25.1. Arbitrage Pricing Theory Introduction

Transcription

1 NPTEL Course Course Ttle: Securty Analyss and Portfolo Management Course Coordnator: Dr. Jtendra Mahakud Module-13 Sesson-25 Arbtrage Prcng Theory Arbtrage Prcng Theory The fundamental prncple of fnance suggests that hgher the rsk, hgher wll be the return. Ths generalsaton of rsk return trade-off challenges the practtoners as well as academc lterature to know what s the approprate measure of rsk. The falure of the conventonal mean varance captal asset prcng model (CAPM) to adequately explan cross-sectonal varaton of rsky asset return has spurred alternatve explanatons of asset prcng. The The Arbtrage Prcng Theory (APT) of Ross (1976) 1, s one such alternatve. Unlke Sharpe (1963) sngle-ndex model, accordng to APT there are multple factors that represent the fundamental rsks n the economy. Each measure of systematc rsk relatve to fundamental rsk factors captures the senstvty of the asset to the correspondng economc factor. The key dfference between these two models les n the way systematc nvestment rsk s defned: CAPM -a sngle, market-wde rsk factor and a few (or several) factors n the APT that capture the of that market-wde rsk Introducton The tradtonal equlbrum model, the CAPM assumes that stock returns are generated by a one-factor model, where the factor represents the market portfolo of all rsky assets. However the emprcal tests of the CAPM across several markets have produced mxed results and more reasonably the fnds conclude that the market factor as the sole determnant of systematc rsk s not approprate one. The crtcal pont n the estmaton of the CAPM s the dffculty of measurng the true market portfolo. As observed by Shanken (1985, p. 1190) n the absence of what are essentally measurement problems, the central mplcaton of the CAPM, that the market portfolo s effcent, may be tested. However, lmted observablty of the unverse of assets and the consequent substtuton of proxes for the true market portfolo make t dffcult to reach 1 Ross, Stephen A., 1976, The arbtrage theory of captal asset prcng, Journal of Economc Theory, vol. 13, pp

2 defnte conclusons from the usual tests. We are really referrng to ths ambguty when we say that the CAPM s not testable. 2 In ths regard APT, developed by Ross (1976) proposes that there are several sources of rsk n the economy such as these rsk factors arse from changes n some fundamental economc and fnancal varables such as nterest rates, nflaton, real busness actvty, a market ndex, nvestor confdence etc. that cannot be elmnated. The arbtrage prcng theory assumes that a securty return s a lnear functon; not only of one, but also a set of common factors and the rsk premum for an asset s related to the rsk premum for each factor. The APT predcted that prces of all rsky assets n the economy conformed to the condton of no arbtrage. No arbtrage mean that an ndvdual holdng a well dversfed portfolo could not earn any addtonal return merely by changng the weghts of the assets ncluded n the portfolo, holdng both systematc and unsystematc rsk constant. In ths regard APT has two basc postulates: returns are generated by a Lnear Factor Model (postulate-1), assume pure arbtrage profts are mpossble (postulate -2) Other Assumptons 1. Captal markets are perfectly compettve. 2. Investors always prefer more wealth to less wealth wth certanty. 3. The stochastc process generatng asset returns can be expressed as a lnear functon of a set of K factors or ndexes. 4. APT assumes that, n equlbrum, the return on a zero-nvestment, zero-systematcrsk portfolo s zero when the unque effects are dversfed away Return Generatng Process Under Equlbrum The APT assumes that the return on any stock s lnearly related to a set of factors also referred to as systematc factors or rsk factors wth followng specfcatons: R... t 1k1 1k1 2k2 3k3 J kj for =1..N (25.1) Where, R t s the return on stock. expected return on stock I f all factors have a value of zero. k J s value of Jth factor whch nfluences the return on stock (j = 1,..J). J s the senstvty of stock s return to the Jth factor. s the random error term whch has a 2 mean of zero and varance of e. 2 Shanken, J. (1985), Mult-Beta CAPM or Equlbrum-APT?: A Reply, The Journal of Fnance, Vol. 40 No. 4, pp

3 Smlar to the CAPM model, the APT assumes that the unque effects (ε) are ndependent and wll be dversfed away n a large portfolo. Further more, the error term s expected to have a mean value of zero,.e., E( ) 0 and s uncorrelated across securtes,.e., E( ) 0for all and j where j. The assumpton of uncorrelated j error term or unque rsk component of the securty wth each other as wth all rsk factors s essental as t s expected that all covarance relatonshps between securtes can be captured through ther relatonshps wth J rsk factors. Gven the above mentoned return generatng process the APT establshes an equlbrum rsk return relatonshp whch gudes the basc dea that, two dentcal portfolos that have the same rsk can not offer dfferent expected returns. If t happens the arbtrageurs wll step n and ther actons wll ensure that the law of one prce holds good. The equlbrum relatonshp accordng to the APT s as follows: ER ( ) J J (25.2) Where, ER ( ) s the expected return on stock, 0 s the return on the rsk-free asset, J s the senstvty f stock to rsk factor J. J s the rsk premum for the type of rsk assocated wth factor J. The factors that have been suggested by the APT lterature wth respected to ther nfluence on securty returns are: the growth rate n ndustral producton ( busness cycle rsk), changes n expected nflaton, changes n market rsk premum as measured by the yeld dfferental between long term corporate bond and government bonds, changes n ol prces, nvestor confdence rsk (smlar to market rsk premum) measured as 20-year corporate bond return mnus 25-year government bond return. In ths regard a postve factor realzaton ( J > 0) mples the yeld spread between rsky corporate bonds and safe government bonds has decreased ndcates ncreased nvestor confdence or postve confdence rsk. Market tmng rsk measured as that part of the S&P 500 return not explaned by the frst four factors. Ths captures the mpacts of earthquakes, poltcs, and pure bull or bear markets. In ths case f exposure would equal CAPM beta f the frst four factor exposures were all zero APT : A Graphcal Illustraton Let us consder a one factor APT model gven as: r.. (25.3) o 1 3

4 Where, o and 1 are constants and returns are generated by one factor. It s the equaton of a straght lne (APT asset prcng lne) meanng that n equlbrum there wll be a lnear relatonshp between expected returns and senstvtes. Suppose there are two assets A and B wth expected return and rsk characterstcs as shown n Fgure Followng the above specfcaton of APT any securty that has factor senstvty and expected return that wll be appear off the lne and wll be msprced. (Fgure 25.1) Accordng to APT theoretcal argument such securty wll present nvestors an opportunty of formng arbtrage portfolos. Here securty A and B are the concerned securtes that gve a smlar framework for analyss. Here both A and B volate the law of one prce snce two assets wth same level of rsk tradng at dfferent prces and offer dfferent expected return. Securty B s relatvely over prced (lower expected return) and securty A s relatvely under prced (hgh expected return) wth equal magntude of rsk. Ths offers an arbtrage opportunty or a chance to make rsk less proft. As per the APT framework the basc premse s that when two or more securtes or portfolos that provde the same pay-off to the nvestors (.e., same level of rsk) are same and must therefore sell at the same prce. Ths premse s also consstent wth the law of one prce. In other words, f there are two securtes or portfolos that have same rsk but dfferent expected returns, nvestors wll arbtrage by buyng securty wth hgher expected return (low prce), and wll sell the securty wth lower expected return ( hgh prce). Ths process of buyng and sellng wll elmnate the prce dfference untl the two securtes wll have same expected returns. The buyng and sellng actvty of nvestors wll cause the prce of the securty of the hgher expected return to rse relatvely to the one wth the lower expected return. APT consders all assets lke A and B, that are n the 4

5 same rsk class to be perfect substtutes and should be traded at the same equlbrum prce level. In fgure 25.1 the nvestor followng APT approach wll go short on securty B and long on securty A. Ths approach wll not nvolve any addtonal commtment of funds. The nvestor wll pay for securty A by usng the sales proceedngs of the securty B. Furthermore, snce securty B and securty A have the same senstvty to the factor, the sellng of securty B and buyng of securty A wll consttute an arbtrage portfolo wth no senstvty to the factor. Fnally, the new arbtrage portfolo wll have a postve expected return because the expected return of securty A s greater than expected return of securty A. The above process wll rase securty A s prce and ts expected return wll fall untl t s located on the APT asset prcng lne Example Let us consder a stuaton where returns are generated by two factors gven by the followng specfcatons: Er ( ) Let us also assume that there are three portfolos wth the followng rsk return characterstcs: Portfolo Expected Return Rsk ( 1 ) Rsk ( 2 ) L M N Here we can see that portfolo L and M are correctly prced whereas portfolo N s overprced. Put t dfferently, expected return on portfolo N s not compatble wth the underlyng rsk measured by 1 and 2. Ths offers a scope for arbtrage. Here t s worth to menton that arbtrage s the process of earnng rsk less profts by takng advantage of dfferental prcng for the same fnancal asset or securty. Here we can construct an arbtrage portfolo S, whch has the same rsk characterstcs as that n portfolo N. Ths new portfolo S can be constructed by combnng portfolos L and M n equal weghts. Thus the rsk return characterstcs of portfolo S wll be: Expected Return or E ( r S ) Rsk ( ) Rsk ( S 1 ) S 2 (0.50 x 15.4)+(0.50 x 16.6) (0.50 x 0.80)+(0.50 x 1.20) (0.50 x 1.20)+(0.50 x 0.80) Here we can see that portfolo S has the same systematc rsk as portfolo N, but offers a hgher return than portfolo N. Because of ths arbtrage opportunty, the arbtrage portfolo can be constructed sellng the relatvely overprced portfolo and 5

6 buyng the under prced portfolo. In other words the nvestor wll try to take a short sellng poston for portfolo N and go for a long on portfolo S. Followng table llustrates how an arbtrage portfolo can be constructed to beneft from the dfferental return gven by portfolo N and S. Suppose by the short sellng portfolo N the nvestor can realse an nflow say Rs. 20,00. It can assumed that the sales proceed s realsed mmedately. Ths can also be treated as negatve value for the nvestors nvestment. Investng the same proceedngs n the arbtrage portfolo S wll make the net nvestment n the arbtrage portfolo zero. Gven the expected return of the portfolo N and S as 14 percent and 16 percent respectvely, the return on portfolo N wll be Rs. 2,800 and on portfolo S wll be +Rs.2,800. The net return n the arbtrage portfolo wll be Rs Lookng at the rsk of arbtrage portfolo, snce beta of both the portfolos N and S are equal n magntude and opposte n sgn, beta or net rsk of the arbtrage portfolo wll be zero. Ths s because beta of a portfolo s equvalent to the sum of weghted average of the betas of the ndvdual assets. Portfolo Investment (Rs.) Return (Rs.) Rsk ( ) Rsk ( 1 ) 2 N -20,000-2, S +20,000 +3, Arbtrage Proft The arbtrage portfolo offers a proft potental wthout any net nvestment and rsk. Ths rsk less proft potental attracts nvestors who shot sell portfolo S and go long on portfolo N. Ths ncreases the prce of portfolo N and decreases the prce of portfolo S. Ths process of arbtrage goes on untl the return dfferental for assets n the same rsk class are elmnated. That s, both portfolos le on the same arbtrage prce lne, ensurng equlbrum rsk return relaton for the securtes Testablty of the APT The emprcal testng of APT has tself not been wthout controversy. The common argument s that, APT s so mprecse that t makes t mpossble ever to test whether the APT s true or false. The most notable fndngs of APT has been proposed by researchers (notably Roll and Ross, 1980) 3 as a testable alternatve and perhaps natural successor, to the CAPM. However, Roll and Ross (1984) acknowledged that the number of rsk factors dffer wth 30 stocks versus 240 but contended that the mportant consderaton s whether the resultng estmates are consstent because t s not feasble to consder all of 3 Rchard Roll and Stephen A. Ross (1980), An Emprcal Investgaton of the Arbtrage Prcng Theory, Journal of Fnance, Vol. 35 No. 5, PP

7 the stocks together. 4 Cho, Elton, and Gruber (1984) tested the APT by examnng the number of factors n the return-generatng process that were prced. 5 An alternatve set of tests of the APT consders anomales lke the small-frm effect and the January effect that are not explaned by CAPM. In ths regard, Renganum (1981) found that APT can account for the dfferences n average returns between small frms and large frms. 6 In another study Gultekn and Gultekn (1987) found that the APT model can adjust for the large returns n January as compared to any other month.e., January effect. 7 However, other researchers have clamed ts applcablty wth cauton as to the nconclusve evdence from the APT theory wth respect to the choce of rsk factors. For nstance Shanken s 8 challenge to the testablty of APT can be outlned as follows: If returns are not explaned by a model, t s not consdered rejecton of a model; however f the factors do explan returns, t s consdered support APT has no advantage because the factors need not be observable, so equvalent sets may conform to dfferent factor structures Emprcal formulaton of the APT may yeld dfferent mplcatons regardng the expected returns for a gven set of securtes Thus, the theory cannot explan dfferental returns between securtes because t cannot dentfy the relevant factor structure that explans the dfferental returns 4 Rchard Roll and Stephen A. Ross (1984) A Crtcal Re-Examnaton of the Emprcal Evdence on the Arbtrage Prcng Theory, Journal of Fnance, Vol. 39 No. 2, pp D. Chnhyung Cho, Edwn J. Elton, and Martn J. Gruber (1984) On the Robustness of the Roll and Ross Arbtrage Prcng Theory, Journal of Fnancal and Quanttatve Analyss, Vol.19 No. 1, pp Mark R. Renganum (1981) The Arbtrage Prcng Theory: Some Emprcal Results, Journal of Fnance, Vol. 36 No.2, pp Mustofa N. Gultekn and N. Bulent Gultekn (1987) Stock Return Anomales and the Tests of APT, Journal of Fnance, Vol.42 No. 5, pp Shanken, J. (1985), Mult-Beta CAPM or Equlbrum-APT?: A Reply, The Journal of Fnance, Vol. 40 No. 4, pp

8 Addtonal Readngs: Alexander, Gordon, J., Sharpe, Wllam, F. and Baley, Jeffery, V., Fundamentals of Investment, 3 rd Edton, Pearson Educaton. Bode, Z., Kane, A, Marcus,A.J., and Mohanty, P. Investments, 6 th Edton, Tata McGraw-Hll. Fsher D.E. and Jordan R.J., Securty Analyss and Portfolo Management, 4th Edton., Prentce-Hall. Jones, Charles, P., Investment Analyss and Management, 9 th Edton, John Wley and Sons. Prasanna, C., Investment Analyss and Portfolo Management, 3rd Edton, Tata McGraw-Hll. Relly, Frank. and Brown, Keth, Investment Analyss & Portfolo Management, 7th Edton, Thomson Soth-Western. Addtonal Questons wth Answers Sesson 25: Arbtrage Prcng Theory 1. Explan the meanng of Arbtrage Prcng Theory (APT) and ts major assumptons? Ans: Arbtrage Prcng Theory: Multple factors expected to have an mpact on all assets. Smlar to the CAPM, the unque effects of multple factors are ndependent and wll be dversfed away n a large portfolo. APT assumes that, n equlbrum, the return on a zero-nvestment, zero-systematc-rsk portfolo s zero when the unque effects are dversfed away. Assumptons: Captal markets are perfectly compettve Investors always prefer more wealth to less wealth wth certanty The stochastc process generatng asset returns can be expressed as a lnear functon of a set of K factors or ndexes R t E b b... b t 1 2 k k 8

9 2. What s the dfference between CAPM and APT? Ans: CAPM s crtczed because of the dffcultes n selectng a proxy for the market portfolo as a benchmark. APT does not assume: A market portfolo that contans all rsky assets, and s mean-varance effcent Normally dstrbuted securty returns Quadratc utlty functon Contrast ths wth CAPM s nsstence that only beta s relevant, APT argues that multple factors expected to have an mpact on all assets: Inflaton Growth n GNP Major poltcal upheavals Changes n nterest rates etc. 3. What s the mplcaton of emprcal evdence wth respect to Arbtrage Prcng Theory? Ans: Implcaton of varous emprcal evdence wth respect to Arbtrage Prcng Theory: Not Conclusve as the number of factors reman unexplaned If returns are not explaned by a model, t s not consdered rejecton of a model; however f the factors do explan returns, t s consdered support APT has no advantage because the factors need not be observable, so equvalent sets may conform to dfferent factor structures Emprcal formulaton of the APT may yeld dfferent mplcatons regardng the expected returns for a gven set of securtes The theory cannot explan dfferental returns between securtes because t cannot dentfy the relevant factor structure that explans the dfferental returns Not able to answer varous anomales persstent n stock market: Small-frm effect, momentum effect, January anomaly APT s dffcult to put nto practce n a theoretcally rgorous fashon. Multfactor models of rsk and return attempt to brdge the gap between the practce and theory by specfyng a set of varables. 9