The Small Noise Arbitrage Pricing Theory

Transcription

1 The Small Nose Arbtrage Prcng Theory S. Satchell Trnty College Unversty of Cambrdge and School of Fnance and Economcs Unversty of Technology, Sydney December 998 Ths aer was wrtten when the Author was vstng the Unversty of Technology, Sydney. He would lke to thank A. D. Hall, J. Knght and S. J. Ln for ther helful advce and comments.

2 Abstract Ths aer resents a small-nose verson of the Arbtrage Prcng Theory (APT) whch allows us to nterret the aroxmate lnearty of the rsk rema n terms of factor exosures for a fxed number of assets. The aroxmaton becomes more accurate as the nose of the system decreases, even though the number of assets stays fxed. JEL Codes: Key Words: G Arbtrage Prcng Theory, Lnear Factor Models

3 Secton Introducton Arbtrage Prcng Theory (APT) stresses the aroxmate lnearty of the rsk rema of assets n terms of the factor loadngs. The basc assumton s that returns are generated by a lnear factor model and that the above lnear aroxmaton becomes ncreasngly accurate as N, the number of assets, ncreases to nfnty. The orgnal aer by Ross (976) has been generalsed n many drectons. The lnear factor model has been dsensed wth by ansal and Vswanathan (993). The roblem has been ut n an equlbrum settng by Connor (984) and has been gven a very general factor structure by Chamberlan and Rothschld (983) and Resman (99). Many other mortant aers have been ut forward, but they all stress the aroxmate lnearty of the rsk rema for large N. It should be sad that there s an nterestng subqueston as to when the lnear APT restrcton holds exactly, see Connor (993) and Huberman and Kandel (987) among many others. The contrbuton of ths aer s to demonstrate that t s not just large economes that matter for the aroxmaton; t s also the magntude of dosyncratc nose. Econometrcans of a certan age wll not be surrsed by ths asserton. In the smultaneous equaton lterature, asymtotc roertes were analysed both as large samle calculatons or, alternatvely, as small sgma (nose) calculatons, see Kadane (97) for examle. Our goal s to construct small nose bounds for the APT. The consequences of ths exercse have some emrcal meanng. In artcular, a market wth a small number of assets may stll be subject to the APT restrctons f most of the rsk to assets n that market s factor rsk. Lkewse, large markets that have substantal dosyncratc rsk may have large devatons from the APT restrctons, a ont that s already understood. A dscusson of relevant lterature s resented n Secton, together wth defntons of a system nose arameter. A small nose APT theorem s resented n Secton 3, together wth dscusson and conclusons. Secton Assume the followng lnear model: x = E + δ + ε where x s a (N x ) vector of asset returns, E( x ) = E, (.) E( δ ) = E( ε ) = and δ s a (k x ) vector of factors, s an (N x k) matrx of loadngs of rank k and ε has a dagonal (N x N) covarance matrx, Σ = (σ ). It s further assumed that δ and ε are uncorrelated. efore roceedng to dscuss how to quantfy system-wde dosyncratc rsk, we dscuss some results n the lterature that ont to lnks between the exact APT restrcton holdng and the absence of dosyncratc rsk. For ths model, the exact APT restrcton s defned as ones and λ and λ are (k+) constants. E = λ + λ where s a (N x ) vector of Frstly, t s well-known, see Ingersoll (987), that f ε =, then the exact APT restrcton holds. Secondly, Connor (984) resents an economy, called an nsurable factor economy, where each nvestor s equlbrum ortfolo conssts of a lnear combnaton of (k+) mutual funds, see Corollory. (g, o ct). These mutual funds are smly ortfolos that fully relcate the factors and have zero dosyncratc varance. He shows that n such an economy, see theorems and 3, the exact APT restrctons holds for fnte N. In ths economy, although dosyncratc rsk exsts, t s not resent n equlbrum. Lkewse Huberman and Kandel (987) resent results whch show that the exact APT

4 restrcton holds f the vector of mutual funds corresondng to the factors ntersect the mean-varance fronter of the assets, see Prooston. The ratonale s the same. Agan, the exstence of the ortfolo of factors/mutual funds on the fronter allows nvestors to hold an otmal ortfolo that contans no dosyncratc rsk. Lastly, the nfnte economy and lmtng economy arguments all have the mlcaton that the dosyncratc rsk should be reduced by dversfcaton, e, by holdng a large enough number of assets. Taken together, these results suggest strongly that the arguments for the APT should be restructured n terms of nose/dosyncratc rsk, and that an alternatve way to thnk about when the APT restrcton holds can be rehrased n terms of dosyncratc nose. We now turn to ssues concerned wth the defnton of a scalar measure of system nose. In equaton (.), Σ has been defned to be dagonal wth th element σ. A useful system measure of nose should have the roerty that as t tends to zero, the σ should all tend to zero. Accordngly, defne σ = d, =, N (.) and set d jj =, where j = arg max (σ ). It follows mmedately that lmσ =. Snce we are takng contnuous lmts here, rather than lmtng on a sequence as s usual n the APT lterature, consderaton needs to be gven to what a contnuum of dfferent economes mght be. Defne a contnuum of economes as r = E + δ + ε where, for + R + R s the set of non-negatve real numbers. Now consder the concet of small arbtrage. Consder a contnuum of ortfolos { x }, x R +. Each ortfolo x has tycal th element x and s of length N, where N s fxed. Assume the followng: Assumton. N (a) x = = (b) x E( ) δ > r j (c) x x cov( r, r ) as. j The above assumtons tell us that; (a), x s a hedge fund, (b) the exected return on the ortfolo s bounded above zero, and (c) the varance of the ortfolo tends to zero as tends to zero. It s arorate to gve some ratonale as to what a contnuum of economes mght mean. The usual assumtons n APT studes are that the return dstrbutons dscussed n () are exogenous. In the case of economes, these dfferent dstrbutons could be thought of as arsng from varous actons by regulators whch nhbt dosyncratc nose, or erhas from the ntroducton of mutual funds that effectvely mmc some of the rsk factors, and whose ntroducton leads to new exogenous dstrbutons through some external equlbratng rocess that leaves the nose comonent less nosy. 3

5 It s straghtforward to demonstrate that an ndvdual wth an ncreasng concave exected utlty functon U wll, ceterus arbus, refer less system-wde nose to more. Let the ndvdual's otmzed return be r where r k = e + j= f + d ε, j j (.3) In the above, ε' s standardzed dosyncratc nose; the standard devaton of the nose beng d. Equaton (.3) s a consequence of equaton (.), the term d s defned by (.). Let the value of the exected utlty be denoted by V = V() where we have wrtten V n terms of the value of the system-wde nose. Intal wealth s denoted by W whch s set to. Thus, V() = E(U(+r )) If V/ s comuted, we see that V = E U ( + r r )) = d E( U ( + r ) ε ) It follows from the concavty of U and the ostve deendence of r on ε' that V/ must be negatve. Thus, a decrease n leads to all nvestors beng better off f ther otmal ortfolo remans factor equvalent to the revous otmum. That such a shft n s actually Pareto mrovng, however, does not follow from the above result wthout further argument. Secton 3 We now rove the followng rooston about small arbtrage: Prooston 3. In a contnuum of economes wth fxed N and no small arbtrage, K lm e λ j j= λ j = Proof: The roof s standard and follows Ingersoll (age 73, o ct.). Consder a best lnear redcton of e, the th element of E, on a constant and the k exosures =, k. Defne the resdual v (wth the suerscrts suressed) as j, j v = e λ K λ j j j=. N It follows that v =, and v j =. = 4

6 We construct our ortfolo as = v / v where v s the Eucldean dstance of v = (v, v,, v n )'. =,... n has the followng roertes: The ortfolo ( ) () =. () ve v e = = = v v v () var( x ) = var( e + ε ) = = v d d / v snce d. It follows that lm var( x ) = ; ths mles that lm E( x ) = lm v = to avod small arbtrage Q.E.D. In concluson, we have shown that smaller economes wth no small arbtrage lead to broadly the same results as when we ncrease the number of assets wth no asymtotc arbtrage. However, the nterretaton here s qute dfferent and, we would argue, more natural. It further rases questons as to the Pareto mrovng nature of a reducton n system nose whch t s hoed to address n further research. More ractcally, a market wth a small number of stocks could be deemed to have rsk rema comuted by the APT n the n the above framework. Tryng to assess ths emrcally for emergng markets should be an nterestng exercse. 5

7 blograhy Rav ansal and S. Vswanathan "No Arbtrage and Arbtrage Prcng: A New Aroach", Journal of Fnance, vol 48, no. 4 (993), 3-6. Gary Chamberlan and M. Rothschld "Arbtrage, Factor Structure, and Mean-Varance Analyss on Large Markets", Econometrca, vol. 5 (983), Gregory Connor "A Unfed eta Prcng Theory", Journal of Economc Theory, vol 34, (984) 3-3. Gur Huberman "A Smle Aroach to Arbtrage Prcng Theory", Journal of Economc Theory, vol. 8 (98), Gur Huberman and Shmuel Kandel "Mean-Varance Sannng", Journal of Fnance, vol 4, no. 4 (Setember 987), Johnathon E. Ingersoll Theory of Fnancal Decson Makng, 987, Rowman and Lttlefeld, New Jersey. J.. Kadane Comarson of K-Class Estmators when Dsturbances are Small, Econometrca, (Se), 97, 6-3. Ham Resman Reference Varables, Factor Structure, Multbeta Reresentaton, Journal of Fnance, vol 47, no. 4 (99), Stehen A. Ross "The Arbtrage Theory of Catal Asset Prcng", Journal of Economc Theory, vol. 3 (976),