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The Fundamental Law of Active Management says that the information ratio of a portfolio is theproduct of the skill of the manager in selecting securities and the breadth of the strategy. In thisconstruct, “skill” is measured by the correlation between the manager’s expected and realizedreturns, and “breadth” is the square-root of the number of independent positions in the portfolio.Grinold and Kahn [1999] state: “The Fundamental Law is designed to give us insight into activemanagement. It isn’t an operational tool.” nevertheless, the usefulness of these insights has ledpractitioners to seek to make it an operational tool. The main advance in this direction were broughtby Clarke, de Silva and Thorley [2002], who introduced the concept of transfer coefficient1 to accountfor the constraints that a manager faces when implementing a particular strategy. They also workedout the application of the Fundamental Law to ex-post performance attribution. However, despitethese advances, the practical application of the Fundamental Law of Active Management remainsdifficult.Firstly, one needs to have access to the manager’s return expectations, which are hard to obtain. Forqualitatively-oriented managers, return expectations are not formalized in a way that allows the 2computation of an information coefficient. Often quantitative managers will not make thisinformation available to external investors in order to avoid potential reverse engineering of themanager’s investment process. Moreover, when assessing a manager, using information that can beindependently verified is always preferable.Secondly, the Fundamental Law relies heavily on the assumption that managers follow the precepts ofthe active portfolio management theory developed in Grinold and Kahn [1999]. Hence, applications ofthe Fundamental Law to managers that do not follow these precepts will lead to questionable results.Finally, by seeking to transform the Fundamental Law into an operational tool, researchers have alsomade it more complex and difficult to interpret (See Clarke, de Silva and Thorley [2006] or Buckle[2004]). One of the key advantages of the analytical framework proposed in this paper is the simplicity 1 The transfer coefficient is the correlation between optimal unconstrained portfolio weights and actual portfolio weights2 The information coefficient is the correlation between the manager expected returns and realized returns September, 2010 2
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of its application. This approach only requires portfolio holdings and realized returns to analyze howthe breadth of the portfolio and the skill of a manager contribute to the information ratio of theportfolio. It does not require any information regarding the manager’s expected returns or assumptionswith respect to the manager’s investment process.A New Approach to Estimating Breadth: The Diversification FactorThe fact that the Fundamental Law is difficult to apply to many portfolios does not mean that it doesnot apply. Indeed, the risk adjusted returns of a portfolio, no matter what the investment process is,will always be a function of a manager’s skill and a strategy’s breadth. In order to apply theFundamental Law to a wider range of strategies, we sidestep the issues outlined in the introduction bystarting from portfolio weights rather than the manager’s expected alphas. The cornerstone of thisapproach consists of computing what would be the information ratio (IR) of the portfolio if there wereno diversification benefits (i.e. if all positions in the portfolio were perfectly correlated). Thedifference between the IR computed assuming that all positions are perfectly correlated and the actualIR represents the benefits from diversification. On the other hand, the IR of the portfolio in theabsence of diversification benefits represents the skill of the manager.If there were no benefit from portfolio diversification, the active risk of the portfolio would be theposition-weighted mean active volatility of the assets included in the portfolio3: wi i iWhere wi is the weight of asset i and i is the active volatility of asset i. The asset active volatility isdefined as the volatility of the return difference between the benchmark and the asset. Hence theimpact of diversification on the active risk of the portfolio can be measured by the ratio of theposition-weighted mean volatility ( ) and the actual portfolio active risk( ). The higher the 3 See appendix 1 for proofSeptember, 2010 3
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ratio is, the larger the impact of diversification. Hence, we define the diversification factor (DI) asfollows: DI= As a result, the active risk of a portfolio can be written as: DIThere are three elements that will affect the diversification of the portfolio: the number of positions,the correlation between these positions and the concentration of assets’ weights and assets’volatilities. The first element is fairly straightforward. More positions mean more diversification.However, not all positions bring the same level of diversification. Adding highly correlated positions tothe portfolio creates little diversification. Hence, the second element that impacts the diversificationof the portfolio is the level of correlation among the positions in the portfolio. The concentration ofthe weights in the portfolio will also impact the diversification factor. Indeed, if we consider aportfolio where one position’s weight is 20 times the weight of all the other positions, thediversification of this portfolio will be lower than that of a similar portfolio where all positions areequally weighted. The same logic applies to volatilities. If one asset’s volatility is 20 times that of theother assets, the portfolio return will be highly dependent on the outcome of this one bet and itsdiversification will be lower than if all volatilities were equal. Concentration of positions’ weights andconcentration of positions’ volatilities have the same impact on diversification. They both reduce it bymaking the portfolio more dependent on fewer positions. We regroup these two variables into onevariable, the concentration of the risk-weights (where risk-weights for position i is equal to ( wi i ) ).We show in appendix 1 that the portfolio diversification can be written as a function of these threefactors: DI= 1 (Equation 1) 1 n (i ) nSeptember, 2010 4
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Where: n is the number of positions in the portfolio is roughly a risk-weighted average correlation among the positions in the portfolios4 i is the risk-weight of position i, (i.e. i (wi i ) ) (i ) is the standard deviation of the risk weights across the portfolio, measuring the concentration in the portfolio.The diversification factor can be interpreted as a number of positions. Let’s illustrate this point withthe example of a portfolio that contains ten positions, where the active returns of these positions areindependent and each position has an equal risk-weight of 10%. In this example, = ( ) =0 andtherefore it follows from equation (1) that the diversification of the portfolio is 10. Hence, for anyportfolio, diversification represents the equivalent number of independent positions with equal risk-weights.A New Approach to Estimating SkillTo assess the impact of diversification on the risk-adjusted returns of a portfolio, we need to divide thenumerator and the denominator of the information ration by : IR * DI Where is the active return of the portfolio. Then we define the quantity as the skill (SK) of the manager which yields:IR SK * DI 4 see exact formula in appendix 1 September, 2010 5
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The Empirical Law of Active Management expresses exactly the same fact as the Fundamental Law,namely, that IR is a product of skill and diversification. However, using the Empirical Law, we do notneed to make any assumptions about the way in which the portfolio is constructed. The onlyinformation that is required to analyze the skill and diversification of the manager is a portfolio’sholdings and realized returns.Skill in the context of the Empirical Law represents the ability of the manager to allocate its risk-capital to securities with high risk-adjusted returns. To demonstrate this point, we need to observethat the portfolio alpha is also the weighted average alpha of each security in the portfolio (i.e. wi i ). Then we can express skill as the risk-weighted average of the information ratios of the iindividual positions in the portfolio: i w i iSK i i i IRi iWhere IRi is the information ratio of position i and i its risk-weight. In this framework, skillrepresents the ability of the manager to select securities that have a strong information ratio on astand-alone basis. In other words, it is a measure of the manager’s ability to maximize the risk-adjusted returns of the portfolio without using the benefit of risk diversification. It is important to notethat in the Empirical Law, the skill coefficient is defined more broadly than the information coefficientof the Fundamental Law. Whereas the information coefficient measures the pure stock picking abilityof the manager, the skill factor, in addition to stock picking, also incorporates the impact of portfolioconcentration and the size of the manager’s opportunity. We can illustrate this point by examining thecovariance between the risk-weights and the information ratio:5 ~ ~ 1 n i ~ ~ 1 n i ~ 1 Cov , IR ~ i * IRi ~ i * ~ IRi n i ~ 5 Note that this decomposition was inspired by Andrew Lo’s AP decomposition. See proposition 1 in Lo [2007]September, 2010 6
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~ ~Where i and IRi are the risk weights and information ratios of each of the securities in themanager’s investment universe and ~ n is the number of securities in the investment universe. Wedefine investment universe as the entire set of securities in which the manager can invest. (The ~denotes that the variable covers the entire universe as opposed to securities included in the portfolioonly.) By including all the securities in the manager’s universe in our covariance computation, we canfully capture the relation between risk-weights and information ratios. Indeed, securities that are inthe manager universe but not in the portfolio do not directly contribute to the risk adjusted return ofthe portfolio; however, they do contain information with respect to the manager’s stock-picking skill. IR 0 . ~Given that the universe is a passive and information-less portfolio, we can assume that i i * IR is our skill factor. ~ ~The expectation of the risk-weights and information ratio products: i i iHence, we can re-write the skill of the manager in function of the covariance between securities riskweights and information ratios: ~ ~ ~ SK n Cov( , IR ~ ~Finally, if n is large enough so that IR ) 1, we obtain: ~ ~ ~ ~SK ( , IR n )This decomposition of the skill factor is helpful to understand the three elements that are important indetermining the skill of a manager: ~ ~ ( , IR : The correlation between risk-weights and securities information ratios is similar to the information coefficient in the Fundamental Law; it represents the ability of the manager to pick-securities. ~ n : The size of the universe of securities that the manager will analyze. It represents the opportunity set from which the manager can create value added for his clients. ~ ) : A measure of the portfolio concentration. This parameter illustrates the fact that a more concentrated portfolio will better leverage the manager’s stock picking ability.September, 2010 7
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This last point illustrates the fact that diversification also forces managers to go down the list of theirinvestment ideas and implement positions with less expected return.Comparison of the Fundamental and Empirical LawsThe Empirical Law and Fundamental Law possess two key differences. First, the Empirical Law sidesteps the concepts of information coefficient and transfer coefficient. The skill of a manager dependson the relation between risk-weight and risk-adjusted performance (see Exhibit 1). As we discussed inthe introduction, for manager-of-managers, the information necessary to compute a transfercoefficient or information coefficient is not available. In practice, fundamental managers cannotcleanly separate transfer coefficient and breadth. The concept of TC makes sense for quantitativemanagers using models that provide expected returns on wide ranges of securities irrespective ofwhether they can be implemented or not. On the other hand, given the cost of bottom-up research,fundamental analysts focus on trades that can be implemented. In an investment firm that runs longonly portfolios, one rarely sees analysts spending much time on overvalued assets. This makes thedistinction between transfer coefficient and diversification less meaningful for fundamental managers.Exhibit 1: Information Triangle Fundamental Law Empirical Law Manager Proprietary Information Ex‐ante Alphas Information Transfer Coefficient Coefficient Performance SK Portfolio Positions (or Risk –Weights) Public Information for Traditional Funds This diagram is inspired from Figure 1 of Clarke de Silva Thorley (2002).September, 2010 8
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Therefore, the Empirical Law does not distinguish between the two. If a position is not in the portfolio,we do not seek to know if this is due to lack of breadth or poor transfer coefficient. However, givenaccess to the unconstrained optimal weights of a quantitative manager, we have the flexibility toreintroduce the concept of transfer coefficient. By recomputing the manager IR, DI and SK with theseoptimal weights (noted IR*, DI* and SK*), we can define our transfer coefficient as:6 IRTC , which yields IR TC * SK * DI * * * IRThe second key difference between the two approaches is that the Fundamental Law makes theassumption that the portfolio manager uses a mean variance optimization to construct the portfolio.The Empirical Law does not rely on any formal assumption regarding the manager’s portfolioconstruction process, and uses the fund’s IR as the yardstick of a manager’s success.Can you get indigestion from the diversification free lunch?Warren Buffett has observed that “Diversification is a protection against ignorance. It makes very littlesense for those who know what theyre doing.”7 Indeed, diversification forces the skilled manager toreduce the portfolio’s concentration in the positions with the highest expected return in order toreduce risk. If one has perfect foresight there is no risk, and there is little to be gained in reducing riskthrough diversification. However, for investment managers with less than perfect foresightdiversification makes a lot of sense. As we noted previously, while diversification also forces portfoliomanagers to add to the portfolio positions with decreasing risk-adjusted return.Moreover, bottom-up stock picking requires specialized knowledge in the securities being analyzed.Thus far, we have implicitly assumed that skill is a fixed quantity that is inherent to the investmentprocess and can be leveraged into as many trades as one can implement. While this somewhat reflects 6 Note than in theory this TC can be greater than one. Since the objective function that produces the optimal weights of themanager is not necessarily the IR, it is conceivable that the constrained IR be greater than the unconstrained IR.7 Source: http://en.wikiquote.org/wiki/Warren_Buffett September, 2010 9
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the way quantitative managers create value, for fundamental managers more diversification means lessdepth of research. Therefore, a fixed SK coefficient does not reflect the tradeoff between quality ofcoverage (or depth) and breadth of coverage, which fundamental managers do face. Fundamentalmanagers could leverage their SK factor across other trades only if they hire more analysts of the samequality; however, producing a good analyst is costly and/or time-consuming. In the long term, it isfeasible to grow a research team, but there are usually increased inefficiencies that come with largerorganizations.8Concentration also has negative consequences for risk adjusted returns. The most obvious disadvantageof concentration is that it will result in a higher risk. Hence, the optimal balance betweendiversification and skill also depends on the client ability to take risk and to diversify this risk. Forinstitutional investors with long investment horizons and the resources to effectively detect skillfulmanagers, diversification is relatively cheap to obtain. In theory, such investors should seekconcentrated portfolios where skill is not diluted by diversification. For retail investors who have tosupport higher transaction costs and cannot perform extensive due diligence, investing in fewdiversified portfolios makes more sense. For a given fund size, more concentration will automaticallyresult in less liquid positions, which, everything else being equal, will have a negative impact onperformance.9Empirical Analysis of the Relation between Diversification and SkillTwo empirical studies have found that mutual funds with high levels of concentration tend tooutperform funds with lower levels of concentration. Kacperczyk et al.[2005, 2007] found that small- 8 Chen et al. [2004], find results consistent with this view. They observe that small sized funds outperform large sized funds andargue that part of this difference in performance is due to organizational diseconomies related to hierarchy costs.9 See Becker and Vaughan [2001] for illustration of this point, see also Chen et al. [2004] who find that a key variable to explaindifference in performance between small funds and large funds is liquidity. September, 2010 10
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cap funds with higher levels of industry concentration were generating greater alpha than funds withlower levels of industry concentration. Cremers and Petajisto [2008] measured the fund concentrationwith its active share and found that funds with high concentration outperformed funds with lowconcentration. The measures of concentration used in these two studies focus on the concentration ofportfolio weights. Unlike the DI coefficient these measures of concentration do not capture the impact 10of concentration in positions’ volatilities and correlations among positions.In order to understand the relation between portfolio diversification and skill, we applied the EmpiricalLaw of Active Management to a universe of 2,798 U.S. mutual funds invested in domestic equities from1980 to 2006. The first obstacle we faced in assessing the skill of a manager was defining thebenchmark. While the concept of benchmark is ubiquitous in today’s asset management industry, itwas not the case in the 80’s and early 90’s. Hence, finding an accurate and consistent definition of afund’s benchmark across all periods is difficult. Moreover, the active returns of a fund measuredagainst its stated benchmark often contain common factor bets such as value, size or momentum thatcan distort the active returns coming purely from stock picking skills. For these reasons, we used factormodels to determine the appropriate benchmark. We labeled as alpha or active return any returns thatcould not be explained by the factors of the model we used. Therefore, the choice of model and thefactors that we included in the model were critical. Our objective was to assess the stock-picking skillof a manager, as a result all known “priced factors” should be included in our model and excluded fromthe manager’s alpha. “Priced factors” are factors for which empirical studies have demonstrated theexistence of positive risk adjusted returns; namely: market, value, size and momentum. To assess afund’s exposure to these factors, we used the same four-factor model as Carhart [1997]. In order toensure that our results were not dependent on the model, we also used two alternative models thatare based on similar factors but that use different techniques to assess the exposures. The first w 10 Kacperczyk et al. define as industry concentration as: portfolio wimarket portfolio 2 (where wi is the weight of industry i in the i iportfolio and in the market portfolio). Cremers and Petajisto use active share, which, is the sum of the absolute deviations of theportfolio weights from the benchmark weights. Active Share w i i portfolio wibenchmarkSeptember, 2010 11
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alternative model is based on the characteristic based benchmarks developed by Daniel et al. [1997]. 11The second alternative model is based on the 25 Fama-French portfolios sorted on value and size.(See appendix 2 for a more precise description of the models and the data).Exhibit 2 shows the average alpha, skill, diversification and number of securities per fund across eachof the periods we examined. Alpha and skill are net of the funds’ expenses ratios. The decline inmutual fund alphas that we observed in this table was very important. The average mutual fund in ouruniverse went from generating a 29 bps positive alpha in the first part of the a 90’s to a negative 1.3%alpha in the second part of the 90’s. This trend is consistent with the four-factor alphas reported byKosowski et al. [2006] for similar periods and is also reported by Fama and French [2009]. Suchvariations in the average alpha and skill of mutual fund managers are surprising considering that sincethe 1990’s, the Investment Company Institute (ICI) has reported a decline in U.S. mutual fund expenseratios.12 The second interesting trend that we observed in Exhibit 2 was a general increase in the 13number of positions per portfolio and therefore in the diversification of the funds. This increase inportfolio diversification among mutual fund managers is consistent with the inroads that modernportfolio theory made during the 80’s and 90’s among practitioners and the development of passivemutual funds. In addition, this period also witnessed a significant development of informationtechnology solutions that drastically reduced the operational costs of running portfolios with a largenumber of positions.Exhibit 2: Number of Funds, Mean Alpha, Skill, Diversification and number of positions perportfolio for each Period.The benchmark of each portfolio is estimated with theCarhart [1997] model. 11 See Daniel et al. [1997] and Wermers [2004] for more details on DGTW portfolio. The DGTW benchmarks are available viahttp://www.smith.umd.edu/faculty/rwermers/ftpsite/Dgtw/coverpage.htm. The Fama-French portfolio are available via:http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html12 See: Investment Company Institute Research Department Staff [2009] and Collins [2009]13 Cremers and Petajisto [2009] report similar results using active share. September, 2010 12
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Number of Number of Period Funds Alpha SK DI Positions 1980‐1984 137 0.58% 0.007 36 71 1985‐1989 270 0.43% 0.004 45 84 1990‐1994 588 0.29% 0.001 62 96 1995‐1999 1519 ‐1.28% ‐0.013 65 125 2000‐2006 2378 ‐1.28% ‐0.008 70 148 1980‐2006 2798 ‐0.84% ‐0.007 63 135We see two non-exclusive explanations for the decline in skill that we observed among equity mutualfunds. First, mutual fund managers were victims of their own success. The net asset value of equitymutual funds as a percentage of the capitalization of the U.S. stock market increased from 3% to 40%between 1980 and 200814. If mutual fund managers are a representative sample of the equity marketinvestors, Sharpe’s [1991] arithmetic of active management entails that as a whole, mutual fundmanagers should have a negative alpha roughly equal to the fees they charge. As the industry growscloser to representing the U.S. stock market, it is more difficult for the industry as a whole tooutperform the market. The second explanation for this phenomenon is that the quality of U.S. Equity mutual funds hasdeclined. The industry would have responded to the increase in demand for mutual funds that tookplace during the 80’s and 90’s by creating sub-par products. The market for asset managementservices, like the Lemons market described by Akerlof [1970], is characterized by an asymmetry ofinformation.15 Since it is difficult for investors to distinguish a skilled from an unskilled manager, thereis a large incentive for unskilled managers to flood the market and distribute low cost investmentstrategies that add no value while charging active management fees. This incentive is even greaterwhen demand expands and new customers with less financial acumen enter the market. Under suchcircumstances, unskilled managers, or “closet indexers”, will want to take the minimum level of riskthat allows them to charge active management fees. On the other hand, skilled managers have anincentive to take relatively more concentrated positions. Such situations should result in a positive 14 To compute these percentages, we used net asset value of equity mutual funds reported in the 2009 Investment Company FactBook (Investment Company Institute Research Department Staff [2009]) and the Wilshire 5000 index capitalization as a proxy forthe capitalization of the U.S. stock market.15 See Foster and Young [2008] September, 2010 13
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skew in the distribution of alphas because skilled managers with concentrated portfolios will generatehigh alphas, while unskilled managers will generate moderate negative alphas. Under this hypothesis,the industry responded to an increasing demand for mutual funds by increasing its “lemon” productionand crafting strategies with little information content and with the look and feel of effective activeinvestment strategies. In order to assess this hypothesis, we examined the cross-sectional relationbetween skill and diversification by splitting our universe of mutual funds into quintiles ofdiversification and compared the skill of the funds located in each quintile.Exhibit 3: Alpha and Skill by Quintiles of DiversificationThe benchmark of each portfolio is estimated with the Carhart [1997] model. Skill by Diversification(DI) Quintile (1980‐2006) Quintile 1: Quintile 5: Low DI Quintile 2 Quintile 3 Quintile 4 High DIMean 0.004 ‐0.007 ‐0.011 ‐0.011 ‐0.011T‐stat (Mean Skill) 2.02 ‐6.34 ‐10.77 ‐11.98 ‐14.50Skill Difference : Quintile 1 ‐ Quintile 2,3,4 or 5 0.012 0.016 0.015 0.015T‐stat (Skill Difference) 4.78 6.45 6.50 6.51Standard Deviation 0.05 0.03 0.02 0.02 0.02Skewness 0.97 ‐0.21 0.15 ‐0.51 ‐0.62Kurtosis 7.06 5.03 5.17 4.46 4.98Number of Observations 560 559 560 559 560In Exhibit 3, we observe that funds located in the first diversification quintile did exhibit significantlymore skill than any of the other quintiles. On average, the skill of these funds was greater than zerowhile the skill in all the other quintiles of the distribution was significantly less than zero. All thedifferences in skill between quintile 1 and each of the four other quintiles are statistically significant.Exhibit 3 also indicates a positive skew in the distribution of the skill of low diversification portfoliomanagers. This fact indicates that our result could be explained by survivorship bias. Lessdiversification implies more risk. Since we required at least 36 months of returns to include a fund inour analysis, it is possible that the low diversification funds that severely underperformed rapidly losttheir clients and did not survive long enough to be included in our analysis. Note that the positive skewin the distribution of alpha can be explained by factors other than survivorship bias. As we observedearlier, the presence of closet indexers or “lemons” in the mutual fund market is consistent with suchdistribution of alphas.September, 2010 14
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In order to test the impact of a possible survivorship bias on our results, we created an artificial biasagainst the top performing funds by eliminating funds in the right tail of the alpha distribution in orderto have a symmetric distribution. We removed from our analysis all funds whose alpha was greater thanthe absolute value of the first percentile of the alpha distribution.After removing these funds, the skew of the distribution became slightly negative; however, the skilldifference between the first quintile of managers and the rest of the universe remained stronglysignificant (see top-panel of exhibit 4). In order to further test the robustness of our results, wedesigned a more conservative artificial survivorship bias, where we removed all funds with alphagreater than the absolute value of the fifth percentile of the alpha distribution. Under this definition ofthe artificial survivorship bias, we eliminated from our sample all the funds with annualized alphagreater than 6.7% per annum. Nevertheless, the bottom panel of exhibit 4 shows that lowdiversification funds still exhibited significantly higher skill under both definitions of the artificialsurvivorship bias. Hence, the relationship that we observed between skill and diversification cannot beexplained away by survivorship bias.Exhibit 4: Alpha and Skill by Diversification Quintiles From 1980 to 2006 Excluding top PerformingFundsOut-performing funds are funds with an alpha greater than the absolute value of the first or fifth percentile of the alphadistribution. The benchmark of each portfolio is estimated with the Carhart model. Skill by Diversification(DI) Quintile (1980‐2006) Excluding Top‐ Performing Funds Quintile 1: Quintile 5: Low DI Quintile 2 Quintile 3 Quintile 4 High DI Mean ‐0.001 ‐0.009 ‐0.011 ‐0.011 ‐0.010 Absolute Value of 1st T‐stat (Mean Skill) ‐0.48 ‐7.55 ‐11.27 ‐12.40 ‐14.27 Skill Difference : Quintile 1 ‐ Quintile 2,3,4 or 5 0.008 0.011 0.010 0.010 Percentile Alpha Alpha less than T‐stat (Skill Difference) 3.61 5.10 5.18 4.94 Standard Deviation 0.04 0.03 0.02 0.02 0.02 Skewness ‐0.04 ‐0.46 ‐0.18 ‐0.54 ‐0.63 Kurtosis 5.35 4.62 3.97 4.43 5.00 Number of Observations 554 554 554 554 554 Mean ‐0.007 ‐0.01 ‐0.012 ‐0.011 ‐0.011 Absolute Value of 5th T‐stat (Mean Skill) ‐4.08 ‐9.28 ‐11.62 ‐12.62 ‐14.53 Skill Difference : Quintile 1 ‐ Quintile 2,3,4 or 5 0.003 0.005 0.005 0.004 Percentile Alpha Alpha less than T‐stat (Skill Difference) 1.80 2.73 2.58 2.31 Standard Deviation 0.04 0.03 0.02 0.02 0.02 Skewness ‐0.59 ‐0.60 ‐0.27 ‐0.55 ‐0.84 Kurtosis 5.53 4.31 3.85 4.42 4.49 Number of Observations 543 542 543 542 543September, 2010 15
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The relation between skill and diversification also held across the three different types of models thatwe used to define the alpha for the 1980-2006 period. Exhibit 5 also shows strongly significant results ifwe use the characteristic based benchmarks or the 25 Fama-French portfolios to establish the alpha ofeach fund. However, the relationship between skill and diversification was not consistent acrossperiods. Breaking up the 1980-2006 periods into 5 distinct sub-periods, we did not find that lowdiversification funds outperformed in each sub-period (see exhibit 5). We did not find any significantrelation between diversification and skill during the eighties and the early nineties.Exhibit 5: Relation between Skill and Diversification across periods and modelsFor each period and each model, we show the mean of the t-stats for the differences in skill between quintile one and quintilestwo to five (Mean t-stat). We also show the smallest of the four t-stats (Min t-stat). For the Carhart model in the period 1980-2006, we can read from table 3 that the four t-stats are 4.78, 6.45, 6.50 and 6.51. The mean of these four t-stats: 6.06 andminimum: 4.78 are displayed in the top left cell of table 5. FF-25 denotes the results that we obtained with the 25 Fama-Frenchportfolios, DGTW indicate that we used the characteristic based benchmarks of Daniel et al. [1997] 1980 2000 1995 1990 1985 1980 to to to to to to Periods: 2006 2006 1999 1994 1989 1984 Mean t‐stat 6.06 7.46 2.11 ‐0.40 0.68 ‐0.28 Carhart Min t‐stat 4.78 6.90 1.84 ‐0.64 0.07 ‐0.71 Mean t‐stat 3.59 2.23 4.08 0.86 ‐0.57 ‐1.52 DGTW Min t‐stat 1.95 1.78 2.54 0.50 ‐2.26 ‐1.98 Mean t‐stat 4.29 3.15 5.33 0.91 1.15 ‐1.03 FF‐25 Min t‐stat 3.66 1.75 4.50 0.04 0.20 ‐1.25Number of Observations per Quintile 559.6 475.6 303.8 117.6 54 27.6Since we could not find a significant relation between diversification and skill in every sub-period, wefound it difficult to conclude that concentrated managers had an intrinsic advantage over lessconcentrated managers. Instead, the result of this study is better interpreted using Lo’s [2004]Adaptative Market Hypothesis. The ability of U.S. mutual funds to beat the market evolved throughtime under the influence of the structure of the U.S. market for mutual funds. Our interpretation ofthese results is based on two elements. First, concentration among mutual fund managers hasdecreased since the 80’s. Second, our analysis shows that since the mid-90’s, skill was more likely tobe found in more concentrated portfolios. These two points taken together indicated that as thedemand for mutual funds increased, the industry responded by producing funds with higherdiversification but lower information content, also known as “closet-indexers.” As a result, the averageSeptember, 2010 16
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skill and alpha of U.S. equity mutual funds decreased. In most industries, increases in demand are metby increases in price; however, no such price increase took place in the asset management industryduring the 90’s.16 We infer from our data that the asset management industry took advantage of thisincrease in demand by cutting the information content of its products while leaving its managementfees unchanged.ConclusionThe Fundamental Law of Active Management played a very important role in the development ofquantitative asset management. The application of this law was designed by quantitative investmentmanagers for quantitative investment managers, and is difficult to apply to other types of processes.However, the result of the Fundamental Law of Active Management applies to all investment managers.The Empirical Law of Asset Management presented in this paper seeks to generalize and extend thistool to a wider range of portfolios while conserving the key insights that made this concept a success.The application of this analytical framework to U.S. equity mutual funds reveals a general decline inskill. We found two explanations for this decrease in skill. Firstly, the mutual fund industry was avictim of its own success, as mutual funds represent a greater share of the market capitalization. It isnow close to impossible for the industry as a whole to outperform the market. Secondly, the decreasein skill was accompanied by an increase in diversification. Since the mid-90’s, we also observe aninverse relationship between skill and diversification. We suggest that the increase in diversificationdoes reflect a decrease in the quality of the information content of U.S. mutual funds. 16 See: Investment Company Institute Research Department Staff [2009]September, 2010 17
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References:Akerlof, George. “The Market for "Lemons": Quality Uncertainty and the Market Mechanism.” TheQuarterly Journal of Economics, Vol. 84, No. 3 (1970), pp. 488-500.Beckers, Stan, G. Vaughan. “Small is Beautiful.” Journal of Portfolio Management, Vol. 27, No. 4(2001), pp. 9-17.Berk, Jonathan, R. Green. “Mutual Fund Flows and Performance in Rational Markets.” Journal ofPolitical Economy, Vol. 112, No. 6 (2004), pp. 1269-1295.Buckle, David. “How to calculate breadth: An evolution of the Fundamental Law of ActiveManagement.” Journal of Asset Management, Vol. 4, No. 6 (2004), pp. 393-405.Daniel, Kent, M. Grinblatt, S. Titman and R. Wermers. “Measuring Mutual Fund Performance withCharacteristic Based Benchmarks.” The Journal of Finance, Vol. 52, No. 3 (1997), pp. 1035-1058. Carhart, Mark M. “On Persistence in Mutual Fund Performance.” The Journal of Finance, Vol. 52, No. 1(1997), pp. 57-82.Chen, Joseph, H. Hong, M. Huang and J. Kubik. “Does Fund Size Erode Mutual Fund Performance? TheRole of Liquidity and Organization.” The American Economic Review, Vol. 94, No. 5 (2004), pp. 1276-1302.Clarke, Roger, H. de Silva, S. Thorley. “Portfolio Constraints and the Fundamental Law of ActiveManagement.” Financial Analysts Journal, September/October 2002, Vol. 58, No. 5 (2002), pp. 48-66.“The Fundamental Law of Active Portfolio Management.” Journal of Investment Management, Vol. 4,No. 3 (2006), pp. 54–72.Collins, Sean. “Trends in the Fees and Expenses of Mutual Funds, 2008.” Investment Company InstituteResearch Fundamentals, Vol. 18, No. 3 (April 2009)Cremers, Martjin, A. Petajisto “How Active is Your Fund Manager? A New Measure that PredictsPerformance.” Review of Financial Studies, Vol. 22, No. 9 (September 2009), pp. 3329-3365.Foster, Dean, P. Young. “The Hedge Fund Game: Incentives, Excess Returns, and Performance Mimics.”University of Oxford, Discussion Paper Series (2008).Grinold, Richard C. “The Fundamental Law of Active Management.” The Journal of PortfolioManagement, Vol. 15, No. 3 (Spring 1989), pp. 30-37.Grinold, Richard C., R. Kahn. Active Portfolio Management: A Quantitative Approach for ProducingSuperior Returns and Controlling Risk, 2nd ed. McGraw-Hill, 1999.Investment Company Institute Research Department Staff, “2009 Investment Company Fact Book”, 49thEdition, (2009).Kacperczyk, Marcin T., C. Sialm, L. Zheng. “On Industry concentration of Actively Managed EquityMutual Funds.” The Journal of Finance, 60 (2005), pp. 1983-2012.Kacperczyk, Marcin T., C. Sialm, L. Zheng. “Industry Concentration and Mutual Fund Performance.”Journal of Investment Management, Vol. 5, No. 1 (2007)Kosowski, Robert, A. Timmermann, R. Wermers, A. White. “Can Mutual fund “Stars” Really Pick Stocks?New Evidence from a Bootstrap Analysis.” Journal of Finance, Vol. 51, No. 6 (2006), pp. 2551-2595.September, 2010 18
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Fama, Eugene, K. French. “Luck versus Skill in the Cross Section of Mutual Fund Alpha Estimates.”working paper (2009)Lo, Andrew. “Risk management for hedge funds: introduction and overview.” Financial AnalystsJournal, Vol. 57, No. 6 (November/December 2001), pp.16-33.“The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective.” Journal ofPortfolio Management, 30 (2004), pp. 15-29.“Where Do Alphas Come From?: A New Measure of the Value of Active Investment Management.”Journal of Investment Management, forthcoming.Sharpe, William. “The Arithmetic of Active Management” Financial Analysts Journal, Vol. 47, No. 1(January/February 1991), pp. 7-9.Shleifer, Andrei, R. Vishny. “The Limits of Arbitrage.” The Journal of Finance, Vol. 52, No. 1 (1997),pp. 35-55.Pollet, Joshua, M. Wilson. “How Does Size Affect Fund Behavior?” working paper (2007).Van Nieuwerburgh, Stijn, L. Veldkamp. “Information Acquisition and Under-Diversification.” workingpaper (2008).Wermers, Russ. “Are Mutual Fund Shareholders Compensated for Active Management "Bets"?” workingpaper (2003).“Is Money Really Smart? New Evidence on the Relation Between Mutual Fund Flows, Manager Behavior,and Performance Persistence.” working paper (2004).September, 2010 19
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Appendix 1: derivation of the Empirical Law of Active Managementwi is the weight of security i in the portfoliow is the vector of all the securities weights in the portfolio. is the covariance matrix of the securities’ active return. If V is a classic covariance matrix of assetsreturns of size (n x n), and B is a (n x n) matrix which contains n times the (n x1) vector of benchmarkweights and I is the identity matrix then can be obtained in the following way: ( I B)V ( I B). 17 The active variance of the portfolio ( w w ) can be decomposed into the risk of the portfolio if allsecurities were perfectly correlated and a second term which represents the benefits fromdiversification.w w wi2 i2 wi w j i j i , j i i j iw w w wi w j i j (1 i , j 1) 2 i i 2 i i j i 2 w w wi i wi w j i j (1 i , j ) i i j iWe separated the risk of the portfolio into two parts: First, the average volatility, which is the risk ifall positions were perfectly correlated. And second, the diversification benefits, which is the decreasein risk due to positions in the portfolio being less than perfectly correlated.Let’s note the average risk of a position in the portfolio as: 17 The I B matrix contains n columns of active weights for n portfolios that are fully invested in one and only one asset: 1 0 ... 0 w1b w1b ... w1b 0 b b 1 ... 0 w2 b w2 ... w2 I B w1active w2 ... wn active active ... ... ... ... ... ... ... ... b b 0 0 ... 1 wn wn ... wn b September, 2010 20
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wi i i wi iLet’s note risk weights: i with these notations the active variance of a portfolio is: 2 w w i 2 i j ( i , j 1) i i j iWe can re-write the active variance of the portfolio in the following way: w w 2 2 i j i , j i j i j i i j i Or: w w 2 1 i j i , j i j i j i i j i The term i j i i j i , j i j can be broken down into two components: i j i- the product of the risk weights = i j i i j (Equation 1)- risk weighted correlation = i j i i j i, j (Equation 2)We can rewrite equation 1 as: 1 i j i i j i i j i j i i i which can be expressed infunction of the variance of the risk weights: 2 1 1 i j i i2 i i i j i i i n n 2 1 1 1 1i i j 1 n 2 i n 2 n i n i j i i i September, 2010 21
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2 1 1i i j 1 n i n i j i n 1 i j i i j n n * Variance(i )Hence the concentration of the risk weights is a direct function of the variance of the risk weights: n 1 i j nVar ( i ) i j i nThe higher the dispersion of the risk weights (that is the dispersion of portfolio weights andvolatilities), the higher the active risk of the portfolio. The second element of this diversification termis the correlations among the positions of the portfolio (equation 2). i j i i j i, j i j i, j i j iWe note i the average correlation of asset i with the other assets in the portfolio, weighted by the 18risk-weight of each asset in the portfolio. i j i , j hence we obtain: i j i , j i i ~ j i i j i iWe call this quantity the risk weighted correlation: i i i i j i i j i, jIf we put everything back together: n2 1 1 w w 2 1 nVar ( ) 2 nVar ( ) n n 18 Note that the sum of the risk-weights excluding asset i does not add-up to one. The correct weighted average should be:i j i , j The correlation term becomes: (1 ) i i i j i (1 i ) iSeptember, 2010 22
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The diversification factor of the portfolio is defined as: 1 DI 1 ~ nVar ( i ) nAppendix 2:The mutual funds quarterly holdings were obtained from the Thomson-Reuters mutual fund database.The funds monthly returns were gathered from the CRSP mutual fund database and the securities’returns were obtained from the CRSP U.S. Stock database. We used the MF Link mapping developed byRuss Wermers to map funds in the Thomson-Reuters database to funds in the CRSP database.We examined separately 6 periods: 1980-1984, 1985-1989, 1990-1994, 1995-1999, 2000-2006 and 1980-2006. To be included in the analysis of a period a fund should have at least 36 months of returns duringthat period. For each period, we used the following steps to compute the diversification and skillfactors. First we computed the return of each fund by taking the asset-weighted average return of allshare-classes. If a share-class asset value was missing on a given date, we took the equal-weightedaverage return of all the share classes.Second, we assessed the fund benchmark using each of the following three pricing models: the four-factor model used by Carhart [1997] (henceforth: “Carhart”); a second pricing model based on theDaniel, Grinblatt, Titman and Wermers characteristic based benchmarks (“DGTW”); and a third pricingmodel based on the 25 Fama-French portfolios sorted on Value and Size (“FF-25”). 19For the Carhart model, we performed a regression of the fund returns on the four factors of the model(market, value, size and momentum). The exposure to the four factors resulting from this regressiondefined the benchmark of the fund. As a result, the active return of the fund was equal to the alpha ofthe fund computed with the Carhart model. 19 See Daniel et al. [1997] and Wermers [2004] for more details on DGTW portfolio. The DGTW benchmarks are available viahttp://www.smith.umd.edu/faculty/rwermers/ftpsite/Dgtw/coverpage.htm. The Fama-French portfolio are available via:http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.htmlSeptember, 2010 23
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For DGTW, we performed a regression of the fund returns on the market portfolio and one of the 125characteristic based benchmarks and the market factor. The DGTW benchmarks are the result of a sortof U.S. stocks on value, size and momentum exposures. We choose a unique DGTW benchmark which,when combined with the market portfolio, maximizes the variance explained by the regression. Themarket portfolio that we used with each of these models was the value-weighted return on all NYSE,AMEX, and NASDAQ stocks (from CRSP), minus the one-month Treasury bill rate. We followed the sameprocedure for the FF-25 portfolios.For each period we computed the historical active volatility of each asset in each fund using thesecurities monthly returns. Asset active volatility is the standard deviation of the difference betweenthe asset returns and the benchmark returns. If a security does not have at least 24 months of returns,we set its active volatility to the equal weighted average volatility of the fund’s assets with more than24 months of returns. Every quarter, we compute the mean weighted volatility. The fund meanweighted volatility for a period is the mean of each of the quarterly mean weighted volatility duringthat period.For a quarterly observation of the portfolio to be included in our analysis, we required that at least 75%of the assets in the portfolio be recognized in the CRSP U.S. Stock database. For a fund to be includedin the analysis of a period, we required at least 8 valid quarters. In addition, we excluded from theanalysis, funds that had an average across all quarters of a period more than 5% of assets notrecognized in the CRSP U.S. Stock database. We excluded funds that, on average across all quarters ofa given period, had fewer than 10 positions, or less than 15 million dollars of asset under management.Using the three models, we computed alpha (active return) and residual risk (active risk) during each ofthe periods we analyzed. We then computed the weighted mean asset volatility of each portfolio usingthe monthly historical return of each asset in the fund.Finally, we took the mean of these quarterly volatilities across the entire period to obtain a proxy forthe fund asset weighted volatility ( ) for the period and used the following decomposition to computethe fund’s skill and diversification.September, 2010 24
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DI= and SK Model ChoicesWhile the Carhart model is the standard pricing model used in the literature, the findings of Cremersand Petajisto [2009] raised questions about the suitability of this model to establish a pertinentperformance benchmark. Unlike Cremers and Patajisto, we obtained strongly significant results withthe Carhart model. However, we still wanted to ensure that our results were not dependent on thetype of model we were using, so we recomputed our results with two alternative models. The DGTWbenchmarks offer a very granular classification of U.S. stocks into benchmarks based on their value,size and momentum characteristics. These DGTW benchmarks have fewer stocks than a typicalbenchmark and therefore more idiosyncratic risk. We also used a coarser asset classificationframework: the 25 Fama-French value and size sorted portfolios. The disadvantage of the Fama-Frenchportfolio is that we did not account for the funds’ momentum exposure. Using these three models, wecomputed alpha (or active return), residual risk (or active risk), skill and diversification of eachportfolio during each of the periods we analyzed.In theory, we should also have excluded from the active returns the returns due all non-priced factorsto which the funds are systematically exposed. However, there is an infinite set of non-priced factorsthat could be included in the benchmark definition. Therefore, using non-priced factors was notpractical. Since exposures are computed based on the historical co-movement of the fund’s returns andthe factor returns, we would have picked up spurious relations between the funds’ returns and thefactors if we used a larger number of non-priced factors. As a result, we would have obtained a verynarrow definition of the fund’s alpha.September, 2010 25
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