The Effects of Systemic Risk on the Allocation between Value and Growth Portfolios

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1 Journal of Mahemaical Finance, 013, 3, hp://x.oi.org/10.436/mf a016 Publishe Online March 013 (hp:// The Effecs of Sysemic Risk on he Allocaion beween Value an Growh Porfolios Gabriel Penagos, Gonzalo Rubio Deparameno e Economía y Empresa, Universia CEU Carenal Herrera, Valencia, Spain gonzalo.rubio@uch.ceu.es Receive November 1, 01; revise December 14, 01; accepe December 7, 01 ABSTRACT Given he sriking effecs of he recen financial urmoil, an he imporance of value an growh porfolios for boh local an inernaional porfolio allocaion, we invesigae he effecs of sysemic umps on he opimal porfolio invesmen sraegies across value an growh equiy porfolios. We fin ha he cos of ignoring sysemic umps is no subsanial, unless he porfolio is highly levere an he average size ampliue of he ump is large enough. From he opimal asse allocaion poin of view, i seems more imporan he effecs of few bu relaively large umps han highly frequen bu small umps. Inee, he perio in which he value premium is higher coincies wih a perio of few, bu large an posiive average size umps for value socks, an negaive an very large average size umps for growh socks. Keywors: Sysemic Risk; Value; Growh; Asse Allocaion; Risk Aversion 1. Inroucion The value premium is one of he mos relevan anomalies iscusse in he asse pricing lieraure. Value socks, which are characerize by high book-o-marke raios, earn higher average reurns han growh socks. In principle, growh opions srongly epen upon fuure economic coniions which sugges ha growh socks shoul have higher beas han value socks. Using monhly aa from January 1963 o December 010, he marke bea of he Fama-French growh porfolio is while he marke bea of he value porfolio is Alhough conrary o he heoreical preicion hese beas are basically he same, i urns ou ha he annualize average reurn of he growh porfolio is 9.6% while he annualize average reurn of he value porfolio is 16.5%. This represens a value premium of 6.9% which is even higher ha he well known marke equiy premium magniue of 5.5% for he same sample perio. 1 A key issue of he research agena of he asse pricing lieraure is o unersan why value socks earn higher average reurns han growh socks. This paper oes no preen o answer his quesion. On he conrary, our paper akes his anomaly as given o invesigae opimal asse allocaion ecisions beween value an growh porfolios. In paricular, he 1 These numbers correspon o he 10 Fama-French monhly porfolios sore by book-o-marke, an using a value-weighe scheme o obain he porfolio reurns. The value premium using an equallyweighe approach is an even much higher 14.0% on annual basis. paper invesigaes he effecs of sysemic umps on he asse allocaion ecision beween hese wo key characerisics of equiy reurns. Recen financial crisis has shown ha he failures of large insiuions can generae large coss on he overall financial sysem. Sysemic risk is one of he main issues o be resolve by he new regulaion of financial markes over he worl. I seems wiely accepe ha previous regulaion focuses excessively on iniviual insiuions ignoring criical ineracions beween insiuions. 3 These ineracions are he leaing source of sysemic risk aroun he worl. From his poin of view, i is imporan o analyze he impac of sysemic risk on porfolio asse allocaion among poenial insiuional invesors. Given ha wo of he mos popular invesmen sraegies of large insiuional invesmen employ value an growh asses, his paper analyzes how relevan is o ignore sys- In [1] i is argue ha cosly reversibiliy an counercyclical marke price of risk cause asses in place o be harer o reuce so hey, in fac become riskier han growh opions. A relae argumen is given in [] which employs nonseparabiliy beween urable an nonurable consumpion o show ha value socks are more procyclical han growh socks. This implies ha hey perform especially baly uring economic ownurns. However, recen evience provie in [3] shows ha he q-heory ynamic invesmen framework fails o explain he value sprea. This is he case espie he fac ha in [4] i is argue ha a simple hree-facor moel inspire on he q-heory of invesmen is able o explain anomalies associae wih shor-erm prior reurns, financial isress, ne sock issues, asse growh, earnings surprises, an some valuaion raios. 3 See [5] for a clarifying iscussion of hese issues. Copyrigh 013 SciRes.

2 166 G. PENAGOS, G. RUBIO emic risk on he ex-pos performance of hese sraegies. We unersan sysemic risk as he risk arising from infrequen bu arbirarily large umps ha are highly correlae across he worl an across a large number of asses. We borrow from he mahemaical ump-iffusion moel evelope in [6] o recognize ha umps occurs a he same ime all over he worl an across all porfolios, bu allowing ha he size of he ump may be ifferen across hem. These auhors erive he opimal porfolio weighs when equiy reurns follow a sysemic umpiffusion process. We calibrae he moel o hree monhly Fama-French book-o-marke porfolios where porfolio one is compose of securiies wih low book-omarke (growh socks), porfolio 5 conains inermeiae book-o-marke asses, an porfolio 10 inclues secureies wih high book-o-marke (value socks). I is well known ha o properly escribe equiy-inex reurns one mus allow for iscree umps. As shown, among ohers, in [7] umps play an imporan role for unersaning of U.S. marke reurns over an above sochasic volailiy an he negaive relaionship beween reurn an volailiy shocks. Reference [8] also shows ha he relevance of umps characerizes he French (CAC), German (DAX), an Spanish (IBEX-35) equiy-inex European reurns. One may herefore expec umps o impac opimal asse allocaion beween value an growh porfolios. Our evience shows ha his is no necessarily he case. We fin ha he effecs of sysemic umps are inee no negligible from 198 o 1997 for low levels of risk aversion (highly levere porfolios) when he frequency of umps is small bu is average size is large. In fac, his perio is characerize by an especially large value premium. These effecs may herefore seem o be paricularly relevan for he allocaion of funs beween growh an value porfolios. When sysemic umps are recognize, he average invesor shoul opimally go long in value an shor in growh in higher proporions han when assuming a pure iffusion process. However, an raher surprisingly, an average invesor woul have no been penalize ignoring sysemic umps from 1997 o 010 when he frequency of umps is much higher bu he average magniue is also smaller. The magniue of he average umps size seems o be very imporan o assess he impac of sysemic umps on asse allocaion beween value an growh porfolios. In he overall perio an in boh sub-perios, inepenenly of using a pure iffusion or a ump-iffusion process, value socks ominae growh socks especially for highly levere porfolios. This paper is srucure as follows. In Secion, we iscuss a moel of equiy reurns ha allows for sysemic risk. Moreover, we also escribe opimal porfolio weighs when equiy reurns have a sysemic risk componen. Secion 3 iscusses he esimaion proceure, an Secion 4 presens he key resuls of he paper using he Fama- French book-o-marke porfolios. Concluing remarks are in Secion 5.. Asse Reurns, Sysemic Risk, an he Opimal Allocaion of Equiy-Reurns This secion firs presen a moel of asse equiy reurns which is base on he asse pricing moel propose in [6]. This moel inrouces sysemic risk by imposing umps ha occur simulaneously across all asses bu also allowing for a varying isribuion of he ump size across all porfolios. Seconly, we iscuss opimal porfolio allocaion given ha he unerlying asses follow a umpiffusion process wih sysemic umps. We compare hese resuls relaive o he case in which asse reurns follow a pure iffusion process..1. Asse Equiy Reurns an Sysemic Risk There is an insananeous riskless asse which follows he reurn process given by, P r P (1) where r is he consan coninuous riskless rae of reurn. Moreover, here are N risky asses in he economy. Each of hem follows a pure-iffusion process given by he well known expression, S Z ; 1,,, N () S where S is he price of asse, Z is a Brownian mo ion, is he rif, an is he volailiy, an superscrip highlighs he pure iffusion characer of he process. We enoe by he NxN covariance marix of he iffusion componens where he ypical componen of his marix is i i i wih i being he correlaion beween he Brownian shocks Z an Z. i In marix noaion, is he N-vecor of expece re urns,, where is he iagonal marix of volailiies, an is he symmeric marix of correlaions. We nex allow for unexpece rare sysemic evens by inroucing a ump o he process given in Equaion (). Following [6], we assume ha he ump arrives a he same ime across all equiy porfolios, an ha all porfolios ump in he same irecion. Then, S Z J 1 Q; 1,,, N S (3) where Q is a Poisson process wih common consan inensiy, an J 1 is he ranom ump magniue ha generaes he percenage change in he price of asse Copyrigh 013 SciRes.

3 G. PENAGOS, G. RUBIO 167 if he Poisson even is observe. I is imporan o noe ha he arrival of umps occur a he same ime for all equiy porfolios. We assume ha he Brownian shock, he Poisson ump, an he ump ampliue J are ine- J ln J has a Normal isribu- penen, an ha ion wih consan mean an variance. Therefore, he isribuion of he ump size is allowe o be ifferen for each porfolio, alhough all umps arrive a he same ime. We efine an as he rif N-vecor an he NxN covariance marix of he iffusion componens of Equaion (3) respecively. I mus be noe ha hey are now he rif an he covariance marix of he iffusion componens when here are umps in he reurn process. In his case, we also have an aiional rif, J, an an aiional covariance marix, J, from he ump componens of he process. Given ha we selec he parameers of he ump-iffusion process in Equaion (3) such ha he firs wo momens for his process mach exacly he firs wo momens of he pure iffusion process in Equaion (), i mus be he case ha, J J.. Porfolio Allocaion Our represenaive invesor maximizes he expece power uiliy efine on erminal wealh, W T, given by 1 he well known expression UWT WT 1, where 0 is he consan relaive risk aversion coefficien. We firs briefly escribe he opimal porfolio weighs using he pure-iffusion process given in Equaion (). Denoing he vecor of he proporions of wealh invese in risky equiy porfolios by, he opimal porfolio problem a is 1 W, max T V W E (5) 1 subec o he ynamic buge consrain W R r Z (6) W where R 1 N r is he N-vecor of excess reurns, is he iagonal marix of volailiies, Z is he vecor of iffusion shocks, an 1N is an N-vecor of ones. The soluion is he vecor of porfolio weighs obaine in [9] corresponing o he sanar iffusion process of Equaion (), 1 1 op R (4) (7) When he process inclues a ump componen as in Equaion (3), he ynamics of wealh for he iniial wealh W0 1 can be wrien as W Rr Z J Q (8) W where R 1 N r is he N-vecor of excess reurns, is he iagonal marix of volailiies, Z is he vecor of iffusion shocks uner a ump-iffusion process, an J J 11, J 1,, J N 1 is he vecor of ranom ump ampliues for he N equiy porfolios. One can employ sochasic ynamic programming o solve for he opimal weighs, 1 0 max UC EV W,, where in his equaion, we can use he generalize umpiffusion Io s lemma o calculae he ifferenial of he value funcion, V. 4 Then, he Hamilon-Jacobi-Bellman equaion is V W, V W, 0 max RrW W 1 V W, W W EV W WJ, V W, The impac of he umps in he reurn process is given by he las erm of Equaion (9). This erm employs he fac ha EQ, an he assumpion of inepen ence of he Poisson ump an he ump ampliue excep for he fac ha he ump size is coniional on he Poisson even happening. As usual in his ype of problems, one can guess he soluion o he value funcion as having he following form:, F V W 1 W 1 (9) (10) Replacing his soluion ino he Hamilon-Jacobi-Bellman equaion we ge: 1 F 0 max 1Rr F 1 1 E 1 J 1 1 (11) By iffereniaing wih respec o we obain he opimal weighs wih sysemic umps as he soluion for each ime o he sysem of N nonlinear equaions which mus be solve numerically: 0 Rop EJ 1 op J (1) 4 Alernaively, we may employ he infiniesimal generaor of he umpiffusion process in [10]. Copyrigh 013 SciRes.

4 168 G. PENAGOS, G. RUBIO I shoul be poine ou ha if we replace Equaion (10) ino he Hamilon-Jacobi-Bellman equaion an evaluae a op we obain: 1 F k, F where, 1 k 1 oprr 1 opop E 1 1 op J 1 We nex employ he bounary coniion FT 1 o obain, kt k F, e e Therefore, he value funcion is given by he expression V W, e k W Cerainy Equivalen Cos (13) As an aiional analysis of he effecs of ignoring sysemic risk on asse allocaion, we can also calculae he cerainy equivalen cos (CEQ hereafer) of following an allocaion sraegy ha ignores he simulaneous umps occurre in he aa. The CEQ gives he aiional amoun in U.S ollars ha mus be ae o mach he expece uiliy of erminal wealh uner he pure-iffusion subopimal allocaion o ha uner he opimal sraegy wih he ump-iffusion process of equiy reurns. In oher wors, as in [6], we calculae CEQ as he marginal amoun of money ha equalizes pure-iffusion expece uiliy wih he ump-iffusion expece uiliy. The compensaing CEQ wealh is herefore compue by equaing he following expressions: V 1 CEQW, ; op V W, ; op (14) Then, using Equaion (10) we have ha F ; op CEQ F ; op The Esimaion Proceure o Analyze he Effecs of Sysemic Risk (15) Le us sar again wih he pure-iffusion case given by Equaion (). The parameers o be esimae are, where an are he N-vecor of expece reurns an he NxN covariance marix of he iffusion componens. The momen coniions for iniviual equiy porfolios are given by S E S (16) S S i E i i Si S i This implies ha momen coniions, can be esimae irecly from he means an he covariance of he acual sample series available. On he oher han, o erive he four unconiional momens of he ump-iffusion process given in Equaion (3), we follow [6] o ienify he characerisic funcion which can be iffereniae o obain he momens of he equiy porfolio reurns process. We nex presen a eaile exposiion of he proceure employe o obain he momens uner he sysemic-ump-iffusion process The Characerisic Funcion We firs wrie he previous process in log reurns wih X ln S. I is well known ha he pure-iffusion process in Equaion () becomes X Z, (17) where 1. As before, he moel in marix noaion is X Z, where is he N-vecor of expece reurns, an is he iagonal marix of volailiies. On he oher han, he ump-iffusion reurn process can be wrien as X Z J Q (18) where 1. Hence, in marix noaion, he coninuously compoune asse reurn vecor for he ump-iffusion moel saisfies he following sochasic ifferenial equaion X Z JQ, where is he N-vecor of expece reurns, is he iagonal marix of volailiies for he ump-iffusion case, an J is he N-vecor of ump ampliues. The N-vecor of he average size of he umps ampliue is enoe by, while he iagonal variance marix of he size of he umps is enoe by. As menione before, he heoreical momens for he ump-iffusion process are calculae using he characerisic funcion which in urn can be erive from he Kolmogorov heorem. We formally work in a probabiliy space,, where, in he ump-iffusion case, boh he process Q an he Brownian moion Z generae he filraion. The coniional characerisic funcion of he process X coniione on is i XT efine as he expece value of e, where 1,, N is he argumen of he characerisic funcion,, given by i XT, XT, T,,, X Ee (19) where T. Copyrigh 013 SciRes.

5 G. PENAGOS, G. RUBIO 169 From he process X, we know ha,, an are consan on X. Therefore, uner some echnical regulariy coniions iscusse in [10], he coniional characerisic funcion has an exponenial affine form given by A,,,, e B X X (0) Accoring o he Feynman-Kac heorem, he coniional characerisic funcion is he soluion o, (1) where is he infiniesimal generaor for he umpiffusion process X given by 1 Trace X X X Y X Y where, as alreay poine ou, is he ump ampliue Normal isribuion. The bounary coniion for he ifferenial equaion is he value of he coniional characerisic funcion a he equiy porfolio horizon i XT T 0,, XT e. Thus, from Equaion (0), i mus be he case ha AT 0, 0 an BT 0, i. In orer o fin ou he expressions for he funcions A, an B,, Equa ion (0) is replace ino Equaion () o obain A B X 1 B, Y B B B e 1 Y where. Thus, he wo orinary ifferenial equaions are, 1 B, e, (), (3) A Y B BB 1 Y B, 0 (4) The secon expression in Equaion (4) implies ha B oes no epen on. Thus, using he bounary coniion, we obain B, i. We now replace B in he firs equaion, we inegrae, an we use he remaining bounary coniion o ge A 1 N (5) i Y, i e 1 Y The inegral in he righ han sie of Equaion (5) can be recognize as he ump ampliue s characerisic funcion, given ha he ranom variable J follows a Normal isribuion: N Therefore, A, J e Y expi i Y J, (6) i expi an, finally, he characerisic funcion is given by 1,, X exp i 1 J expi ix 3.. Unconiional Momens (7) Using he characerisic funcion one can erive he K comomens hroughou he following expression:,, X E X X X K k1 k kn K 1,,, N i k1 k kn 1 N 0, (8) N where K k 1. The graien an he Hessian, an are use o fin he firs an secon momens respecively. For he one-perio invesmen horizon, 1, an using he conversion from he noncenral o cenral momens we obain he mean, covariance, coskewness, an excess kurosis: N I (9) (30) 1 3 I 1N 1N (31) (3) N In he expressions above, I is he NxN marix of ones, an enoes he N-imes elemen-by-elemen muliplicaion. I shoul be noe ha 1 an 4 are N- vecors, an an 3 are NxN marices. If we now compare he mean an covariance for he 5 A echnical appenix is available from he auhors upon reques. The Appenix erives he unconiional momens using alernaive proceures. These erivaions exen an complee he mahemaical echnicaliies suggese in [6]. Copyrigh 013 SciRes.

6 170 G. PENAGOS, G. RUBIO ump-iffusion process above wih hose for he pureiffusion processes for 0, we observe ha J (33) J I (34) Then, he iffusion momens of he ump-iffusion process are rerieve using he expressions in Equaion (4) as: (35) I (36) From he momen coniions in Equaions (9)-(3) he parameers o be esimae are,,,,. For he universe of N asses here are N ump ampliue means an N ump ampliue volailiies. This represens N+1 parameers o be esimae incluing he Poisson inen- siy. On he oher han, here are N co-skewness momens an N excess kurosis momens for a oal of NN 1 momen coniions o be employe in he generalize meho of momen (GMM) esimaion proceure Sampling Momens The sampling mean an co-momens of a variable X wih respec o variable Y are: m X 1 T T 1 X 1 m X m Y T XY X Y T 1 m 1 r 1 T XY X Y r r 1 m X X XY for r 3 an X m m, where r is he ausmen erm for he unbiaseness correcion. 4. Opimal Allocaion for Value an Growh Porfolios wih Sysemic Risk: Empirical Resuls We esimae he moel using 3 porfolios from he 10 monhly book-o-marke sore porfolios aken from Kenneh French s web page. Porfolio 1 conains he companies wih low book-o-marke, while porfolio 10 inclues asses wih high book-o-marke. We refer o porfolio 1 as he growh porfolio, an porfolio 10 as he value porfolio. We also employ porfolio 5 enoe as he inermeiae porfolio. In orer o pay special aenion o hese characerisics we use he equallyweighe scheme of he iniviual socks raher han he more popular value-weighe porfolios. This weighing approach also amplifies he value effec anomaly Parameer Esimaes for he Jump-Diffusion Reurn Process Table 1 repors he escripive saisics of he hree monhly Fama-French book-o-marke porfolios for he full sample perio from January 198 o Ocober 010, Table 1. Descripive saisics for he fama-french growh an value porfolios. (a) Momens of he monhly reurns a ; (b) Correlaion coefficiens b. (a) Jan 8-Oc 10 Growh BE-ME 5 Value Mean % Volailiy % Skewness Kurosis Mar 97-Oc 10 Mean % Volailiy % Skewness Kurosis Jan 8-Feb 97 Mean % Volailiy % Skewness Kurosis (b) Jan 8-Oc 10 Growh BE-ME 5 Value Growh BM-ME Value Mar 97-Oc 10 Growh BE-ME 5 Value Growh BM-ME Value Jan 8-Feb 97 Growh BE-ME 5 Value Growh BM-ME Value a Panel A of his able repors he firs four mome ns of he monhly reurns for he gr owh an value porfolios consruce by Fama-French using heir en book-o-marke sore porfolios, where he firs porfolio is he growh porfolio, porfolio BE-ME 5 is he inermeiae porfolio, an he las porfolio is he value porfolio. The aa for he full perio goes from January 198 o Ocober 010, while he sub-perios conain aa from January 198 o February 1997 an from March 1998 o Ocober 010. b Panel B of he able gives he correlaion coefficiens among he monhly reurns for he growh, inermeiae, an value porfolios. Copyrigh 013 SciRes.

7 G. PENAGOS, G. RUBIO 171 an wo sub-perios from January 198 o February 1997, an from March 1997 o Ocober 010. Boh sub-perios conain episoes wih large negaive shocks. The firs sub-perio inclues he marke crash of Ocober 1987, he Gulf War I in Augus 1990, an he Mexican crisis in December On oher han, he secon sub-perio conains he Asian crisis of July 1997, he Russian crisis of Augus 1998, he bursing of he o.com bubble, he erroris aack of Sepember 001, he oubreak of he Gulf War II in March 003, he beginning of he subprime crisis, an he Lehmann Brohers efaul in Sepember 008. Panel A of Table 1 shows ha he use of he equallyweighe growh an value porfolios leas o an impres- sive value premium of 15.8% on annual basis for he full sample perio. Moreover, he value premium is 13.0% an 18.4% for he secon an firs sub-perios respecively. As expece, he annualize volailiy of he growh porfolio is higher han he corresponing volailiy of he value porfolio is all hree sample perios. On annual basis, he growh volailiy premium is 4.7% for he full perio, an 5.6% an 3.9% for he secon an firs sub-perios respecively. Inee, he growh porfolio seems o be riskier han he value porfolio. The problem is, of course, he enormous average reurn of he value porfolio. The hir an fourh momens of boh porfolios also presen an ineresing behaviour. For he full perio, he growh porfolio has posiive skewness, while he value porfolio has a slighly negaive skewness. Moreover, growh socks have lower kurosis han he value porfolio. The overall behaviour is he consequence of wo very ifferen sub-perios. From 1997 o 010, he value porfolio has negaive skewness, an, on he conrary, growh socks have posiive skewness. The excess kurosis is similar for boh porfolios. However, from 198 o 1997, we repor a very high negaive skewness for he growh porfolio, an a posiive skewness for value socks. Similarly, he behaviour of excess kurosis is also surprising. Value socks presen a much higher kurosis han he growh porfolio. The changes observe in boh skewness an kurosis from one sub-perio o he oher for growh an value porfolios are sriking an eserve furher aenion. For compleeness, Figure 1 shows he ensiy funcions for boh sub-perios of he growh, inermeiae an value porfolios, an he QQ plos o assess he eviaions of heir reurns from he Normal isribuion. Panel B of Table 1 repors he correlaion coefficiens among he hree book-o-marke porfolios. The resuls show ha from 1997 o 010 he value an growh por- folios are less correlae han in he previous sub-perio. Panel A of Table conains he parameer esimaes obaine by he generalize meho of momens wih he Table. Parameer esimaes for he reurns processes wih umps. (a) Parameer esimaes for he ump process a ; (b) Comparison beween higher orer sample momens of reurns an he umps process heoreical higher orer momens b. (a) Perio Jan 8 - Oc 10 M ar 97 - Oc 10 Jan 8 - Feb Avg Avg λ Year (s) (b) Jan 8 - Oc 10 Growh BE-ME 5 Value Sample Skewness Moel Skewness Sample Kurosis Moel Kurosis Mar 97-Oc 10 Growh BE-ME 5 Value Sample Skewness Moel Skewness Sample Kurosis Moel Kurosis Jan 8-Feb 97 Growh BE-ME 5 Value Sample Skewness Moel Skewness Sample Kurosis Moel Kurosis a Panel A of his able repors esimaes of he para meer for he ump-iffu- sion porfolio reurns obaine by minimizing h e square of he ifference beween he heoreical momen coniions an he momens implie by he aa., for = 1,, 3, refers o he mean of he ump size for he growh, inermeiae, an value porfolios respecively., for = 1,, 3, is he volailiy of he ump size for he growh, inermeiae, an value porfolios respecively. These esimaes are given in percenage erms. Avg gives he average magniue of he average an volailiy ump sizes across porfolios, represens he frequency of he umps, an Year (s) is he number of years necessary o observe a ump accoring o he ump-iffusion process. b Panel B conains he reconsruce momens ha are obaine by subsiuing he parameer esimaes in he heoreical moel, an he sample momens. Copyrigh 013 SciRes.

8 17 G. PENAGOS, G. RUBIO (a) (b) Figure 1. Densiy funcions an QQ plos; (a) March 1997-Ocober 010; (b) January 198-February ieniy marix using Equaions (9)-(3). The esimae value for of 0.15 for he full sample perio implies ha on average he chance of a simulaneous ump in any monh across book-o-marke porfolios is abou 15%, or one ump is expece every 6.6 monhs or, equivalenly, 0.55 years. The average size of he ump across hree porfolios is 1.307, an i seems much higher (in absolue value) for he value porfolio han for he growh socks. Alhough, he inermeiae porfolio has he highes average size of he ump, he volailiies of he size of he umps are higher for he exreme growh an value porfolios. As before, he more ineresing resuls come from he changing behavior of he growh an value socks across sub-perios. The esimae value for of from 1997 o 010 implies ha on average he chance of a simulaneous ump in any monh across book-o-marke porfolios is almos 94%, or one ump i s expece every 1.1 monhs or 0.09 years. Hence, he frequency of simulaneous umps in he las sub-perios is exremely high. Copyrigh 013 SciRes.

9 G. PENAGOS, G. RUBIO 173 However, uring his sub-perio, he average size of he ump is 0.43 which is a relaively low relaive o he esimae for he full sample perio. This relaively small average size is parly ue o he compensaion beween he posiive average size for growh socks, an he negaive average size for value socks. Hence, form 1997 o 010, growh socks have on average posiive umps, while value socks presen on average negaive umps. A he same ime, he volailiy of he size of he umps seems o be lower han he volailiy for he full sample perio. The crisis episoes of he las ecae or so, seems o affec much more negaively value han growh socks. This is clearly consisen wih he negaive (posiive) skewness for he value (growh) socks from 1997 o 010. The economic implicaion is ha value socks seem o be more pro-cyclical han growh socks. Once again, he esimaion resuls are very ifferen for he firs sub-perio from 198 o The esimae value for of from 198 o 1997 implies ha, on average, he chance of a simulaneous ump in any monh across book-o-marke porfolios is abou 3%, or one ump is expece every 3.3 monhs or.7 years. Hence, he frequency of simulaneous umps in he firs subperio is much lower han in he mos recen sub-perio. This is, by iself, a relevan resul. During he las foureen years, here seems o be many more sysemic ump episoes han uring he previous fifeen years. Somehow surprisingly, however, he average size of he ump from 198 o 1997 is srongly negaive an much larger in absolue value han uring he las sub-perio. Also, he volailiy of he size of he umps is much larger han he volailiy repore from 1997 o 010. Ineresingly, he average size of he ump is posiive (negaive) for value (growh) socks which are precisely he opposie repore from 1997 o 010. Again, his is consisen wih he negaive (posiive) skewness for he growh (value) porfolio shown in Table 1. I is also imporan o poin ou ha i is precisely in his sub-perio when he value premium is as high as 18.4% on annual basis. The umps of he firs sub-perio are herefore much less frequen bu of larger magniue han he umps observe from 1997 o 010. The relaively few bu larger umps affec more negaively growh socks, while very frequen alhough smaller umps impac more negaively value socks. Again, i shoul be recalle ha he value premium is much larger from 198 o 1997 han from 1997 o 010. Panel B of Table compares he reconsruce momens ha are obaine by subsiuing he parameer esimaes in he heoreical ump-iffusion moel an he sample momens. As before, he comparison exercise is performe for each of he hree sample perios. Overall, he ump-iffusion moel capures very well he sample excess kurosis for all ime perios an porfolios. How- ever, he moel seems o have more problems fiing he magniues of he asymmery of he isribuion of reurns. This is especially rue for he value porfolio, where he heoreical process oversaes he magniue of skewness. This resul comes basically from he ba performance of he moel capuring he skewness of value socks from 1997 o 010. The ump-iffusion process generaes much more negaive skewness han he one observe in he aa. 4.. Porfolio Weighs We wan o solve numerically Equaion (1) o obain he opimal weighs when he reurn process follows he ump-iffusion moel of Equaion (3). The parameers we employ for he reurn process are hose repore in Panel A of Table. We also assume ha he annualize risk-free rae is equal o 6%, an we solve for opimal weighs assuming 10 alernaive values of he relaive risk aversion coefficien. As he benchmark case, we use he pure-iffusion moel where he opimal weighs are given by he well known expression in Equaion (7). Panel A of Table 3 conains he equiy porfolio allocaion resuls for he benchmark case, while Panel B repors he resuls for he ump-iffusion moel. The resuls show ha, inepenenly of he sample perio employe an he level of risk aversion, value socks receive a higher invesmen proporion of funs han growh socks. The opimal allocaion implies in all cases o go long in he value porfolio an shor on he growh porfolio. This is also rue wheher we recognize simulaneous umps across asses or no. This is, of course, wha a zero-cos invesmen on he HML porfolio in [11] precisely oes. Surprisingly, for he full sample perio, he inermeiae book-o-marke porfolio ominaes he value porfolio ue o he large weighs his porfolio ges from 198 o For his firs sub-perio, i mus be poine ou ha, when we incorporae umps, he value porfolio inee ges more proporion of wealh han he inermeiae porfolio bu only for he mos levere posiion. I is ineresing o observe he imporan impac ha umps have in his case for he nonconservaive invesor. The proporion invese in he inermeiae asses ecreases from 8.7% wihou umps o 17.4% wih umps, while he same proporions go from 14.0% o 18.5% for he value porfolio. Finally, he value porfolio ominaes he invesmen in he risky componen of he opimal asse allocaion for he 1997 o 010 sub-perio inepenenly of recognizeing umps or no. From Panel C of Table 3, where we repor he ifferences in weighs beween he benchmark case an he ump-iffusion moel, we observe ha, for he overall sample perio, he recogniion of umps iminishes he i fferences in he sense of increasing he long posiion on Copyrigh 013 SciRes.

10 174 G. PENAGOS, G. RUBIO Table 3. Porfolio Weighs. (a) Diffusion Weighs a ; (b) Jump-iffusion (Sysemic) Weighs b ; (c) Diffusion-Jump Weigh Differences c. a (a) γ G 5 V Riskless Risky γ G 5 V Riskless Risky γ G 5 V Riskless Risky Jan 8-Oc 10 Mar 97- Oc 10 Jan 8-Feb 97 (b) γ G 5 V Riskless Risky γ G 5 V Riskless Risky γ G 5 V Riskless Risky Jan 8-Oc 10 Mar 97- Oc 10 Jan 8-Feb 97 (c) γ G 5 V Riskless Risky γ G 5 V Riskless Risky γ G 5 V Riskless Risky Jan 8-Oc 10 Mar 97-Oc 10 Jan 8-Feb 97 Panel A of his able gives he porfolio weighs for an invesor who selecs invesmens in hree equiy porfolios (growh, inermeiae, value) an he riskless asse o maximize expece power uiliy of erminal wealh wih consan relaive risk aversio n. The invesor opimi zes expece uiliy ignoring s ysemic um ps an assumes a pure iffusion process for porfoli o reurns. is he relaive risk aversion coefficien, G, 5, an V inicae he growh, inermeiae b porfolio, an value porfolios respecively. Risky is he oal weigh given o all hree equiy porfolios. Panel B repors he opimal weighs when he invesor recognize sysemic umps. c Panel C conains he ifferences beween he opimal weighs for pure iffusion an he weighs for he ump-iffusion process. Copyrigh 013 SciRes.

11 G. PENAGOS, G. RUBIO 175 he value porfolio while, a he same, i suggess no shor-selling as much on he growh porfolio. 6 The recogniion of umps implies o pu relaively fewer funs in he risky porfolio for all levels of risk aversion, alhough his seems o be especially he case for he nonconservaive invesors an, herefore, for he mos levere posiions. However, he main poin is ha he value porfolio ges more weighs for all. A he en, a leas from 198 o 010, he use of sysemic umps in he opimal allocaion of funs, increases he proporion of value socks, reuces he amoun of shor-selling in he growh socks, an iminishes he proporion of funs invese in he inermeiae porfolio. As in oher cases, he resuls of he full sample perio seem o be he consequences of wo ifferen sub-perios. The effecs of umps from 1997 o 010 are, o all effecs, negligible. The ifferences of opimal weighs beween he benchmark case an he ump-iffusion moel are herefore basically zero excep, if anyhing, for he highly levere posiion. The main effecs of umps come from he firs sub-perio. From 198 o 1997, here were very few umps bu wih a relaively very large average size. In fac, he average ump size for he value porfolio is posiive, which makes he proporion of funs invese in value o increase imporanly an, especially, for he mos levere porfolios. These effecs can be seen irecly from Panel C of Table 3 an hey are also reflece in Figure. By recognizing sysemic umps, opimal asse allocaion increases significanly for value socks an for all levels of risk aversion, alhough more for he less conservaive more levere case. A he same ime, umps reuce consierably he proporion invese in he inermeiae porfolio. As a consequence of recognizing umps, here is also a reucion in he shor-selling proporions of growh socks. The same conclusion is obaine by he resuls shown in Table 4. We repor he percenages of he growh, inermeiae, an value porfolios in he oal proporion invese in risky asses. We observe ha he sprea be- ween he proporions invese in value an growh asses is higher in he firs han in he secon sub-perio. I is also he case, ha he sprea in he firs sub-perio is paricularly imporan for he mos levere porfolios, alhough he relevance of risk aversion is negligible in he las sub-perio. We may conclue ha sysemic umps across porfolios seem o be imporan for asse allocaion as long as he average magniue of he umps is large enough. From he poin of view of asse allocaion, i is herefore more imporan he ampliue han he frequency of he 6 A negaive sign for he value porfolio in Panel C inicaes ha he recogniion of simulaneous umps makes he invesor o allocae more funs in he value porfolio. A negaive sign on he growh porfolio implies ha he recommenaion woul be o shor-sale growh socks in less proporion han he case of he pure-iffusion benchmark. Table 4. Composiion of he Risky Porfolios. (a) Jumpiffusion (Sysemic) Weighs a ; (b) Diffusion Weighs b. (a) γ Growh BE-ME 5 Value Sprea Mar 97-Oc 10 γ Growh BE-ME 5 Value Sprea Jan 8-Feb 97 (b) Growh BE-ME 5 Value Sprea Mar 97-Oc 10 Growh BE-ME 5 Value Sprea Jan 8-Feb 97 a Pan el A of his able c onains he composiion of h e risky porfolios for alernaive levels of relaive risk aversion, which is obaine by iviing he weigh for each porfolio by he oal amoun invese in risky porfolios. I assumes he ump-iffusion process. b Panel B conains he same resuls assuming he pure iffus ion process. I shoul be noe ha he weighs o no epen on he level of relaive risk ave rsion when he simple iffusion process is employe in he esimaion. umps. In our specific sample, umps urn ou o be relevan for value socks given he combinaion of a large an posiive average ump size. Jumps also seems o fa- Copyrigh 013 SciRes.

12 176 G. PENAGOS, G. RUBIO vor value socks even wih a negaive average size ump as in he full sample perio, bu his is relaively less imporan an, in any case, he possible effecs are us concenrae in highly levere porfolios Cerainy Equivalen Coss We nex analyze he effec on uiliy of he opimal porfolio sraegy ha recognizes he occurrence of sysemic umps across book-o-marke asses relaive o he sraegy ha ignores hese simulaneous umps. We employ he CEQ given in Equaion (15) ha calculaes he aiional wealh per $1000 of invesmen neee o raise he expece uiliy of erminal wealh uner he nonopimal porfolio sraegy o ha uner he opimal invesmen sraegy. We consier he effecs for invesmen horizons of 1 o 5 years an for levels of risk aversion of o 10. Table 5 repors he resuls. As before, he effecs of frequen bu relaively small umps observe from 1997 o 010 is negligible. Even he U.S. ollar consequences of he nonopimal sraegy over he full sample perio are very small. For highly risk-aken invesors an over a 5 years horizon, he cos of ignoring umps is only abou $.00. All he relevan effecs come, once again, from he 198 o 1997 sub-perio in which he average size ump seems o be high enough o impac he porfolio allocaion of equiy porfolios. As we observe, he CEQ ecreases as risk aversion increases. This suggess ha, as he becomes more risk averse, he invesor hols a smaller proporion of his wealh in risky porfolios an, herefore, boh he exposure o simulaneous umps an he effecs (a) (b) Figure. Porfolio weighs an relaive risk aversion. (a) Pure-iffusion process; (b) Jump-iffusion process; This figure shows he porfolio weighs for he growh, inermeiae an value Fama-French porfolios for alernaive levels of relaive risk aversion. Panel A conains he weighs for he pure-iffusion process, while Panel B gives he weighs for he ump-iffusion process. The firs figure of each panel correspons o he las sample perio from March 1997 o Ocober 010, an he secon figure of each panel correspons o he firs sample perio from January 198 o February Copyrigh 013 SciRes.

13 G. PENAGOS, G. RUBIO 177 on CEQ are smaller. However, when he invesor is willing o accep higher risks, his opimal porfolio becomes much more levere o buy aiional risky asses. Then, he consequences of ignoring sysemic umps shoul be higher. This is exacly wha we observe from Table 5 Table 5. Cerainy equivalen cos of ignoring sysemic umps a. γ One year Two years Three years Four years Five years Jan 8-Oc 10 an Figure 3. For highly levere porfolios, he CEQ goes from $48.40 for a 1-year horizon o $66.50 for he longes horizon per $1000 of invesmen. I shoul no be surprising ha he higher he leverage posiion of an invesmen is, he higher he impac of umps on porfolio sraegies. I shoul be recognize however, ha he ollar effecs iminish very rapily as he invesor becomes more risk averse. For 4, even for he longes horizon of 5 years, he CEQ is $30.4 per $1000 which oes no seem o be subsanial. γ One year Two years Three years Four years Five years Mar 97-Oc 10 γ One year Two years Three years Four years Five years Jan 8-Feb 97 a This able repors he c erainy equivalen coss (CEQ) of ignoring sysemic umps calculae as he aiional wealh per $1,000 of invesmen neee o raise he expece uiliy of erminal wealh uner he subopimal porfo- lio sraegy o ha uner he opimal invesmen sraegy. The able conains he CEQ for invesmen horizons of 1 o 5 years, an for levels of relaive risk aversio from o 10. n Figure 3. Cerainy equivalen cos of ignoring sysemic umps. This figure shows he cerainy equivalen coss (CEQ) of ignoring sysemic umps calculae as he aiional wealh per $1,000 of invesmen neee o raise he expece uiliy of erminal wealh uner he subopimal porfolio sraegy o ha uner he opimal invesmen sraegy. The figure conains he CEQ for invesmen hori- zons of 1 o 5 years, an for levels of relaive risk aversion from o 10. The firs figure correspons o he las sample perio from March 1997 o Ocober 010, an he secon figure of each panel correspons o he firs sample perio from January 198 o February Copyrigh 013 SciRes.

14 178 G. PENAGOS, G. RUBIO 4.4. Sample Average Riskless Rae In he previous iscussion, we impose a 0.5% monhly riskless rae for he full sample perio an for boh subperios. Alhough his is a reasonable riskless rae for he firs sub-perio, i may be oo high for he secon subperio. For his reason, we esimae again he moel from 1997 o 010 imposing he acual average riskless rae of 0.5% per monh or 3% per year. I mus be noe ha he ineres rae affecs he parameer esimaes an, herefore, i may have consequences for he general conclusions abou he opimal porfolio allocaion uring he secon sub-perio. Table 6 repors he resuls affece by he riskless rae. The empirical evience is almos ienical o he evience conaine in Tables, 3 an 5. The average size of umps is even slighly lower, an he frequency of he umps is now 1.31 relaive o he previous esimae of Thus, he frequency of he simulaneous umps is higher han he one repore in Table. Once again, his sub-perio is characerize by many umps of small average ampliue. The effecs of umps on he porfolio weighs an on he cos of ignoring umps are very similar o he previously repore resuls. As expece, given ha now he risk-free invesmen offers a lower rae, he opimal amoun of he risky porfolios is higher wih respec o he allocaion shown in previous ables. However, he effecs abou he isribuion of resources among book-o-marke porfolios wih or wihou umps are negligible. All our previous conclusions remain he same Table 6. Jump iffusion parameer an weigh esimaes March 1997-Ocober 010 for he sample average annualize riskless rae of 3%. (a) Parameer esimaes for he ump process a ; (b) Comparison beween higher orer sample momens of reurns an he umps process heoreical higher orer momens b ; (c) Diffusion weighs, ump-iffusion weighs an cerainy equivalen coss (CEQ) c. (a) η 1 η η 3 Avg Avg λ Year (s) (b) Growh BM - ME 5 Value Sample Skewness Moel Skewness Sample Kurosis Moel Kurosis (c) Diffusion weighs G 5 V Riskless Risky Jump Diffusion Weighs G 5 V Riskless Risky Cerainy Equivalen Cos G 5 V Riskless Risky a Panel A repors he esimaes of he parameers for he ump-iffusion porfolio reurns for an annualize riskless rae of 3%. b Panel B conains he reconsruce momens an he sample momens. c Panel C hese ables gives he porfolio weighs for an invesor who ignoring sysemic umps, he opimal weighs when he invesor recognize sysemic umps an he cer- ainy equivalen coss (CEQ) of ignoring sysemic umps calcula e as he aiional wealh per $ 1,000 of inve smen neee o raise he expece uiliy of erminal wealh uner he subopimal po rfolio sraegy o ha uner he opimal invesmen sraegy. even wih a much lower riskless rae. Copyrigh 013 SciRes.

15 G. PENAGOS, G. RUBIO Specificaion Tess w ih a Pre-Spe cifie Weighing Marix Up o now, he sa isical perform ance of he moel has be en very informal. Inenionally, our iscussion has been base mainly on economic inuiion raher han saisical formaliy. We finally wan o es he overall fi of he moel. The es-saisic is he GMM es of overienificaion resricions. We en oe by f he K-v ecor of mom en coni- ions conaini ng he pricing errors generae by he ump-iffusion m oel a ime, a n by he se of pa- rame ers o be esi mae. The correspo ning sample av- erages are enoe by gt. Then, he GMM esimaor proceure minim izes he quaraic form gt WTg T where WT is a weighing square marix. The evaluaion of he moel perform ance is car- rie ou by esing he null hypohesis T Dis 0, Dis g W g where he weighing ma- wih T T T rix, W, is in our case he ieniy marix. If he weighing marix is opi mal, T Dis ˆ is asympoically is- ribue as a Chi-s quare wih K P 1 egrees of freeom, where P is he number of param eers. However, for any oher weigh ing m arix (incluing he ieniy m arix), he isribuion of he es saisic is unk nown. Reference [1] shows h a, in his case, T Dis ˆ is asympoi- cal ly isribue as a weighe s um of K P 1 inepen- en Chi-squares ranom variables wih one egree of free om. Tha is ˆ T Dis i i 1 7) (3 where i, for i 1,,, K P1, are he posiive eigenvalues of he following marix: KP1 i1 A S W I W D D W D T T K T T T T T DW W S T T T T 1 in which X means he upper-riangular marix from he Choleski ecomposiion of X, an I K is a K-imensional ieniy marix. Moreover, S T an DT are given by, S T T 1 T f T f f 1 DT T Therefore, in o rer o es he ifferen moels we esimae, we procee in he following way. Firs, we esi-, mae he marix A an compue is nonzero K P 1 eigenvalues. Secon, we generae hi, h 1,,,100, i 1,,, K P1, inepenen ranom raws from a 1 ribuion. For each h, KP1 uh i 1 i hi is is compue. Then we compue he number of cases for which u T Dis ˆ h. Le p e- noe he percenage of his number. We repea his proceure 1000 imes. Finally, he p-value for he specificaion es of he moel is he average of he p values for he 1000 replicaions. I urns ou ha we are no able o reec he umpiffusion moel in any of he alernaive sample perios employe in he esimaion. The p-values for he full sample perio, he firs sub-perio, an he secon subperio are 0.57, 0.103, an respecively. These resuls sugges ha he ump-iffusion moel fis he acual aa beer from 1997 o 010 han from 198 o This is he case espie he poor reconsrucion of he acual skewness for he value porfolio from 1997 o Conclusions Given he enency of globalizaion an increasing inegraion of financial markes, i is generally accepe ha equiy porfolios of ifferen characerisics an also inernaional equiies are characerize by simulaneous umps. We invesigae he effecs of hese umps on opimal porfolio allocaion using value an growh socks an wo ifferen sub-perios. I seems ha he effecs of sysemic umps may be poenially subsanial as long as marke equiy reurns experimen very large average (negaive) sizes. However, i oes no see m o be relevan ha sock markes experience very frequen umps if hey are no large enough o impac he mos levere porfolios. All poenially relevan effecs are concenrae in porfolios finance wih a consierable amoun of leverage. In fac, for conservaive invesors wih low leverage posiions he poenial effecs of sysemic umps on he opimal allocaion of resources are no subsanial even uner large average size umps. Final ly, he value premium is paricularly high when he average size of he umps of value socks is posiive, large an relaively infrequen, while he average size of growh socks is also very large bu negaive. I seems herefore plausible o conclue ha he magniue of he value premium is closely relae o he characerisics of he umps experi- ence by value an growh socks. 5. Acknowlegemens Gonzalo Rubio acknowleges financial suppor from Minisry of Economía an Compeiivia gran MEC Copyrigh 013 SciRes.