Noise Trading and Asset Pricing Factors∗

Shiyang Huang Yang Song Hong Xiang

May 3, 2020

Abstract

We demonstrate that a broad set of asset pricing factors (anomalies) are

significantly exposed to “noise trader risk,” and the noise trader risk is priced

in factor premia. Since mutual fund investors are ignorant of asset pricing

factors, we confirm that mutual funds’ flow-induced trades of factors are un-

informed as they have a large price impact on factor returns, followed by a

complete reversal. We then show asset pricing factors are subject to noise

trader risk in that expected variation and covariation of flow-induced noise

trading strongly forecast variance and covariance of factor returns. Impor-

tantly, we find that factor premium is higher when the flow-driven noise trader

risk is more salient.

∗We thank Bronson Argyle, Itzhak Ben-David, Philip Bond, Jaewon Choi, Lauren Cohen, Zhi Da,Larry Harris, Paul Irvine, Lukas Kremens, Hanno Lustig, Dong Lou, Stephan Siegel, Yangru Wu, YaoZeng, and seminar participants at University of Southern California, University of Washington, Universityof Colorado Boulder, Monash University, Deakin University, the University of Technology Sydney, Ts-inghua University (SEM), Renmin University of China, Shanghai Advanced Institute of Finance (SAIF),CUHK Shenzhen, Fudan Fanhai and the Central University of Economics and Finance, as well as partici-pants at the Lancaster Factor Investing Conference and the 7th Annual Melbourne Asset Pricing Meeting.Huang and Xiang are with Faculty of Business and Economics, the University of Hong Kong, and Songis with Foster School of Business, University of Washington. Email: huangsy@hku.hk, songy18@uw.edu,and hongxiang18@hku.hk. A previous version of this manuscript was circulated under the title “Flow-Induced Trades and Asset Pricing Factors.”

1 Introduction

Asset pricing literature has documented a broad set of stock market factors/anomalies

that explain cross-sectional variations in stock returns.1 Understanding the source and

variations of these factors has been arguably one of the main themes of asset pricing

research over the past several decades.2 In this paper, we provide a new perspective on

asset pricing factors. Specifically, we show that asset pricing factors are heavily exposed

to noise trader risk, which arises from uninformed capital allocations of mutual fund

investors. We further show that the flow-driven noise trader risk is significantly priced in

factor premia, corroborating the theory of De Long, Shleifer, Summers, and Waldmann

(1990) at the factor level.

We start by confirming our premise that the demand shift of mutual fund investors

is largely uninformed in that mutual funds’ trades induced by fund flows generate large

short-term price impact on factor returns, which reverts entirely afterward.3 Then, we

show that asset pricing factors are heavily exposed to noise trader risk as variation and

covariation of flow-induced factor trading strongly predict the variance-covariance struc-

ture among the factor returns. Importantly, when using variations of flow-induced trading

to quantify noise trader risk, we find that noise trader risk is significantly priced in factor

premium by arbitrageurs and other investors: in the time series, average premium across

factors is higher when the aggregate flow-driven noise trader risk is expected to be higher;

cross-sectionally, the return of a factor is higher when its flow-induced trading is expected

to be more correlated with the aggregate flow-driven demand.

Our analysis is motivated by recent observations that mutual fund investors are igno-

rant of asset pricing factors (Berk and van Binsbergen, 2016; Barber, Huang, and Odean,

1We use factors and anomalies interchangeably in this paper.2See, for example, Cochrane (2011), Nagel (2013), McLean and Pontiff (2016), Harvey, Liu, and Zhu

(2016), Hou, Xue, and Zhang (2018), and Harvey and Liu (2019).3Because the construction of flow-induced trading does not use contemporaneous stock returns, our

results are not subject to the critique in Wardlaw (2019).

1

2016) and respond to uninformative signals (Ben-David, Li, Rossi, and Song, 2019), when

they allocate capital among equity mutual funds. This is not surprising since the ma-

jority of mutual fund investors are households with limited information readily available

to them. For example, according to the 2011 ICI Fact Book, 93.7% of mutual fund

assets in the U.S. were held by households. Moreover, an extensive literature has also

documented that mutual fund investors exhibit behaviors that are generally considered

unsophisticated.4

As argued by Grinblatt and Titman (1989) and Pastor and Stambaugh (2002), in a

fully rational world, mutual fund investors should consider all factors that explain cross-

sectional variation in fund performance regardless of whether the factors are priced or

not, and only reward mutual fund managers with “real” alphas. In such a world, fund

flows would have little influence on factor returns. Given mutual fund investors’ actual

behaviors and their large holding of the stock market, however, we hypothesize and verify

that mutual funds’ flow-driven trades of factors are mostly uninformed, and the flow-

driven noise trading is an important state variable that is priced in factor premia.

To analyze the relationship between noise trader risk and factor premia,5 we use 70

characteristic-based stock market factors, including the Fama-French five factors and the

momentum factor. To construct a characteristic-based factor, we sort all NYSE-AMEX-

NASDAQ stocks into quintile portfolios based on the NYSE breakpoints and measure

factor returns as the spreads between the value-weighted returns of the top-quintile and

the bottom-quintile stocks.

We use a bottom-up approach to measure mutual funds’ flow-driven demand of a given

4For example, mutual fund investors prefer funds that report holdings of recent winners and lotterystocks (Solomon, Soltes, and Sosyura (2014) and Agarwal, Jiang, and Wen (2018)); invest in funds thatadvertise a lot (Jain and Wu (2000)) or appear in the media (Kaniel and Parham (2017)); prefer fundsthat recently experienced an extremely positive monthly return (Akbas and Genc (2020)); and time themarket poorly (Frazzini and Lamont (2008) and Akbas, Armstrong, Sorescu, and Subrahmanyam (2015)).Song (2019) shows that investors’ unsophisticated behaviors lead to a mismatch between managerial skilland scale of active funds.

5While our approach applies smoothly to individual stocks, in this paper we focus on asset pricingfactors as understanding source and variations of factors is one of the key topics of asset pricing research.

2

factor at a quarterly frequency. That is, we estimate mutual fund flow-induced trading

(FIT) for individual stocks in a given quarter following Lou (2012). In a nutshell, FIT

measures the magnitude of flow-driven trading by the aggregate mutual fund industry

on a particular stock over a quarter. Then, for a given factor, we use stocks’ portfolio

weights within the factor to aggregate the stock-level FIT to the factor level.6 In total,

we use 208,419 fund-quarter observations with 4,999 active equity mutual funds in the

US from 1980 to 2017.

To demonstrate that noise trader risk is priced in factor premia, we organize our

analysis in three steps. To start, we confirm that flow-induced trading of factors (FITOF)

is mostly uninformed. Specifically, we find that flow-induced trades generate large short-

term price impact on factor returns, and the flow-driven price pressure reverts entirely

afterward. For example, in a given quarter, the top 20 factors by FITOF over the same

quarter significantly outperform the bottom 20 factors by 3.75% on average (15% on an

annual basis) in terms of the Fama-French five-factor (FF5) alpha. However, when we

track their performance over a longer horizon (beyond the portfolio formation quarter),

the short-term price impact completely reverts over the following two years. This full

reversal confirms the earlier evidence that mutual fund investors are ignorant of asset

pricing factors (Berk and van Binsbergen, 2016; Barber et al., 2016; Ben-David et al.,

2019).7

In the second step of our analysis, we demonstrate that asset pricing factors are

significantly exposed to the flow-driven noise trader risk because expected variations (co-

variations) of flow-induced trading strongly forecast variations (covariations) of factor

returns. To this end, we use fund ownership and flows to estimate expected variance

and expected covariance of flow-induced trading of factors by extending the approach

6In the robustness checks in Appendix B, we also isolate fund flows that are driven by fund alphasand re-construct our measures with the “alpha-free” fund flows. We obtain similar results using the“alpha-free” fund flows.

7Moreover, we rule out reverse causality by showing that there are no return patterns when we sortfactors based on past factor returns. As a placebo test, we also show in Appendix B that the non-flow-induced trades of mutual funds do not generate factor return reversal.

3

of Greenwood and Thesmar (2011), which we refer to as “factor fragility” and “factor

co-fragility,” respectively.

We find that when mutual fund trading of a factor is expected to be more volatile,

the return volatility of this factor is indeed higher; when flow-induced trades between

factors are expected to be more correlated, factor returns covary more with each other.

For example, in the Fama-MacBeth regression of quarterly frequency, a one-standard-

deviation increase in factor co-fragility predicts an increase of 46% of a standard deviation

in factor return covariance over the next quarter, even after controlling for lagged factor

return covariance. The results remain largely unchanged when excluding the crisis periods.

Taken together, our findings indicate that the asset pricing factors are heavily exposed to

noise trader risk. In this case, the noise trader risk arises from the uninformative capital

allocation of mutual fund investors.

Our goal is to understand whether and to what extent the flow-driven noise trader

risk is “priced” by arbitrageurs and other sophisticated investors who trade these factors.8

According to De Long et al. (1990), noise trader risk should affect arbitrageurs’ willingness

to trade these factors: when noise trader risk is higher, arbitrageurs are less willing to

trade, and consequently, these factors have higher returns. In other words, arbitrageurs

and other investors shall demand higher compensation to trade these factors when the

noise trader risk is expected to be more salient.

To this end, we use fragility of the aggregate factor portfolio, that is, the expected

variation of flow-induced trading of the equal-weighted factor portfolio, to proxy for the

aggregate flow-driven noise trader risk. Based on De Long et al. (1990), the average

premium across factors should be higher when the aggregate fragility is higher. Similarly,

in the cross-sections, the required return of a factor should also be higher if flow-induced

trading of that factor is expected to be more correlated with the aggregate flow-driven

demand. Specifically, we measure this trading covariation by the co-fragility between the

8Hanson and Sunderam (2014) and McLean and Pontiff (2016), among others, show that arbitrageurs,such as hedge funds or quant funds, have widely exploited asset pricing anomalies.

4

factor and the aggregate factor portfolio.

Consistent with the intuition, aggregate factor fragility significantly and positively

forecasts future average factor premia. For example, in the time-series regression, a one-

standard-deviation increase in aggregate fragility of the 70 factors forecasts an increase

of about 60 bps in average factor premium over the next quarter. This is economically

significant as the average premium across the set of factors is about 78 bps per quarter.

The results continue to hold after controlling for the market sentiment of Baker and

Wurgler (2006), the average value spread of factors, past average factor return, and the

average factor return covariance, suggesting that the average factor co-fragility indeed

captures information beyond these predictors. Moreover, the out-of-sample (OOS) tests of

Welch and Goyal (2007) further confirm the strong predictive power of aggregate fragility

on future factor premia.

In the cross-sectional tests, we also find that the required premium of a factor is

significantly higher when its flow-driven noise trading is more correlated with the ag-

gregate flow-driven demand. On average, a factor’s expected return increases by 51 bps

per quarter with a one-standard-deviation increase in its co-fragility with the aggregate

factor portfolio. In sum, both the time-series and cross-sectional results indicate that the

flow-driven noise trader risk is priced in factor premia.

To further corroborate our claim that arbitrageurs price the flow-driven noise trader

risk, we explore hedge funds’ trading activities. Specifically, we decompose each factor

into two “sub-factors” based on hedge fund trading volume. We find that the flow-driven

risk is priced in factor premia mostly through stocks that are actually traded by hedge

funds, consistent with that arbitrageurs require compensation for bearing noise trader

risk. In contrast, other measures of sentiment (e.g. Baker and Wurgler (2006)) mostly

affect factor premia through small-cap stocks and stocks that are often not traded by

hedge funds and other institutions. This sharp difference also suggests that flow-induced

noise trading is orthgonal to existing measures of market sentiment.

5

Our paper is closely related to the literature that studies the role of non-fundamental

trading on asset prices. Examples include Shiller (1981), Lee, Shleifer, and Thaler (1991),

Campbell and Cochrane (2000), Lettau and Ludvigson (2001), and Baker and Wurgler

(2006). We complement this literature in two aspects: First, we offer the first analysis of

noise trader risk on asset pricing factors and find strong evidence that noise trader risk

is priced in factor premia. Second, unlike the prior studies that either propose measures

of market sentiment or analyze how levels of market sentiment affect stock returns, we

directly quantify noise trader risk for tested assets and find that, both in the time-series

and in the cross-sections, higher noise trader risk is associated with a higher expected

return. Moreover, while market sentiment (e.g., Baker and Wurgler, 2006)) has larger

effects on small-cap stocks, the flow-induced noise trader risk is priced mostly through

large-cap stocks, which are more likely to be traded by hedge funds and other institutional

investors.9

Another strand of literature shows that investor demand unrelated to fundamentals

can affect asset prices. For example, Coval and Stafford (2007), Frazzini and Lamont

(2008), and Lou (2012) show that mutual fund flow-induced demand shocks have a con-

siderable price impact on individual stock prices. Teo and Woo (2004) and Li (2019)

document that mutual fund flows negatively predict style-level stock returns, consistent

with the “style investing” hypothesis in Barberis and Shleifer (2003). This paper comple-

ments this literature by comprehensively analyzing the impact of uninformative demand

shocks on a large collection of asset pricing factors. Our results highlight that asset pricing

factors are heavily exposed to “noise trader risk,” which we further show is significantly

priced by arbitrageurs.

Our paper is also closely related to the recent literature that investigates the high

dimensionality of cross-sectional asset pricing models.10 We offer a new perspective on

9Stambaugh, Yu, and Yuan (2012) compare the performance of asset pricing anomalies following theperiods of high and low sentiment. Specifically, they argue that the presence of short-sale constraintdrives anomalies, particularly the short legs of anomalies, stronger following high levels of sentiment.

10Examples include Harvey, Liu, and Zhu (2016), Harvey (2017), McLean and Pontiff (2016), Kozak,

6

asset pricing factors by demonstrating the important influence of noise trading on factor

premia and the variance-covariance structure among these factors.

The rest of the paper is organized as follows. Section 2 introduces the dataset, the set of

asset pricing factors, and our measure of flow-induced trading of factors (FITOF). Section

3 confirms that flow-induced factor trades are nonfundamental. Section 4 quantifies the

influence of flow-induced trading on return volatilities and return comovements among

the factors. Section 5 shows that the flow-driven noise trader risk is significantly priced

in factor premia. Section 6 concludes. Robustness checks and supplementary results are

reported in the appendices.

2 Data and Methodology

In this section, we describe the data, the construction of the 70 asset pricing factors

(anomalies), and how we estimate mutual fund flow-induced trading of factors.

2.1 Factor Construction

We use the CRSP and Compustat datasets to construct 70 asset pricing factors. Table

C.1 shows the list of factors, which include size, book-to-market ratio, profitability, and

momentum, among many others.

Our sample stocks include all ordinary common shares (CRSP share code 10 or 11)

listed on NYSE, AMEX, and NASDAQ. To be included in our sample in a given quarter,

the stock is required to be held by at least one mutual fund with non-missing holding data

from Thomson Reuters CDA/Spectrum database and valid fund flow (calculated using

CRSP mutual fund database) in that quarter.

The universe of factors consists of 70 annually or quarterly rebalanced factors based on

Nagel, and Santosh (2019), Hou, Xue, and Zhang (2018), Kelly, Pruitt, and Su (2019), Feng, Giglio, andXiu (2019), among others.

7

firm characteristics.11 To construct a characteristic-based factor, we follow Hou, Xue, and

Zhang (2018) and use NYSE breakpoints of the characteristic to form quintile portfolios.

For annually rebalanced factors, at the June-end of each calendar year, we sort all stocks

into quintiles based on the NYSE breakpoints of sorting variables (e.g., book-to-market

ratio) at the fiscal year ending in the previous calendar year. We then track the value-

weighted portfolio returns from July to next June. For quarterly rebalanced factors that

rely on Compustat quarterly fundamentals data, we skip one quarter between the portfolio

formation date and the start of the portfolio holding period to ensure that all information

is available upon portfolio formation. Specifically, at the end of each quarter, we form

quintile portfolios based on sorting variables as of the fiscal quarter ending in the previous

calendar quarter, and hold the portfolios in the next calendar quarter. More details of

the factor construction are provided in Appendix C.

2.2 Estimate Mutual Fund Flow-Induced Trades of Factors

To measure flow-induced trades of each of the 70 factors, we first estimate flow-induced

trades of individual stocks. To this end, we merge the Thomson Reuters CDA/Spectrum

database with the CRSP Survivorship-bias-free mutual fund database. In particular, we

obtain mutual funds’ holding data from the CDA/Spectrum database. Mutual funds’

total net assets (TNA), monthly net returns (after fee), and annual expense ratios are

from the CRSP database. For mutual funds with multiple share classes, we use the sum

of TNA across all share classes as the TNA of the fund, and we take TNA-weighted

average net returns and expense ratios across all share classes. We compute mutual fund

monthly gross returns (before fee) as the sum of monthly net returns and 1/12 of the

annual expense ratio.

We focus on actively-managed equity mutual funds. Specifically, we filter out non-

11We transform several typical monthly rebalancing factors (e.g., momentum) into quarterly rebal-ancing factors to match the quarterly mutual fund holdings data. Our results do not change even afterexcluding those monthly rebalancing factors.

8

equity funds based on investment objective codes reported in the CDA/Spectrum database

and the CRSP mutual fund database.12 In addition, we require the ratio of common stock

holdings to TNA to be between 80% and 105% on average over the sample period. Finally,

we exclude fund-quarter observations with less than $1 million TNA. Our fund sample

includes 4, 999 distinct US domestic equity funds with 208, 419 fund-quarter observations

during 1980-2017.

We take two steps to construct the stock-level flow-induced trading. We first calculate

quarterly mutual fund flow, defined as the percentage change of total net assets after

adjusting for the appreciation of fund holdings (Sirri and Tufano, 1998):

Flowk,t =TNAk,t − TNAk,t−1 × (1 +Rk,t)

TNAk,t−1

,

where TNAk,t is the total net assets of fund k at the end of quarter t and Rk,t is the gross

return of fund k in quarter t.

Second, we measure quarterly aggregate mutual fund trading of an individual stock in

response to fund flows. Specifically, we follow Lou (2012) and estimate the flow-induced-

trading (FIT) measure as follows :

FITj,t =

∑k Sharesk,j,t−1 × Flowk,t × PSFk,t∑

k Sharesk,j,t−1

, (1)

where Sharesk,j,t−1 is the number of shares of stock j held by fund k at the end of quarter t−

1, Flowk,t is the percentage flow of fund k in quarter t, and PSF is the partial scaling factor.

12We mainly follow Kacperczyk, Sialm, and Zheng (2008) to screen funds in the following steps. First,we screen funds by investment objectives reported by CDA/Spectrum database. We exclude funds withInvestment Objective Codes in 1, 5, 6, or 7, in the CDA/Spectrum database. Then, we screen funds byinvestment objectives reported by CRSP mutual funds database. For funds with non-missing “Type ofSecurities Mainly Held by Fund” variable (policy variable), we remove those with policy in C&I, Bal,Bonds, Pfd, B&P, GS, MM, or TFM. We then require remaining funds to have Lipper ClassificationCode in EIEI, G, LCCE, LCGE, LCVE, MCCE, MCGE, MCVE, MLCE, MLGE, MLVE, SCCE, SCGE,SCVE, or Missing. For funds with missing Lipper Classification Code, we require them to have StrategicObjective Insight Code in AGG, GMC, GRI, GRO, ING, SCG or missing. If a fund has both missingLipper Classification Code and Strategic Objective Insight Code, we screen them through WiesenbergerFund Type Code and retain funds with objective codes in G, G-I, AGG, GCI, GRI, GRO, LTG, MCG,SCG, or Missing.

9

The scaling factor reflects how fund managers, on average, increase and liquidate their

holdings in response to capital inflows and outflows, respectively. Lou (2012) estimates

PSFk,t to be 0.970 for outflows, and 0.858 for inflows. We use the same estimates of the

partial scaling factor in our study.13 Moreover, we use FIT rather than the entire realized

trading of mutual funds because FIT only captures those trades that are driven by the

demand shifts from mutual fund investors, which are largely uninformative (Ben-David,

Li, Rossi, and Song, 2019). For robustness, in Appendix B, we re-construct FIT using

mutual fund flows that are not driven by fund alpha components, and the results remain

largely unchanged.

Based on stock-level flow-induced trading, we measure flow-induced trading of a factor

π as the value-weighted average FIT of stocks in the factor’s long leg minus the value-

weighted average FIT of stocks in the short leg. That is,

FITOFπ,t =∑j∈NπL

µπj,t−1FITj,t −∑j∈NπS

µπj,t−1FITj,t, (2)

where N πL and N π

S are the set of stocks consisting of the long-leg and short-leg of factor π

at time t, respectively, and µπj,t−1 is the weight of stock j in factor π. In short, FITOF mea-

sures the flow-induced trading of the long-leg stocks relative to the flow-induced trading

of the short-leg stocks.

[Table 1 Here]

Table 1 reports the basic statistics of our sample. As shown in Panel A, The sample

coverage steadily rises as the relative size of the mutual fund sector grows substantially

over time (from 2.62% to 20.07%). At the beginning of the sample period, our sample

covers 46.30% in terms of the number of stocks and 94.45% in terms of market capital-

ization, indicating that mutual funds tend to avoid tiny stocks, which is consistent with

Frazzini and Lamont (2008). At the end of the sample period, our sample covers 94.34%

13Our results are not sensitive to the choices of PSF.

10

in terms of the number of stocks and 99.15% in terms of market capitalization. It is also

worthy to note that because we use the value-weighted scheme to construct factor returns

or factor-level flow-induced trading, including stocks that are not held by mutual funds

(e.g., tiny stocks) in the construction of factors has little influence on our results.

As shown in Panel B of Table 1, the average quarterly return of the 70 factors is 0.78%,

with a standard deviation of 6.54%. The 25th and 75th percentiles of the stock-level FIT

are −1.95% and 3.02%, respectively. This suggests that, in response to retail investors’

demand shifts, mutual funds adjust their stock holdings relative to their existing holdings

at a scale between −1.95% and 3.02% within a quarter in the 25th to 75th percentile

range. The 25th and 75th percentiles of FITOF are −0.53% and 0.55%, respectively.

3 Flow-Induced Factor Trading is Uninformed

In this section, we justify the premise that mutual funds’ flow-induced trades of factors

are largely uninformed.14 We find that flow-induced trades significantly and positively

influence contemporaneous factor returns, followed by full reversals over longer horizons.

This return pattern confirms the prior findings that mutual fund flows are mostly unin-

formed and ignorant of systematic factors (Barber, Huang, and Odean, 2016; Ben-David,

Li, Rossi, and Song, 2019).

To examine the return pattern associated with flow-induced trading of factors (FITOF),

at each quarter-end, we sort the 70 factors into three groups based on FITOF over the

same quarter, with 20/30/20 factors in each group respectively. We then track each group

with equal-weighted factor returns in the next 12 quarters. Table 2 reports the monthly

(adjusted) returns of the three portfolios of factors sorted by FITOF.

[Table 2 Here]

14While some earlier work, such as Coval and Stafford (2007), Frazzini and Lamont (2008), and Lou(2012), shows that mutual fund flows generate price pressure on individual stocks, it is not clear whetherthe flow-induced price impact would cancel out at the factor level, as factors are constructed to bediversified long-short portfolios.

11

The first pattern to note is that FITOF generates a strong price impact on factors

over the contemporaneous quarter (Qtr 0). For example, Panel D of Table 2 reports that

the low-FITOF group earns an average monthly Fama-French five-factor (FF5) alpha of

−0.42% in the formation quarter, while the high-FITOF group earns a monthly FF5 alpha

of 0.82%. The high-minus-low return spread is associated with an average monthly FF5

alpha of 1.25% with a t-statistic of 5.93.

Second, the short-term return effect strongly reverts in the subsequent two years. For

example, when we adjust the returns for the FF5 factors (Panel D), the high-minus-low

spread is indistinguishable from zero in the first post-formation year, but it is −0.33%

per month in the second year. In the third year, there is no further reversal associated

with portfolios ranked by FITOF.

[Figure 1 Here]

To visualize the return pattern, Figure 1 plots the cumulative FF5 alpha of a long-short

portfolio that longs the high-FITOF group and shorts the low-FITOF group. As one can

see, the positive FF5 alpha of the long-short portfolio in the formation quarter almost

fully reverts by the end of the second year. This return pattern validates the premise

that flow-induced trades of factors are mostly uninformed. In addition, by exploiting

the reversal effect associated with FITOF, we find that a tradable strategy that longs

the factors with low flow-induced trades in the past eight quarters and shorts the factors

with high past FITOF generates annualized alphas of 7.3% to 11.6% depending on the

benchmark (see Table B.6).

We conduct several placebo tests in Appendix B. First, to ensure that the above return

dynamics are not driven by the mean reversion of factor returns, we conduct a similar

portfolio-sorting exercise but instead of sorting on FITOF, we use factor returns as the

sorting variable. We do not find any return patterns (see Panels A and B of Table B.3).

We also analyze the influence of mutual fund trades of factors that are not driven by fund

12

flows.15 We find that non-flow-induced trades do not experience factor return reversals

(see Panel C of Table B.3). These placebo tests highlight the uniqueness of the return

patterns associated with flow-induced factor trades.

In addition to the placebo tests, we conduct a comprehensive robustness check in

Appendix B. First, to account for time and factor fixed effects that might be correlated

with factor returns, we estimate panel regressions of factor returns on flow-induced trading

of factors. The regression exercise confirms the strong influence of fund flows on factor

returns (see Table B.2). Furthermore, we conduct the regression exercise in the first-

and second-half sample periods (1982-1998 versus 1999-2017), respectively. We find that

the effects of FITOF on factor returns are stronger in the second-half sample, which is

consistent with the rapid growth of the mutual fund industry over time.

In summary, the results in this section and in Appendix B demonstrate that the flow-

driven demand shifts are statistically significant and economically strong drivers of factor

returns. More importantly, our findings confirm the mounting evidence that mutual fund

flows are largely uninformative.

4 Factors are Exposed to Noise Trader Risk

In this section, we estimate the extent to which asset pricing factors are exposed to

noise trader risk. Specifically, we quantify noise trader risk by estimating the variance-

covariance matrix of flow-induced trading among the set of factors. We find that (i) the

expected volatility of flow-induced trading strongly forecasts future factor return volatility

and (ii) the expected covariance of flow-induced trading of factors strongly forecasts future

factor return covariance. Together with the results in Section 3 that mutual funds’ flow-

induced factor trades are uninformed, we conclude that these 70 asset pricing factors are

15The non-flow-induced trades are the difference between mutual funds’ realized trades and the flow-induced trades. To calculate mutual funds’ realized trades of factors, we first compute mutual funds’aggregate realized trades on each stock (RT) in a similar way as we calculate FIT. Then we computemutual funds’ realized trades of a factor as portfolio-weighted average RT on stocks that constitute thatfactor.

13

significantly exposed to noise trader risk, which arises from the uninformative fund flow

movements (De Long et al., 1990).

4.1 Estimate (Co)variation of Flow-Induced Factor Trading

We first describe how we estimate the variance-covariance matrix of flow-induced trad-

ing of factors. Following the spirit of Greenwood and Thesmar (2011), the expected vari-

ance of flow-induced trading of a given factor π over quarter t + 1, which we refer to as

“factor fragility,” can be estimated by

Gπt = W π

t

′Et(Ωt+1)W π

t . (3)

Here, Et(Ωt+1) is the conditional variance-covariance matrix of mutual fund flows in quar-

ter t + 1 and W πt =

(wπ1,t, . . . , w

πK,t

)′is the vector of mutual fund weights in factor π. In

particular, the weight of mutual fund k in factor π is calculated as

wπk,t =∑j

µπj,tSharesk,j,tShroutj,t

,

where µπj,t is the weight of stock j in factor π,16 and Shroutj,t is the number of shares

outstanding of stock j.

Likewise, we estimate the expected covariance of flow-induced trading of factors π1

and π2 over quarter t+ 1, referred to as “factor co-fragility,” by

Gπ1,π2t = W π1

t

′Et(Ωt+1)W π2

t . (4)

As one can see, factor fragility and factor co-fragility depend on mutual fund ownership

concentration and the expected variance-covariance matrix of mutual fund flows. To

estimate Et(Ωt+1), we calculate the variance-covariance matrix of mutual fund flows using

16For a long-leg stock, µπj,t equals its original weight in the long leg. For a short-leg stock, µπj,t is itsoriginal weight in the short leg multiplied by negative one.

14

observations in the past two years, and the results are not sensitive to this choice. We

report the detailed estimation in Appendix A.

To illustrate the role of flow variation on factor return variation, we plot the return

volatility and the square root of lagged factor fragility of the Fama-French size and value

factors in Figure 2. At first glance, there is a clear positive correlation between future

factor return volatility and volatility of flow-induced trading of factors. We formally

explore the association between factor fragility (co-fragility) and factor volatility (co-

variance) in the next subsection.

[Figure 2 Here]

4.2 Variations of Flow-Induced Trading and Return Variations

We now examine how factor fragility and factor co-fragility, our measures of expected

variance and expected covariance of flow-induced trades of factors, respectively, can fore-

cast future variance and covariance of factor returns.

To this end, we first estimate the following Fama-MacBeth regression:

σπ1,π2t+1 = α + βGπ1,π2t + τσπ1,π2t + γZπ1,π2

t + επ1,π2t+1 , (5)

where σπ1,π2t+1 is the return covariance between factors π1 and π2 in quarter t + 1 and is

estimated based on weekly factor returns, and Gπ1,π2t is the estimated co-fragility between

factors π1 and π2 in equation (4). To account for persistence in factor return covariance, we

include lagged factor return covariance. We also include a set of control variables, Zπ1,π2t ,

that comprises the factor-pairwise difference in size, book-to-market, and momentum as

in Anton and Polk (2014).17 For easy interpretation, all variables are normalized to have

17We construct factor pairwise characteristics difference as follows. First, following Anton and Polk(2014), we construct a stock-level NYSE percentile ranking of characteristics. Second, we take value-weighted NYSE percentile rankings for each of the long and short legs and compute factor-level NYSEpercentile ranking as the long-short difference. The factor pair-level difference is the absolute value ofthe difference in factor-level NYSE percentile ranking of characteristics.

15

a standard deviation of one.

[Table 3 Here]

Panel A of Table 3 reports the results. The expected covariance of flow-induced trades

between factors strongly predicts future factor return comovements. In the univariate

regression, a one-standard-deviation increase of factor co-fragility predicts an average in-

crease of 80% of a standard deviation of factor return covariance over the next quarter.

Even after controlling for lagged factor return covariance, a one-standard-deviation in-

crease of factor co-fragility still predicts an increase of 46% of a standard deviation of

factor return covariance. In columns (3) and (4), we exclude the crisis periods (from 2000

to 2001 and from 2007 to 2008), and confirm that our results are not driven by crisis.

In fact, we get even stronger effects after excluding crisis periods. These results indicate

that the covariation in flow-induced trading is an important determinant of factor return

comovement.

We then estimate the predictability of factor fragility on future factor return variation

through the following Fama-MacBeth regression:

σπt+1 = a+ b√Gπt + cσπt + ηt+1. (6)

Here, the dependent variable σπt+1 is the one-quarter-ahead factor volatility, estimated as

the standard deviation of weekly factor returns over the next quarter. The independent

variable of interest is the square root of factor fragility. We also control for past factor

return volatility.

Panel B of Table 3 reports the results. Across all specifications, we find that the square

root of factor fragility,√Gt, positively and significantly forecasts the one-quarter-ahead

factor return volatility. For example, in the univariate regression excluding the crisis

period (column 3), a one-standard-deviation increase in√Gt predicts an increase of 37%

of a standard deviation of factor volatility in the next quarter. After controlling for past

16

volatility, a one-standard-deviation increase in√Gt still leads to an average increase of

17% of a standard deviation of factor return volatility (column 4).

In summary, the results in Sections 3 and 4.2 indicate that flow-induced factor trad-

ing, while being driven by uninformative capital allocation of mutual fund investors,

significantly determines average returns, variations, and covariations among the broad

set of asset pricing factors (anomalies). In other words, these asset pricing factors are

significantly exposed to noise trader risk, which arises from the uninformative fund flow

movements. We further explore the asset pricing implications of the flow-driven risk in

the next section.

5 Is the Flow-Driven Noise Trader Risk Priced?

We have shown that mutual fund flow movements are largely uninformative, yet drive

a large portion of variations and covariations of factor returns. In this section, we explore

whether the flow-driven noise trader risk is priced in factor premia by arbitrageurs and

other investors.

Through both in-sample and out-of-sample time-series tests, we find that average

premia across factors are much higher when the aggregate flow-driven risk is expected

to be more salient. Through cross-sectional tests, we find that the required return of a

factor is higher when its flow-induced trading is expected to be more correlated with the

aggregate flow-driven risk. We also find that these “pricing” effects are mainly driven

by stocks that are traded extensively by hedge funds and driven by large-cap stocks that

institutional investors tilt towards (Gompers and Metrick, 2001). In sum, our findings

suggest that flow-driven noise trader risk is indeed priced by arbitrageurs, which confirms

the theory of De Long et al. (1990) at the factor level.

17

5.1 In-Sample Time-Series Tests

According to De Long et al. (1990), noise trader risk should affect arbitrageurs’ will-

ingness to trade factors: when noise trader risk is higher, arbitrageurs are less willing to

trade, and consequently, factors have higher returns. In this section, we show that the

average expected return across factors is indeed higher when the aggregate flow-driven

risk is higher.

Specifically, we estimate the following predictive time-series regression:

1

70

70∑i=1

rπi,t+1 = α + β Fragilityt

(1

70

70∑i=1

πi

)︸ ︷︷ ︸

Aggregate Fragility

+γ′Kt + ηt+1. (7)

Here, the dependent variable is the equal-weighted return of the 70 factors in quarter t+1.

The independent variable of interest is the fragility of the agggregate portfolio of the 70

factors, which measures the expected variation of flow-induced trading of the aggregate

factor portfolio. Kt is a vector of controls that potentially predicts future factor premium,

including the investor sentiment measure of Baker and Wurgler (2006) (BW), the average

value spread of factors (Ilmanen et al., 2019), and past average factor returns.

To set the stage, Table 4 reports the correlations between aggregate factor fragility,

Fragilityt(∑70

i=1 πi/70), and those predictors in the control Kt of equation (7). The corre-

lation between aggregate factor fragility and the BW sentiment is 0.15, and the correlation

with the average value spread is 0.30. These fairly low correlations indicate that aggregate

fragility shall capture information beyond these other predictors.

[Table 4 Here]

Table 5 presents the time-series regression of future average factor premia on aggregate

factor fragility in equation (7). For easy interpretation, all independent variables are

normalized to have a standard deviation of one. From column (1), aggregate factor

18

fragility positively and significantly forecasts future average factor premia (t = 3.22). In

terms of the economic magnitude, a one-standard-deviation increase in aggregate factor

fragility is associated with an increase of 60 bps in the average factor premia over the

next quarter. Considering that the average factor premia is about 78 bps per quarter in

our sample, the effect of noise trader risk on factor premia is economically important.

[Table 5 Here]

For comparison, columns (2)-(4) of Table 5 report the predictability of other predictors

on future factor premia. In column (2), we find that average factor return covariance also

positively forecasts future average factor returns. This result is consistent with the finding

of Pollet and Wilson (2010) that the average comovement of individual stock returns

predicts average stock returns. Columns (3) and (4) show that the BW sentiment and the

average value spread of factors are also strong predictors of future average factor premia,

consistent with Stambaugh, Yu, and Yuan (2012) and Ilmanen et al. (2019).

To confirm that the predictability of aggregate factor fragility is not driven by the

correlation with other predictors, we further conduct pairwise horse-race regressions in

columns (5) to (7) of Table 5. Column (5) shows that the predictive power of aggregate

fragility is not subsumed by average factor return covariance. In contrast, the lagged

average return covariance becomes statistically insignificant with the presence of factor

fragility. Meanwhile, aggregate fragility remains statistically significant with mild reduc-

tions in coefficient estimates when controlling for the BW sentiment or the average value

spread. Finally, we put all these predictors together into the predictive regression. Col-

umn (8) shows that only aggregate fragility remains statistically significant, while other

predictors lose the power of predicting future average factor premia.

To adjust for potential small-sample bias in the predictive regressions (Stambaugh,

1999), we also apply the bias-reduction estimation approach proposed in Amihud and

Hurvich (2004). Table 6 reports the results. Panel A shows that aggregate factor fragility

has very low auto-correlation, and Panel B shows that the innovation in the predictive

19

regression of future factor premia on aggregate fragility is almost uncorrelated with the

innovation in the AR(1) regression of aggregate fragility. Hence, the predictive regression

does not suffer from the small sample bias. Panel C confirms that the results are almost

unchanged after bias-correction.

[Table 6 Here]

We further conduct several robustness tests. First, we detrend aggregate factor

fragility to alleviate the effect of time trend on our regression analysis.18 We find that

the results are almost unchanged (columns (3)-(6) of Table 7). Second, we confirm that

the results are not specific to the particular way that we construct factors. Specifically,

we form factors with the NYSE decile breakpoints of characteristic variables and repeat

the in-sample regression.19 Under this alternative factor construction method, aggregate

fragility remains to be a statistically significant predictor of future average factor premia,

and its magnitude is even higher (columns (1) and (2) of Table 7).20

[Table 7 Here]

In summary, the predictive time-serires regressions indicate that when the aggregate

flow-driven risk is expected to be higher, other investors indeed require higher average

premia for these factors.

5.2 Out-of-sample Time-Series Tests

Welch and Goyal (2007) point out that many variables with in-sample forecasting

power cannot forecast returns out-of-sample. To further validate the role of aggregate

18In our sample period 1982Q1-2017Q4, we consider residuals from the OLS regression of aggregatefragility on the year-quarter indicator as linearly-detrended aggregate fragility. Similarly, quadratic trendis excluded through regressing aggregate fragility on both level and square terms of the year-quarterindicator.

19All factor-related variables are also re-constructed based on the decile portfolios.20 In Appendix Table B.8, we show that aggregate fragility cannot predict stock market returns and

bond market returns. In contrast, the BW sentiment index or the stock market variance can significantlypredict the long-term yield or the default yields. This indicates that the “noise trader” risk arising fromequity mutual fund flows does not capture the information contained in variables like the BW sentimentor stock market variance.

20

factor fragility in predicting future factor premia, in this section, we conduct out-of-

sample (OOS) tests as in Welch and Goyal (2007). In particular, our OOS test uses only

real-time data of a given predictor to forecast future average factor premia. Then the

OOS predictive power of the predictor is evaluated against that of the historical average

factor premia. We find that aggregate factor fragility strongly forecasts future average

factor premia in a series of OOS tests.

Specifically, we conduct the OOS test by estimating the following predictive regression

recursively:

Rt+1 = α + βAt + εt. (8)

Here, Rt+1 is the average factor return over quarter t + 1, and At refers to a specific

return predictor (e.g., aggregate factor fragility in equation (7)). Starting with an initial

in-sample estimation period, we estimate the above predictive regression and obtain the

OLS estimates (αt, βt) of (α, β). We then forecast average factor premia over the next

quarter by

Rt+1 = αt + βtAt. (9)

At each quarter, we expand the estimation window by one quarter until we reach the end

of our sample period.

To evaluate the OOS performance for a given predictor of future average factor premia,

we follow Welch and Goyal (2007) and compute the following OOS statistics:

R2OOS = 1− MSEA

MSEN

and ∆RMSE =√

MSEN −√

MSEA.

Here,

MSEA =T−1∑t=n

(Rt+1 − Rt+1

)2

, MSEN =T−1∑t=n

(Rt+1 − Rt+1

)2,

where T is the total number of quarters in our sample, n is the number of in-sample

quarters used for the first forecast, Rt+1 is the actual average factor premium, Rt+1 is the

21

forecast of average factor premium in (9), and Rt+1 is the historical mean of the average

factor premia. Intuitively, R2OOS and ∆RMSE are positive when the forecast errors based

on the predictor At, MSEA, are smaller than the forecast errors of the historical mean,

MSEN . To test the hypothesis, we also compute the MSE-F statistic by (T − h + 1) ×

((MSEN −MSEA)/MSEA) , and we compare it against the asymptotic critical values in

McCracken (2007).21

Table 8 reports the OOS performance of aggregate factor fragility and other predictors

of average factor premia used in Table 5. We choose a long enough evaluation period from

1992Q1 to 2017Q4, which starts ten years after the first quarter in our sample.22 As one

can see, aggregate factor fragility, the BW sentiment, and average value spread all have

superior OOS predictive power for future average factor premia. For instance, R2OOS

of aggregate fragility is 8.07% in the evaluation period. By comparison, average factor

return covariance fails to beat the historical mean in the OOS tests, although it has

positive in-sample predictability (Table 5).

[Table 8 Here]

Taken together, the results in Tables 5 to 8 indicate that aggregate factor fragility

positively forecasts future average factor premia both in-sample and out-of-sample. As

higher aggregate fragility implies higher noise trader risk, this evidence indicates that

arbitrageurs and other investors demand higher average premia trading these factors

when the flow-driven noise trader risk is more salient.

5.3 Explore Arbitrageurs’ Trading Activities

In this section, we further corroborate the claim that flow-driven “noise trader” risk

is priced by arbitrageurs and other sophisticated investors.

21h is the degree of overlap. Here, h = 1 for no overlap.22Hansen and Timmermann (2012) suggest that the power of forecast evaluation tests is stronger with

longer out-of-sample periods. Our choice of evaluation period ensures that the out-of-sample period islong enough relative to the initial estimation period. We also report OOS test results based on alternativechoices of evaluation periods in Table B.9, and the results are robust.

22

Specifically, we explore arbitrageurs’ trading activities. Our hypothesis is that noise

trader risk should affect factor premia only through stocks that are indeed traded by

arbitrageurs and other investors. In other words, if we construct factors using stocks

with similar factor-characteristics but different likelihood to be traded by arbitrageurs,

we should expect to find a stronger relationship between noise trader risk and factor

premia, when factors are constructed with stocks that are more likely to be traded by

arbitrageurs.

As hedge funds are typical arbitrageurs who often make use of long-short equity strate-

gies, we use hedge funds’ trading volume of a stock to gauge whether this stock is inten-

sively traded by arbitrageurs. In addition, motivated by previous literature that insti-

tutional investors trade larger stocks more than smaller stocks (Gompers and Metrick,

2001), we also construct factors using large-cap stocks and small-cap stocks separately in

a similar exercise.

Specifically, for each factor, we further sort the stocks in the long leg or the short leg

of that factor into two even groups based on their hedge fund trading volume (HF-Trade)

in the previous year.23 We then reconstruct two “sub” factors by only using stocks in

the low or the high HF-Trade group. Take the momentum factor as an example. We

construct the so-called “Low-HF-Trade” momentum factor by longing the winner stocks

in the low HF-Trade group and shorting the loser stocks that are also in the low HF-Trade

group. Similarly, the “High-HF-Trade” momentum factor is constructed by longing the

winner stocks in the high HF-Trade group and shorting the loser stocks that are also in

the high HF-Trade group.

Within the low HF-Trade factors or the high HF-Trade factors, we repeat the in-

sample and out-of-sample predictability tests of aggregate fragility on future average factor

23We calculate hedge funds’ trading volume of a stock in a given quarter using the quarterly changein hedge funds’ holdings reported in the 13F files. Then we aggregate the absolute dollar trading volumeof all hedge funds on each stock in each year. Missing hedge fund trading volume is set to zero.

23

premia. That is, similar to equation (7), we estimate

1

70

70∑i=1

rhigh HFπi,t+1 = α + βFragilityt

(1

70

70∑i=1

πhigh HFi

)+ γ′Kt + ηt+1, (10)

and

1

70

70∑i=1

rlow HFπi,t+1 = α + βFragilityt

(1

70

70∑i=1

πlow HFi

)+ γ′Kt + ηt+1, (11)

where the factors are constructed by stocks with high and low hedge fund trading volumn,

respectively.

Panel A of Table 9 reports the in-sample predictive regressions in equations (10) and

(11). We see a sharp contrast in the coefficient estimates of aggregate fragility between

the low and high HF-Trade factors (columns (1)-(4)). For example, in the univariate

regressions, the coefficient of aggregate fragility is 0.05 (t = 0.25) for low HF-Trade group,

while it is 0.59 (t = 2.74) for high HF-Trade group. This comparison beomes more striking

when we take into account the fact that the unconditional factor returns are higher for

factors constructed by stocks with low hedge fund trading volume.24

[Table 9 Here]

Panel B of Table 9 reports the out-of-sample tests in Table 8 for the low and the high

HF-Trade group of factors, respectively. Consistent with the in-sample regression results,

the out-of-sample performance of aggregate fragility is positive and significant in the high

HF-Trade group while it is slightly negative and insignificant, for the low HF-Trade group.

In a similar exercise, we also use small-cap stocks and large-cap stocks to re-form the

factors. By comparing factors formed with small and large stocks, we find the significant

relationship between aggregate fragility and average factor premia only exists among

24Average factor returns in low and high HF-Trade groups are 1.16% and 0.76% per quarter, respec-tively.

24

factors in large market capitalization group (columns (5)-(8)), consistent with Gompers

and Metrick (2001) that institutional investors tend to trade large-cap stocks.25

In short, the results in Table 9 indicate that the influence of flow-driven risk on future

factor premia is mostly through large-cap stocks and through stocks that are heavily

traded by hedge funds. By contrast, the predictability of the BW sentiment on factor

returns is mostly through stocks with high limits-to-arbitrage (e.g., stocks with low hedge

fund trading volume or small-cap stocks).

5.4 Evidence from Cross-Sectional Factor Returns

The time-series analysis so far shows that arbitrageurs require higher average factor

premia when the aggregate flow-driven noise trader risk is higher. Cross-sectionally, when

a factor’s flow-induced trading is expected to be more correlated with the aggregate flow-

driven demand, arbitrageurs should also require a higher premium for trading that factor.

We formally test this cross-sectional prediction in this section.

To measure the expected covariance of a factor’s flow-driven trading and the aggregate

noise trading, we compute the co-fragility between the factor and the aggregate factor

portfolio by

Co-Fragilityπi,t ≡ Co-Fragilityt

(πi,

1

70

70∑j=1

πj

). (12)

That is, a higher Co-Fragility implies that a factor is expected to experience more flow-

induced trading when the aggregate flow-driven demand is higher.

With this, we test the cross-sectional relationship between Co-Fragility and expected

factor returns through the Fama-MacBeth regression:

rπi,t+1 = α + βCo-Fragilityπi,t + γYπi,t + επi,t+1. (13)

25We also sort factor long/short leg into terciles by market capitalization and conduct the in-sampleand out-of-sample tests within each size tercile. Figure 4 shows that the correlation between averageco-fragility and future average factor premia monotonically increases with size tercile ranking, and suchrelation is only statistically significant in the largest size tercile.

25

Here, the dependent variable is the return of factor πi in quarter t+1, and the independent

variable of interest is the average co-fragility between factor πi and the aggregate factor

porfolio at the end of the prior quarter, as defined in equation (12). For control variables

Yπi , we include the average return covariance between factor πi and the factor portfolio

in the prior quarter (dubbed by Covariance), the past-one-quarter factor returns, the

factor value spread at the previous quarter-end, and the average FITOF in the past eight

quarters.

Table 10 reports the regression results. For easy interpretation, all independent vari-

ables are normalized to have a standard deviation of one. Across all specifications, higher

co-fragility between a factor and the aggregate factor portfolio predicts higher future

factor returns. Column (1) of Table 10 shows that a one-standard-deviation increase in

Co-Fragility is associated with an increase of 51 bps in factor return per quarter, and the

effect is statistically significant at the 1% level.26 This magnitude is also economically

significant, given that our sample average quarterly factor return is around 78 bps. Fur-

thermore, the predictive power of Co-Fragility is unchanged after including other control

variables (column (3)).

[Table 10 Here]

To alleviate the concerns of outliers, we also use the quintile ranking of Co-Fragility

in each quarter as the regressor in columns (2) and (4) of Table 10. The coefficients

remain statistically significant. To provide an easy interpretation of the economic mag-

nitude, in column (5), we only retain factors assigned to the lowest or highest quintile of

Co-Fragility each quarter and regress factor returns on a dummy variable, which indicates

the highest quintile ranking (Dummy Rank5). Column (5) shows that the coefficient of

Dummy Rank5 is 1.59 (t = 2.70), which means a long-short strategy that longs (shorts)

26In the Fama-MacBeth regression, we standardize all independent variables, except Rank andDummy Rank5, by their standard deviation in each time period. Hence, the coefficient reported hererepresents the change in monthly factor returns associated with a one-cross-sectional-standard-deviationincrease in the independent variable.

26

factors in the highest (lowest) quintile of Co-Fragility produces returns of 159 bps per

quarter.

We also show the positive return predictability of Co-Fragility through portfolio anal-

ysis. At each quarter-end, we sort factors into quintiles based on Co-Fragility in the

quarter and hold the quintile portfolios in the next quarter. As shown in Table B.10,

the CAPM alpha (FFC four-factor alpha or FF five-factor alpha) increases monotoni-

cally with Co-Fragility. For example, the monthly CAPM alpha increases from 0.02 % to

0.69% from the bottom to the top quintile. The spread of monthly CAPM alpha is 0.67%

(t = 3.73). This return pattern is also shown in Figure 3.

[Figure 3 Here]

In summary, both the time-series and the cross-sectional analyses in this section indi-

cate that the flow-driven noise trader risk on factors is an important state variable that

is priced by arbitrageurs, consistent with the theoretical predictions of De Long, Shleifer,

Summers, and Waldmann (1990).

6 Conclusion

Stock market factors (anomalies) are one of the building blocks of asset pricing re-

search. In this paper, we provide a new perspective on asset pricing factors. That is, we

demonstrate that asset pricing factors are heavily exposed to noise trader risk, where the

noise trader risk arises from uninformative demand shifts of retail mutual fund investors.

We also show that the flow-driven noise trader risk is priced in factor premia, indicating

that the predictions of De Long, Shleifer, Summers, and Waldmann (1990) also apply to

asset pricing factors.

Specifically, we take a bottom-up approach to measure mutual funds’ flow-induced

trading of asset pricing factors. We first show that the flow-induced trades have large

price impacts on contemporaneous factor returns, which fully revert afterward. This

27

evidence justifies that the flow-induced trades of factors are mostly uninformed. We then

show that the uninformative flow movements strongly forecast variation and covariation

among the asset pricing factors, indicating that factors are subject to noise trader risk.

More importantly, in both the time-series and the cross-sections, we find that arbitrageurs

and other investors require higher factor premia when the flow-driven noise trader risk is

expected to be more salient.

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33

Table 1: Summary Statistics

This table reports the summary statistics of mutual funds, stocks, and factors inour sample. Panel A reports the summary statistics of the US equity mutual funds inour study. # Funds is the number of distinct mutual funds in each period. TNA is theaverage fund total net assets (in million $). % Coverage of stock (EW) is the number ofdistinct stocks held by mutual funds in our sample, divided by the total number of CRSPstocks. % Coverage of stock (VW) is the total market capitalization of distinct stocksheld by mutual funds in our sample, divided by total market capitalization of CRSPstocks. % Market is the average percentage of the US common stocks held by the mutualfunds in our sample. Panel B reports the stock and factor characteristics. Size andbook-to-market ratio of our sample stocks are shown in NYSE percentiles. Stock-levelflow-induced trading (FIT) is defined in (1). Flow-induced trading of factor (FITOF) isdefined as the value-weighted FIT of a factor’s long-leg stocks minus that of the short-legstocks in (2). The definitions of factor-level square root of fragility (

√Fragility) and

factor pairwise co-fragility are in Section 4.1. The list of factors is in Table C.1.

Panel A: Summary statistics of mutual funds

Period # Funds TNA % Coverage of stock % Market

Median Mean EW VW

1980-1984 370 64.43 159.62 46.30 94.45 2.621985-1989 610 79.18 264.52 58.58 97.15 4.331990-1994 1,453 71.81 299.02 65.76 98.51 7.511995-1999 2,699 110.79 698.13 72.56 98.40 13.192000-2004 3,461 120.31 837.97 86.77 99.54 15.352005-2009 3,636 172.73 1,097.92 92.77 99.61 18.522010-2014 2,875 297.49 1,664.22 91.29 98.44 18.612014-2017 2,216 479.88 2,757.40 94.34 99.15 20.07

Panel B: Summary statistics of stocks and factors

Variables Mean SD Q1 Median Q3

Stock level:Size 0.3105 0.2923 0.0547 0.2143 0.5190Book-to-Market 0.4839 0.3033 0.2088 0.4801 0.7534FIT 0.0157 0.1196 −0.0195 0.0017 0.0302Factor level:Quarterly Ret 0.0078 0.0654 −0.0270 0.0048 0.0388FITOF 0.0003 0.0177 −0.0053 0.0001 0.0055SD of Daily Ret 0.0060 0.0041 0.0037 0.0049 0.0068√

Fragility 0.0012 0.0014 0.0005 0.0008 0.0015Factor-pair level:Cov of Daily Ret (10−6) 0.4683 6.0821 −0.5313 0.1116 0.8814Co-Fragility (10−6) 0.0097 0.7377 −0.0377 0.0021 0.0514

34

Tab

le2:

Retu

rnpatt

ern

of

flow

-in

duce

dtr

adin

gof

fact

ors

This

table

rep

orts

the

per

form

ance

offa

ctor

por

tfol

ios

sort

edby

flow

-induce

dtr

adin

gof

fact

or(F

ITO

F).

FIT

OF

isth

ep

ortf

olio

-wei

ghte

dflow

-induce

dtr

adin

g(F

IT)

ofa

fact

or’s

long-

leg

stock

sm

inus

that

ofth

esh

ort-

leg

stock

s(s

eeeq

uat

ion

(2))

.In

each

quar

tert,

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the

70fa

ctor

sin

toth

ree

grou

ps

bas

edon

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rF

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ance

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ter

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ith

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an(1

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turn

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tfol

ios

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eren

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ters

.P

anel

sA

toD

rep

ort

aver

age

mon

thly

raw

retu

rns,

mon

thly

CA

PM

alpha,

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rench

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orte

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vela

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Portfolio

Qtr

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1-4

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5-8

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9-12

Portfolio

Qtr

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1-4

Qtr

5-8

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9-12

PanelA:M

onth

lyRaw

Retu

rnPanelB:M

onth

lyCAPM

Alpha

Low

−0.2

20.4

1∗∗

∗0.4

9∗∗

∗0.

27∗∗

∗L

ow−

0.10

0.55∗∗

∗0.6

3∗∗

∗0.

42∗∗

∗

(−1.

61)

(3.4

2)(3

.86)

(2.7

0)

(−0.

70)

(4.1

2)

(4.8

7)

(4.7

9)

Mid

0.2

6∗∗

∗0.

25∗∗

∗0.2

7∗∗

∗0.

25∗∗

∗M

id0.

35∗∗

∗0.

34∗∗

∗0.3

5∗∗

∗0.

35∗∗

∗

(4.5

5)(4

.48)

(4.5

7)

(4.1

8)

(6.6

5)

(6.4

9)

(6.3

0)

(5.7

1)

Hig

h0.7

6∗∗

∗0.

16∗∗

0.0

50.2

4∗∗

∗H

igh

0.85∗∗

∗0.

23∗∗

∗0.1

3∗∗

0.33∗∗

∗

(8.0

4)(2

.21)

(0.8

1)

(3.4

1)

(8.5

3)

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0)

(2.0

0)

(3.7

3)

H−

L0.

97∗∗

∗−

0.25

∗−

0.44∗

∗∗−

0.03

H−

L0.

95∗∗

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0.32∗

−0.

50∗

∗∗−

0.08

(5.1

8)(−

1.74

)(−

2.87)

(−0.

27)

(4.5

1)

(−1.

90)

(−3.

04)

(−0.

66)

PanelC:M

onth

lyFFC4Alpha

PanelD:M

onth

lyFF5Alpha

Low

−0.1

9∗

0.3

6∗∗

∗0.3

9∗∗

∗0.

26∗∗

∗L

ow−

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∗∗0.2

0∗∗

∗0.3

6∗∗

∗0.

23∗∗

∗

(−1.

72)

(4.9

8)(4

.13)

(2.8

5)

(−4.

43)

(2.8

3)

(3.3

8)

(2.6

8)

Mid

0.2

1∗∗

∗0.

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∗0.

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id0.

18∗∗

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∗0.1

8∗∗

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∗

(4.5

4)(5

.30)

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7)

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(3.1

6)

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(4.3

5)

(3.4

2)

Hig

h0.6

5∗∗

∗0.

100.0

50.1

8∗∗

∗H

igh

0.82∗∗

∗0.

19∗∗

∗0.0

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12∗

(5.6

6)(1

.57)

(0.8

7)

(3.2

6)

(6.5

5)

(2.5

9)

(0.5

1)

(1.7

5)

H−

L0.

84∗∗

∗−

0.26

∗∗−

0.33∗∗

−0.

08

H−

L1.

25∗∗

∗−

0.01

−0.

33∗

∗−

0.12

(3.9

9)(−

2.40

)(−

2.48)

(−0.

65)

(5.9

3)

(−0.

05)

(−2.

25)

(−0.

89)

35

Table 3: Predicting factor return covariance and volatility

Panel A reports the Fama-MacBeth regressions of one-quarter-ahead factor pair-wise return covariance (σπ1,π2t+1 ) on the factor pairwise co-fragility (Gπ1,π2

t ). σπ1,π2t+1 is thecovariance of weekly returns between factors π1 and π2 in quarter t+1. Gπ1,π2

t is theco-fragility between factors π1 and π2 measured at the end of quarter t, following thedefinition in equation (4). The control variables include the one-quarter lagged pairwisereturn covariance and pairwise differences in size, book-to-market, and momentum. PanelB reports the Fama-MacBeth regressions of one-quarter-ahead factor return volatility(σt+1) on the square root of factor fragility (

√Gt). σt+1 is the standard deviation of

weekly factor returns in quarter t+1 and√Gt is the square root of factor fragility, which

is defined in equation (3). In columns (1)-(2), we report the estimates of the full sampleperiod. In columns (3)-(4), we exclude observations in the crisis period (from 2000 to2001 and from 2007 to 2008). For easy interpretation, all variables are standardizedto have unit variance. *, **, *** indicate significance at the 10%, 5%, and 1% level,respectively.

Panel A: Predict pairwise factor return covariance

DepVar: σπ1,π2t+1 (1) (2) (3) (4)

Full Sample Exclude Crisis PeriodGπ1,π2t 0.80∗∗∗ 0.46∗∗∗ 1.06∗∗∗ 0.60∗∗∗

(4.96) (5.64) (4.55) (5.51)σπ1,π2t 0.53∗∗∗ 0.53∗∗∗

(16.18) (15.53)Size Diff −0.03∗∗∗ −0.04∗∗∗

(−7.26) (−7.14)BM Diff 0.01 0.02∗∗

(1.50) (2.27)MOM Diff −0.00 −0.01

(−0.40) (−1.38)

No. Obs. 347,760 347,760 309,120 309,120Adj. R2 0.10 0.34 0.10 0.34

Panel B: Predict factor return volatility

DepVar: σt+1 (1) (2) (3) (4)

Full Sample Exclude Crisis Period√Gt 0.27∗∗∗ 0.12∗∗∗ 0.37∗∗∗ 0.17∗∗∗

(7.49) (5.10) (9.26) (5.77)σt 0.63∗∗∗ 0.62∗∗∗

(25.54) (26.04)

No. Obs. 10,080 10,080 8,960 8,960Adj. R2 0.07 0.41 0.07 0.39

36

Table 4: Correlation between aggregate fragility and other sentiment measures

This table reports the pairwise correlations among aggregate fragility and severalsentiment measures. Aggregate fragility is the fragility calculated based on the equal-weighted portfolio of 70 factors in each quarter. Avg Covariance is the average pairwisereturn covariance of the factors in a given quarter. BW Sentiment is the investorsentiment index from Baker and Wurgler (2006) in the last month of a given quarter.Avg Value Spread is the average value spread of the factors at the end of a givenquarter. The value spread of a factor is computed as the log difference between theportfolio-weighted book-to-market ratio of the long-leg and the short-leg. Avg FactorRet is the equal-weighted average quarterly returns of the factors in a given quarter.

(1) (2) (3) (4) (5)

(1) Aggregate Fragility 1.00

(2) Avg Covariance 0.23 1.00

(3) BW Sentiment 0.15 0.78 1.00

(4) Avg Value Spread 0.30 0.41 0.25 1.00

(5) Avg Factor Ret 0.27 0.33 0.16 0.75 1.00

37

Table 5: Aggregate fragility and future average factor premia

This table reports the estimation results from the predictive regressions of averagefactor premia on aggregate fragility. The dependent variable is the equal-weightedaverage quarterly returns (in percent) of the 70 factors in quarter t+ 1. The independentvariables include the fragility calculated on the equal-weighted portfolio of the 70 factorsin quarter t (Aggregate Fragility), the average pairwise daily return covariance of thefactors in quarter t (Avg Covariance), the investor sentiment index from Baker andWurgler (2006) in the last month of quarter t (BW Sentiment), the average value spreadof the factors at the end of quarter t, and average quarterly returns of the factors inquarter t. The sample period is from 1982Q1 to 2017Q4. For easy interpretation,all independent variables are standardized to have unit variance. The t-statistics inparentheses are computed based on standard errors with Newey-West correction of fourlags. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively.

(1) (2) (3) (4) (5) (6) (7) (8)

Aggregate Fragility 0.60∗∗∗ 0.56∗∗∗ 0.45∗∗∗ 0.42∗∗∗ 0.43∗∗

(3.22) (3.11) (2.71) (3.00) (2.41)Avg Covariance 0.32∗ 0.19 −0.10

(1.83) (1.43) (−0.44)BW Sentiment 0.70∗∗∗ 0.58∗∗∗ 0.25

(3.60) (2.93) (0.82)Avg Value Spread 0.74∗∗∗ 0.62∗∗∗ 0.42

(3.02) (2.90) (1.03)Avg Factor Ret 0.15

(0.65)

No. Obs. 144 144 144 144 144 144 144 144Adj. R2 0.08 0.02 0.11 0.12 0.08 0.15 0.15 0.15

38

Table 6: Bias adjustment of the predictive regression in Table 5

This table reports the analyses of small-sample bias in the predictive regression ofaverage factor premia on aggregate fragility. The dependent variable is the equal-weighted average quarterly returns (in percent) of the 70 factors in a given quarter t+ 1,and the independent variable is the fragility calculated on the equal-weighted portfolioof the 70 factors in quarter t (Aggregate Fragility). The sample period is from 1982Q1to 2017Q4. Panel A reports the OLS estimates from the following two equations: AvgFactor Rett+1 = a + b × Aggregate Fragilityt + ut+1 and Aggregate Fragilityt+1 = c + d× Aggregate Fragilityt + vt+1. Panel B reports correlations or standard deviations(shown in brackets) of the innovations in the two regressions above. Panel C reports thecoefficient estimates and t-statistics (shown in parentheses) of the predictive regressionbased on the bias-reduction estimation approach in Amihud and Hurvich (2004). *, **,*** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Original OLS estimates

a b c d

0.58∗∗∗ 0.60∗∗∗ 0.45∗∗∗ −0.07

(3.19) (3.63) (5.00) (−0.86)

Panel B: Correlation [SD]

u v

u [1.98] 0.06

v [1.00]

Panel C: Bias-adjusted estimates

ac bc

0.58∗∗∗ 0.60∗∗∗

(3.19) (3.62)

39

Table 7: Robustness checks for predicting average factor premia

This table reports several robustness checks for the predictive regressions of aver-age factor premia on factor co-fragility. First, we re-construct factors through formingvalue-weighted long-short portfolios with NYSE decile breakpoints of characteristicvariables. All factor-level variables are defined following Table 5 but are constructedusing NYSE decile long-short portfolios. The regression results are reported in columns(1) and (2). We also detrend the key independent variable Aggregate Fragility in theregressions. Specifically, in the sample period of 1982Q1-2017Q4, we regress AggregateFragility on a year-quarter time indicator and use the residuals as linear-detrended AvgCo-Fragility. Similarly, we regress Aggregate Fragility on a year-quarter time indicatortogether with its square term and use the residuals as quadratic-detrended AggregateFragility. Regression results with the detrended Aggregate Fragility are reported incolumns (3)-(6). All independent variables are standardized to have unit variance. Thet-statistics in parentheses are computed based on standard errors with Newey-Westcorrection of four lags. *, **, *** indicate significance at the 10%, 5%, and 1% level,respectively.

(1) (2) (3) (4) (5) (6)NYSE Decile Portfolio Detrend-Linear Detrend-Quadratic

Aggregate Fragility 0.74∗∗∗ 0.51∗∗ 0.61∗∗∗ 0.43∗∗ 0.60∗∗∗ 0.43∗∗

(3.22) (2.39) (3.25) (2.39) (3.36) (2.43)Avg Covariance −0.01 −0.09 −0.09

(−0.03) (−0.43) (−0.39)BW Sentiment 0.55∗ 0.25 0.23

(1.71) (0.82) (0.74)Avg Value Spread 0.30 0.41 0.43

(0.75) (1.02) (1.07)Avg Factor Ret 0.05 0.14 0.15

(0.23) (0.64) (0.65)

No. Obs. 144 144 144 144 144 144Adj. R2 0.08 0.15 0.08 0.15 0.08 0.15

40

Table 8: Out-of-sample tests of forecasting average factor premia

This table reports the statistics of the out-of-sample forecast errors for averagefactor premia at a quarterly frequency. We calculate the out-of-sample test statistics,R2

OOS and ∆RMSE, following Welch and Goyal (2007) to compare the predictiveregression forecast against the unconditional mean forecast. A star next to the estimatesof R2

OOS is based on the critical values of the MSE-F statistic given by McCracken (2007).The MSE-F statistic tests the equivalence of MSE of the unconditional mean forecastand the conditional forecast. The definition of each predictor is the same as in Table5. All numbers are in percent. *, **, *** indicate significance at the 10%, 5%, and 1%level, respectively.

Predictor R2OOS ∆RMSE MSE F

Aggregate Fragility 8.07∗∗∗ 0.09 12.55

Avg Covariance −16.11 −0.18 −19.84

BW Sentiment 9.47∗∗∗ 0.11 14.96

Avg Value Spread 8.03∗∗∗ 0.09 12.48

Avg Factor Ret 0.11 0.00 0.15

41

Tab

le9:

Pre

dic

ting

avera

ge

fact

or

pre

mia

for

fact

ors

form

ed

wit

hsu

bse

tsof

stock

s

This

table

rep

orts

the

resu

lts

from

the

pre

dic

tive

regr

essi

ons

ofav

erag

efa

ctor

pre

mia

inT

able

s5

and

8,w

her

efa

ctor

sar

eco

nst

ruct

edusi

ng

stock

sw

ith

diff

eren

thed

gefu

nd

trad

ing

volu

mes

orm

arke

tca

p.

Inco

lum

ns

(1)-

(4),

we

sort

the

stock

sin

the

long

orsh

ort

leg

ofa

fact

orin

totw

oev

engr

oups

bas

edon

thei

rhed

gefu

nd

trad

ing

volu

me

inth

epre

vio

us

year

.F

orea

chfa

ctor

,w

ere

-con

stru

ctva

lue-

wei

ghte

dlo

ng-

shor

tp

ortf

olio

susi

ng

stock

sin

the

low

and

hig

hhed

ge-f

und-t

radin

g-vo

lum

egr

oup,

resp

ecti

vely

.Sim

ilar

ly,

inco

lum

ns

(5)-

(8),

we

sort

the

long-

leg

orsh

ort-

leg

stock

sin

tosm

all-

cap

and

larg

e-ca

pgr

oups

bas

edon

thei

rm

arke

tca

pit

aliz

atio

nat

the

pre

vio

us

year

-end

and

re-c

onst

ruct

fact

ors

usi

ng

smal

l-ca

pan

dla

rge-

cap

stock

s,re

spec

tive

ly.

All

fact

or-l

evel

and

fact

or-p

air

leve

lva

riab

les

are

re-c

onst

ruct

edfo

llow

ing

the

defi

nit

ions

inT

able

5.F

orea

syin

terp

reta

tion

,al

lin

dep

enden

tva

riab

les

are

stan

dar

diz

edto

hav

eunit

vari

ance

.T

het-

stat

isti

csin

par

enth

eses

are

com

pute

dbas

edon

stan

dar

der

rors

wit

hN

ewey

-Wes

tco

rrec

tion

offo

ur

lags

.*,

**,

***

indic

ate

sign

ifica

nce

atth

e10

%,

5%,

and

1%le

vel,

resp

ecti

vely

.

PanelA:In

-sample

Regre

ssions

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Low

HF

Tra

de

Hig

hH

FT

rad

eS

mal

lM

ktc

apL

arg

eM

ktc

ap

Agg

regate

Fra

gil

ity

0.0

5-0

.03

0.59∗∗∗

0.42∗∗

−0.

16−

0.21∗

0.58∗∗∗

0.4

2∗∗

(0.2

5)(-

0.19

)(2

.74)

(2.1

4)(−

1.23

)(−

1.94)

(2.6

2)

(2.0

9)

Avg

Cov

aria

nce

−0.0

4−

0.07

−0.

39∗

−0.0

8(−

0.4

4)(−

0.29

)(−

1.83)

(−0.3

6)B

WS

enti

men

t0.4

4∗∗

0.20

0.6

8∗∗∗

0.24

(2.5

5)(0

.67)

(3.0

0)(0

.79)

Avg

Val

ue

Sp

read

0.3

4∗0.

440.

50∗

0.42

(1.8

7)(1

.07)

(1.7

0)(1

.05)

Avg

Fact

orR

et−

0.0

00.

13−

0.26

0.1

4(−

0.0

0)(0

.60)

(−0.

97)

(0.6

2)

No.

Ob

s.144

144

144

144

144

144

144

144

Ad

j.R

2−

0.01

0.08

0.07

0.14

0.00

0.16

0.07

0.1

4

PanelB:Out-of-sa

mple

Tests

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

Low

HF

Tra

de

Hig

hH

FT

rad

eS

mal

lM

ktc

apL

arg

eM

ktc

ap

R2 O

OS

∆R

MS

ER

2 OO

S∆

RM

SE

R2 O

OS

∆R

MS

ER

2 OO

S∆

RM

SE

Agg

regate

Fra

gil

ity−

1.19

−0.0

17.2

5∗∗∗

0.08

−0.

300.0

07.

27∗∗∗

0.09

42

Table 10: Cross-sectional predictive regressions of factor returns

This table reports the results of the cross-sectional Fama-MacBeth regressions.The dependent variable is the return (in percent) of a factor in quarter t+1. Co-Fragilityis the co-fragility between a given factor and the equal-weighted portfolio of 70 factorsin the prior quarter. Rank is the quintile ranking of Co-Fragility in a given quarter.Dummy Rank5 is a dummy that equals one for factors in the highest Co-Fragilityquintile and zero otherwise. Covariance is the daily return covariance between a givenfactor and equal-weighted portfolio of the 70 factors in the prior quarter. Other controlsinclude factor returns in quarter t, the value spread of a given factor at the end of theprior quarter, and the average FITOF over the past eight quarters. We standardize theindependent variables by their cross-sectional standard deviations each quarter, exceptfor Rank and Rank5. Columns (1) to (4) report the regression results in the full sample.Columns (5) to (6) report the regression results among factors in the lowest and highestCo-Fragility quintile each quarter. t-statistics in parentheses are computed based onstandard errors with Newey-West correction of 12 lags. *, **, *** indicate significanceat the 10%, 5%, and 1% level, respectively.

(1) (3) (2) (4) (5) (6)

Full Sample Extreme Quintiles

Co-Fragility 0.51∗∗∗ 0.30∗∗

(2.57) (2.22)Rank 0.33∗∗ 0.18∗∗

(2.42) (2.09)Dummy Rank5 1.59∗∗∗ 0.99∗∗

(2.70) (2.42)

Covariance 0.18 0.18 0.06(0.89) (0.83) (0.26)

Past one-quarter return 0.24 0.24 0.27(1.41) (1.47) (1.50)

Value Spread 0.03 0.06 0.03(0.08) (0.28) (0.20)

Past eight-quarter FITOF −0.27∗ −0.27∗ −0.39∗

(−1.89) (−1.86) (−1.91)

Adj. R2 0.12 0.11 0.44 0.43 0.19 0.49

43

PortfolioFormation

-1 0 1 2 3 4 5 6 7 8

Event Quarter

0%

1%

2%

3%

4%

5%

6%C

um

ula

tive

FF

5 A

lph

aFITOF Hedge Portfolio

Figure 1: Cumulative FF5 alpha of the FITOF-hedge portfolio. This figure plotsthe cumulative FF5 alpha of the long-short portfolio ranked by FITOF. FITOF measuresthe mutual fund flow-induced trading for a given factor (see equation (2)). At eachquarter-end, the 70 factors are sorted into three groups based on FITOF in ascendingorder (20/30/20 factors are assigned into the low/mid/high group, respectively). Eachfactor is given equal weight in the portfolios, and the portfolios are held for three years.The long-short portfolio goes long in the high FITOF group and short in the low FITOFgroup. Quarter 0 is the portfolio formation quarter.

44

1982Q1 1990Q1 2000Q1 2010Q1 2017Q4

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

SM

B V

ola

tilit

y

0.000

0.002

0.004

0.006

SM

B F

rag

ility

SMB FragilitySMB Volatility

1982Q1 1990Q1 2000Q1 2010Q1 2017Q4

0.005

0.010

0.015

0.020

HM

L V

ola

tilit

y

0.000

0.001

0.002

0.003

0.004

0.005

HM

L F

rag

ility

HML FragilityHML Volatility

Figure 2: Factor volatility and square root of lagged factor fragility. This figureplots the return volatility and the square root of lagged factor fragility of the Fama-Frenchsize and value factors. Factor return volatility is measured as the standard deviation ofweekly factor returns over a given quarter, and factor fragility is defined in equation (3).

45

Q1 Q2 Q3 Q4 Q5 Q5-Q1

Quintile and Hedge Portfolios

0.00%

0.25%

0.50%

0.75%

1.00%M

on

thly

CA

PM

Alp

ha

Figure 3: Monthly CAPM alpha of portfolios sorted on factor-level Avg Co-Fragility. Each quarter-end, we sort the 70 factors into quintiles based on their averageco-fragility with the equal-weighted portfolio of the 70 factors (Co-Fragility) in the quar-ter, and we hold the portfolios in the next quarter. This figure shows the average monthlyCAPM alpha of the quintile portfolios (leftmost five bars) and the “quintile five minusquintile one” long-short portfolio (rightmost bar) in the holding period of January 1982to December 2017. Limit lines around the bars show the 95% confidence intervals forestimates of monthly CAPM alpha.

46

1 2 3

Size Terciles

-0.05

0.10

0.25

0.40

0.55

Co

effi

cien

t E

stim

ates

-0.3

0.6

1.5

2.4

3.3

T-s

tati

stic

T-statisticCoefficient Estimates

1 2 3

Size Terciles

-3.0%

-1.5%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

OO

S R

-Sq

uar

ed

-0.03%

-0.02%

0.00%

0.02%

0.03%

0.05%

0.06%

0.08%

Del

ta R

MS

E

Delta RMSEOOS R-Squared

Figure 4: Predicting average factor premia for factors formed by stocks withdifferent market cap. We sort the stocks in the long or short leg of a factor into tercilesbased on their market capitalization at previous year-end (Size tercile 1 to 3 from smallestto largest) and re-construct factors using stocks in each of the size terciles separately. Wethen conduct the in-sample and out-of-sample tests across the size terciles (see Table 9for example). The top panel shows the coefficient estimate of Aggregate Fragility in theunivariate regression of predicting one-quarter-ahead average factor premia across sizeterciles. The bottom panel shows OOS test statistics (R2

OOS and ∆RMSE) across sizeterciles.

47

Appendix

A Derivation of Factor fragility and factor co-fragility

In this section, we describe how we derive the measures of factor fragility (co-fragility).

Similar to Greenwood and Thesmar (2011) (GT), we assume the following relationship

between mutual fund flow-induced trading and return of stock j:

rj,t = αj + λj

∑k Sharesk,j,t−1fk,tPSFk,t∑

k Sharesk,j,t−1

+ εj,t. (14)

Here, rj,t is the return of stock j in quarter t, Sharesk,j,t−1 is the number of shares of

stocks j held by fund k at the end of quarter t− 1, fk,t is the percentage flow of fund k in

quarter t, and PSF is the partial scaling factor as in (1). αj and λj are two parameters.

In our implementation, we assume that λj = λ∑

k Sharesk,j,t−1/Shroutj,t−1, where λ is

the unconditional price impact factor and Shroutj,t−1 is shares outstanding of stock j at

the end of quarter t− 1. The residual term, εj,t, has a conditional mean of zero and may

capture other sources of variation of returns (e.g., news about fundamentals).

Factors are value-weighted portfolios of stocks and thus the return of factor π can be

expressed as:

rπ,t =∑j

µπj,t−1rj,t, (15)

where µπj,t−1 is the weight of stock j in factor π in quarter t.27 Combining (14) and (15),

we get

rπ,t =∑j

µπj,t−1αj + λ

(∑k

wπk,t−1fk,tPSFk,t

)+∑j

µπj,t−1εj,t, (16)

where wπk,t−1 =∑

j µπj,t−1Sharesk,j,t−1/Shroutj,t−1 can be regarded as the weight of mutual

fund k in factor π in quarter t.

27For a long-leg stock, µπj,t equals its original weight in the long leg. For a short-leg stock, µπj,t is itsoriginal weight in the short leg multiplied by negative one.

48

Based on equation (16), the conditional variance and covariance of rπ,t+1 at the end

of quarter t are

Vart(rπ,t+1) = λ2W πt

′Et(Ωt+1)W π

t + Vart

(∑j

µπj,tεj,t+1

)(17)

and

Covt(rπ1,t+1, rπ2,t+1) = λ2W π1t

′Et(Ωt+1)W π2

t + Covt

(∑j

µπ1j,tεj,t+1,∑j

µπ2j,tεj,t+1

), (18)

respectively. Here, Et(Ωt+1) is the conditional variance-covariance matrix of mutual fund

flows in quarter t + 1 and W πt =

(wπ1,t, . . . , w

πK,t

)is the vector of mutual fund weights in

factor π.

Similar to GT, we define “factor fragility” of factor π in quarter t as

Gπt = W π

t

′Et(Ωt+1)W π

t . (19)

Likewise, we define co-fragility between factor π1 and factor π2 to be

Gπ1,π2t = W π1

t

′Et(Ωt+1)W π2

t . (20)

To estimate Et(Ωt+1), we calculate the variance-covariance matrix of mutual fund flows

using observations in the most recent eight quarters (including quarter t). The summary

statistics of fragility and co-fragility are reported in Table 1.

B Additional Results

This section provides supplementary information for our main results.

Table B.1 reports the quarterly transition matrix for the factor quintile portfolios

sorted on FITOF. The probability that a factor stays in the same quintile over two

49

consecutive quarters is 31% to 50%. This suggests that FITOF is not highly persistent.

Table B.2 shows the relation between FITOF and factor returns in Panel and Fama-

MacBeth regressions. After accounting for factor and time fixed effects, we still find a

strong positive relation between factor return and contemporaneous FITOF and a sizable

negative relation between factor return and average FITOF in past five- to eight-quarter.

This result confirms our findings in Table 2. Furthermore, we run the regressions of

factor returns on contemporaneous or past FITOF in the first- and second-half sample

periods separately. We find that the price effect of FITOF is stronger in the second-half

sample period (1999-2017). This is consistent with the dramatic growth of mutual fund

industries, which is also documented in the summary statistics (see Table 1).

Table B.3 examines the return pattern of factor portfolios sorted on factor returns or

mutual funds’ non-FIT trade. The portfolio analysis procedure is the same as that in

Table 2, and the only difference is the sorting variable. When sorting on factor returns,

we do not find reversals in the three years after portfolio formation. This ensures that

the reversal pattern associated with FITOF is not driven by the mean-reversion of factor

returns. We also sort on mutual funds’ non-FIT trade. We take a bottom-up approach

to compute mutual funds’ non-FIT trade of factors. First, at stock-level, we calculate

mutual funds’ non-FIT trade on a stock as mutual funds’ aggregate realized trade scaled

by the total number of shares held by mutual funds (dubbed by RT) minus flow-induced

trading (see 1). Second, we compute mutual funds’ non-FIT trade of a factor as the

portfolio-weighted average non-FIT trade of stocks that constitute the factor. We find no

statistically significant return patterns associated with mutual funds’ non-flow-induced

trades on factors. This highlights the unique non-fundamental feature of mutual funds’

flow-induced trading on factors.

In Table B.4 and Table B.5, we isolate fund flows that are driven by fund alphas and

re-construct our key measures (FITOF, factor fragility, and factor co-fragility) with the

alpha-isolated fund flows. In these robustness tests, the return patterns of FITOF and the

50

predictive power of factor fragility (co-fragility) on factor return volatility (covariance)

are close to those in the main tables.

Table B.6 shows the performance of a trading strategy that longs factors with low

past FITOF and shorts factors with high past FITOF. We compute past FITOF for a

given factor as follows. For each stock in the long-short portfolio of the factor in a holding

period that belongs to quarter t, we compute its average FIT during Qtr t − 5 to Qtr

t− 8. Then we compute portfolio-weighted average past FIT for each factor based on its

portfolio composition in quarter t as past FITOF. At the beginning of each quarter, we

sort the 70 factors into quintiles by their past FITOF and long (short) the lowest (highest)

past FITOF quintile for one quarter. Each factor is given equal weight in the portfolio,

and the portfolios are rebalanced quarterly. During April 1982 to December 2017, such a

trading strategy generates an average monthly raw returns of 0.79% (t-statistic = 3.05)

and a CAPM alpha of 0.97% (t-statistic = 3.53).

We conduct several robustness tests for Table 5. First, we decompose the aggregate

fragility of the 70-factor portfolio into average pair-wise co-fragility of the 70 factors (Avg

Co-Fragility) and average fragility of the 70 factors (Avg Fragility). We then re-conduct

the regression analysis as in Table 5 but with the two new variables. Table B.7 shows that

only the Avg Co-Fragility can positively and significantly predict future average factor

premia. Second, in Table B.8, we conduct a placebo test using average co-fragility to

predict stock market returns and bond market returns. We find no results in placebo

tests.

Table B.9 considers alternative evaluation periods for the out-of-sample test and finds

that the OOS performance is robust. It is worthy to note that the OOS performance of

aggregate fragility is stronger in the later periods.

Finally, we conduct two robustness checks for Table 10. First, we use portfolio sorting

analysis to show the positive return predictability of Co-Fragility. At each quarter-end,

we sort factors into quintiles based on Co-Fragility in the quarter and hold the quintile

51

portfolios in the next quarter. Table B.10 shows that factor returns (alphas) increase

monotonically with factor-level average co-fragility. Second, we decompose Co-Fragility

into a factor’s average co-fragility with the rest of the factors and the factor’s own fragility,

and we re-conduct the regression analysis in Table 10 using these two variables as key

independent variables seperately. Table B.11 shows that only a factor’s average co-fragility

with the rest of the factors can positively and significant forecasts its future returns.

Table B.1: Transition matrix of FITOF quintile portfolios

In each quarter, we sort the 70 factors into quintiles based on the FITOF in thatcurrent quarter. This table reports the quarter-to-quarter transition likelihood for theFITOF quintile ranking.

Rank Qtr t+ 1 →Rank Qtr t ↓ 1 2 3 4 5

1 0.49 0.22 0.12 0.08 0.08

2 0.22 0.33 0.22 0.13 0.10

3 0.11 0.23 0.31 0.23 0.11

4 0.08 0.13 0.22 0.35 0.20

5 0.09 0.09 0.12 0.20 0.50

52

Table B.2: Regression analysis of factor return on FITOF

This table reports the regression of factor return on contemporaneous and pastFITOF. The dependent variable is the monthly factor return (in percent) in a givenmonth. Panel A reports the regressions of monthly factor returns on the flow-inducedtrading of factor (FITOF) in the contemporaneous quarter. FITOF is the value-weightedflow-induced trading (FIT) of a factor’s long-leg stocks minus that of the short-legstocks. Panel B reports the regressions of monthly factor returns on past FITOF. In agiven quarter, past FITOF refers to the average FITOF in the period of qtr t− 5 to qtrt − 8. Columns (1)-(2) report the regression results based on the full sample period ofApr 1982 to Dec 2017. Columns (3)-(6) report the regression results in two sub-periods:1982-1999 (first-half) and 1999-2017 (second-half). Regression method “Panel” refersto panel regression where time and factor fixed effects are included, and t-statistics arecomputed based on standard errors double clustered by factor and time. The regressionmethod “FM” refers to the Fama-MacBeth regression. *, **, *** indicate significance atthe 10%, 5%, and 1% level, respectively.

Panel A: Factor return and contemporaneous FITOF(1) (2) (3) (4) (5) (6)

Full Sample First Second First Second

Contemporaneous FITOF 0.24∗∗∗ 0.44∗∗∗ 0.15∗∗∗ 0.53∗∗ 0.27∗∗∗ 0.59∗∗∗

(3.29) (3.97) (3.14) (2.48) (3.33) (3.01)

Factor FE Yes Yes YesYear-Month FE Yes Yes YesRegression Method Panel FM Panel Panel FM FM

No. Obs. 30,030 30,030 14,070 15,960 14,070 15,960Adj. R2 0.09 0.17 0.07 0.11 0.15 0.09

Panel B: Factor return and past FITOF(1) (2) (3) (4) (5) (6)

Full Sample First Second First Second

Past FITOF −0.20∗∗∗ −0.35∗∗∗ −0.15∗∗∗ −0.28∗∗ −0.29∗∗∗ −0.41∗∗∗

(−3.29) (−4.03) (−2.65) (−2.13) (−3.28) (−2.81)

Factor FE Yes Yes YesYear-Month FE Yes Yes YesRegression Method Panel FM Panel Panel FM FM

No. Obs. 30,030 30,030 14,070 15,960 14,070 15,960Adj. R2 0.08 0.10 0.06 0.09 0.19 0.08

53

Table B.3: Return pattern of portfolios sorted by factor returns or non-FITtrades. This table reports the performance of factor portfolios sorted by factor returnor non-FIT trades. We conduct the following portfolio analysis with different sortingvariables across Panels A to C: At the end of each quarter t, we sort the 70 factors intothree groups based on a given sorting variable, with 20/30/20 factors in the low/mid/highgroup respectively. Each factor is given equal weight in the respective portfolio. Theportfolios are rebalanced every quarter and held for three years. Qtr 0 is the portfolioformation quarter. We track the monthly calendar-time returns of factor portfolios fromQtr 1 to Qtr 12. We deal with overlapping portfolios in each holding month followingJegadeesh and Titman (1993). Monthly FF5 alphas (%) are reported. The t-statisticsin parentheses are computed based on standard errors with Newey-West correction fortwelve lags. In Panel A, the sorting variable is the factor return in Qtr 0. In Panel B, thesorting variable is the cumulative factor return from Qtr −3 to Qtr 0. In Panel C, thesorting variable is mutual funds’ non-FIT trade of factors. At stock-level, we calculatemutual funds’ non-FIT trade on a stock as mutual funds’ aggregate realized trade scaledby the total number of shares held by mutual funds (dubbed by RT) minus flow-inducedtrading (see equation 1). Then we compute mutual funds’ non-FIT trade of a factor asthe portfolio-weighted average non-FIT trades of stocks that constitute the factor.

Portfolio Qtr 1-4 Qtr 5-8 Qtr 9-12

Panel A: Sort on current-quarter ret

Low0.10∗∗ 0.18∗∗∗ 0.21∗∗∗

(2.51) (2.72) (3.37)

Mid0.18∗∗∗ 0.19∗∗∗ 0.19∗∗∗

(3.82) (4.54) (3.80)

High0.30∗∗∗ 0.21∗∗∗ 0.13∗∗∗

(2.79) (3.67) (2.79)

H−L0.20 0.03 −0.08

(1.44) (0.42) (−1.08)

Panel B: Sort on past four-quarter ret

Low0.09 0.24∗∗∗ 0.21∗∗

(1.52) (2.81) (2.55)

Mid0.18∗∗∗ 0.21∗∗∗ 0.17∗∗∗

(4.17) (4.64) (3.53)

High0.31∗∗∗ 0.12∗ 0.16∗∗∗

(2.73) (1.80) (2.71)

H−L0.23 −0.11 −0.06

(1.45) (−0.98) (−0.49)

Panel C: Sort on mutual funds’ non-FIT trades

Low0.25∗∗∗ 0.20∗∗∗ 0.16∗∗

(2.78) (2.70) (2.20)

Mid0.17∗∗∗ 0.21∗∗∗ 0.19∗∗∗

(4.58) (3.86) (3.36)

High0.16∗∗∗ 0.15∗∗∗ 0.19∗∗∗

(3.75) (4.10) (5.54)

H−L−0.09 −0.05 0.03

(−0.90) (−0.60) (0.42)

54

Tab

leB

.4:

Retu

rnpatt

ern

offlow

-in

duce

dtr

adin

goffa

ctors

:excl

ude

fund

flow

com

pon

ents

dri

ven

by

fun

dalp

has

To

isol

ate

fund

flow

sth

atar

edri

ven

by

fund

alpha

com

pon

ents

,w

ees

tim

ate

afu

nd-q

uar

ter

pan

elre

gres

sion

wit

hti

me

fixed

effec

ts:

The

dep

enden

tva

riab

leis

fund

flow

inquar

tert,

and

the

indep

enden

tva

riab

les

incl

ude

FF

C4

alpha

ofth

efu

nd,

esti

mat

edusi

ng

obse

rvat

ions

inth

e24

mon

ths

pri

orto

quar

tert,

and

four

lags

offu

nd

flow

from

quar

tert−

4to

quar

ter

t−

1.W

ith

the

alpha-

excl

uded

fund

flow

s,w

ere

-con

stru

ctF

ITO

Fan

dco

nduct

the

sam

ple

exer

cise

asin

Tab

le2.

Pan

elA

toP

anel

Dre

por

tsm

onth

lyra

wre

turn

s,m

onth

lyC

AP

Mal

pha,

mon

thly

Fam

a-F

rench

-Car

har

tfo

ur-

fact

oral

pha,

and

mon

thly

Fam

a-F

rench

five

-fac

tor

alpha,

resp

ecti

vely

.T

he

retu

rns

and

alphas

are

rep

orte

din

per

cent.

Thet-

stat

isti

csin

par

enth

eses

are

com

pute

dbas

edon

stan

dar

der

rors

wit

hN

ewey

-Wes

tco

rrec

tion

for

twel

vela

gs.

Portfolio

Qtr

0Qtr

1-4

Qtr

5-8

Qtr

9-12

Portfolio

Qtr

0Qtr

1-4

Qtr

5-8

Qtr

9-12

PanelA:M

onth

lyRaw

Retu

rnPanelB:M

onth

lyCAPM

Alpha

Low

−0.

28∗∗

0.41

∗∗∗

0.49∗∗

∗0.

28∗∗

∗L

ow−

0.18

0.53∗∗

∗0.

61∗∗

∗0.

41∗∗

∗

(−2.

38)

(3.3

8)(4

.01)

(2.9

7)

(−1.

54)

(3.9

3)

(4.7

6)

(4.7

2)

Mid

0.29

∗∗∗

0.26

∗∗∗

0.26∗∗

∗0.

26∗∗

∗M

id0.

39∗∗

∗0.

35∗∗

∗0.

36∗∗

∗0.

36∗∗

∗

(4.3

7)(4

.60)

(4.3

3)

(4.2

0)

(6.1

0)

(6.8

0)

(6.2

1)

(5.7

2)

Hig

h0.

77∗∗

∗0.

15∗∗

0.0

60.2

2∗∗

∗H

igh

0.87∗∗

∗0.

23∗∗

∗0.1

4∗∗

0.32∗∗

∗

(7.7

5)(2

.10)

(0.8

9)

(2.9

9)

(8.5

9)

(3.4

3)

(2.2

3)

(3.6

6)

H−

L1.

05∗∗

∗−

0.26

∗−

0.43∗

∗∗−

0.06

H−

L1.

05∗∗

∗−

0.31∗

−0.

47∗

∗∗−

0.09

(5.8

0)(−

1.73

)(−

2.89)

(−0.

57)

(5.4

9)

(−1.

75)

(−2.

83)

(−0.

70)

PanelC:M

onth

lyFFC4Alpha

PanelD:M

onth

lyFF5Alpha

Low

−0.

22∗∗

0.37

∗∗∗

0.39∗∗

∗0.

27∗∗

∗L

ow−

0.43∗∗

∗0.

21∗∗

∗0.

35∗∗

∗0.

23∗∗

∗

(−2.

14)

(4.9

1)(4

.32)

(2.8

5)

(−4.

13)

(3.1

2)

(3.6

2)

(2.7

8)

Mid

0.23

∗∗∗

0.22

∗∗∗

0.22∗∗

∗0.

23∗∗

∗M

id0.

20∗∗

∗0.

19∗∗

∗0.

18∗∗

∗0.

19∗∗

∗

(4.8

2)(5

.33)

(4.5

5)

(5.0

4)

(3.2

1)

(3.9

8)

(3.8

4)

(3.5

4)

Hig

h0.

65∗∗

∗0.

090.0

60.1

6∗∗

∗H

igh

0.80∗∗

∗0.

18∗∗

0.0

40.1

0(5

.85)

(1.3

5)(0

.99)

(3.0

4)

(6.1

3)

(2.2

9)

(0.6

3)

(1.6

0)

H−

L0.

87**

*−

0.28

∗∗−

0.33∗

∗−

0.11

H−

L1.2

2***

−0.

03

−0.

31∗

∗−

0.13

(4.3

1)(−

2.41

)(−

2.50)

(−0.

85)

(5.4

5)

(−0.

29)

(−2.

28)

(−1.

08)

55

Table B.5: Predicting factor return covariance and volatility: exclude fund flowcomponents driven by fund alphas. We exclude fund flow components driven by fundalphas as in Table B.4. With the alpha-excluded fund flows, we re-construct factor fragilityand co-fragility and repeat the analyses in Table 3. Panel A reports the Fama-MacBethregressions of one-quarter-ahead factor pairwise return covariance (σπ1,π2t+1 ) on the factorpairwise co-fragility (Gπ1,π2

t ). Panel B reports the Fama-MacBeth regressions of one-quarter-ahead factor return volatility (σt+1) on the factor fragility (

√Gt). In columns

(1)-(2), we report Fama-MacBeth estimates in the full sample period from 1981Q1 to2017Q4. In columns (3)-(4), we exclude observations in the crisis period (years 2000,2001, 2007, and 2008). For easy interpretation, all variables are standardized to have unitvariance. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Predict pairwise factor return covariance

DepVar:σπ1,π2t+1 (1) (2) (3) (4)

Full Sample Exclude Crisis PeriodGπ1,π2t 0.41∗∗∗ 0.24∗∗∗ 0.52∗∗∗ 0.30∗∗∗

(8.77) (8.27) (7.80) (7.47)σπ1,π2t 0.52∗∗∗ 0.52∗∗∗

(15.75) (15.16)Size Diff −0.03∗∗∗ −0.04∗∗∗

(−5.56) (−6.44)BM Diff 0.01 0.01∗∗

(0.87) (1.99)MOM Diff −0.00 −0.01

(−0.04) (−1.00)

No. Obs. 347,760 347,760 309,120 309,120Adj. R2 0.13 0.35 0.12 0.34

Panel B: Predict factor return volatility

DepVar: σt+1 (1) (2) (3) (4)

Full Sample Exclude Crisis Period√Gt 0.19∗∗∗ 0.07∗∗∗ 0.28∗∗∗ 0.11∗∗∗

(10.13) (5.70) (11.20) (6.34)σt 0.63∗∗∗ 0.62∗∗∗

(25.44) (26.59)

No. Obs. 10,080 10,080 8,960 8,960Adj. R2 0.07 0.41 0.08 0.39

56

Table B.6: Trading strategy based on flow-induced trades of factors.

This table reports the performance of factor portfolios sorted by past FITOF. Wecompute past FITOF for a given factor in quarter t as follows. For each factor compo-nent stock in a holding period that belongs to quarter t, we compute its average FITduring Qtr t − 5 to Qtr t − 8. Then we compute portfolio-weighted average past FITfor each factor based on its portfolio composition in quarter t as past FITOF. At thebeginning of each quarter, we sort the 70 factors into quintiles by their past FITOFand hold the equally-weighted portfolio for one quarter. This table reports the monthlyrisk-adjusted returns for each factor portfolio in the holding period of April 1982 toDecember 2017. t-statistics are computed based on standard errors with Newey-Westcorrection for twelve lags.

Portfolio Excess CAPM FFC4 FF5

1 (L)0.63∗∗∗ 0.81∗∗∗ 0.53∗∗∗ 0.48∗∗∗

(3.66) (4.58) (3.81) (3.07)

20.46∗∗∗ 0.61∗∗∗ 0.38∗∗∗ 0.33∗∗∗

(3.83) (5.19) (4.09) (3.66)

30.28∗∗∗ 0.38∗∗∗ 0.19∗∗∗ 0.17∗∗

(3.85) (4.89) (3.87) (2.04)

40.14∗∗∗ 0.22∗∗∗ 0.14∗∗ 0.12∗

(2.59) (4.14) (2.52) (1.74)

5 (H)−0.16 −0.16 −0.12 −0.13

(−1.51) (−1.43) (−1.15) (−1.07)

1− 50.79∗∗∗ 0.97∗∗∗ 0.64∗∗∗ 0.61∗∗

(3.05) (3.53) (2.82) (2.24)

57

Table B.7: Predict average factor premia by Avg Co-Fragility and Avg Fragility

This table reports the estimation results from the predictive regressions of averagefactor premia on the two components of Aggregate Fragility: Avg Co-Fragility and AvgFragility. The dependent variable is the equal-weighted average quarterly returns (inpercent) of the 70 factors in quarter t + 1. In Panel A, the key independent variableis the average pairwise co-fragility of the 70 factors in quarter t (Avg Co-Fragility). InPanel B, the key independent variable is the average fragility of the 70 factors in quartert (Avg Fragility). Control variables and sample period are the same as Table 5. Foreasy interpretation, all independent variables are standardized to have unit variance.The t-statistics in parentheses are computed based on standard errors with Newey-Westcorrection of four lags. *, **, *** indicate significance at the 10%, 5%, and 1% level,respectively.

Panel A: Avg Co-Fragility as Predictor

(1) (2) (3) (4) (5)

Avg Co-Fragility 0.54∗∗∗ 0.50∗∗∗ 0.40∗∗∗ 0.37∗∗∗ 0.38∗∗

(3.14) (3.21) (2.65) (2.89) (2.47)Avg Covariance 0.23 −0.07

(1.60) (−0.34)BW Sentiment 0.61∗∗∗ 0.27

(2.95) (0.88)Avg Value Spread 0.64∗∗∗ 0.42

(2.82) (1.02)Avg Factor Ret 0.13

(0.60)

No. Obs. 144 144 144 144 144Adj. R2 0.07 0.08 0.15 0.16 0.17

Panel B: Avg Fragility as Predictor

(1) (2) (3) (4) (5)

Avg Fragility 0.16 0.14 0.11 0.11 0.10(0.52) (0.47) (0.42) (0.47) (0.41)

Avg Covariance 0.31∗ −0.02(1.83) (-0.09)

BW Sentiment 0.69∗∗∗ 0.31(3.75) (1.05)

Avg Value Spread 0.73∗∗∗ 0.49(3.16) (1.09)

Avg Factor Ret 0.07(0.32)

Observations 144 144 144 144 144Adj. R2 0.00 0.01 0.11 0.12 0.11

58

Table B.8: Placebo test: Predicting other returns

This table reports the predictive regressions of one-quarter-ahead quarterly S&P500excess return, long-term yield, and T-bill rate on aggregate fragility. Definitions of thedependent variables follow Welch and Goyal (2007). *, **, *** indicate significance atthe 10%, 5%, and 1% level, respectively.

(1) (2) (3)DepVar: S&P 500 Long-term yield T-bill rate

Aggregate Fragility −1.09 −0.00 −0.00(−1.61) (−1.56) (−0.92)

Avg Covariance −1.56∗ −0.01∗∗∗ −0.01∗∗∗

(−1.89) (−5.84) (−5.38)BW Sentiment −0.22 0.02∗∗∗ 0.02∗∗∗

(−0.30) (10.92) (9.64)Stock market variance 0.52 0.01∗∗∗ 0.00∗

(0.68) (2.93) (1.95)

No. Obs. 144 144 144Adj. R2 0.05 0.46 0.40

Table B.9: OOS tests of forecasting average factor premia with differentevaluation periods

This table reports the OOS performance of average co-fragility in the forecasts ofone-quarter-ahead average factor premia. We report OOS performance in three differentevaluation periods, as indicated in the first column. All numbers are in percent. *, **,*** indicate significance at the 10%, 5%, and 1% level, respectively.

OOS Period R2OOS ∆RMSE MSE F

1987Q1-2017Q4 (5 years from sample start) 7.42∗∗∗ 0.08 11.47

1997Q1-2017Q4 (15 years from sample start) 8.31∗∗∗ 0.11 12.96

2000Q1-2017Q4 8.81∗∗∗ 0.12 13.82

59

Table B.10: Performance of factor portfolios sorted on Co-Fragility

This table reports the performance of factor portfolios sorted by past Co-Fragility.The definition of Co-Fragility follows Table 10. At the beginning of each quarter, wesort the 70 factors into quintiles by their Co-Fragility in previous quarters and hold theportfolios for one quarter. Each factor is given equal weight in the portfolio. This tablereports the average monthly raw returns, CAPM alpha, Fama-French-Carhart four-factoralpha, and Fama-French five-factor alpha for each factor portfolio in the holding periodof January 1982 to December 2017. t-statistics are computed based on standard errorswith Newey-West correction for twelve lags.

Portfolio Excess CAPM FFC4 FF5

1 0.01 0.02 -0.04 0.00(0.15) (0.27) (−0.50) (−0.03)

2 0.24∗∗∗ 0.30∗∗∗ 0.16∗∗∗ 0.20∗∗∗

(3.65) (4.30) (3.00) (2.70)3 0.26∗∗∗ 0.38∗∗∗ 0.21∗∗∗ 0.18∗∗∗

(3.48) (5.94) (3.42) (3.14)4 0.32∗∗∗ 0.46∗∗∗ 0.28∗∗∗ 0.19∗∗∗

(3.76) (6.02) (4.34) (2.84)5 0.52∗∗∗ 0.69∗∗∗ 0.52∗∗∗ 0.39∗∗∗

(4.31) (6.10) (5.10) (3.43)

5− 1 0.51∗∗∗ 0.67∗∗∗ 0.56∗∗∗ 0.39∗∗

(2.74) (3.73) (3.24) (2.01)

60

Table B.11: Cross-sectional predictive regressions of factor returns

This table reports the results of a similar Fama-MacBeth regression as in Table10 with different independent variables. In Panel A, the key independent variable,

Co-Fragility, is the average pairwise co-fragility between a given factor and the rest of

the 70 factors in the prior quarter. Rank is the quintile ranking of Co-Fragility in agiven quarter. Dummy Rank5 is a dummy that equals one for factors in the highest

Co-Fragility quintile and zero otherwise. In Panel B, the key independent variable,Fragility, is the fragility of a given factor in the prior quarter. Rank and Dummy Rank5are defined following Panel A but based on factor fragility in the quarter. Controlvariables are the same as those in Table 10. We standardize the independent variablesby their cross-sectional standard deviations for each cross-section, except for Rank andRank5. Columns (1) to (4) report the regression results in the full sample. Columns(5) to (6) report the regression results among factors in the lowest and highest quintileeach quarter. t-statistics in parentheses are computed based on standard errors withNewey-West correction of 12 lags. *, **, *** indicate significance at the 10%, 5%, and1% level, respectively.

Panel A: Average co-fragility with other factors

(1) (2) (3) (4) (5) (6)

Full Sample Extreme Quintiles

Co-Fragility 0.48*** 0.30**(2.63) (2.30)

Rank 0.33** 0.18**(2.48) (2.17)

Rank5 1.56*** 0.87**(2.76) (2.15)

Controls No No Yes Yes No YesAdj. R2 0.11 0.44 0.11 0.43 0.19 0.49

Panel B: Fragility as regressor

(1) (2) (3) (4) (5) (6)

Full Sample Extreme Quintiles

Fragility 0.09 0.06(1.16) (1.17)

Rank 0.03 0.03(0.31) (0.69)

Rank5 0.06 0.06(0.27) (0.33)

Controls No No Yes Yes No YesAdj. R2 0.01 0.01 0.40 0.40 0.02 0.43

61

C List of factors

Table C.1 lists the 70 factors studied in our paper and reports their average monthly

raw returns, CAPM alphas, and Fama-French five factor alphas during January 1980-

December 2017. We compute the sorting variables for the 70 factors (anomalies) follow-

ing Hou, Xue, and Zhang (2018), Linnainmaa and Roberts (2018), and Arnott, Clements,

Kalesnik, and Linnainmaa (2019). Since our study requires the factor long-short port-

folio to be rebalanced quarterly or annually to match with the quarterly mutual fund

holdings data, we convert several typically monthly rebalanced factors into quarterly re-

balanced ones. These factors include the 52-week high, industry momentum, intermediate

momentum, long-term reversals, maximum daily returns, momentum, customer momen-

tum, geographic momentum, industry lead-lag, segment momentum, residual momentum,

Frazzini-Pedersen beta, and idiosyncratic volatility. For these variables, we use the latest

possible sorting variables at each quarter-end to form portfolios and hold the portfolios

in the next quarter.

62

Table C.1: List of factors

Factor Raw Ret CAPM FF5 Factor Raw Ret CAPM FF5

52-week high 0.51 1.01 0.50 Industry momentum 0.44 0.26 0.36Abnormal capital investment 0.18 0.12 0.25 Intermediate momentum 0.47 0.45 0.62Accruals 0.21 0.33 0.08 Investment growth 0.19 0.30 0.02Advertising expense 0.39 0.46 −0.01 Investment-to-assets 0.13 0.12 −0.10Altman’s Z-score 0.07 −0.08 0.25 Investment-to-capital 0.08 0.40 −0.28Amihud illiquidity 0.30 0.22 0.18 Long-term reversals 0.25 0.26 0.03Analyst earnings forecast Revision 0.26 0.30 0.24 M/B and accruals 0.34 0.43 0.19Asset Growth 0.25 0.39 −0.15 Maximum daily return 0.30 0.93 0.09Book-to-june-end-market 0.20 0.28 −0.10 Momentum 0.18 0.28 0.26Book-to-market 0.19 0.28 −0.08 Net operating assets 0.33 0.31 0.27Capital turnover 0.19 0.14 0.23 Net payout yield 0.34 0.68 0.01Cash-based profitability 0.18 0.29 0.52 Number of earnings increase 0.23 0.91 0.48Cashflow-to-price 0.24 0.46 −0.01 Ohlson’s O-score 0.16 0.32 0.31Change in asset turnover 0.14 0.17 0.01 One-year share issuance 0.37 0.57 0.05Change in long-term NOA 0.29 0.35 0.00 Operating cash flow-to-price 0.31 0.46 −0.03Customer momentum 0.55 0.66 0.56 Operating leverage 0.34 0.43 0.19Debt issuance 0.14 0.17 0.01 Operating profitability 0.35 0.60 0.17Discretionary accruals 0.27 0.20 0.42 Organizational capital-to-book 0.31 0.47 0.17Distress risk 0.61 0.89 0.68 Percent accruals 0.22 0.31 0.09Earnings forecast to Price 0.37 0.61 0.09 Piotroski’s F-score 0.06 0.10 0.06Earnings persistence 0.39 0.53 0.46 Profit margin -0.05 0.16 −0.10Earnings timeliness 0.06 −0.06 0.07 QMJ profitability 0.40 0.49 0.29Earnings-to-price 0.32 0.59 0.01 R&D expense 0.44 0.26 0.36Enterprise multiple 0.25 0.25 0.41 Real estate ratio 0.30 0.22 0.42Firm age −0.01 −0.32 0.28 Residual momentum 0.56 0.67 0.55Five-year share issuance 0.40 0.54 0.21 Return on assets 0.35 0.58 0.27Frazzini-Pedersen beta 0.23 0.91 0.48 Return on equity 0.35 0.56 0.15Geographic momentum 0.32 0.37 0.37 Sales growth −0.14 −0.35 0.17Gross profitability 0.18 0.24 0.32 Sales-minus-inventory growth 0.15 0.15 0.09Growth in Inventory 0.38 0.48 0.21 Sales-to-price 0.35 0.33 −0.18Growth score 0.11 0.27 0.31 Segment momentum 0.28 0.35 0.29Idiosyncratic volatility 0.36 0.88 0.22 Size 0.17 0.05 0.04Industry adjusted CAPX growth 0.18 0.29 0.06 Sustainable growth 0.28 0.42 0.04Industry concentration 0.26 0.16 0.54 Tax expense change 0.23 0.16 0.24Industry lead-lag 0.32 0.37 0.37 Total external financing 0.25 0.53 0.04

Average 0.26 0.37 0.23

63