Transcript of "False discoveries in mutual fund performance presentation by me"
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Laurent BarrasMcGill University - Faculty of ManagementO. ScailletUniversity of Geneva - HEC; Swiss Finance InstituteRuss WermersUniversity of Maryland - Robert H. Smith School ofBusinessPresentation: Chinbat.DLecture: Dr. Tony Chieh-tse Hou30th May 2011Working Paper No. RHS-06-043
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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
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introductionthis working paper lead to False discoveries in Mutualfunds measures a alpha. 2076 fund but it is notsignificant number on this working paper Footer Text 12/10/2011 4
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To control for “False discoveries” of mutual funds that exhibitsignificant 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.
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This paper have new approach to controlling for FD in a multiple fund settingusing 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 2004Footer Text 12/10/2011 6
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1.The impact of luck on mutual fund performanceFooter Text 12/10/2011 7
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the large cross-section of funds in our database makes these estimatedproportions very accurate estimators of thetrue values, even when funds arecross-sectionally correlated. Monte Carlo simulations, that our simpleapproach 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 skillsFooter Text 12/10/2011 8
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To begin suppose that a population of M actively managed mutualfunds is composed of three distinct performance categories, where performace is due to stock-selection skills.Footer Text 12/10/2011 9
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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- HypothesisFooter Text 12/10/2011 10
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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 trueFooter Text 12/10/2011 11
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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.Footer Text 12/10/2011 12
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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-alphaFooter Text 12/10/2011 13
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How do to measure the frequency of FD in cross-sectional t-distribution Using this to determine expected proportion of skilled fundsing 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 Footer Text 12/10/2011 14
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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 themajority of skilled funds lie in the extreme right tail, then increasing the value of 7from 0.10 to 0.20 in Equation (3) would result in a very small increase in E(Tγ+), theproportion 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.. Footer Text 12/10/2011 15
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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 Footer Text 12/10/2011 16
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P-values larger than a sufficiently high threshold λ=0.6 show in the figureFooter Text 12/10/2011 17
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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 Footer Text 12/10/2011 18
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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 estimatesFooter Text 12/10/2011 19
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• 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. Footer Text 12/10/2011 20
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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 equalsThe average value of the FDR estimatorThe ‗‘fulltracks approach which assumes that πo=1, closely luck‘‘ true population value denoted by E(Fγ-) Footer Text 12/10/2011 21
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• 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 Footer Text 12/10/2011 22
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Mutual funds can have correlated residual if they ―herd‖ in theirWermers (1999) stockholdings or hold similar industry allocationKTVVW show that a complicated bootstrap 13 necessary to test the significance ofperformance of a fund located at a particular alpha rank, since this test depends on thejoint distribution of all fund estimated alphas—cross-correlated fund residuals must bebootstrapped simultaneously.However, in order to explicitly verify the properties of our estimators, we run aMonte-Carlo simulation. In order to closely reproduce the actual pairwise correlationsbetween funds in our dataset. we estimate the residual covariance matrix directly fromthe data, then use these dependencies in our simulations. In further simulations, weFooter Text 12/10/2011 23
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In this case, all fund p-value would be the same, and the p-valuehistogram 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*.Footer Text 12/10/2011 24
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Variable DescriptionR i,t Is month (t) excess return of fund (i) over the riskfreeR 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)Footer Text 12/10/2011 26
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Unconditional four –factor model for time-varying expose the market portfolio Variable Description 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 bondsFooter Text 12/10/2011 27
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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 matchedTime periodJanuary 1975 Footer Text December 2006 12/10/2011 28
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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 stockFooter Text 12/10/2011 29
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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 populationThis is simply due to very lucky outcomes forsmall proportion of the 1565 zero-alpha funds in the populationFooter Text 12/10/2011 31
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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 skillsFooter Text 12/10/2011 32
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• 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) Footer Text 12/10/2011 33
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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
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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-valuesFooter Text 12/10/2011 35
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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-termFooter Text result of left tail funds are truly unskilled. 12/10/2011 36
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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 modelFooter Text 12/10/2011 37
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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)
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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 fundsFooter Text 12/10/2011 39
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• 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 Footer Text 12/10/2011 40
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this performance advantage declines during later years,when the proportion of skilled funds decreasessubstantially, making them much tougher to locate.Therefore, find that the superior performance of the FDRportfolio is tightly linked to the prevalence of skilled fundsin the population. 41
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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 fundsFooter Text 12/10/2011 42
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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/Footer Text 12/10/2011 43
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FRD measure also has natural Bayesian interpretation Variable Description Gi Random variable which takes the value of (i) (-1,0+0) FDR+ Fdrγ+ Ti Positive and significant ofFooter Text 12/10/2011 44
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Gi also provides some relevant information for modeling the fund alpha prior distribution in an empirical Bayes settingWBMW (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. Footer Text 12/10/2011 45
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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 availableFooter Text 12/10/2011 46
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