An Application of Returns-based Style Analysis to Investigating the Disappearance of the Size Premium

An application of returns-basedstyle analysis to investigating the

disappearance of the sizepremium

Julia SawickiAdvance Research Centre, Instituto Superior de Economia e Gestao,

Universidade Tecnica de Lisboa, Lisbon, Portugal

Abstract

Purpose – The purpose of this study is to investigate explanations for the behaviour of the sizepremium using measures of large and small stock holdings of mutual funds.

Design/methodology/approach – Returns-based style analysis is used to measure asset classexposure by regressing equity fund returns on asset class returns over the period 1965 to 2003. Thecoefficients estimate portfolio asset allocation indicating a fund’s investment styles. The estimatesfrom 36-month rolling regressions of US equity fund returns on various asset classes are aggregatedand used as measures of investors’ exposure to small stocks. The patterns are analyzed in the contextof the behaviour of the abnormal returns to small stocks.

Findings – The results indicate the importance of the 1974-1975 bear market to the historical sizepremium and support an overreaction and reversal argument. Exposure to small stocks dropsdramatically between 1975 and 1977, suggesting a sell-off of small stocks. Fund exposuresubsequently increases rapidly to its highest levels between 1982 and the market crash of 1987. Thesepatterns are consistent with pricing pressure that would lead to the initial undervaluation andsubsequent overvaluation driving returns to small stocks over this period.

Originality/value – The study introduces the application of the returns-based style analysismethodology to studying an asset-pricing phenomenon and demonstrates important insights that canbe obtained from the use of this methodology in new contexts and at an aggregate level.

Keywords Stocks, Returns, Unit trusts

Paper type Research paper

1. IntroductionThere is substantial empirical evidence of a size premium in equity markets: portfoliosconstructed on the basis of size (market capitalization) generate excess risk-adjustedreturns. The evidence that small stocks outperform large stocks first documented byBanz (1981) spawned an enormous literature investigating the small stock premium.Explanations for this apparent market anomaly include: correlation with other returnanomalies (notably the January effect), transactions costs, analyst neglect,measurement errors, default risk and model misspecification.

Interestingly, after the publication of Banz’s (1981) results, the size effect seems tohave disappeared. Apart from the recessionary periods of 1991-1993 and 2001-2002, thepremium is negative with small stocks underperforming large stocks. This apparentdisappearance has led to suggestions that the size effect may have been arbitragedaway after the popularization of the small stock premium and the introduction ofsmall-cap funds designed to capture the advantages of investing in small stocks

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

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Journal of Modelling in ManagementVol. 4 No. 3, 2009pp. 216-234q Emerald Group Publishing Limited1746-5664DOI 10.1108/17465660911006459

(Cochrane, 1999a, 2001; Campbell, 2000). A different perspective is offered byHirshleifer (2001) who argues that small stocks had been overpriced by the mid-1980sdue to the huge run-up in the prices of small-caps in the 1975-1983 period, resulting inthe subsequent underperformance of the small stocks relative to the large stocks.Historical estimates of the size premium are very sensitive to the 1975 to 1983 periodwhen small stocks outperformed large stocks by 20 per cent per annum. Siegel (2002)notes that the small stock premium can be almost entirely attributed to the phenomenalsmall stock performance during this period.

This study investigates these explanations with a unique application ofreturns-based style analysis (RBSA) to measure mutual fund exposure to smallstocks over the 40-year period, 1962-2002. Introduced by Sharpe (1988, 1992), RBSA isused extensively, both as a commercial software tool and an empirical researchspecification, to identify managed funds’ investment styles. The analysis uses a factormodel to estimate portfolio exposures to asset classes, where the factors are returns tobenchmark portfolios representing different investment styles, such as fixed income,small cap, value, growth or country. Time series data of mutual fund and benchmarkreturns form a constrained linear regression where the least-squares regressioncoefficients under linear equality constraints are an estimation of the portfolio styleweights. This typical quadratic programming problem with a closed-form solution andknown distribution for the estimator has become standard practice. Using aconstrained regression of fund returns on size-sorted benchmark portfolio returns, weare able measure the exposure of managed funds to small stocks and gain insight intothe explanations for the behavior of the size premium over time.

The primary contribution of this study is the introduction of the application ofreturns-based style analysis (RSBA) to investigating the behavior of an importantempirical regularity, the size premium. This methodology is used extensively inassessing the investment styles of individual funds. This investigation demonstratesits application and usefulness at a different level. The results provide striking evidenceof a dramatic change in fund exposure during the 1975-1983 period and importantinsight into explanations for the virtual disappearance of the small-stock premiumafter 1983. Small factor loadings pre-1975 are similar to post-1985 loadings. This isinconsistent with a notion of the size premium as compensation for a narrowly-heldrisk. The evidence supports Hirshleifer’s (2001) proposition of small stock oversellingin the vicious 1974-1975 bear market, giving rise to a rebound in the subsequent1975-1983 period.

The paper proceeds with a review of explanations for the size effect and itsdisappearance in Section 2. Section 3 outlines the data and methodological specificationused in this research. Section 4 analyses the empirical results, Section 5 discusseslimitations and extensions, and Section 6 concludes.

2. DiscussionThe research on the anomalous returns to certain investment strategies exploitingapparent contradictions of the efficient market hypothesis for the past two decades hasbeen very fruitful with the emergence of various efficient- and behavioral-basedtheories explaining anomalous empirical regularities[1]. Prominent among the marketefficiency anomalies is the return advantage to investing in small stocks. As withmany anomalies, the size effect is not consistent, leading to questions of its nature as a

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risk premium and allegations of its disappearance (Horowitz et al., 2000; Hirshleifer,2001). Surprisingly, the apparent disappearance of the size effect has not been studiedin detail[2], despite its serious implications for the common application of ad-hoc riskand return factor models in financial research, which usually incorporate firm size asone of the explanatory factors[3].

A basic premise of efficient markets is that investors respond rapidly to investmentopportunities. Abnormal returns disappear and as prices adjust to levels providingexpected returns commensurate with risk. Schwert (2003) provides evidence thatseveral anomalies including the size effect seem to have weakened or disappeared afterpapers highlighting them have been published, which raises the possibility thatanomalies are more apparent than real. He further considers that even if the anomaliesexisted in the sample period in which they were first identified, the implementation ofstrategies to exploit abnormal returns can cause the anomalies to disappear. Similarly,Dimson and Marsh (1999) observe that once an anomaly is publicized it disappears orgoes into reverse, citing the six-percent historical small stock premium in the UK whichbecame a 6 percent discount after its publication.

One explanation of the size premium which is relevant to understanding thedisappearance of the small stock premium is posited by Banz (1981) and Cochrane(1999b) who invoke the notion of a “narrowly-held risk”. A particular risk could benarrowly-held because that risk is not well-known, not well-understood, or simply notavailable to the average investor. Owing to its characteristics, a narrowly-held risksuch as size tends to be illiquid as well, potentially resulting in mispricing of smallstocks. Cochrane (1999b) believes that efficient arbitrage of a previously“narrowly-held small-cap risk” is behind the demise of the size effect.

Campbell (2000) argues that investor sophistication and industry innovationsenable more investors to participate in “narrowly-held risks”. Small-cap funds make itpossible for investors to get the higher returns without facing the risk of holdingilliquid small-caps, since they can move in and out of funds, depending on theirinvestment-consumption decisions. An illustration of this is Barberis and Shleifer(2003) who show that prices of risky assets deviate from fundamental values as certaininvestment styles become popular or unpopular. In their model, investors are assumedto allocate funds based on relative past performance allocating money to investmentstyles that have performed well in recent years, and they withdraw funds from stylesthat have performed poorly to finance this shift in investment style. As more fundsflow into the popular investment style, the prices of the assets belonging to this stylewill rise until the style collapses, due to arbitrage or bad fundamental news.

An alternative view of the small stock premium is behavioral. Daniel et al. (2002)consider the view of market efficiency unrealistic because the process of exploitingpredictable patterns is itself erratic and prone to under- and over-reaction. Arbitrage isalso subject to a coordination problem. Daniel and Titman (1999) propose a weakerdefinition of efficiency that recognizes behavioral biases of most market participantswhere patterns of predictability persist or are overexploited.

Several researchers have tied the size premium to overreaction. DeBondt and Thaler(1987) posit a “loser-effect” where investors overreact to poor performance which tendsto be concentrated in low market capitalization firms. Subsequent reversal of thisundervaluation results in abnormal performance. Demirtas and Guner (2008) contendthat the weakening of the size premium following its discovery indirectly supports a

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behavioral explanation and provide evidence that at least part of the effect is driven bymispricing. On a broad level Siegel (2002) proposes undervaluation driven by thepessimism and overselling during 1973-1975 recession, and subsequent reversal ofreturns as an explanation for the behavior of the small stock premium.

We investigate the emergence and disappearance of the size effect by measuring theexposure of mutual funds to small stocks using an asset class factor model. Inferencesabout relative proportions of small versus large stocks provide insights important tounderstanding the size premium and distinguishing between explanations of itsbehavior over time. The following section describes data and methodology used inexamining mutual fund exposure to small stocks over the period 1964 to 2002 usingreturns-based style analysis. The estimates of the loadings on portfolios of size-sortedfactor portfolios allow us to compare the investment styles over time with temporalbehavior of the small stock premium and thus evaluate the explanations considered inthis study.

3. Data and methodologyThe sample used for calculating returns consists of all firms listed on the Center forResearch in Security Prices (CRSP) data files from January 1962 to December 2002.Prior to December 1972, the CRSP data files include NYSE and AMEX firms only;NASDAQ firms are added to the sample from December 14, 1972. Size portfolios areconstructed increasing the size discrimination. Small and big are partitioned by themedian market capitalization. Lo, Med and High portfolios are formed with the lowest30 per cent, middle 40 per cent and top 30 per cent market capitalization. Quintile anddecile portfolios are also formed. Descriptive statistics in Table I show that theportfolios performance for different size portfolios.

The small-caps outperform the portfolios representing large-caps for the 1962-2002period. The small-cap portfolios are also riskier, with higher standard deviation andrange values.

The fund data comprise diversified equity funds, balanced funds, and flexiblefunds. We conduct style analysis for the entire set of funds (referred to as EnlargedEquity) and a subset of diversified equity funds consisting of equity funds only. Thebalanced, flexible, and flexible funds must fulfill a criterion whereby their holdings ofcommon stocks must be at least 50 per cent of the total assets[4].

The investigation is aimed at comparing two explanations for the disappearance ofthe small stock premium: arbitrage of a narrowly-held risk and a behavioralexplanation of overreaction leading to undervaluation and subsequent reversal.Arbitrage of the size effect is not expected to be immediate (Cochrane, 1999b). Anincrease in the participation rate of investors would initially result in a period of highaverage returns of small stocks as investors bid the prices of small stocks up toequilibrium levels. Subsequently, the abnormal returns of the size effect wouldgradually decline as the prices of the small stocks reach the equilibrium levels. Thisleads to the expectation of a sigmoid-shaped small stock loading over the time periodreflecting the popularization of the size effect in the 1980s and the introduction ofsmall-cap funds[5]. The behavioral explanation of overselling in the bear market andreversal suggests a fairly rapid decrease in the small stock holdings during the1973-1975 recession followed by a rebound in loadings reflecting the recognition of the


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under-pricing. The pattern predicted by this scenario would be U-shaped where adecline coincides with the recession and a subsequent increase.

The empirical tests in this study use a proxy for the participation rate of investors:the level of small stock holdings of mutual funds. Empirical patterns of theparticipation rate are compared with the hypothesized patterns. The two most commonapproaches to evaluate mutual funds’ investment style over time are portfolio-basedstyle analysis (PBSA) and returns-based style analysis (RBSA). The PBSA approachdetermines the investment style of a mutual fund by analyzing the characteristics ofthe holdings over a period of time. Sharpe (1988, 1992) pioneers the RBSA methodologyin which a factor model is used to explain fund returns. RBSA regress fund returnsagainst a set of predetermined benchmark indices, representing asset classes orinvestment styles that approximate the behavior of the fund. The factor coefficientsestimate the exposure to the asset classes specified as factors.

Chan et al. (2002) provide empirical evidence indicating that the PBSA approach ismore accurate in forecasting risk and return of a particular fund. On the other hand,Cummisford and Lummer (1996) argue that the lack of timely portfolio holdings data(only available on a semi-annual or annual basis) makes PBSA less reliable. Theunavailability of up-to-date holdings data is problematic to the extent that theirportfolio holdings change during the year. Horst et al.’s (2004) theoretical model showsthat the RBSA approach gives a better estimate of the actual fund’s investment stylebecause of cross-correlations between the asset classes and the likelihood that the fund

Variable Mean SD Sum Minimum Maximum Range

Small 0.0118 0.0584 5.7863 20.2916 0.2718 0.5634Big 0.0098 0.0431 4.8288 20.2086 0.1625 0.3711Lo 0.0114 0.0621 5.5960 20.2919 0.2678 0.5597Med 0.0106 0.0536 5.1939 20.2694 0.2277 0.4971Hi 0.0087 0.0441 4.2992 20.2076 0.1771 0.3847Qnt1 0.0114 0.0637 5.6250 20.2933 0.2727 0.5660Qnt2 0.0111 0.0593 5.4538 20.2921 0.2490 0.5411Qnt3 0.0105 0.0542 5.1725 20.2682 0.2263 0.4945Qnt4 0.0103 0.0511 5.0670 20.2486 0.2022 0.4508Qnt5 0.0086 0.0438 4.2374 20.2034 0.1802 0.3836Dec1 0.0116 0.0647 5.7209 20.2876 0.2900 0.5776Dec2 0.0113 0.0635 5.5656 20.3001 0.2841 0.5842Dec3 0.0114 0.0607 5.5946 20.2887 0.2573 0.5460Dec4 0.0109 0.0587 5.3475 20.2947 0.2416 0.5363Dec5 0.0112 0.0562 5.4896 20.2770 0.2496 0.5266Dec6 0.0100 0.0532 4.9345 20.2606 0.2083 0.4689Dec7 0.0105 0.0520 5.1745 20.2601 0.2246 0.4847Dec8 0.0101 0.0509 4.9889 20.2403 0.1897 0.4300Dec9 0.0095 0.0465 4.6577 20.2232 0.1812 0.4044Dec10 0.0085 0.0437 4.1619 20.1973 0.1800 0.3773

Note: This table reports the descriptive statistics for various size-sorted portfolios of CRSP datasetstocks. Small and Big are separated by the median market capitalization. Lo, Med and Hi are portfoliosof the lowest 30 per cent, middle 40 per cent and largest 30 per cent stocks based on marketcapitalization. The stocks are also sorted by quintiles and deciles, where Qnt1 and Dec1 are thesmallest stocks, and Qnt5 and Qnt10 are the largest stocks

Table I.Descriptive statistics forthe size-sorted portfolios

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manager choose to invest in assets with relatively high or low betas relative to thebenchmark index. Their empirical tests support the superior accuracy of RBSA indetermining the actual factor exposures that are relevant in predicting future returnsand identifying the risk exposures of the fund.

The unavailability of mutual fund holdings data for much of the 1960s and 1970sprecludes the use of the PBSA in this study. Following Sharpe (1992) we use the RBSAapproach to analyze the level of small-cap investing by mutual funds over the period1964 to 2002.

The factor loadings of RBSA are based on arbitrage pricing theory in which thatreturns are generated by a factor structure:

ri ¼ bi1F1 þ bi2F2 þ :::þ biNFK þ 1i ð1Þ

where ri is the return on asset and F1 is the value of factor 1 of K systematic factorsdetermining returns. The factor coefficients, bi,1-K, are the sensitivities of the return onasset i to the factor realization. An important assumption of the model is that theidiosyncratic portion of returns, ei, is independent of the non-factor returns of all otherassets, ej.

Sharpe (1992) assumes that the factors can be represented by asset classes. Themodel can be thus rewritten:

ri;t ¼ aþk

XbkFk;t þ 1i;t ð2Þ

and the factors can be represented by a small number of asset classes, with valuesestimated by an index portfolio designed to represent the systematic realization of thefactor value in each period t. RBSA proposes that the best explanation of the returngenerated by the portfolio is given by the style factors, bk, which minimize the residualvariance of the model subject to two constraints reflecting leverage and short-sellingrestrictions. The constraints are:

Portfolio constraint where 100 per cent of the portfolio must be invested

XN

k¼1

bk ¼ 1ð3Þ

Positivity constraint where short selling is prohibited bk $ 0; k ¼ 1; . . .;N : ð4Þ

RBSA applies a quadratic programming algorithm to determine a fund’s exposures tochanges in the returns of major asset classes. A more exact solution can be obtainedusing Markowitz’s (1987) method.

Sharpe (1992) identifies the key contribution of the model as its ability to separatethe performance of a portfolio into two components: style as driven by the factors, andselection as captured by the idiosyncratic term. Our application of the model allows usto estimate holdings of small stocks by specifying an index of stocks with the lowestmarket capitalization. The 36-month rolling RBSA constrained quadratic regression, isestimated with the following model:


221

rt235:t ¼XN

k¼1

bkI k;t235:t þ et235:t ; t ¼ 36; 37; . . .;T ð5Þ

where Rt235:t denotes the 36-month mutual fund returns, N is the number of stylefactors, bk represents the sensitivity of the fund’s return to the style factor k, I k;t235:t

refers to the 36-month returns of style factor k, and et235:t is the error term. Our tests areconducted with various index constructions depending on the definition of smallversus large, control for the value effect and inclusion of fixed income indices.

The 36-month rolling regressions have the dual objective of reducing betameasurement errors (since statistical theory suggest at least 30 observations arenecessary for a regression to yield consistent betas) and trying to maximize the numberof rolling regression. Style is not constant over time. We thus model changes inexposures by minimizing the objective function:

wt235:tðet235:t 2 �eÞ2; t ¼ 36; 37; . . .;T ð6Þ

where wt235:t represents the 36-month adjusted weights et235:t denotes the 36-monthresiduals, and �e is the average of the 36-month residuals. Change in asset allocation orinvestment style over time is referred to as style drift. In the presence of style drift,RBSA estimates is the average of the actual factor exposures over time rather than themost recent estimate. An adjustment to improve style factor estimates by placinggreater weights on recent months than on the more distant months. This adjustmentminimizes the weighted tracking variance by assigning a weight of one to the firstmonth of the square of the difference between the observed residual value and themean of the residuals,ðet 2 �eÞ2, and subsequently assigning a weight equal to 2^(1/35)

times the weight assigned to the previous month. This “36-month half-life” adjustmentresults in each month receiving slightly over 2 per cent more weight than itspredecessor, and the weight assigned to month t is twice of that assigned to montht 2 35. An alternative method is to model time variation of style weights using Kalmanfilter as in Swinkels and van der Sluis (2006).

4. Empirical resultsWe begin with documentation of the size effect over time. To test the robustness of thesize effect, we use different definitions of the size premium: with and without the valueeffect, small stock returns versus large stock returns, and small stock returns versusthe market returns. Table II reveals that the size effect exhibits periodic fluctuations.

The historical small stock returns are consistent with those documented in otherstudies. Small stocks outperformed the large stocks in the 1960s and 1970s, andsubsequently underperformed the large stocks in the 1980s and 1990s. The negative sizepremium for much of the 1980s and 1990s has led many researchers to infer that the sizeeffect has disappeared over time. The small stock premium very high from the mid-1960sto mid-1980s, particularly in the 1965-1968 period and the 1975-1983 period. Fama andFrench (1992) document a relation between stock returns and firm size which is muchstronger in the 1960s and 1970s than weaker in the 1980s. Subsequent to 1983, it is positiveonly during recessionary periods, of 1991-1993 and 2001-2002. The disappearance of thesize effect reported here is consistent with Dichev (1998) and Amihud (2002) who report nosize premium for the respective periods 1981-1995 and 1980-1997.

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Year Lo-Hi Qnt1-Qnt5 Dec1-Dec10 SMB Lo-MK Qnt1-MK Dec1-MK Small-MK

1962 20.0219 20.0260 20.0079 20.0901 20.0095 20.0098 0.0067 20.05271963 20.0222 20.0411 20.0344 20.0630 20.0128 20.0276 20.0189 20.03361964 0.0347 0.0358 0.0680 20.0179 0.0310 0.0295 0.0656 20.00051965 0.2459 0.2862 0.3685 0.2160 0.2188 0.2472 0.3111 0.22561966 0.0305 0.0293 0.0142 0.0276 0.0251 0.0176 20.0093 0.02251967 0.7181 0.8163 0.9358 0.5085 0.6551 0.7475 0.8686 0.47981968 0.3602 0.4071 0.5230 0.2418 0.3237 0.3610 0.4576 0.25541969 20.2074 20.2413 20.2715 20.1437 20.1759 20.2044 29.2206 20.13621970 20.1759 20.2040 20.2121 20.1163 20.1466 20.1749 20.1858 20.07361971 0.0309 0.0380 0.0232 0.0631 0.0170 0.0170 0.0042 0.04411972 20.1737 20.1983 20.2159 20.1205 20.1601 20.1712 20.1695 20.12371973 20.2170 20.2381 20.2530 20.2360 20.1788 20.1931 20.1971 20.16301974 20.0008 0.0029 0.0120 20.0067 0.0044 0.0076 0.0130 0.02251975 0.2207 0.2469 0.2517 0.1480 0.2005 0.2147 0.2090 0.21231976 0.2433 0.2657 0.2763 0.1425 0.2099 0.2197 0.2136 0.21621977 0.3186 0.3555 0.3956 0.2309 0.2766 0.3041 0.3342 0.23081978 0.1656 0.1866 0.2271 0.1409 0.1407 0.1600 0.2030 0.11261979 0.2094 0.2280 0.2622 0.2036 0.1589 0.1644 0.1706 0.16781980 0.0684 0.0724 0.0881 0.0573 0.0596 0.0685 0.0839 0.01891981 0.0488 0.0460 0.0364 0.0699 0.0381 0.0247 0.0014 0.10801982 0.0787 0.0936 0.0843 0.0928 0.0680 0.0743 0.0706 0.11011983 0.1347 0.1400 0.1435 0.1364 0.1103 0.1147 0.1187 0.13201984 20.1557 20.1855 20.2289 20.0800 20.1119 20.1338 20.1745 20.04151985 20.0192 20.0359 20.0770 20.0003 20.0068 20.0235 20.0699 0.00641986 20.1350 20.1444 20.1622 20.0991 20.1107 20.1169 20.1348 20.06641987 20.1566 20.1816 20.1911 20.1069 20.1362 20.1576 20.1583 20.09801988 0.0428 0.0211 0.0042 0.0592 0.0353 0.0105 20.0065 0.07111989 20.1866 20.2318 20.2585 20.1221 20.1539 20.1939 20.2141 20.10411990 20.1993 20.2486 20.2693 20.1422 20.1736 20.2113 20.2159 20.14061991 0.1487 0.1734 0.1617 0.1585 0.1366 0.1511 0.1240 0.13091992 0.1180 0.1420 0.2013 0.0735 0.1071 0.1204 0.1641 0.11631993 0.0977 0.1078 0.1866 0.0604 0.0751 0.0749 0.1433 0.07741994 20.0517 20.0658 20.0687 20.0144 20.0368 20.0460 20.0425 20.01371995 20.0506 20.0536 20.0756 20.0744 20.0280 20.0193 20.0364 20.05761996 20.0568 20.0666 20.0950 20.0238 20.0425 20.0455 20.0618 20.02631997 20.1116 20.1064 20.1284 20.0484 20.0821 20.0648 20.0824 20.03551998 20.3534 20.3934 20.4927 20.2521 20.2783 20.2955 20.3585 20.25141999 0.1203 0.1929 0.1588 0.1468 0.0840 0.1476 0.1201 20.00022000 0.0390 0.0432 0.0291 20.0164 0.0599 0.0533 0.0168 0.17412001 0.3601 0.4234 0.4661 0.1834 0.3446 0.4010 0.4324 0.24142002 0.0696 0.1068 0.2018 0.0345 0.0592 0.0937 0.1784 0.0345

Notes: This table reports the annual returns for various size-sorted portfolios. MK is the marketportfolio comprising all NYSE, AMEX and NASDAQ stocks. Lo and Hi are portfolios comprising the30 per cent smallest and largest stocks based on market capitalization in the CRSP database. Portfoliosare also formed on the basis of quintiles and deciles where Qnt1 and Dec1 comprise the smallest stocksand Qnt5 and Dec10 comprise the largest stocks. The portfolios are constructed at the end of June, thevalue weighted returns are from January to December. The returns are the differences between variousportfolios. The highlighted areas indicate positive small stock excess returns

Table II.The annual size effect


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Figure 1 illustrates the pattern of the size effect: highly positive in mid- and late-1960sand from the mid-1970s to the mid-1980s.

There is a dramatic drop in the stock market returns in the early 1970s. In a “flightto quality” scenario, investors move into “safer” securities (large, growth stocks)during time of economic distress and avoid riskier investments including small stocks.Figure 1 is consistent with an overselling of small stocks. The 12-month movingaverages of SMB returns were negative during the early 1970s to mid-1970s period andbecame highly positive in the mid-1970s to mid-1980s period. The huge run-up in theprices (and returns) of small stocks in the mid-1970s to mid-1980s period and thesubsequent the negative size premium in the mid-1980s to the late-1990s is consistentwith speculation that small stocks may have been overpriced by the mid-1980s.

Table III reports the descriptive statistics for the mutual funds’ subset of equityfunds only, and the full data set which also includes balanced and flexible funds.

The diversified group’s higher returns reflect the higher holdings of equitiescompared to the balanced funds spread of investments across lower risk asset classes.The average returns to the small stock portfolio (Small, comprising stocks with marketcapitalization below the median) is greater than the average returns to the large stockportfolio (Big, comprising stocks with market capitalization above the median) in alltime periods except 1984-2002 when Big exceeds Small.

The style analysis estimates using the Small and Big portfolios and the equity fundsonly are depicted in Figure 2[6].

Small cap loadings average about 25 per cent in the 1960s indicating significantexposure to small capitalization stocks. This is not consistent with the notion of size asa narrowly-held risk. The dramatic fall in the small loadings coinciding with the onsetof 1973 recession is quite remarkable. The SMB of 223.6 per cent that year is thelargest underperformance of the study period. The decline in loadings is consistentwith rapid, large shifts out of small stocks. The Small loadings fall to zero in 1975 andmysteriously remain there until a rebound in 1977. The excess return to small stocks issubstantial in 1975 and 1976 (SMB approximately 14 per cent) yet the patterns indicatethat the fund returns are not explained by these returns. The subsequent increase inthe small-cap loading indicates a response to the robust positive size effect of prioryears. Also interesting to note is that small-cap loadings increase dramatically duringthe 1981-1982 recession contrary to an expectation that small-cap loadings would fall.

Results for the RBSA estimates using the enlarged equity sample are reported inFigure 3.

The pattern is similar to that demonstrated by the loadings estimated with thesub-set of equity-only funds in finding that the small stock loadings are negligibleduring the 1970s recession. The pre-recession exposure to small stocks is lower whenestimating the model with the entire set of funds compared with the estimates usingthe equity-only funds. The 1964-1974 small stock exposure averages around 20 percent for the entire sample, compared with 25 per cent for the equity funds.Post-recession, the equity-only funds re-establish their holdings of small stocks morerapidly but the estimates converge after about 1982 and the magnitude and behavior isvery similar.

In order to separate size and value effects we estimate the model using factors forthe six size- and value-sorted portfolios that make up the Small and Big portfolios. Thehigh correlations suggest that multi-collinearity may skew the estimates. Nevertheless,

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Figure 1.The size effect and market

portfolio returns


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1962

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1.79

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Table III.Descriptive statistics forthe diversified andenlarged fund portfolios

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226

Figure 2.Diversified equity funds

Small and Big styleanalysis


227

Figure 3.Enlarged equity fundsSmall and Big styleanalysis

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228

due to the lack of portfolio holdings data to run PBSA, we run different size- anddouble-sorted style analysis to confirm the RBSA estimates, and the results arecomparatively consistent. The results not reported here are similar to the estimatesbased on the size factors alone in Figures 2 and 3.

The analysis is repeated by estimating factor loadings for the Lo (bottom 30 percent), Med (middle 40 per cent) and Hi (top 30 per cent) portfolios. The results arereported in Figure 4.

Although the general loading pattern of rapid decline, bottoming and return tosmall stocks over the 1975-1977 period is similar to that of the Small and Big analysis,there are very important differences. The Med loading usurps both the Small and Bigfrom the previous analyses. Prior to 1975, the Lo loading is negligible, however theexposure is evident beginning in 1982. Prior to 1975 the Med loading absorbs thepreviously-estimated Small exposure (Figure 3). This demonstrates the importance ofthe benchmark specification in RBSA and provides evidence consistent with the notionof small stock risk as a narrowly held risk in the past.

Overall, one important result emerges from the analysis: the plunge and recovery ofsmall-cap exposure in the mid-1970s. This pattern and the timing are distinct andsignificant. The pattern clearly indicates a large, rapid sell-off of small stocks matchedby a rapid return to holding this asset class. Equally clear is the suggestion ofover-reaction followed by reversal.

The 1973-1975 bear market is a well-researched event. Many studies attribute the1970’s stock market crash to the burst of the “Nifty Fifty” bubble (Malkiel, 1990). The“Nifty-Fifty”, a group of 50 large growth stocks with stellar earnings and dividend growthrecords, had become institutional darlings in the early 1970s. Their meltdown in the bearmarket is often used as an example of irrational exuberance (unwarranted optimism overthe prospects of growth stocks). Siegel (1995) argues that the Nifty-Fifty were fairly pricedand that the high price-earnings ratios of these stocks are justified, even at the marketpeak in December 1972, due to their future realized returns. He calculates that the prices ofthese stocks were almost 35 per cent below their true values, based on their future returns.Corroborating this is Shiller (1984) who uses S&P Composite Index data to show that realstock prices decline during the 1974-1975 bear market despite the fact that real earningsand dividends are relatively stable during that period. These results point to investors’sudden pessimism rather than irrational optimism during this period. Small stockssuffered a larger fall in prices compared with the large stocks, consistent with investors’preference for safer investments during periods of economic distress – the classic “flightto quality” phenomenon. Since most equity fund managers are mandated to invest acertain percentage of the fund’s assets in stocks, the money managers would increasetheir large-cap investments and simultaneously decrease their small-cap investments in abearish stock market environment.

The small-cap loadings over this period arguably reflect irrational pessimism andrapid selling in 1973 and 1974. Loadings rebound equally rapidly in 1977 increasingsteadily, even during the 1981-1982 recessionary period. The pattern demonstrates thevalidity of behavioral explanations of the size premium.

5. Limitations and extensionThis approach to investigating the historical size premium is fraught with caveatswhich need to be considered in interpreting the results and future applications.


229

Figure 4.Diversified equity fundsLo, Med and Hi styleanalysis

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The RBSA factor loadings provide an estimate of average exposure to asset classes.The regression is essentially explaining the fund returns with returns of the factorportfolios. The interpretations of the factor exposures are especially important duringperiods of extreme performance of the asset classes. Fund performance is less likely tofollow the extremes and the coefficients will load on the less extreme asset classes.Notably here are 1975 and 1976 when small stocks returned 59 and 48 per cent,compared to large stock returns of 41 and 34 per cent respectively. In general, animportant drawback to RBSA is that the constraints “force” an estimate which may notbe reflective of the underlying asset allocation. Also, this work uses only size indices inthe regression. (Fixed income loadings were insignificant and not reported.)

The definition of “small” is very important as demonstrated by the change in resultsfrom the Small/Big segmentation to the Lo/Med/Hi. Also, including other asset classesand in the model is an important extension and analysis of the sensitivity of the resultsto different specifications should be carried out.

The size and importance of mutual funds changes substantially across the timespan of this study. The participation rate for which the factor exposures proxy isarguably much more relevant after 1985. This suggests that the application of thismethodology is more appropriate in studies focusing on the post-1985 period whenassets managed represent substantial proportions of common stock holdings.

Estimating style exposure can reveal some interesting, important patterns asdemonstrated here. A more rigorous investigation would examine confidence intervals forthe factors. Another level of rigour would be to test directly for a relationship between thesmall stock premium and factor loadings estimated with RBSA. For example, RBSAfactor loadings would be estimated using index portfolios (e.g. market, small, large, value,growth, bonds). The question of how well the small stock premium explains the exposureto small stocks could be investigated through a regression analysis of these variables.

6. ConclusionThis study introduces the application of returns-based style analysis, a quadraticprogramming algorithm, to investigate the behavior of a stock market anomaly. Whilethis approach is widely used in investment decisions and empirical research into theperformance of investment funds, this is a novel application.

Anomalies are not expected to persist in efficient markets. Investors trading on aneffect that has been shown to produce superior returns will cause prices to adjust andeliminate the “free lunch”. The disappearance of the premium to investing in smallstocks shortly after its documentation by Banz (1981) is arguably a case in point. Onthe other hand, a behavioral explanation of over-reaction and recovery has beenapplied to explaining the small stock premium and provides another perspective forconsidering its disappearance.

This paper investigates this using a Sharpe’s (1992) constrained style regression tostudy the relationship between the popularity of an investment styles and the relativereturns of the underlying assets. The argument posited by Cochrane (1999a, 2001) andCampbell (2000), suggests that the size effect may have been arbitraged away after theintroduction of small-cap funds designed to capture the advantages of investing in smallstocks. We measure changes in mutual funds’ portfolio exposure to small stocks usingSharpe’s (1992) constrained regression analysis known as returns-based style analysis toinvestigate an arbitrage explanation of the disappearance of the small stock premium.


231

Our results reveal a very interesting picture of a dramatic decline in the mutualfunds’ exposure to small cap stocks during the years 1973 to 1978, indicating importantavenues for future research. The factor loadings drop rapidly with the onset of the 1973recession falling to zero between 1974 and 1977. The drop and subsequent increase inthe loadings from 1979 to a peak in 1984 is consistent with an alternative explanationwhich attributes the appearance and disappearance of the premium to the recoveryfrom the dramatic recessionary sell-off as conjectured by Siegel (2002).

When the RSBA model is estimated with the factors estimated with the medianseparating small and large indices, the exposure to small stocks pre-1973 is similar tothat in the latter years. This is inconsistent with a notion that small caps were anarrowly held risk prior to popularization of the premium. However, a finerpartitioning of size (quintiles, deciles or small-med-big) produces a different picture.Prior to the 1980s the factor loadings on the smallest three deciles are negligible,appearing after 1984 and averaging around ten percent. This pattern supportsCochrane’s (1999b) hypothesis of the factor as a narrowly-shared risk in the past whichhas been arbitraged away through increased exposure.

The salient aspect of our results is the dramatic decline in the mutual funds’exposure to small cap stocks during the years 1973 to 1978 regardless of how the indexportfolios are formed. The factor loadings drop rapidly with the onset of the 1973recession falling to zero between 1974 and 1977. The extent to which the behavior ofthe size premium can be explained by irrationality of fundamental factors is animportant ongoing investigation. These results provide new evidence to consider inefforts to distinguish between the alternative explanations.

Notes

1. Rational-based explanations include: Berk (1995), Berk et al. (1999), Fama (1996), and Poterbaand Weisbenner (2001). See Kahneman and Tversky (1979), Barberis et al. (1998) and Danielet al. (1998) for behavioral-based explanations.

2. Van Dijk (2007) finds it remarkable that hardly any research has addressed the decrease inthe size premium since the 1980s. An exception to this is Gompers and Metrick (2001) whoreport evidence that institutional investors doubled their share of the stock market between1980 and 1996. They argue that this represents a shift of increased (decreased) demand forlarge (small) firms which explains part of the reversal of the historical size premium after1980.

3. The most popular model is Fama and French’s (1993) three-factor model, on which otherversions are built.)

4. Funds identified with an investment mandate inconsistent with the RBSA assumptions suchas short-selling, leveraged investment and option-writing were eliminated.

5. Small capitalization stock exposure measured with RSBA factor loadings is used as a proxyfor the participation rate of investors/arbitrageurs and compared with the historicalbehavior of the size effect.

6. We included the returns of Lehman Brothers Aggregate Bond index and the returns ofBlume-Keim high-yield index in the style analysis. However, these two variables are notstatistically significant and results are thus reported for the size index regressions.

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Further reading

Chan, K.C. and Chen, N.F. (1991), “Structural and return characteristics of small and large firms”,Journal of Finance, Vol. 46, pp. 1467-84.

Lakonishok, J., Shleifer, A. and Vishny, R.W. (1994), “Contrarian investment, extrapolation andrisk”, Journal of Finance, Vol. 49, pp. 1541-78.

La Porta, R. (1996), “Expectations and the cross-section of stock returns”, Journal of Finance,Vol. 51, pp. 1715-42.

Corresponding authorJulia Sawicki can be contacted at: [email protected]

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An Application of Returns-based Style Analysis to Investigating the Disappearance of the Size Premium

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