THE JOURNAL OF FINANCE • VOL. LXVIII, NO. 3 • JUNE 2013
Reverse Survivorship Bias
JUHANI T. LINNAINMAA∗
ABSTRACT
Mutual funds often disappear following poor performance. When this poor performance is partly attributable to negative idiosyncratic shocks, funds’ estimated alphas
understate their true alphas. This paper estimates a structural model to correct for
this bias. Although most funds still have negative alphas, they are not nearly as low
as those suggested by the fund-by-fund regressions. Approximately 12% of funds have
net four-factor model alphas greater than 2% per year. All studies that run fund-byfund regressions to draw inferences about the prevalence of skill among mutual fund
managers are subject to reverse survivorship bias.
THE TYPICAL SURVIVORSHIP BIAS argument starts from the observation that
mutual funds often disappear following poor performance. Thus, a study
that conditions on fund survival overstates performance.1 In this paper, I
show that the correlation between performance and survival induces another bias with the opposite sign: mutual fund alphas estimated from a survivorship bias-free data set are biased downward relative to the true alpha
distribution.
The mechanics of this bias are transparent in an example in which investors
learn about fund alphas. Suppose, first, that investors have a prior belief that
a fund’s alpha is zero; second, that the true alpha is fixed; and third, that
investors abandon a fund (and the fund shuts down) when their posterior belief about the fund’s alpha is lower than −ᾱ. Investors update their beliefs
by observing risk-adjusted returns. The posterior mean always lies between
∗ Juhani Linnainmaa is at University of Chicago Booth School of Business. I thank Alan Bester,
John Cochrane, Phil English, Gene Fama, Andrea Frazzini, Bobbie Goettler, Mark Grinblatt,
John Heaton, Markku Kaustia, Bryan Kelly, Matti Keloharju, Ralph Koijen, Annette Larson
(Morningstar), Robin Lumsdaine, Toby Moskowitz, Robert Novy-Marx, Ľuboš Pástor, Antti
Petäjistö, Alexi Savov, Clemens Sialm (NBER discussant), Rob Stambaugh, Matt Taddy, Luke
Taylor, Pietro Veronesi, Z. Jay Wang (WFA discussant), and Russ Wermers for helpful discussions.
I also thank seminar and conference participants at the University of Chicago, the Norwegian
School of Economics and Business Administration (Bergen), American University, BI Norwegian
School of Management (Oslo), Copenhagen Business School, Aalto University (Helsinki), the University of Texas at Austin, the University of Illinois at Chicago, the University of Missouri, the
NBER Asset Pricing working group conference (Winter 2010), and the 2010 Western Finance Association meetings for comments on earlier drafts. Finally, I am especially grateful for the comments
of an anonymous referee, an Associate Editor, and the Editor, Campbell Harvey.
1 See, for example, Brown et al. (1992), Elton, Gruber, and Blake (1996), Carpenter and Lynch
(1999), and Carhart et al. (2002).
DOI: 10.1111/jofi.12030
789
The Journal of FinanceR
790
α
Prior distribution about alpha
Posterior mean: E [α rte, rte – 1 ,..., prior]
0
t
Fund disappears
–αˉ
Reverse
survivorship
bias
Average risk-adjusted return
Figure 1. Illustrating reverse survivorship bias. This figure illustrates the downward bias
in estimated alphas. Investors have a prior distribution about a fund’s alpha centered around zero
and they abandon a fund if the posterior mean falls below −ᾱ. Investors learn about alphas from
risk-adjusted returns. Because the posterior mean is the weighted average of the prior mean and
the signal, a fund can only disappear if the average risk-adjusted return (= the estimated alpha)
is strictly less than −ᾱ. The gap between the true alpha and the estimated alpha is the reverse
survivorship bias.
the prior mean and the average risk-adjusted return. As a consequence, if the
posterior mean ever hits the −ᾱ boundary and the fund disappears, then the
estimated alpha must have been strictly lower than −ᾱ. If not, the posterior
mean could not have crossed the threshold. I call this positive gap between the
true and estimated alphas reverse survivorship bias.2 This bias corresponds
to a counterfactual absent from the data: if the disappearing funds were allowed to survive, their average alpha would increase, converging toward −ᾱ.
Figure 1 illustrates this example.
Reverse survivorship bias arises from the correlation between performance
and fund survival. It does not matter whether this correlation involves learning
or not. To appreciate this bias as a statistical result, suppose a mutual fund
has a fixed alpha α and that its risk-adjusted returns are r̃te ≡ α + ε̃t , where ε̃t
is mean zero and independently distributed. A fund survives for a random T̃
number of months. The expected sum of ε̃t s is zero:3
⎡
E⎣
T̃
⎤
ε̃t ⎦ = 0.
(1)
t=1
2 Pástor, Taylor, and Veronesi (2009) make a similar argument in the IPO literature. If a firm
has an IPO only if the posterior mean about its profitability hits some threshold, then its pre-IPO
profitability must have strictly exceeded this threshold. The market thus rationally expects the
firm to experience a post-IPO drop in profitability.
3 See, for example, Williams (1991).
Reverse Survivorship Bias
791
It follows immediately from equation (1) that the expected average riskadjusted return is
⎡
⎤
⎛
⎞
T̃
T̃
1
1
E⎣
(2)
r̃te ⎦ = α + cov ⎝ ,
ε̃t ⎠ .
T̃ t=1
T̃ t=1
If the survival probability is increasing in the risk-adjusted return, the average
risk-adjusted return is a downward-biased measure of the fund’s true alpha,
T̃ e
r̃t ] < α.
E[ T̃1 t=1
Intuitively, the bias arises from the ambiguity about whether a realized
return is low because the true alpha is low or because the idiosyncratic shock,
ε̃t , is low. A fund manager may have a small positive alpha, but negative
idiosyncratic shocks result in low returns. If such a fund dies, it leaves behind
an alpha estimate that is too low. No mechanism eliminates just-lucky mutual
funds to offset this bias.
This bias is related to, but not an example of, the “baby-boy fallacy”: if parents
stop having children after their first son, there will not be more boys than girls
in the population. The difference between this fallacy and reverse survivorship
bias is the same as the difference between the sums and averages in equations
(1) and (2): do we count the number of boys and girls, or do we compute the
average fraction of girls in each family? If parents follow a stop-at-a-boy rule,
the expected number of boys and girls is the same but the expected fraction
of girls is less than 12 . The fraction of girls will be zero in 12 of families, 12
in 14 of families (girl−boy), 23 in 18 of families (girl–girl–boy), and so forth.
Continuing ad infinitum, the expected fraction of girls in a family is then
( 12 )1 10 + ( 12 )2 12 + · · · = 0.307. Analogously, if mutual funds are more likely to
disappear following “bad” outcomes then, conditional on a fund dying, realized
returns do not constitute a random sample. These returns contain on average
T̃
ε̃t ] < 0, and this negative average
more negative than positive errors, E[ T̃1 t=1
error bleeds into the alpha estimate.
This paper adjusts for reverse survivorship bias by estimating a structural
model with heterogeneous alphas and a correlation between performance and
survival. In estimation I match, between the data and model, the average alphas of both surviving and disappearing funds, the changes in these averages
as funds mature, mutual fund survival rates, the mean and standard deviation of the distribution of estimated alphas, and the volatility of idiosyncratic
shocks. The estimates indicate that reverse survivorship bias is economically
important. Whereas the average true four-factor model alpha is −0.49% after
accounting for reverse survivorship bias, the simple average alpha estimated
from the data is −0.92% per year. The difference, 43 basis points, is the estimate of the size of the reverse survivorship bias in the mean. An alternative regression-based approach places the bias in the mean around 62 basis
points. Because idiosyncratic shocks are very volatile compared to the crosssectional variation in alphas, var(ε̃ j,t ) var(α̃ j ), funds with positive alphas also
frequently disappear because of a run of bad luck. As a consequence, the
792
The Journal of FinanceR
distribution of estimated alphas shows less skill in its right tail than what
there actually is. The structural model estimates suggest, for example, that
12% of funds have true (after-fees) four-factor model alphas greater than 2%
per year.
It is important to understand both the scope and limits of reverse survivorship bias. For example, the return on a strategy that invests the same amount
in each actively managed fund is unaffected by this bias. Its profitability does
not depend on the counterfactual of how well a fund would have performed had
it not disappeared. Reverse survivorship bias is a concern, however, when we
draw inferences about the prevalence of skill among mutual fund managers.
Because the field has converged, by my reading, to the view that the average
dollar invested in actively managed funds probably has a negative alpha,4 its
focus has shifted toward studying heterogeneity in skill. Reverse survivorship
bias becomes an issue when we estimate fund-by-fund regressions to draw
inferences about this heterogeneity.
Several recent papers measure the prevalence of skill (or lack thereof) among
mutual fund managers.5 Kosowski et al. (2006) find that a “sizable minority”
of managers pick stocks well enough to more than cover their costs. Barras,
Scaillet, and Wermers (2010) suggest that the number of negative-alpha funds
(24%) far outweighs the number of positive-alpha funds (0.6%). Fama and
French (2010) set estimated alphas to zero and run simulations to separate luck
from skill. They find only weak evidence of superior skill in the extreme right
tail of the net alpha distribution. Related to my study, they find that the actual
alpha distribution’s left tail is thicker than that of the simulated distribution.
Although this thick tail could arise from either negative stock-picking skills
or high trading costs, it may also signify the presence of reverse survivorship
bias. The average alpha estimate is too low for funds that disappear and so
they end up thickening the distribution’s left tail.
The paper is organized as follows. Section I describes the data and examines
the correlation between survival and performance. Section II estimates the bias
in the mean of the alpha distribution and uses simulations to demonstrate the
distortion in the shape of this distribution. Section III estimates a structural
model to measure the prevalence of skill among actively managed mutual
funds. Section IV concludes and discusses the broader implications of reverse
survivorship bias.
I. Data and Correlation between Performance and Survival
A. Data and Performance Measurement
I use mutual fund data from the Center for Research in Security Prices
(CRSP). Following French (2008), Fama and French (2010), and Barras,
Scaillet, and Wermers (2010), I first include only actively managed funds that
4
See, for example, Fama and French (2010) for a discussion.
Although here I discuss the recent literature, the tradition of estimating alphas fund by fund,
and not for portfolios of funds, goes back to Jensen (1968).
5
Reverse Survivorship Bias
793
invest in U.S. common stocks and then combine different share classes of the
same fund into a single fund. I obtain information on funds’ objectives and
multiple share classes by merging the CRSP database with the Thomson/CDA
database using the MFLINKS product of Wharton Research Data Services.
Barras, Scaillet, and Wermers (2010) suggest that the Thomson fund investment objective information is more consistent over time compared to the CRSP
codes. Moreover, identification of multiple share classes using the CRSP data
set is inadvisable because it is based on an examination of error-prone fund
names. The downside of the Thomson/CDA database is that, if a fund exists
only for a year or two, it may never get added to these data.6
I restrict the sample to mutual funds that start on January 1984 or later
because the earlier data are unreliable.7 A mutual fund enters the sample
after its combined net asset value across all share classes exceeds $5 million
in December 2006 dollars. Once a fund has exceeded this threshold, I keep
the fund in the sample no matter what happens to its net asset value to avoid
selection bias. This procedure guards against the incubation bias of Evans
(2010).8 If a monthly return is missing, I delete the following-month return
because this represents the cumulative return over the reporting gap in the
CRSP database. Similarly, I delete any leading-zero return observations from
fund returns because these appear to be incorrect entries in the CRSP database.
I measure performance throughout this study using the CAPM alpha, the
three-factor model alpha (Fama and French (1993)), and the four-factor model
alpha (Carhart (1997)) by estimating regressions such as
r j,t − rf ,t = α j + b j (Rm,t − rf ,t ) + s j SMBt + h j HMLt + mj MOMt + e j,t ,
(3)
where r j,t is fund j’s month-t return, rf ,t is the risk-free rate, Rm,t is the month-t
return on the value-weighted CRSP index, and SMBt , HMLt , and MOMt are
month-t returns on long-short portfolios for size, value, and momentum. I require a fund to have at least 8 months of return data to estimate its alpha.
The CRSP mutual fund files contain monthly returns up to the end of December 2010. My sample contains data on 1,853 mutual funds that have at least
8 months of return data.
6 I account for this possibility when estimating the structural model by choosing moment conditions that are not influenced by the omission of some funds that do not survive for more than
2 years. I thank Russ Wermers for his notes on the issues surrounding the CRSP mutual fund data
set, MFLINKS, and the Thomson/CDA database.
7 See, for example, Elton, Gruber, and Blake (2001) and Fama and French (2010). The average
returns in the pre-1984 part of the data are significantly higher for mutual funds that report
monthly returns than for those that report annual returns.
8 Incubation bias arises when fund management companies provide seed money to new private
funds and then selectively open the funds with the best histories to the public. When funds can
backfill their return histories for the incubation period, a positive wedge emerges between the
fund’s past performance, which is selected to be good, and its expected performance. Fama and
French (2010) use the same $5 million threshold as their main specification.
794
The Journal of FinanceR
B. Correlation between Performance and Survival
The correlation between fund returns and disappearance is important for
the size of the reverse survivorship bias (see equation (2)). Although a number
of papers9 find a positive correlation between performance and survival, I also
study this correlation because, first, my sample differs from those used in these
earlier studies, and second, these estimates serve as inputs for the structural
model estimation.
Table I reports on average alphas for mutual funds that either survive or do
not survive through the T th year of their lives, conditional on having survived
through year T − 1. If a fund disappears by the end of year T , I compute its
alpha using all available return data. If the fund is still alive at the end of year
T , I compute its alpha using data up to the end of year T . Each column in
Table I reports on average alphas for funds that either survive or disappear in
year T . The four-factor model estimates show, for example, that the average
alpha is −4% per year for a fund that disappears during its fifth year but 0.19%
for those that survive through this year.
The last column reports the cumulative survival rates. If a mutual fund is
T years old at the end of the CRSP mutual fund data set, I count the fund
as a survivor in year 1, . . . , T computations but ignore it in year T +1, . . . , 10
computations. Ten percent of mutual funds disappear before reaching the age
of 4 and approximately one-third disappear by the end of the 10th year. The
proper treatment of dead funds in the estimation of alphas is thus important.
The high survival rates for the first 2 years are consistent with the view that
the Thomson/CDA database may not have data on some of the most short-lived
funds. Whereas the conditional disappearance probabilities for years 3 through
) to 0.051, these probabilities are just 0.005 and
10 range from 0.034 (= 1 − 0.943
0.976
0.019 for the first 2 years.
The alpha estimates in Table I suggest that mutual funds that disappear
perform considerably worse than surviving mutual funds. This conclusion is
not sensitive to the year in which the comparison is made or to the choice of the
asset pricing model used to estimate alphas. The disappearance of poorly performing funds is potentially a significant factor in performance evaluation via
the reverse survivorship bias channel. For example, the reverse survivorship
bias argument suggests the four-factor model alpha estimate of −4% for the
funds that die in their fifth year probably understates these funds’ true alphas.
This average may be low not only because these funds’ true alphas are low but
also because some of them experienced bad luck. It is possible that the average
expected alpha for these funds at the time of disappearance was higher than
−4%. That is, had these funds been allowed to survive, their alpha estimates
would probably have shown some recovery.
Figure 2 plots cumulative risk-adjusted returns for surviving and disappearing funds to show how these returns vary from year to year leading up to fund
9 See, for example, Brown and Goetzmann (1995), Khorana (1996, 2001), Chevalier and Ellison
(1999), and Lynch and Musto (2003).
Reverse Survivorship Bias
795
Table I
Mutual Fund Alpha Estimates Conditional on Fund Survival,
Percentage Points per Year
This table reports mutual fund alpha estimates conditional on a fund surviving or not surviving
through year T . The sample includes 1,853 actively managed mutual funds that (1) invest primarily
in U.S. common stocks, (2) first appear in the CRSP data in January 1984 or later, and (3) have at
least 8 months of return data. The sample period ends in December 2010. Different share classes
of the same fund are combined into a single fund using net asset value weights. I estimate alphas
using a time-series regression that includes monthly returns up to the end of year T (if the fund
survives) or all available data (if the fund disappears). A fund that disappears during year T is
assigned into the year-T “Do Not Survive” pool of funds; funds still alive at the end of year T are
assigned into the year-T “Survive” pool. Alphas are estimated using the CAPM, the three-factor
model, and the four-factor model. I report t-values in parentheses. The last column reports the
fraction of funds that survive for at least T years. These survival rate computations also include
those funds that do not have 8 months of return data.
Fund
Age
1
2
3
4
5
6
7
8
9
10
Survive
Do Not Survive
CAPM
FF
FF+MOM
CAPM
FF
FF+MOM
− 0.62
(− 2.11)
− 1.16
(− 4.97)
− 0.75
(− 3.89)
− 0.11
(− 0.67)
0.15
(0.98)
0.35
(2.67)
0.54
(4.58)
0.73
(6.73)
0.72
(7.03)
0.75
(7.66)
0.08
(0.32)
− 0.21
(− 1.03)
0.02
(0.13)
0.26
(1.66)
0.35
(2.40)
0.32
(2.55)
0.26
(2.30)
0.30
(2.82)
0.21
(2.02)
0.18
(1.78)
− 0.09
(− 0.37)
− 0.36
(− 1.92)
− 0.15
(− 0.91)
0.12
(0.82)
0.19
(1.45)
0.17
(1.48)
0.10
(0.99)
0.15
(1.48)
0.05
(0.54)
0.03
(0.34)
− 5.92
(− 3.77)
− 3.66
(− 2.68)
− 5.42
(− 6.11)
− 4.25
(− 3.84)
− 4.37
(− 6.25)
− 2.73
(− 2.73)
− 2.31
(− 4.96)
− 2.83
(− 4.40)
− 1.62
(− 3.94)
− 2.14
( − 6.04)
− 5.87
(− 3.96)
− 2.98
(− 3.40)
− 3.88
(− 5.67)
− 4.28
(− 4.86)
− 4.48
(− 7.07)
− 2.72
(− 2.90)
− 2.20
(− 5.34)
− 2.90
(− 5.51)
− 1.75
(− 4.74)
− 1.89
( − 5.42)
− 4.57
(− 4.41)
− 2.88
(− 3.02)
− 3.49
(− 5.44)
− 4.74
(− 5.83)
− 4.03
(− 6.78)
− 2.66
(− 3.08)
− 1.97
(− 5.01)
− 2.88
(− 5.11)
− 1.73
(− 4.46)
− 2.04
( − 5.62)
Survival
Rate
0.995
0.976
0.943
0.908
0.868
0.830
0.789
0.754
0.717
0.681
disappearance (or survival). I compute risk-adjusted returns using both the
CAPM and the four-factor model. The three-factor model estimates are similar to the four-factor model estimates and thus are not reported. The month-t
risk-adjusted return from the four-factor model is
r̂ ej,t = R j,t − rf ,t − b̂ j (Rm,t − rf ,t ) − ŝ j SMBt − ĥ j HMLt − m̂j MOMt ,
(4)
where b̂ j , ŝ j , ĥ j , and m̂j are fund j’s loadings on the market, size, value, and
momentum factors. If a fund disappears in year T , I estimate its factor loadings by using all data up to the month of disappearance. If the fund survives, I use all data up to the end of year T . Figure 2 reports on cumulative
risk-adjusted returns for funds that survive for at least 2, 4, 6, 8, or 10 years,
The Journal of FinanceR
796
Panel B: Four-Factor Model
Cumulative Risk-Adjusted Return (%)
Cumulative Risk-Adjusted Return (%)
Panel A: CAPM
Do Not Survive
Survive
Fund Age
Do Not Survive
Survive
Fund Age
Figure 2. Cumulative risk-adjusted returns conditional on fund survival. I estimate
CAPM and four-factor model alphas for all mutual funds that either survive or do not survive
through year T . This figure uses these estimates to plot the cumulative risk-adjusted returns up
to the end of year T or the month in which the fund disappears. This figure plots estimates only for
even-numbered years. The sample includes 1,853 actively managed mutual funds that (1) invest
primarily in U.S. common stocks, (2) first appear in the CRSP data in January 1984 or later, and
(3) have at least 8 months of return data. The sample period ends in December 2010. Different
share classes of the same fund are combined into a single fund with net asset value weights. Solid
(dashed) lines denote average cumulative risk-adjusted returns of surviving (disappearing) funds.
For example, the solid (dashed) year-10 line represents average cumulative risk-adjusted returns
of funds that survive through (disappear during) year 10.
as well as for funds that disappear in years 2, 4, 6, 8, or 10. The estimates
suggest mutual funds often disappear following a string of low returns. Mutual
funds that disappear during their sixth year, for example, cumulatively lose
approximately 15% on a risk-adjusted basis in both specifications.
C. Cox Proportional Hazards Model of Mutual Fund Survival
Table II reports Cox proportional hazards model estimates that measure how
sensitive mutual fund survival is on performance. Each fund enters the sample
for every month of its existence. The time-varying covariate in Panel A is the
fund’s alpha (the first three columns) or the t-value associated with the fund’s
alpha (the last three columns), estimated using monthly returns up to the
month of the observation. Panel B reports on otherwise identical regressions
but lags the covariates by 1 year. Panel C includes Morningstar ratings and
assets under management as additional covariates. I cluster standard errors
by fund to adjust for within-fund correlation in observations.10
10 These regressions are not free of the errors-in-variables problem because the regressor is
a first-stage alpha estimate. The average alphas reported in Table I, however, suggest that the
errors-in-variables problem does not materially influence the inferences.
Reverse Survivorship Bias
797
Table II
Cox Proportional Hazards Model for Mutual Fund Survival
This table reports Cox regressions that measure sensitivity of mutual fund survival to performance,
as measured by funds’ alphas or t-values associated with these alphas. The sample includes 1,853
actively managed mutual funds that (1) invest primarily in U.S. common stocks, (2) first appear
in the CRSP data in January 1984 or later, and (3) have at least 8 months of return data. The
sample period ends in December 2010. Different share classes of the same fund are combined into
a single fund using net asset value weights. I exclude the first 7 months of data for each fund
because only funds that survive at least this long enter the sample. The data consist of 240,784
fund-month observations. Panel A estimates fund alphas and t-values using all data up to the
current month. Panel B lags the alpha estimates 1 year. Alphas are estimated using the CAPM,
the three-factor model (FF), and the four-factor model (FF+M). Alpha estimates are multiplied by
100 so that “1” stands for an annualized alpha of 1%. Panel C reports regressions estimated using
a Morningstar database (see text for details). These regressions include, as additional covariates,
dummy variables for funds’ Morningstar ratings (the five-star category is the omitted dummy) and
the log-value of assets under management (AUM). Standard errors, reported in square brackets,
are clustered by fund.
Panel A: Current Alpha Estimates as Time-Varying Covariates
Covariate: α̂
Hazard ratio
SE
Pseudo R2
Covariate: t(α̂)
CAPM
FF
FF+M
CAPM
FF
FF+M
0.913
[0.005]
0.019
0.900
[0.006]
0.020
0.891
[0.006]
0.022
0.642
[0.019]
0.024
0.713
[0.016]
0.015
0.707
[0.018]
0.015
Panel B: 1-Year Lagged Alpha Estimates as Time-Varying Covariates
Covariate: α̂
Hazard ratio
SE
Pseudo R2
Covariate: t(α̂)
CAPM
FF
FF+M
CAPM
FF
FF+M
0.938
[0.005]
0.013
0.927
[0.005]
0.013
0.922
[0.006]
0.014
0.641
[0.019]
0.024
0.710
[0.016]
0.015
0.704
[0.018]
0.015
Panel C: Fund Survival in the Morningstar Database
Covariate
(1)
t(CAPM α̂)
(2)
(3)
0.616
[0.014]
t(FF α̂)
0.660
[0.013]
(6)
★✩✩✩✩
★★★✩✩
★★★★✩
14.144
[2.909]
8.704
[1.760]
4.218
[0.860]
1.634
[0.359]
ln(AUM, millions)
0.034
0.027
(7)
0.870
[0.022]
0.649
[0.014]
★★✩✩✩
Pseudo R2
(5)
0.862
[0.026]
t(FF + Mα̂)
Morningstar rating:
(4)
0.023
0.023
3.774
[0.850]
3.369
[0.721]
2.198
[0.460]
1.083
[0.247]
0.661
[0.017]
0.091
3.979
[0.887]
3.446
[0.723]
2.201
[0.457]
1.073
[0.244]
0.659
[0.017]
0.091
0.851
[0.021]
3.855
[0.859]
3.356
[0.700]
2.155
[0.445]
1.062
[0.241]
0.659
[0.017]
0.091
798
The Journal of FinanceR
Table II, Panel A, shows that the probability a mutual fund disappears
decreases significantly as the fund’s alpha estimate increases. To put these
estimates in perspective, consider mutual funds that have survived for 4 years.
The unconditional probability that a fund disappears sometime during its fifth
year is 0.044 and the standard deviation of the distribution of estimated CAPM
alphas of these funds is approximately 5%. The hazard ratio of 0.913 in Panel
A for the CAPM alpha suggests that, if a fund’s alpha is one standard deviation above the average, the disappearance probability decreases to 0.028
(= 0.044 × (0.913)5 ). Conversely, a one standard deviation decrease in the alpha estimate increases the disappearance probability to 0.07. The results are
similar when alpha estimates are replaced with the t-values associated with
these alphas. For example, if the t-value associated with the CAPM alpha increases by two, the disappearance probability decreases from its unconditional
value of 0.044 to 0.018. Panel B shows the results are largely the same when
current alpha estimates are replaced with estimates lagged by 1 year. The correlation between performance and survival is thus not only due to exceptionally
poor performance just before a fund disappears.
Panel C includes Morningstar ratings and the log-amount of assets under
management as additional covariates. I use a January 1984 through December 2010 Morningstar mutual fund database for this analysis. The sampleconstruction rules described in Section I.A apply to these data as well. For
example, I again delete a fund’s return history until it has at least $5 million
in assets, and funds have to start their operations in 1984 or later. If a fund
has multiple share classes, the combined fund’s star rating is the (rounded)
value-of-assets weighted average of individual classes’ star ratings. Although
the Morningstar sample is constructed using the same rules as the main sample, they are not identical because the Morningstar and CRSP fund populations
do not overlap perfectly—each database has some funds not available in the
other.11 Columns two through four replicate Panel A’s “t-values” regressions
using the Morningstar sample for reference.
Panel C shows that, in addition to alphas, Morningstar ratings predict the
disappearance of mutual funds. The hazard ratio increases monotonically as
the star rating decreases. While the average monthly disappearance rate is
1.5% for one-star funds in the data, it is just 0.1% for five-star funds. The
first column’s regression with the star ratings has a slightly higher explanatory power than the second column’s regression, which uses the t-value associated with the CAPM alpha. Both the star ratings and alpha t-values remain
significant in the regressions that contain both covariates simultaneously.12
The regressions that include everything—the star ratings, alpha t-values, and
11 The databases diverged in 2008 when CRSP rebuilt its mutual fund database using Lipper
Global Holdings Feed as the data source. The discrepancies and efforts to rebuild the database are
detailed in the database release notes, available at www.crsp.com.
12 Morningstar computes its ratings using up to 10 years of style-specific risk-adjusted returns.
Funds younger than 3 years old are not rated. See www.morningstar.com/help/data.html for additional details. A referee suggests that 401(k) plan sponsors review Morningstar (and Lipper)
Reverse Survivorship Bias
799
log-value of assets under management—have significantly higher pseudo R2
values than the regressions with just some of the covariates.
Panel A’s results are the most relevant for understanding the reverse survivorship bias. This bias arises from the “raw” correlation between the performance evaluation regression’s error term and survival, not from the correlation
that remains after conditioning on other attributes of survival. The other attributes matter for the bias only indirectly through their correlations with
estimated alphas. Suppose, for example, that we run the four-factor model regression for each fund and study the distribution of estimated alphas. The bias
in this distribution is due to the correlation between this regression’s error
term and survival. Because the usual performance evaluation regressions do
not include any variables other than the pricing factors, the partial correlations of survival with variables such as Morningstar ratings do not influence
the bias. The results in Panel C, however, remind us that one could think of this
bias as an omitted variable problem. If there is a variable that correlates with
survival but not with funds’ true alphas, we could use such a variable to reduce
the bias. The fundamental problem, similar to the one we face when searching
for instrumental variables, is that such variables are not easy to find. Morningstar ratings and assets under management, for example, are unsuitable for
this purpose as they probably also correlate with the true alphas.
The Cox regression estimates are similar across the three asset pricing models in all three panels. Table II indicates that the t-values associated with
the CAPM alphas have the most predictive power of fund disappearances, as
measured by the pseudo R2 . Two effects may make it difficult, however, to empirically distinguish between these models. First, as the asset pricing model
becomes more complicated, more data are needed to maintain the precision of
the alpha estimates. If the amount of data is fixed, then the addition of a new
factor decreases the precision at which alphas are estimated. For example, even
if the four-factor model was the model used by the market to evaluate funds,
the noisiness of the four-factor model alphas relative to the CAPM alphas could
tilt the Cox regressions to favor the CAPM. Second, if fund disappearance is
related to investors’ inferences about alphas, the relevant question is not which
model is the true asset pricing model but rather which model was used by investors during the sample period. Before Carhart (1997), for example, mutual
fund performance studies did not typically include the momentum portfolio as
a risk factor (or as a passive benchmark). Cremers, Petäjistö, and Zitzewitz
(2008), moreover, note that practitioners commonly evaluate fund managers
by comparing their returns against long-only benchmark indices. Nevertheless, the estimates in Tables I and II indicate that, no matter which asset
pricing model we choose, performance correlates significantly with fund survival. These findings in turn suggest that reverse survivorship bias may distort
inferences about the prevalence of skill among fund managers.
performance reports as the basis for putting funds on a “watch” list or dropping them. The fact
that Morningstar ratings are significant determinants of disappearance after controlling for alphas
is consistent with the view that the market uses additional information to make decisions about
investing in mutual funds.
The Journal of FinanceR
800
II. Measuring Reverse Survivorship Bias
A. Bias in the Mean of the Distribution
Because some good managers experience poor returns because of bad luck,
and their funds die, the estimated alpha distribution does not show enough
skill. Before using a structural model to extract the full distribution of alphas,
it is instructive to first measure the bias in the mean of the estimated alpha
distribution. The benefit here is that this gap can be measured with relatively
few assumptions about fund returns. In contrast, more structure is needed to
extract the full distribution of alphas from the data.
The shift in the mean of the alpha distribution can be estimated by comparing
two sets of alpha estimates. I assume that funds’ returns conform to a factor
model:
r j,t − rf ,t = α j + Ft β j + ε j,t ,
(5)
where Ft is a 1 × K vector of date-t factor realizations and β j is a K × 1 vector
of fund j’s risk loadings. I assume that neither the fund’s alpha nor its K × 1
risk loadings change over time. If equation (5) describes fund returns, the
N
average fund alpha N1
j=1 α j can be estimated in two ways: (1) run one timeseries regression for each fund to estimate α̂ j s and compute the average alpha,
1 N
j=1 α̂ j , or (2) compute the average fund return for each month and estimate
N
one time-series regression for this average. That is, average equation (5) across
funds,
N
N
N
N
1 1 1 1 r j,t −rf ,t =
α j +Ft
βj +
ε j,t ,
N
N
N
N
j=1
j=1
j=1
j=1
r̄t
ᾱ
β̄
(6)
ε̄t
and estimate the average fund alpha, ᾱ, by regressing the average return
against the factors.
Both methods should yield consistent estimates of the average alpha given
equation (5). The reverse survivorship bias argument, however, suggests that
the fund-by-fund regression approach will produce a lower estimate. Because
fund j’s time series is sometimes cut short because ε j,t is low, the average
T
dead fund’s returns contain more negative than positive errors, E[ T1 t=1
ε j,t ] <
0, and this biases the alpha estimate downward. A comparison of these two
methods can thus be used to measure the size of the reverse survivorship bias
in the mean.
Because the sample only includes funds that start on January 1984 or later,
only a handful of funds are present for the first few years. To facilitate a fair
comparison between the two methods, I begin the sample here in January 1987.
If a fund is born before this date, I use only post-January 1987 return data to
compute its alpha. This approach ensures that the monthly average return in
equation (6) is computed over at least 209 funds, which is the number of funds
present in January 1987.
Reverse Survivorship Bias
801
The two procedures yield very different estimates of the average alpha. The
average annualized CAPM, three-factor, and four-factor model alpha estimates
from the fund-by-fund regressions are −0.69% (t-value = −6.4), −1.10% (t-value
= −12.0), and −1.14% (t-value = −13.2), respectively.13 The alpha estimates
based on monthly returns averaged across funds are notably higher: −0.16%
(t-value = −0.3), −0.51% (t-value = −1.4), and −0.53% (t-value = −1.4). These
estimates thus place the bias in the mean of the distribution around 0.46%
to 0.62% per year, depending on the asset pricing model. Although this comparison indicates that reverse survivorship bias is an economically significant
consideration, it is uninformative about the distortion in the right tail of the
distribution where the skilled managers reside.
B. Simulation Evidence on the Magnitude of the Bias
I run simulations to investigate the distortion in the shape of the alpha distribution. By fixing the true alpha distribution, I can quantify reverse survivorship using a variety of metrics. I assume that funds’ true annual alphas are
normally distributed with mean zero and standard deviation of 1.25%, which
is the same estimate as that used by Fama and French (2010) to frame their
discussion about the amount of skill among managers. I assume that mutual
fund returns are again governed by the factor model in equation (5) and that
investors back out risk-adjusted returns, r ej,t , from raw returns,
r̃ ej,t = α j + ε̃ j,t ,
(7)
where ε̃ j,t ∼ N(0, σ j2 ) and σ j is fund j’s idiosyncratic volatility. I use the CRSP
data to generate a distribution for σ j2 . I estimate monthly CAPM regressions
for all sample funds and record their estimated idiosyncratic variances. I then
fit a gamma distribution to this empirical distribution via maximum likelihood,
and draw σ j2 s from this distribution in the simulations. The average fund has
an annualized idiosyncratic volatility of 7.1%. The 10th and 90th percentiles
are 3.1% and 12.0%, respectively.
A fund’s survival is related to the market’s belief about the fund’s alpha.
Funds are initially indistinguishable from each other and hence the rational
prior distribution about each fund’s alpha is the same as the distribution from
which alphas are drawn, α̂ j,0 ∼ N(0, 0.01252 ). The market knows each fund’s
idiosyncratic volatility, σ j , and uses Bayes’s rule to update from the prior distribution to the posterior distribution based on each return realization:
mj,t+1 = (1 − w j,t )mj,t + w j,t r ej,t+1 ,
(8)
v j,t+1 = σ j2 w j,t ,
(9)
13 The t-values here assume independence across funds and should thus be interpreted cautiously. See Fama and French (2010) for a discussion on across-fund correlations in residuals. The
estimates reported here are higher than those reported in Fama and French (2010, Table II) for an
equal-weighted portfolio of funds because of differences in sample construction and time period.
The Journal of FinanceR
802
where w j,t =
v j,t
v j,t +σ j2
is the weight Bayes’s rule places on month t + 1’s risk-
adjusted return.
I consider two alternative rules of fund disappearance. Under the first rule
a fund disappears if its posterior mean, mj,t , falls below some fixed threshold. I
choose this threshold so that the fraction of funds surviving for at least 10 years
is the same between the simulations and data, 68% (see Table I). This rule sets
the critical threshold at ᾱ = −0.55%. Under the second rule the market uses a
statistical test to determine fund survival. A fund disappears if the probability
that the fund is “good enough” falls below 0.05. To match the 10-year survival
rates between the simulations and the data, I set the threshold at ᾱ = 1.3%,
that is, a fund disappears if the probability that its alpha is at least 1.3% falls
below 0.05.
I simulate data for 10 million funds as follows. I first draw funds’ alphas
independently from each other from the true alpha distribution and generate
sequences of risk-adjusted monthly returns. I then process this sequence and
check for fund survival using one of the two exit rules described above. If a fund
T j e
r j,t , where T j is the
disappears, I record the fund’s estimated alpha, T1j t=1
month fund j disappears. For each fund, I generate a maximum of 10 years of
data. I also run a second set of simulations to generate a randomized-exit alpha
distribution. In these simulations, which use the same draws of alphas and riskadjusted monthly returns as the actual simulations, funds disappear randomly.
When a fund disappears in the actual simulation, a randomly selected fund
closes in the shadow simulation. The funds in the shadow simulations thus
disappear from the data at exactly the same rate as they do in the actual
simulations.
Figure 3 quantifies the reverse survivorship bias in the simulated data using
three measures. Because the results are similar for the two exit rules, here I
only discuss those based on the first rule. The left-hand panel compares the distribution of estimated alphas to the randomized-exit alpha distribution. This
comparison is useful because, even with 10 years of data, alpha estimates are
very noisy. However, the true and randomized-exit distributions have the same
amount of noise, so they are comparable. The left panel shows how the correlation between performance and survival pushes mass toward the distribution’s
left tail. The resulting distortions are significant. First, the mean of the realized
distribution is −1.1% as opposed to the true mean of zero. Whereas 50% of funds
have positive true alphas, the fraction of positive estimated alphas is 45.9%.
Second, the amount of mass that disappears from the distribution’s right tail is
nontrivial. For example, 7.3% of the funds in the randomized-exit alpha distribution have estimated alphas greater than 5% per year. This fraction is 4.2% in
the actual estimated alpha distribution. These numbers are suggestive of the
amount of skill we may not see in the distribution of estimated alphas due to
reverse survivorship bias. In this example, 43% of the mass of the distribution
is missing above the 5% threshold.
While the left panel is useful in suggesting that the realized alpha distribution is indeed distorted, it is less useful in revealing how significantly the
Reverse Survivorship Bias
803
Panel A: Fund disappears if mj,t <−0.55%
Realized Alpha Distribution
Estimated Alpha (%)
All Funds
Survive > 1 Year
All Funds
Survive > 1 Year
Fraction of Alphas Understated (%)
Observed minus True Alpha (%)
Density
Fraction of Alphas Understated
Reverse Survivorship Bias by True Alpha
Observed
Randomized
True Alpha (%)
True Alpha (%)
Panel B: Fund disappears if Pr(αj >1.3% | mj, t , vj, t )< 0.05
Estimated Alpha (%)
Reverse Survivorship Bias by True Alpha
All Funds
Survive > 1 Year
True Alpha (%)
Fraction of Alphas Understated
Fraction of Alphas Understated (%)
Density
Observed
Randomized
Observed minus True Alpha (%)
Realized Alpha Distribution
All Funds
Survive > 1 Year
True Alpha (%)
Figure 3. Reverse survivorship bias in simulated fund data. This figure illustrates the
reverse survivorship bias using simulated data for 10 million funds. Each fund’s alpha is drawn
from N(0, 0.01252 ) and the idiosyncratic volatility distribution is matched to the CRSP mutual
fund data for U.S. equity funds. Upon observing each month’s risk-adjusted return, the market
uses Bayes’s rule to update beliefs about each fund’s alpha, α j , starting from the prior of α j ∼
N(0, 0.01252 ). In Panel A, a fund disappears if the posterior mean about its alpha falls below
−0.55%; in Panel B, a fund disappears if the probability that its alpha is above 1.3% falls below
0.05. These cut-offs are chosen to match the 10-year survival rate between the simulations and data.
The left-hand panel on each row compares the distribution of estimated alphas to a distribution
that would obtain if funds were to disappear randomly at exactly the same rate as they do in
simulations. The middle panel plots the difference between the estimated alpha and the true
alpha, α̂ j − α j , as a function of the true alpha. The right panel measures the frequency with which
the estimated alpha is lower than the true alpha, α̂ j < α j . If the alpha estimates were unbiased,
the line in the middle panel would be flat at zero and the one in the right panel would be flat at
50%. In the middle and right-hand panels, the dashed line excludes funds that disappear before
turning 1 year old.
observed alpha distribution deviates from the true alpha distribution after the
former is stripped clean of estimation noise. The middle and right panels in
Figure 3 focus on this comparison. The middle panel plots the average difference between the estimated and true alphas, α̂ j − α j , as a function of the true
alpha. The logic of this computation is as follows. If we pick α j = 0 on the
x-axis, we consider generating many zero-alpha funds and putting them
The Journal of FinanceR
804
through the simulations. For each fund, we obtain a noisy alpha estimate,
α̂ j . If these estimates were unbiased, the average estimated alpha would conN
verge to the true alpha, that is, lim N→∞ N1
j=1 α̂ j = 0 for zero-alpha funds.
However, as the middle panel shows, the bias is substantial. The average zeroalpha fund’s alpha estimate, for example, is −1.2%. The bias decreases as we
move toward the right tail of the distribution. As we increase the true alpha,
α j , a fund is less likely to die because of bad luck and thus have too low an
alpha estimate. The bias does not, however, disappear. Even for funds with a
true alpha as high as 2%—a value that is 1.6 standard deviations above the
mean—the bias is −0.54%.
The reason the bias exists even in the right tail of the distribution is rooted
in the relative amounts of luck and skill. Whereas the standard deviation of the
true alpha distribution is just 1.25%, the average fund’s idiosyncratic volatility
is almost 10% per year. It is thus not too uncommon an occurrence that a fund
with a positive alpha experiences sufficiently bad luck that it disappears.
The rightmost panel in Figure 3 reports how often the estimated alpha is
lower than the true alpha, α̂ j < α j . If the estimated alphas were unbiased,
they would fall evenly above and below the true alphas. The estimated alpha
is too low for 6% of the zero-alpha funds and for 2% of the funds with a true
alpha of 1% per year. These are funds that disappear because of bad luck. Their
estimated alphas are far lower than their true alphas—if not, they could not
have crossed the disappearance threshold.
The dashed lines in the middle and right-hand panels include funds that
survive for at least a year. Although reverse survivorship bias decreases, its
importance is unchanged. The problem with trying to control for reverse survivorship bias by throwing out short-lived funds is that one then introduces
standard survivorship bias. These simulations suggest that reverse survivorship bias is an important consideration when measuring the prevalence of skill
among mutual fund managers. In addition to having to strip out estimation
noise from the distribution of estimated alphas, we also need to adjust for the
fact that individual alpha estimates may be biased downward.
III. Simulated Method of Moments Estimation
A. Structural Model
In this section, I estimate a structural model of mutual fund survival using
the CRSP mutual fund data. The Internet Appendix constructs and calibrates
an alternative model that endogenizes fund survival.14 This structural model
is related to simulations given in Section II. The difference is that, instead of
making assumptions about the alpha distribution and the disappearance of
funds, I use the model to extract these objects from the mutual fund data.
I assume that each mutual fund’s alpha is drawn from a normal distribution with mean μ and variance σ 2 . The market knows the parameters of this
14
An Internet Appendix may be found in the online version of this article.
Reverse Survivorship Bias
805
distribution. Funds are initially indistinguishable from each other and hence
the rational prior distribution about each fund’s alpha is the same as the distribution from which alphas are drawn, α̂ j,0 ∼ N(μ, σ 2 ). Each mutual fund generates monthly return observations. The market uses some asset pricing model to
obtain an estimate of the fund’s monthly risk-adjusted return, r̃ ej,t = α j + ε̃ j,t ,
where ε̃ j,t is normally distributed with mean zero and variance σe2 . Idiosyncratic variances are the same across funds for simplicity. The market’s asset
pricing model here is correct in that E[r̃ ej,t ] = α j .
The market uses each return realization to update from the prior distribution
to the posterior distribution. Because both the alphas and returns are normally
distributed, the mean (mj,t ) and variance (vt ) of the posterior distribution about
fund j’s alpha evolve according to
where wt =
vt
.
vt +σ j2
mj,t+1 = (1 − wt )mj,t + wt r ej,t+1 ,
(10)
vt+1 = σe2 wt ,
(11)
The posterior variance does not have the fund subscript j
because its process is deterministic and shared by all funds.
I assume that the probability a fund disappears is a function of the fund’s
posterior distribution. I construct the hazard rate as follows. First, let ᾱ denote
some critical level of alpha and record the probability p j,t that fund j’s alpha
ᾱ
is below this level, p j,t ≡ −∞ φ(α; mj,t , vt )dα, where φ() is the normal density.
The term p j,t can be interpreted as a summary statistic about fund quality. A
fund disappears after month t with probability
(12)
H( p j,t , t) = Pr ũ j,t ≤ γ0 + γ1 p j,t + γ2 p2j,t + γ3 t ,
where ũ is an i.i.d. draw from a uniform distribution on the interval [0, 1].
This hazard rate approach avoids the need to specify the economic mechanism
that drives the disappearance of mutual funds. Moreover, because equation
(12) does not force the exit probability to decrease in alpha, it can permit some
successful funds to close as well.15 The structural model can, in fact, “undo”
the built-in learning process if the data suggest such a reversal is warranted.
The eight structural parameters of the problem are the mean and variance of
the alpha distribution, μ and σ 2 ; the variance of risk-adjusted returns, σe2 ; and
the hazard rate parameters, ᾱ, γ0 , γ1 , γ2 , and γ3 .
B. Identifying Structural Parameters
I use 15 moment conditions to identify these eight structural parameters.
The first eight moment conditions are the average alphas of funds that either
survive or do not survive through years 4, 6, 8, and 10. These are the same
15 Kostovetsky (2009), for example, finds evidence of a flight from top-performing mutual funds
to the hedge fund industry. Deuskar et al. (2011), however, caution that it is predominantly the poor
performers who leave the mutual fund industry for smaller and younger hedge fund companies.
806
The Journal of FinanceR
statistics as those reported in Table I except I aggregate the data to the biennial
frequency to reduce noise and I omit the first 2 years because, as discussed
in Section I.A, the fund data here may not be representative. The next four
moment conditions are the drop-out rates for the same 2-year periods, that is,
the fraction of funds that disappear in years 3 and 4, 5, and 6, 7 and 8, and 9
and 10. These first 12 moments conditions are instructive about the hazardrate function parameters. Funds have to drop out from the model at the correct
rate and the drop outs (and survivors) have to have alphas that match those
observed in the data.
The next two moment conditions are the mean and standard deviation of the
estimated alpha distribution.16 These two moments instruct the model to push
the distribution of estimated alphas close to the distribution in the data. The final moment condition is the average standard deviation of idiosyncratic shocks,
which is represented by σe2 in the model. This moment tells the model to make
sure that risk-adjusted returns (and in turn alpha estimates) are just as noisy
as they are in the data. I use these 15 moments to estimate the eight structural
parameters (μ, σ 2 , σe2 , ᾱ, γ0 , γ1 , γ2 , γ3 ) to obtain an overidentified model.
Because the structural model does not yield closed-form estimation equations, I use simulated method of moments (SMM) for indirect inference. I draw
random funds from the true alpha distribution, generate monthly risk-adjusted
returns, update the market’s beliefs using equations (10) and (11), and then
compute the probability that a mutual fund disappears from the data. The
SMM estimator θ̂ is
θ̂ = arg min( M̂ − M(θ ))
W( M̂ − M(θ )),
θ
(13)
where θ is an eight-by-one vector of structural parameters, M̂ is the vector of estimated moments from the data, M(θ ) is the vector of model-implied moments,
and W is an arbitrary positive-definite weighting matrix. I use the optimal
weighting matrix, which is the inverse of the estimated covariance of moments
M.17
The downside of SMM is that the simulated moments inherit simulation
noise. Instead of applying the simulated annealing method to minimize (13), I
exploit the problem’s specific structure to smooth the simulated moments. If the
simulations are for N funds, I first generate N × (120 + 1) draws of standard
normal variables. Every time I evaluate equation (13) for some parameter
16 I compute each fund’s alpha using up to 10 years of data. Because some funds in the CRSP
data start within the last 10 years of the sample, I run the simulations so that the simulated funds
have the same probability of getting truncated as they do in the data. For example, if 10% of funds
in the data can appear in the data for at most 4 years, I let 10% of funds in the simulations appear
for at most 4 years.
17 Some elements of cov(M) cannot be estimated from the data. Funds, for example, do not
have both “survive” and “do not survive” alphas for the same period. Because many off-diagonal
elements, such as those corresponding to covariances between different drop-out rates, are also
identically zero, I diagonalize the estimated cov(M). This procedure, unlike the alternative that
constructs the matrix from pairwise covariances, ensures that the estimated covariance matrix is
positive definite.
Reverse Survivorship Bias
807
vector θ , I recall this set of draws to generate the N alphas and the N × 120
monthly returns. For example, as the structural parameters μ and σ 2 change
slightly, the same primitive draws now correspond to (slightly) different fund
alphas. This approach thus smooths the simulated moments, improving the
behavior of the objective function in equation (13). Because fund exits are
probabilistic, every fund can also be used to generate two observations for each
month, one for the fund that survives through this month and other for the
fund that disappears. I combine these 2 × 120 distinct per-fund observations
with their respective probabilities of being realized to compute M(θ ) in equation
(13). I estimate the model by simulating data for 100,000 funds.
C. Estimation Results
Table III compares the moment conditions between the data and the structural model (Panel A) and reports on the estimates of the eight structural
parameters (Panel B). The structural model matches the salient features of the
mutual fund data. First, the observed alphas are low for funds that disappear
early on but then increase monotonically as funds grow older. For example,
the average four-factor model alpha for funds that disappear in years 3 or 4 is
−4.1% in the data and −4.6% in the model. These averages increase to −1.9%
and −1.5% for funds that disappear in years 9 or 10. Second, the alphas of
the surviving funds also behave similarly in the data and simulations. Third,
the drop-out rates are very close to each other in the model and the data. The
discrepancies between the model and data are never greater than 0.7%. The
last three moment conditions indicate that the model also matches the mean
and volatility of the distribution of estimated alphas and that it is able to keep
the amount of idiosyncratic volatility in line with the data.
The first two parameters in Panel B describe the shape of the true alpha
distribution. Both the mean and volatility of this distribution are higher in the
CAPM than they are in the four-factor model. I limit my discussion here to the
four-factor model results because the CAPM estimates are usually deemed
inappropriate for evaluating funds’ stock-picking abilities.18 The estimated
volatility of the true alpha distribution in the four-factor model, 2.1%, is close
to the upper bound suggested by Fama and French (2010).
Panel C of Table III reports on the means and percentiles of three different alpha distributions. The “Data” row of each block reports the empirical
distribution of alphas in the data. The average alpha over all sample mutual
funds is −0.92% per year in the four-factor model. The second row reports the
distributions of estimated alphas in the model. The mean, −0.91%, is close to
that seen in the data. In addition to the mean, the percentiles of the data- and
model-based distributions are very similar. The 5th and 95th percentiles, for
example, are −7.7% and 5.2% in the data and −8.0% and 4.7% in the model.
18 See, for example, Carhart (1997) and Fama and French (2010). The latter, for example,
comment on this issue by writing that “the CAPM tests are a lesson about how failure to account
for common patterns in returns and average returns can affect inferences about the skill of fund
managers” (p. 1946).
The Journal of FinanceR
808
Table III
Estimates of the Structural Learning Model
This table reports the estimates of a structural model of mutual funds. The sample includes
1,853 actively managed mutual funds that (1) invest primarily in U.S. common stocks, (2) first
appear in the CRSP data in January 1984 or later, and (3) have at least 8 months of return
data. The sample period ends in December 2010. Different share classes of the same fund are
combined into a single fund using net asset value weights. In the model each mutual fund’s alpha
is drawn from a normal distribution with mean μ and variance σ 2 and generates monthly riskadjusted returns r̃ ej,t = α j + ε̃ j,t , where ε̃ j,t ∼ N(0, σe2 ). The market updates its beliefs about α j
from each return realization. The hazard rate for fund j disappearing after month t is H( p j,t , t) =
Pr(ũ j,t ≤ γ0 + γ1 p j,t + γ2 p2j,t + γ3 t), where ũ j,t is an i.i.d. draw from a uniform distribution, p j,t ≡
ᾱ
−∞ φ(α; mj,t , vt )dα, φ() is the normal density, and mj,t and vt are the mean and variance of the
posterior distribution. The eight structural parameters of the problem are the mean and variance
of the alpha distribution, μ and σ 2 ; the variance of risk-adjusted returns, σe2 ; and the hazard
rate parameters, ᾱ, γ0 , γ1 , γ2 , and γ3 . These parameters are estimated using Simulated Method
of Moments with the optimal weighting matrix. The 15 moment conditions are: (1) four moment
conditions for the average alphas of funds that disappear in years 3 or 4, 5 or 6, 7 or 8, and 9 or
10; (2) four moment conditions for the average alphas of the funds that survive through years 4,
6, 8, and 10; (3) four moment conditions for the fraction of funds that disappear in years 3 or 4,
5 or 6, 7 or 8, and 9 or 10; (4) two moment conditions for the mean and standard deviation of the
distribution of estimated alphas, where alphas are estimated using at most 10 years of data and
the censoring in the model is set to match that in the data; and (5) one moment condition for the
average standard deviation of the idiosyncratic shock. All moment conditions and parameters μ,
σ , ᾱ, and σe are annualized in this table. I estimate the model using moments computed from the
CAPM, the three-factor model, and the four-factor model. Panel A compares simulated moments to
sample moments, Panel B reports the structural parameter estimates, and Panel C tabulates the
distribution of estimated alphas in the data (“data” row) and the model (“model, observed” row),
and the distribution of true alphas in the model (“model, true” row).
Panel A: Actual Moments versus Simulated Moments
CAPM
Moment Condition
Data
Model
Alpha Estimates for Disappearing Funds
Years 3–4
− 4.830
− 5.040
Years 5–6
− 3.583
− 3.401
Years 7–8
− 2.552
− 2.414
Years 9–10
− 1.881
− 1.719
Alpha Estimates for Surviving Funds
Years 3–4
− 0.112
0.269
Years 5–6
0.355
0.451
Years 7–8
0.726
0.596
Years 9–10
0.754
0.697
Fraction of Funds Disappearing
Years 3–4
6.760
6.301
Years 5–6
8.143
8.194
Years 7–8
8.543
8.852
Years 9–10
9.056
9.020
Observed Alpha Distribution
Mean
−0.546
−0.587
Standard Deviation
4.835
4.999
SD of Idiosyncratic Risk
9.357
9.340
Three-Factor
Four-Factor
Data
Model
Data
Model
− 4.081
− 3.640
− 2.521
− 1.817
− 4.681
− 3.159
− 2.220
− 1.561
− 4.124
− 3.374
− 2.396
− 1.883
− 4.562
− 3.026
− 2.131
− 1.502
0.265
0.323
0.302
0.176
− 0.016
0.154
0.279
0.360
0.120
0.172
0.146
0.031
− 0.128
0.023
0.132
0.200
6.760
8.136
8.534
9.059
6.775
8.309
8.513
8.370
6.767
8.139
8.537
9.058
6.730
8.289
8.494
8.368
−0.847
4.318
7.355
−0.816
4.304
7.366
−0.920
4.031
7.131
−0.910
4.007
7.142
(Continued)
Reverse Survivorship Bias
809
Table III—Continued
Panel B: Structural Parameter Estimates
CAPM
Parameter
μ
σ
ᾱ
σe
γ0
γ1
γ2
γ3
J-test, χ 2 ( p-value)
EST
−0.129
2.756
−2.846
9.340
0.286
−0.035
0.001
−0.043
9.78
Three-Factor
SE
(0.079)
(0.452)
(3.931)
(0.125)
(1.044)
(0.102)
(0.000)
(0.120)
(0.202)
Four-Factor
EST
SE
EST
SE
−0.406
2.478
−2.173
7.366
0.664
−0.058
0.001
−0.111
13.26
(0.132)
(0.566)
(4.448)
(0.097)
(2.828)
(0.176)
(0.001)
(0.400)
(0.066)
−0.494
2.112
−2.098
7.142
0.767
−0.073
0.002
−0.119
12.92
(0.092)
(0.375)
(2.195)
(0.095)
(1.968)
(0.126)
(0.001)
(0.249)
(0.074)
Panel C: Distributions of Estimated and True Alphas
Percentiles
Model
CAPM
Data
Model
Observed
True
Three-Factor Model
Data
Model
Observed
True
Four-Factor Model
Data
Model
Observed
True
Mean
1%
5%
25%
50%
75%
95%
99%
−0.547
−16.048
−8.568
−2.567
−0.242
1.908
5.969
9.725
−0.587
−0.129
−15.284
−6.539
−9.293
−4.662
−3.261
−1.988
−0.188
−0.129
2.615
1.730
6.662
4.404
9.915
6.282
−0.847
−13.292
−7.719
−2.937
−0.758
1.169
5.720
9.804
−0.816
−0.406
−13.393
−6.170
−8.369
−4.482
−3.051
−2.077
−0.444
−0.406
1.908
1.266
5.369
3.670
8.343
5.359
−0.920
−11.889
−7.665
−2.848
−0.767
0.956
5.202
8.993
−0.910
−0.494
−12.851
−5.407
−7.988
−3.968
−2.921
−1.919
−0.539
−0.494
1.606
0.930
4.746
2.980
7.533
4.419
Given that the distributions of estimated alphas are similar between the data
and model, it is reasonable to ask how the model’s true alpha distribution differs
from the distribution of estimated alphas. That is, what underlying true distribution of alphas is responsible for the distribution of estimated alphas? The
third row of each block, which reports on the true distribution, shows that the
differences between the true and observed distributions are economically significant. In the four-factor model, the true mean alpha is −0.49% per year as opposed to the alpha estimate of −0.92% in the data. The gap between these two,
43 basis points, is comparable to the regression-based estimate in Section II.A.
The estimates of the true alpha distribution suggest that a minority of mutual
managers may pick stocks well enough to cover the fees they impose on their
investors. In the four-factor model, 24% of managers have true alphas greater
810
The Journal of FinanceR
than 1% per year and 12% have alphas greater than 2% per year. These estimates of the size of the skilled-manager group lie between the estimates in
Kosowski et al. (2006) (who give a higher estimate) and Barras, Scaillet, and
Wermers (2010) and Fama and French (2010) (who give lower estimates). My
estimates suggest that a minority of managers may be able to pick stocks well
enough to cover the costs they impose on their investors.
Although my estimates suggest that some fund managers have skill, they offer a rather bleak assessment about one’s ability to find them using past returns
alone. The problem is that the average fund has a negative alpha (−0.49% per
year in the four-factor model) and returns are so noisy compared to the amount
of cross-sectional variation in alphas. As a consequence, the market resolves
uncertainty about alphas very slowly. In the underlying normal–normal updating problem, the variance of the date-T posterior is
vT =
v0 σe2
.
T v0 + σe2
(14)
The posterior variance thus reaches some level vT in T = ( v1T − v10 )σe2 years,
an amount of time that increases linearly in the variance of returns. Based
on the four-factor model estimates, it takes 11.4 years for the variance of the
belief distribution to decrease by half. My results offer no advice on how to
find these skilled managers. To do so, one may have to study fund managers’
characteristics (Chevalier and Ellison (1999)), the way they tilt their portfolios
(Coval and Moskowitz (2001)), and the social networks they may be able to
benefit from (Cohen, Frazzini, and Malloy (2008)).
IV. Conclusions
Reverse survivorship bias affects the measurement of mutual fund managers’
stock-picking abilities. Because mutual funds often disappear following poor
performance, some funds disappear because of bad luck and not because their
true alpha is low. A fund that disappears because of a negative idiosyncratic
shock leaves behind an alpha estimate that is too low. Because no mechanism
eliminates just-lucky mutual funds, the observed distribution of alphas gives
too pessimistic a view about the prevalence of skill among actively managed
funds. In this paper I estimate a structural model to measure the size of the
bias using CRSP mutual fund data.
The size of the reverse survivorship bias in the mean, around 60 basis points
per year, is similar in magnitude but opposite in direction to that of the direct
survivorship bias that plagued early mutual fund databases.19 Thus, if we
take a database that excludes all dead funds and then start adding them back
19 Grinblatt and Titman (1989) estimate that the direct (upward) survivorship bias is “relatively
small” and between 10 and 30 basis points per year; Brown and Goetzmann (1995) get estimates
between 20 and 80 basis points; Elton, Gruber, and Blake (1996) find estimates between 71 and 91
basis points based on three-factor model alphas; and Carhart et al. (2002) show that the bias can
be as large as 1% in samples longer than 15 years.
Reverse Survivorship Bias
811
in, the (positive) direct survivorship bias decreases but the (negative) reverse
survivorship bias begins to drag fund-specific alpha estimates down. When all
dead funds are in, the mean alpha estimate is too low by approximately 60
basis points per year.
The question of whether any active fund managers have valuable information
is a central question in the field of empirical asset pricing. Although the average actively managed dollar must lose money because of the market-clearing
constraint and transaction costs,20 nothing rules out the possibility that some
active investors profit at the expense of others. Some mutual funds, for example, could be privy to a stream of signals that lets them trade profitably against
other mutual funds, individual investors, or pension funds. Reverse survivorship is important in examining whether any mutual fund managers trade on
valuable information. To properly measure the prevalence of skill among mutual fund managers, we need to consider the counterfactual of what would have
happened had no fund disappeared following poor performance.
I do not consider the equilibrium effects of money flows on alphas. In Berk
and Green (2004), alphas get pushed to zero because investors can perfectly
diversify all nonmarket risk and because the model’s mutual fund monopolistically sets its fee to choose its size. In Pástor and Stambaugh (2012), by contrast,
equilibrium alphas do not equate to zero because, first, investors cannot diversify away all risk, second, the active management industry is competitive, and
third, there is a finite number of mutual fund investors. Nevertheless, even
if expected alphas are sensitive to money flows, equilibrium considerations do
not alter the premise of reverse survivorship bias. As long as poorly performing mutual funds still shut down, as they do in the data, estimated alphas are
biased downward relative to the true alphas.
Reverse survivorship bias has implications beyond assessing the abilities
of mutual fund managers. First, hedge funds, pension funds, and individual
investors also often cease trading following poor performance.21 The reverse
survivorship bias argument applies to inferences about these investors’ abilities
as well. Individual investors’ true alphas, for example, are not necessarily as
low as what the distribution of estimated alphas suggests. Second, two-stage
inferences about the determinants of skill become problematic. Consider, for
example, a study that seeks to understand variation in CEO skill. If CEOs’
survival correlates with performance, a panel regression of performance with
CEO fixed effects22 will produce coefficient estimates that exhibit the same
reverse survivorship bias. This bias plays a further role if these estimates are
then used as inputs to investigate determinants of ability. If there is a variable
that correlates with the sensitivity of CEO survival to bad performance, this
variable will seem to explain variation in CEOs’ abilities because of the errorsin-variables problem. The same concern extends to any two-stage analysis of
mutual funds, hedge funds, and individual investors.
20
See French (2008) for a discussion.
See, for example, Brown, Goetzmann, and Park (2001), Seru, Shumway, and Stoffman (2010),
and Linnainmaa (2011).
22 See, for example, Bertrand and Schoar (2003).
21
812
The Journal of FinanceR
No simple resolution to reverse survivorship bias seems to exist. One cannot
resolve this bias, for example, by omitting funds that survive for fewer than T
years. The issue is that, although fund performance improves as T increases,
we do not know a priori what T should be. Moreover, even if we could choose
T so that the average estimated alpha coincides with the true mean, the distribution of estimated alphas would still be distorted. This distortion would
render inferences about the prevalence of skill invalid. A viable resolution to
the bias would be to estimate fund alphas by using as little data as possible
from the very beginning of funds’ lives, before any funds shut down. However,
even though in principle this approach circumvents the bias, the problem is
that these alpha estimates would be based on just a handful of data points and
thus would be too noisy to be of any use. One could also approach this bias as
an omitted variable problem and search for a variable that correlates with survival but not with true alphas. It is not obvious, however, that such a variable
exists. Another partial remedy might be to study the distribution of t-values
associated with the alpha estimates and not the alpha estimates themselves.
However this approach is similar to discarding short-lived funds. By construction, short-lived funds’ t-values are pulled toward zero because their alphas are
estimated with greater noise. It seems necessary to impose some structure to
draw valid inferences about the true alpha distribution. The problem of finding
a remedy that is not only effective and robust but also simple to implement is
left for future research.
Initial submission: May 9, 2011; Final version received: February 28, 2012
Editor: Campbell Harvey
REFERENCES
Barras, Laurent, Olivier Scaillet, and Russ Wermers, 2010, False discoveries in mutual fund
performance: Measuring luck in estimated alphas, Journal of Finance 65, 179–216.
Berk, Jonathan B., and Richard C. Green, 2004, Mutual fund flows and performance in rational
markets, Journal of Political Economy 112, 1269–1295.
Bertrand, Marianne, and Antoinette Schoar, 2003, Managing with style: The effect of managers
on firm policies, Quarterly Journal of Economics 118, 1169–1208.
Brown, Stephen J., and William N. Goetzmann, 1995, Performance persistence, Journal of Finance
50, 679–698.
Brown, Stephen J., William Goetzmann, Roger G. Ibbotson, and Stephen A. Ross, 1992, Survivorship bias in performance studies, Review of Financial Studies 5, 553–580.
Brown, Stephen J., William N. Goetzmann, and James Park, 2001, Careers and survival: Competition and risk in the hedge fund and CTA industry, Journal of Finance 56, 1869–1886.
Carhart, Mark M., 1997, On persistence in mutual fund performance, Journal of Finance 52,
57–82.
Carhart, Mark M., Jennifer N. Carpenter, Anthony W. Lynch, and David K. Musto, 2002, Mutual
fund survivorship, Review of Financial Studies 15, 1439–1463.
Carpenter, Jennifer N., and Anthony W. Lynch, 1999, Survivorship bias and attrition effects in
measures of performance persistence, Journal of Financial Economics 54, 337–374.
Chevalier, Judith, and Glen Ellison, 1999, Career concerns of mutual fund managers, Quarterly
Journal of Economics 114, 389–432.
Reverse Survivorship Bias
813
Cohen, Lauren, Andrea Frazzini, and Christopher Malloy, 2008, The small world of investing:
Board connections and mutual fund returns, Journal of Political Economy 116, 951–979.
Coval, Joshua D., and Tobias J. Moskowitz, 2001, The geography of investment: Informed trading
and asset prices, Journal of Political Economy 109, 811–841.
Cremers, Martijn K. J., Antti Petäjistö, and Eric Zitzewitz, 2008, Should benchmark indices have
alpha? Revisiting performance evaluation, Critical Finance Review, forthcoming.
Deuskar, Prachi, Joshua Matthew Pollet, Zhi Jay Wang, and Lu Zheng, 2011, The good or the bad?
Which mutual fund managers join hedge funds? Review of Financial Studies 24, 3008–3024.
Elton, Edwin J., Martin J. Gruber, and Christopher R. Blake, 1996, Survivorship bias and mutual
fund performance, Review of Financial Studies 9, 1097–1120.
Elton, Edwin J., Martin J. Gruber, and Christopher R. Blake, 2001, A first look at the accuracy of
the CRSP mutual fund database and a comparison of the CRSP and Morningstar mutual fund
databases, Journal of Finance 56, 2415–2430.
Evans, Richard B., 2010, Mutual fund incubation, Journal of Finance 65, 1581–1611.
Fama, Eugene F., and Kenneth French, 1993, Common risk factors in the returns of stocks and
bonds, Journal of Financial Economics 33, 3–56.
Fama, Eugene F., and Kenneth French, 2010, Luck versus skill in the cross section of mutual fund
returns, Journal of Finance 65, 1915–1947.
French, Kenneth R., 2008, Presidential address: The cost of active investing, Journal of Finance
63, 1537–1573.
Grinblatt, Mark, and Sheridan Titman, 1989, Mutual fund performance: An analysis of quarterly
portfolio holdings, Journal of Business 62, 393–416.
Jensen, Michael C., 1968, The performance of mutual funds in the period 1945–1964, Journal of
Finance 23, 389–416.
Khorana, Ajay, 1996, Top management turnover: An empirical investigation of mutual fund managers, Journal of Financial Economics 40, 403–426.
Khorana, Ajay, 2001, Performance changes following top management turnover: Evidence from
open-end mutual funds, Journal of Financial and Quantitative Analysis 36, 371–393.
Kosowski, Robert, Allan Timmermann, Russ Wermers, and Hal White, 2006, Can mutual fund
“stars” really pick stocks? New evidence from a bootstrap analysis, Journal of Finance 61,
2551–2595.
Kostovetsky, Leonard, 2009, Human capital flows and the financial industry, Working paper,
University of Rochester.
Linnainmaa, Juhani, 2011, Why do (some) households trade so much? Review of Financial Studies
24, 1630–1666.
Lynch, Anthony W., and David K. Musto, 2003, How investors interpret past fund returns, Journal
of Finance 58, 2033–2058.
Pástor, Ľuboš, and Robert F. Stambaugh, 2012, On the size of the active management industry,
Journal of Political Economy 102, 740–781.
Pástor, Ľuboš, Lucian A. Taylor, and Pietro Veronesi, 2009, Entrepreneurial learning, the IPO
decision, and the post-IPO drop in firm profitability, Review of Financial Studies 22, 3005–
3046.
Seru, Amit, Tyler Shumway, and Noah Stoffman, 2010, Learning by trading, Review of Financial
Studies 23, 705–739.
Williams, David, 1991, Probability with Martingales (Cambridge University Press, Cambridge,
MA).
Supporting Information
Additional Supporting Information may be found in the online version of this
article at the publisher’s web site:
Appendix S1: Internet Appendix