Performance Analysis of Equally weighted Portfolios: USA and Hungary

Transcription

1 Aca Polyechnica Hungarica Vol. 9, No., 1 Performance Analysis of Equally weighed Porfolios: USA and Hungary András Urbán, Mihály Ormos Deparmen of Finance, Budapes Universiy of Technology and Economics, Magyar udósok körúja, H-1117 Budapes, Hungary urban@finance.bme.hu ormos@finance.bme.hu Absrac: Invesigaing U.S. equally weighed porfolios, one can measure posiive abnormal reurns (Jensen alphas) according o he classical equilibrium models. Applying he Carhar four-facor model, we show ha excess reurns generaed by he equally weighed muli-period invesmen sraegy are neiher caused by he small-firm effec, nor by he book-o-marke equiy, nor even by persisence. We documen ha his phenomenon canno be observed in he Hungarian sock marke, where he equally weighed rebalancing sraegy neiher achieves significan abnormal reurn, nor ouperforms he value weighed index in erms of mean reurn. This laer resul suggess ha, from his poin of view, he Hungarian capial marke exhibis a higher level of efficiency han is US counerpar. Keywords: equally weighed porfolio, performance measure, marke efficiency 1 Inroducion We invesigae a simple muli-period invesmen sraegy using equally weighed porfolios by comparing he performance of a value weighed marke index o equally weighed porfolios. An equally weighed porfolio akes every asse ino accoun wih he same weigh, while in a value weighed porfolio, he marke capializaion deermines he weigh of a sock. In he case of he U.S. sock marke, we documen posiive abnormal reurns for he equally weighed porfolios using he Capial Asse Pricing Model (CAPM) by Sharpe (196), Linner (196), and Mossin (1966), and he Carhar (1997) Four-Facor Model. We argue ha he excess reurn is neiher due o he small firm effec documened by Banz (1981) and Reinganum (198, 1981), nor he book o marke equiy facor documened by Basu (1983). This phenomenon canno be observed in he Hungarian sock marke; furhermore, he equally weighed porfolio does no ouperform he value weighed marke index in erms of mean reurn. 1

2 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary We even sae ha he negaive auocorrelaion caused by he mean revering behavior of sock reurns (see French and Roll (1986), Fama and French (1988), Poerba and Summers (1989) or De Bond and Thaler (198, 1987)) has no effec on he reurn of he equally weighed porfolios. Raher, holding a porfolio compiled by rebalancing differen random processes gains higher reurns by he naure of he sochasic processes. Opposie o he Budapes Sock Exchange (BSE), on equally weighed porfolios formed from U.S. socks, one can measure much higher reurns han ha of a value weighed porfolio. A large number of explanaions can be found in he lieraure which ry o give some heoreical background for he significan difference. If an equally weighed porfolio is invesigaed, he firs argumen is conneced o small firm effec. As Roll (1981) saes, "a value weighed index such as he S&P is obviously more heavily invesed in large firms han is an equally weighed index. Thus, comparing he behavior of wo such indexes will enable us o sudy, wih very lile effor, he size effec." In oher words he argues ha he difference beween he reurns of similar risky porfolios in ha he behavior comes from he size differences. Roll argues ha he small firm effec is he resul of a measuremen problem and rading infrequency seems o be a powerful cause of bias in risk assessmens wih shor-inerval daa. Raher horrendous bias is induced in daily daa and he bias is sill large and significan wih reurns measured over inervals as long as one monh. In our analysis, we use monhly reurns insead of daily ones. This argumen is similar o Banz s (1981) resuls. The oher reasoning according o he higher reurn is concenraed on he auocorrelaion in he process; however, he auocorrelaion of daa has a ime varying behavior (see Li and Yen, 11). In he shor run, one can measure a mean revering price behavior, which in urn means negaive serial auocorrelaion; see e.g. Dyl and Maxfield (1987), Bremer and Sweeney (1988) or Brown e al. (1988). On longer ime inervals, for weekly reurns, Howe (1986) or Lehmann (1988) measures also negaive auocorrelaions. Similarly, for monhly reurns, Rosenberg e al. (198), Jagedeesh (199), Brown and Harlow (1988) measure negaive auocorrelaion. For even longer inervals, for welve monhs, Jagedeesh (199) documens posiive auocorrelaion. However, invesigaing much longer inervals, e.g. DeBond and Thaler (198, 1987), Poerba and Summers (1989) or Fama and French (1988) repor again a negaive serial correlaion in marke reurns over observaion inervals of hree o five years. In he case of an equally weighed porfolio, like e.g. he S&P equally weighed index (S&P EWI), which is compiled from he larges U.S. socks, hese auocorrelaions have a high impac on he reurn. However, he sudies sae ha he abnormal reurn of he S&P EWI is due o he small firm effec, neglecing he findings according o he auocorrelaions. The S&P EWI is a quarerly rebalanced porfolio; herefore, he negaive auocorrelaion measured for his inerval has a posiive effec on he reurn because he equally weighed porfolio increases he weighs of he "pas losers" and a he same ime decreases he weighs of he "pas winners", where he pas means hree monhs performance. However, if he process is mean-revering, he 16

3 Aca Polyechnica Hungarica Vol. 9, No., 1 reurn generaed by his conrarian invesmen sraegy mus be higher han he reurn of he value weighed index. Of course, because of is naure, i is no difficul o see ha i would be helpful o have such processes o achieve beer performance. However, regarding Mulvey and Kim (8), he ruh is ha meanreversion is no necessary for he fixed mix o accomplish superior performance. Sein e al. (9) invesigae he diversificaion and rebalancing of Emerging Marke counries porfolios. They show ha even hough Emerging Markes suffer high ransacion coss and unreliable informaion, pragmaic porfolio implemenaions such as equally weighed rebalancing wih relaively lile rading sill promise excess performance. However, our findings on BSE sock porfolio does no suppor his issue. The remaining quesion is wheher he difference in he reurn, if i exiss, can be explained by he sandard equilibrium models (he CAPM and Four-Facor Model) or no. If no, i.e., significan posiive alpha can be measured especially by he four facor model, his means ha he sraegy promises excess reurn, above he equilibrium, where he small-firm-effec, he book-o-marke equiy effec and he effec of persisency is already managed. In fac, our resuls for survivorship biased daase using he componens of he Sandard and Poor s index componens in he rebalancing sraegy gains significan posiive or nonsignifican bu posiive alphas. The same sraegy formed from BUX index (he main Hungarian equiy index) componens only provide non-significan posiive alpha. Sock Marke Model The model of sock marke invesigaed in his secion is he one considered, among ohers, by Luenberger (1998), Mulvey and Kim (8). Consider a marke of d asses whose mean reurn vecor is r where rr d. Le SR d d be a covariance marix. Assuming normaliy for he reurn s join disribuion, a saic porfolio s reurn according o he porfolio vecor w R d is also normal wih mean w,r and variance w,sw, where, denoes inner produc. Le us invesigae a consanly rebalanced porfolio made of he same socks and rebalanced in each insananeous momen according o w. We can wrie he following sochasic differenial equaion for he price process of asse i as dp p 1 ( ri i ) d dbi where r i and i are he reurn and variance of asse i, respecively. B is a geomeric Brownian moion. Thus, for he value of a consanly rebalanced porfolio we have (1) 17

4 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary dp dp 1 P d d wi wi ( r ) i i d dbi 1 1 i p i The porfolio s growh rae is he weighed average of he individual asse s rae, ha is, we can wrie dp 1 wr w P d P dw for he porfolio s growh rae, where P denoes he porfolio s variance and W is an elemen of a sandardized Wiener process. Thus, for he consanly rebalanced porfolio s mean reurn for a uni period we have r P wr w P wr w wsw ha is, he consanly rebalanced porfolio s mean is larger han he saic case by he facor of ½w, -w,sw, which is he so-called rebalancing gain (see Mulvey and Kim (8)). Since w, is he weighed sum of he porfolio consiuens variances, is value is equal o he porfolio s variance (w,sw) if and only if he consiuens are absoluely correlaed. In any oher case, he consanly rebalanced porfolio ouperforms is saic counerpar in erms of mean reurn respec o ha (1) boh porfolios consis of he same socks and () he saic porfolio s iniial capial allocaion vecor is idenical o he rebalancing sraegies w. Furhermore, he consanly rebalanced porfolio s reurns are also normal wih he same variance as he saic porfolio (w,sw). In he nex secion we invesigae wo ypes of equally weighed porfolios o ge empirical evidence wheher hese aracive heoreical properies are manifesed in abnormal reurns in erms of an equilibrium model or no, and wheher hese excess reurns infer higher risks. () (3) () 3 Empirical Resuls 3.1 Porfolio Consrucion We invesigae porfolios which are formed on he basis of Sandard & Poor s large-cap index for 1-year-long periods. We launch a new porfolio a he beginning of each year from 197 o 1999 in he following way: The porfolios are reviewed each monh o map exacly he acual Sandard & Poor s consiuens. Socks included or excluded from he index no a he beginning of a monh are considered for he whole monh. These porfolios are no free of survivorship bias (NFB) since hey follow he performance of he acually larges companies. To ease he noaion of hese porfolios, we refer o hem as S&P 18

5 Aca Polyechnica Hungarica Vol. 9, No., 1 EW. We rebalance all porfolios on he firs rading day of each monh according o a weigh vecor which divides he accumulaed wealh equally among consiuens. For each porfolio a 1-year-long holding period is invesigaed. The firs porfolio is launched in January 197 and ends in December 198. Similarly, new porfolios are formed a he beginning of each year unil January We use daa of U.S. sock reurns from Cener for Research in Securiy Prices (CRSP) daabase. The reurns are merged o Sandard & Poor s consiuens lis from he Compusa Norh America daase. Similarly we form an equally weighed porfolio on he basis of he BUX index. Due o he limied availabiliy of daa, five-year-long periods are invesigaed; ha is, we launch a new porfolio a he beginning of each year from 1999 o in he following way: The porfolios are reviewed each monh o map exacly he acual BUX consiuens. Alhough he BSE reopened in 199, he mauriy of he marke and he limied availabiliy of consisen daa provide he faciliy o form porfolios in he above-menioned way. For beer comparison, we launch each porfolio in exacly he same way as ha of he S&P; however, in saisical erms, he oucome of his analysis is no represenaive. Before he analysis we modify by splis, dividends and we recalculae he reurns in U.S. dollars, by which we ge resuls ha are comparable o he U.S. marke. The applied mehodology, as in he previous case, is also no free of survivorship bias. The BSE equally weighed porfolio is referred as BUX EW. In his case he capializaion weighed BUX index is used for he comparison of he equally and capializaion weighed porfolios. 3. Analysis of Pas Performance In Figures 1 and we presen he wealh levels of he inroduced S&P EW and BUX EW sraegies agains ime. The solid lines are he EW porfolios wealh. The capializaion weighed indices wealh are capured by dashed lines. On he U.S. marke one can see ha boh ypes of he proposed porfolios ouperform he capializaion weighed S&P index in he sense of final wealh (and almos always in sense of any ineremporal wealh level); however, he equally weighed sraegies are more volaile. The more volaile reurn induces higher expeced reurn in an equilibrium seing; herefore, if one would like o compare he wo syles of porfolio creaion, he difference beween he reurns should be exended wih sysemaic risk measures. Formalizing he mehod of performance measuremen, wo equilibrium models are consruced, he classical Capial Asse Pricing Model (CAPM) (see Sharpe 196, Liner 196, Mossin 1966), and a Four-Facor Model (see Carhar [6]). More precisely, one can esimae he reurn of he sraegies in he following ways sequenially: r r ( r r ) l f l l m f l () 19

6 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary Figure 1 Wealh levels accumulaed by he S&P Noes: EW porfolios and he capializaion weighed S&P marke proxy. EW porfolios are launched each January from 197 unil 1999, and rebalanced on he firs rading day of each monh according o a weigh vecor which divides he accumulaed wealh equally among Sandard & Poor s 16

7 Aca Polyechnica Hungarica Vol. 9, No., 1 index consiuens which appear in he index anyime in he monh for he 1-year long period. CRSP-VW is a capializaion based index and needs no rebalancing Figure Wealh level accumulaed by he BUX 161

8 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary Noes: Figure shows he EW and he capializaion weighed BUX porfolio. BUX index porfolio is launched in January 1999, and rebalanced on he firs rading day of each monh according o a weigh vecor which divides he accumulaed wealh equally among BUX index consiuens which appear in he index anyime in he monh for he 1-year long period. The pure BUX index is a capializaion weighed and needs no rebalancing. r r ( r r ) s SMB h HML mom MOM l f l l m f l l l l where l,, r f, (r m r f ), sand for asse l, ime, risk free rae, marke premium and esimaion residuals, respecively. The risk free rae is he rae of reurn of he onemonh Treasury-bill obained from Ibboson and Associaes. We use he capializaion weighed S&P index as marke proxy. According o Fama and French (1993) SMB (small-minus-big) measures he average reurn difference beween small and large capializaion asses, while HML (high-minus-low) is he average reurn difference beween high and low book-o-marke equiy (B/M) companies. MOM is he one-year momenum facor (see Carhar 1997), which shows he average excess reurn of he las one year s winners above he reurn of las one year s loser securiies 1. The regression coefficiens,, SMB, HML and MOM were esimaed based on equaion () and (6). The alpha parameer measures he excess reurn of an invesmen above or below he risk adjused equilibrium value. According o he Efficien Marke Hypohesis of Fama (196, 197) and Jensen s (1968) research on abnormal performance of muual funds, one can achieve repeaedly posiive alpha only by chance. Thus, a significan alpha in he model is agains he marke equilibrium assumpions since i implies sysemaic abnormal performance in he pas reurns. We invesigae 1- year-long periods in he case of he S&P and -year-long periods for he Budapes Sock Exchange BUX, because on he one hand, he reurn anomalies in his long periods are diminishing, i.e. hese inervals are as long ha he probabiliy of beaing he equilibrium reurn only by chance is small, and on he oher hand, we have he opporuniy o run saisical analyses on he reurns comparing he periods invesigaed. Tables 1 and presen he resuls of he regression analysis based on equaions () and (6) for all EW porfolios and for various periods. Alhough each period covered differen economic environmens, he coefficiens do no show remarkable differences in he various periods for he U.S. marke excep in he case of he small-firm facor loading, which is much smaller in he more recen periods. As he ables show, he majoriy of sensiiviy coefficiens are significan a. level; ha is, each facor has unambiguous loading on he EW (6) 1 SMB, HML and MOM facor porfolio reurns are obained from Kenneh French s homepage 16

9 Aca Polyechnica Hungarica Vol. 9, No., 1 performance. This is no he case in he Hungarian marke, where only he is significan. Regarding our analysis, however, he mos imporan observaions are he values of he alphas, which are no always significan, bu heir significan values are always posiive. The monhly abnormal reurns in he CAPM are beween -.9% and.%. Regarding he Four-Facor Model heir values in he U.S. scaer from monhly.% o.9%, where he laer is 3.8% in annum. An ineresing fac is ha he R -s are consisenly lower on he closer invesing inervals as he porfolios excess reurns are also lower for he more recen periods. Table 1 S&P EW and BUX EW Summary Saisics for he CAPM CAPM for USA Period r SP EW-r f Sd Dev R adj R %.19% 1.1.% %.76% 1.6.3% %.77% 1..8% %.6% 1.7.9% %.37% 1.6.3% %.7% 1.6.6% %.% % %.7% % %.19% % %.13% 1.1.9% %.% % %.91% % %.7% 1.1.% % 3.81% % %.16% % %.18% % % 3.93%.86.8% %.8%.8.3% %.6%.9.8% %.8%.9.3% %.78%.9.3% %.8%.93.36% %.76%.9.36% %.68%.93.3% %.8%.99.3% Average.87%.8% 1.3.% CAPM for Hungary Period r BUX EW-r f Sd Dev R adj R % 7.3%.3.6% % 7.%..6% % 6.93%.1 -.1% % 6.8%.9.1% % 6.%. 1.% % 6.36% % % 7.1%.9.6% % 7.%. 1.1%

10 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary Noes: Table 1 presens he S&P EW and BUX EW Summary Saisics for he CAPM. Porfolios are launched each January from 197 unil We rebalance he porfolios on he firs rading day of each monh according o a weigh vecor which divides he accumulaed wealh equally among Sandard & Poor s and BUX index consiuens which appear in he index anyime in he monh. r SP EW r f and r BUX EW r f are he average U.S. Dollar denominaed reurn of he S&P EW and BUX EW porfolios in excess of he one-monh U.S. Treasury-bill reurn. Sd Dev refers o he sandard deviaion of he excess reurn. and are parameers of he OLS regression model (). Parameer measures he average abnormal reurn (significan alphas a. level are in bold). R is he coefficien of deerminaion. The model selecion crieria is he adjused R. Table S&P EW and BUX EW Summary Saisics for he Four-Facor Model Four-Facor Model for USA Period r SP EW-r f Sd Dev SMB HML MOM R adj R %.19% % %.76% % %.77% % %.6% % %.37% % %.7% % %.% % %.7% % %.19% % %.13% % %.% % %.91% % %.7% % % 3.81% % %.16% % %.18% % % 3.93% % %.8% % %.6% % %.8% % %.78% % %.8% % %.76% % %.68% % %.8% %.9.93 Average.87%.8% % Four-Facor Model for Hungary Period r BUXEW-r f Sd Dev SMB HML MOM R adj R % 7.3% % % 7.% % % 6.93% % % 6.8% % % 6.% % % 6.36% % % 7.1% % % 7.% %

11 Aca Polyechnica Hungarica Vol. 9, No., 1 Noes: Table shows he S&P EW and BUX EW Summary Saisics for he Four-Facor Model. Porfolios are launched each January from 197 unil We rebalance he porfolios on he firs rading day of each monh according o a weigh vecor which divides he accumulaed wealh equally among Sandard & Poor s and BUX index consiuens which appear in he index anyime in he monh. r SP EW r f and r BUX EW r f are he average U.S. Dollar denominaed reurn of he S&P EW and BUX EW porfolios in excess of he one-monh U.S. Treasury-bill reurn. Sd Dev refers o he sandard deviaion of he excess reurn., SMB, HML, MOM and are parameers of he OLS regression model (6). Parameer measures he average abnormal reurn (significan alphas a. level are in bold). R is he coefficien of deerminaion. The model selecion crieria is he adjused R. The beas are slighly higher han one for each porfolio; ha is, over-weighing relaively smaller firms agains he abou gians which dominaed he value weighed index appreciably raised he porfolio risk, alhough he CAPM s beas are slighly lower in more recen periods. According o he model selecion crieria (adjused R ) he addiional facors raise he explanaory power of model (6), especially in he more recen daes. I is worh noing, however, ha hese porfolios consised of large-cap firms; he SMB facor had a small, bu significanly posiive loading on EW premia. Posiive HML coefficiens imply ha over-weighing smaller companies also suppors invesing in high book-o-marke equiy socks. The negaive loading on reurn momenum is he naural aendan of an equally weighed sraegy, which gives relaively larger porions for socks which performed below he average in he previous invesmen period. Firs and las, invesors who had preferred an equally weighed mixure of U.S. large-cap socks could achieve exra yield on every 1-year-long period in conras o he value weighed index, which is no he case in he Hungarian marke. For he BUX EW porfolio, boh models have very low explanaory power, he facor loadings are insignifican, and he equally weighed index slighly underperforms he capializaion weighed BUX index in erms of final wealh. On he one hand, we argue ha he Hungarian capial marke exhibis a higher level of marke efficiency from his poin of view, as he rebalancing sraegy gains no significan abnormal reurns. This resul may conradic is US counerpar; however i confirms he weak form of marke efficiency. On he oher hand, as he explanaory power of he equilibrium model is very low, one would sugges ha besides he BUX value weighed reurns, oher parameers should have been used o proxy he marke. However, if we accep he arbirage pricing heory by Ross (1976) all efficien porfolio can be used as a reference. Conclusions and Furher Research Direcions We show ha he equally weighed porfolios higher reurn compared o he capializaion weighed marke index canno be explained by he well-known equilibrium model. We use survivorship biased porfolio seing, and hese significanly over-perform he equilibrium models. Using he Four-Facor Model, we show ha he excess reurn generaed by he U.S. equally weighed muliperiod invesmen sraegy is neiher caused by he small-firm effec, nor by he book-o-marke equiy, nor by he persisence. Conrary o he resuls for he U.S. markes, on he Budapes Sock Exchange we canno measure excess reurn wih 16

12 A. Urbán e al. Performance Analysis of Equally weighed Porfolios: USA and Hungary he periodically equally weighing rebalancing sraegy. Alhough he explanaory power of he Four-Facor model is low, we sae ha, from his poin of view, a very high level of marke efficiency can be measured on he Hungarian capial marke. There are several direcions from which he equally rebalanced porfolio sraegy can be furher invesigaed. On he one hand, one can argue ha he variance or sandard deviaion or any oher risk parameer which is compiled by using hese measures is no adequae. One may use he variance of he log reurns for calculaing he variance, because in his seup he "penaly", i.e. he increase in he variance in he case of a posiive rigger, is lower han ha of a negaive one. This assumpion is even more reasonable if, insead of uiliy maximizaion, one would use he loss-aversion approach. On he oher hand he equilibrium model se up suggess a one-period world. The goal of he one-period porfolio heory (Markowiz 19) and he rival equilibrium models (like CAPM, APT, Fama- French, or Carhar model), is he opimizaion of he asse allocaion in order o achieve he opimal rade-off beween expeced one-period reurn and risk. This supposes a world where he invesors opimize heir consumpions and invesmen sraegies for ha given one period. However, mos of he mean-variance analysis handles only saic models, conrary o he expeced uiliy models, whose lieraure is rich in muli-period models, supposing an individual wih longer inerval han simply one-period hinking. One could suppose ha a Daa Envelopmen Analysis (see Gokgoz, 1) or fracal analyses (see Bohdalová and Greguš, 1) would increase he accuracy of our esimaions. In he muli-period models he invesors are allowed o rebalance heir porfolios in each rading period, and herefore heir invesmens may be characerized in differen ways in one and muliple periods due o he muliplicaive effec of consecuive reinvesmens. There is a hird direcion which seems o be worh a closer look; his is he volailiy. The quesion arises as o wha is he mahemaical connecion in a discree world beween he volailiy of he single securiies and he reurn of he porfolio. Acknowledgmens We are hankful o he anonymous reviewer for he valuable commens and suggesions. References [1] Bohdalová, M and Greguš, M.: Fracal Analysis of Forward Exchange Raes, Aca Polyechnica Hungarica, Vol. 7, 1, pp [] Bremer, M. A. and Sweeney R. J.: The Reversal of Large Sock-Price Decreases, Journal of Finance, Vol. 6, 1991, pp [3] Brown, K. C., Harlow, W. V. and Tinic S. M.: Risk Aversion, Uncerain Informaion, and Marke Efficiency, Journal of Financial Economics, Vol., 1988, pp

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