decisions he or she is making, but perhaps more important, incorrect conclusions _ may be reached about how well managers are performing. ' CONCLUSION in this chapter, we have reviewed 1. Modern concepts of arbitrage pricing. 2, Alternative approaches to estimating arbitrage pricing models. 3. Some uses of arbitrage pricing models: Considerable evidence continues to be prbduccd on the usefulness of arbitrage pricing models. ' R; 3 RF +06,- + llbfi + fight-2 +A3bf3 + o--+b,-1Ii+b5212 +bi313 +v--+e£ APPENDfX A A. StMPLE EXAMPLE OF FACTOR ANALYSIS In order to prov'pgle the reader who has never used any form of factor anaiysis with a demonstration of how it works, we include a simple exampie in this appendix. We choose to use principal component analysis for the example, because this leads to a solution that is easiest to interpret.29 We choose 10 years of monthly data on the Morgan Stanley Capital International stock indexes for each of four countries: the United States, Canada, France, and Belgium. Remember that pn‘ncipai components analysis extracts from this data the index that expiains as much as posdbie of the correlation in returns between the four countries and then finds a second index that explains as much as possible of the correlation in returns not expiained by the first index.” The indexes produced by principal components are formed by combining (weighting) the time ' series of return for each country with the mean return for each country extracted. Before we perform principal component anaiysis, iet‘s think about what we would expect the results to iook like. We might hypothesize that the first index would be some sort of measure of how stocks in general did, that is, some general aggregation of the returns under Study. In thinking about the problem, one would expect Canada and the United States to act somewhat alike and France and Beigium to act somewhat alike, whereas we would expect the differences between these paired countries to be greater. In fact, the correlations - between the four countries as shown in Table 16.2 bear out this speculation. where or; is the extra return the security analyst predicts on security i. Think of this equation for each of two portfolios: portfolio L is a portfolio of long pas tions and portfolio S is a portfoiio of short positions. Furthermore, assume that the portfo4 lios are formed so that by + [75} = O for ali js. Then, combining the preceding equation for each of the two portfoiios, we get a risk neutrai (or more specifically systematic risk neu: ital) portfolio denoted by N with a return given by RN 2 RF+LXL 441.3 and with a risk given by EN = EL +65 Burtueister, Roll, and Ross (1994) examined the payoff from such a model from the: period Aprii 1991 to March i992. assuming as could be correctly identified. They found that over this period the We? index has a return of “57% per year and a standard devia- tion of 18.08%. Their factor—neutral portfolio had a return of 38.04% per year and a stan: dard deviation of 6.26% per year. While these are obviously optimistic figures, for they assume perfect foresight, they do indicate the ability of factor—neutral portfolios to tower risk and, if forecasting ability exists, increase return. ' Although one can perform the same type of analysis with a single—index model rather than a multiwindex model, the overall risks of the portfolio will be greater and the user likely to find he or she is undertaking factor bets (inflation, interest rate, etc.) rather than pure security selection bets. Table T62 Correlation Coefficient Between Returns in Four Countries Belgium France Canada U.S. .'M_—wmwmmmmww Performance Measurement and Attribution New 1.0 I . France 955 L9 The last use of mold-index and APT models we should examine is in the area of portfolio Canada 933 (Ml 1.0 performance evaluation. It is difficult to discuss the use of APT in performance measurement - United States 0.41 0.43 0.72 1.9 . . . . . . . . - W and evaluation Without revzewmg the whole literature in this area. Because of this, we W19 leave a detailed discussion and the continuation of the example we started in this chapter. until Chapter 25. HOWever, consideration of the model we have discussed shows that the expected performance of any portfolio is not just a function of the portfolio’s sensitiVity t5: the market but also a function of the portfolio’s sensitivity to saies grovvth and inflation. I”Sce Elton and Gruber (3394} for a detailed discussion on the use of factor analysis in mold-index models. :fi’rincipal components then extract a third and fourth index. In this case, we only report the first two, since the llnrd and fourth are not statistically significant. 390 PART 3 MODELS OF EOUIUBRHJM N THE CAPITAL WXET CHAPTER 16 THE ARBITRAGE FRlCiNG MODEL APT—ad NEW APPROACH TO EXleNENG ASSET PRJCES 'I Now assume a very well~diversified portfolio called m. For this portfolio residual risk The indexes that are the first two principal components estimated from this data are m _ ‘ aWrenches zero and“ a seated below. Remember that in performing principal components analysis, we do no specify the indexes we expect to find; we simply let the data determine the indexes. J h ‘ d T e In axes are: Rm! x Km "1" RF! '4" F}! "3' Fit! 1,, = 0.67(R3, — EB) + 0.76(R,., w E.) d“ 0.76(Rc, ~— Ec)+ 0.77(R,,, m 1?“) '“1 m _. __ __ .r 12, = ——t}.40(R3, — RB) — o.37(n,., w RF) + 0.73%, m R,)+ 0.410%“: M R”) where it," 2 2 small. + M .31 where J h ‘ I . Burmeister and McElroy assume the market portfolio has no resrdual risk em, = 0, this Fr, 5 the unobserved error term. 4 «i ' However, Ft, can be estimated by the residual of an Ordinary Least Squares (0L5) time-series {egressiofi of Rm, on the observed variabies asih Equation (13.4} or rearranging (8.4) yields J' Fk: :{Rmr “RFJ” Am +2bmjfh (3-5) .51 IE, and 12, are the two indexes extracted from the data. The Rs are monthly returns, and the subscripts B, C, F, U, and t represent Beiginm Canada, France, United States, and time. Note that the first index is very close to 'an eqtzaily weighted index of all four markets: and thus meets our expectation that the index that would explain as much as possible of' returns is the genera] return index. The second index is iong in North America and short in Europe. It meets our expectation that the second index should capture the fact that North American and European markets are less associated with each other than with markets" within their own region. " To see how well these two indexes work, we can regress the retnrns from each country. against the two indexes. When we did so, the I??- were: 0.81 for Beigium, 0.95 for Canada, 0.84 for France, and 9.74 for the United States. McElroy and Burmeister (19§8) show that 15}, is an unbiased estimate of the common stocks Fm. Thus substituting FM for FR, in (8.3} and adjusting the residual yields in J J' Rs: 3 RF: "l" 2557“; "’9’ bikfi‘k "’r Ebola} 4‘ bikar 4%} (33.5} i=1 i=1 To estimate this equation McEEroy and Burmeiser [1988) first use time-series anaiysis to estimate Equations {BA and B5) and then nonlinear seemingly unrelated regressions to estimate (33.6}.32 In doing so, they make one more interesting change in this modei. They allow for the possibility that although APT correctly prices every security in their sample, it may not correctly price every security in the highiy diversified portfolio. APPENDiX B SPECIFICATION OF THE APT Wi'FH AN UNOBSERVED MARKET FACTOR This appendix is a brief recapping of the procedures put forth in a series of articles by Burmeister, McElroy, and others. For further details see {1987, 1988, i986) and (i988). :- We can represent a return-generating process (multidndex model} with observable indices pius an unobservable index designated by index k as ' _ QUESTIONS AND PROBLEMS 1. Assume that the foliowing two—index model describes returns RE “*2 at +bil‘rl + [9:212 + 9: m J' " Assume that the foliowing three portfolios are observed. RR = 12,, +21913FJif + smirk, + 5,, (Bl) . . . .. . . .. . _. . . . _ _ . .=I Portfolio Expected Return bu be Making the no arbitrage assumption of AFT, expected return is approximately given by t; a J C 12.0 3 was Rh 3 7“Or + 2h};ij + bikhkt 032? ' Find the equation of the plane that must describe equiiibrinm returns. ' 2. Referring to the results of Problem 1, illustrate the arbitrage opportunities that would exist if a portfoiio called D with the foliowing properties were observed. E0219 b13122 [70sz We wiii make the assumption of McElroy and Butmeister (1988} that ail ha, equai the risk-free rate and ail other As are constant over time, substiteting (Bl) into (B.l}. _ “The assumption is made that the unobserved variabie Ft, is t‘ecalicd to have a Beta of one with the portfolio in. . 32Conditions on the relationship between an and 5;, are delineated in McEimy and BunneisterUQSS). J .I R,, = RF, + Emilia-219513;, + akik +b5kfi, +e,, (s3) i=1 i=1 392 3. Portfolio WWW-"WM . A 22 1.0 12' e rs 1.5 13 C 17 0 5 —.3 ‘ I 4. Referring to the results of Problem 3,111ustrate the arbitrage opportunities that weal E4 exist if a portfolio called D with the following characteristics were observed. ' ' Ely—"15 bpizl by)sz 5 .l v 5. If we accept the Sharpe model as a description of expected returns, using the data) Table 16.1 find the expected return on a stock in the construction industry with the fq' 15v lowing characteristics. Assume a riskless rate of 8%. 17 Beta 3 1.2 ' Yield 3 6 - 13. '. m .4 ' 3176 0 19‘ Bond Beta m 0.2 Alpha m I I 20. 6. Return to Problem 1. If {R}, — RF) = 4, find the values for the following variables th 2, would make the expected returns from Problem 1 consistent with equilibrium dete . mined by the simple {Sharpe—Lintneerossin) CAPM. 22' A» 1311 353d 312 23' B. 13,, for each of the three portfolios C. R,» 24. 25. BIBLIOGRAPHY I i. Admati, Anat, and Pfleiderer, Paul. “Interpreting the Factor Risk Pternia in Arbitrage Pricing 26' Theory," Journal of Economic Theory, 35 (Feb. 1985), pp. 191495. 2. Berry, Michael, Burmeistcr, Edwin, and McEiroy, Marjorie. “Sorting Out Risks Using Known: _ APT Factors,” Financial Analysts Journal (March 1988}, pp. 29412. _ . . 27‘ 3. Black. F. “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, 45- _ (July 1972). pp. 444454. _ 28~ 4. Black, Fisher, Jensen, Nick, and Scholes, Myron. 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