1. The Empirical Law of Active Management
Perspectives on the Declining Skill of U.S. Fund Managers
Edouard Senechal, CFA
Forthcoming Journal of Portfolio Management Fall 2010
Abstract:
This paper proposes a new analytical framework for assessing the breadth (or diversification) and skill
of a portfolio manager: the Empirical Law of Active Management. This framework requires no
assumption regarding the manager’s asset returns expectations or investment process. It generalizes
Grinold’s [1989] Fundamental Law of Active Management, and creates an analytical framework for
measuring skill and diversification in a consistent manner for large cross sections of funds. We applied
this framework to analyze the evolution of skill and diversification since 1980 for 2,798 U.S. mutual
funds and found that skill has been declining among U.S. mutual funds while diversification has been
increasing. We put forward two explanations for this decrease in skill. First, the growth in mutual fund
assets has made it more difficult for the industry as a whole to outperform the market. Second, our
analysis of the relationship between skill and diversification led us to conclude that the U.S. mutual
fund industry responded to an increase in the demand for its products by creating funds with less
information content.
This is the submitted version of the following article: “The Empirical Law of Active Management; Perspectives on the
Declining Skill of U.S. Fund Managers”, Edouard Sénéchal, Journal of Portfolio Management, Fall 2010, Copyright © 2010,
Institutional Investor, Inc. which will be published in its final form at: http://www.iijournals.com/toc/jpm/current
The author would like to thank Brian Singer for his encouragements, inputs and constructive critics, Russell Wermers for
helpful suggestions, Eugene Fama and the participants at the spring 2009 “Research Project: Finance” course at the
University of Chicago Booth School of Business, Edwin Denson, Greg Fedorinchick and Mabel Lung for helpful suggestions.
September, 2010 1

2. The Fundamental Law of Active Management says that the information ratio of a portfolio is the
product of the skill of the manager in selecting securities and the breadth of the strategy. In this
construct, “skill” is measured by the correlation between the manager’s expected and realized
returns, 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 active
management. It isn’t an operational tool.” nevertheless, the usefulness of these insights has led
practitioners to seek to make it an operational tool. The main advance in this direction were brought
by Clarke, de Silva and Thorley [2002], who introduced the concept of transfer coefficient1 to account
for the constraints that a manager faces when implementing a particular strategy. They also worked
out the application of the Fundamental Law to ex-post performance attribution. However, despite
these advances, the practical application of the Fundamental Law of Active Management remains
difficult.
Firstly, one needs to have access to the manager’s return expectations, which are hard to obtain. For
qualitatively-oriented managers, return expectations are not formalized in a way that allows the
2
computation of an information coefficient. Often quantitative managers will not make this
information available to external investors in order to avoid potential reverse engineering of the
manager’s investment process. Moreover, when assessing a manager, using information that can be
independently verified is always preferable.
Secondly, the Fundamental Law relies heavily on the assumption that managers follow the precepts of
the active portfolio management theory developed in Grinold and Kahn [1999]. Hence, applications of
the 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 also
made 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 weights
2
The information coefficient is the correlation between the manager expected returns and realized returns
September, 2010 2

3. of its application. This approach only requires portfolio holdings and realized returns to analyze how
the breadth of the portfolio and the skill of a manager contribute to the information ratio of the
portfolio. It does not require any information regarding the manager’s expected returns or assumptions
with respect to the manager’s investment process.
A New Approach to Estimating Breadth: The Diversification Factor
The fact that the Fundamental Law is difficult to apply to many portfolios does not mean that it does
not 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 the
Fundamental Law to a wider range of strategies, we sidestep the issues outlined in the introduction by
starting from portfolio weights rather than the manager’s expected alphas. The cornerstone of this
approach consists of computing what would be the information ratio (IR) of the portfolio if there were
no diversification benefits (i.e. if all positions in the portfolio were perfectly correlated). The
difference between the IR computed assuming that all positions are perfectly correlated and the actual
IR represents the benefits from diversification. On the other hand, the IR of the portfolio in the
absence 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 the
position-weighted mean active volatility of the assets included in the portfolio3:
   wi i
i
Where wi is the weight of asset i and  i is the active volatility of asset i. The asset active volatility is
defined as the volatility of the return difference between the benchmark and the asset. Hence the
impact of diversification on the active risk of the portfolio can be measured by the ratio of the
position-weighted mean volatility (  ) and the actual portfolio active risk(  ). The higher the  
3
See appendix 1 for proof
September, 2010 3

4. ratio is, the larger the impact of diversification. Hence, we define the diversification factor (DI) as
follows:

DI=

As a result, the active risk of a portfolio can be written as:


DI
There 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 to
the portfolio creates little diversification. Hence, the second element that impacts the diversification
of the portfolio is the level of correlation among the positions in the portfolio. The concentration of
the weights in the portfolio will also impact the diversification factor. Indeed, if we consider a
portfolio where one position’s weight is 20 times the weight of all the other positions, the
diversification of this portfolio will be lower than that of a similar portfolio where all positions are
equally weighted. The same logic applies to volatilities. If one asset’s volatility is 20 times that of the
other assets, the portfolio return will be highly dependent on the outcome of this one bet and its
diversification will be lower than if all volatilities were equal. Concentration of positions’ weights and
concentration of positions’ volatilities have the same impact on diversification. They both reduce it by
making the portfolio more dependent on fewer positions. We regroup these two variables into one
variable, 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 three
factors:
DI=
1 (Equation 1)
1
   n  (i )
n
September, 2010 4

5. 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 with
the example of a portfolio that contains ten positions, where the active returns of these positions are
independent and each position has an equal risk-weight of 10%. In this example,  =  ( ) =0 and
therefore it follows from equation (1) that the diversification of the portfolio is 10. Hence, for any
portfolio, diversification represents the equivalent number of independent positions with equal risk-
weights.
A New Approach to Estimating Skill
To assess the impact of diversification on the risk-adjusted returns of a portfolio, we need to divide the
numerator 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

6. 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 not
need to make any assumptions about the way in which the portfolio is constructed. The only
information that is required to analyze the skill and diversification of the manager is a portfolio’s
holdings 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 observe
that 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
i
individual positions in the portfolio:
i

 w  i i
SK   i i
  i IRi
  i
Where IRi is the information ratio of position i and i its risk-weight. In this framework, skill
represents the ability of the manager to select securities that have a strong information ratio on a
stand-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 note
that in the Empirical Law, the skill coefficient is defined more broadly than the information coefficient
of the Fundamental Law. Whereas the information coefficient measures the pure stock picking ability
of the manager, the skill factor, in addition to stock picking, also incorporates the impact of portfolio
concentration and the size of the manager’s opportunity. We can illustrate this point by examining the
covariance 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

7. ~ ~
Where i and IRi are the risk weights and information ratios of each of the securities in the
manager’s investment universe and ~
n is the number of securities in the investment universe. We
define 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 portfolio
only.) By including all the securities in the manager’s universe in our covariance computation, we can
fully capture the relation between risk-weights and information ratios. Indeed, securities that are in
the manager universe but not in the portfolio do not directly contribute to the risk adjusted return of
the 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
i
Hence, we can re-write the skill of the manager in function of the covariance between securities risk
weights 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 in
determining 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

8. This last point illustrates the fact that diversification also forces managers to go down the list of their
investment ideas and implement positions with less expected return.
Comparison of the Fundamental and Empirical Laws
The Empirical Law and Fundamental Law possess two key differences. First, the Empirical Law side
steps the concepts of information coefficient and transfer coefficient. The skill of a manager depends
on the relation between risk-weight and risk-adjusted performance (see Exhibit 1). As we discussed in
the introduction, for manager-of-managers, the information necessary to compute a transfer
coefficient or information coefficient is not available. In practice, fundamental managers cannot
cleanly separate transfer coefficient and breadth. The concept of TC makes sense for quantitative
managers using models that provide expected returns on wide ranges of securities irrespective of
whether 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 long
only portfolios, one rarely sees analysts spending much time on overvalued assets. This makes the
distinction 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

9. 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, given
access to the unconstrained optimal weights of a quantitative manager, we have the flexibility to
reintroduce the concept of transfer coefficient. By recomputing the manager IR, DI and SK with these
optimal weights (noted IR*, DI* and SK*), we can define our transfer coefficient as:6
IR
TC  , which yields IR  TC * SK * DI
* *
*
IR
The second key difference between the two approaches is that the Fundamental Law makes the
assumption 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 portfolio
construction 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 little
sense for those who know what they're doing.”7 Indeed, diversification forces the skilled manager to
reduce the portfolio’s concentration in the positions with the highest expected return in order to
reduce risk. If one has perfect foresight there is no risk, and there is little to be gained in reducing risk
through diversification. However, for investment managers with less than perfect foresight
diversification makes a lot of sense. As we noted previously, while diversification also forces portfolio
managers 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 investment
process 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 the
manager 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

10. the way quantitative managers create value, for fundamental managers more diversification means less
depth of research. Therefore, a fixed SK coefficient does not reflect the tradeoff between quality of
coverage (or depth) and breadth of coverage, which fundamental managers do face. Fundamental
managers could leverage their SK factor across other trades only if they hire more analysts of the same
quality; however, producing a good analyst is costly and/or time-consuming. In the long term, it is
feasible to grow a research team, but there are usually increased inefficiencies that come with larger
organizations.8
Concentration also has negative consequences for risk adjusted returns. The most obvious disadvantage
of concentration is that it will result in a higher risk. Hence, the optimal balance between
diversification and skill also depends on the client ability to take risk and to diversify this risk. For
institutional investors with long investment horizons and the resources to effectively detect skillful
managers, diversification is relatively cheap to obtain. In theory, such investors should seek
concentrated portfolios where skill is not diluted by diversification. For retail investors who have to
support higher transaction costs and cannot perform extensive due diligence, investing in few
diversified portfolios makes more sense. For a given fund size, more concentration will automatically
result in less liquid positions, which, everything else being equal, will have a negative impact on
performance.9
Empirical Analysis of the Relation between Diversification and Skill
Two empirical studies have found that mutual funds with high levels of concentration tend to
outperform 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 and
argue 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 explain
difference in performance between small funds and large funds is liquidity.
September, 2010 10

11. cap funds with higher levels of industry concentration were generating greater alpha than funds with
lower levels of industry concentration. Cremers and Petajisto [2008] measured the fund concentration
with its active share and found that funds with high concentration outperformed funds with low
concentration. The measures of concentration used in these two studies focus on the concentration of
portfolio weights. Unlike the DI coefficient these measures of concentration do not capture the impact
10
of concentration in positions’ volatilities and correlations among positions.
In order to understand the relation between portfolio diversification and skill, we applied the Empirical
Law of Active Management to a universe of 2,798 U.S. mutual funds invested in domestic equities from
1980 to 2006. The first obstacle we faced in assessing the skill of a manager was defining the
benchmark. While the concept of benchmark is ubiquitous in today’s asset management industry, it
was not the case in the 80’s and early 90’s. Hence, finding an accurate and consistent definition of a
fund’s benchmark across all periods is difficult. Moreover, the active returns of a fund measured
against its stated benchmark often contain common factor bets such as value, size or momentum that
can distort the active returns coming purely from stock picking skills. For these reasons, we used factor
models to determine the appropriate benchmark. We labeled as alpha or active return any returns that
could not be explained by the factors of the model we used. Therefore, the choice of model and the
factors that we included in the model were critical. Our objective was to assess the stock-picking skill
of a manager, as a result all known “priced factors” should be included in our model and excluded from
the manager’s alpha. “Priced factors” are factors for which empirical studies have demonstrated the
existence of positive risk adjusted returns; namely: market, value, size and momentum. To assess a
fund’s exposure to these factors, we used the same four-factor model as Carhart [1997]. In order to
ensure that our results were not dependent on the model, we also used two alternative models that
are 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
i
portfolio and in the market portfolio). Cremers and Petajisto use active share, which, is the sum of the absolute deviations of the
portfolio weights from the benchmark weights. Active Share  w i
i
portfolio
 wibenchmark
September, 2010 11

12. alternative model is based on the characteristic based benchmarks developed by Daniel et al. [1997].
11
The 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 each
of the periods we examined. Alpha and skill are net of the funds’ expenses ratios. The decline in
mutual fund alphas that we observed in this table was very important. The average mutual fund in our
universe 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 by
Kosowski et al. [2006] for similar periods and is also reported by Fama and French [2009]. Such
variations in the average alpha and skill of mutual fund managers are surprising considering that since
the 1990’s, the Investment Company Institute (ICI) has reported a decline in U.S. mutual fund expense
ratios.12 The second interesting trend that we observed in Exhibit 2 was a general increase in the
13
number of positions per portfolio and therefore in the diversification of the funds. This increase in
portfolio diversification among mutual fund managers is consistent with the inroads that modern
portfolio theory made during the 80’s and 90’s among practitioners and the development of passive
mutual funds. In addition, this period also witnessed a significant development of information
technology solutions that drastically reduced the operational costs of running portfolios with a large
number of positions.
Exhibit 2: Number of Funds, Mean Alpha, Skill, Diversification and number of positions per
portfolio for each Period.
The benchmark of each portfolio is estimated with the
Carhart [1997] model.
11
See Daniel et al. [1997] and Wermers [2004] for more details on DGTW portfolio. The DGTW benchmarks are available via
http://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.html
12
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

13. 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 63 135
We see two non-exclusive explanations for the decline in skill that we observed among equity mutual
funds. First, mutual fund managers were victims of their own success. The net asset value of equity
mutual 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 market
investors, Sharpe’s [1991] arithmetic of active management entails that as a whole, mutual fund
managers should have a negative alpha roughly equal to the fees they charge. As the industry grows
closer to representing the U.S. stock market, it is more difficult for the industry as a whole to
outperform the market.
The second explanation for this phenomenon is that the quality of U.S. Equity mutual funds has
declined. The industry would have responded to the increase in demand for mutual funds that took
place during the 80’s and 90’s by creating sub-par products. The market for asset management
services, like the Lemons market described by Akerlof [1970], is characterized by an asymmetry of
information.15 Since it is difficult for investors to distinguish a skilled from an unskilled manager, there
is a large incentive for unskilled managers to flood the market and distribute low cost investment
strategies that add no value while charging active management fees. This incentive is even greater
when demand expands and new customers with less financial acumen enter the market. Under such
circumstances, unskilled managers, or “closet indexers”, will want to take the minimum level of risk
that allows them to charge active management fees. On the other hand, skilled managers have an
incentive 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 Fact
Book (Investment Company Institute Research Department Staff [2009]) and the Wilshire 5000 index capitalization as a proxy for
the capitalization of the U.S. stock market.
15
See Foster and Young [2008]
September, 2010 13

14. skew in the distribution of alphas because skilled managers with concentrated portfolios will generate
high 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” production
and crafting strategies with little information content and with the look and feel of effective active
investment strategies. In order to assess this hypothesis, we examined the cross-sectional relation
between skill and diversification by splitting our universe of mutual funds into quintiles of
diversification and compared the skill of the funds located in each quintile.
Exhibit 3: Alpha and Skill by Quintiles of Diversification
The 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 DI
Mean 0.004 ‐0.007 ‐0.011 ‐0.011 ‐0.011
T‐stat (Mean Skill) 2.02 ‐6.34 ‐10.77 ‐11.98 ‐14.50
Skill Difference : Quintile 1 ‐ Quintile 2,3,4 or 5 0.012 0.016 0.015 0.015
T‐stat (Skill Difference) 4.78 6.45 6.50 6.51
Standard Deviation 0.05 0.03 0.02 0.02 0.02
Skewness 0.97 ‐0.21 0.15 ‐0.51 ‐0.62
Kurtosis 7.06 5.03 5.17 4.46 4.98
Number of Observations 560 559 560 559 560
In Exhibit 3, we observe that funds located in the first diversification quintile did exhibit significantly
more skill than any of the other quintiles. On average, the skill of these funds was greater than zero
while the skill in all the other quintiles of the distribution was significantly less than zero. All the
differences 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 portfolio
managers. This fact indicates that our result could be explained by survivorship bias. Less
diversification implies more risk. Since we required at least 36 months of returns to include a fund in
our analysis, it is possible that the low diversification funds that severely underperformed rapidly lost
their clients and did not survive long enough to be included in our analysis. Note that the positive skew
in the distribution of alpha can be explained by factors other than survivorship bias. As we observed
earlier, the presence of closet indexers or “lemons” in the mutual fund market is consistent with such
distribution of alphas.
September, 2010 14

15. In order to test the impact of a possible survivorship bias on our results, we created an artificial bias
against the top performing funds by eliminating funds in the right tail of the alpha distribution in order
to have a symmetric distribution. We removed from our analysis all funds whose alpha was greater than
the absolute value of the first percentile of the alpha distribution.
After removing these funds, the skew of the distribution became slightly negative; however, the skill
difference between the first quintile of managers and the rest of the universe remained strongly
significant (see top-panel of exhibit 4). In order to further test the robustness of our results, we
designed a more conservative artificial survivorship bias, where we removed all funds with alpha
greater than the absolute value of the fifth percentile of the alpha distribution. Under this definition of
the artificial survivorship bias, we eliminated from our sample all the funds with annualized alpha
greater than 6.7% per annum. Nevertheless, the bottom panel of exhibit 4 shows that low
diversification funds still exhibited significantly higher skill under both definitions of the artificial
survivorship bias. Hence, the relationship that we observed between skill and diversification cannot be
explained away by survivorship bias.
Exhibit 4: Alpha and Skill by Diversification Quintiles From 1980 to 2006 Excluding top Performing
Funds
Out-performing funds are funds with an alpha greater than the absolute value of the first or fifth percentile of the alpha
distribution. 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 543 542 543 542 543
September, 2010 15

16. The relation between skill and diversification also held across the three different types of models that
we used to define the alpha for the 1980-2006 period. Exhibit 5 also shows strongly significant results if
we use the characteristic based benchmarks or the 25 Fama-French portfolios to establish the alpha of
each fund. However, the relationship between skill and diversification was not consistent across
periods. Breaking up the 1980-2006 periods into 5 distinct sub-periods, we did not find that low
diversification funds outperformed in each sub-period (see exhibit 5). We did not find any significant
relation between diversification and skill during the eighties and the early nineties.
Exhibit 5: Relation between Skill and Diversification across periods and models
For each period and each model, we show the mean of the t-stats for the differences in skill between quintile one and quintiles
two 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 and
minimum: 4.78 are displayed in the top left cell of table 5. FF-25 denotes the results that we obtained with the 25 Fama-French
portfolios, 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 ‐1.25
Number of Observations per Quintile 559.6 475.6 303.8 117.6 54 27.6
Since we could not find a significant relation between diversification and skill in every sub-period, we
found it difficult to conclude that concentrated managers had an intrinsic advantage over less
concentrated 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 through
time under the influence of the structure of the U.S. market for mutual funds. Our interpretation of
these results is based on two elements. First, concentration among mutual fund managers has
decreased since the 80’s. Second, our analysis shows that since the mid-90’s, skill was more likely to
be found in more concentrated portfolios. These two points taken together indicated that as the
demand for mutual funds increased, the industry responded by producing funds with higher
diversification but lower information content, also known as “closet-indexers.” As a result, the average
September, 2010 16

17. skill and alpha of U.S. equity mutual funds decreased. In most industries, increases in demand are met
by increases in price; however, no such price increase took place in the asset management industry
during the 90’s.16 We infer from our data that the asset management industry took advantage of this
increase in demand by cutting the information content of its products while leaving its management
fees unchanged.
Conclusion
The Fundamental Law of Active Management played a very important role in the development of
quantitative asset management. The application of this law was designed by quantitative investment
managers 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 this
tool 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 in
skill. We found two explanations for this decrease in skill. Firstly, the mutual fund industry was a
victim of its own success, as mutual funds represent a greater share of the market capitalization. It is
now close to impossible for the industry as a whole to outperform the market. Secondly, the decrease
in skill was accompanied by an increase in diversification. Since the mid-90’s, we also observe an
inverse relationship between skill and diversification. We suggest that the increase in diversification
does 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

18. References:
Akerlof, George. “The Market for "Lemons": Quality Uncertainty and the Market Mechanism.” The
Quarterly 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 of
Political Economy, Vol. 112, No. 6 (2004), pp. 1269-1295.
Buckle, David. “How to calculate breadth: An evolution of the Fundamental Law of Active
Management.” 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 with
Characteristic 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? The
Role 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 Active
Management.” 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 Institute
Research Fundamentals, Vol. 18, No. 3 (April 2009)
Cremers, Martjin, A. Petajisto “How Active is Your Fund Manager? A New Measure that Predicts
Performance.” 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 Portfolio
Management, Vol. 15, No. 3 (Spring 1989), pp. 30-37.
Grinold, Richard C., R. Kahn. Active Portfolio Management: A Quantitative Approach for Producing
Superior Returns and Controlling Risk, 2nd ed. McGraw-Hill, 1999.
Investment Company Institute Research Department Staff, “2009 Investment Company Fact Book”, 49th
Edition, (2009).
Kacperczyk, Marcin T., C. Sialm, L. Zheng. “On Industry concentration of Actively Managed Equity
Mutual 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

19. 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 Analysts
Journal, Vol. 57, No. 6 (November/December 2001), pp.16-33.
“The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective.” Journal of
Portfolio 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.” working
paper (2008).
Wermers, Russ. “Are Mutual Fund Shareholders Compensated for Active Management "Bets"?” working
paper (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

20. Appendix 1: derivation of the Empirical Law of Active Management
wi is the weight of security i in the portfolio
w 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 assets
returns of size (n x n), and B is a (n x n) matrix which contains n times the (n x1) vector of benchmark
weights 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 all
securities were perfectly correlated and a second term which represents the benefits from
diversification.
w' w   wi2 i2   wi w j  i j  i , j
i i j i
w' 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 i
We separated the risk of the portfolio into two parts: First, the average volatility, which is the risk if
all positions were perfectly correlated. And second, the diversification benefits, which is the decrease
in 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

21.    wi i
i
wi i
Let’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 i
We 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 in
function 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 1
i i j  1    n 2     i  n    2 n  i  n 
i j i   i

 i  
September, 2010 21

22. 2
1  1
i  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 n
The higher the dispersion of the risk weights (that is the dispersion of portfolio weights and
volatilities), the higher the active risk of the portfolio. The second element of this diversification term
is the correlations among the positions of the portfolio (equation 2).
   i j i
i j  i, j   i   j  i, j
i j i
We note  i the average correlation of asset i with the other assets in the portfolio, weighted by the
18
risk-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 i
We call this quantity the risk weighted correlation:          
i
i i
i j i
i j i, j
If 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 ) i
September, 2010 22

23. The diversification factor of the portfolio is defined as:
1
DI 
1 ~
   nVar ( i )
n
Appendix 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 by
Russ 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 during
that period. For each period, we used the following steps to compute the diversification and skill
factors. First we computed the return of each fund by taking the asset-weighted average return of all
share-classes. If a share-class asset value was missing on a given date, we took the equal-weighted
average 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 the
Daniel, Grinblatt, Titman and Wermers characteristic based benchmarks (“DGTW”); and a third pricing
model based on the 25 Fama-French portfolios sorted on Value and Size (“FF-25”). 19
For 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 regression
defined the benchmark of the fund. As a result, the active return of the fund was equal to the alpha of
the 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 via
http://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.html
September, 2010 23

24. For DGTW, we performed a regression of the fund returns on the market portfolio and one of the 125
characteristic based benchmarks and the market factor. The DGTW benchmarks are the result of a sort
of 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. The
market 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 same
procedure for the FF-25 portfolios.
For each period we computed the historical active volatility of each asset in each fund using the
securities monthly returns. Asset active volatility is the standard deviation of the difference between
the 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 than
24 months of returns. Every quarter, we compute the mean weighted volatility. The fund mean
weighted volatility for a period is the mean of each of the quarterly mean weighted volatility during
that 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 included
in the analysis of a period, we required at least 8 valid quarters. In addition, we excluded from the
analysis, funds that had an average across all quarters of a period more than 5% of assets not
recognized in the CRSP U.S. Stock database. We excluded funds that, on average across all quarters of
a 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 of
the periods we analyzed. We then computed the weighted mean asset volatility of each portfolio using
the 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 for
the fund asset weighted volatility (  ) for the period and used the following decomposition to compute
the fund’s skill and diversification.
September, 2010 24

25.  
DI= and SK 
 
Model Choices
While the Carhart model is the standard pricing model used in the literature, the findings of Cremers
and Petajisto [2009] raised questions about the suitability of this model to establish a pertinent
performance benchmark. Unlike Cremers and Patajisto, we obtained strongly significant results with
the Carhart model. However, we still wanted to ensure that our results were not dependent on the
type of model we were using, so we recomputed our results with two alternative models. The DGTW
benchmarks 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 typical
benchmark and therefore more idiosyncratic risk. We also used a coarser asset classification
framework: the 25 Fama-French value and size sorted portfolios. The disadvantage of the Fama-French
portfolio is that we did not account for the funds’ momentum exposure. Using these three models, we
computed alpha (or active return), residual risk (or active risk), skill and diversification of each
portfolio during each of the periods we analyzed.
In theory, we should also have excluded from the active returns the returns due all non-priced factors
to which the funds are systematically exposed. However, there is an infinite set of non-priced factors
that could be included in the benchmark definition. Therefore, using non-priced factors was not
practical. Since exposures are computed based on the historical co-movement of the fund’s returns and
the factor returns, we would have picked up spurious relations between the funds’ returns and the
factors if we used a larger number of non-priced factors. As a result, we would have obtained a very
narrow definition of the fund’s alpha.
September, 2010 25