The Elusive “Stock-Picker’s Market”: Dispersion and Mutual Fund Performance

Jacob Epstein1

Professor Emma Rasiel, Faculty Advisor

Duke University

Durham, North Carolina

2020

1 Jacob graduated from Duke University in May 2020 with High Distinction in Economics, a second major in Statistical Science and a minor in Computer Science. Beginning in June 2020, Jacob will be joining The Blackstone Group in their Real Estate group. He can be reached at jacob.epstein0@gmail.com.

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Acknowledgments

I would like to thank my faculty advisor, Dr. Emma Rasiel, for her guidance and feedback during the research process and her mentorship throughout my entire undergraduate career. I am also very grateful for all of my professors and teaching assistants in the Economics, Statistical Science and Computer Science departments for providing a fantastic undergraduate experience.

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Abstract

This paper explores the relationship between active mutual fund performance and market dispersion from January 1990 to December 2018. It expands on previous analysis of dispersion and mutual fund performance by incorporating more recent data, utilizing 4-factor alpha rather than net return as the performance metric, and allowing the relationship to vary by decade. I find a significant positive relationship between dispersion and 4-factor alpha overall, providing some evidence of managerial skill. There are large differences in this relationship by decade and fund selectivity. The results suggest active mutual funds were able to take advantage of stock-picking opportunities during the 1990s and 2000s, particularly the most active subset of funds. However, I find a significant negative relationship between dispersion and alpha for funds in the 2010s, indicating this relationship has changed over time. I discuss several possible explanations for this reversal, which could present interesting avenues for further research.

JEL classification: C30, G12, G23 Keywords: Mutual Funds, Active Management, Dispersion, Cross-Sectional Volatility

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1. Introduction

In 2008, Warren Buffett famously made a $1 million bet against asset manager Ted Seides that

the S&P 500 would outperform actively managed funds over a period of 10 years. By early 2017,

it was clear Buffett had won; the S&P had returned 7.1% compounded annually, compared to a

mere 2.2% achieved by a basket of active funds hand-picked by Seides2. While that contest was

squarely settled, the larger debate over active versus passive management remains. Actively

managed mutual funds, run by managers who try to outperform the broader market by picking

stocks, have long been one of the most popular ways for retail investors to invest. In recent years,

however, passive mutual funds, and other passive instruments, such as ETFs, have risen

dramatically in popularity. These funds aim to mirror the returns of specific market indices and

charge much lower fees than charged by active funds. A visualization of these trends is shown in

figure 1. Some proponents of passive investing argue that active managers’ performance is a

function of luck rather than skill. In his popular 1973 book, A Random Walk Down Wall Street,

Princeton economist Burton Malkiel claimed, “a blindfolded monkey throwing darts at a

newspaper's financial pages could select a portfolio that would do just as well as one carefully

selected by experts.” Since then, a large body of research has explored whether active managers

add value. In this paper, I evaluate whether active mutual fund managers demonstrate skill by

examining whether actively managed mutual funds are able to outperform when their

opportunity set is larger—periods which could be referred to as “stock-pickers’ markets”.

2https://fortune.com/2017/12/30/warren-buffett-million-dollar-bet/

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Figure 1: Trends in active and passive management

I use dispersion, the cross-sectional volatility of the market, as a proxy for the extent to which

there are stock picking opportunities. At high levels of dispersion, stock returns are very different

from each other, providing ample opportunity for a stock picker to identify winners and losers.

At the other extreme, when dispersion is low, all stocks perform similarly, so there is limited

opportunity for stock pickers to outperform. If active managers are skilled at stock picking, one

would expect to see a positive relationship between fund performance and dispersion. I first

examine whether there is evidence of such a relationship. I then examine whether this

relationship has changed over time. Finally, I evaluate whether a fund’s performance during

periods of high dispersion provides meaningful insights from the perspective of an individual

considering an investment in that fund.

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2. Literature Review

2.1) Performance Evaluation Framework

While there exist many frameworks for evaluating mutual fund performance, one of the most

commonly used and empirically tested frameworks is the four-factor model developed by

Carhart (1997) and Fama and French (1992, 1993). This approach estimates a portfolio’s excess

returns using the market’s excess returns, as well as size, value and momentum factors. I use this

framework to evaluate the performance of mutual funds in this paper.

2.2) Active vs. Passive Management

Graham and Dodd’s Security Analysis (1934) makes a strong case for active management,

arguing that rigorous security-level analysis allows investors to outperform the market. That said,

many studies suggest that active managers tend to underperform passive managers. In one of the

first studies on the topic, Jensen (1968) examined mutual fund returns from 1945-1964 and

found little evidence of any individual fund outperforming the market more often than what

would be expected from random chance alone. Similarly, Davis (2001) used a Fama-French

three-factor model to analyze the returns of around five thousand actively managed mutual funds

from 1968 – 1998 and found no evidence of positive abnormal returns. These results held when

grouping by fund style and evaluating each style individually. Jones and Wermer (2011)

surveyed literature on the value of active management and concluded that the average active

manager does not outperform but a significant minority do add value. They also argue that

comparing active funds to a particular index directly is not a fair comparison, as one cannot

invest in the index directly. Instead, an investor would have to invest in a passive fund or ETF

which tracks the corresponding index and charges a small fee to do so. After taking this fee into

account, Jones and Wermer find that active and passively managed funds perform approximately

the same.

Cremers and Petajisto (2009) propose a new measure to quantify the extent to which a mutual

fund is actively managed and argue that this measure positively predicts fund performance. The

new metric, “active share”, is calculated as the sum of absolute differences between weightings

of stocks in a portfolio with stock weightings in the corresponding benchmark index. Frazzini,

Friedman and Pomorski (2016) replicate this study but report their results separately for each

benchmark index and find that this metric is as likely to correlate positively with returns as it is

to correlate negatively when grouping by benchmark. Regardless, the authors note that this

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metric is commonly used by institutional money managers in determining fund allocations. This

metric is difficult to calculate systematically because it requires knowledge of a fund’s target

benchmark and weighting of each stock within that benchmark. Amihud and Govenko (2013)

propose a different proxy for fund activity, 1-R2: the percentage variability in fund return not

explained by market returns. This metric identifies fund variance due to idiosyncratic factors;

funds that are more active will inherently have more idiosyncratic variance. Since this metric is

easily systematically applied to many funds, I use this 1-R2 metric, which I refer to as selectivity,

as a proxy for fund activity.

2.3) Dispersion

Several papers have explored cross-sectional volatility of stock returns, or dispersion,

particularly in the context of comparison to cross-sectional volatility of fund returns. This

measure is calculated as either the value-weighted or equal-weighted standard deviation of

security returns in the market in a given period and quantifies the variability of security returns

within a given portfolio. Ankrim and Ding (2002) and de Silva, Sapra and Thorley (2001) both

comment on dispersion of securities as a driver of large variation in mutual fund returns during

late 1990s and early 2000s. Petajisto (2013) explores another aspect of dispersion, using it as a

proxy for attractiveness of a stock picker’s opportunity set. He argues that periods of high

dispersion represent “stock-pickers’ markets,” where large differences in returns between the

best and worst-performing stocks provide ample opportunity for active managers to identify

winners and losers. He then finds a modest positive relationship between dispersion and excess

return on a benchmark for the most active subset of funds between 1990 and 2009, supporting

the idea of a stock-picker’s market. When including all funds, not just the most active subset, this

relationship was not significant. This paper expands on Petajisto’s analysis by incorporating

more funds into the analysis as well as more recent data on these funds, using a four-factor alpha

instead of simple outperformance, and allowing differences in this relationship over time.

2.4) Survivorship Bias

As with any study of mutual funds, survivorship bias is an important topic to address before

conducting analysis. Carhart, Carpenter, Lynch and Musto (2002) discuss at length sources of

bias in studies of mutual funds. In particular, they demonstrate that mutual funds tend to

disappear following poor multi-period performance, resulting in a positive bias of approximately

1% of average annual returns for studies of 15 years or more. This form of bias is unavoidable,

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as one cannot prevent poorly-performing mutual fund from shutting down. A more explicit form

of bias is introduced when “dead” funds, or funds which are not present the end of the sample,

are excluded from a sample. Many papers, such as Petajisto (2013), remark that samples which

include all dead funds are essentially survivorship-bias free, as they avoid introducing this form

of bias. As such, I use a sample that includes both live and dead funds in order to minimize

survivorship bias.

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3. Data and Methodology

3.1) Mutual Fund Data

Data on mutual fund returns come from The Center for Research in Security Prices (CRSP)

Mutual Funds database. Excluding index funds, his database includes monthly data on returns

and net asset values for 61,979 funds in existence for at least one month between January 1990

and December 2018. The average fund life was 103 months, resulting in a data set of 6.4 million

fund-month observations. As per CRSP’s methodology, different classes of the same mutual

fund are associated with different unique identifiers, and thus are treated as separated funds in

this paper. In order to exclude funds not benchmarked to equity indices, I then calculated Carhart

four-factor betas for each fund over their entire lives and excluded all funds with market betas

lower than 0.8 or above 1.2. I also limited the analysis to funds with lifespans of three years or

greater, because at least that much data is necessary in order to calculate three-year rolling

metrics. Subsetting criteria are discussed further in section 3.3. With these criteria, the final

dataset includes data on 19,433 funds, with an average life of 132 months, resulting in 2.6

million fund-month observations.

3.2) Supplementary Data

Dispersion data was generated as described in section 3.4 using S&P 500 constituent data from

Compustat - Capital IQ’s Index Constituents database and stock returns from CRSP’s Stock /

Security Files database. It includes data on the cross-sectional volatility of the S&P 500,

calculated monthly for a total of 348 months. Other volatility data, including monthly VIX,

comes from the Wharton Research and Data Services’s (WRDS) CBOE Indexes database. Fama-

French factor data comes from the WRDS Fama-French Portfolio and Factors database, which in

turn draws from Kenneth French’s website. These data include monthly values for the risk-free

rate and excess return of the market, as well as size, momentum and value factors. These are as

defined by Carhart (1997) and Fama and French (1993) and reproduced below:

Risk free rate (rf) – one month U.S. Treasury bill rate

Market return (rmkt) – value-weighted returns of all NYSE, AMEX, and NASDAQ stocks

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Size (rsmb) – difference in returns between small-cap and large-cap stocks. Calculated as

the average return of three small-cap portfolios minus the average return of three large-

cap portfolios.

Momentum (rumd) – difference in return between high prior-period-return stocks and low

prior-period-return stocks.

Value (rhml) – difference in return of value stocks and growth stocks. Calculated as the

average return of two value portfolios minus the average return of two growth portfolios.

3.3) Sample Identification

First, I created a subset of relevant funds using the market beta cutoffs mentioned in section 3.1.

In order to estimate betas for each fund, I used a Carhart four-factor model over each fund’s

entire life. Its formulation shown below, where rj,t represents the return of fund j in period t, and

rmkt,t, rsmb,t, rhml,t, and rumd,t are as described in the section 3.2:

After estimating betas for each fund, I included only funds whose betas were between 0.8 and

1.2, since it is reasonable to assume these funds are benchmarked to U.S. equity indices. While

this approach is somewhat arbitrary, it provides an intuitive way to exclude funds uncorrelated

with U.S. equity markets, is and is easy to apply systematically as it does not require knowledge

of each fund’s holdings. Furthermore, the results of this paper are not sensitive to small changes

in cutoff values, as can be seen in Appendix 1. A distribution of market betas for each fund is

shown in figure 2. I also filtered out any funds designated as ETFs by the CRSP database, and

any funds with less than 36 months of data, because it would be impossible to calculate three-

year rolling metrics for these funds. By including all funds that were extant for at least three

years, I avoid introducing survivorship bias.

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Figure 2: Distribution of market betas for all funds

3.4) Dispersion Calculation

Concurrently, I used data on S&P 500 constituents and returns to generate values for dispersion,

or cross-sectional volatility of the S&P 500, equal weighted. I used S&P 500 components to

calculate market dispersion rather than, say, all publicly listed equities because historical data is

more readily available on these securities and these are the most commonly held stocks by

mutual funds. Though Petajisto (2013) calculated dispersion on a value-weighted basis, I

calculated dispersion on an equally weighted basis as it is more intuitively compatible with the

idea of stock picker’s markets. Large variance in stock returns present opportunities for active

stock pickers regardless of the size of the stocks. Dispersion is calculated as follows, where nt is

the number of stocks in the S&P 500 in period i, rj,t is the return of stock j in period t and is the

average return of S&P 500 stocks in period i:

For context, a time series of dispersion, compared to VIX is shown below. The two are relatively

strongly correlated (r = 0.59). Dispersion measures a different angle of market volatility than

0

2500

5000

7500

−0.5 0.0 0.5 1.0 1.5 2.0Market Beta

Num

ber o

f Fun

dsHistogram of Fund Market Betas

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Dispersiont =q

1nt�1

Pnt

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12

VIX; dispersion corresponds to cross-sectional variance of returns whereas VIX corresponds to

time-series volatility. Historical data for both metrics are shown in figure 3.

Figure 3: Historical data for VIX and Dispersion.

3.5) Fund Alpha

I then calculated Carhart four-factor alphas for each of the funds on a three-year rolling basis.

For each month a fund was in the sample, I calculated its three-year rolling alpha, market beta,

size beta, value beta and momentum beta using the following model:

where MKT, HML, SMB and UMD are defined in section 3.2. I then calculated monthly alpha,

aj,t for each fund (not to be confused with three-year rolling alpha, aj), defined as:

Or equivalently:

In other words, a fund’s monthly alpha is its outperformance in a given month relative to the

fund’s expected return given market return, risk free rates, size, value and momentum.

20

40

60

10

20

30

1990 2000 2010 2020Date

VIX

Dispersion (%

)

Parameter Vix

Dispersion

VIX and Dispersion Over Time

rj,t � rf,t = ↵j + �mkt,j(rmkt,t � rf,t) + �hml,jrhml,t + �smb,jrsmb,t<latexit sha1_base64="VLNLjtfKo3M//TDQ8W3pTIh6mSg=">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</latexit><latexit sha1_base64="VLNLjtfKo3M//TDQ8W3pTIh6mSg=">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</latexit><latexit sha1_base64="VLNLjtfKo3M//TDQ8W3pTIh6mSg=">AAACa3icbZHNTttAFIXHpkBIKQ2woCosRg1UqQiRzQY2lSJ1w5JKDSAlkXU9GSeTzNjWzDVSZHnDI7LjDdjwDh0nFi0/Vxrp0znnauwzYSqFQc97cNyVD6tr67WN+sfNT1ufG9s7VybJNOM9lshE34RguBQx76FAyW9SzUGFkl+Hs1+lf33LtRFJ/AfnKR8qGMciEgzQSkHj7lAH+bSNBT2hlqI2tfiTDkCmEwim9JgOQo4Q5GqGbTotWrrCFxs//uUmSpY5XZGNPVtGhZW1ICwO6+UEjabX8RZD34JfQZNUcxk07gejhGWKx8gkGNP3vRSHOWgUTPKiPsgMT4HNYMz7FmNQ3AzzRVcFPbLKiEaJtidGulD/38hBGTNXoU0qwIl57ZXie14/w+h8mIs4zZDHbHlRlEmKCS2LpyOhOUM5twBMC/utlE1AA0P7PGUJ/utffgtXpx3f6/i/T5vd71UdNbJPvpEW8ckZ6ZILckl6hJFHZ8vZc744T+6u+9U9WEZdp9rZJS/GPfoLAQC1Lw==</latexit><latexit sha1_base64="VLNLjtfKo3M//TDQ8W3pTIh6mSg=">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</latexit>

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↵j,t = rj,t�rf,t��mkt,j(rmkt,t�rf,t)��hml,jrhml,t��smb,jrsmb,t��umd,jrumd,t<latexit sha1_base64="Bx2ywLzwuMHsOx3W7OeUQnY+BJg=">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</latexit><latexit sha1_base64="Bx2ywLzwuMHsOx3W7OeUQnY+BJg=">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</latexit><latexit sha1_base64="Bx2ywLzwuMHsOx3W7OeUQnY+BJg=">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</latexit><latexit sha1_base64="Bx2ywLzwuMHsOx3W7OeUQnY+BJg=">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</latexit>

↵j,t = ↵j + ✏t<latexit sha1_base64="6pHsJQD1WhSzUZw9vzLoiPCNtkA=">AAACE3icbVDLSgMxFM3UV62vUZdugq0iKmWmG90IBTcuK9gHdIYhk6Zt2syD5I5Qhv6DG3/FjQtF3Lpx59+YaWeh1gOBk3POJbnHjwVXYFlfRmFpeWV1rbhe2tjc2t4xd/daKkokZU0aiUh2fKKY4CFrAgfBOrFkJPAFa/vj68xv3zOpeBTewSRmbkAGIe9zSkBLnnlacYiIh8RLR+cwxVc4v47wGXZYrLjQKaiUNDyzbFWtGfAisXNSRjkanvnp9CKaBCwEKohSXduKwU2JBE4Fm5acRLGY0DEZsK6mIQmYctPZTlN8pJUe7kdSnxDwTP05kZJAqUng62RAYKj+epn4n9dNoH/ppjyME2AhnT/UTwSGCGcF4R6XjIKYaEKo5PqvmA6JJBR0jVkJ9t+VF0mrVrWtqn1bK9eP8zqK6AAdohNkowtURzeogZqIogf0hF7Qq/FoPBtvxvs8WjDymX30C8bHNxjPm6A=</latexit><latexit sha1_base64="6pHsJQD1WhSzUZw9vzLoiPCNtkA=">AAACE3icbVDLSgMxFM3UV62vUZdugq0iKmWmG90IBTcuK9gHdIYhk6Zt2syD5I5Qhv6DG3/FjQtF3Lpx59+YaWeh1gOBk3POJbnHjwVXYFlfRmFpeWV1rbhe2tjc2t4xd/daKkokZU0aiUh2fKKY4CFrAgfBOrFkJPAFa/vj68xv3zOpeBTewSRmbkAGIe9zSkBLnnlacYiIh8RLR+cwxVc4v47wGXZYrLjQKaiUNDyzbFWtGfAisXNSRjkanvnp9CKaBCwEKohSXduKwU2JBE4Fm5acRLGY0DEZsK6mIQmYctPZTlN8pJUe7kdSnxDwTP05kZJAqUng62RAYKj+epn4n9dNoH/ppjyME2AhnT/UTwSGCGcF4R6XjIKYaEKo5PqvmA6JJBR0jVkJ9t+VF0mrVrWtqn1bK9eP8zqK6AAdohNkowtURzeogZqIogf0hF7Qq/FoPBtvxvs8WjDymX30C8bHNxjPm6A=</latexit><latexit sha1_base64="6pHsJQD1WhSzUZw9vzLoiPCNtkA=">AAACE3icbVDLSgMxFM3UV62vUZdugq0iKmWmG90IBTcuK9gHdIYhk6Zt2syD5I5Qhv6DG3/FjQtF3Lpx59+YaWeh1gOBk3POJbnHjwVXYFlfRmFpeWV1rbhe2tjc2t4xd/daKkokZU0aiUh2fKKY4CFrAgfBOrFkJPAFa/vj68xv3zOpeBTewSRmbkAGIe9zSkBLnnlacYiIh8RLR+cwxVc4v47wGXZYrLjQKaiUNDyzbFWtGfAisXNSRjkanvnp9CKaBCwEKohSXduKwU2JBE4Fm5acRLGY0DEZsK6mIQmYctPZTlN8pJUe7kdSnxDwTP05kZJAqUng62RAYKj+epn4n9dNoH/ppjyME2AhnT/UTwSGCGcF4R6XjIKYaEKo5PqvmA6JJBR0jVkJ9t+VF0mrVrWtqn1bK9eP8zqK6AAdohNkowtURzeogZqIogf0hF7Qq/FoPBtvxvs8WjDymX30C8bHNxjPm6A=</latexit><latexit sha1_base64="6pHsJQD1WhSzUZw9vzLoiPCNtkA=">AAACE3icbVDLSgMxFM3UV62vUZdugq0iKmWmG90IBTcuK9gHdIYhk6Zt2syD5I5Qhv6DG3/FjQtF3Lpx59+YaWeh1gOBk3POJbnHjwVXYFlfRmFpeWV1rbhe2tjc2t4xd/daKkokZU0aiUh2fKKY4CFrAgfBOrFkJPAFa/vj68xv3zOpeBTewSRmbkAGIe9zSkBLnnlacYiIh8RLR+cwxVc4v47wGXZYrLjQKaiUNDyzbFWtGfAisXNSRjkanvnp9CKaBCwEKohSXduKwU2JBE4Fm5acRLGY0DEZsK6mIQmYctPZTlN8pJUe7kdSnxDwTP05kZJAqUng62RAYKj+epn4n9dNoH/ppjyME2AhnT/UTwSGCGcF4R6XjIKYaEKo5PqvmA6JJBR0jVkJ9t+VF0mrVrWtqn1bK9eP8zqK6AAdohNkowtURzeogZqIogf0hF7Qq/FoPBtvxvs8WjDymX30C8bHNxjPm6A=</latexit>

13

3.6) Selectivity

Following Amihud and Goyenko (2013), I estimated 1-R2 using the S&P 500 as the market

benchmark. This 1-R2 measure can be interpreted as the percentage variability in fund return that

is not explained by market returns; thus, we use this measure as a proxy for fund selectivity.

First, I estimate each fund’s selectivity proxy as follows:

I then categorize fund as high, middle or low selectivity based on this metric, both aggregated as

well as for each decade. The 1990s decade is defined as January 1990-December 1999, and

2000s and 2010s are defined similarly. The motivation behind having two different selectivity

labels is to allow for fund selectivity to vary over time, as one could imagine would happen as a

fund shifts management or style. Low selectivity funds are defined as those in the bottom

quintile of 1-R2, high selectivity funds are those in the top quintile, and middle selectivity as

those in the middle 3 quintiles. Cutoff values for each group are shown below in table 1. Percent

of fund-month observations in each category are shown in table 2. 85% of observations had the

same selectivity designation whether calculated overall or by decade, indicating that funds

generally tend to maintain similar levels of selectivity over time.

Table 1: Selectivity cutoffs by decade

Table 2: Percent of fund-month observations in each selectivity category

3.7) Model

I then estimated three different models to understand the relationship between fund alpha and

dispersion. First, I estimate the model across the entire time period, allowing for interaction with

1�R2j = 1� Cov(Rj ,RS&P )2

V ar(RS&P )V ar(Rj)<latexit sha1_base64="TFUGKwNN5JGRbTNVuQfvo8VSA0c=">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</latexit><latexit sha1_base64="TFUGKwNN5JGRbTNVuQfvo8VSA0c=">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</latexit><latexit sha1_base64="TFUGKwNN5JGRbTNVuQfvo8VSA0c=">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</latexit><latexit sha1_base64="TFUGKwNN5JGRbTNVuQfvo8VSA0c=">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</latexit>

Low Middle HighOverall (0, 0.08) (0.08, 0.35) (0.35, 0.99)1990s (0, 0.14) (0.14, 0.54) (0.54, 0.99)2000s (0, 0.08) (0.08, 0.34) (0.35, 0.94)2010s (0, 0.07) (0.07, 0.31) (0.31, 0.99)

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Decade: High Decade: Middle Decade: LowOverall: High 16% 5% 0%Overall: Middle 4% 54% 4%Overall: Low 0% 2% 15%

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14

overall selectivity. VIX is also included as a covariate because, as Low (2000) points out, it

represents a proxy for investor fear in the market. This model is explicitly defined below:

More simply:

Low selectivity is the default selectivity designation in this formulation and dispersion is mean

centered to allow for easier interpretation of b0. I then allowed for variation over time by adding

in interaction terms for decade and used decade-by-decade measures for selectivity. The time-

varying model is defined below:

Again, more simply:

I then explored dispersion in fund alpha as a function of market dispersion, comparing a simple

linear relationship to a log-log relationship. Fund dispersion is defined similarly to market

dispersion. Its formulation is shown below, where nt is total number of funds in a given month,

and is average alpha of all funds in a given month:

The linear model is defined as:

And the log-log model is defined as:

↵j,t = �0+�1V IXt+�2Dispersiont+�3(Middle Selectivity)+�4(High Selectivity)+�5(Dispersiont : Middle Selectivity)+�6(Dispersiont : High Selectivity)+✏t

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↵j,t ⇠ V IXt + Selectivityj +Dispersiont + Selectivityj ⇤Dispersiont<latexit sha1_base64="pGoMORnffGqrpqwI8HjDpt9ZA2M=">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</latexit><latexit 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↵j,t = �0+�1V IXt+�2(V IXt : 2000s)+�3(V IXt : 2010s)+�4Dispersiont+�52000s+�62010s+�7(Middle Selectivity)+�8(High Selectivity)+�9(Dispersiont :2000s) + �10(Dispersiont : 2010s) + �11(Dispersiont : Middle Selectivity) +�12(Dispersiont : High Selectivity)+�13(Middle Selectivity : 2000s)+�14(Middle Selectivity :2010s) + �15(High Selectivity : 2000s) + �16(High Selectivity : 2010s) +�17(Dispersiont : Middle Selectivity : 2000s)+�18(Dispersiont : Middle Selectivity :2010s) + �19(Dispersiont : High Selectivity : 2000s) + �20(Dispersiont :High Selectivity : 2010s) + ✏t

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↵j,t ⇠ V IXt+V IXt ⇤Dispersiont+Selectivityj+Dispersiont+Decadet+Selectivityj ⇤Dispersiont + Selectivityj ⇤Decadet +Decadet ⇤Dispersiont +Dispersiont ⇤ Selectivityj ⇤Decadet

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↵̄t<latexit sha1_base64="uk+qffGtJhKIHm2wdrGmPfOgHBw=">AAAB/3icbVBNS8NAEN3Ur1q/ooIXL8FW8VSSXvRY8OKxgv2AJoTJdtMu3WzC7kYoMQf/ihcPinj1b3jz37hpc9DWBwOP92aYmRckjEpl299GZW19Y3Orul3b2d3bPzAPj3oyTgUmXRyzWAwCkIRRTrqKKkYGiSAQBYz0g+lN4fcfiJA05vdqlhAvgjGnIcWgtOSbJw03AJG5wJIJ+JnK80ZNwzfrdtOew1olTknqqETHN7/cUYzTiHCFGUg5dOxEeRkIRTEjec1NJUkAT2FMhppyiIj0svn9uXWulZEVxkIXV9Zc/T2RQSTlLAp0ZwRqIpe9QvzPG6YqvPYyypNUEY4Xi8KUWSq2ijCsERUEKzbTBLCg+lYLT0AAVjqyIgRn+eVV0ms1Hbvp3LXq7Ysyjio6RWfoEjnoCrXRLeqgLsLoET2jV/RmPBkvxrvxsWitGOXMMfoD4/MH1FGUjw==</latexit><latexit sha1_base64="uk+qffGtJhKIHm2wdrGmPfOgHBw=">AAAB/3icbVBNS8NAEN3Ur1q/ooIXL8FW8VSSXvRY8OKxgv2AJoTJdtMu3WzC7kYoMQf/ihcPinj1b3jz37hpc9DWBwOP92aYmRckjEpl299GZW19Y3Orul3b2d3bPzAPj3oyTgUmXRyzWAwCkIRRTrqKKkYGiSAQBYz0g+lN4fcfiJA05vdqlhAvgjGnIcWgtOSbJw03AJG5wJIJ+JnK80ZNwzfrdtOew1olTknqqETHN7/cUYzTiHCFGUg5dOxEeRkIRTEjec1NJUkAT2FMhppyiIj0svn9uXWulZEVxkIXV9Zc/T2RQSTlLAp0ZwRqIpe9QvzPG6YqvPYyypNUEY4Xi8KUWSq2ijCsERUEKzbTBLCg+lYLT0AAVjqyIgRn+eVV0ms1Hbvp3LXq7Ysyjio6RWfoEjnoCrXRLeqgLsLoET2jV/RmPBkvxrvxsWitGOXMMfoD4/MH1FGUjw==</latexit><latexit sha1_base64="uk+qffGtJhKIHm2wdrGmPfOgHBw=">AAAB/3icbVBNS8NAEN3Ur1q/ooIXL8FW8VSSXvRY8OKxgv2AJoTJdtMu3WzC7kYoMQf/ihcPinj1b3jz37hpc9DWBwOP92aYmRckjEpl299GZW19Y3Orul3b2d3bPzAPj3oyTgUmXRyzWAwCkIRRTrqKKkYGiSAQBYz0g+lN4fcfiJA05vdqlhAvgjGnIcWgtOSbJw03AJG5wJIJ+JnK80ZNwzfrdtOew1olTknqqETHN7/cUYzTiHCFGUg5dOxEeRkIRTEjec1NJUkAT2FMhppyiIj0svn9uXWulZEVxkIXV9Zc/T2RQSTlLAp0ZwRqIpe9QvzPG6YqvPYyypNUEY4Xi8KUWSq2ijCsERUEKzbTBLCg+lYLT0AAVjqyIgRn+eVV0ms1Hbvp3LXq7Ysyjio6RWfoEjnoCrXRLeqgLsLoET2jV/RmPBkvxrvxsWitGOXMMfoD4/MH1FGUjw==</latexit><latexit sha1_base64="uk+qffGtJhKIHm2wdrGmPfOgHBw=">AAAB/3icbVBNS8NAEN3Ur1q/ooIXL8FW8VSSXvRY8OKxgv2AJoTJdtMu3WzC7kYoMQf/ihcPinj1b3jz37hpc9DWBwOP92aYmRckjEpl299GZW19Y3Orul3b2d3bPzAPj3oyTgUmXRyzWAwCkIRRTrqKKkYGiSAQBYz0g+lN4fcfiJA05vdqlhAvgjGnIcWgtOSbJw03AJG5wJIJ+JnK80ZNwzfrdtOew1olTknqqETHN7/cUYzTiHCFGUg5dOxEeRkIRTEjec1NJUkAT2FMhppyiIj0svn9uXWulZEVxkIXV9Zc/T2RQSTlLAp0ZwRqIpe9QvzPG6YqvPYyypNUEY4Xi8KUWSq2ijCsERUEKzbTBLCg+lYLT0AAVjqyIgRn+eVV0ms1Hbvp3LXq7Ysyjio6RWfoEjnoCrXRLeqgLsLoET2jV/RmPBkvxrvxsWitGOXMMfoD4/MH1FGUjw==</latexit>

FundDispersiont =q

1nt�1

Pnt

j=1(↵j,t � ↵̄t)2<latexit sha1_base64="eJi9tKEVH6sG1SWZlRhd2DVGTWk=">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</latexit><latexit sha1_base64="eJi9tKEVH6sG1SWZlRhd2DVGTWk=">AAACTXicbZDPaxQxFMcz64/WbbWrHr0Ety0V7DLZS3spFBTxWMFtCzvb4U0m002byUyTN8IS8g96KfTmf+HFgyJiZncFbf1C4JvPe4+XfLNaSYtx/CXq3Lv/4OHK6qPu2vrjJxu9p8+ObdUYLka8UpU5zcAKJbUYoUQlTmsjoMyUOMku37T1k0/CWFnpjzirxaSEcy0LyQEDSnv55rtG52+lrRdNKdIDmtgrgy4pDHDHvAtwl/nENmXqLg6YP2uJpzsJqHoKgb2m4bpLkwyM+wPR+1dnQ7/ZDUp7/XgQz0XvGrY0fbLUUdq7SfKKN6XQyBVYO2ZxjRMHBiVXwneTxooa+CWci3GwGkphJ26ehqdbgeS0qEw4Gumc/j3hoLR2VmahswSc2tu1Fv6vNm6w2J84qesGheaLRUWjKFa0jZbm0giOahYMcCPDWymfQggRQ7ZtCOz2l++a4+GAxQP2Ydg/3F7GsUpekJdkhzCyRw7Je3JERoSTz+Qr+U5+RNfRt+hn9GvR2omWM8/JP+qs/AaDaLM3</latexit><latexit sha1_base64="eJi9tKEVH6sG1SWZlRhd2DVGTWk=">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</latexit><latexit sha1_base64="eJi9tKEVH6sG1SWZlRhd2DVGTWk=">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</latexit>

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log(FundDispersiont) = �0 + �1log(Dispersiont) + ✏t<latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit>

15

4. Results

4.1) Initial Model

The initial model is shown below, and its corresponding output is shown in table 3:

where VIX and dispersion are both mean-centered.

Table 3: Initial model coefficients

The R-squared of this regression indicates that about 0.2% of the variability in alpha can be

explained by variability in the predictors. The low R-squared is likely due to the noisy nature of

alphas, since these alphas represent the components of returns not already explained by general

market movement, size, value and momentum. Despite this low R-squared, many of the

predictors are highly significant in explaining alpha. Given the statistical significance of

coefficients, we can conclude that this model provides useful information about the mean alphas

for funds given selectivity and dispersion; however, the low explanatory power of this model

indicates it is not particularly informative for each specific fund. A visualization of the model

output is shown in figure 4, with the vertical gray bar representing mean dispersion levels.

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N R-Squared F P-Value1,865,799 0.002 674.9 0.000

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Estimate Standard Error T P-Value

Intercept (0.27) 0.04 (7.09) 0.00

VIX (0.17) 0.00 (62.64) 0.00

Dipsersion 34.06 1.45 23.54 0.00

High Selectivity (0.35) 0.05 (6.76) 0.00

Middle Selectivity (0.23) 0.04 (5.32) 0.00

Dispersion x High Selectivity 3.76 1.81 2.08 0.04

Dispersion x Middle Selectivity 1.52 1.55 0.98 0.33<latexit sha1_base64="laM8Z5CUKUf7pQcUwsubpeVTIOM=">AAAE2HichVPLbtNAFHUbA8U8msKSzYiIKllg+REn6a4SrdQukILapBFxFE3Gk2TU8UMz44rIisQChNjyaez4Cb6Bcey6OE3Uuzpz78yZc67unUSUcGEYf3Z2K+qjx0/2nmrPnr94uV89eNXnYcwQ7qGQhmwwgRxTEuCeIILiQcQw9CcUX02uP6T1qxvMOAmDS7GI8MiHs4BMCYJCpsYHu3/dCZ6RIBFQPlkOR9rdOaaQLROaxlJz5+kXGnggDsEpF8SHAkt4IWDgQeaBU8ZCJhOXxa3u+z6kMQauC3Lm80Bg6SkSW5nrhm61Gyto6EazXGvrxlEjLxlpznW1/vngAbGS0ryjNMq1lqW3mo0y5QmJeNbObZR2UzdaGTT1plOqWbbuNMGayjMym4MLTDES5IaIxUaVtlOoLFPWW3q7tW78I/E8ireTZr20t/XS0W1rnfKE8Cg3/gVskCyNSyHg1njHLBvXjQ4o/bZOuUFyyuNYBaVTNm7oRwWlbWeUt7Pk4sArJrg4yfnWxtVaaioNcB+YOagpeXTH1d+uF6LYx4FAFHI+NI1IjBLIBEEpoRtzHEF0DWd4KGEAfcxHyWoxl+CdzHhgKmd/GgYCrLL/v0igz/nCn8ibcmXmfL2WJjfVhrGYdkYJCaJY4ABlH01jCkQI0i0HHmGyjXQhAUSMSK0AzSGDSO4YT5tgrlu+D/qWbsr1+GTVjg/zduwpb5S3Sl0xlbZyrJwpXaWnoEqvklS+Vb6rn9Wv6g/1Z3Z1dyd/81ophfrrHx3FM3w=</latexit><latexit sha1_base64="laM8Z5CUKUf7pQcUwsubpeVTIOM=">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</latexit><latexit sha1_base64="laM8Z5CUKUf7pQcUwsubpeVTIOM=">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</latexit><latexit sha1_base64="laM8Z5CUKUf7pQcUwsubpeVTIOM=">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</latexit>

16

Figure 4: Initial model visualization

Low, middle and high selectivity funds had average alphas of -0.27%, -0.50% and -0.62%, all

significantly different from 0 and from each other. The coefficient for VIX indicates that a 1-

point increase in VIX is associate with a 17 basis point decrease in alpha. Low and middle

selectivity funds had a significantly positive relationship with dispersion; a 1 percentage point

increase in dispersion is associated with a 34 basis point increase in alpha for these funds. Highly

selective funds have an even larger positive relationship between alpha and dispersion (38 basis

points of alpha per percentage point of dispersion), suggesting that highly selective funds, are

able to take advantage of high dispersion environments slightly more effectively than their less

selective peers. This result is consistent with Petajisto (2013). The lower overall alpha for these

funds may indicate the increased transaction costs associated with being highly active are not

offset by their advantage in periods of high dispersion. This analysis implicitly assumes that fund

alpha, and the relationship between dispersion and outperformance does not change over time.

This assumption is likely flawed, as the investment environment has changed significantly over

the past three decades. In order to account for changes in investment environment over time, the

next iteration of the model allows for variation by decade.

4.2) Time-Varying Model

After adding decade as a predictor, the new model is shown below, and its output is in shown in

table 4.

−8

−4

0

4

8

0 5 10 15Dispersion (%)

Aver

age

Alph

a (%

)

SelectivityLow

Middle

High

Alpha as a Function of Dispersion and Selectivity

↵j,t ⇠ V IXt+V IXt ⇤Dispersiont+Selectivityj+Dispersiont+Decadet+Selectivityj ⇤Dispersiont + Selectivityj ⇤Decadet +Decadet ⇤Dispersiont +Dispersiont ⇤ Selectivityj ⇤Decadet

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17

where VIX and dispersion are again both mean-centered.

Table 4: Time-varying model coefficients

The R-squared of this regression indicates that about 0.8% of the variability in alpha can be

explained by variability in the predictors. The model now explains about 3 times as much of the

variance in alpha as before. Again, this low R-squared is likely due to the noisy nature of

monthly alphas mentioned earlier. Given the interaction terms, this model is equivalent to

running regressions separately for each of the decades. For ease of interpretation, coefficients

and t-values for regressions run separately for each decade are shown in Table 5. Note that all

coefficients are identical in Tables 4 and 5 except for the intercept. This is due to mean-centering

of dispersion and VIX. In Table 4, the intercept can be interpreted as average alpha for a low

selectivity fund in the 1990s at mean levels of overall dispersion and VIX, whereas the intercepts

in Table 5 can be interpreted as average alpha for a low selectivity fund at mean levels of

dispersion and VIX for that particular decade. In order to draw conclusions from the model,

relationships are interpreted by decade. The model output is displayed graphically in Figures 5, 6

and 7, where the vertical gray bar on each graph corresponds to mean dispersion in that decade.

N R-Squared F P-Value1,865,799 0.008 720.1 0.000

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Estimate Standard Error T P - Value

Intercept 0.04 0.15 0.27 0.79

VIX (0.24) 0.01 (21.36) 0.00

VIX x 2000s 0.04 0.01 3.16 0.00

VIX x 2010s 0.10 0.01 8.27 0.00

Dispersion (25.40) 5.67 (4.48) 0.00

2000s (0.38) 0.17 (2.27) 0.02

2010s (1.20) 0.17 (6.93) 0.00

Highly Selective 1.62 0.22 7.32 0.00

Middle Selective 0.37 0.18 2.12 0.03

Dispersion x 2000s 80.57 6.02 13.38 0.00

Dispersion x 2010s 4.05 7.04 0.58 0.56

Dispersion x Highly Selective 294.67 7.75 38.03 0.00

Dispersion x Middle Selective 74.74 6.28 11.90 0.00

2000s x Highly Selective (1.91) 0.24 (7.92) 0.00

2010s x Highly Selective (6.48) 0.25 (25.56) 0.00

2000s x Middle Selective (0.92) 0.19 (4.75) 0.00

2010s x Middle Selective (1.60) 0.20 (8.01) 0.00

Dispersion x 2000s x Highly Selective (306.49) 8.12 (37.77) 0.00

Dispersion x 2010s x Highly Selective (527.92) 9.84 (53.68) 0.00

Dispersion x 2000s x Middle Selective (70.85) 6.62 (10.70) 0.00

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18

Table 5: Model coefficients for each decade

4.2.1) 1990s

Figure 5: 1990s time-varying model visualization

In the 1990s, low selectivity funds averaged an alpha of -0.01% at mean levels of dispersion and

VIX for the decade, compared to 0.81% and 3.40% for middle and high selectivity funds

respectively. These differences are statistically significant. Each of the selectivity groups had

significant and large differences in the relationship between dispersion and alpha. Highly

selective firms had the largest positive relationship between dispersion and alpha: a one

percentage point increase in dispersion is associated with an increase of 269 basis points in

monthly alpha. For middle and low selectivity funds, the expected change in alpha for a one

percentage point increase in dispersion is positive 49 basis points and negative 25 basis points

respectively. In other words, highly selective funds outperformed their peers in the 1990s,

especially during times of high dispersion. This evidence is consistent with the existence of

“stock pickers’ markets” in the 1990s, as the most active funds were able to outperform relative

to what was expected when there existed high variability in stock returns.

1990s 2000s 2010s

Variable Coe�cient T Coe�cient T Coe�cient T

Intercept (0.01) (0.07) 0.14 2.00 (0.45) (10.20)

VIX (0.24) (15.69) (0.20) (52.20) (0.14) (34.55)

Dispersion (25.40) (3.29) 55.16 26.99 (21.35) (5.46)

High Selectivity 3.41 11.99 (0.47) (5.35) (1.01) (15.35)

Middle Selectivity 0.82 3.65 (0.49) (6.36) (0.23) (4.52)

Dispersion x High Selectivity 294.67 27.95 (11.83) (4.83) (233.26) (40.93)

Dispersion x Middle Selectivity 74.74 8.75 3.89 1.82 (60.38) (13.57)<latexit sha1_base64="ihVha0S6rVejnvRc+FVDyzuYa9k=">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</latexit><latexit sha1_base64="ihVha0S6rVejnvRc+FVDyzuYa9k=">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</latexit><latexit sha1_base64="ihVha0S6rVejnvRc+FVDyzuYa9k=">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</latexit><latexit sha1_base64="ihVha0S6rVejnvRc+FVDyzuYa9k=">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</latexit>

−8

−4

0

4

8

0 5 10 15Dispersion (%)

Aver

age

Alph

a (%

)

SelectivityLow

Middle

High

Alpha as a Function of Dispersion and Selectivity, 1990s

19

4.2.2) 2000s

Figure 6: 2000s time-varying model visualization

In the 2000s, low selectivity funds averaged an alpha of 0.14% at mean levels of dispersion and

VIX for the decade, compared to -0.35% and -0.33% for middle and high selectivity funds

respectively. For highly selective funds, a 1 percentage point increase in dispersion is associated

with a 43 basis point increase in alpha, compared to expected 59 and 55 basis point increases for

middle and low selectivity funds respectively. Low selectivity funds outperformed middle and

high selectivity funds at mean levels of dispersion and had a very similar relationship between

outperformance and dispersion during this period. This result seems surprising, given that

general intuition is that more active funds would outperform during periods of crisis, and this

period included the worst financial crisis of the last 75 years.

−8

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0

4

8

0 5 10 15Dispersion (%)

Aver

age

Alph

a (%

)

SelectivityLow

Middle

High

Alpha as a Function of Dispersion and Selectivity, 2000s

20

4.2.3) 2010s

Figure 7: 2010s time-varying model visualization

In the 2010s, low selectivity funds averaged an alpha of -0.45% at mean levels of dispersion and

VIX, compared to -0.69% and -1.46 % for middle and high selectivity funds respectively. These

differences are statistically significant. Each of the selectivity groups had large differences in the

relationship between dispersion and alpha. For highly selective funds, a 1 percentage point

increase in dispersion is associated with a 254 basis point decrease in alpha, compared to an 82

basis point decrease for middle selectivity funds and a 21 basis point decrease for low selectivity

funds. These differences are statistically significant. In other words, highly selective funds

underperformed relative to their peers during periods of high dispersion. This result may be

driven by the low-volatility bull market during this period which favored passive investing over

active investing.

4.3) Fund Dispersion vs Market Dispersion

I then tested whether dispersion in security returns is associated with dispersion in fund alpha. To

do so, I estimated dispersion of fund alphas as cross-sectional standard deviation of fund alphas

for every month in the data. I created two different models to explain fund dispersion with

market dispersion: a simple linear model, and a model with log transformations of both fund

dispersion and market dispersion. I chose to explore this log transformed model in addition to the

linear model because the data displayed a fanning pattern, and thus violated the

heteroskedasticity condition of linear regression. The two models are shown below, and their

outputs are shown in tables 6 and 7. Scatterplots for each model are shown in figure 8.

−8

−4

0

4

8

0 5 10 15Dispersion (%)

Aver

age

Alph

a (%

)

SelectivityLow

Middle

High

Alpha as a Function of Dispersion and Selectivity, 2010s

21

Linear:

Table 6: Linear dispersion vs. dispersion model output

Log-Log:

Table 7: Log-log dispersion vs. dispersion model and output

Figure 8: Scatterplots for linear and log-log models

The log-log model has an r-squared that is larger than that of the simple linear model (0.30 vs

0.12), and it better satisfies the condition of heteroskedasticity (as seen in the figure above. Since

there appear to be two outliers, I also considered using the robust regression technique of M-

estimation according to Maronna et al. (2019); however, the output of the robust regression,

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sha1_base64="K50/q0KVaGajYLqGHH+jf1oJmOU=">AAACLHicbVBNSwMxFMz6bf1a9egl2CqCUHZ70YsgKOJRwdpCW5Zs+mpDs9kleSuUxR/kxb8iiAdFvPo7zLZ7sK0DgcnMPJI3YSKFQc/7dObmFxaXlldWS2vrG5tb7vbOvYlTzaHOYxnrZsgMSKGgjgIlNBMNLAolNMLBRe43HkEbEas7HCbQidiDEj3BGVopcC8qV6nqXgqTjEMB0jPaDgFZ4NHjgvkTvlUhMULml0rJInDLXtUbgc4SvyBlUuAmcN/a3ZinESjkkhnT8r0EOxnTKLiEp1I7NZAwPmAP0LJUsQhMJxst+0QPrNKlvVjbo5CO1L8TGYuMGUahTUYM+2bay8X/vFaKvdNOJlSSIig+fqiXSooxzZujXaGBoxxawrgW9q+U95lmHG01eQn+9Mqz5L5W9b2qf1srnx8WdayQPbJPjohPTsg5uSY3pE44eSav5IN8Oi/Ou/PlfI+jc04xs0sm4Pz8AkKfpZk=</latexit><latexit sha1_base64="K50/q0KVaGajYLqGHH+jf1oJmOU=">AAACLHicbVBNSwMxFMz6bf1a9egl2CqCUHZ70YsgKOJRwdpCW5Zs+mpDs9kleSuUxR/kxb8iiAdFvPo7zLZ7sK0DgcnMPJI3YSKFQc/7dObmFxaXlldWS2vrG5tb7vbOvYlTzaHOYxnrZsgMSKGgjgIlNBMNLAolNMLBRe43HkEbEas7HCbQidiDEj3BGVopcC8qV6nqXgqTjEMB0jPaDgFZ4NHjgvkTvlUhMULml0rJInDLXtUbgc4SvyBlUuAmcN/a3ZinESjkkhnT8r0EOxnTKLiEp1I7NZAwPmAP0LJUsQhMJxst+0QPrNKlvVjbo5CO1L8TGYuMGUahTUYM+2bay8X/vFaKvdNOJlSSIig+fqiXSooxzZujXaGBoxxawrgW9q+U95lmHG01eQn+9Mqz5L5W9b2qf1srnx8WdayQPbJPjohPTsg5uSY3pE44eSav5IN8Oi/Ou/PlfI+jc04xs0sm4Pz8AkKfpZk=</latexit><latexit sha1_base64="K50/q0KVaGajYLqGHH+jf1oJmOU=">AAACLHicbVBNSwMxFMz6bf1a9egl2CqCUHZ70YsgKOJRwdpCW5Zs+mpDs9kleSuUxR/kxb8iiAdFvPo7zLZ7sK0DgcnMPJI3YSKFQc/7dObmFxaXlldWS2vrG5tb7vbOvYlTzaHOYxnrZsgMSKGgjgIlNBMNLAolNMLBRe43HkEbEas7HCbQidiDEj3BGVopcC8qV6nqXgqTjEMB0jPaDgFZ4NHjgvkTvlUhMULml0rJInDLXtUbgc4SvyBlUuAmcN/a3ZinESjkkhnT8r0EOxnTKLiEp1I7NZAwPmAP0LJUsQhMJxst+0QPrNKlvVjbo5CO1L8TGYuMGUahTUYM+2bay8X/vFaKvdNOJlSSIig+fqiXSooxzZujXaGBoxxawrgW9q+U95lmHG01eQn+9Mqz5L5W9b2qf1srnx8WdayQPbJPjohPTsg5uSY3pE44eSav5IN8Oi/Ou/PlfI+jc04xs0sm4Pz8AkKfpZk=</latexit>

N R-Squared F P-Value312 0.118 41.29 0.000

<latexit sha1_base64="ggq62n3ZtryBbLux3x08/0ft0bk=">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</latexit><latexit sha1_base64="ggq62n3ZtryBbLux3x08/0ft0bk=">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</latexit><latexit sha1_base64="ggq62n3ZtryBbLux3x08/0ft0bk=">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</latexit><latexit sha1_base64="ggq62n3ZtryBbLux3x08/0ft0bk=">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</latexit>

Estimate Standard Error T P-ValueIntercept 0.01 0.00 6.73 0.00Dispersion 0.10 0.02 6.43 0.00

<latexit sha1_base64="tF881KF5B2KhOav8T15Hxmsi2CY=">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</latexit><latexit sha1_base64="tF881KF5B2KhOav8T15Hxmsi2CY=">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</latexit><latexit sha1_base64="tF881KF5B2KhOav8T15Hxmsi2CY=">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</latexit><latexit sha1_base64="tF881KF5B2KhOav8T15Hxmsi2CY=">AAADKXichVJNb9NAEF2br2I+mtIjlxURFRciO0UtNyrRSnALokkrxVY0Xq+TVddra3eMFFn5O1z4K1xAoipc+SOsa5M2LRVz8ds3M2933jgupDDo+z8d99btO3fvrd33Hjx89Hi9s/FkZPJSMz5kucz1cQyGS6H4EAVKflxoDlks+VF88rbOH33i2ohcHeK84FEGUyVSwQAtNdlw3oQxnwpVIdiWxTjyLs6lBL2oZB0LL5zVV3j0P7FFDwyKDJBb+BFBJaATeqB1ri1xuKwavByBLDkNQ9oqv1fI7UwF3qjs9/yg+fgX5E5vd5teYsPQ2xemaEa+WSnwm57+Cr/Te7W9ovX3cSFXydKS5cka5k063bq8DnodBC3okjYGk873MMlZmXGFTIIx48AvMKpAo2C1YFgaXgA7gSkfW6gg4yaqzje9oM8tk9DUmpnmCuk5e7mjgsyYeRbbSruDmbmaq8l/5cYlpq+jSqiiRK5Yc1FaSoo5rX8bmgjNGcq5BcC0sG+lbAYamF2aqU0Iro58HYz61vZe8KHf3dtq7VgjT8kz8oIEZJfskXdkQIaEOZ+dr84P59T94n5zz9xfTanrtD2bZCXc338AdWPd+Q==</latexit>

log(FundDispersiont) = �0 + �1log(Dispersiont) + ✏t<latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit><latexit sha1_base64="JjKPFttnWjAI6k1gLvfdCJ8v8NY=">AAACNnicbVBNSwMxFMzWr1q/Vj16CbZKRSi7XvQiCIp4ERSsFtqyZNPXNpjNLslboRR/lRd/h7dePCji1Z9gtt2DVgcCk5l5JG/CRAqDnjdyCjOzc/MLxcXS0vLK6pq7vnFr4lRzqPNYxroRMgNSKKijQAmNRAOLQgl34f1p5t89gDYiVjc4SKAdsZ4SXcEZWilwLysy7lXPU9U5EyaZBAPco8e0FQKywKP7OfOz4O+QtSAxQma3SskicMtezRuD/iV+Tsokx1XgvrQ6MU8jUMglM6bpewm2h0yj4BIeS63UQML4PetB01LFIjDt4XjtR7pjlQ7txtoehXSs/pwYssiYQRTaZMSwb6a9TPzPa6bYPWoPhUpSBMUnD3VTSTGmWYe0IzRwlANLGNfC/pXyPtOMo+0mK8GfXvkvuT2o+V7Nvz4on+zmdRTJFtkmVeKTQ3JCLsgVqRNOnsiIvJF359l5dT6cz0m04OQzm+QXnK9vbdepIw==</latexit>

N R-Squared F P-Value312 0.295 129.8 0.000

<latexit sha1_base64="qnSm1kWLsf/7WnO6yaU6qaYaKj4=">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</latexit><latexit sha1_base64="qnSm1kWLsf/7WnO6yaU6qaYaKj4=">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</latexit><latexit sha1_base64="qnSm1kWLsf/7WnO6yaU6qaYaKj4=">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</latexit><latexit sha1_base64="qnSm1kWLsf/7WnO6yaU6qaYaKj4=">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</latexit>

Estimate Standard Error T P-ValueIntercept (2.70) 0.13 (21.59) 0.00Log(Dispersion) 0.56 0.05 11.39 0.00

<latexit sha1_base64="i8hK3jm4mBQfmB3oEPeO/6lYrdk=">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</latexit><latexit sha1_base64="i8hK3jm4mBQfmB3oEPeO/6lYrdk=">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</latexit><latexit sha1_base64="i8hK3jm4mBQfmB3oEPeO/6lYrdk=">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</latexit><latexit sha1_base64="i8hK3jm4mBQfmB3oEPeO/6lYrdk=">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</latexit>

22

which can be found in Appendix 2, is not meaningfully different from the ordinary least squares

regression. Accordingly, I will report results for the log-log model using ordinary least squares

regression.

This model indicates that market dispersion has a statistically significant positive relationship

with dispersion of mutual fund alpha. A 10% increase in market dispersion is associated with

about a 6% increase in dispersion of mutual fund alphas. This result is consistent with de Silva,

Sapra and Thorley (2001), who demonstrated that market dispersion is associated with dispersion

of mutual fund returns. This result also suggests that it is unwise for an investor to make

allocations to a particular fund based on that fund’s alpha during a period of high market

dispersion. Since times of high market dispersion are associated with high dispersion of fund

alphas, it is difficult to discern whether that alpha is the result of manager skill or random

chance.

23

5. Conclusions and Further Research

This paper suggests that dispersion can be a significant predictor of mutual fund alpha, but there

are large differences in this relationship over time. During the 1990s, the most selective funds

had a very significant positive relationship between dispersion and alpha, more so than their less

selective peers. This result is intuitive and consistent with the idea of a stock picker’s market; the

most active stock pickers have the most to benefit from periods of high dispersion. In the 2000s,

this relationship was positive for all funds, providing more evidence in favor of the existence of

stock picker’s markets. Additionally, high selectivity funds performed roughly in line with their

peers at all levels of dispersion during this period. Since this period was dominated by two

financial crises: the dot-com bubble and the global financial crisis of 2008, this result seems to

contradict a popular selling point for active investing, that it is better to have money actively

managed during times of turmoil. The 2010s provided the most surprising results, as funds of all

selectivity levels demonstrated significant negative relationships between dispersion and alpha.

This result is even more surprising when considering the explosive rise in the market share of

passively managed investment vehicles. One might expect that a smaller pool of active managers

would result in more mis-pricings in the market, allowing the remaining active managers to

outperform, but this is not what the data have shown.

There are several possible explanations for these results. One possible explanation is that the

stock market has simply gotten more efficient over time. In the 1990s, managers could have

taken advantage of inefficient pricing in the market, and the most active managers would have

the most opportunity to generate alpha. By the 2010s, there could have been fewer inefficiencies

for active managers to exploit. Managers would incur transaction costs when picking stocks

during times of high dispersion, but not generate outsized returns, resulting in a negative

relationship between dispersion and alpha. This explanation relies on the assumption that funds

would transact more often during times of high dispersion. This question could be interesting to

explore in further research.

Another possible explanation for the results is that mutual fund managers have gotten less skilled

relative to the average market participant over the past few decades. That is not to say that active

managers have become less skilled overall, but the average market participant may have gotten

more sophisticated. It could be the case that in the 1990s, active mutual funds had a stock

picking advantage over less-sophisticated retail investors. Active funds in high dispersion

24

markets would have the most opportunity to exploit this edge. As more sophisticated

participants, such as quantitative hedge funds, have entered the market and grown in size in

recent years3, active managers could now be at a disadvantage relative to the average market

participant. Funds which transact the most would therefore be expected to perform the worst,

explaining the near complete reversal in trends between the 1990s and 2010s. Further evidence is

needed to evaluate whether the average market participant has actually gotten more

sophisticated, so this could be an interesting topic for future research.

From a practical perspective, this analysis does not recommend using a fund’s performance

during times of high dispersion to determine whether to make allocations to a particular fund.

While the results are statistically significant, they apply to funds on average, and are difficult to

apply to a single fund. As seen by the particularly low R-squared values in the regressions, there

is a huge amount of noise in monthly alphas. Furthermore, dispersion in the market is strongly

correlated with dispersion of fund alphas, meaning it is very difficult to discern whether a

particular fund’s strong performance in a time of high dispersion is the product of managerial

skill or luck.

3 https://www.ft.com/content/ff7528bc-ec16-11e7-8713-513b1d7ca85a

25

Appendix 1: Sensitivity of Model to Beta Cutoffs

●

●

●

●

●

●

●

(Intercept)

Dispersion

Dispersion x Low Selectivity

Dispersion x Middle Selectivity

Low Selectivity

Middle Selectivity

VIX

−20 0 20 40 60Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Aggregated Model to Cutoff Values (Wide View)

●

●

●

●

(Intercept)

Dispersion

Dispersion x Low Selectivity

Dispersion x Middle Selectivity

Low Selectivity

Middle Selectivity

VIX

−1.0 −0.5 0.0 0.5 1.0Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Aggregated Model to Cutoff Values (Zoomed View)

●

●

●

●

●

●

●

(Intercept)

Dispersion

Dispersion x Low Selectivity

Dispersion x Middle Selectivity

Low Selectivity

Middle Selectivity

VIX

−20 0 20 40 60Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Aggregated Model to Cutoff Values (Wide View)

●

●

●

●

(Intercept)

Dispersion

Dispersion x Low Selectivity

Dispersion x Middle Selectivity

Low Selectivity

Middle Selectivity

VIX

−1.0 −0.5 0.0 0.5 1.0Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Aggregated Model to Cutoff Values (Zoomed View)

26

●

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●

●

(Intercept)2000s

2000s x Dispersion2000s x Dispersion x Low Selectivity

2000s x Dispersion x Middle Selectivity2000s x Low Selectivity

2000s x Middle Selectivity2010s

2010s x Dispersion2010s x Dispersion x Low Selectivity

2010s x Dispersion x Middle Selectivity2010s x Low Selectivity

2010s x Middle SelectivityDispersion

Dispersion x Low SelectivityDispersion x Middle Selectivity

Low SelectivityMiddle Selectivity

VIXVIX x 2000sVIX x 2010s

−600 −300 0 300 600Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Time−Varying Model to Cutoff Values (Wide View)

●

●

●

●

●

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●

●

●

●

●

●

(Intercept)2000s

2000s x Dispersion2000s x Dispersion x Low Selectivity

2000s x Dispersion x Middle Selectivity2000s x Low Selectivity

2000s x Middle Selectivity2010s

2010s x Dispersion2010s x Dispersion x Low Selectivity

2010s x Dispersion x Middle Selectivity2010s x Low Selectivity

2010s x Middle SelectivityDispersion

Dispersion x Low SelectivityDispersion x Middle Selectivity

Low SelectivityMiddle Selectivity

VIXVIX x 2000sVIX x 2010s

−10 −5 0 5 10Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Time−Varying Model to Cutoff Values (Zoomed View)

●

●

(Intercept)

log(Dispersion)

−3 −2 −1 0Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Dispersion vs Dispersion Model to Cutoff Values (Log−Log transform)

●

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(Intercept)2000s

2000s x Dispersion2000s x Dispersion x Low Selectivity

2000s x Dispersion x Middle Selectivity2000s x Low Selectivity

2000s x Middle Selectivity2010s

2010s x Dispersion2010s x Dispersion x Low Selectivity

2010s x Dispersion x Middle Selectivity2010s x Low Selectivity

2010s x Middle SelectivityDispersion

Dispersion x Low SelectivityDispersion x Middle Selectivity

Low SelectivityMiddle Selectivity

VIXVIX x 2000sVIX x 2010s

−600 −300 0 300 600Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Time−Varying Model to Cutoff Values (Wide View)

●

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●

●

●

(Intercept)2000s

2000s x Dispersion2000s x Dispersion x Low Selectivity

2000s x Dispersion x Middle Selectivity2000s x Low Selectivity

2000s x Middle Selectivity2010s

2010s x Dispersion2010s x Dispersion x Low Selectivity

2010s x Dispersion x Middle Selectivity2010s x Low Selectivity

2010s x Middle SelectivityDispersion

Dispersion x Low SelectivityDispersion x Middle Selectivity

Low SelectivityMiddle Selectivity

VIXVIX x 2000sVIX x 2010s

−10 −5 0 5 10Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Time−Varying Model to Cutoff Values (Zoomed View)

●

●

(Intercept)

log(Dispersion)

−3 −2 −1 0Estimate

Cov

aria

te

Cutoffs● (0.7, 1.3)

(0.8, 1.2)

(0.9, 1.1)

Sensitivity of Dispersion vs Dispersion Model to Cutoff Values (Log−Log transform)

27

Appendix 2: Robust Regression of Market Dispersion vs. Fund Dispersion

The model output for robust regression of market dispersion vs fund dispersion is shown above. The response variable in this regression is log-transformed dispersion of fund alphas, measured in percentage points. This robust regression utilizes an M-estimator and is fitted using iterated re-weighted least squares (IWLS). It is implemented using the MASS package in R.

Estimate Standard Error T P-ValueIntercept (2.63) 0.10 (25.25) 0.00Log(Dispersion) 0.59 0.04 14.44 0.00

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