# Momentum and Contrarian Stock-Market Indices

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www.business.unsw.edu.au School of Economics Australian School of Business UNSW Sydney NSW 2052 Australia http://www.economics.unsw.edu.au Momentum and Contrarian Stock-Market Indices Jon Eggins and Robert J. Hill School of Economics Discussion Paper: 2008/07 The views expressed in this paper are those of the authors and do not necessarily reflect those of the School of Economic at UNSW. ISSN 1323-8949 ISBN 978 0 7334 2640 7 Momentum and Contrarian Stock-Market Indices∗,∗∗Jon EgginsRussell Investment Group, Level 17, MLC Center, Sydney 2000, Australia(E-mail: [email protected])Robert J. HillDepartment of Economics, University of Graz, Universit¨atstrasse 15/F4, 8010 Graz,Austria (E-Mail: robert.hill@uni-graz-at)School of Economics, Australian School of Business, University of New South Wales,Sydney 2052, Australia (E-Mail: [email protected])April 22, 2008Abstract:We propose a new class of investable momentum and contrarian stock-market indicesthat partition a benchmark index, such as the Russell 1000. Our momentum indicesoverweight stocks that have recently outperformed, while our contrarian indices under-weight these same stocks. Our index construction methodology is extremely ﬂexible, andallows the index provider to trade-oﬀ the distinctiveness of the momentum/contrarianstrategies with portfolio turnover. Momentum investment styles in particular typicallyentail a high level of turnover, and hence high associated transaction costs. The creationof momentum and contrarian indices and exchange traded funds (ETFs) based on ourmethodology would allow investors to access these styles at lower cost than is currentlypossible. Our indices also provide performance benchmarks for momentum/contrarianinvestment managers, and good proxies for a momentum factor. Over the period 1995-2007 we ﬁnd that short term momentum and long term contrarian indices outperformthe reference Russell 1000 index. We also document the changing interaction betweenthe momentum/contrarian and value/growth styles. (JEL C43, G11, G23)KEY WORDS: Momentum index; Contrarian index; Performance measurement; Turnover;Momentum factor; Behavioral ﬁnance∗The views expressed in this article are those of the authors and do not necessarilyreﬂect those of the Russell Investment Group.∗∗We thank Daniel Buncic for excellent research assistance.1. IntroductionIn recent years, the eﬀectiveness of momentum and contrarian investment strate-gies has been the subject of much discussion. A contrarian investment strategy earnsrisk-adjusted excess returns when investors overreact to news, while a momentum strat-egy earns excess risk-adjusted returns when investors underreact. DeBondt and Thaler(1985, 1987) ﬁnd evidence of overreaction over periods of a few years, while Jegadeeshand Titman (1993, 2001) ﬁnd evidence of underreaction over periods of a few months.These ﬁndings are consistent with the experimental results of Kahneman and Tversky(1982), who observed systematic violations of Bayes Rule amongst subjects when re-vising their beliefs in the presence of new information [see also Barberis, Shleifer andVishny (1998)].When attempting to explain stock returns and investment manager performance,researchers generally turn to the 3 factor Fama and French (1992, 1993, 1996) or 4factor Carhart (1997) model. This is because a signiﬁcant body of research evidenceexists demonstrating that the market return, ﬁrm size, valuation and momentum areimportant drivers of stock returns. Each of these factors, excluding momentum, has astandard market proxy in the form of a stock market index (e.g., a market cap weightedindex, large and small cap indices, value and growth indices). The missing ingredientin the market is a momentum index.This is particularly important given that large numbers of investors follow con-trarian and momentum strategies [see for example Goetzmann and Massa (2002)]. Thesame is true of mutual funds. Grinblatt, Titman and Wermers (1995) ﬁnd that 77 per-cent of the 155 mutual funds in their sample engage in momentum investing. Similarly,Menkhoﬀ and Schmidt (2005) use survey data to show that momentum and contrar-ian strategies are common among German fund managers. Given the extensive use ofthese strategies, it is surprising that no explicitly momentum and contrarian indices arecurrently computed by any of the major index providers.There is therefore a clear need for momentum and contrarian indices. Such in-dices would provide useful performance benchmarks for active managers following these1strategies. Active managers should not be unduly rewarded or punished for perfor-mance attributable to their innate factor tilts [Mulvey and Kim (2008) ﬁnd that thebest performing managers tend to have a momentum tilt]. The development of momen-tum and contrarian indices could therefore enhance the performance evaluation of activemanagers following these styles. Furthermore, these indices could facilitate the develop-ment of momentum and contrarian exchange traded funds (ETFs), providing investorswith a lower cost option for gaining access to the momentum and contrarian modesof investing (we return to this issue shortly). Given the ﬁndings of deBondt-Thalerand Jegadeesh-Titman, for example, there might be particular demand for momentumfunds with horizons of less than a year and contrarian funds with longer horizons.The vast majority of stock-market indices are market-cap weighted. This is becausemarket-cap weighting is conceptually straightforward, consistent with a passive buy-and-hold strategy, has relatively low transaction costs (due to low turnover rates andthe focus on larger cap and hence more liquid stocks), and when followed by all investorsdoes not violate market clearing [see Bailey (1992), Siegel (2003) and Arnott, Hsu andMoore (2005)]. In recent years there has been a proliferation of stock-market indices, forexample value and growth indices, small and mid-cap indices, and sector (e.g., banks,health, information technology) indices. Small, mid-cap and sector indices are usuallyalso market-cap weighted over their chosen domain, while value and growth indices areusually constructed by splitting a market-cap-weighted portfolio into two more or lessequal halves (in terms of market capitalization).Momentum indices are conspicuous in their absence. The only example of a con-trarian index currently available is an equal-weighted index that is rebalanced period-ically. The act of rebalancing back to equal weights requires buying past losers andselling past winners. In recent years, equal-weighted versions of the S&P 500, Dow-Jones Wilshire 5000 and Nasdaq 100 indices have appeared [see for example Standardand Poor’s and Rydex Global Advisors (2003)]. Value Line also produces an equal-weighted index (VLA) deﬁned on about 1620 stocks. It should be noted, however,that these indices are all advertised as equal-weighted indices. They are contrarian by2accident rather than by design.One reason for the lack of momentum and contrarian indices in the public domain isthe way momentum portfolios are typically constructed. The literature generally mimicsthe momentum factor by constructing portfolios that go long the best performing stocksand short the worst performers [see for example Jegadeesh and Titman (1993)]. Suchinvestment strategies are not practical for investors since they require shorting of assets,which many investors are either not able or not permitted to do. Also, by excludingmost stocks in a benchmark, such indices will tend to have high turnover (and hencehigh transaction costs) and low diversiﬁcation. Further, they do not provide a measureof momentum relative to the market return (which is available currently in the valueand growth dimension but not in the momentum and contrarian dimension) and theirhigh levels of concentration make them less able to adequately evaluate the performanceof investment managers. See Bailey (1992) for a list of properties that a performancebenchmark should ideally possess.In this article we propose a new class of ﬂexible momentum/contrarian indices thatdoes not require shorting and allow the index provider to trade-oﬀ turnover against thedistinctiveness of the momentum/contrarian strategy. Like many value/growth indices,our momentum/contrarian indices are benchmarked to existing indices – usually of themarket-cap-weighted variety although this is not necessary. For example, momentumand contrarian versions of equal-weighted indices or of fundamental indices [Arnott,Hsu and Moore (2005)] are also possible. Here we use the Russell 1000 index as ourbenchmark.We calculate the performance, risk and turnover characteristics of our momentumand contrarian indices over a 12 year period. Our momentum and contrarian indicespartition the benchmark index into momentum and contrarian portfolios that whencombined yield the original index portfolio. This approach should be familiar to in-vestors since value/growth indices are typically constructed in a similar way. Also,this approach has a distinct advantage over the traditional long minus short momen-tum factors in that it better approximates the actual behavior of active investment3managers.We begin by constructing a partition in the momentum/contrarian domain basedon the past price performance of stocks relative to the benchmark index over a speciﬁedtime horizon. We then propose a new and ﬂexible approach for transforming this basepartition that makes use of the beta function. By varying the parameter β in the betafunction the index provider has considerable ﬂexibility with regard to the design of theﬁnal partition actually used to construct the momentum and contrarian portfolios. Thisbeta function transformation is a useful innovation in its own right, and represents anattractive method for constructing passive factor portfolios. Although here we focus onthe price momentum factor, it could equally well be applied to other factors such asvalue and growth, earnings momentum, dividend yield or price acceleration (the rate ofchange of momentum).During the “tech” boom we ﬁnd a close aﬃnity between momentum indices andgrowth indices, and between contrarian indices and value indices. This relationship isconsistent with the ﬁndings of Assness (1997), but it reverses after 2001. Thereafterthe momentum indices move more in line with value indices, and contrarian indiceswith growth indices. In the more recent history we ﬁnd little relationship betweenmomentum/contrarian and growth/value styles. These ﬁndings highlight the fact thatmomentum/contrarian strategies are not simply a proxy for growth and value, andrepresent unique investment strategies in their own right.We also compute the turnover of our momentum and contrarian indices. Althoughmomentum strategies inevitably lead to indices with higher turnover than the refer-ence Russell 1000 index and corresponding value/growth indices, this turnover can becontrolled to an acceptable level without sacriﬁcing the distinctiveness of the momen-tum and contrarian styles. Furthermore, the more relevant comparisons of turnover forthese types of indices are with traditional (long minus short) momentum factors andinvestment managers following momentum investment strategies. Our indices comparefavorably on both counts, and hence could potentially allow investors to access themomentum style of investing at lower cost than is currently possible.4Momentum/contrarian ETFs based on our indices therefore are viable and couldbe of interest to investors aiming to tactically or strategically tilt towards or away frommomentum at various horizons, or as a passive factor exposure in a multi-managerstructure (e.g. if the fund has an unwanted bias away from momentum or if a managerwants to tilt towards it). These strategies are not currently implementable at low cost.We explore the impact of varying the time horizon (formation and holdings peri-ods) on the performance and turnover of momentum and contrarian indices. The timehorizons considered range from six months to three years. Our ﬁndings are consistentwith those of the behavioral ﬁnance literature. Momentum strategies tend to work wellover shorter horizons, while contrarian strategies outperform over longer horizons. Wealso show how our momentum and contrarian indices can be used to construct proxiesthat correlate very closely with the momentum factor in a Carhart (1997) factor model.The rest of the article is structured as follows. In the next section we describe ourmomentum/contrarian index methodology. Section 3 provides an empirical demonstra-tion of our momentum/contrarian indices benchmarked to the Russell 1000 index overthe period 1995-2007. Our main ﬁndings are summarized in the conclusion.2. Constructing momentum/contrarian stock-market indicesWe propose a class of momentum/contrarian indices that partitions a benchmarkportfolio. That is, a momentum and contrarian allocation is allotted to each stockin the benchmark index, such that the allocations sum to one. For example, it couldbe decided that stock X is 80 percent momentum and 20 percent contrarian. Theseallocations are then used to construct momentum and contrarian index portfolios, whichwhen combined exactly equal the portfolio underlying the benchmark index.The momentum/contrarian allocations are constructed in two stages. In the ﬁrststage, the momentum and contrarian allocations for each stock are determined as fol-lows:5Stage 1 Allocations:pMomentum : µM =t,n/pt−k,n,t,n(pt,n/pt−k,n) + (It/It−k)IContrarian : µC =t/It−k,(1)t,n(pt,n/pt−k,n) + (It/It−k)where µM and µC denote the fraction of total shares from the reference index of stockt,nt,nn allocated to the momentum and contrarian portfolios, pt,n denotes the price of stockn and It denotes the level of the benchmark index. The allocations µM and µC aret,nt,nfunctions of the price relatives pt,n/pt−k,n. When a stock n in the benchmark portfolioin period t is not listed in period t − k (i.e., pt−k,n is not available), we simply setµM = µC= 0.5. That is, this stock is split equally between the momentum andt,nt,ncontrarian portfolios.The benchmark index It here excludes dividends. This is not a requirement. Thestage 1 formulas could be reformulated to include dividend payments. Also, otherperformance measures such as earnings per share (EPS) growth could be used in placeof price to construct a diﬀerent factor portfolio/index that may be useful as a benchmarkand investment vehicle for say an EPS growth strategy.Both µM and µC are bounded between 0 and 1. Also, by construction, µM +µC =t,nt,nt,nt,n1. When a stock n rises by the same proportion as the index itself (i.e., pt,n/pt−k,n =It/It−k), it follows that the stock is allocated equally between the momentum andcontrarian portfolios (i.e. µM = µC = 0.5). When a stock rises by more than thet,nt,nreference index, µM > 0.5 and µC < 0.5.t,nt,nA distinction must be drawn between the formation and holding periods of amomentum/contrarian strategy. The formation period is the period over which pricesare observed to determine the momentum/contrarian allocations for each stock. Theformation period here is from period t − k to t, and hence the formation horizon is kperiods. The holding period is the period over which a momentum/contrarian portfoliois held, i.e., it measures the frequency with which the index is rebalanced.Rebalancing of the momentum/contrarian indices should not be confused withreconstitution of the benchmark index itself. For example, the Russell 1000 index is6reconstituted (i.e., stocks are added and deleted) on an annual basis. Using these deﬁni-tions, note that equal-weighted, fundamental, value/growth and momentum/contrarianindices all need to be both rebalanced and reconstituted periodically, whereas market-cap weighted indices generally only require reconstitution.To provide an easy way to distinguish the various momentum/contrarian indicespresented in the empirical section, we use the notation (f,h), where f=formation pe-riod and h=holding period. For example, a (6,12) index would determine the momen-tum/contrarian allocations based on 6 months of price performance, and then hold theresulting portfolios for 12 months, at which time the process is repeated.To give the index provider some ﬂexibility with regard to the distinctiveness of themomentum/contrarian strategies, the stage 1 allocations are then transformed usingthe regularized incomplete beta function with its two parameters α and β set equal toeach other.1 The stage 2 transformed momentum and contrarian allocations θM andt,nθC are determined as follows:2t,nStage 2 AllocationsB(β)µMθM =t,n,(2)t,nB(β)whereµMt,nB(β) =xβ−1(1 − x)β−1dx,µMt,n0B(β)µCθC =t,n= 1 − θM ,t,nB(β)t,nandµCt,nB(β) =xβ−1(1 − x)β−1dx,µCt,n01B(β) =xβ−1(1 − x)β−1dx.01Tables of the beta function can be easily accessed via the internet, and are also available in moststatistical software packages such as Matlab and Ox. The properties of the beta function are discussedat the following website:http://functions.wolfram.com.2The parameter β used in the Stage 2 allocations refers to the beta function only, and is unrelatedto the β typically associated with the CAPM. Unless otherwise stated, throughout this paper β refersto the parameter from the beta function, which is selected by the index compiler to gain the desiredtrade-oﬀ between the turnover and distinctiveness of the momentum/contrarian indices.7The objective here is to provide the index provider with a means of varying theallocations in a way that preserves the ranking across stocks. For example, for stocksm and n, suppose µM > µM . This means that stock m is more momentum than stockt,mt,nn. Hence a greater proportion of the total holding of m is allocated to the momentumportfolio than is the case for n in period t. The transformation of allocations using theregularized incomplete beta function will preserve this ranking. That is, irrespective ofthe choice of β (as long as it is positive and ﬁnite), when in stage 1 µM > µM it willt,mt,nalso be the case for the stage 2 transformed allocations that θM > θM .t,mt,nAnother important feature of the beta function as deﬁned above is that it has aﬁxed point at 0.5. It follows from this and the fact that it is a monotonically increasingfunction that µM > 0.5 implies that θM > 0.5, and that µM < 0.5 implies thatt,nt,nt,nθM < 0.5. That is, if the original µ values identify an asset as more (less) momentumt,nthan contrarian, the transformed θ allocations will do likewise. Also, the transformedmomentum and contrarian allocations will still sum to one for each asset. That is,µM + µC = 1 and θM + θC = 1 for all assets n in the portfolio.t,nt,nt,nt,nOnce a positive value for β has been chosen, the transformed allocations θM andt,nθC can be derived from µM and µC , respectively, using tables of the beta function.t,nt,nt,nThe impact of changes in β on the relationship between µM and θM is graphed int,nt,nFigure 1. Figure 1 shows that increasing the value of β leads to more distinct momen-tum/contrarian allocations.Insert Figure 1 HereThe momentum/contrarian portfolios are obtained by multiplying the benchmarkportfolio Qt,n (for example that of the Russell 1000 index) by the stage 2 momen-tum/contrarian allocations:Momentum portfolio :qM = θM × Qt,nt,nt,n,Contrarian portfolio :qC = θC × Qt,nt,nt,n,(3)where Qt,n denotes the number of units of stock n in the benchmark index in period t,and qM and qC denote the number of units of stock n in the momentum and contrariant,nt,n8Document Outline

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