Sentiment and Beta Herding

Soosung Hwang and Mark SalmonSchool of Economics, SKKU, Seoul and Warwick Business

School

Prepared forWarwick Seminar

October 2010

Other Related ResearchSentiment:

Sentiment and Price Formation: The Impact of Non-Linearity(with Karakatsani N.)

Sentiment and Price Formation: Interactions and RegimeShifts (with Karakatsani N.)

Sentiment and Asset Pricing, (with Matus Mrazik),

Knightian Uncertainty :

On Uncertainty, Market Timing and the Predictability of Tickby Tick Exchange Rates, (with Roman Kozhan)

The Impact of Parameter Risk and Model Uncertainty onportfolio credit risk modelling ( with Yile Wu)

Estimating Ambiguity in asset prices using confidence sets(with Roman Kozhan)

Forward Premium Puzzle Under Ambiguity: Is it Still aPuzzle? (with Roman Kozhan)

Uncertainty aversion in a heterogeneous agent model offoreign exchange rate formation,(with Roman Kozhan),

Loss Aversion:

Performance Measurement with Loss Aversion (with G.Gemmill and S. Hwang)

Herding:

Market Stress and Herding,(with Soosung Hwang),

Introduction

Herding arises when investors decide to imitate the observeddecisions of others or movements in the market rather thanfollow their own beliefs and information.

One diffi culty in the study of herding resides in the failure ofstatistical methods to differentiate between a rational reactionto changes in fundamentals and irrational herding behavior.

It is critical to discriminate empirically between these twoforces, since the former simply reflects an effi cient reallocationof assets, whereas the latter potentially leads to marketineffi ciency.

Our notion of “herding”measures the behaviour of investorswho follow the performance of specific factors such as themarket index or particular sectors, styles, or macroeconomicsignals, and thus buy or sell individual assets at the same timedisregarding their underlying risk-return relationship.

We focus on herding toward the market portfolio, which wecall ‘beta herding’.

The cross-sectional variation of betas, the market-widerisk-return relationship, can reflect irrational pricing owing tosentiment or herding.Our notion of beta herding measures the bias in asset returns,which arises from bias in the systematic risk - biased beliefsIt’s a non-parametric measure.It captures the aggregate effect of herding in the market ratherthan herding by individuals or by small groups of investors.

We extend the model of Hwang and Salmon (JEF 2004) (HS) by incorporating the interaction between sentiment andherding.

Sentiment is considered a market wide phenomenon thatevolves over time, whereas herding will in principle affectindividual assets within the market.

Herding can be highly persistent and slow-moving over time

Behavioural forces are often thought to be phenomena thatarise rapidly, and thus most studies of herding have eitherexplicitly or implicitly examined herding over very short timeintervals.but a clear example of slow-moving noise is the growth of a‘bubble’, which may not be completed within days, weeks oreven months.

The Tulip Bubble in seventeenth-century Holland,the real estate bubble in Japan in the late 1980s,the recent dot-com bubble

So we now use monthly data rather than higher frequencydata.

Our empirical results using individual stocks from the USequity market

Beta herding is effectively independent of- robust to businesscycle and stock market states.As a crisis appears, beta herding weakens, - investors becomemore concerned with fundamentals rather than overall marketmovements.Beta herding becomes more apparent when investors areconfident (homogeneous) regarding the future direction of thestock market whether it be a bull or a bear market.We demonstrate that beta herding increases with market-widesentiment.

Implications of our results for cross-sectional asset pricing

Fama and French (1992, 1993, 1996) show that beta is notpriced (unconditionally).Our results show that betas have predictive power conditionalon the level of beta herding.Clear impact of herding on returns of large and small betastocksthe returns of high beta portfolios are higher following anadverse herding state than those following a herding state,whereas the returns of low beta portfolios are lower followingadverse herding than those following herding states.

Sentiment

Baker and Wurgler (2006): the returns of certain firms (i.e.,young, small, volatile, unprofitable, non-dividend paying,growth and distressed firms) are relatively more affected bymoves in market sentiment, as they are diffi cult to value andarbitrage.

The predictive power of sentiment increases as the forecastinghorizon increases due to the slow reversion of these firms.

The majority of firm characteristics have significantrelationships with beta except for profitability (earnings).Over shorter horizons, our lagged beta herding measurepredicts the performance of portfolios sorted on sales growthand the proportion of tangible assets.

Beta Herding

CAPM in equilibrium; Et (rit ) = βimtEt (rmt ).

Cross-sectional beta herding within the market (by HS)

E bt (rit )Et (rmt )

= βbimt = βimt − hmt (βimt − 1). (1)

Herding hmt βbimt E bt (r it )Perfect 1 1 Et (rmt )

Normal 0 < hmt< 1βimt< βbimt< 11 < βbimt< βimt

Et (r it ) < Ebt (r it )

E bt (r it ) < E t (r it )None 0 βimt Et (r it )

Adverse hmt< 0βbimt< βimt< 11 < βimt< βbimt

E bt (r it ) < E t (r it )Et (r it ) < E

bt (r it )

hmt moves around zero, mean reverting.

Et (rmt ) is treated as given in this framework.

Sentiment and Beta Herding

Let δmt and δit represent sentiment impact on the marketportfolio and asset i respectively.

Then an investors’biased expectation in the presence ofsentiment is

E st (rit ) = Et (rit ) + δit

E st (rmt ) = Et (rmt ) + δmt

where δmt = Ec (δit ) and Ec (.) represents cross-sectionalexpectation, and the superscript s represents the bias due tothe sentiment.

Then we have

βsimt =E st (rit )E st (rmt )

=βimt + sit1+ smt

,

where smt = δmtEt (rmt )

and sit =δit

Et (rmt )represent sentiment in

the market portfolio and asset i relative to the expectedmarket return, i.e., the degree of optimism or pessimism.

Consider how beta is biased in the presence of sentiment inindividual assets and/or market;

βsimt =

βimt + sit when δit 6= 0 and δmt = 0,

βimt1+smt

when δit = 0 and δmt 6= 0,βimt+sit1+smt

when δit 6= 0 and δmt 6= 0.

Assumesit = smt − hmt (βimt − 1) +ωit ,

The implied beta in the presence of herding and sentiment is

βsimt = 1+1

1+ smt[(1− hmt )(βimt − 1) +ωit ] .

When there is neither herding nor sentiment, βsimt = βimt .

For given smt , a positive hmt (herding) make βsimt movetowards 1, and a negative hmt (adverse herding) make βsimtmove away from 1.

For given hmt , βsimt moves toward 1 as smt increases, and viceversa.

Measure of Beta Herding:Individual Stocks

When βimt is independent of ωit , we have

Varc (βsimt ) =

1(1+ smt )2

[(1− hmt )2Varc (βimt ) + Varc (ωit )

].

(2)

Two reasonable assumptions

Varc (βimt ) is constant over timeVarc (ωit ) is constant over time

A reduction in Varc (βsimt ) when there is beta herding and

positive market-wide sentiment.

A decrease in Varc (βsimt ) from an increase in smt is more likelyduring bull markets rather than bear markets.A decrease in Varc (βsimt ) from an increased hmt is possibleany time.

Portfolios

There are several benefits from using portfolios instead ofindividual stocks.

First, the idiosyncratic sentiment of a portfolio spt could moreeasily be zero. i.e., ωpt = 0.

spt = smt − hmt (βpmt − 1),

Varc (βspmt ) =(1− hmt )2(1+ smt )2

Varc (βpmt ).

Therefore under the assumption that Varc (βpmt ) is invariantover time, we observe herding by measuring Varc (βspmt ).

Second, the estimation error will be reduced.

As the number of equities in the portfolio increases we havep lim β̂

spmt = βspmt .

Definition of Beta Herding

DefinitionThe degree of herding towards the market portfolio is given by

Hmt =1Nt

Nt

∑i=1(βsimt − 1)

2 , (3)

where Nt is the number of stocks at time t. Herding towards themarket portfolio therefore decreases with Hmt .

beta herding increases as Hmt gets smaller.

Estimating and Testing Beta Herding

One major obstacle in calculating the herd measure is thatβsimt is unknown and needs to be estimated.

HOmt =1Nt

Nt

∑i=1(bsimt − 1)

2 , (4)

bsimt is the estimate of beta for the market portfolio for stock iat time t.

However, HOmt could be affected by insignificant estimates ofβsimt’s.

The significance of the OLS estimates of the betas couldchange over time, affecting HOmt even though βsimt wasconstant.

DefinitionStandardised beta herding is defined using

H∗mt =1Nt

Nt

∑i=1

(bsimt − 1

σ̂bsimt

)2, (5)

where σ̂bsimt is the robust standard error of bsimt .

This new measure of herding is distributed as 1/Nt times thesum of non-central χ2 distributions and a constant:

Var [H∗mt ] =2N 2t

[R + 2δ∗Rm

].

Estimation of Beta and Data

Multi-factor unconditional models with rolling windows of 60monthly observations

Betas are estimated in the presence of Fama-French threefactors and momentum.

rit = αsi + βsimrmt + βsis rsmbt + βsihrhmlt + βsimm rmmt + εit .

Data

Individual stocks whose prices are higher than $5 (nonpennystocks) at the beginning of the estimation period and forwhom 60 past monthly observations are available

To minimise nonsynchronous trading and microstructurebiases and thin trading

Portfolios: Fama-French 25 and 100 value weighted portfoliosformed on size and book-to-market, and 49 industry portfoliosThe top and bottom 1% of beta estimates and standardizedbeta estimates ((bsimt − bsmt )/std(bsimt )) are omitted in ourcalculations of the herd measure.

Data Sources

The monthly data file from the Center for Research in SecurityPrices (CRSP) from July 1963 to December 2006

Ordinary common stocks listed on the NYSE, AMEX andNASDAQ.

For excess market returns the CRSP value weighted marketportfolio returns and 1 month treasury bills are used, andFama-French’s size (Small minus Big, SMB), book-to-market(High Minus Low, HML), and momentum (MM) are used.

Beta-based and Standardised Herd Measures

All herd measures are highly non-normal: they are positivelyskewed and leptokurtic.

Beta needs to be estimated with the other factors.

Table 1 and Figure 1

A considerable difference between the beta-based (HOmt ) andstandardized (H∗mt ) herd measures

80% of the beta-based herd measure is explained by theestimation error.Portfolios formed on t-statistics have the same cross-sectionalpattern in betas.

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Irrational Herding vs Rational Herding

Table 2 and Figure 1

Five macroeconomic variables; one-month Treasury bill rate(TBt ), the difference between one-month and three-monthTreasury bill rates (DTBt ), the term spread (TSt , thedifference between the US ten year and one year Treasury bondrate), the credit spread (CSt , the difference between Moody’sAAA and BAA rated corporate bonds), and the dividend yield(DYt , the dividend yield of S&P500 index)Two market variables: market return and volatilityThe dynamics of the cross-sectional dispersion of the betas isnot well explained by the macroeconomic variables and the twomarket variablesThe vast majority of the dynamics in the standardised herdmeasure arises from irrational herding.

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Beta Herding and Economic Events

Herding occurs in both bull and bear markets.

During crises herding disappears

The herd measure shows that herding occurs when investorsare confident about the outlook for the market.If the direction toward which the market is heading is assured,herding begins to increase, regardless of whether it is a bull orbear market.

Our evidence does not support the notion that herding occurswhen financial markets are in stress (or in crisis).

Figure I clearly shows that it is the estimation error that leadsto sharp decreases in the beta-based herd measure during the1987 Crash and the 1998 Russian Crisis.

The Relationship between Market Sentiment and BetaHerding

From Equation (??) we have

lnHmt = α+ βSmt + νt , (6)

with a negative coeffi cient on β.

Sentiment Indices

The direct sentiment index by Investors IntelligenceThe index on business conditions for the next 12 months,Michigan Consumer Sentiment index.Sentiment index of Baker and Wurgler (2007)

Table 3 and Figure 4

Sentiment indices are indeed negatively related to our herdmeasures.R2 values are around 10%.

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Cross-sectional Asset Returns Conditional on Beta Herding

The performance of high beta stocks relative to low betastocks may depend on beta herding.

When beta herding is strong, cross-sectional asset returns willnot differ significantly.When beta herding does not exist or adverse beta herdingexists (and thus high betas are upward biased (higher) and lowbetas are downward biased (lower)) the cross-sectionalvariation in asset returns would increase.

Panel A of Table 4

Decile portfolios sorted on standardized betas (i.e., t-statisticsof estimated betas for individual stocks,(bsimt − bsmt )/std(bsimt )) with NYSE breakpoints, and thencalculate the following month’s equally weighted return foreach of these portfolios, depending on the previous month’sherding states.

Three herding states are obtained from the herd measure, i.e.,herding (bottom 30% of herd measure), no herding (middle40% of herd measure), and adverse herding (top 30% of herdmeasure).

As in Fama and French (1992, 1993, 1996), beta does notforecast the following month’s cross-sectional returnsunconditionally.

However, the returns of high beta portfolios are higherfollowing an adverse herding state than those following aherding state, whereas the returns of low beta portfolios arelower following adverse herding than those following herdingstates.

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Panel B of Table 4

In order to test if the cross-sectional return difference betweenhigh and low beta portfolios change significantly depending onthe beta herding, we conduct the following regressions

r βHigh,t+h − r

βLow ,t+h = c0 + c1H

∗mt + εit+h,

and

r βHigh,t+h− r

βLow ,t+h = c0+ c1H

∗mt + c2rSMBt+h+ c3rHMLt+h+ c4rMMt+h+ εit+h.

The high and low beta portfolio returns, r βHigh,t+h and

r βLow ,t+h , are obtained by equally weighting the top andbottom 30% of stocks formed on the standardized betascalculated at time t with NYSE breakpoints.

Over shorter horizons, beta does forecast cross-sectionalreturns conditioning on the level of herding

The positive coeffi cients on the herd measure confirm thatbetas become less priced when investors have confidence in thedirection toward which the market is heading.

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Beta Herding and Firm Characteristics

Baker and Wurgler (2006) show that for firms that arediffi cult to price, i.e., newer, smaller, volatile, unprofitable,non-dividend paying, and distressed firms, asset returns areaffected more profoundly by sentiment.

Table 5

All these firm characteristics except for Profitability showsignificant relationships with standardized betas.

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Table 6

Time series regression

rHigh,t+h − rLow ,t+h = c0 + c1H∗mt + c2Smt + c3rmt+h+c4rSMBt+h + c5rHMLt+h + c6rMMt+h + εit+h.

The effects of herding on these hedge portfolios are weak.The effects are significant over shorter horizons for Tangibilityand Sales Growth.Firms with higher levels of tangible assets tend to performbetter than those with lower tangible assets following adversebeta herding, against expectation.

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Conclusions

We have proposed a measure of (irrational) herding byfocusing on changes in systematic risk which are not explainedby fundamentals.

Contrary to a common belief that beta herding is significantwhen the market is under stress, we find that beta herdingbecomes more apparent when investors feel confidentregarding the future direction of the market.

Once a crisis appears, beta herding weakens substantially asconcern for fundamentals takes over.

We have also explained how sentiment affects beta herding.Our empirical results confirm the positive relationship betweensentiment and beta herding, but sentiment explains only 10%of beta herding. There are separate forces at work.

Implications of beta herding for asset pricing

We have found that beta matters conditionally: high betastocks are priced higher than low beta stocks after adversebeta herding.We have weak evidence that beta herding predicts theperfromance of firm characteristic-sorted portfolios.

This approach can also be applied at the sector (industry)level and different herding behavior may well be found indifferent sectors such as IT or old economy stocks or on ageographical basis, etc.