False discoveries in mutual fund performance presentation by me

Laurent Barras
McGill University - Faculty of Management
O. Scaillet
University of Geneva - HEC; Swiss Finance Institute
Russ Wermers
University of Maryland - Robert H. Smith School of
Business

Presentation: Chinbat.D
Lecture: Dr. Tony Chieh-tse Hou
30th May 2011

Working Paper No. RHS-06-043

Introduction
1952 Harry Markowitz he came with idea fund manager have to
look at Risk

1964 Willian Sharpe CAPM introduced a risk-adjusted measure
portfolio performance.
[Rp-RF]/SD=excess return/risk

Then look at definition of Beta measures the volatility a portfolio
versus market portfolio
Then look at definition of Beta came up it measures the
Also managers outperform market return that called alpha if
volatility a portfolio versus market portfolio B=1
manager that ability outperform market alpha > 0 positive
however manager underperform market alpha< 0 negative
Alpha is a risk-adjusted measure of active managers
performances. the return of a benchmark is subtracted in order
to consider relative performance, which yields Jensen alpha.

Footer Text

introduction
this working paper lead to False discoveries in Mutual
funds measures a alpha. 2076 fund but it is not
significant number on this working paper

Footer Text 12/10/2011 4

To control for “False discoveries” of mutual funds that exhibit
significant alphas by luck alone.
separates fund into
• 1 Unskilled
• 2 zero-alpha
• 3 skilled even dependencies in cross-fund estimated
alphas.

75% of Funds a zero-alpha consistent with the Berk and Green 2004
equilibrium. Prior to 1996 find a significant proportion skilled positive
alpha but almost none by 2006 also show that controlling for false
discoveries substantially improves the ability to find with persistent
performance.

This paper have new approach to controlling for FD in a multiple fund setting
using a econometric tests
• Estimated alpha t-statistic /truly negative or positive alphas /
• Determine the frequency of FD /proportion of zero-alpha/
• P-value for individual fund
• Monte-Carlo experiment accurate partition of mutual fund into zero-alpha
unskilled, and skilled funds
• Cross-sectional dependencies among fund estimated alpha

The monthly return of 2076 actively
Measure estimate managed U.S open-end, domestic-equity
mutual funds between 1975-2006
Long-term performance 75.4% are zero-
alpha fund managers having stockpicking
ability
24.0% are unskilled (true a <0) while only
0.6 are skilled (true a>0) Berk and Green
2004

1.The impact of luck on mutual fund
performance


the large cross-section of funds in our database makes these estimated
proportions very accurate estimators of thetrue values, even when funds are
cross-sectionally correlated. Monte Carlo simulations, that our simple
approach is quite robust to cross-fund dependencies.

High proportion of unskilled funds prior to measure flows

These skilled funds are concentrated in the extreme right tail of
cross-sectional estimated alpha distribution which indicates that a
very low p-value is accurate of short-run fund manager skill

Aggressive Growth Highest proportion of skilled managers

Growth& Income No funds exhibit skills


To begin suppose that a population of M actively managed mutual
funds is composed of three distinct performance categories, where
performace is due to stock-selection skills.


Each of the above skill groups from performance estimates for
individual fund? suppose first use the T-statistic as performance
measure

This procedure, simultaneously applied across all funds is multiple-
Hypothesis


Level of 5%, should expect that 5% of these zero-alpha funds will have
significant estimated alphas-some of them unlucky (α<0) while other are
lucky significant with (α>0) but all will be FD funds with significant
estimated alphas, but zero-alpha true


Panel a shows the distribution of the fund t-statistic across the tree skill
group. The true four factor alpha equal to (-3.2%) and +3.8% per year for
the unskilled and skilled funds are centered at -2.5 and +3

the left and right tails of the cross-sectional estimated alpha determine the
frequency of FD the only parameter needed is proportion of zero-alpha
funds in population π0.

Does this area consist merely
of skilled funds as definedshaded region in left
The
above? overestimates the proportion of
unskilled

The same applies to Clearly not because some funds can
Panel B displays the very that the positive and significant region the three
unskilled The probability exhibit
funds that t-statistic distribution it is a mixture of
distribution this example set 75%, -23%, 2% to matchA average estimated value
lucky of the right tail of Panel zero alpha
estimated t-stat of skilled fund
funds positive and significant
over final 5 years of sample
is lower that ti=-1.65 is less thatestimated t-stat
0.001%

Measure performance with a limited sample data, therefore unskilled and
skilled funds cannot easily distinguished from zero-alpha


How do to measure the frequency of FD in cross-sectional t-distribution

Using this to determine
expected proportion of
skilled fund

sing equation that E(Fγ)=3.75
(πo) =75% zero-alpha funds
Exhibits luck equal
expected proportion of
γ/2=10%
lucky funds

Using a simple Monte-Carlo experiment demonstrate that
approach provides a much more accurate partition of mutual
funds into zero-alpha, unskilled and skilled funds


this paper-determining the location of truly skilled (or unskilled) funds in the tails of the cross-sectional t-distribution—
can only be achieved by evaluating Equations (3) and (4) at several different values of 7. For instance, if the
majority of skilled funds lie in the extreme right tail, then increasing the value of 7
from 0.10 to 0.20 in Equation (3) would result in a very small increase in E(Tγ+), the
proportion of truly skilled funds, since most of the additional significant funds, E(Sγ-), would be lucky funds.

Probability of including a zero-alpha in the portfolio equals
2.5% (85%) in population 2*85=1.7, 75*2.5=1.8 the lucky funds
equal to ((1.7/3.5))*3.8=1.8 per year..


Measuring luck in a group setting, show as equation (2) is the estimator
of the proportion πo, of zero-alpha funds in population

• The recent estimation approach developed by Storey (2002) called False
discovery rate
• The FDR approach is very straightforward, as its sole input are (two-sided)
p-values associated with the (alpha) t-statistic of each of the M funds.
• FDR uses information from the center of the cross-sectional t-distribution
/which dominated by zero-alpha/

FDR technique is to estimate these
frequencies-from the sample t-statistics

P-values larger than a sufficiently
high threshold λ=0.6 show in the
figure


measure the proportion of total
area

Is close to 75% which is the true value
of π0

Bootstrap procedure introduced by Storey 2002, it minimizes the
estimated mean-squared error (MSE) of zero-alpha funds

Using equation (6) the estimated proportion of
unskilled and skilled funds equal to


Finally estimate the proportions of unskilled and skilled funds in the
entire population as

This method is entirely data-
driven, some flexibility in choice of
γ*, as long as it sufficiently high

Select with a bootstrap procedure which
minimizes the estimated MSE of skilled and
unskilled alphas denoted by

Simply setting γ*, to prespecified values
0.35-0.45 produces similar estimates


• The previous section has followed two alternative
approaches when estimating the proportion of unskilled
and skilled funds
• Panel A of figure 1 in the proportions π0,πA-,and πA+.
for each zero-alpha fund the ratio (0.23/2) is held fixed
to11.5 in figure 1, to assure that the proportion of skilled
funds remains low compared to the unskilled funds
• Second uses these sampled t-statistics to estimate the
proportion of unlucky, lucky and skilled, unskilled funds
under each approach
• First two steps 1000 times then compare the average
value of each estimator with true population value.


Assuming that πo=0, the “no luck”
approach consistently underestimates
Panel C,D the true value propotion of true proportion of zero-alpha funds
the unskilled, skilled funds
decrease by construction when πo=75% no luckis higher(πlarge
exhibits a o >0)
upward bias estimate the total proportion of unskilled, skilled funds
E(Tγ-)+E(Tγ+) underestimates

Panel B are exactly same since
proportion of true values equals

The average value of the FDR estimator
The ‗‘fulltracks approach which assumes that πo=1,
closely luck‘‘ true population value
denoted by E(Fγ-)


• In addition to the bias properties exhibited by FDR
estimators, their variability is low because of the large
cross-section of funds (M-2,076)
• Proportion estimator that depends on proportion of p-
values higher than significant λ*, the law of Large
Numbers drives it close to its true value with large
sample size
• Λ*=0.6 threshold and π=75%the standard deviation of
75% is low as 2.5% with independent p-value


Mutual funds can have correlated residual if they ―herd‖ in their
Wermers (1999) stockholdings or hold similar industry allocation

KTVVW show that a complicated bootstrap 13 necessary to test the significance of
performance of a fund located at a particular alpha rank, since this test depends on the
joint distribution of all fund estimated alphas—cross-correlated fund residuals must be
bootstrapped simultaneously.

However, in order to explicitly verify the properties of our estimators, we run a
Monte-Carlo simulation. In order to closely reproduce the actual pairwise correlations
between funds in our dataset. we estimate the residual covariance matrix directly from
the data, then use these dependencies in our simulations. In further simulations, we


In this case, all fund p-value would be the same, and the p-value
histogram would not converge to the true p-value distribution, as shown in Figure 2.
Clearly, we would make serious errors no matter where we set λ*.


Variable Description
R i,t Is month (t) excess return of fund (i) over the riskfree
R m,t Month (t) excess return on (CRSP NYSE/AMEX/NASDAQ
value-weighted market portfolio
(Rsmb,t) Month (t) return on zero-investment factor-mimicking
(Rhml,t) portfolios for size, book-to-matket, and momentum
(Rmom,t)


Unconditional four –factor model for time-varying expose the market portfolio

Zt-1 Denotes the Jx1 vector of predictive variables measure
at the end of month (t) 1975-2006
Bʹ Is the Jx1 vector coefficient
Four variables One month T-bill yield: dividend yield of CRSP Value
weighted NYSE/AMEX stock index
The term spread, peroxide by the difference between
yield on 10-year Treasury and three month T-bill, and
the default spread proxied by the yield difference
between Moody’s Baa-rate and Aaarated corporate
bonds


2076 open-end, domestic equity
mutual funds existing for 60
months

Growth (1304 funds)

Aggressive Growth (388 funds)

Growth & Income (384 funds)

Two data base are
matched

Time period
January 1975
Footer Text
December 2006
12/10/2011 28

Estimated annualized alpha

Panel A,B estimated alphas for
each category are negative
from -0.45%to-0.60%

Aggressive Growth funds tilt
toward small capitalization, book-
to-market,momentum stock


However significant alpha does not always meancomprised of unskilled
That left-tail funds are overwhelmingly
and not merely manager‖
―skilled fund unlucky funds have a relatively many significant alpha
There are long fund life
12.7 years on average funds in the right tail 8.2
(170funds) in total population

This is simply due to very lucky outcomes for
small proportion of the 1565 zero-alpha funds
in the population


Growth funds show similar results to overall
universe of funds 76.5% have zero-alpha
(1565 funds) 23.5% are unskilled
Long-term
existence of this
G&I funds consist of largest proportion of category of
unskilled funds (30.7%) actively-
managed funds,
which includes
―value funds‖
and ‗‘core
funds‘‘ these
poor results.

A-Growth funds, 3,9% of them show true skills


• Entire period 1975-2006 may not accurately describe the
performance generated by industry before this rapid
expansion

• At the end of each year from 1989-2006, estimate the
proportion of unskilled and skilled funds using the entire
return history for each fund up to that point time

• On December 31, 1989 to December 2006 15year funds

• 1975-1989 (427 funds) basically in 32 years 75-06 (2076
funds)


Unskilled funds rises from
9.2% to 24.0% of the entire
universe of fund
1989to 2006, skilled funds
declines from 14.4% to 0.6%`

During the 1990‘s generate very poor performance The growth industry has also affected the
that 24% of them are unskilled, while none are
skilled
alpha of the older funds created before Jan
1990

During 1997-2006 34.8% of these older
funds are truly unskilled

Panel B shows the yearly count of funds included in the
Footer Text estimated proportion 12/10/2011 34

To test for short-run mutual fund performance in five years, beginning
from 1977-1981 ending with 2002-2006 sub period have 60 monthly
return observations'

Five years records together across all time periods to represent the
average experience of an investor in randomly chosen fund during a
randomly chosen five-year period total of 3311 p-values


Superior performance is rare but
does exist compare to long-term

In left tail unskilled and not merely
Almost entirely addition
unlucky zero-alpha funds is 5
zero-alpha funds are lucky
times in proportion of unlucky
Center of the distribution
produces almost no funds
additional skilled funds

The short-term result are similar to the long-term
Footer Text result of left tail funds are truly unskilled. 12/10/2011 36

The BG model implies that larger and older funds should exhibit lower alphas,
since they have presumably grown (or survived) to the point where they
provide no superior alphas, net of fees—partly due to flows that followed past
superior performance

BG also implies that flow should disproportionately move to truly skilled funds
and that these funds should exhibit the largest reduction in future skill

The result are strongly supportive of BG model


Previous analysis reveals that only 2.4% of the funds are skilled over short-
term Can it detect these skilled funds over time, in order to capture their
superior alphas?

Expected proportion of lucky
funds included in portfolio at
significance level γ:

FDR+ target level z+, in persistence test : z+= 10%,20%,50%,70%and
90%

Storey (2002) implement the following straightforward estimator of the FDR

Portfolio formation date is Dec 1979 to
Dec 2005 (5years return observed)

Higher FDR target means increase in the proportion of funds
included

Result reveal that FDR portfolios successfully detect fund with short-
term skills

IR=p-value/STD

The result sharply illustrate the short-term nature of truly
outperforming funds


• How the estimate alpha of the portfolio FDR10% evolves
over time using expanding windows.
• The initial value 1989 Dec 31 yearly of out-of-sample /α/
• Measure over the period 1980-1989,
• Final value, 2006 Dec 31 is the yearly out-of-sample
alpha
• Entire 1980-to-2006 measured over


this performance advantage declines during later years,
when the proportion of skilled funds decreases
substantially, making them much tougher to locate.
Therefore, find that the superior performance of the FDR
portfolio is tightly linked to the prevalence of skilled funds
in the population.

41

This result indicates that only a small fraction
of fund managers have stock picking skill
/24%to 4.5/

The proportion of pre-expense
unskilled funds remains equal to
zero until end 2003

Poor skill cannot explain
unskilled funds

F.F model have substantial risk premium
over the period /9.4%/

CAPM model have substantial loading on the size and the book-to-market
factor positive premium over sample period /3.7% and 5.7%per year/


FRD measure also has natural Bayesian interpretation

Gi Random variable which takes the value
of (i) (-1,0+0)
FDR+ Fdrγ+
Ti Positive and significant of


Gi also provides some relevant information for modeling the fund alpha
prior distribution in an empirical Bayes setting
WBMW (2001)

A full Bayesian estimation of fdr* requires to posit prior distributions for
the proportions -1,0 and +1. and for the distribution parameters of Ti for each skill
group. This method, based on additional assumptions (including independent p-
values) as well as intensive numerical methods, is applied by Tang. Ghosal, and
Roy (2007) to estimate the traditional FDR in a genomics study.


FDR technique to show that proportion of skilled fund managers has
diminished rapidly over 20 years, while the proportion of unskilled funds
has increased substantially

This paper also shows that Long-term actively managed mutual fund
underperformance due to long-term survival of truly underperforming
fund

Most active managed funds provide positive zero net of expense alphas,
putting them at least on passive funds. But it is still puzzling

Most key concept is econometric method in this paper work so far
unskilled, zero-alpha, skilled in our data decreased by 2006 potentially
wide applications in finace. It can be used to control luck in any setting
in which a multiple-hypothesis test run and a large sample is available


False discoveries in mutual fund performance presentation by me

False discoveries in mutual fund performance presentation by me

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