A Review of Capital Asset Pricing Models

Transcription

1 A Revew of Captal Asset Prcng Models Don U.A.Galagedera * Department of Econometrcs and Busness Statstcs Monash Unversty PO Box 197 Caulfeld East Vctora 3145 Australa Abstract Ths paper provdes a revew of the man features of asset prcng models. The revew ncludes sngle-factor and multfactor models, extended forms of the Captal Asset Prcng Model (CAPM) wth hgher order co-moments, and asset prcng models condtonal on tme-varyng volatlty. Key words: Asset prcng, CAPM, sngle-factor and multfactor models 1. Introducton The foundatons for the development of asset prcng models were lad by Markowtz (1952) and Tobn (1958). Early theores suggested that the rsk of an ndvdual securty s the standard devaton of ts returns a measure of return volatlty. Thus, the larger the standard devaton of securty returns the greater the rsk. An nvestor s man concern, however, s the rsk of hs/her total wealth made up of a collecton of securtes, the portfolo. Markowtz observed that () when two rsky assets are combned ther standard devatons are not addtve, provded the returns from the two assets are not perfectly postvely correlated and () when a portfolo of rsky assets s formed, the standard devaton rsk of the portfolo s less than the sum of standard devatons of ts consttuents. Markowtz was the frst to develop a specfc measure of portfolo rsk and to derve the expected return and rsk of a portfolo. The Markowtz model generates the effcent fronter of portfolos and the nvestors are * Correspondence to: Tssa.Galagedera@buseco.monash.edu.au

2 expected to select a portfolo, whch s most approprate for them, from the effcent set of portfolos avalable to them. The computaton of rsk reducton as proposed by Markowtz s tedous. Sharpe (1964) developed a computatonally effcent method, the sngle ndex model, where return on an ndvdual securty s related to the return on a common ndex. The common ndex may be any varable thought to be the domnant nfluence on stock returns and need not be a stock ndex (Jones, 1991). The sngle ndex model can be extended to portfolos as well. Ths s possble because the expected return on a portfolo s a weghted average of the expected returns on ndvdual securtes. When analysng the rsk of an ndvdual securty, however, the ndvdual securty rsk must be consdered n relaton to other securtes n the portfolo. In partcular, the rsk of an ndvdual securty must be measured n terms of the extent to whch t adds rsk to the nvestor s portfolo. Thus, a securty s contrbuton to portfolo rsk s dfferent from the rsk of the ndvdual securty. Investors face two knds of rsks, namely, dversfable (unsystematc) and non-dversfable (systematc). Unsystematc rsk s the component of the portfolo rsk that can be elmnated by ncreasng the portfolo sze, the reason beng that rsks that are specfc to an ndvdual securty such as busness or fnancal rsk can be elmnated by constructng a well-dversfed portfolo. Systematc rsk s assocated wth overall movements n the general market or economy and therefore s often referred to as the market rsk. The market rsk s the component of the total rsk that cannot be elmnated through portfolo dversfcaton. 2

3 The CAPM developed by Sharpe (1964) and Lntner (1965), dscussed n the followng secton, relates the expected rate of return of an ndvdual securty to a measure of ts systematc rsk. Snce then, a varety of models have been developed to predct asset returns. These are dscussed n Secton 3. A bref summary s gven n Secton 4. 2 The captal asset prcng model The CAPM conveys the noton that securtes are prced so that the expected returns wll compensate nvestors for the expected rsks. There are two fundamental relatonshps: the captal market lne and the securty market lne. These two models are the buldng blocks for dervng the CAPM. Even though they are not new, t s llustratve to dscuss them here brefly. Further, snce one of the ams of ths thess s to nvestgate varous forms of CAPM, these models deserve some attenton n ths paper. 2.1 Captal market lne The captal market lne (CML) specfes the return an ndvdual nvestor expects to receve on a portfolo. Ths s a lnear relatonshp between rsk and return on effcent portfolos that can be wrtten as: where, ( R ) E m R f E( R p ) = R f + σ p (1) σ m R p R f R m = portfolo return, = rsk-free asset return, = market portfolo return, σ p = standard devaton of portfolo returns, and σ m = standard devaton of market portfolo returns. 3

4 Accordng to (3.2.1), the expected return on a portfolo can be thought of as a sum of the return for delayng consumpton and a premum for bearng rsk nherent n the portfolo. The CML s vald only for effcent portfolos and expresses nvestors behavour regardng the market portfolo and ther own nvestment portfolos Securty market lne The securty market lne (SML) expresses the return an ndvdual nvestor can expect n terms of a rsk-free rate and the relatve rsk of a securty or portfolo. The SML wth respect to securty can be wrtten as: f { E( R ) R } E( R ) = R + β (2) m f where, ( R, R ) σ rm cov m β = =, (3) σ σ and r = the correlaton between securty return, R and market portfolo return. The m m 2 m β can be nterpreted as the amount of non-dversfable rsk nherent n the securty relatve to the rsk of the market portfolo. Equaton (2) s a verson of the CAPM. The set of assumptons 1 suffcent to derve the CAPM verson of (2) are the followng: () () () the nvestor s utlty functons are ether quadratc or normal, all dversfable rsks are elmnated and the market portfolo and the rsk-free asset domnates the opportunty set of rsky assets. The SML s applcable to portfolos as well. Therefore, SML can be used n portfolo analyss to test whether securtes are farly prced, or not. 1 See Snclar (1987) for a descrpton of these assumptons. 4

5 3 Asset prcng models 3.1 Sngle-factor CAPM In order to test the valdty of the CAPM researchers always test the SML gven n (2). The CAPM s a sngle-perod ex ante model. However, snce the ex ante returns are unobservable, researchers rely on realsed returns. So the emprcal queston arses: Do the past securty returns conform to the CAPM? The beta n such an nvestgaton s usually obtaned by estmatng the securty characterstc lne (SCL) that relates the excess return on securty to the excess return on some effcent market ndex at tme t. The ex post SCL can be wrtten as: R t ft ( Rmt R ft ) ε t R = η + b + (4) where, η s the constant return earned n each perod and b s an estmate of β n the SML. The estmated β s then used as the explanatory varable n the followng crosssectonal equaton: R γ b + u (5) t = 0 + γ 1 t to test for a postve rsk return trade-off. The coeffcent γ 0 s the expected return of a zero beta portfolo, expected to be the same as the rsk-free rate and γ 1 s the market prce of rsk (market rsk premum), whch s sgnfcantly dfferent from zero and postve n order to support the valdty of the CAPM. When testng the CAPM usng (4) and (5), we are actually testng the followng ssues: () b s are true estmates of hstorcal β s, () the market portfolo used n emprcal studes s the approprate proxy for the effcent market portfolo for measurng hstorcal rsk premum and () the CAPM specfcaton s correct (Radclffe, 1987). 5

6 Early studes (Lntner, 1965; Douglas, 1969) on CAPM were prmarly based on ndvdual securty returns. Ther emprcal results were dscouragng. Mller and Scholes (1972) hghlghted some statstcal problems encountered when usng ndvdual securtes n testng the valdty of the CAPM. Most studes subsequently overcame ths problem by usng portfolo returns. Black, Jensen and Scholes (1972), n ther study of all the stocks of the New York Stock Exchange over the perod , formed portfolos and reported a lnear relatonshp between the average excess portfolo return and the beta, and for beta >1 (<1) the ntercept tends to be negatve (postve). Therefore, they developed a zero-beta verson of the CAPM model where the ntercept term s allowed to change n each perod. Extendng the Black, Jensen and Scholes (1972) study, Fama and MacBeth (1973) provded evdence () of a larger ntercept term than the rsk-free rate, () that the lnear relatonshp between the average return and the beta holds and () that the lnear relatonshp holds well when the data covers a long tme perod. Subsequent studes, however, provde weak emprcal evdence on these relatonshps. See, for example, Fama and French (1992), He and Ng (1994), Davs (1994) and Mles and Tmmermann (1996). The mxed emprcal fndngs on the return-beta relatonshp prompted a number of responses: () The sngle-factor CAPM s rejected when the portfolo used as a market proxy s neffcent. See 2, for example, Roll (1977) and Ross (1977). Even very small devatons from effcency can produce an nsgnfcant relatonshp between rsk and expected returns (Roll and Ross, 1994; Kandel and Stambaugh, 1995). () Kothar, Shanken and Sloan (1995) hghlghted the survvorshp bas n the data used to test the valdty of the asset prcng model specfcatons. 2 Also see Fama and MacBeth (1973), Black (1993) and Chan and Lakonshok (1993) and the references theren. 6

7 () Beta s unstable over tme. See, for example, Bos and Newbold (1984), Faff, Lee and Fry (1992), Brooks, Faff and Lee (1994) and Faff and Brooks (1998). (v) There are several model specfcaton ssues: For example, (a) Km (1995) and Amhud, Chrstensen and Mendelson (1993) argued that errors n varables mpact on the emprcal research, (b) Kan and Zhang (1999) focused on a tme-varyng rsk premum, (c) Jagannathan and Wang (1996) showed that specfyng a broader market portfolo can affect the results and (d) Clare, Prestley and Thomas (1998) argued that falng to take nto account possble correlatons between dosyncratc returns may have an mpact on the results. 3.2 Multfactor models A growng number of studes found that the cross-sectonal varaton n average securty returns cannot be explaned by the market beta alone and showed that fundamental varables such as sze (Banz, 1981), rato of book-to-market value (Rosenberg, Red and Lansten, 1985; Chan, Hamao and Lakonshok, 1991), macroeconomc varables and the prce to earnngs rato (Basu, 1983) account for a szeable porton of the cross-sectonal varaton n expected returns. Fama and French (1995) observed that the two non-market rsk factors SMB (the dfference between the return on a portfolo of small stocks and the return on a portfolo of large stocks) and HML (the dfference between the return on a portfolo of hgh-book-to-market stocks and the return on a portfolo of low-book-to-market stocks) are useful factors when explanng a cross-secton of equty returns. Chung, Johnson and Schll (2001) observed that as hgherorder systematc co-moments are ncluded n the cross-sectonal regressons for portfolo returns, the SMB and HML generally become nsgnfcant. Therefore, they argued that SMB and HML are good proxes for hgher-order co-moments. Ferson and Harvey (1999) clamed 7

8 that many multfactor model specfcatons are rejected because they gnore condtonng nformaton. Another possblty s to construct multfactor arbtrage prcng theory (APT) models ntroduced by Ross (1976). APT models allow for prced factors that are orthogonal to the market return and do not requre that all nvestors are mean-varance optmsers, as n the CAPM. Groenewold and Fraser (1997) examned the valdty of these models for Australan data and compared the performance of the emprcal verson of APT and the CAPM. They concluded that APT outperforms the CAPM n terms of wthn-sample explanatory power. 3.3 CAPM wth hgher-order co-moments It s clear from well-establshed stylsed facts that the uncondtonal securty return dstrbuton s not normal (see, for example, Ané and Geman, 2000 and Chung, Johnson and Schll, 2001) and the mean and varance of returns alone are nsuffcent to characterse the return dstrbuton completely. Ths has led researchers to pay attenton to the thrd moment skewness 3 and the fourth moment kurtoss. Many researchers nvestgated the valdty of the CAPM n the presence of hgher-order comoments and ther effects on asset prces. In partcular, the effect of skewness on asset prcng models was nvestgated extensvely. For example, Kraus and Ltzenberger (1976), Frend and Westerfeld (1980), Sears and We (1985) and Faff, Ho and Zhang (1998), among others, 3 Early studes examned the emprcal relaton of ex post returns to total skewness (see, for example, Ardtt, 1967). Subsequent studes argued that systematc skewness s more relevant to market valuaton rather than total skewness (see, for example, Kraus and Ltzenberger, 1976) refutng the usefulness of quadratc utlty as a bass for postve valuaton theory. The expermental evdence that most ndvduals have concave utlty dsplayng absolute rsk averson also supports ncluson of hgherorder co-moments n rsk-return analyss (see, for example Gordon, Parads and Rorke, 1972). 8

9 extended the CAPM to ncorporate skewness n asset valuaton models and provded mxed results. Harvey and Sddque (2000) examned an extended CAPM, ncludng systematc coskewness. Ther model ncorporates condtonal skewness. The extended form of CAPM s preferred as the condtonal skewness captures asymmetry n rsk, n partcular downsde rsk 4, whch has recently become consderably mportant n measurng value at rsk. Harvey and Sddque reported that condtonal skewness explans the cross-sectonal varaton of expected returns across assets and s sgnfcant even when factors based on sze and book-to-market are ncluded. A few studes have shown that non-dversfed skewness and kurtoss play an mportant role n determnng securty valuatons. Fang and La (1997) derved a four-moment CAPM and t was shown that systematc varance, systematc skewness and systematc kurtoss contrbute to the rsk premum of an asset. See, also, Chrste-Davd and Chaudhry (2001) who show that the thrd and fourth moments explan the return-generatng process n futures markets well. Investors are generally compensated for takng hgh rsk as measured by hgh systematc varance and systematc kurtoss. Investors also forego the expected returns for takng the beneft of a postvely skewed market. It also has been documented that skewness and kurtoss cannot be dversfed away by ncreasng the sze of portfolos (Ardtt, 1971). 4 Downsde rsk s the rsk of loss or underperformance that s consdered as the approprate measure of rsk. Varance, as a measure of rsk, ncludes returns above and below the average return, n the same ven. Ths has led to crtcsm of varance as a measure of rsk. 9

10 3.4 Condtonal asset prcng models Testng for the nstablty of beta and the valdty of the return-beta relatonshp s not new. Followng the suggeston made by Levy (1974) to compute separate betas for bull and bear markets, Fabozz and Francs (1977) were the frst to formally estmate and test the stablty of betas over the bull and bear markets. They found no evdence supportng beta nstablty. However, n an emprcal analyss of the cross-sectonal relatonshp between the expected returns and beta, Fabozz and Francs (1978) concluded that nvestors lke to receve a postve premum for acceptng downsde rsk, whle a negatve premum was assocated wth the up market beta, suggestng that downsde rsk as measured by the beta correspondng to the bear market may be a more approprate measure of portfolo rsk than the conventonal sngle beta. Prompted by Fabozz and Francs (1978), several studes tested for randomness of beta. Km and Zumwalt (1979) extended the Fabozz-Francs desgn to analyse the varaton of returns on securty and portfolos n up and down markets. They used three alternatve measures to determne what consttuted an up and down market. Up market consttuted those months n whch the market return exceeded () the mean market return, () the mean rsk-free rate or () zero. Km and Zumwalt concluded that downsde rsk mght be a more approprate measure of portfolo rsk than the conventonal sngle beta. Chen (1982) allowed beta to be nonstatonary n an examnaton of the rsk-return relatonshp n the up and down markets and concluded that () under the condton of ether constant or changng beta, nvestors seek compensaton for assumng downsde rsk and () as n the Km and Zumwalt (1979) study, the down market beta s a more approprate measure of portfolo rsk than the sngle beta. Bhardwaj and Brooks (1993) observed that the systematc rsks n bull and bear tme perods are statstcally dfferent. Ther classfcaton of bull and bear markets s based on whether the market return exceeds the medan market return or not. Studes have consdered three-beta 10

11 models as well. For example, Faff and Brooks (1998), notng that there s no reason to beleve that beta s constant, especally over long estmaton perods, defned three regmes relatng to two major past events. Ferson and Harvey (1991), on the other hand, n ther study of US stocks and bond returns, revealed that the tme varaton n the premum for beta rsk s more mportant than the changes n the betas themselves. Ths s because equty rsk premums were found to vary wth market condtons and busness cycles. Schwert (1989) attrbuted dfferental rsk prema between up and down markets to varyng systematc rsk over the busness cycle. Pettengll, Sundaram and Mathur (1995) hghlghted that the weak and ntertemporally nconsstent results of studes testng for a systematc relaton between return and beta s due to the condtonal nature of the relaton between the beta and the realsed return. They argued that when realsed returns are used, the relaton between the beta and the expected return s condtonal on the excess market return. They postulated a postve (negatve) relaton between the beta and returns durng an up (down) market. See Secton for more detals. Ther study of US stocks sampled over the perod reported the exstence of a systematc condtonal relaton between the beta and the return for the total sample perod, as well as across sub-sample perods. Followng Pettengll, Sundaram and Mathur (1995), Crombez and Vander Vennet (2000) analysed the condtonal relatonshp between stock returns and beta on the Brussels Stock Exchange over the perod They observed that the beta factor s a strong and consstent ndcator of both upward potental n bull markets and downsde rsk n bear 11

12 markets. They found the results to be robust for varous defntons 5 of beta and dfferent specfcatons 6 of up and down markets. Further, they hghlghted that nvestors could mprove the performance of ther portfolos by usng up and down market betas n ther asset selecton practce. A common feature n the above studes s the use of monthly data. As far as we are aware only one study has adopted the Pettengll, Sundaram and Mathur approach to nvestgate an extended CAPM wth hgher-order co-moments. Postulatng that the systematc rsks correspondng to varance, skewness and kurtoss are dfferent for up and down markets, Galagedera and Slvapulle (2002) examned the relatonshp between the returns and hgher-order systematc co-moments n the up and down markets. They found strong emprcal evdence to suggest that n the pr4esence of skewness n the market returns dstrbuton, the expected excess rate of return s related not only to beta but also to systematc co-skewness. 3.5 CAPM condtonal on tme-varyng volatlty 5 Beta computed usng dfferent market ndces. 6 Up market defned as months n whch market return s non-negatve and other strong crtera: () market return exceeds the average value of postve market returns and () market return exceeds the average value of postve market returns plus a factor (0.5 and 0.75) of the standard devaton of postve market returns. 12

13 Snce the ntroducton of ARCH/GARCH 7 -type processes by Engle (1982) and others, testng for, and modellng of, tme-varyng volatlty (varance/covarance) of stock market returns (and hence the tme-varyng beta) have been gven consderable attenton n the lterature. See Bollerslev, Engle and Wooldrdge (1988) the frst study to model the beta n terms of tmevaryng varance/covarance and the survey paper by Bollerslev, Engle and Nelson (1994). The ARCH-based emprcal models appear to provde stronger evdence, though not convncngly, of the rsk-return relatonshp than do the uncondtonal models. Usng monthly data from the Unted Kngdom market from 1975 to 1996, Fraser, Hamelnk, Hoesl and MacGregor (2000) compared the cross-sectonal rsk-return relatonshp obtaned wth an uncondtonal specfcaton of the asset s betas wth betas obtaned through Quanttatve Threshold ARCH (QTARCH 8 ) and GARCH-M 9 models. In all specfcatons, they allowed for possble negatve return-rsk relatonshps when excess return on the market s negatve. Fraser, Hamelnk, Hoesl and MacGregor observed that CAPM holds better n downward movng markets than n upward markets and suggested that beta as a rsk measure s more approprate n the bear markets. They observed that the QTARCH specfcaton, n 7 The ARCH model allows the current condtonal varance to be a functon of the past squared error terms. Ths s consstent wth volatlty clusterng. Bollerslev (1986) later generalsed the ARCH (GARCH) model such that the current condtonal varance s allowed to be a functon of the past condtonal varance and past squared error terms. The return-generatng process can be wrtten as: ARMA(m,n) mean: R m n t = + α Rt + β jε t j = 1 j= 1 µ + ε t, where ε t 2 ( 0 σ ) Ω, s t 1 ~, t Ω t 1 the nformaton set avalable at tme 2 t 1, and the condtonal varance, s defned as: σ t p q = t = 1 j= 1 2 GARCH(p,q): σ δ δ ε δ σ. t 8 See Goureroux and Monfort (1992) for detals. 9 Due to Bollerslev, Engle and Wooldrdge (1988). 2 t j 13

14 whch they allowed for asymmetres n the frst and second moments of returns, yelds a sgnfcant beta wthout havng to account for up and down markets. Recently, several studes nvestgated the effect of good and bad news (leverage effects), as measured by postve and negatve returns on beta. See, for example, Braun, Nelson and Suner (1995) (BNS hereafter) and Cho and Engle (1999) (CE hereafter) and the references theren. BNS nvestgated the varablty of beta 10 usng bvarate Exponental GARCH (EGARCH 11 ) models allowng market volatlty, portfolo-specfc volatlty and beta to respond asymmetrcally to postve and negatve market and portfolo returns. CE, on the other hand, used a two-beta model wth an EGARCH varance specfcaton and daly stock returns of ndvdual frms. CE concluded that news asymmetrcally affects the betas whle the BNS study that used monthly data on portfolos dd not uncover ths relatonshp. An alternatve approach to capture market movements s through varous market volatlty regmes. Galagedera and Faff (2003) examned the approprateness of a condtonal three-beta model as a securty return generatng process. Havng modelled the market return volatlty as a GARCH(1,1) process, they defned three volatlty regmes based on the sze of the condtonal volatltes. Even though ther results overwhelmngly suggest that the betas n the low, usual and hgh volatlty regmes are postve and sgnfcant, most of the securty/ portfolo betas were not found to be sgnfcantly dfferent n the three regmes. 10 See also Huang (2000) for the use of a Markov regme-swtchng model to nvestgate the nstablty of beta. 11 Due to Nelson (1991). 14

15 4 Conclusons For the CAPM to hold, normalty of returns s a crucal assumpton, and f the CAPM holds, then only the beta should be prced. Several studes have shown that securty returns are nonnormal and ths s evdent especally n hgh frequency data. When returns are normal, the mean and the varance are suffcent to descrbe the return dstrbuton. On the other hand, an adequate descrpton of a non-normal return dstrbuton requres statements on hgher-order moments such as skewness and kurtoss. Prompted by the mxed results of the sngle-factor CAPM studes and the non-normal nature of return dstrbuton, the CAPM wth hgher-order co-moments was proposed n the lterature as an alternatve to the sngle-factor CAPM. These emprcal studes, too, reported mxed results. Because of the falure of market beta alone to explan cross-sectonal varaton n securty returns, multfactor models emerged. These models ncorporate fundamental varables such as sze and the prce-to-earnngs rato n addton to market beta. Pettengll, Sundaram and Mathur (1995) argued that the studes on the beta and crosssectonal returns relatonshp that used realsed return as a proxy for the expected returns mght have produced based results due to the aggregaton of postve and negatve market excess returns. They postulated that when the market return n excess of the rsk-free return s negatve, an nverse relatonshp between beta and portfolo returns s expected. Ther test for a systematc condtonal relatonshp between the realsed returns and the beta n an emprcal nvestgaton of US data revealed a postve rsk premum n the up market and a negatve rsk premum n the down market. Other studes that adopted Pettengll, Sunderam and Mathur s condtonal model to test the beta rsk-return relatonshp on dfferent data sets reported stronger results than they would obtan otherwse. 15

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