Journal of Applied Finance & Banking, vol.1, no.1, 2011, 53-81 ISSN: 1792-6580 (prin version), 1792-6599 (online) Inernaional Scienific Press, 2011 Marke iming and saisical arbirage: Which marke iming opporuniies arise from equiy price buss coinciding wih recessions? The Swedish sock marke in he financial crises 2008 Klaus Grobys 1 Absrac Even hough a random walk process is from a saisical poin of view no predicable, some movemens can be correlaed wih specific evens concerning oher variables. Then, predicable paerns may arise being dependen on his join even. There is evidence given ha equiy price buss being associaed wih recessions coninue unil he economy swiches from he sae of recession o an economic pick-up. The following conribuion akes ino accoun he Swedish sock index OMX 30 and 25 preseleced socks. The ou-of-sample period runs from Sepember 12, 2008 March 12, 2009, whereas on Sepember 11, 2008 he official press release was issued ha European economies face a recession. This sudy suggess a marke iming opporuniy resuling in a maximum saisical arbirage opporuniy corresponding o a profi of 19% p.a. wih an empirical probabiliy of 50.14%. The opimal defensive sraegies, however, exhibi excess 1 E-mail: Klaus.grobys@klarna.com Aricle Info: Revised: March 5, 2011. Published online : May 31, 2011
54 Marke iming and saisical arbirage reurns of 15.12% p.a. above he benchmark wih a marginal lower volailiy as he benchmark, respecively, 28.08% p.a. wih 7.99 percen unis higher volailiy as he benchmark. JEL classificaion numbers: C13, C22, G11, G12 Keywords: Saisical arbirage, Financial crises, equiy price buss, Coinegraion 1 Inroducion The percepion ha sock prices already reflec all available informaion being ofen referred o as he efficien marke hypohesis is widely discussed in he academic lieraure and ress upon sudies by Kendall (1953). As soon as here is any news available indicaing ha a sock is underpriced, Bodie, Kane and Marcus (2008) highligh ha raional invesors would buy his sock immediaely and, hence, bid up is price o a level where only ordinary reurns can be expeced o gain. Consequenly, he efficien marke hypohesis suggess ha sock prices follow random walk processes involving ha prices changes are random and hus unpredicable. Sudies by Chan, Gup and Pan (1997) who examine eigheen naional sock markes by using uni roo ess figure ou ha he world equiy markes are weak-form efficien and, hence, suppor he efficien marke hypohesis. Uni-roo ess as applied by Chan, Gup and Pan (1997) hough ake only informaion ino accoun which is involved in he univariae daa generaing processes. Agains his, Claessens, Kose and Terrones (2009) provide a comprehensive empirical characerizaion of linkages beween key macroeconomic and financial variables around business and financial cycles. Their sudies involve 21 OECD counries and cover over 47 years from 1960-2007. Thereby, hey ake ino accoun 122 recessions, 113 credi conracions and 245 episodes of equiy prices declines, whereas 61 of hese equiy price declines are referred o as price buss. Equiy price buss are in accordance o he definiion of Claessens, Kose and
K. Grobys 55 Terrones (2009) peak-o-rough declines in equiy prices which fall wihin he op quarile of all price declines. Their findings show ha equiy price buss overlap abou one-hird of he recession episode. Furhermore, given he even ha he economy faces a recession, in 60% of all cases equiy price declines occur a he same ime. Moreover, Claessens, Kose and Terrones (2009) conclude ha recessions end o coincide wih conracions in domesic credi and declines in asse prices and in mos advanced counries. Thereby, he ypical duraion of en equiy price bus is wice ha of a recession, bu ends a he same ime when he associaed recession is ending. Bu wha implicaions do equiy price buss offer concerning asse allocaions if he bus coincides wih a recession? If marke paricipaors expec sock prices o fall in furher periods, hey will rebalance heir sock porfolios such ha he expeced loss will be minimized. As an alernaive, invesors could consruc arbirage porfolios while going shor on he index and long on an equiy porfolio exhibiing defensive properies and herewih ouperforms if he sock markes declines in fuure periods. In he following conribuion he equiy price bus being associaed wih he financial crises in 2008 is analyzed wih respec o marke-iming opporuniies. Furhermore, he opimizaion problem being associaed wih an advanageous asse allocaion condiional on he sae of he economy is examined. Thereby, he Swedish leading sock index OMX 30 is aken ino accoun. 50 differen porfolios are esimaed which rack arificial indices corresponding o defensive invesmen sraegies. The opimizaion procedure accouns for 20 socks corresponding o he companies exhibiing he highes marke capializaion being line wih Alexander and Dimiriu (2005). While holding he opimal weighs consan wihin a six-monh period ou-of-sample (i.e. Sepember 12, 2008-March 12, 2009), evidence is given for saisical arbirage opporuniies. The esimaed opimal asse allocaions as suggesed here dominae he index in boh, he Reward-o-Risk raio and he Reward-o-Risk-Difference raio. During he six monhs period being
56 Marke iming and saisical arbirage examined boh evens occur a he same ime, he economy faces a recession and he financial marke faces an equiy price bus. Consequenly, a raional invesor who expecs he equiy prices o fall in fuure periods will selec a defensive asse allocaion in order o minimize expeced losses as soon as he recession is ascerained. Opimal defensive sraegies, as suggesed here, exhibi, given he considered Swedish sock marke condiions, reurns of -19.19% p.a. and -6.13% p.a. involving a volailiy of 53.50%, respecively, 61.50% p.a. The OMX 30 hough had a reurn of -34.16% p.a. and exhibied a volailiy being equal o 53.51% under he same period of consideraion (i.e. Sepember 12, 2008-March 12, 2009). 2 Background Even hough he efficien marke hypohesis holds when esing sock markes price movemens of mos advanced counries wheher he even "equiy price bus" occurs, given he even ha he economy faces a recession, predicable paerns may evolve. Claessens, Kose and Terrones (2009) who consider a large daa se of recessions, equiy price declines and credi conracions wihin OECD counries, argue ha a ypical episode of an equiy price decline, respecively, an equiy price bus ends o resul in a 24% and 51% fall in equiy prices. Thereby, he duraion s mean is 6.64 and, respecively 11.79 quarers where he laer figure is saisically significan even on a 1% significance level. Furhermore, recessions ha coincide wih equiy price buss las for 3.79 quarers on average, whereas recessions ha do no coincide wih equiy price buss las 3.49 quarers on average. Claessens, Kose and Terrones (2009) conclude ha equiy price declines overlap wih abou one in hree recessions. If a recession coincides wih an equiy price bus, he recession can sar as lae as four o five quarers afer he asse bus has sared. However, he equiy price bus ypically ends wih he end of is
K. Grobys 57 corresponding recession bu can coninue for wo o nine quarers afer he recession has ended. Agains i, he minimum duraion of a recession is in accordance o Claessens, Kose and Terrones (2009) wo quarers, whereas a ypical recession lass abou four quarers. The laer fac clearly exhibis marke iming poenial: A raional invesor who expecs sock prices o fall during he nex wo quarers will choose a defensive asse allocaion in order o minimize he expeced loss in fuure periods. Furhermore, invesors could exploi from equiy price declines by shoring he index and aking a posiion in socks which response defensively. Once he recession is ascerained, a raional marke paricipan will rebalance he equiy porfolio in order o anicipae a furher fall in equiy prices. Even hough Aroa and Buza (2003) menion ha recessions are no periodic and ha hey differ in duraion, inensiy and occurrence, here are sill similariies in he sequence of evens and circumsances ha ypically occur over he course of a business cycle. The same is he case wih respec o sock marke crashes. Each sock marke crash is preceded by a bubble formaion as argued by Aroa and Buza (2003) where bubbles, respecively, bull markes are usually associaed wih a period of prosperiy, when he fuure seems brigh and invesors have easy access o money. Agains his, excessive pessimism follows his exuberance and creaes as a consequence he sock marke crash, respecively, he bear marke. In accordance o Aroa and Buza (2003), he same mass psychology evoking he expecaion ha every do-com company will be profiable and, hence, creaed he boom in he sock marke during 1995-1999, was accounable for he crash in he NASDAQ in January-March 2000. The U.S economy began o slow down during he second half of he year 2000, and he res of he world followed, resuling in a worldwide recession. If he marke sands in a bear marke, crisis evens which can be generalized as bad news exacerbae he sock marke s downurn movemens as menioned by Aroa and Buza (2003). Of course, he downurn will no end as long as he majoriy of news which arrive he marke will be evaluaed as good news
58 Marke iming and saisical arbirage from he marke paricipiians' poin of view. Hence, press releases such as issued on Sepember 11, 2008 from he German Insiue for Worldeconomics (IfW Kiel) declaring ha European counries face a recession will consequenly be associaed wih an expecaion ha sock prices will coninue o decline even in fuure periods. 2 For insance, he Swedish leading sock index OMX 30 los already 36.91% compared o is peak on July 16, 2007 on he day where he official press release was issued (i.e. Sepember 11, 2008). Considering a period of wo quarers hereafer i could be observed ha he OMX 30 fell by addiional 17.08% (i.e. from Sepember 12, 2008 March 12, 2009). Hence, he equiy price bus began more abou 14 monhs before he recession was ascerained and coninued aferwards. The same paerns could be observed in 2001. On November 26, 2001, he Naional Bureau of Economic Research issued a press release declaring he recession began in March 2001. 3 Marke observers recognized similar paerns: From November 26, 2001 unil May 26, 2002 he Swedish leading sock index OMX 30 fell by 18.85%. However, he recession was also anicipaed by an equiy price bus where he OMX 30 los already 44.27% from is peak on March 7, 2000 unil he day where he recession was declared by he Naional Bureau of Economic Research (i.e. on November 26, 2001). The same paerns could be invesigaed concerning oher European sock markes. For insance, he German's leading sock index DAX fell in he period November 26, 2001 unil May 26, 2002 by 36.89%, whereas he sock index addiionally fell by 4.20% from beween November 26, 2001 and May 26, 2002. Considering he financial crises in 2008 and he associaed equiy price bus which again anicipaed he recession, he DAX los 23.77% from July 16, 2007 unil Sepember 11, 2008 and only addiional 1.54% from Sepember 12, 2008 March 12, 2009. 2 See hp://www.ifw-kiel.de/media/press-releases/2008/pr11-09-08b.
K. Grobys 59 However, no all socks paricipae in booms, respecively, bull markes. In accordance o Aroa and Buza (2003) railroad socks were excluded from he boom of 1928-1929, whereas overinvesing in uiliies caused his speculaive bubble formaion. The bubble formaion during 1995-1999 showed an overpricing of he elecommunicaion and inerne secor as sudied by Jensen (2005) and Harmanzis (2004), whereas a similar mass psychology caused he overpricing concerning he financial secor during 2004-2008 as described by Baker (2008) and Soros (2008). Poerba and Summers (1989) and Cecchei, Lam and Mark (1990) poin ou ha he secor will adjus he sronger he more excessive he speculaive bubble has been. Therefore, a raional invesor who expecs he marke o decline in furher periods will allocae he asses o a porfolio exhibiing defensive properies during he equiy price decline. As a consequence of he equiy price bus which sared March 2000, Aroa and Buza (2003) menion ha invesors had moved he money ino energy and healh care company socks during 2000 and 2001 since hese secors were expeced o response defensively in bear markes. Bu do defensive asse allocaion sraegies being buil on hisorical sochasic movemens of arificial indices exhibi robusness wihin he ou-of-sample period? This conribuion hrows ligh on he following issues: Firs, 50 differen asse allocaions will be esimaed which rack consruced arificial indices assuming o exhibi defensive properies if he invesors expec he marke o decline in furher periods. Thereby, he Swedish sock index OMX 30 will be employed in order o consruc arificial indices and a se of 20 preseleced socks will be used in order o esimae opimal asse allocaion weighs. Second, based on hese porfolios racking defensive arificial indices i will be deermined which would be he opimal asse allocaion, given he ou-of-sample risk-reurn esimaes. Thereby, wo differen opimizaion calculi will be aken ino accoun. The hird issue is ha 3 See hp://www.nber.org/cycles/november2001/ (accessed on December 02, 2010 21:25).
60 Marke iming and saisical arbirage i will be discussed how his marke-iming approach can be applied for boh, funds managemen and hedge funds managemen. 3 Economeric Mehodology In order o esimae weigh allocaions exhibiing defensive sock porfolios, hree years of hisorical daily daa is aken ino accoun. Following Alexander and Dimiriu (2005), he coinegraion approach is employed where hree years of daily daa is necessary o esimae robus coinegraion opimal allocaion weighs. The day where he press release is issued will in he following be denoed as rec, whereas he day where he sock index exhibis he highes noaion during he laes bubble formaion will in he following be denoed as max. In line wih Grobys (2010) a linear rend is added o he hisorical index reurns ha swich he direcion on he day where he price bus begins. Since he exac day is unknown, i will be assumed ha he price bus akes place on day max since on he laer day he sock marke noaion shows he maximum difference beween max and rec wihin he las hree years. Arora and Buza (2003) repor ha no all socks paricipae in bubble formaions. Hence, esimaing porfolios ha do no follow he marke s exaggeraion are expeced o decline less han he marke during he crash and can consequenly be employed o esimae porfolios involving defensive asse allocaions. In line wih Grobys (2010) he linear rend is firs subraced o he marke reurns and swiches a ime poin max he direcion. Subracing a linear rend erm unil max and adding he erm from observaion max onwards resuls in an arificial index being below he benchmark unil max and exhibiing higher reurns from max onwards as he bubble disperses. Then, he inegraed ime series corresponding o he arificial indices are given by δ for OMX i= 1 i= 1 p = c + R δ i max = 1,..., (1) for OMX δ = max + max + max δ δ i= i= p p R i max = + 1,..., rec, (2)
K. Grobys 61 where δ denoes he facor ha is subraced, respecively, added o he index in daily erms and OMX R denoes ordinary index reurns a ime. Hence, for each δ = δ,..., 1 δm differen inegraed arificial indices p δ can be generaed. Figure (1) shows he index and he arificial index for he facor δ = 0.10 (i.e. 25% in annual erms) for he in-sample period. The inegraed ime series of socks being employed o rack he arificial indices are in line wih Grobys (2010) calculaed such ha where k rec i= 1 p = c+ R, (3) R k denoes he reurn of sock k a ime and c is a consan erm. In order o esimae coinegraion opimal weigh allocaions, he maximum-likelihood opimizaion procedure is employed being in line wih Grobys (2010) and given by where T T 2 1 log L( θ,, δ) = log ( 2 π) logσ 2 2 2 K ε δ = pδ a k 1 δk p = δk k ( ε ) 2 δ (4) 2 T σ. In accordance o van Monefor, Visser and Fijn van Draa (2008) i is usual o impose weigh resricions. In he following i is sufficien hough o resric he weighs o sum up o one and o be posiive being given by K = 1 (5) a i= k δ k a δ > 0 for k = 1,..., K. (6) k
62 Marke iming and saisical arbirage Figure 1: The OMX 30 and he an arificial index wihin he in-sample period The weighs being esimaed a day rec are hold consan wo quarers ahead as he marke decline is in accordance o Claessens, Kose and Terrones (2009) expeced o end wih he recession while he minimum recession akes per definiion wo quarers. Each opimal weigh allocaion is sored in a vecor and employed o esimae M differen ou-of-sample porfolio processes depending on δ = δ,..., 1 δm. The opimal defensive sraegies can be deermined by opimizing he wo following opimizaion problems being differen from each oher: Firs, he Reward-o-Risk raio can be maximized, given by T T OMX ( rec R rec δ R = = ) max δ T T 1/ N rec R 1/ N rec R = δ = δ, (7) where R ˆ ˆ δ = aδ1 R1 +... + aδk RδK denoes he esimaed reurns of he porfolio δ ha racks he arificial index p δ, OMX R denoes he ordinary index reurns rec and he ou-of-sample window runs from = + 1,..., T while rec N = T
K. Grobys 63 denoes he rading days wihin he ou-of-sample period. In equaion (7) i is calculaed how much does an addiional reurn above he benchmark cos in erms of volailiy and ress upon he Reward-o-Risk raio being inroduced by Sharpe (1964). The maximum value is opimal in he sense ha i depics he asse allocaion ha generaes he highes reurn for each uni porfolio volailiy wih respec o he ou-of-sample ime window. However, he opimal asse allocaion can be anoher one if he excess reurns are relaed o he increase of volailiy: In his case a raional invesor would prefer o inves in porfolio p l insead of p m if he increase of excess reurns exceeds he increase in porfolio volailiy. Then, he opimizaion problem is in conras o equaion (7) given by T T OMX ( rec R rec δ R = = ) max δ T T N OMX T 1/ N rec R 1/ N rec R rec R 1/ N rec R = δ = δ = = As he volailiy of he opimal porfolio can be lower compared o he sock marke s volailiy, he absolue amoun has o be maximized. The consruced porfolios are esed wheher he maximum likelihood esimaion provides weigh allocaions ha exhibi a coinegraion relaionship wih he arificial indices being racked concerning he in-sample period. In line wih Alexander and Dimiriu (2005) he ADF-es will be employed, given by L Δ ˆ ε = γ ˆ ε + α Δ ˆ ε + δ δ δ 1 δl δ l δ l= 1 OMX (8) u (11) Thereby, he null hypohesis esed is of no coinegraion, i.e. γ δ = 0., agains he alernaive of γ δ < 0. 4 Wheher he null hypohesis of no coinegraion is rejeced, he coinegraion-opimal racking porfoliod based on he maximum likelihood procedure of equaion (4) is expeced o have similar sochasic paerns 4 The criical values for he -saisic of y are obained using he response surfaces provided by MacKinnon (1991).
64 Marke iming and saisical arbirage as he arificial indices concerning he in-sample period. The error vecor ˆ ε δ comes from an auxiliary regression from he inegraed porfolio ime series on he inegraed arificial marke index imes series such ha ( ) 1 ε = p p ˆ β where, ˆδ δ k δ δ ˆ = p ' p p ' p k (see equaions (1) and (2)). If ˆ ε δ is saionary, he β δ δ δ δ δ esimaed porfolios are said o be coinegraion opimal. Since he arificial indices are via consrucion coinegraed wih he benchmark, he porfolios can be considered as being coinegraed wih he ordinary benchmark, oo. Furhermore, he ou-of-sample porfolios are priced firs of all by running OLS-regressions as following: where R = α + β R + u, (11) OMX δ δ δ δ u δ is assumed o be a whie noise process and ( ) R = p p 100 / p. Equaion (11) is ofen referred o as ordinary index δ δ δ 1 δ 1 model (see Bodie, Kane and Marcus 2008) and is usually employed o deermine wheher he porfolio bea (i.e. β δ ) is above or below he marke bea being equal o one. Thereby, a bea being larger han one indicaes an offensive asse allocaion while a bea below one usually indicaes a defensive one. Furhermore, if he porfolio alpha (i.e. α δ ) is saisically significan higher han zero, he porfolio is said o generae abnormal reurns and, hence, involves saisical arbirage opporuniies. However, Grobys (2010) menions ha he resuls of regressions such as formalized by equaion (11) can be misleading as he saisical arbirage is cached in he rend-saionary sochasic process being inegraed in he porfolio processes. Moreover, regressions ha ake ino accoun only he derended series, such as he porfolio reurns, do no accoun for his issue as menioned by Alexander (1999). In line wih Bondarenko (2003) a saisical arbirage opporuniy arises when he expeced payoff of a zero-cos rading sraegy is posiive and negaive reurns occur only sochasically. Therefore, i will be analyzed how much he empirical probabiliy is ha an esimaed porfolio
K. Grobys 65 exhibis reurns being above he benchmark, ha is ( ( ) ( 1 ) ( OMX rec δ δ ) 1,..., ) P E R + E R = + T. Finally, a regression is performed in order o figure ou how well he enhancemen facors predic he ou-of-sample excess reurns, respecively, ou-of-sample performance: T T OMX ( = δ = ) δ (12) 1/ rec rec N R R = c + δ + u 2 where u δ is assumed o be a whie noise error erm wih u δ ( 0, σ u ) consan erm and ( T rec R T rec ) OMX δ R = =, c is a denoes he excess reurn of porfolio corresponding o he enhancemen facors δ = δ,..., 1 δm wihin he ou-of-sample period. If he daa sugges a breakpoin, equaion (12) is augmened by a dummy variable accouning for a break in he parameers given by where T T OMX ( = δ = ) 1 2 1 2 δ (13) 1/ rec rec N R R = c + c d + δ + δ d + υ υ δ is assumed o be a whie noise error erm d = 0 before he break and d = 1 oherwise. 4 Resuls In his work, he OMX 30 is employed which is he leading sock index in Sweden and accouns for socks of he larges 30 companies in accordance o heir marke capializaion. The daa concerning he in-sample and ou-of-sample periods can be downloaded for free on he index provider s websie www.nasdaqomxnordic.com. In order o rack he consruced arificial indices, 25 socks exhibiing he highes marke capializaion (see he appendix) on Sepember 2008 are preseleced in order o esimae he maximum likelihood funcions. This sock selecion approach is in line wih Alexander and Dimiriu (2005) who also selec sock in accordance o heir marke capializaions. The
66 Marke iming and saisical arbirage German Research Insiue for Worldeconomis (IfW, Kiel) issued on Sepember 11, 2008 an official press release where i was repored ha he Euro area faces a recession 5. A he same ime he Swedish leading sock index OMX 30 los already 36.91% compared o is peak on July 16, 2007 (i.e. Since OMX max =1.311,87). rec is in his sudy Sepember 11, 2008, 750 days before he laer dae have o be aken ino accoun in order o esimae he maximum-likelihood funcion corresponding o high frequened daily daa from Sepember 21, 2005 Sepember, 11, 2008. The asse allocaion akes place on Sepember 12, 2008 and he allocaion weighs are held consan from Sepember 12, 2008 unil March 12, 2009 corresponding o N = 124 rading days ou-of-sample. Claessens, Kose and Terrones (2009) denoe such price declines such as he OMX 30 exhibied during he in-sample period as buss. Equiy price buss which anicipae recessions are much sronger compared o ordinary prices declines and end wih he recession, earlies hough wo quarers aferwards (see Claessens, Kose and Terrones (2009) for deailed informaion). The arificial indices are in accordance o equaions (1)-(2) consruced wih δ = 0.0040,0.0080,...,0.2000 (i.e. corresponding o δ = 1%,2%,...,50% in annual erms) uniformly disribued over ime so ha 50 differen asse allocaions could be esimaed which is also in line wih Alexander and Dimiriu (2005). Exhibi 1 gives an overview concerning he saisical properies, whereas in he appendix, he asse allocaions are given wih respec o all esimaed porfolios. The opimizaion procedure concerning equaion (7) sugges an asse allocaion corresponding o porfolio 29 (see ables 1a-d and figure 2) which racks an arificial index being consruced wih δ = 0.1160 (i.e. 29% in annual erms). The hree main posiions join 94.40% of he overall weigh allocaion and are invesed in he indusrial machinery secor, heavy elecrical equipmen indusry annual 5 See IfW Press Release Sepember 11, 2008.
K. Grobys 67 and he clohing indusry. However, opimizing wih respec o equaion (8) suggess a more diversified asse allocaion. Porfolio 14 racking an arificial index wih δ = 0.0560 (i.e. 14% in annual erms, see ables 1a-d) invess only 40.53% in he same business secors as porfolio 29. The OMX 30 declined from Sepember 12, 2008 unil March 12, 2009 by 17.08%, whereas porfolio 14 declined only by 9.52% and performed consequenly 7.56% beer in comparison o he benchmark while exhibiing a marginal lower volailiy of 53.50% p.a. compared o he benchmark s volailiy being 53.51%. Exhibi 1 shows ha he bea is close o he marke bea (i.e. ˆ β 14 = 0.98). Porfolio 29 which is opimal wih respec o he opimizaion procedure concerning equaion (7) exhibis wihin he ou-of-sample window a loss of only 3.04% (i.e. corresponding o excess reurns of 28.08% p.a.) while he volailiy is 7.99 per cen unis higher in comparison o he benchmark s volailiy. Again, he OLS regression being in line wih he ordinary index model indicaes ha he bea (i.e. ˆ β 29 = 1.03 ) is quie close o he marke bea. Tesing for coinegraion shows ha all esimaed weigh allocaions exhibi a coinegraion relaionship wih he arificial indices up o an enhancemen facor equal o δ = 0.1320 (see able 2 in he appendix) on a 5% significance level. Agains i, porfolios racking arificial indices involving rends being larger δ = 35% do no exhibi a coinegraion relaionship wih he arificial annual benchmark. Exhibi 1 shows ha he abnormal reurns being esimaed in accordance o he ordinary index model (see equaion (11)) are saisically no significan concerning he ou-of-sample period. As he inegraed arificial indices are via consrucion coinegraed wih he benchmark (see equaions (1) and ((2)), i can be concluded ha he esimaed porfolio exhibi a coinegraion relaionship wih he ordinary benchmark, oo, while involving a saionary rend swiching a max he direcion. Figure 2 shows clearly he saisical arbirage opporuniy since on 77% of all days, he coinegraion opimal porfolio 29 ouperforms he
68 Marke iming and saisical arbirage index while exhibiing an excess reurn equal o 14.04% afer 124 rading days ou-of-sample. Furhermore porfolios 23 and 25 exhibi maximum saisical arbirage opporuniies as heir reurns are 19% above he index reurns or higher wih an empirical probabiliy of 50.14%. Even if porfolio 29 which racks an arificial index being enhanced by he facor δ = 0.1160 (i.e. 29% in annual erms) is opimal wih respec o is Reward-o-Risk raio, i generaes reurns of 29% p.a. above he index or higher wih an empirical probabiliy of 45.53%. The saisical arbirage opporuniies are in accordance o he definiion of Bondarenko (2003) wih respec o porfolio 29 limied up o excess reurns of 9% as he empirical probabiliy ha he porfolio generaes reurns of 9% or higher han he index is 50% for he laer figure. Figure 3 plos he Reward-o-Risk raios and shows an increasing rend on a decreasing rae while afer he maximum, corresponding o porfolio 29, he raio is declining. Esimaing he forecas adequacy concerning he maximum likelihood opimal weigh allocaions gives he resuls of equaion (12). Thereby, equaion (12) akes only he elemens 1,,33 ino accoun, as firs, a visual inspecion of he vecor δ clearly shows a changed slope beween δ = 0.1320 and δ = 0.1360. The second indicaor for a break is ha coinegraion opimaliy does only hold for he sample δ = 0.0040 unil δ = 0.1320 (see able 2 in he appendix). Therefore, equaion (13) akes also ino accoun he break in he slope parameer (-saisics in parenhesis): T T OMX ( rec rec δ ) = + δ = = annual annual ( 6.21) ( 18.64) T T OMX ( rec rec = = ) ( 6.48) ( 6.98) ( 19.45) ( 10.49), for 1,...,33 1/ N R R 7.96 1.35 δ = (15) (16) 1/ N R R 7.96 37.30 d 1.35 annual 1.52 d δ = + + δ δannual for δ annual = 1,...,50, wih d = 0 for δ annual < 33 and d = 1 else. Equaions (15) and (16) show ha all parameer esimaes are saisically significan. The R-squared of 0.9254 concerning equaion (15) and 0.9124 wih respec o equaion (16) sugges ha he forecas capabiliy of coinegraion opimal weigh allocaions is high. Equaions (15) and (16) esimae for porfolio 29 an excess
K. Grobys 69 reurn of 15.60% afer he six monh period ou-of-sample whereas he realized excess reurn was only 1.56 percen unis lower (i.e. 14.04%) in he end of he forecas period. In oher words, coinegraion opimal porfolios which rack hese arificial indices are relaively sable even wihin he ou-of-sample period up o a premium of δ = 0.1160 in daily (i.e. equaion (15) and (16) ake ino accoun he annual premium being added o he index). 5 Discussion If a recession is anicipaed by an equiy price bus, invesors would expec he equiy prices o fall furher in fuure periods as longs as he economy faces he sae of recession. If he majoriy of news ha arrive he marke paricipaors do no change he marke paricipaors mind such ha he fuure seems brigh again, here may be no raional reasons for a breakup concerning he equiy price decline. Table 1a: Saisical properies of he porfolios ou-of-sample Facor p.a. in % Annual mean in % Annual volailiy in % Rewardo-Risk raio Rewardo ΔRiskraio Bea -saisic of bea 1.00-37.25 57.38-0.05-0.73 1.07 8.52 2.00-43.29 58.32-0.15-1.84 1.08 7.29 3.00-43.75 58.12-0.16-2.02 1.07 6.76 4.00-42.05 57.36-0.13-1.98 1.06 7.50 5.00-42.07 57.59-0.13-1.87 1.07 7.24 6.00-39.98 57.18-0.10-1.51 1.06 7.39 7.00-32.47 55.39 0.04 1.05 1.03 9.65 8.00-32.04 55.27 0.04 1.36 1.03 8.22 9.00-28.28 54.52 0.11 6.08 1.01 7.95 10.00-25.88 54.18 0.16 12.84 1.00 7.12 11.00-25.27 54.11 0.17 15.28 1.00 6.27 12.00-21.89 53.61 0.23 128.33 0.99 5.76 13.00-20.28 53.34 0.27-85.26 0.98 5.19 14.00-19.19 53.50 0.29-1099.85 0.98 4.66 15.00-18.24 53.55 0.30 377.68 0.97 4.15
70 Marke iming and saisical arbirage 16.00-17.55 53.83 0.31 52.06 0.97 3.78 17.00-16.96 54.83 0.32 13.20 0.99 3.60 18.00-16.18 55.46 0.33 9.36 0.99 3.35 19.00-14.54 55.41 0.36 10.45 0.99 3.19 20.00-14.10 56.23 0.36 7.47 1.00 3.04 21.00-13.36 57.39 0.37 5.44 1.01 2.85 22.00-12.56 57.73 0.38 5.18 1.01 2.74 23.00-12.10 58.93 0.38 4.12 1.03 2.57 24.00-9.99 59.25 0.41 4.26 1.02 2.42 25.00-9.56 59.41 0.42 4.21 1.02 2.37 Table 1b: Saisical properies of he porfolios ou-of-sample Facor p.a. in % Annual mean in % Annual volailiy in % Rewardo-Risk raio Rewardo ΔRiskraio Bea -saisic of bea 26.00-9.65 60.36 0.41 3.62 1.03 2.27 27.00-7.53 60.46 0.44 3.87 1.03 2.17 28.00-6.75 60.67 0.46 3.87 1.03 2.12 29.00-6.13 61.50 0.46 3.54 1.03 2.02 30.00-7.46 62.84 0.43 2.89 1.05 1.98 31.00-11.35 65.89 0.35 1.86 1.09 1.91 32.00-6.83 62.48 0.44 3.08 1.04 1.95 33.00-7.33 63.25 0.43 2.78 1.05 1.91 34.00-9.41 64.84 0.39 2.21 1.07 1.89 35.00-11.47 66.78 0.34 1.73 1.10 1.85 36.00-12.60 66.74 0.33 1.65 1.10 1.86 37.00-15.20 65.81 0.29 1.56 1.09 1.90 38.00-17.82 65.54 0.25 1.38 1.09 1.95 39.00-18.70 65.56 0.24 1.31 1.09 1.95 40.00-9.58 69.17 0.36 1.59 1.10 1.63 41.00-9.10 68.70 0.37 1.67 1.10 1.67 42.00-10.42 67.48 0.36 1.72 1.10 1.78 43.00-11.42 67.84 0.34 1.61 1.10 1.73 44.00-12.23 66.92 0.33 1.66 1.10 1.81 45.00-13.11 67.08 0.32 1.57 1.09 1.78 46.00-14.01 67.03 0.30 1.51 1.09 1.77 47.00-12.11 67.27 0.33 1.62 1.10 1.78
K. Grobys 71 48.00-10.04 67.79 0.36 1.71 1.10 1.75 49.00-14.44 68.05 0.29 1.38 1.09 1.66 50.00-15.06 67.54 0.29 1.38 1.09 1.70 Table 1c: Saisical properies of he porfolios ou-of-sample Facor p.a. in % Annual alpha in % -saisic of alpha Tracking-Error volailiy p.a. in % 1.00-0.57-0.06 73.88 2.00-6.11-0.54 87.59 3.00-6.74-0.56 93.96 4.00-5.46-0.51 83.71 5.00-5.36-0.48 87.00 6.00-3.51-0.32 84.73 7.00 2.99 0.37 63.05 8.00 3.27 0.35 73.71 9.00 6.53 0.68 75.20 10.00 8.65 0.81 83.22 11.00 9.12 0.75 94.11 12.00 12.10 0.93 101.26 13.00 13.44 0.94 111.40 14.00 14.48 0.91 123.90 15.00 15.27 0.86 138.64 16.00 15.94 0.81 152.12 17.00 17.04 0.82 162.17 18.00 18.03 0.80 175.02 19.00 19.49 0.83 183.13 20.00 20.28 0.81 193.91 21.00 21.49 0.79 210.11 22.00 22.35 0.79 218.76 23.00 23.24 0.76 236.20 24.00 25.26 0.79 249.49 25.00 25.68 0.78 254.69
72 Marke iming and saisical arbirage Table 1d: Saisical properies of he porfolios ou-of-sample Facor p.a. in % Annual alpha in % -saisic of alpha Tracking-Error volailiy p.a. in % 26.00 25.91 0.75 268.48 27.00 27.81 0.77 279.18 28.00 28.55 0.78 286.32 29.00 29.35 0.76 301.11 30.00 28.63 0.71 313.33 31.00 26.23 0.61 336.98 32.00 28.95 0.71 315.15 33.00 28.74 0.69 323.84 34.00 27.47 0.64 334.90 35.00 26.34 0.58 350.68 36.00 25.21 0.56 349.72 37.00 22.30 0.51 337.86 38.00 19.71 0.46 330.12 39.00 18.86 0.44 330.13 40.00 28.34 0.55 400.31 41.00 28.86 0.58 389.05 42.00 27.42 0.58 365.56 43.00 26.39 0.55 374.95 44.00 25.48 0.55 356.99 45.00 24.54 0.53 362.62 46.00 23.55 0.50 364.17 47.00 25.63 0.55 364.08 48.00 27.85 0.58 371.30 49.00 23.10 0.46 387.28 50.00 22.41 0.46 378.17
K. Grobys 73 Figure 2: The OMX 30 and a coinegraion opimal porfolio wihin he ou-of-sample Figure 3: Reward-o-Risk Raio depending on δ annual As a consequence, raional invesors expecing equiy prices o fall will rebalance
74 Marke iming and saisical arbirage heir porfolios and choose a more defensive asse allocaion sraegy in order o minimize he expeced loss in he nearer fuure. Thereby, he opimizaion procedures concerning he asse allocaion depends on he invesors individual Reward-o-Risk preferences. Two differen opimizaion procedures are considered. Boh opimal asse allocaion sraegies dominae he underlying index and exhibi wihin he ou-of-sample period excess reurns wih low volailiies. Given he daa se being employed, maximum-likelihood esimaion gives robus parameer esimaes providing adequae forecass concerning he porfolios ou-of-sample performance. However, he forecas adequacy depends on he sock daa se being employed. The higher he arificial indices are enhanced, he less sable will be he forecas reliabiliy of he parameer esimaes as fewer socks are available in order o mimic he consruced sochasic processes, respecively, o rack hese defensive sraegies. The marke iming opporuniy as suggesed here depends exclusively on he informaion being provided by public insiues. The major price decline hough is hardly predicable and occurred before he press release was issued. The laer empirical fac holds for boh, equiy price buss being menioned earlier (i.e. 2000-2001 and 2008-2009). Moreover, he German sock marke, for insance, showed only marginal addiional price declines afer he press releases (i.e. on November 26, 2001 and, respecively, on Sepember 11, 208) were issued. However, he S&P 500 showed similar paerns like he OMX 30 and fell by 17.13% from March 7, 2000 unil November 26, 2001, whereas he price decline was addiional 6.36% beween November 26, 2001 and May 26, 2002. During he financial crises in 2008, he S&P 500 fell by 19.31% beween July 16, 2007 and Sepember 11, 2008 and addiional 39.88% beween Sepember 11, 2008 and March 12, 2009. Thus, he U.S. Index exhibied he same good marke iming opporuniies like he Swedish sock marke during he financial crises period. However, Aroa and Buza (2003) who consider a large daa se accouning for 20 bear markes wihin he las 102 years figured ou ha bear marke duraions have
K. Grobys 75 been reduced from 25.89 monhs for he firs 15 bear markes o 14.28 monhs for he las 5 bear markes (excluding he las bear marke in 2008). Furhermore, marke iming sraegies as inroduced here can be employed for acive funds managemen who is aiming a minimizing losses in down-marke movemens. However, marke-iming is wo-sided: Mainaining a defensive asse allocaion in bull markes can resul in lower porfolio reurns compared o he underlying sock index. Hence, he funds managemen faces he problem o define a rule when he defensive asse allocaion sraegy should be changed o an offensive one, for insance. Moreover, he German Insiue for Worldeconomics in Kiel (IfW) issued already on March 13, 2008 a press release where i was repored ha he world economic growh has slowed significanly owards he end of 2007 in response o he housing marke crisis in he US. Besides i was menioned ha he problems in he financial secor would coninue o weigh on he real economy and ha he risk of he US slipping ino recession was subsanial. 6 A his ime he OMX 30 los already 28.61% (i.e. beween July 16, 2007 and March 13, 2008) and a price bus could have already been ascerained. Since he laer price bus ook eigh monhs (i.e. from July 2007 unil March 2008) marke paricipaors could have expeced he bus o coninue for a leas addiional six monhs on average. Aroa and Buza (2003) menion ha he sock marke crashes in Ocober 1929, Ocober 1987 and March 2000 had in common, ha all hree periods were preceded by periods of increased volailiies. Considering he financial crises in 2008 hough, an increase in volailiy could also be ascerained. Significan changes in sock marke volailiy could herefore also ac as an indicaor for a forhcoming equiy price bus. As a consequence, he day when he recession is officially declared may be he las chance for an acive funds-managemen o minimize losses during he coninuing bear marke. 6 See hp://www.ifw-kiel.de/media/press-releases/2008/pr13-03-08a.
76 Marke iming and saisical arbirage Alexander and Dimiriu (2005) consruc six plus/minus benchmarks by adding and subracing annual reurns of 5%, 10% and 15% o and from he reconsruced DJIA reurns, uniformly disribued over ime. Their findings ha coinegraion opimal porfolios can be found even if he arificial benchmarks diverge significanly from he benchmark can be suppored in his sudy, oo. The consruced porfolios exhibi a coinegraion relaionship wih he arificial index up o an enhancemen facor of 33% in annual erms. In conras o Alexander and Dimiriu (2005) who argue ha reurns are significanly more volaile wihou any compensaion of addiional reurns as he spread beween he benchmarks racked widens, i is shown here ha he porfolio racking an arificial index being enhanced by 29% in annual erms exhibis he highes Reward-o-Risk raio. Alexander and Dimiriu s (2005) sudy sugges, however, ha he bes performance is produced by sraegies racking narrow spreads such as 5% hedged wih he porfolio racking he arificial benchmark. The laer issue canno be suppored in his sudy: The resuls (see able 1a-d) show ha all sraegies racking narrow spreads such as 1%-5% hedged wih he porfolio racking he arificial indices exhibi reurns below he benchmark and volailiies above he benchmark s volailiy. However, saisical arbirage opporuniies as defined by Bondarenko (2003) should exhibi an expeced payoff being nonnegaive. Given a required excess reurn of 10% above he sock marke, an invesor has o selec a porfolio racking an enhancemen facor of 25% in order o gain he required excess reurns wih he highes probabiliy (i.e. 52.85%) wih respec o each rading day ou-of-sample. As porfolio 25 exhibis a bea of 1.02 which is close o he marke bea, i can be employed o consruc a zero-cos rading sraegy while going shor on he index and long on porfolio 25, six monhs ahead while he expeced profi is 17% p.a. (corresponding o he empirical probabiliy of 50.14% which means ha he porfolio reurns are expeced o be 17% above he index or higher on each rading day ou-of-sample). However, he realized excess reurn of porfolio 25 was
K. Grobys 77 12.30% (i.e. 24.60 % p.a.) above he OMX 30 on he las day of he ou-of-sample period under consideraion (i.e. Sepember 12, 2008 - March 12, 2009). The laer fac holds even hough he empirical probabiliy ha he porfolio reurns exceed he enhanced benchmark reurns wih he daily enhancemen facor δ = 0.1000, corresponding o δ annual = 25%, is 47.15%. Grobys (2010) consrucs arificial indices irrespecive of he underlying economy s sae. Thereby, four years of daily daa are aken ino accoun in order o consruc arificial indices while he underlying sock marke s sochasic process is modified (i.e. S&P 500) by subracing a linear rend wihin he firs wo years in-sample and adding a linear rend wih he same slope parameer wihin he hird and fourh year concerning he in-sample period. The idea is in ha sudy o mimic rends in differen business secors while assuming ha business secors behave differen during a business cycle. This mehodology resuls in a sable coinegraion relaionship concerning he ou-of-sample period and saisical significan abnormal reurns of 6.83% p.a. while he volailiy is one hird lower in comparison o he benchmark. In conras o Grobys (2010) sudies where eleven differen muual funds are aken ino accoun, in he sudies presened here, socks are employed, only. The marke iming opporuniy does no res upon an assumpion abou cyclical paerns concerning some business secors bu on official press releases and he empirical fac ha an equiy price bus ha coincides wih a recession ends he earlies when he associaed recession is ending bu can even coninue aferwards. Consequenly, his marke iming opporuniy canno be exploied by ieraive working compuer programs bu requires a criical observaion of he sock marke as well as he sae of he economies. Unlike Grobys (2010) he ou-of-sample porfolio processes canno be priced by applying Vecor-Error-Correcion models. Therefore, empirical probabiliies are esimaed in his sudy for each porfolio, given differen enhancemen facors. Saisical arbirage opporuniies could be ascerained. However, he empirical probabiliies sugges ha he saisical arbirage
78 Marke iming and saisical arbirage opporuniies are below he expeced mean. In oher words, porfolio 29, for insance, may be expeced o generae ou-of-sample abnormal reurns in line wih he enhancemen facor being equal o 9% in annual erms. The empirical probabiliy ha abnormal reurns are gained wih a probabiliy of a leas 50% holds only wih respec o excess reurns being equal o 9%. 6 Concluding Remarks Even hough a random walk process is from a saisical poin of view no predicable, some movemens can be correlaed wih some specific evens concerning oher variables. Then, predicable paerns may arise being dependen on his join even. There is evidence given ha equiy price buss being associaed wih recessions coninue unil he economy swiches from he sae of recession o an economic pick-up. Such join evens can be exploied for boh, rebalancing equiy porfolios in order o selec defensive weigh allocaions or o exploi his marke condiions o consruc a saisical arbirage porfolio. There are incenives from an invesor s poin of view o shor he index and ake a defensive porfolio posiion being evened up afer six monhs, for insance. In he lieraure is repored ha sock marke crashes ofen happen in news vacuum (see Arora and Buza (2003), for insance). Anicipaing a sock marke crash is hardly possible even hough an increasing volailiy can be considered as one kind of indicaor ha some marke condiions can be changing. However, increasing volailiies can also be observed in bull markes. I remains sill unclear when he marke is changing from a bull o bear marke and vice versa. Invesing in a porfolio exhibiing defensive properies may be a good choice if in fac he marke coninues o decline furher. In conras, excess reurns can be diminished if he marke moves upwards earlier han expeced. Thus, here is sill need of research concerning he opimal rebalancing momen. No all sock markes are falling during recessions o he same exen as he Swedish index OMX 30. However, sock marke inegraion drives inernaional sock markes of mos
K. Grobys 79 advanced counries o exhibi similar properies. Advanageous asse allocaions do no only require o employ algorihmic opimizaion mehods being based on hisorical daa bu o include also psychological effecs driving he marke such as herding behavior as well as speculaive aacks and accouning for marke paricipaors expecaions concerning fuure periods. The laer iems hough require a good undersanding of he linkages beween macroeconomic, financial and psychological variables References [1] C. Alexander, Opimal Hedging Using Coinegraion, Philosophical Transacions of he Royal Sociey Series A, 357, (1999), 2039-2058. [2] C. Alexander and A. Dimiriu, Indexing and Saisical Arbirage, The Journal of Porfolio Managemen, 31, (2005), 50-63. [3] H.K. Arora and M.P. Buza, Unied Saes Economy & The Sock Marke, Journal Of Business And Economics Research, 1(1), (2003), 107-116. [4] D. Baker, The housing Bubble and he Financial Crises, Real-World Economics Review, 46, (2008), 73-81. [5] Z. Bodie, A. Kane and A.J. Marcus, Invesmens, Sevenh ediion, McGrawHill, New York, 2008. [6] O. Bondarenko, Saisical Arbirage and Securiies Prices, The Review of Financial Sudies, 16, (2003), 875-919. [7] S.G. Cecchei, P. Lam and N.C. Mark, Mean Reversion in Equilibrium Asse Prices, NBER Working Paper W2762, (1990). [8] K.C. Chan, B.E. Gup and M.-S. Pan, Inernaional Sock Marke Efficiency and Inegraion: A Sudy of Eigheen Naions, Journal of Business Finance & Accouning, 24, (1997), 803 813. [9] S. Claessens, M.A. Kose and M.E. Terrones, Wha happens during recessions, crunches and buss?, Economic Policy, 24, (2009), 653 700.
80 Marke iming and saisical arbirage [10] K. Grobys, Do Business Cycles exhibi Beneficial Informaion for Porfolio Managemen? An Empirical Applicaion of Saisical Arbirage, The Review of Finance and Banking, 2(1), (2010), 43-59. [11] F. Harmanzis, Inside he Telecom Crash: Bankrupcies, Fallacies and Scandals A Closer Look a he WorldCom case, working paper, Sevens Insiue of Technology Wesley J. Howe School of Technology Managemen, 2004. [12] M.C. Jensen, Overpricing elecoms in 2000, Financial Managemen, 34, (2005), 5-19. [13] M. Kendall, The Analysis of Economic Time Series, Par I: Prices, Journal of he Royal Saisical Sociey, 96, (1953), 11-24. [14] J.M. Poerba and L.H. Summers, Mean Reversion in Sock Prices: Evidence and Implicaions, NBER Working Paper, W2343, (1989). [15] G. Soros, The crises and wha o do abou i, Real-World Economics Review, 48, (2008), 312-318.
K. Grobys 81 Appendix Table 2: Tesing for coinegraion Facor p.a. in % ADF-saisic in-sample Facor p.a. in % ADF-saisic in-sample 1-16.00* 26-3.49* 2-14.97* 27-3.36* 3-15.14* 28-3.23* 4-15.58* 29-3.07* 5-14.88* 30-2.94* 6-14.74* 31-2.88* 7-14.18* 32-2.76* 8-13.36* 33-2.76* 9-12.26* 34-2.55* 10-11.20* 35-2.36 11-10.24* 36-2.33 12-9.42* 37-2.10 13-8.68* 38-1.87 14-8.06* 39-1.86 15-7.56* 40-2.28 16-7.09* 41-2.22 17-6.57* 42-2.17 18-6.14* 43-1.97 19-5.73* 44-1.94 20-5.27* 45-1.98 21-4.83* 46-1.94 22-4.52* 47-1.89 23-4.20* 48-1.91 24-3.94* 49-1.99 25-3.72* 50-1.88