2010 09 the empirical law of active management

September, 2010 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 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

computation of an information coefficient. 2 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 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:

i

iiw

Where iw 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 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 )( iiw ).

We show in appendix 1 that the portfolio diversification can be written as a function of these three

factors:

DI= (Equation 1)

)(1

1

inn

September, 2010 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. )( iii w )

)( 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 :

DIIR

*

Where is the active return of the portfolio. Then we define the quantity

as the skill (SK) of the

manager which yields:

DISKIR *

4 see exact formula in appendix 1

September, 2010 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.

i

iiw ). Then we can express skill as the risk-weighted average of the information ratios of the

individual positions in the portfolio:

Where iIR 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

i

ii

ii

ii RInn

RIn

RI,Cov~

~1

*~~1~

*~~1~

~

5 Note that this decomposition was inspired by Andrew Lo’s AP decomposition. See proposition 1 in Lo [2007]

i

iii i

iii

IR

w

SK

September, 2010 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.

Given that the universe is a passive and information-less portfolio, we can assume that 0~

iiRI .

The expectation of the risk-weights and information ratio products: i

ii RI~

*~ is our skill factor.

Hence, we can re-write the skill of the manager in function of the covariance between securities risk

weights and information ratios: RICovnSK~

,~(~

Finally, if n~ is large enough so that )~RI 1, we obtain:

)~~~,~( nRISK

This decomposition of the skill factor is helpful to understand the three elements that are important in

determining the skill of a manager:

RI~

,~( : 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 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

Ex‐ante Alphas

Performance Portfolio Positions(or Risk –Weights)

Transfer Coefficient

Information Coefficient

SK

Public Information for Traditional Funds

Manager Proprietary Information

Fundamental Law

Empirical Law

This diagram is inspired from Figure 1 of Clarke de Silva Thorley (2002).

September, 2010 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

IRTC , which yields ** ** DISKTCIR

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 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 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

of concentration in positions’ volatilities and correlations among positions. 10

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

10 Kacperczyk et al. define as industry concentration as:

i

portfoliomarketi

portfolioi ww

2 (where wi is the weight of industry i in the

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 i

benchmarki

portfolioi ww

September, 2010 12

alternative model is based on the characteristic based benchmarks developed by Daniel et al. [1997].

The second alternative model is based on the 25 Fama-French portfolios sorted on value and size. 11

(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

number of positions per portfolio and therefore in the diversification of the funds. 13 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 13

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]

Period

Number of

Funds Alpha SK DI

Number of

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

September, 2010 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.

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.

Quintile 1:

Low DI Quintile 2 Quintile 3 Quintile 4

Quintile 5:

High DI

Mean 0.004 ‐0.007 ‐0.011 ‐0.011 ‐0.011

T‐stat (Mean Skil l) 2.02 ‐6.34 ‐10.77 ‐11.98 ‐14.50

Skil l 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

Skil l by Diversification(DI) Quintile (1980‐2006)

September, 2010 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.

Quintile 1:

Low DI Quintile 2 Quintile 3 Quintile 4

Quintile 5:

High DI

Mean ‐0.001 ‐0.009 ‐0.011 ‐0.011 ‐0.010

T‐stat (Mean Skil l) ‐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

T‐stat (Skil l Difference) 3.61 5.10 5.18 4.94


Skewness ‐0.04 ‐0.46 ‐0.18 ‐0.54 ‐0.63

Kurtosis 5.35 4.62 3.97 4.43 5.00


Mean ‐0.007 ‐0.01 ‐0.012 ‐0.011 ‐0.011

T‐stat (Mean Skil l) ‐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

T‐stat (Skil l Difference) 1.80 2.73 2.58 2.31


Skewness ‐0.59 ‐0.60 ‐0.27 ‐0.55 ‐0.84

Kurtosis 5.53 4.31 3.85 4.42 4.49


Skil l by Diversification(DI) Quintile (1980‐2006) Excluding Top‐

Performing Funds

Alpha less than

Absolute Value of 1st

Percentile Alpha

Alpha less than

Absolute Value of 5th

Percentile Alpha

September, 2010 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]

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

Periods:

1980

to

2006

2000

to

2006

1995

to

1999

1990

to

1994

1985

to

1989

1980

to

1984

Mean t‐stat 6.06 7.46 2.11 ‐0.40 0.68 ‐0.28

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

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

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

Carhart

DGTW

FF‐25

September, 2010 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 18

References:

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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.

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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 19

Fama, Eugene, K. French. “Luck versus Skill in the Cross Section of Mutual Fund Alpha Estimates.” working paper (2009)

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“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).

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September, 2010 20

Appendix 1: derivation of the Empirical Law of Active Management

iw 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: )()'( BIVBI

. 17 The active variance of the portfolio ( ww ' ) 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.

i ijjijiji

iii

i ijjijiji

iii

wwwww

wwwww

)11('

'

,22

,22

i ijjijiji

iii wwwww )1(' ,

2

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 BI matrix contains n columns of active weights for n portfolios that are fully invested in one and only one asset:

activen

activeactive

bn

bn

bn

bbb

bbb

www

www

www

www

BI ...

...

............

...

...

1...00

............

0...10

0...01

21222

111

September, 2010 21

i

iiw

Let’s note risk weights: ii

i

w with these notations the active variance of a portfolio is:

i ijjiji

iiww )1(' ,

2

2

We can re-write the active variance of the portfolio in the following way:

i ijji

i ijjijiww ,

22'

Or:

i ijji

i ijjijiww ,

2 1'

The term

i ij

jii ij

jiji , can be broken down into two components:

- the product of the risk weights =

i ij

ji (Equation 1)

- risk weighted correlation = i ij

jiji , (Equation 2)

We can rewrite equation 1 as: i

iii ij

jii ij

ji 1 which can be expressed in

function of the variance of the risk weights:

iii

iii

i ijji nn

22 11

i ii

ii

i ijji nnnn

112

111

2

2

September, 2010 22

ii

i ijji nn

211

1

)(*1

ii ij

ji Variancenn

n

Hence the concentration of the risk weights is a direct function of the variance of the risk weights:

n

nnVar i

i ijji

1)(

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 ij

jijii ij

jiji ,,

We note i the average correlation of asset i with the other assets in the portfolio, weighted by the

risk-weight of each asset in the portfolio. 18

ij

jiji ,~ hence we obtain:

iii

i ijjiji ,

We call this quantity the risk weighted correlation:

i i ij

jijiii ,

If we put everything back together:

)(

11)(1' 2

22 nVar

nn

nnVarww

18 Note that the sum of the risk-weights excluding asset i does not add-up to one. The correct weighted average should be:

ij

jii

ji ,)1(

The correlation term becomes:

iiii )1(

September, 2010 23

The diversification factor of the portfolio is defined as:

)~(1

1

inVarn

DI

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 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 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.

Economy & Finance

2010 09 the empirical law of active management