Download Empirical Research: The Discontinuity in Pooled Distribution

Empirical Research: The Discontinuity in Pooled Distribution
of Mutual Fund Monthly Returns
ZHANG Li, ZHANG Shuguang
The University of Science and Technology of China, 230026
[email protected]
Abstract: The pooled cross-sectional, time-series distribution of mutual fund returns in our country
features a discontinuity at zero: the number of small gains far exceeds the number of small losses. We
discuss how do this discontinuity generate and give empirical evidence of the probability that mutual
fund managers purposely avoid reporting losses.
Keywords: Mutual funds, pooled distribution, discontinuity, kink, misreport
1. Introduction
When studying the shape of the pooled distribution of monthly mutual fund returns, we have taken
notice of the jump around zero: the number of small gains far exceeds the number of small loses. Then
we try to interpret this phenomenon and investigate the probability of misreporting by empirical
research. In this article, our analysis doesn’t rely on a factor model to recognize abnormal returns, which
is important because the existing literature has not come to a consensus on appropriate risk adjustment.
2. Data
We obtain our data from Wind Information mutual fund database for the period from January 2002 to
March 2009. The initial sample contains all open-end mutual funds in the January 2002 to March 2009
period. From this initial sample, we exclude all funds that we cannot confidently describe as being
diversified, domestic equity mutual funds. Thus, we remove money market, bond and income, and
specialty mutual funds, such as sector or international funds, to obtain our sample of diversified,
domestic equity mutual funds. After applying these exclusionary criteria, 220 mutual funds and 8347
mutual fund return observations remain in our sample.
We use histograms of mutual fund returns to test whether the underlying densities possess significant
discontinuities. In selecting the bin width, we minimize the mean squared error between the true
distribution and the histogram and follow Silverman (1986) by setting bin width
b = α x1.364 min(σ , Q / 1.340) N −1/5
(1)
where σ is the empirical distribution’s standard deviation, Q is its interquartile range, N is the number
of observations, and α is a scalar that depends on the type of underlying distribution assumed.
Devroye (1997) shows through simulation that the definition in (1) is robust to alternative distributional
assumptions. Here we set α = 0.776, corresponding to a normal distribution.
Figure 1 displays a histogram of the mutual fund returns. The two solid black vertical bars include
observations just below and just above zero. The feature is readily apparent: there is a jump in the
distribution at zero, that is, the bar just above zero is significantly higher than expected and the bar just
below zero appears deflated.
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Figure 1.Full sample (2002.1~2009.3),n=8347,Bin size 99 bps
While the histogram provides a visual evaluation of our hypothesis, we require a rigorous statistical test
for a discontinuity in the distribution. We use the approach proposed by Nicolas P.B. Bollen and
Veronika K. Pool (2009) to calculate the value of a standard normal test statistic to determine whether
the actual number of observations around zero is significantly different than expected.
i
Serving nonparametric kernel density which captures the features of the empirical distribution as a
reference, and using it to estimate the expected number of observations in the bins around zero of
the histogram, under the null hypothesis that no discontinuity exists. Figure 2 shows the kernel
density.
ii
Using a Gaussian kernel to fit the kernel density, that is
）
）
Λ
f ( a) =
1
N
∑Φ
Nb i =1
( )
xi − a
,
(2)
b
where b is the bandwidth of the kernel that is identical to the optimal bin width above, xi is the
data in the bin, Φ is the standard normal density function, and N is the number of observations.
Figure 2. Kernel density
） Integrating the kernel density along the boundary of the two bins around zero to get the probability
iii
that an observation will reside in it. We denote this probability as p. Then we calculate the value of
a standard normal test statistic
xi − Npi / Npi (1 − pi )
(3)
Table 1 shows the statistics of the two bins around zero which we focus on. We can see the value of the
bin just above zero is significant at the 1% level. Thus the underlying distribution isn’t smooth, and
there is a jump around zero.
Table 1
The bin
Just above
Just below
Number of
observations
784
447
proportion
9.39%
5.36%
3. Analysis
429
Test
statistics
20.99
-0.19
1% level
reject
accept
3.1. Inducement Searching
The discontinuity at zero is actually exists. One explanation for this result is that some mutual fund
managers purposely avoid reporting losses. The “cockroach theory” implies that investors will overreact
to the slightest bit of bad news, such as a negative monthly mutual fund return, because they fear that
more bad news lurks. Faced with redemption pressure, mutual fund managers must take care of their
fund’s returns. We examine the relation between investor fund flows and reported losses by running the
regression (4):
'
'
'
Fi , t = β 0 + β 0 Ii , t − 1 + ( β 1 + β 1 Ii , t − 1) Ri , t − 1 + ( β 2 + β 2 Ii , t − 1) NPi , t − 1 + ε i , t
4
Fi , t = [TNAi , t − TNAi , t − 1)(1 + Ri , t )] / TNAi , t − 1
(5)
Where Fi , t is the percentage fund flow for fund i in year t, Ii , t − 1 is an indicator variable that equals
one if fund i was age three years or less in year t-1 and zero otherwise, Ri , t − 1 is the cumulative
（）
annual return, and
xi − Npi / Npi (1 − pi )
is the number of months with positive returns. TNAi , t
is the total net assets of fund i in year t.
Table 2
Parameter
β1
β 1'
β2
β 2'
Estimate
t value
1.83745
0.695
0.04416
1.07876
0.382
0.61879
1.245
-0.07201
-0.189
Adjusted R
2
Table 2 shows the results. For funds of all ages there is a positive relation between fund inflows and
months that fund returns are positive. So the desire to consistently achieve positive returns in any market
environment may induce mutual fund managers not to report losses. They may report small positive
returns when they actually suffer losses.
3.2. Further Empirical Evidence
Before accepting the discontinuity as evidence of misreporting and altering data, we must exclude that it
is generated by mutual fund managers’ skill of avoiding losses. If managers create the discontinuity
through their skill, then we might expect some degree of performance persistence. The difficulties in
testing this hypothesis are that we have no way of knowing the timing and magnitudes of overstatements
likely vary across funds and over time. Figure 3 is the histogram of bimonthly returns. It is quite smooth,
and is qualitatively different from that of monthly returns in Figure 1.This result is consistent with some
overstatements are reversed the following month under the assumption that skill persists from month to
month.
Figure 3. Bimonthly returns
Then we recognize the mutual funds that have abnormal monthly returns by a statistic characteristic
named “kink”, which is proposed by Nicolas P.B. Bollen and Veronika K. Pool (2009).They define
“kink” as the percentage of a fund’s observations that are between 0 and 50 basis points minus the
percentage of observations that are negative and no less than minus 50 basis points. The null hypothesis
is that the probability of drawing a return from either bin is the same, that is, the value of kink is zero.
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We use a standard test of proportions that controls for the number of observations. We depart funds into
two groups: funds that feature a significantly positive kink are labeled abnormal funds and others are
labeled normal funds. Table 3A are groups of funds sorted by kink computed over the first half of each
fund’s life. Table 3B are groups of funds sorted by kink computed over the second half of each fund’s
life.
Table 3A.Devided by pre kink
Table 3B.Devided by post kink
Listed in table 3 are cross-sectional means of summary statistics: average monthly return µ , standard
deviation of monthly return σ , Sharpe ratio Sharpe, skewness Skew, excess kurtosis Kurt, and a
measure of discontinuity kink. For abnormal funds in Table 3A, the Sharpe ratio is -0.2202 in pre and
0.2160 in post. The relevant kink falls from 0.2300 to 0.0329. For abnormal funds in Table 3B, the
Sharpe ratio is 0.3584 in pre and 0.0896 in post. The relevant kink rises from 0.0122 to 0.1402.Two
points can be drawn from the table. First, for the funds in the abnormal groups, their average kink is
significant when their performance is poor, that is, the average Sharpe ratio is low. As the two statistics
exhibit a negative correlation, we couldn’t regard fund managers’ skill as the reason of the generation of
abnormal small gains. If so, the value of kink should exhibit somewhat persistence during the fund’s life
and likely be a positive relation with Sharpe ratio. This result is consistent with that part of managers
purposely avoids reporting losses to beautify the performance when the fund is in a rainy day. They may
write back overstatements with larger losses during a month with a loss or smaller gains during a month
with a gain.
Second, if pay attention to the ages of funds that feature discontinuities, we can find that most kinky
values distribute mainly in years 2007 and 2008, which is an acutely fluctuant period of security market
in china. We know that fund managers are confronted with redemption pressure, especially for young
funds. When the market fluctuates acutely, they must try harder to steady their funds’ returns and make
the fund’s performance look tolerable. Otherwise chain reactions in stock market and the fund’s net
value caused by redemption will make situation worse.
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4. Conclusion
We have demonstrated the statistical significance of the discontinuity by comparing the actual number
of observations around zero to an expected number given by an underlying smooth density. Of the 220
funds, 46 funds have featured distortion data during different period of their lives. This result is at least
partly due to mutual fund managers purposely avoiding reporting losses, when fund returns are at their
discretion and when their reported returns are not closely monitored. As a result, investors may
underestimate the potential risk of losses in the future and in aggregate allocate more capital to mutual
funds than is warranted. Regulators should supervise mutual funds’ operation more closely to ensure
investors’ interests and guide the healthy development of whole industry.
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