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1 DOCUMNTS D TRBALL D LA FACULTAT D CIÈNCIS CONÒMIQUS I MPRSARIALS Colleccó d conoma The relatonsh of catalzaton erod length wth market ortfolo comoston and betas Jord steve Comas (*) Dídac Ramírez Sarró (**) Unversty of Barcelona Adreça corresondènca: Deartament de Matemàtca conòmca, Fnancera Actuaral Facultat de Cènces conòmques mresarals Unverstat de Barcelona Av. Dagonal Barcelona (San) Tel.: Fax: mal.- esteve@ub.edu; dramrezs@ub.edu (*) Full rofessor at the Unversty of Barcelona. Hs doctoral thess was enttled An aroach to ortfolo theory startng from lnear ndex models. Alcaton to Sansh mutual funds. Member of the IAFI Grou. (**) Professor at the Unversty of Barcelona. Head of the Deartment of conomc, Fnancal and Actuaral Mathematcs. Member of the Sansh Royal Academy of conomc and Fnancal Scences. Head of the IAFI Grou.

2 Abstract: Beta coeffcents are not stable f we modfy the observaton erods of the returns. The market ortfolo comoston also vares, whereas changes n the betas are the same, whether they are calculated as regresson coeffcents or as a rato of the rsk remums. The nstantaneous beta, obtaned when the catalzaton frequency aroaches nfnty, may be a useful tool n ortfolo selecton. JL Classfcaton: G, G2 Keywords: CAPM, Perod Catalzaton, Beta, Portfolo Comoston, Instantaneous Beta. Resumen: Los coefcentes beta no son estables s se modfca la duracón de los erodos en los que se mde la rentabldad de los actvos. La comoscón de la cartera de mercado tambén varía. Los cambos en las betas son los msmos s éstas han sdo calculadas como coefcentes de regresón o como cocentes de rmas de resgo. La beta nstantánea obtenda cuando la frecuenca de catalzacón tende a nfnto uede ser utlzada como herramenta en la seleccón de carteras.

3 . INTRODUCTION The beta of an asset n relaton to market ortfolo M admts two defntons: as a lnear regresson coeffcent: β (LR) Covarance between returns on asset and returns on market M Varance n returns from market ortfolo M and as a quotent of the rsk remums: () β xected return asset - (Return rsk-free asset) xected return M - (Return rsk-free asset) Rsk remum asset Rsk remum M M The Catal Asset Prcng Model (CAPM) underles the second way of calculatng the beta coeffcent above. The model assumes that betas of the dfferent assets have been obtaned accordng to () and uses a set of hyotheses to demonstrate that the rsk remum on asset s equal to the beta of asset multled by the rsk remum on the market ortfolo M. Thus, mathematcally: ( LR) ( ) β β (3) So, from ths exresson, the followng one s obtaned: ( LR β ) M (4) Wth the CAPM, t does not matter whch of the two formulae () or (2) calculates the beta. Thus, two dfferent calculaton rocedures exst for any sngle beta. As a consequence of (3), one usually seaks of the beta wthout worryng about whch of the two rocedures s used. () (2)

4 Whether () or (2) s emloyed, the hstorcal data on the dstrbuton of the return on the asset need to be known. Accordng to CAPM assumtons, for any asset, these returns wll refer to a sngle erod,.e. quarterly or monthly or so forth. The erod length s constant. However, t has been roven that a change n the erod length modfes the value of the betas. Thus, for nstance, n Meucc, A. (2005), the beta s defned deendng on the erod nterval. Snce the betas vary, we wondered whether, n theoretcal terms, t was ossble to fnd a functonal relatonsh between the betas and the erod. Furthermore, no emrcal observatons exst on market ortfolo comoston varaton as a functon of erod length. We wondered f ths varaton dd ndeed exst and whether t was ossble to fnd a functonal relatonsh theoretcally. Fnally, we wondered about the lmt of such a functonal relatonsh when the erod length aroaches 0. In accordance wth the CAMP assumtons and the statonarty and ndeendence of the return dstrbutons n dfferent ntervals, the urose of ths aer s as follows: Frst, we establsh the functonal deendence between market ortfolo comoston and erod length (see theorem ). Second, we demonstrate that there s a functonal deendence between erod length and betas (see theorem 2). Thrd, we ntroduce the concets of nstantaneous beta and nstantaneous market ortfolo and we obtan ther formulae (see secton The CAPM when erod length aroaches 0 ). Although ths aer has a theoretcal character, the aforementoned results have ractcal consequences for nvestors. For nstance, n ther daly ractce they have to For an emrcal study of ths knd aled to shares n the Sansh stockmarket ndex, the Ibex35, see Fernandez, P. (2004). 2

5 consder the way the betas are calculated so as not to oerate wth non-homogeneous magntudes. Ths aer s dvded nto fve arts. Frst of all we state the mathematcal notatons and ror assumtons. Next we obtan the market ortfolo comoston and, n the thrd art, betas are obtaned as a functon of deendence on the erod length. In the fourth art, we develo the CAPM when erod length aroaches 0 and, secfcally, we obtan nstantaneous betas. Fnally, we brng together the results and suggest some ossble avenues for future develoment. 2. MATHMATICAL NOTATIONS AND PRIOR ASSUMPTIONS Gven a grou of N rsky assets ( I {,2,..., N} ) and a rsk-free asset 0, for assets I, let r ( N ) be the random varable for the return on asset n any erod length. Let 0 be the catalzaton erod length, assumed to be constant and exressed n years. We also assume that the dstrbuton functons of the corresondng returns ( 0 r ) r satsfy the followng hyotheses: () They are known. () For a concrete asset I, they are dentcal for dfferent tme ntervals of length 0. () They have fnte varance. (v) For unconnected tme ntervals of erod length 0, they are ndeendent (both for a sngle asset and for varous assets). 3

6 Hyothess () suoses that, as n the CAPM model, all nvestors oeratng n the market have the same exectatons; () mles that these exectatons are statonary,.e. they do not vary from one erod to another; () requres the varance of returns to be fnte, snce otherwse () could not be calculated; (v) mles that assets behave n accordance wth the random - walk. 2 Let: A ( ) ( + r + r ) be the exected fnancal factor for a erod length, corresondng to asset. 0 Let A A ( ) ( ) Let r r 0 be the effectve return on a rsk-free asset referrng to a erod length ( ) ( ) Let A A 0 + r 0 be the certan fnancal factor Let t/ 0 (>0 s the new erod length under consderaton) It follows that for a erod length : ( ) ( ) / 0 A + r ( ) ( ) 0 o + r o + r o t A 0 t (5) Let () A () -A () Nx 0 be the column vector of the N rsk remums. Nx 2 In addton, note that () gves rse to two versons of the study, deendng on whether the hyothess s alcable or not to the market ortfolo M. The frst verson wll be the subect of another artcle. Here we dscuss the second verson. Hyothess () wll only be alcable to M f t admts the entrance of new assets and the ext of others. Otherwse, the return of M could not be statonary, snce the assets wth a hgher exected return would be more lkely to ncrease ther weght n the ortfolo than those assets wth a lower exected return. 4

7 From (38) (see aendx), the vector column corresondng to the mean rsk remums for a erod length s: () At At Nx 0 (6) Nx Let be the covarance matrx of the N rsky assets, where: NxN cov( r, r ) cov( r, r ) ( N, N) + + are the covarances of the returns on assets and corresondng to one sngle erod of length. Let V be the varance of asset for an nterval wth length. Let X () M, be the market ortfolo comoston vector. Nx 3. MARKT PORTFOLIO COMPOSITION AS A FUNCTION OF PRIOD LNGTH In the roof of the CAPM equaton, the relatve weghts n the market ortfolo of the N assets are roortonal to the vector resultng from the multlcaton of the nverse of the covarance matrx by the column vector of the rsk remums. 3 Dvson of the resultng vector by the sum of ts comonents gves a vector wth the sum of ts comonents equal to. Ths vector rovdes the relatve weghts of the dfferent assets n the market ortfolo. If we aly the CAPM to a erod length, the result s: 4 3 For ths roof, see Jaqullat (989: 53-56). 4 Comments on exresson (7): 5

8 () () () NxN X Nx M, (7) Nx ( () () ) xn NxN Nx xresson (7) shows that the market ortfolo comoston deends on the covarance matrx and on the vector of the rsk remums. However, both the matrx and the vector deend on (see (38) and (39) and bear n mnd that t/ 0 ). As a consequence, the market ortfolo comoston deends on the length of the erod that we are consderng. Theorem : The market ortfolo comoston s determned by the followng exresson as a functon of : S () X () ( 2,,...N) (8) M, N S () where:.. n.. S 2 2 2n (9) n n nn a) The result of oeratng the numerator s an Nx vector whose comonents are roortonal to the relatve weghts of the N assets n the market ortfolo. b) () Nx reresents a row vector whose comonents are all equal to. c) The result of oeratng the denomnator s a x matrx (n fact t s a real number). It s ossble to rove that ths number s the sum of the comonents of the vector obtaned n the numerator. d) As a consequence, the fnal result s a vector whose comonents add u to. 6

9 s the determnant of the matrx resultng from the relacement, n the covarance matrx, of the vector of the rsk remums by the th column. 5 Proof: Let us consder the system of N+ equatons n N unknowns: ( ) ( ) ( e ) ( ) X xn NxN M, Nx N X í ( ) ( ) ( ), ( ) ( ) ( M X X ) M, xn NxN M, Nx (, 2...N) N X M, In the above exresson, (e ) s the th vector of the canoncal base of R n (.e., t s a vector that has all ts comonents equal to zero excet the th comonent, whch s equal to ). The frst N equatons are a consequence of the CAPM equaton (the fracton s the beta and the arenthess s the market rsk remum). The last equaton establshes that the sum of the weght of the N assets must be. Startng from the revous equaton system, by dvdng equatons 2 to N by the frst equaton, the followng lnear system s obtaned: (0) 5 Comment on rooston: a) What haens f alcaton of formula (8), whch rovdes the market ortfolo comoston, gves a negatve weght (X <0)? Concrete examles demonstrate that ths s mathematcally ossble, but whle an ndvdual ortfolo can have some negatve weghts, the market ortfolo must have, by defnton, no negatve weghts. b) A ossble soluton to the roblem outlned n ont a) s to dscard the assets that rovde negatve weghts (thus, X 0 for the assets wth ntally negatve weghts) and to reeat of the calculaton of the formulas of theorem, wthout the rows and columns for the assets wth ntally negatve weghts. The am s to fnd the best ossble aroach for choosng the most arorate soluton from the nonnegatve solutons that satsfy systems (0) and (). 7

10 N X M, í ( 2...N) ( ) N X M, N X M, Soluton of ths system gves the solutons determned by exressons (8). It can easly be confrmed that these solutons satsfy equatons systems (0) and (), f one bears n mnd the followng result: () N S NxN (2) 4. TH BTA AS A FUNCTION OF PRIOD LNGTH Theorem 2: a) Betas of the assets also deend on. b) For any, the betas of the dverse assets are roortonal to the resectve rsk remums. Ths can be exressed as follows: ( ) ( ) () ( ) xn NxN Nx β (3) xn ( ) ( ) xn xn NxN Nx Proof: We begn wth the exresson: 8

11 β M V ( ) M ( ) ( e ) NxN Nx xn NxN ( ) ( ) () xn NxN Nx Nx NxN ( ) NxN Nx NxN ( ) ( ) xn () xn NxN Nx NxN Nx () (4) Smlfcaton of the revous exresson gves: ( ) ( ) () ( ) xn NxN β Nx (5) xn NxN Nx xresson (5) shows that the betas of the dverse assets are roortonal to the resectve rsk remum. xressng (5) vectorally results n (3). Corollary : The beta of the market ortfolo M s equal to for any real >0. Proof: For any, the sum of the revous betas, weghted by the assets, s equal to. In effect, (7) and (3) gve: 9

12 N ( ) X β X M, β M, xn Nx ( ) ( ) () xn ( ) NxN Nx ( ) ( ) ( ) xn xn NxN Nx NxN Nx β ( ) M ( ) () xn NxN Nx (6) Corollary 2: For any real we have β (7) M Proof: From (7), the rsk remum on the market ortfolo s obtaned by weghtng the rsk remum of the N assets: N X X M. M M, xn Nx ( ) NxN Nx xn ( ) ( ) () xn NxN Nx ( ) ( ) ( ) xn NxN Nx (8) ( ) M ( ) ( ) ( ) xn NxN Nx By alyng (8) to (5), we obtan (7) 0

13 Corollary 3: For any real ostve the beta obtaned as a regresson coeffcent concdes wth the beta obtaned as a quotent of the rsk remums. Proof: It s ossble to calculate the beta as a regresson coeffcent drectly as a quotent of the rsk remums: ( ) ( ) ( ) ( ) β (9) ( ) N ( ) ( ) X M M, X M, xn Nx Alcaton of (7) gves: ( ) ( ) β ( ) NxN Nx xn ( ) ( ) () xn NxN Nx ( ) ( ) () xn NxN Nx xn NxN Nx whch s dentcal to (5) (20) 5. TH CAPM WHN PRIOD LNGTH APPROACHS 0 If we calculate the followng lmts:

14 ( ) ( ) ( ) ( ) xn NxN Lm β Lm Nx 0 0 (2) xn NxN Nx ( ) NxN Lm Nx X Lm 0 M, (22) Nx 0 ( ) ( ) ( ) xn NxN Nx we obtan the nstantaneous beta β ( INST) and the nstantaneous market ortfolo comoston X ( INST), resectvely. When calculatng the lmt when aroaches 0 n (9) and alyng L Hôtal s rule, we obtan: ( INST) ρ ρ ρ ρ β 0 0 (23) N X ( INST) ρ ρ ρ ρ M 0 M, 0 where: ρ Ln( A ) ; ρ Ln( A ) o 0 ; ρ N X ( INST ) M M, (24) In addton, we obtan the nstantaneous market ortfolo comoston by calculatng the lmt, when aroaches 0, through alyng both L Hôtal s rule N tmes and the dervaton rules for determnants on (8): S ( INS) ( ) X INS (,2,...N) N S ( INS) (25) 2

15 where: A Ln... Ln... Ln N + A2 + A A A 0 N S ( INS) A Ln Ln 2... Ln + 2N A A A A A N (26) A Ln + N... Ln N... NN Ln A A A + 2 N 0 A N Usng the nstantaneous market ortfolo and nstantaneous betas has the advantage that the value of the above mentoned magntudes does not deend on the ntal erod 0. Thus, the lmts mentoned above enable us to unfy crtera when dealng wth the varables related to the CAPM. 6. CONCLUSIONS When the CAPM s aled to a grou of N assets assumed to have statonary and ndeendent return dstrbutons, for dfferent erods, the results obtaned deend on the erod length. In short, when vares, the followng varables also change: The market ortfolo comoston vector. The rsk remum on the N assets. The rsk remum market return. Betas of each of the N assets. Therefore, we have as many CAPM models as ostve values of. 3

16 The market ortfolo comoston vector and the vector of the betas of the dverse assets vary, nasmuch as the erod length vares n whch the returns on the assets are measured. In artcular, exressons (7) and (3) show these results, whch rovde, as a functon of, the market ortfolo comoston and the vector of the betas resectvely. In both exressons, the nverse of the covarance matrx and the vector of the rsk remums ntervene. The lmt when aroaches 0 s relevant. In ths case, we wll obtan nstantaneous betas and the nstantaneous market ortfolo comoston. The followng requres further study: a) The vectoral functon: R+ R N β (27) Nx as well as each of the comonents of ths vectoral functon (they are real functons of a real varable). b) The vectoral functon: R+ R N X (28) M, Nx as well as each of the comonents of ths vectoral functon (they are real functons of a real varable). c) It s artcularly mortant to study the ntervals n whch the comonents ncrease and decrease, as well as the lmts at zero and nfnty relatng to each comonent of the two revous vector functons, usng the rsk remums and the covarances between the varous assets. It would be nterestng to analyse whether the evoluton of the varables lnked to the assets wth a beta below s qualtatvely 4

17 dfferent from that of the assets wth a beta larger than. It would also be useful to analyse the behavor of ossble negatve betas. The greatest dffculty n achevng these three obectves les n obtanng, for any, the general exresson of the nverse of the covarance matrx and, n artcular, ts lmt when aroaches 0. 5

18 APPNDIX Set out below are the statstcal roertes of the means, varances and covarances when modfyng the erod length n whch returns are measured. In the roertes A) and A2) of ths aendx we do not suose that the returns on assets are statonary. A) Proertes for erod length K 0 (K ostve nteger). 0 K ) ( + r (K ) ) ( (r )) + (29), ( ( r 0 ) s the return on asset n the erod of length, 0 ) Proof: Snce the random varables are ndeendent of each other, followng Cramer (962:23) the exectaton of the roduct s a roduct of exectatons, as a result of whch the roerty s demonstrated. ) corollary: If the random varables corresondng to every erod of length 0 have the same dstrbuton, A, A, then we have the followng result: ( 0 ( ) K + r K ) A AK (30) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0 2) K K K A ),, A - A,, A + (3), ( ) ( 0 s the covarance between the returns on assets and n the erod wth, length 0 ) 6

19 Proof: ( K 0) ( 0) ( 0) ( 0) ( 0 K K K K ) ( + r A, )( r A ) +, ( 0) ( 0) K ( 0) K ( 0 K r r K A ) + +, A, K ( 0) ( 0 ( r )( ) K +, r ) + ( 0) K 0 ( ), A,, A K ( 0) ( 0) ( 0) ( 0 ( )( K K r ) ) +, + r A,, A, K ( 0) ( 0) ( 0) ( 0) ( 0 K ) + A,, A A,, A, 2 ) corollary: If the random varables are statonary -.e. f for every erod of length they have the same dstrbuton - we have: ( 0) K K K K K ( A, A ) A, A ( A A ) ( A A +,,, ) t t t t t t + t (32) ( 0 3) V K) (V + A 2K ) -A2K (33) Proof: Ths s the result of roerty 2, makng B) Proertes for erod length 0 /m (m ostve nteger). In ths secton t s assumed that the dstrbutons of returns on the varous assets are statonary for erods of length 0 /m. 7

20 0 ) ( + r ( /m) ) A/m (34) Proof: On alyng roerty of secton A), takng Km, we have: m ( 0/ ) ( 0 ( m r ) ( r ) A ( r / m) ) A/ m ( 0/ ) ( 0 2) m ( ) AA ) / m-(aa ) / m + (35) Proof: On alyng roerty 2 of secton A), makng Km, and roerty B), we have: ( 0) 0 ( ( /m) A /m A /m ) m -(A /m A/m) m + (36) 0 ( / m) On solvng we have the equalty that we sought to demonstrate. ( 0/ ) ( 0 3) V m (V ) A 2/ ) m-a2/ m + (37) Proof: Ths results drectly from roerty 2) above, takng. C) Proertes for erod length (t 0, t ostve real). In ths secton we assume that the dstrbutons are statonary. By combnng the roertes of A) and B) n ths aendx, we fnd that, for any ostve real t/ 0, the followng occurs: ) ( + r ) t A 2) ( + A A ) t ( A A ) t 3) V ( V A2) t A2t + ( 38) ( 39) ( 40) 8

21 RFRNCS Cramer, H. (962), Random varables and robablty dstrbutons, Cambrdge Unversty Press, age 23. Fernández, P. (2004), Valoracón de emresas, Gestón Madrd. Jaqullat, B. B. Solnk (989), Marchés Fnancers, Dunot, Pars, Meucc, A. (2005), Rsk and Asset Allocaton, Srnger-Verlag, ,