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Essays in tail risks

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Essays in tail risks

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Content ESSAYS IN TAIL RISKS
by
Jerchern Lin
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BUSINESS ADMINISTRATION)
August 2012
Copyright 2012 Jerchern Lin
ii
Acknowledgements
I am grateful to Wayne Ferson, Fernando Zapatero, Chris Jones, Pedro Matos, David Solomon, and
to participants at the 5th Professional Asset Management Conference, 28th International Confer-
ence of French Finance Association (AFFI), SMU-ESSEC Symposium on Empirical Finance and
Financial Econometrics, 6th International Finance Conference, 4th V olatility Institute Conference
at New York University, Norwegian School of Economics, Simon Fraser University, Tulane Univer-
sity, State University of New York at Buffalo, and USC Finance and Business Economics Seminar
for comments and discussions.
iii
Table of Contents
Acknowledgements ii
List of Tables v
List of Figures vi
Abstract vii
Chapter 1: Tail Risks across Investment Funds 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 How Fund Strategies Impact Tail Risks . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Comparisons across Investment Funds . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Return Dynamics and Tail Dependence . . . . . . . . . . . . . . . . . 8
1.4.2 Characterization of Compensation Structure and Optimization Problem 13
1.4.3 Monte Carlo Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Empirical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.1 Frequency of Tail Returns . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.2 Systematic and Idiosyncratic Tail Risk . . . . . . . . . . . . . . . . . . 31
1.7 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.8 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.8.1 An Application of the Model on Mutual Funds . . . . . . . . . . . . . 59
1.8.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.8.3 Exogenous Systematic Factors . . . . . . . . . . . . . . . . . . . . . . 65
1.8.4 Year 1996-2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 2: Fund Convexity and Tail Risk-Taking 72
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Why Should Investors Move Beyond V olatility and Care About Skewness Risk? 75
2.3 Risky or Skewed Bets? A View from the Literature . . . . . . . . . . . . . . . 77
2.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5 Empirical Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.1 Changes in Tail Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.2 Multivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
iv
2.5.3 Convexity and Tail Risks . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Bibliography 132
Appendix A: The Numerical Procedure for the Optimization Problem 139
Appendix B: Conditioning Biases and Benchmarks 140
Appendix C: Open-ended Fund Styles 143
v
List of Tables
Table 1.1: Parameters across Fund Types . . . . . . . . . . . . . . . . . . . . . . . . . 14
Table 1.2: Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Table 1.3: Frequency of Tail Returns across Fund Types . . . . . . . . . . . . . . . . . 27
Table 1.4: Summary of Higher Moment Covariance Risks . . . . . . . . . . . . . . . . 36
Table 1.5: Skewness Decomposition by Equal-weighted Portfolios . . . . . . . . . . . 45
Table 1.6: Kurtosis Decomposition by Equal-weighted Portfolios . . . . . . . . . . . . 52
Table 1.7: Skewness and Kurtosis Decomposition of HFs Adjusted for Autocorrelations 63
Table 1.8: Skewness Decomposition by Beta-weighted Exogenous Factors . . . . . . . 66
Table 1.9: Kurtosis Decomposition by Beta-weighted Exogenous Factors . . . . . . . . 68
Table 2.1: Cross-Sectional Distribution of Fund Skewness and Kurtosis . . . . . . . . . 84
Table 2.2: Comparison of Differences in Average Fund Skewness . . . . . . . . . . . . 94
Table 2.3: The Sensitivity of Fund Skewness to Lagged Relative Performance . . . . . 96
Table 2.4: Fund Skewness to Lagged Relative Performance across Groups . . . . . . . 101
Table 2.5: Regression of Fund Skewness on the Fractional Rank of Relative Performance 106
Table 2.6: Impact of Firm Characteristics on Skewed Bets . . . . . . . . . . . . . . . . 110
Table 2.7: Convexity Impact on Tail Risks in CEFs -Premiums/Discounts . . . . . . . . 114
Table 2.8: Convexity Impact on Tail Risks in OEFs -Tournaments . . . . . . . . . . . . 116
Table 2.9: Convexity Impact on Tail Risks in OEFs -Flow-Performance . . . . . . . . . 121
Table 2.10: Convexity Impact on Tail Risks in OEFs -Tournament and Flow-Performance 123
Table 2.11: Convexity Impact on Tail Risks in HFs -High-Water Marks . . . . . . . . . . 128
vi
List of Figures
Figure 1.1: The Optimal Weight of the Benchmark and Big Bet . . . . . . . . . . . . . . 17
Figure 1.2: Return Decomposition and Convexity Effects on Benchmark and Big Bet . . 17
Figure 1.3: The Optimal Fund Skewness and Kurtosis . . . . . . . . . . . . . . . . . . . 18
Figure 1.4: The Optimal Weight of the Benchmark and Big Bet . . . . . . . . . . . . . . 60
Figure 2.1: Differences in Skewness Between High- and Low-Performing Groups -CEFs 91
Figure 2.2: Differences in Skewness Between High- and Low-Performing Groups -OEFs 92
Figure 2.3: Differences in Skewness Between High- and Low-Performing Groups -HFs . 93
vii
Abstract
This dissertation is comprised of two essays.
The first essay is titled “Tail Risks across Investment Funds.” Managed portfolios are subject to
tail risks, which can be either index level (systematic) or fund-specific. Examples of fund-specific
extreme events include those due to big bets or fraud. This paper studies the two components in
relation to compensation structure in managed portfolios. A simple model generates fund-specific
tail risk and its asymmetric dependence on the market, and makes predictions for where such risks
should be concentrated. The model predicts that systematic tail risks increase with an increased
weight on systematic returns in compensation and idiosyncratic tail risks increase with the degree
of convexity in contracts. The model predictions are supported with empirical results. Hedge funds
are subject to higher idiosyncratic tail risks and Exchange Traded Funds exhibit higher systematic
tail risks. In skewness and kurtosis decompositions, I find that coskewness is an important source
for fund skewness, but fund kurtosis is driven by cokurtosis, as well as volatility comovement and
residual kurtosis, with the importance of these components varying across fund types. Investors are
subject to different sources of skewness and fat tail risks through delegated investments. V olatility
based tail risk hedging is not effective for all fund styles and types.
The second essay, titled “Fund Convexity and Tail Risk-Taking,” studies how a fund manager
takes skewed bets in two dimensions. First, the fund manager constantly reexamines fund perfor-
mance relative to his or her peers and takes a position with respect to skewness risk. I show that
when a fund manager underperforms peers, he or she will gamble on trades with lottery-like returns.
On the other hand, when a fund outperforms peer funds, the fund manager will take negatively
skewed trades. The results are robust to different econometric specifications. Second, I examine
viii
how convexity in incentives affects tail risks across and within different types of investment funds.
The literature has documented different forms of convexity that a fund manager faces: discounts
in closed-end funds, tournaments and fund flow-performance relation in open-ended funds, and
high-water mark provisions in hedge funds. Sorting funds by the degree of convexity and compar-
ing skewness between the group with the most convexity and the group with the least convexity, I
conclude that convexity affects fund tail risks. This result suggests that both implicit and explicit
convexities provide incentives for fund managers to take systematic and idiosyncratic bets with tail
risks.
1
Chapter 1
Tail Risks across Investment Funds
1.1 Introduction
It is well-known that financial asset returns exhibit asymmetry and fat-tailedness. Mandelbrot
(1963) and Fama (1965) provide theoretical arguments and empirical evidence that price changes
follow stable Paretian distributions. Along with the observation of time-varying volatility, asymmet-
ric volatility, and volatility clustering by Bekaert and Wu (2000) and others, financial economists
have been trying to find sources that contribute to the skewness and kurtosis in returns data, both
conditionally and unconditionally. Facts about non-normality and jumps in returns and volatility
reinforce the importance of higher order moments. Most importantly, financial markets do crash,
as in 1929, Black Tuesday in 1987, the Asian financial crisis in 1997, Long-Term Capital Manage-
ment in 1998, the dot-com bubble burst in 2000, and the recent financial crash of 2008. Tail risks
are important and relevant.
Tail risks are of central importance to investors. A large negative event can significantly reduce
portfolio value and the literature has tried to model this.
1
Large drawdowns in wealth due to extreme
events in the last decade lead investors to fear another market crisis. To cope with investors’ fears
for extreme events, the fund industry has recently developed volatility-based tail risk hedging funds.
1
In recent literature on portfolio choice and delegated principal-agent problems, many models incorporate a VaR
constraint to limit downside risk. The motivation behind downside risk is that investors are concerned with losses in
extreme events and thus they will demand compensation for such extreme, but rare risks, and consider these risks in their
investment decisions.
2
Managed futures have also become a popular alternative investment class as investors seek broad
diversification.
Tail risks can complicate investors’ economic decisions. Samuelson (1970) points out that
mean-variance efficiency becomes inadequate when higher moments matter for portfolio alloca-
tion. Harvey, Liechty, Liechty, and M¨ uller (2010) emphasize the importance of higher moments in
portfolio allocation. Cvitanic, Polimenis, and Zapatero (2008) show that ignoring higher moments
in portfolio allocation can imply welfare losses and overinvestment in risky assets. If investors have
preference for higher moments, they will demand a higher rate of return to compensate for negative
tail risks.
A lack of diversification in investor holdings due to trading constraints or market frictions sug-
gests that investors will care about not only systematic tail risks, but also idiosyncratic tail risks in
their portfolio returns. Idiosyncratic risk is theoretically uncorrelated with market risk. However,
higher moments of idiosyncratic shocks can be correlated with systematic shocks. Similarly, the
covariance risk between the higher moments of systematic shocks and idiosyncratic shocks can be
priced.
Given that most investors delegate their wealth to fund managers and care about tail risks, it
is important to understand the structure of tail risks in managed portfolios and look for solutions
to prevent extreme downside risk. For example, if investors are not aware of tail risks hidden in
managed portfolios, dynamic trading and negatively skewed trading strategies can improve fund
performance in view of mean and variance, but induce great downside risk.
The investment funds in this study include closed-end funds (CEFs), exchange-traded funds
(ETFs), open-ended funds (OEFs), and hedge funds (HFs). In the finance literature, few have looked
at the link between tail risks and returns across different types of funds. However, different fund
types are subject to different rules and regulations. Importantly, different fund types are subject
to different compensation schemes and agency costs. These differences lead to different tail risk
exposures.
Conventionally, investors regard HFs as high risk investment products due to the lack of trans-
parency and loose regulation. Hedge fund managers often claim that certain hedge fund strategies
3
can be used to hedge tail risks. This paper addresses four questions: 1. Are tail risks in hedge funds
systematically different from other types of investment funds? 2. Are tail risks in managed portfo-
lios well diversified? 3. Do hedge funds offer an alternative for investors to hedge tail risks? 4. Can
compensation structure explain the heterogeneity in the sources of tail risks across fund types?
I use two empirical methods to document differences in tail risks across investment funds. First,
I count the frequency of monthly returns exceeding 3 and 5 standard deviations from the mean
(“three and five sigma” events). The results show that the probabilities of tail returns exceed those
under normal distributions. The frequencies across fund types are not statistically different. These
results imply that on average, investors suffer from the occurrence of a “three sigma” event every
two years, regardless of fund types. Second, I use skewness and kurtosis as tail risk measures.
Empirical findings support the presence of conditional skewness and kurtosis in financial assets.
2
Except fixed income ETFs, all fund types have negative skewness and excess kurtosis.
I decompose skewness and kurtosis into systematic versus idiosyncratic tail risks. I find that
HFs are subject to higher idiosyncratic tail risks, but ETFs exhibit higher systematic tail risks.
The decomposition of skewness shows that coskewness is an important source of skewness across
fund types. Kurtosis for ETFs and OEFs mainly comes from cokurtosis, but CEFs and HFs have
the largest components in volatility comovement and residual kurtosis, respectively. Thus, the de-
composition reveals that there are interesting differences in tail risks across fund types that is not
revealed by counting outliers. Idiosyncratic cokurtosis is consistently the least important contribut-
ing factor to kurtosis across fund styles and types. Overall, the combined contribution of cokurtosis
and volatility comovement exceeds more than 50% of kurtosis across fund types.
The decomposition results suggest that (1) investors cannot diversify tail risks in traditional
investment funds, including HFs, because most of their skewness and tail risks come from coskew-
ness, cokurtosis, and volatility comovement; (2) an effective tail risk hedging mechanism should
consider fund performance relative to extreme market movements in return, volatility, and skew-
ness. A volatility-based tail risk hedging fund or a fund offering negative correlation with broad
2
See e.g. Hansen (1994), Harvey and Siddique (1999), and Jondeau and Rockinger (2003).
4
asset classes is not likely to be sufficient; (3) The decomposition of tail risks may reflect the trading
strategies undertaken by a fund type.
This paper further ties fund managers’ compensation schemes with tail risks and tries to under-
stand the decomposition of tail risks across fund types. The literature on agency costs, incentive
contracts and the fund flow-performance relationship examine fund managers’ risk-taking behav-
ior. Brennan (1993) proposes an agency based model with relative performance and suggests that
option-like compensation can induce skewness in fund returns. Motivated by relative performance
measures and convex payoff structures, fund managers may take fund-specific tail risks (big bets)
endogenously.
I use a simple model to illustrate how fund managers adjust systematic and idiosyncratic tail
risks in response to the weight on compensation relative to a benchmark (the return decomposition
effect) and to the importance of incentive compensation (the convexity effect). I model a nor-
mal shock for the benchmark, a negatively skewed shock for the fund-specific big bet, and their
asymmetric tail dependence by the copula to generate nonzero covariance risks between the higher
moments of the two assets. The model predicts the following: First, the more the compensation
depends on systematic returns, the more systematic risk the fund managers would take. This action
would increase total fund skewness and decrease total fund kurtosis. Second, when the weight on
the incentive contract increases, the increased convexity encourages fund managers to take big bets
and funds exhibit lower skewness and higher kurtosis.
The rest of the paper proceeds as follows. Section 1.2 explains how fund strategies affect tail
risks. Section 1.3 offers descriptions of and comparisons across different types of investment funds.
Section 1.4 describes the model to produce tail returns and risks in response to the weight between
systematic/idiosyncratic risk and the convexity in compensation across fund types. Section 1.5
outlines the data. Section 1.6 explains empirical methods. Section 1.7 presents empirical results.
Section 1.8 presents a robustness analysis. Section 1.9 concludes.
5
1.2 How Fund Strategies Impact Tail Risks
Two strategies that traditional fund managers use to outperform benchmarks or peers are stock
picking and beta timing. These two strategies have their own implications for fund tail risks. If
market factors are skewed and fund managers use aggressive bets on beta timing, fund returns can
be skewed.
3
Time-varying betas can induce time-varying systematic skewness risk. Alternatively,
a fund can follow a strategy of holding asset classes or compositions of assets different from the
benchmark and achieve good stock selection to have better performance. If a fund manager relies
on stock selection to generate alpha, idiosyncratic tail risk of the fund reflects the tail risks of the
stocks the fund focueses on. The turnover of individual stocks in managed portfolios can also cause
time-varying fund tail risks.
Fund risk can be decomposed into systematic and idiosyncratic components. Funds’ systematic
tail risk comoves with the market. Kraus and Litzenberger (1976) provide theoretical and empirical
evidence that unconditional systematic skewness matters for market valuation. Harvey and Sid-
dique (2000) extend the study to conditional skewness. Dittmar (2002) concludes that conditional
systematic kurtosis is relevant to the cross-section of returns. If fund managers want to increase
funds’ systematic coskewness, in expectation of an upswing in the market, they can add positively
coskewed financial assets. Adding an asset with positive coskewness, such as out-of-money op-
tions, makes the fund more right skewed. Buying or selling options on the market or individual
security options will affect the skewness of the managed portfolio relative to the market (Leland
(1999)). Harvey and Siddique (2000) document that abnormal returns from momentum strategies
result from buying assets with negative coskewness (winners) and shorting assets with positive
coskewness (losers). Therefore, a contrarian trading strategy, i.e. buying losers and sell winners,
can increase fund skewness. Similarly, fund managers can increase portfolio kurtosis by adding
assets with high cokurtosis.
Another mechanism that fund managers can use to increase overall portfolio skewness and kur-
tosis operates through idiosyncratic skewness and kurtosis. Some financial assets with specific
3
In an ICAPM setting with conditional volatility, Engle and Mistry (2007) study negative skewness in priced risk
factors - Fama and French factors and Carhart’s momentum factor.
6
characteristics, such as small-cap stocks, illiquid foreign securities, convertible bonds, may have
more skewed distributions. Adding these assets can make investment funds more skewed. Like-
wise, foreign currencies have fatter tails than stocks or bonds. Currency fund managers can adjust
the level of kurtosis via currency exposure.
In addition to what a fund manager trades (where), trading strategies (how) can also result in
fund tail risks. However, trade positions in fund holdings disclosure may disguise the magnitude
of skewness and fat tail risks. For example, a fund manager bets on two assets to converge to one
price. A merger arbitrage manager bets on the completion of a merger by buying the target firm and
selling the bidding firm. An event driven manager trades on corporate events that can affect share
prices, such as restructurings, recapitalizations, spin-offs, etc. A pairs trading strategy is based on
relative mispricings of two assets in the same sector. A statistical arbitrage trade captures pricing
inefficiencies between securities. These strategies create a short position on a synthetic put option,
i.e. if desired events do not occur, the loss can be substantial.
The short volatility trades above are one type of negatively skewed bet. A negatively skewed
trade is characterized by a concave function of the underlying price level, which delivers steady
profits with low volatility most of the time. For example, a fund manager can collect premiums by
shorting put options. However, extreme events can wipe out all those gains. Examples are covered
call writing, short derivative positions, short vega option strategies, leveraged positions, illiquid
trades, etc. Dynamic trading strategies of a HF manager can improve Sharpe ratios at the expense
of significant tail risks (Leland (1999)). Goetzmann et al. (2007) argue that fund managers can
manipulate performance through dynamic trading.
1.3 Comparisons across Investment Funds
Financial institutions offer a wide variety of financial products to meet investors’ needs. This study
examines four fund types: CEFs, ETFs, OEFs, and HFs. An OEF issues and redeems shares at net
asset value (NA V) at market close each day in response to investors’ demands. The NA V of an OEF
is calculated directly from the prices of stocks or bonds held in the fund. An OEF is required to
7
report its NA V by 4 pm Eastern Standard Time, and trades on OEFs can only be legally executed
end of the day when NA Vs are determined.
Unlike an OEF, a CEF has a finite number of shares traded on an exchange. A fixed number of
shares are sold at the initial public offering (IPO) and investors are not allowed to redeem shares after
the IPO. Due to a set amount of shares traded on the exchanges, a CEF can be traded at a premium or
a discount relative to the value of its portfolio. Numerous studies have attributed unrealized capital
gains, the liquidity of the assets held, agency costs, and irrational investor sentiment as possible
reasons for the CEF discount. Since redemptions of shares are restricted, a CEF is able to invest
in less liquid securities than an OEF. About 80% of CEFs are income oriented and most CEFs are
leveraged (Cherkes, Sagi, and Stanton (2009)). A CEF can borrow additional investment capital by
issuing auction rate securities, preferred shares, long-term debt, reverse-repurchase agreements, etc.
Therefore, a CEF can have higher risks and earn higher returns from illiquidity premiums, active
management, and leverage.
ETFs, like CEFs, are traded on a stock exchange. However, market prices of an ETF diverge
from its NA V in a very narrow range. Since major market participants can redeem shares of an
ETF for a basket of underlying assets, if the prices of an ETF deviate too much from its NA V , an
arbitrage opportunity takes place. Moreover, most ETFs passively track their target market indices.
But some ETFs, in contrast to mutual funds, are designed to provide 2 or 3 times leverage on the
benchmarks. Leveraged ETFs have return characteristics similar to options in terms of amplifying
investment returns, but no preset expiration dates.
Mutual funds and ETFs are under SEC regulations, but HFs face minimal regulations by the
SEC. Only HFs with more than $100,000,000 in assets are required to register as investment ad-
visors and report holding information through 13-F filings. Therefore, HF managers are generally
free to employ dynamic trading strategies (Fung and Hsieh (1997)). Management fees on HFs are
between 1.5% and 2% of assets under management and performance fees are asymmetric and on
average 20%. Like CEFs, HFs can invest in illiquid assets due to lockups and redemption noti-
fication periods (Aragon (2007)). HFs further suffer from smoothed returns (Asness, Krail, and
Lieu (2001)). Getmansky, Lo, and Makarov (2004) show that serial correlation in HF returns can
8
be explained by illiquidity exposure and smoothed returns. In addition, HF managers use lever-
age to increase capital efficiency and investment returns. In short, illiquidity, leverage, high-water
marks, investment flexibility, asymmetric performance fees, lack of transparency, and redemption
requirements may increase HFs’ tail risk exposures.
Convexity affects tail risks. HF managers are compensated by high-water mark contracts. The
compensation is calculated as 20% of profits in excess of high-water marks only if previous losses
are fully recovered. This option-like compensation can induce HF managers to take idiosyncratic
bets to turn around fund performance. An OEF manager receives compensation based on assets
under management. Sirri and Tufano (1998) and Chevlier and Ellison (1997) find a nonlinear re-
lationship between fund flow and past performance. Asymmetric return chasing by investors can
create incentives for OEF managers to take big bets to improve returns relative to the markets. In ad-
dition, relative performance evaluation to a benchmark or peers can motivate a mutual fund manager
to take idiosyncratic bets to climb up in the rankings. The compensation for ETF managers depends
more on systematic fund returns because they are generally evaluated based on how closely they
track the benchmarks. As such, systematic tail risks are more important for ETF managers. Overall,
the compensation structure can impact on a fund manager’ tail risk taking behavior and induce fund
tail risks from heterogeneity in asset classes.
In summary, differences in fund characteristics, such as active management, redemptions, reg-
ulations, transparency to investors, agency costs, etc., may lead to differences in tail distributions
across fund types. Most importantly, I propose that heterogeneity in compensation structure can
explain heterogeneity in tail risks across fund types because compensation structure is linked to a
fund manager’s tail risk taking and optimal allocation among asset classes and risks.
1.4 The Model
1.4.1 Return Dynamics and Tail Dependence
I model a fund manager facing an exogenous compensation structure. The model predicts how
the compensation structure can induce systematic and idiosyncratic skewness and kurtosis in fund
9
returns. The manager chooses an optimal allocation between a benchmark and a negatively skewed
bet on idiosyncratic returns. The model predictions are used to explain tail risks across fund types.
Suppose that a fund manager faces a stylized portfolio choice problem today at timet between a
benchmark and a big bet. The benchmark exposure captures market timing and the big bet captures
selectivity and tail-risk management. Assume the joint distribution of returns of the two assets are
independent and identically distributed (i.i.d) through time and their complete moments and joint
distribution are observable before the allocation is updated. Thus for j= 1...t, the fund’s return
dynamics is modeled as follows:
R
i,t+1
= w
∗
R
p,t+1
+(1−w
∗
)R
BB,t+1
(1.1)
whereR
i,j
is the return at timet+1 for fundi. R
p,j
andR
BB,j
are the returns of the benchmark
and the big bet at time j, respectively. w
∗
is the optimal weight that maximizes expected wealth
at time t and w
∗
∈ [0,1].
4
For simplicity, I drop subscript j and t + 1 in the following analysis.
A fund manager’s strategies on beta timing and security selection do not only affect the magnitude
of systematic and idiosyncratic components of returns. Even if both components are uncorrelated,
the higher moments of one component and the mean and variance of the other component are not
necessarily uncorrelated, and I model this correlation below.
The benchmark represents the systematic risk of a fund and suffers from macroeconomic shocks.
The benchmark is assumed to follow a normal distribution and satisfy zero residual tail risks.
5
In the
empirical work, I construct equal-weighted portfolios of funds by using funds within the same style
and beta-weighted exogenous factors as proxies for benchmarks. The weight on the benchmark
captures a fund manager’ market timing strategy at timet.
The big bet reflects fund-specific risk or microeconomic shocks. Fund managers often engage
in security selection, undertaking idiosyncratic risk to generate alpha. Simonson (1972) provides
4
For robustness, I also test the model withw∈[−1,1] to allow a fund manager to short sell.
5
The benchmark can also be assumed to be positively or negatively skewed, as long as the tail risks from the bench-
mark are lower than the big bet. The benchmark has limited tail risks since a underperforming firm in the benchmark will
be replaced and investors do not observe benchmarks to blow up. Leverage on the benchmark will not yield downside
risk as severe as individual assets.
10
evidence for speculative behavior of mutual fund managers. HF managers commonly engage in
negatively skewed bets (Taleb (2004)). A negatively skewed bet is characterized as a trade that has
a large chance of making gains but a very small chance of losing big money. Examples are arbitrage
trading strategies, leveraged trades, short (derivatives) positions, illiquid assets, credit related instru-
ments, syndicated loans, pass-through securities, etc. Big bets can endogenously generate tail risks
and induce asymmetric payoffs in investment funds. Moreover, trades that endogenously generate
left tail risks can help fund managers manipulate performance measurement (Goetzmann, Ingersoll,
Spiegel, and Welch (2007)).
Additional motivations to model the big bet as a negatively skewed bet are the following. First,
fraud or ponzi schemes follow negatively skewed distributions. For instance, Benard Madoff’s
hedge funds made a succession of considerable gains, but once he was charged with fraud, fund
performance plummeted. The return distribution is negatively skewed. Second, due to the negative
price of risk for skewness, the big bet captures exposure to a non-benchmark asset that are possibly
rewarded with a positive expected return. Third, the negatively skewed shock captures left-tail risk
or crash risk. Crash risk arises from a low probability event that produces large negative returns.
Fourth, the combination of the benchmark and the big bet under aforementioned assumptions can
assure fund returns to be close to normal or negatively skewed. This is consistent with what we
observe in the data.
Big bets are idiosyncratic because if a fund manager wants to camouflage a fund’s trading, s/he
will use a trading strategy or an asset isolated from market movement. For example, frauds are fund-
specific. Moreover, greater tail risks are associated with higher risk premiums. Fund managers have
a wide variety of securities to select for negatively skewed trades, compared to some benchmarks,
based on their expertise and research. For instance, illiquidity premiums are associated with stock
options due to wider bid-ask spreads than index options. The downside risk of short volatility trades
on individual securities is higher than the benchmarks because of higher idiosyncratic volatility. Due
to compensation structure, fund managers may have incentives to camouflage fund alpha by taking
idiosyncratic big bets with significant tail risks. Titman and Tiu (2010) find that HFs deviating from
systematic factors provide abnormal returns or higher Sharpe ratios.
11
The literature on pay-performance well documents managerial risk-taking behavior in response
to performance relative to a benchmark (e.g. Murphy (1999)). Brown, Harlow, and Starks (1996)
find that mid-year losers tend to increase fund risk in the latter part of the year. Chevalier and Ellison
(1997) conclude that mutual fund managers alter fund risk towards the end of year due to incentives
to increase fund flows. Kempf and Ruenzi (2008) find that mutual funds adjust risk according to
their relative ranking in a tournament within the fund families.
To capture the bet having a low probability of blowing up, but a large chance of winning, I use
the skewed t-distribution to model the big bet.
6
In this study, the marginal distribution of the big
bet follows the skewed t-distribution withλ = −0.6 (skewness) andν = 7 (degree of freedom) to
generate negative skewness and excess kurtosis. Both parameters are in the reasonable range from
the aforementioned empirical papers. Since only unexpected shocks matter for unexpected returns,
both the benchmark and the big bet are standardized to be mean zero and variance one.
There are alternatives to endogenously generate fund tail risks with an idiosyncratic big bet. For
instance, one can add jumps in asset prices and volatility to generate skewness and kurtosis. Another
approach is to model a mixture of normal distributions in returns and volatility. Both approaches
require more assumptions on parameter specifications than the skewed t. To my knowledge, the
parameter values for funds are not well documented. For example, there is little evidence on the
frequency of jumps and jump sizes in investment funds.
The dependence structure between the benchmark and the big bet can impact fund tail risks. The
change of the moments and the return distribution of a fund depends on the covariance, coskewness,
and cokurtosis risk between the benchmark and the big bet. For example, Boguth (2010) models
state-dependent idiosyncratic variance and its correlation with the mean and variance of a systematic
factor to induce fund skewness and kurtosis. Recent studies have also documented asymmetric tail
dependence among financial assets (Longin and Solnik (2001) and Ang and Chen (2002)).
6
The generalized skewed t-distribution is first suggested by Hansen (1994) and is applied to model time-varying
asymmetry and fat-tailedness by Jondeau and Rockinger (2003) and Patton (2004). Theodossiou (1998) and Daal and Yu
(2007) show that the skewed t-distribution provides a better fit for financial asset returns in both the U.S. and emerging
markets than GARCH-jump models. Recent studies also adopt the skewed t-distribution to model asset returns and extend
its applications in asset allocation, risk management, credit risk, and option pricing (e.g. Aas and Haff (2006), Dokov,
Stoyanov, Rachev (2007)).
12
I model the tail dependence between the benchmark and the big bet by a T-Copula.
7
The bi-
variate copula is the joint distribution of two marginal distributions. Financial asset returns tend to
comove together more strongly in bad economic states than good ones. The copula models asym-
metric joint risks among financial assets. Its application includes credit default risk, catastrophic
risk for insurers, systemic risk among financial institutions, etc.
8
I focus on T-Copula because of its
prominence in the tail dependence literature. Results are based on tail dependent parameterκ = 0.
9
The model setup follows Patton (2004). He studies the optimal conditional weight between a
big-cap and a small-cap portfolio under various tail dependence structures. To solve the optimal
weight for two given assets, it is necessary to estimate the conditional mean and variance. Unlike
his study, my focus is on the unconditional weight and I do not restrict the benchmark and the big
bet to be any specific financial assets. Because I want to emphasize the differences in tail risks
between these two assets, I adopt two arbitrary standardized financial assets.
10
If I am interested in
two specific financial assets, such as S&P 500 and a stock option on Citibank, I can multiply the
standardized time-series by their respective volatilities and add back their respective means to derive
the optimal unconditional weight of these two specific assets. I show one example with mutual fund
data in the robustness analysis section.
This allocation problem reflects a fund manager’s ability to adjust systematic and idiosyncratic
tail risk. For example, market-neutral HFs have low systematic tail risk but high idiosyncratic tail
risk. ETF or index funds have high systematic tail risk, but relatively low idiosyncratic tail risk.
In daily fund management, fund managers can adopt market-timing or stock-picking strategies to
decide the allocation between systematic and idiosyncratic returns. In a multi-period setting, a
fund manager can disguise fund performance by betting on negatively skewed assets or investing
strategies.
7
I also test with Normal and Rotated Gumbel copula for a robustness check. Normal copula has zero tail dependence
and Rotated Gumbel copula has lower tail dependence only.
8
See e.g. Frey, McNeil, and Nyfeler (2001), McNeil, Frey, and Embrechts (2005).
9
Results hold forκ = 0.5 and 0.9, reflecting different levels of covariance, coskewness, cokurtosis risk between the
benchmark and the big bet.
10
I follow Kan and Zhou (1999) to standardize the systematic factor to simulate asset returns.
13
1.4.2 Characterization of Compensation Structure and Optimization Problem
I consider a combination of a linear and a convex compensation contract. The linear contract is
based on a fund manager’s systematic and fund-specific returns with the nonnegative allocation
weightα and1−α, respectively:
11
W
linear
= α(wR
p
)+(1−α)((1−w)R
BB
)
whereα is specified in the incentive contract. The return decomposition parameterα reflects the
weight of the systematic component on the compensation. For largerα, the manager’s compensation
depends more on the systematic component of returns.
A fund manager’s total compensation may also depend on the convex payoff W
opt
= 1 +
max(φ(R
i
+K),0) andW
linear
, weighted by nonnegativeg and1−g, respectively:
W = gW
opt
+(1−g)W
linear
(1.2)
= g(max(φ(R
i
+K),0)) (1.3)
+(1−g)[α(wR
p
)+(1−α)((1−w)R
BB
)]
where the incentive fee φ is subject to high-water marks and commonly quoted as 20% in the
HF industry. Fund managers receive incentive fees only if fund value exceeds the highest value the
fund has previously achieved. The convexity parameter g is exogenously given and varies across
fund types. The larger theg, the more convex the compensation. K measures the cumulative losses
up to timet and is modeled asK
t
=min(0,K
t−1
+R
t
).
I directly model the option-like payoff like HFs, instead of using an arbitrary fixed K. An
arbitrary K may reflect implicit convexity faced by fund managers, such as tournaments or fund-
flow performance relations, but it is too arbitrary to justify a specific value toK. To my knowledge,
there are no empirical studies that estimate the range of K across funds. Furthermore, incentive
11
Ramakrishnan and Thakor (1984) show that in the presence of moral hazard, contracts will depend on both systematic
and idiosyncratic risks.
14
fees in the mutual fund industry are calculated based on cumulative performance over previous
periods as well. Elton, Gruber, and Blake (2003) show that fulcrum fees can always be converted to
non-negative incentive fees. Nonetheless, I also use a fixedK = 1% as a robustness check.
This setup for managerial compensation is very stylized so that it can be applied to different
types of investment funds. HF managers are measured against high-water marks and thus g = 1.
For ETFs and index funds, tracking errors are critical in performance measurement and no convex
payoff applies to compensation.
12
Therefore, α and g are 1 and 0, respectively. Because actively
managed OEFs are subject to implicit optionality, such as fund-flow performance relations and
“tournaments”, the compensation should depend on on a combination of total fund returns and
fund-specific returns (0 < α,g < 1). CEFs are subject to discounts, which can be regarded as the
moneyness of an option that investors sell to the management. Both α and g are between 0 and 1
for CEFs. The setup implicitly captures relative performance in ETFs, CEFs, OEFs, and absolute
performance in HFs. The order of the magnitude ofα (index tracking) across fund types is ETFs,
CEFs or OEFs, and HFs; the effect ofg (convexity) is in the order of HFs, OEFs or CEFs, and ETFs.
In summary, Table 1.1 shows how I apply the model for the different fund types:
Table 1.1: Parameters across Fund Types
ETFs Index Active OEFs CEFs HFs
α 1 1 ∈ (0,1) ∈ (0,1) ∈ (0,1)
g 0 0 ∈ (0,1) ∈ (0,1) 1
K NA NA Cumulative
∗
NA Cumulative
φ NA NA < 1%
∗
NA 20%
∗
if applicable (Elton, Gruber, and Blake (2003))
Following Patton (2004), I assume that fund managers optimize his/her wealth for the period
t+1 using returns observed up to timet to form expectations. Under the assumption of i.i.d returns,
the optimal weight can be solved by maximizing the sum of utility functions up-to-date.
w
∗
≡ argmaxE
t
[U(W
t+1
)] (1.4)
= argmax
1
t
t
X
j=1
U(W
j
)
12
Kim (2010) shows that the flow-performance relation is weak for index funds.
15
whereW
j
is the manager’s total compensation at timej. For simplicity, I drop the subscripj in
the following notation.
The non-normal fund returns and option-like compensation structure lead to nonlinearity and
non-normality of total wealthW . The utility below follows Mitton and V orkink (2007) and Boguth
(2010) and captures the higher moments of wealth.
U(W) = E(W)−
1
2τ
2
Var(W)+
1
3τ
3
Skew(W)−
1
12τ
4
Kurt(W) (1.5)
where τ
2
, τ
3
, and τ
4
are risk tolerance for the second, third, and fourth moments of W . The
central moments are defined as Var(W) =E[W −E(W)]
2
, Skew(W) =E[W −E(W)]
3
, and
Kurt(W) =E[W −E(W)]
4
- 3Var(W)
2
. The main results use τ
2
= 1.5, τ
3
= 0.15, and τ
4
=
0.015. The parameters of risk tolerance for the second, third, and fourth moments under this utility
is translated into relative risk aversion between 5 and 10 under the power utility
13
According to Kane
(1982), the skewness ratio and kurtosis ratio for the power utility are equal to 1+γ and (1+γ)(2+γ),
respectively. γ is the relative risk aversion and skewness (kurtosis) ratio reflects preference for the
third (fourth) moment relative to aversion to variance. Thus, the range of skewness ratio is between
6 and 11 and kurtosis ratio is between 42 and 132 forγ = 5 and 10. Parameters for risk tolerance
used in the model suggest skewness ratio and kurtosis ratio to be 10 and 100, respectively. The
initial wealth is set to be 1 because the optimal allocation does not depend on the initial wealth
under this utility.
The positive sign of the third term denotes the manager’s preference for skewness. The negative
sign of the fourth term corresponds to the manager’s dislike of kurtosis. This type of utility captures
the manager’s concern for skewness and kurtosis relatively to dispersion.
Since the distribution of fund returns in this model is not solely determined by mean and variance
and managerial compensation is convex, the utility taking account of the probability distribution
of wealth up to the fourth moments is used. Fund managers are assumed to value skewness and
kurtosis. A convex contract is not desirable for a fund manager who is neutral to risks or cares only
13
.
16
about mean and variance. Hemmer, Kim, and Verrecchia (2000) show that the incentive contract
should be more convex when skewness is increased, and the amount of convexity depends on the risk
aversion. The return generating process and asymmetric dependence structure guarantees skewness
and kurtosis in wealth. Fund skewness and kurtosis cannot be diversified away in this model. The
preference for higher moments ensures fund managers consider tail risks in the asset allocation
between the benchmark and the big bet according to compensation structure.
Career concern and “tournament” also support the preference for higher moments. As Taleb
(2004) states, “Does one gamble dollars to win a succession of pennies (negative skewness) or one
risks a succession of pennies to win dollars (positive skewness)?” Although the conventional utility
theory suggests that a rational manager would prefer positive skewness and dislike excess kurtosis,
most funds are negatively skewed and fat-tailed. One reason can be career concerns. If a fund
manager takes a positively skewed bet, the probability of failures is too high to stay in the business.
From the “tournament” perspective, if a fund underperforms its peer, the fund manager may choose
to gamble with a large probability of considerable losses, but a tiny probability of huge gains. Large
losses can blow up the fund. On the other hand, an outperforming fund may take a negatively
skewed bet instead because of a very tiny probability of losses and frequent gains.
1.4.3 Monte Carlo Results
Since the optimization problem above has no closed-form solution, I following Patton (2004) to
numerically solve the asset allocation problem. The details are in the Appendix A.
Figure 1.1 presents the optimal weights of the benchmark and the big bet. Figure 1.2 shows the
snapshot of the optimal weights with respect toα andg, i.e. the return decomposition and convexity
effect. Figure 1.3 displays the optimal skewness and kurtosis of a fund.
The model predicts that as convexity in the contract increases (i.e. g increases), fund managers
will increase weights on the idiosyncratic big bet and thus reduce fund skewness and increase fund
kurtosis. On the other hand, if a fund managers’ compensation ties more to the systematic returns
(i.e. α increases), more weight will be allocated to the benchmark to increase fund skewness and
reduce fund kurtosis.
17
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
alpha: Return decomposition
The Return Decomposition and Convexity Effect on Benchmark
g: Convexity
Benchmark weight
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
alpha: Return decomposition
The Return Decomposition and Convexity Effect on Big Bet
g: convexity
Big bet weight
Figure 1.1: The Optimal Weight of the Benchmark and Big Bet
The return decomposition parameterα and the convexity parameterg are the weight of the systematic return
and convex payoff in managerial compensation, respectively. z-axis is the optimal weight on the benchmark.
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
alpha: Return decomposition
Benchmark weight
0 0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
alpha: Return decomposition
Big bet weight
0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
g: Convexity
Benchmark weight
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
g: Convexity
Big bet weight
Figure 1.2: Return Decomposition and Convexity Effects on Benchmark and Big Bet
The graphs on the top panel show the return decomposition effect on the benchmark (left) and the big bet
(right). The graphs on the bottom panel show the convexity effect on both assets. The snapshot is taken by
averaging weights across allg andα for eachα on the x-axis andg on the y-axis, respectively.
18
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
−1.5
−1
−0.5
0
alpha: Return decomposition
g: Convexity
Optimal skewness
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
2
4
6
8
alpha: Return decomposition
g: Convexity
Optimal kurtosis
Figure 1.3: The Optimal Fund Skewness and Kurtosis
The graphs show the fund skewness and kurtosis based on optimal weights from Figure 1.1.
The incentive to take the idiosyncratic big bet is to risk the possibility of negatively skewed
outcomes in exchange for improving the fund’s expected alpha for the next period. Consider two
types of fund managers in the economy: conservative and aggressive. A fund manager whose com-
pensation depends more on the systematic component of returns (i.e. a larger α) can be viewed
as the conservative one. An ETF fund manager is one example. The conservative fund manager
face a linear contract and tail risks have symmetric impact on managers. Thus, s/he simply trades
the benchmark and has no incentive to improve alpha and trade idiosyncratic big bets since trading
big bets does not increase utility. On the contrary, when a fund manager is endowed with a more
convex compensation scheme (i.e. a largerg), fund managers care about the upside and downside
differently. An aggressive fund manager prefers idiosyncratic big bets that improve or camouflage
the short-term performance at the cost of increased left tail risks. HFs are the example. Convex-
ity generally increases skewness, but the introduction of a negatively skewed bet can mitigate the
convexity effect.
19
One intriguing implication from the model is that if the compensation structure depends mostly
on idiosyncratic returns with little convexity (i.e. α andg are both very low) , the model suggests
that a fund manager will invest mostly in the idiosyncratic big bet to increase expected returns and
undertake tail risks. However, it is hard to find this type of compensation structure since the compen-
sation structure should be based on any signals that informs about managers’ actions (Holmstrom
(1979)). Most funds’ compensation relies on convexity and systematic returns to some degrees.
In summary, Figure 1.1 shows the predictions for the tail risks for the different fund types. HFs’
skewness and kurtosis come mostly from the idiosyncratic component of returns because the convex
compensation is associated with g = 1. The increased weight on the idiosyncratic big bet lowers
the skewness and raises the kurtosis of a HF. ETFs, represented by higherα and lowerg, are subject
to higher systematic tail risks. Figure 1.3 shows that ETFs exhibit less negative skewness and lower
kurtosis. OEFs and CEFs are associated withα andg between 0 and 1. As such, their weights of
the idiosyncratic components in total fund skewness and kurtosis are between HFs and ETFs.
1.5 The Data
The ETFs, OEFs, CEFs, and HFs in this study are investment funds managed in the U.S. The list
of ETFs and CEFs domiciled in the U.S. are screened from the Morningstar database, including
both live and dead funds. Monthly returns of ETFs and CEFs from the CRSP monthly stock return
table are merged with the list of funds from Morningstar database by dates and tickers. ETFs and
CEF returns start from 1993 and 1929, respectively. Monthly OEF returns are from CRSP U.S.
survivorship-free mutual fund database and start in 1962. The HF sample is constructed from the
HFR database, starting in 1996. The data period for all four fund types ends in 2008.
20
I form groups of funds by styles for analysis. ETFs and CEFs are grouped by Morningstar
styles.
14
OEFs are grouped by CRSP style codes.
15
HFs are grouped by HFR main strategies.
16
Table 1.2 summarizes univariate statistics of “average” funds across fund styles and types. By
“average”, it means that statistics for individual funds in the same group are averaged to represent
“average” or individual fund statistics.
HFs are the most negatively skewed. ETFs are the least negatively skewed and fixed income
ETFs have positive skewness. The level of skewness in OEFs and CEFs is between HFs and ETFs.
The kurtosis of HFs (ETFs) is close to that of CEFs (OEFs). The model fully predicts the tail
risks in HFs and ETFs. The tail risks in HFs increase because increased convexity in compensation
motivates fund managers to take more big bets (negatively skewed bets). The tail risks in ETFs
declines because the increased weight on compensation relative to the benchmark induces a ETF
manager to increase loadings on the benchmark, which bears lower left tail risk. There are variations
in tail risks across fund styles within the same fund type. It can be observed from the variation of
the significance level of the Jarque-Berra test.
14
Equity ETFs: Global, Currency, Sector, Balanced, Bear Market, Commodities, Large/Mid/Small Cap, Growth/Value,
and Others. Fixed Income ETFs: Global, Sector, Long Term, Intermediate Term, Short Term, Government, High Yield,
and Others. Equity CEFs are Global, Balanced, Sector, Commodities, Large/Mid/Small Cap, Growth/Value, and Others.
Fixed Income CEFs are Global, Sector, Long Term, Intermediate Term, Short Term, Government, High Yield, and Others
15
Equity funds are classified as Index, Commodities, Sector, Global, Balanced, Leverage and Short, Long Short, Mid
Cap, Small Cap, Aggressive Growth, Growth, Growth and Income, Equity Income, and Others. Fixed income funds
are classified as Index, Global, Short Term, Government, Mortgage, Corporate, and High Yield. The classification
methodology is in Appendix C.
16
Equity Hedge, Event-Driven, Fund of Funds, HFRI Index, HFRX Index, Macro, and Relative Value. Descriptions of
these investment strategies are available available from HFR at http://www.hedgefundresearch.com.
21
Table 1.2: Summary Statistics
This table reports summary statistics for average funds across fund styles and types. Nofunds is the total number of funds. Nobs is
the average number of nonmissing time series observations of average funds. Each statistic for a style is reported as the cross-sectional
average of statistics of individual funds in the same style. Mean is the average mean, std is the average standard deviation, skew is the
average skewness, kurt is the average excess kurtosis,ρ
1
is the average first order sample autocorrelation,ρ
2
is the average second order
sample autocorrelation, andρ
3
is the average third order sample autocorrelation. Reported statistics are in percentage per month. JB is
the Jarque Bera p-value for the test for normality. JB test statistic is
Nobs
6
(Skew
2
+
Kurt
2
4
). LQ is the Ljung-Box q statistics for the test
of lag-3 autocorrelation. LQ test statistic isNobs(Nobs+2)
P
3
j=1
ρj
Nobs−j
. FI Average is the average of statistics across fixed-income
fund styles. EF Average is the average of statistics across equity fund styles. Group Average is the average of statistics across all fund
styles. CEFs/ETFs/OEFs/HFs refer to closed-end funds/exchange-traded funds/open-ended funds/hedge funds, respectively.
Style Nofunds Nobs Mean Std Skew Kurt Min Max JB ρ1 ρ2 ρ3 LQ
Panel A: CEFs
FI Global 26 133 0.185 5.953 −0.602 6.080 −26.00 20.33 <0.05 −0.043 −0.002 −0.122 0.2713
FI Sector 29 151 0.309 6.214 −0.399 4.611 −22.48 22.33 <0.05 −0.033 0.028 −0.057 0.4299
FI Long Term 26 129 −0.580 7.356 −1.224 8.322 −31.50 19.77 <0.05 −0.065 0.091 −0.020 0.1992
FI Int Term 25 259 0.803 5.759 0.203 5.443 −19.15 26.50 <0.05 −0.097 −0.025 −0.045 0.145
FI Short Term 6 114 0.584 3.777 −0.912 4.361 −15.76 10.52 <0.001 −0.061 −0.002 0.058 0.1801
FI Government 13 120 0.451 2.963 −0.185 2.305 −9.37 9.78 0.0679 −0.067 −0.012 0.028 0.3736
FI High Yield 47 142 −0.377 6.545 −0.620 3.708 −24.05 19.95 <0.05 0.075 0.048 −0.003 0.357
FI Others 23 73 −0.685 4.672 −1.656 6.415 −19.71 10.70 <0.001 0.259 0.186 0.013 0.0643
FI Average 24 140 0.086 5.405 −0.675 5.156 −21.00 17.49 <0.05 −0.004 0.039 −0.019 0.2526
EF Balanced 46 110 −0.695 7.329 −0.780 4.193 −24.99 20.72 0.0693 0.112 0.021 −0.058 0.3177
EF Global 92 122 −0.145 9.955 −0.059 2.725 −29.53 31.94 0.1199 0.060 0.017 −0.031 0.4727
EF Sector 17 157 0.153 7.142 −0.162 4.424 −23.74 28.19 <0.05 0.064 −0.002 −0.035 0.2411
EF Commodities 20 125 −1.214 9.109 −1.136 4.279 −32.94 18.95 <0.05 0.056 0.188 0.010 0.2806
EF Large Cap 59 145 −0.272 6.438 −0.698 5.530 −23.44 20.71 0.0828 0.104 0.001 −0.024 0.361
EF Mid Cap 11 152 0.169 7.198 −0.258 4.565 −22.40 20.36 <0.05 0.038 −0.071 −0.056 0.3546
EF Small Cap 6 161 0.611 12.124 −0.138 3.321 −31.95 51.28 <0.05 0.110 0.017 −0.019 0.4631
EF Growth 18 97 0.200 8.482 −0.480 4.707 −24.89 28.51 <0.05 0.101 −0.030 −0.042 0.3682
EF value 19 63 −0.584 6.251 −0.996 3.886 −21.98 14.63 0.053 0.099 0.025 −0.073 0.4109
EF Others 32 65 −1.318 10.111 −1.426 5.649 −38.08 20.11 0.1106 0.164 −0.050 −0.004 0.3635
Continued on next page
22
Table 1.2 (Continued)
Style Nofunds Nobs Mean Std Skew Kurt Min Max JB ρ1 ρ2 ρ3 LQ
EF Average 32 120 −0.310 8.414 −0.613 4.328 −27.39 25.54 0.0645 0.091 0.011 −0.033 0.3633
Group Average 29 129 −0.134 7.077 −0.640 4.696 −24.55 21.96 <0.05 0.049 0.024 −0.027 0.3141
Panel B: ETFs
FI Global 2 14 −0.218 6.383 −0.314 2.910 −15.61 12.82 0.2702 0.084 −0.341 −0.209 0.1995
FI Sector 3 22 0.677 1.680 0.826 1.841 −2.61 4.98 0.1073 0.360 −0.367 −0.287 0.0708
FI Long Term 2 49 0.631 3.407 0.945 7.720 −8.81 13.33 <0.001 0.223 −0.475 −0.297 <0.01
FI Int Term 6 28 0.464 2.001 0.585 3.537 −4.08 6.13 <0.05 0.227 −0.392 −0.183 0.1196
FI Short Term 3 21 0.437 0.969 −0.252 2.156 −2.00 2.66 0.1579 0.163 −0.207 −0.062 0.2098
FI Government 12 44 0.927 3.990 0.526 1.444 −8.12 11.55 0.4384 0.115 −0.050 0.024 0.2719
FI High Yield 2 17 −1.562 6.988 0.654 2.636 −13.97 16.36 0.2087 0.051 −0.233 −0.504 0.0681
FI Others 2 40 0.409 2.477 −1.078 3.699 −7.78 5.74 <0.01 0.176 −0.239 −0.130 0.2464
FI Average 4 29 0.221 3.487 0.236 3.243 −7.87 9.20 0.1517 0.175 −0.288 −0.206 0.1484
EF Balanced 5 14 −1.728 4.575 −0.215 1.988 −10.87 7.97 0.1928 0.150 −0.351 −0.284 0.0631
EF Global 108 56 −1.117 7.917 −0.733 2.404 −24.24 15.27 0.192 0.268 −0.046 −0.059 0.2524
EF Sector 111 42 −0.939 6.346 −0.726 1.687 −18.51 11.23 0.2064 0.161 −0.148 −0.093 0.2482
EF Commodities 47 38 −0.939 10.759 −0.892 1.430 −29.54 16.56 0.1325 0.253 0.129 −0.003 0.2758
EF Large Cap 82 51 −0.956 5.537 −1.310 2.842 −18.78 7.67 <0.05 0.284 −0.116 −0.028 0.1816
EF Mid Cap 55 40 −1.496 7.450 −1.194 2.648 −23.77 10.13 0.1149 0.298 −0.116 −0.077 0.1537
EF Small Cap 35 50 −1.269 6.874 −1.327 2.731 −24.21 8.64 <0.05 0.231 −0.138 −0.175 0.1783
EF Growth 46 47 −1.457 7.245 −1.068 1.884 −22.61 10.35 0.1382 0.290 −0.017 −0.099 0.2181
EF value 46 51 −0.843 5.699 −1.413 3.579 −20.47 8.19 <0.05 0.239 −0.189 −0.050 0.1512
EF Bear Market 40 22 1.624 10.706 0.676 1.142 −15.53 26.78 0.2369 0.170 −0.288 −0.206 0.1844
EF Currency 10 29 0.197 3.486 −0.670 3.786 −10.02 7.36 0.0801 0.284 0.091 0.016 0.3396
EF Others 82 48 −1.555 8.767 −0.737 1.749 −24.44 16.49 0.1675 0.162 −0.172 −0.118 0.2887
EF Average 55.583 41 −0.873 7.113 −0.801 2.323 −20.25 12.22 0.1294 0.233 −0.113 −0.098 0.2113
Group Average 34.95 36 −0.436 5.663 −0.386 2.691 −15.30 11.01 0.1383 0.209 −0.183 −0.141 0.1861
Panel C: OEFs
Continued on next page
23
Table 1.2 (Continued)
Style Nofunds Nobs Mean Std Skew Kurt Min Max JB ρ1 ρ2 ρ3 LQ
FI Index 32 95 0.508 1.252 −0.034 1.378 −3.14 4.21 0.2657 0.136 −0.096 0.065 0.2406
FI Global 303 87 0.381 2.367 −0.556 3.739 −8.00 6.47 0.1495 0.181 −0.114 −0.038 0.1798
FI Short Term 645 76 0.287 0.763 −0.890 5.023 −2.07 2.28 0.2501 0.242 0.084 0.124 0.1684
FI Government 727 93 0.464 1.124 −0.138 1.311 −2.88 3.60 0.1831 0.210 0.016 0.117 0.2543
FI Mortgage 219 81 0.399 0.898 −0.420 2.018 −2.33 2.69 0.2976 0.191 0.002 0.138 0.2589
FI Corporate 798 74 0.421 1.285 −0.580 2.578 −3.81 3.63 0.2704 0.142 −0.090 0.043 0.2166
FI High Yield 944 87 0.361 2.338 −1.174 5.553 −9.41 6.04 0.1161 0.224 −0.112 −0.094 0.1069
FI Others 619 79 0.623 1.006 0.286 3.412 −2.32 3.60 0.2316 0.425 0.378 0.334 0.1669
FI Average 536 84 0.431 1.379 −0.438 3.126 −4.24 4.07 0.2205 0.219 0.009 0.086 0.199
EF Index 838 79 −0.177 5.149 −0.966 2.850 −17.75 10.37 0.1072 0.199 −0.066 −0.013 0.2642
EF commodities 238 78 0.340 8.826 −0.516 1.585 −27.06 20.58 0.1587 0.085 0.050 0.065 0.2728
EF Sector 1331 69 −0.322 6.933 −0.371 1.268 −18.82 15.72 0.2876 0.137 −0.051 −0.059 0.3842
EF Global 3373 78 −0.115 5.884 −0.796 2.245 −19.52 12.36 0.1489 0.237 0.024 0.008 0.2393
EF Balanced 988 80 0.134 3.096 −1.098 3.332 −11.16 6.03 0.1121 0.192 −0.065 0.003 0.274
EF Lev/Short 675 62 −0.099 6.231 −0.287 1.597 −16.50 14.93 0.2483 0.119 −0.051 −0.059 0.336
EF Long Short 80 22 −1.774 4.877 −0.980 1.439 −14.30 5.24 0.1843 0.293 −0.061 −0.109 0.2205
EF Mid Cap 1331 63 −0.335 5.942 −0.919 2.579 −19.61 11.80 0.1253 0.229 −0.050 −0.051 0.2213
EF Small Cap 1970 68 −0.183 6.207 −0.741 1.930 −19.66 13.03 0.1336 0.170 −0.053 −0.132 0.2247
EF AG 247 46 1.493 6.072 −0.734 2.067 −17.52 13.05 0.1803 0.059 −0.094 −0.036 0.5208
EF Growth 4586 73 −0.141 5.145 −0.812 2.114 −16.50 10.24 0.1826 0.186 −0.047 −0.023 0.3083
EF GI 2459 70 −0.160 4.494 −0.883 2.169 −14.83 8.48 0.1484 0.165 −0.073 −0.027 0.3286
EF EI 509 55 −0.208 4.233 −0.782 2.041 −13.09 8.05 0.2366 0.140 −0.095 −0.026 0.327
EF Others 1758 65 1.267 5.362 −0.498 1.994 −15.42 12.90 0.2147 0.062 −0.044 −0.093 0.4216
EF Average 1456 65 −0.020 5.604 −0.742 2.086 −17.27 11.63 0.1763 0.163 −0.048 −0.039 0.3102
Group Average 1121 72 0.144 4.067 −0.631 2.465 −12.53 8.88 0.1924 0.183 −0.028 0.006 0.2698
Panel D: HFs
Equity Hedge 2367 48 0.399 5.051 −0.299 2.001 −12.87 12.30 0.2928 0.102 0.015 −0.019 0.3381
Event-Driven 585 56 0.381 3.241 −0.618 4.071 −9.32 8.23 0.1682 0.262 0.102 0.057 0.209
Continued on next page
24
Table 1.2 (Continued)
Style Nofunds Nobs Mean Std Skew Kurt Min Max JB ρ1 ρ2 ρ3 LQ
Fund of Funds 1194 46 0.091 2.625 −0.981 2.951 −7.83 4.82 0.2097 0.272 0.159 0.064 0.2403
HFRI 75 137 0.612 2.925 −1.115 6.471 −11.69 9.49 <0.05 0.302 0.148 0.073 0.1028
HFRX 27 46 −0.425 2.337 −1.709 6.048 −9.27 2.80 0.1688 0.273 0.182 0.049 0.1318
Macro 810 45 0.470 4.562 0.012 2.006 −10.38 11.74 0.3267 0.054 −0.052 −0.044 0.3471
Relative Value 786 46 0.241 2.991 −1.208 7.001 −10.18 5.84 0.146 0.265 0.100 0.073 0.2289
Group Average 835 60 0.253 3.390 −0.845 4.364 −10.22 7.89 0.1945 0.219 0.093 0.036 0.2283
25
1.6 Empirical Design
1.6.1 Frequency of Tail Returns
If an investment fund is well diversified, the distribution of returns should be close to normal, i.e. its
skewness is zero and kurtosis is 3. However, Table 1.2 suggests that tail returns and risks do exist in
investment funds. One direct approach is to measure the frequency of tail returns in a given fund.
Tail returns of an individual fund are defined as its monthly returns above or below a cutoff
stated in terms of a number of standard deviations from the mean.
¨
Ait-Sahalia (2004) estimates the
probability of observing one jump conditional on a large log-return. He concludes that as far into the
tail as 3.5 standard deviations, a large observed log-return can still be produced by Brownian noise.
A large log-return above 3.5 standard deviations in a finite time would help identify at least one
jump. A fund with a high frequency of monthly returns exceeding 5 standard deviations suggests
that jumps can be identified in the fund returns. As such, I use 3 and 5 standard deviations as
thresholds to determine tail returns.
I decompose funds’ monthly returns into systematic and idiosyncratic components and com-
pute the percentage of monthly systematic and idiosyncratic returns exceeding 3 and 5 standard
deviations of the means of respective distributions.
Let COUNT
i,t
i
be one if fund i’s monthly return on month t
i
is greater than 3 or 5 standard
deviations from the mean. I derive the test statistics of the frequency of tail returns for fund i by
assuming that COUNT
i,t
i
follows the Bernoulli distribution and the sequence of COUNT
i,t
i
is
independent and identically distributed, i.e. COUNT
i,t
i
is 1 with probabilityp and 0 otherwise on
each month. Thus, at the individual fund level, the frequency of tail returns and its test statistics can
be represented as follows:
X
i
=
1
T
i
T
i
X
t
i
=1
COUNT
i,t
i
∼ N(p,
p(1−p)
T
i
)
26
where T
i
is the number of monthly returns for fund i and t
i
= (1,2,...,T
i
) ∈ T
i
. At the style or
type level,
Y
s
=
1
N
s
Ns
X
i=1
X
i
∼ N(p,
1
N
2
s
(
X
i
p(1−p)
T
i
+
X
i
X
j6=i
ρ
tail
s
p(1−p)
T
i
s
p(1−p)
T
j
))
where N
s
is the number of funds in the style or type s. ρ
tail
is calculated as follows. If the
returns of different funds in the same style or type s are jointly within 3 standard deviations from
their respective means in montht, i.e. COUNT
i,t
= 0 for all fundi in the style or types in month
t, those returns are dropped to compute correlations. Then I average correlations between different
funds in the same style or type to deriveρ
tail
. ρ
tail
reflects correlation between funds at the extreme
states.
To compare any two fund styles or types (Y
s
andY
r
) at the aggregate level:
Y
s
−Y
r
∼ N(0,var(Y
s
)+var(Y
r
)−2cov(Y
s
,Y
r
))
cov(Y
s
,Y
r
) =
1
N
s
N
r
X
i
X
j
ρ
tail
s
p(1−p)
T
i
s
p(1−p)
T
j
Table 1.3 presents the frequencies of monthly returns exceeding 3 and 5 standard deviations
from the mean across fund types. The frequency of raw tail returns ranges from 1.78% (CEFs)
to 1.10% (OEFs) and 0.13% (CEFs) to 0.01% (ETFs) for the 3 and 5 standard deviations, respec-
tively.
17
Both ranges exceed the probability of 3 and 5 sigma events under the normal distribution,
i.e. 0.27% and less than 0.0001%, respectively. This result substantiates the presence of tail risks in
managed portfolios.
17
Results for 2 standard deviations are also available upon request. Across fund types, the frequency of raw tail returns
ranges from 4.74% and 5.6%; the frequencies of both systematic and idiosyncratic tail returns are very close to 5%.
27
Table 1.3: Frequency of Tail Returns across Fund Types
Tail returns are defined as monthly returns exceeding (+/−)5 and (+/−)3 standard deviations from the means. The frequency of tail
returns of a fund is calculated as the count of tail returns divided by its total number of monthly returns. The test statistics is calculated by
assuming the distribution of the counts of tail returns to be Bernoulli and i.i.d. Total fund returns are further decomposed into systematic
and idiosyncratic components to calculate the frequency of systematic and idiosyncratic tail returns. Results are reported in three rows
for each fund type. The first row is the frequency of total tail returns. The second row is the frequency of systematic tail returns. The
third row is the frequency of idiosyncratic tail returns. The cross cell by the same fund type represents the average frequency of tail
returns across funds in that fund type. The cross cell of two different fund types is the difference in frequency of tail returns between two
fund types. T-values are in the parenthesis based on the test hypothesis of zero frequency. CEFs/ETFs/OEFs/HFs refer to closed-end
funds/exchange-traded funds/open-ended funds/hedge funds, respectively.
CEFs ETFs OEFs HFs
5std 3std 5std 3std 5std 3std 5std 3std
Panel A: All Funds
CEFs
Total
0.132 1.775 0.123 0.617 0.086 0.672 0.055 0.609
(-0.47) (-1.54) (0.21) (1.33) (0.04) (0.45) (0.03) (0.50)
Sys
0.139 1.885 0.137 0.462 0.109 0.814 0.093 0.788
(-0.47) (-1.46) (0.23) (0.99) (0.06) (0.55) (0.06) (0.64)
Id
0.095 1.076 0.087 0.426 0.035 0.212 0.032 0.156
(-0.49) (-2.02) (0.15) (0.92) (0.02) (0.14) (0.02) (0.13)
ETFs
Total
0.009 1.158 -0.037 0.056 -0.068 -0.008
(-0.34) (-1.25) (-0.06) (0.10) (-0.04) (-0.01)
Sys
0.002 1.423 -0.028 0.352 -0.044 0.326
(-0.34) (-1.13) (-0.05) (0.72) (-0.02) (0.22)
Id
0.007 0.650 -0.052 -0.214 -0.055 -0.270
(-0.34) (-1.47) (-0.09) (-0.45) (-0.03) (-0.18)
OEFs
Total
0.046 1.102 -0.031 -0.063
(-0.39) (-1.53) (-0.08) (-0.21)
Sys
0.030 1.071 -0.016 -0.026
(-0.40) (-1.55) (-0.03) (-0.01)
Id
0.059 0.864 -0.003 -0.056
(-0.39) (-1.66) (-0.00) (-0.22)
HFs
Total
0.077 1.165
Continued on next page
28
Table 1.3 (Continued)
CEFs ETFs OEFs HFs
5std 3std 5std 3std 5std 3std 5std 3std
(-0.50) (-1.95)
Sys
0.046 1.097
(-0.51) (-2.00)
Id
0.063 0.920
(-0.50) (-2.12)
Panel B: Fixed Income Funds
CEFs
Total
0.199 1.967 0.159 1.173 0.052 0.867
(-0.49) (-1.59) (0.05) (0.48) (0.02) (0.36)
Sys
0.234 2.054 0.193 1.715 0.127 0.930
(-0.47) (-1.53) (0.06) (0.70) (0.04) (0.38)
Id
0.146 1.265 0.106 -0.288 -0.016 -0.034
(-0.52) (-2.14) (0.03) (-0.12) (-0.01) (-0.01)
ETFs
Total
0.041 0.793 -0.107 -0.306
(-0.28) (-1.21) (-0.03) (-0.13)
Sys
0.041 0.339 -0.066 -0.785
(-0.28) (-1.38) (-0.02) (-0.33)
Id
0.041 1.553 -0.122 0.254
(-0.28) (-0.92) (-0.04) (0.11)
OEFs
Total
0.147 1.099
(-0.26) (-1.13)
Sys
0.106 1.124
(-0.27) (-1.12)
Id
0.162 1.299
(-0.26) (-1.05)
Panel C: Equity Funds
CEFs
Total
0.085 1.642 0.078 0.464 0.065 0.552
Continued on next page
29
Table 1.3 (Continued)
CEFs ETFs OEFs HFs
5std 3std 5std 3std 5std 3std 5std 3std
(-0.41) (-1.35) (0.40) (3.06) (0.04) (0.48)
Sys
0.074 1.769 0.074 0.285 0.064 0.724
(-0.42) (-1.28) (0.38) (1.88) (0.04) (0.64)
Id
0.059 0.946 0.053 0.346 0.026 0.191
(-0.42) (-1.75) (0.28) (2.28) (0.02) (0.17)
ETFs
Total
0.007 1.178 -0.013 0.088
(-0.34) (-1.22) (-0.01) (0.13)
Sys
0.000 1.484 -0.010 0.439
(-0.34) (-1.09) (-0.01) (0.54)
Id
0.006 0.600 -0.028 -0.156
(-0.34) (-1.47) (-0.03) (-0.20)
OEFs
Total
0.020 1.091
(-0.36) (-1.35)
Sys
0.010 1.044
(-0.36) (-1.37)
Id
0.033 0.756
(-0.35) (-1.51)
30
For all fund types, the null hypothesis that a 3 (5) standard deviation event occurs 4% (1%)
per month is not rejected at 1% significance level. This suggests that on a monthly basis, all four
fund types are subject to a 3 (5) sigma event with 4% (1%) probability. In view of economic
significance, investors who delegate investment decisions to fund managers still face 3 “sigma”
event approximately every two years.
The frequencies of idiosyncratic tail returns are less varied across fund types than systematic
tail returns. At the 3 standard deviations, CEFs have the highest frequency of tail returns on both
return components.
18
ETFs show high frequency of systematic tail returns, but lowest frequency of
idiosyncratic tail returns. The frequencies of both systematic and idiosyncratic tail returns at the 5
standard deviations follow the same order as raw tail returns. The test statistics associated with the
hypothesis that the occurrence of systematic/idiosyncratic returns exceeding 3 (5) standard devia-
tions from the mean equals to 4% (1%) per month are not significant at 1% significance level. The
classic portfolio theory suggests that idiosyncratic tail risks can be diversified away by increasing
the number of assets. It is interesting to see that managed futures suffer from both systematic and
idiosyncratic tail risks at similar frequency.
Investors suffer more systematic risks by investing in ETFs, but more idiosyncratic risks in HFs
and OEFs. The high frequencies of idiosyncratic tail returns in CEFs and HFs imply that both fund
types have high tracking errors, and their managers trade on individual assets with high idiosyncratic
risks to increase performance. ETFs exhibit higher frequency of systematic tail risks than HFs and
OEFs since tracking errors or idiosyncratic risks should be minimized for ETFs.
T-tests of differences in frequencies of tail returns (raw, systematic, and idiosyncratic) fail to
reject the hypothesis that funds in different fund types have the same frequency at 1% significance
level, except for equity CEFs and ETFs at the 3 standard deviations. This indicates that investors
should be aware of 3 and 5 sigma events not only for HFs, but for all four types of investment funds.
18
One concern is that the recording of the last return due to delisting varies across data vendors. One reason for CEFs
to have higher a frequency may be due to traded price discounts. However, the order of frequencies across fund types still
hold if the last observation is removed from the analysis.
31
I further break down the frequencies of tail returns by right and left tails. The striking finding is
that most tail returns come from the left tails. This evidence supports the importance of downside
risk and the prevalence of negative skewness and leptokurtosis across fund types.
1.6.2 Systematic and Idiosyncratic Tail Risk
1.6.2.1 The Benchmarks
Different fund styles and types have different levels of systematic risk and are exposed to different
risk factors. Therefore, a broad-based index is not the appropriate benchmark to decompose risk
into systematic and idiosyncratic components across fund styles and types. CEF returns are subject
to discounts and Lee, Shleifer, and Thaler (1991) show that changes in discounts are correlated with
small firm returns. The discounts resemble market-to-book ratios and Thompson (1978) show that
discounts predict the expected returns of CEFs. ETFs track market indexes and are most sensitive to
market factors directly associated with the benchmarks they track. Because OEFs follow long-only
strategies, standard asset classes may be appropriate market factors. HFs have no benchmarks, and
fund managers tend to maximize total fund returns due to high watermark provisions. In addition,
different HF styles pursue different directional/nondirectional trades and dynamic trading strategies,
and differ in option-like payoffs. These HF characteristics lead to distinctive risk profiles among
HFs, compared to other fund types.
Inappropriate factors may lead to a misleading measure of systematic and idiosyncratic risk
decomposition. If the chosen market factors don’t appropriately explain the variations of systematic
components of returns, too much idiosyncratic risk is mistakenly identified. Then empirical results
will spuriously show fund skewness and kurtosis mostly come from the idiosyncratic component of
returns.
I use the equal-weighted portfolios of funds to decompose systematic and idiosyncratic compo-
nents of returns. This follows many studies on fund performance (e.g. Grinblatt and Titman (1994),
Brown, Goetzmann, and Ibottson (1999), and Ackermann, McEnally, Ravenscraft (1999)).
The advantages of using portfolios of funds within the same style as a benchmark include the
following: Portfolios of funds are readily observable and capture diversification effects to isolate
32
idiosyncratic returns of funds within the style.
19
Second, many fund managers in the same style
make similar bets or share similar trading strategies. Therefore, funds in the same style may be ex-
posed to the same common factors (Hunter, Kandel, Kandel, and Wermers (2010)). The benchmark
can capture a common component in the variation over time and across funds within the group.
In addition, return characteristics and distributions differ across fund styles and types and the
portfolios of funds capture distinctive differences. For example, HFs exhibit nonlinearities in returns
and the magnitudes of nonlinearities differ across HF styles. An index constructed of the funds in
the same style captures style-specific returns.
Third, a fund manager is regarded as providing valuable services when the investment oppor-
tunity set is expanded by the trading strategies of the fund. Therefore, a benchmark should share
common assets with the fund. For example, if the Janus Balanced Fund trades growth stocks and
U.S. Treasuries, both types of securities should be included in the benchmark. The portfolios of
funds represent a joint set of reference assets for funds with the same trading strategy.
Fourth, portfolios of funds create a peer group of managers who pursue the same style. Thus,
portfolios of funds have the highest correlations with funds in the same style and represent asset
classes in that style. Fund managers are increasingly evaluated relative to a benchmark specific to
their styles, instead of a broad-based benchmark. An inappropriate benchmark can induce incorrect
measurement of relative performance. For example, a small-cap fund manager may underperform
relative to a broad market index, but overperform relative to a small stock benchmark.
1.6.2.2 The Decomposition
I run the following regression to decompose the systematic and idiosyncratic components of risks:
R
i,t
−E(R
i
) = β
i
(R
p,t
−E(R
p
))+u
i,t
(1.6)
R
i,t
andR
p,t
are returns for fundi and portfolios of fundsp at timet. The portfolios of funds are
constructed based on the investment styles outlined in section 1.5. β
i
(R
p,t
−E(R
p
)) andu
i,t
stand
19
Thek
th
order moment of portfolios of funds isO(
1
n
k−1
). Asn→∞,E[Rp−E(Rp)]
k
=E[
1
n
P
Ri−
1
n
P
E(Ri)]
k
=
1
n
k
E[
P
Ri−
P
E(Ri)]
k
≤
n
n
k
33
for the systematic and idiosyncratic component of de-meaned returns for fundi. Both components
are orthogonal to each other.
The simple linear regression in (1.6) is advantageous to study systematic and idiosyncratic tail
risks.
20
Under the single factor model, the skewness ofr
i
can be decomposed as follows:
E(r
3
i
) = E[(β
i
r
p
+u
i
)
3
]
= β
2
i
cov(r
i
,r
2
p
)+2β
2
i
cov(u
i
,r
2
p
)
| {z }
COSKEW
+3β
i
cov(u
2
i
,r
p
)
| {z }
ICOSKEW
+ E(u
3
i
)
|{z}
RESSKEW
(1.7)
wherer
i
andr
p
are de-meaned returns, i.e. r
i
=R
i
−E(R
i
) andr
p
=R
p
−E(R
p
). According
to (1.7), the skewness decomposition consists of three parts: coskewness (COSKEW), idiosyn-
cratic coskewness (ICOSKEW), and residual skewness (RESSKEW). Since both COSKEW and
ICOSKEW contain β and covary with the market, they are different forms of systematic skew-
ness. RESSKEW represents idiosyncratic tail risk. Note that coskewness in this study is defined
as the sum of two covariance terms - the covariance of fund returns with market volatility and the
covariance of fund residuals with market volatility. The latter is small under the assumption of
orthogonality between the systematic and idiosyncratic components in the one-factor regression.
Moreno and Rodr´ ıguez (2009) show that coskewness is managed and the coskewness policy
is persistent over time. In their remark, “managing coskewness” refers to having a specific pol-
icy regarding the assets incorporated into the fund’s portfolio to achieve higher or lower portfolio
coskewness. If a manager consistently adds assets with negative coskewness to reduce fund skew-
ness, the fund will exhibit negative coskewness and investors will demand a higher risk premium.
The idiosyncratic coskewness, i.e. the covariance between idiosyncratic volatility and market
returns, is advocated by Chabi-Yo (2009). Chabi-Yo (2009) proves that idiosyncratic coskewness is
20
If I add the quadratic terms to (1.6), i.e. Ri,t−E(Ri) = αi +βi(Rp,t−E(Rp))+γi(Rp,t−E(Rp))
2
+it, the
skewness decomposition becomesE(r
3
i
) =β
3
i
E(r
3
p
) +3βiE(rp
2
i
) +E(
3
i
) +[3β
2
i
γiE(r
4
p
) +3βiγ
2
i
E(r
5
p
) +3γiE(r
2
p

2
i
) +
3γ
2
i
E(r
4
p
i) +6βiγiE(r
3
p
i) +γ
3
i
E(r
6
p
)] = COSKEW + ICOSKEW + RESSKEW + other higher moments. Similarly, the
kurtosis decomposition expands asE(r
4
i
) =β
4
i
E(r
4
p
) +4βiE(rp
3
i
) +E(
4
i
) +4β
3
i
γiE(r
5
p
) +6β
2
i
γ
2
i
E(r
6
p
) +4βiγ
2
i
E(r
5
p
)
+ γ
4
i
E(r
8
p
) + 4i[3β
2
i
γiE(r
4
p
) + 3βiγ
2
i
E(r
5
p
) + γ
3
i
E(r
6
p
)] + 6
2
i
[2βiγiE(r
3
p
) + γ
2
i
E(r
4
p
)]+4[βiE(rp
3
i
) + γiE(r
2
p

3
i
)] =
COKURT + ICOKURT + RESKURT + VOLCOMV + other higher moments. The components in this study can also be
extracted under the quadratic assumption.
34
equivalent to a weighted average of individual security call and put betas. He shows that in a single
factor model, during market upswings (r
p
> 0), ICOSKEW is positive and the idiosyncratic risk
premium is negative; during market downswings (r
p
< 0), ICOSKEW is negative and the idiosyn-
cratic risk premium is positive. In other words, stocks whose option betas with high sensitives to
market returns have low average returns because they hedge against market upswings and down-
swings. Out-of-money options written on these stocks have large betas or higher sensitivities with
market returns. Investors prefer options written on stocks with lottery-like returns. The idiosyn-
cratic coskewness explains two market anomalies. First, Ang, Hodrick, Xing, and Zhang (2006,
2009) document that stocks with high idiosyncratic volatility have low expected returns. Second,
idiosyncratic coskewness helps explain the empirical finding that distressed stocks have low returns
(Chabi-Yo and Yang (2009)).
Note that cov(u
2
i
,r
p
) is equivalent to cov[E(u
2
i
|r
p
),r
p
] orE[E(u
2
i
|r
p
)r
p
]. This decomposition
implies that the sign and the magnitude of ICOSKEW depends on the risk-return relation and the
level of conditional heteroscedasticity. Skewed fund returns can be generated through conditional
heteroscedasticity. If an asset has high idiosyncratic conditional heteroscedasticity, negatively cor-
related with market returns, adding this asset to a fund will impart negative skewness through a large
negative ICOSKEW.
Mitton and V orkink (2007) and Barberis and Huang (2008) document that idiosyncratic skew-
ness is priced and its relation with expected returns is negative. Boyer, Mitton, and V orkink (2009)
empirically test the negative relation between idiosyncratic skewness and expected returns.
The decomposition of kurtosis is derived as follows:
E(r
4
i
) = E[(β
i
r
p
+u
i
)
4
]
= β
3
i
cov(r
i
,r
3
p
)+3β
3
i
cov(u
i
,r
3
p
)
| {z }
COKURT
+6β
2
i
E(r
2
p
u
2
i
)
| {z }
VOLCOMV
+4β
i
cov(u
3
i
,r
p
)
| {z }
ICOKURT
+ E(u
4
i
))
| {z }
RESKURT
(1.8)
This decomposition displays four sources of fund kurtosis: cokurtosis (COKURT), comove-
ments of volatility (VOLCOMV), idiosyncratic cokurtosis (ICOKURT), and residual kurtosis (RES-
KURT). COKURT, VOLCOMV , and ICOKURT are exposed to the market and are classified as sys-
35
tematic tail risks. RESKURT is considered as idiosyncratic tail risk. The importance and validity
of cokurtosis on asset returns are documented by Dittmar (2002).
The cokurtosis of an asset can impact the total kurtosis of the fund. Investors dislike fat-tails in
returns and thus demand a positive risk premium on an asset with large kurtosis. Such an asset will
increase the total kurtosis of the fund. If a manager constantly adopts the strategy of buying positive
cokurtosis assets, the fund will show a large weight on cokurtosis in the kurtosis decomposition.
In addition, since cokurtosis reflects the covariance between market skewness and individual fund
returns, a fund with positive cokurtosis indicates a positive relation between the fund return and the
skewness of the market returns.
The VOLCOMV term is the comovement of shocks to fund conditional volatility and market
volatility. The negative relationship between these two shocks can reduce the kurtosis level of funds.
Since investors prefer assets with lower kurtosis, fund managers can add assets, whose volatility
moves oppositely to market volatility to achieve this goal. For example, a fund manager can engage
trades on variance swaps, VIX options, or VIX futures to reduce exposure to market volatility in
extreme markets.
The concept of comovement of volatility is often applied across international markets.
21
The
comovement of volatility between the market and a fund can be interesting as well. Fund managers
are known to use market-timing and market volatility timing strategies (eg. Treynor and Mazuy
(1966), Henriksson and Merton (1981) and Busse (1999)). From the hedging perspective, if an
investor’s portfolio is exposed to the market, adding a fund which comoves with market volatility
can be suboptimal due to kurtosis. Since kurtosis is the variance of the variance, a fund manager
can add assets with high volatility comovements with the market to increase the kurtosis of the
fund. When a fund exhibits a large VOLCOMV component, it is inferred that using comovements
of volatility is a common strategy for the fund.
Following Chabi-Yo (2009), I refer to the covariance between idiosyncratic skewness and mar-
ket returns as idiosyncratic cokurtosis. Like idiosyncratic coskewness, idiosyncratic cokurtosis can
21
See, for example, Hamao, Masulis, and Ng (1990) and Susmel and Engle (1994).
36
be interpreted as a weighted average of individual security call and put betas. For a single factor
model, market upswings imply positive option betas and thus positive idiosyncratic cokurtosis.
cov(u
3
i
,r
p
) can be rewritten ascov[E(u
3
i
|r
p
),r
p
] orE[E(u
3
i
|r
p
)r
p
]. The idiosyncratic cokurto-
sis is implicitly embedded with a skewness-return relation and the magnitude of conditional het-
eroskewticity. Conditional heteroskewticity is a property of residual returns and kurtosis in fund
returns can be induced by conditional heteroskewticity from different assets. If fund managers
prefer funds being less fat-tailed, in expectation of an increase in market returns, they can add as-
sets with high idiosyncratic skewness covarying negatively with market returns. A trading strategy
involving small cap stocks is one example.
Chabi-Yo (2009) extends his analysis to higher moments and concludes that risk premium on
higher moments is driven by individual security call and put betas. Although the risk premium
on idiosyncratic kurtosis is not well documented in the literature, a fund with a larger weight on
idiosyncratic kurtosis implies that the manager has more flexibility in what and how to trade. For
example, since HF managers constantly use high leverage and dynamic strategies, and are able to
invest in a wider class of assets, HFs should exhibit a larger weight on RESSKEW and RESKURT.
The components in skewness and kurtosis decompositions are summarized below:
Table 1.4: Summary of Higher Moment Covariance Risks
Components Economic Type of Risk Likely to be Driven by
Interpretation Compensation/ Fund Type
22
COSKEW Covariance between fund
Systematic Systematic (α=1)/ ETFs
returns and market volatility
ICOSKEW Covariance between fund
volatility and market returns
RESSKEW Idiosyncratic skewness Idiosyncratic Convex (g=1)/ HFs
held in the fund
COKURT Covariance between fund
Systematic Systematic (α=1)/ ETFs
returns and market skewness
VOLCOMV Covariance between fund
volatility and market volatility
ICOKURT Covariance between fund
skewness and market returns
RESKURT Idiosyncratic kurtosis Idiosyncratic Convex (g=1)/ HFs
held in the fund
∗
α (g) is the weight in compensation relative to benchmark (convex payoff).
Like beta risk, investors should concern themselves with different sources of tail risks. Investors
fear those “black swans” that cause widespread disruption, and the components from skewness and
kurtosis decompositions can help them identify the sources of tail risks in their portfolios. Market
37
crashes cause not only spikes in market volatility, but also declines in market returns and skewness.
COSKEW and VOLCOMV (ICOSKEW and ICOKURT) measure fund movement against market
volatility (market returns). COKURT refers to the relation between fund performance and market
skewness.
Investors always try to diversify risks across styles or types of funds. If investors want to hedge
their investments against “black swans”, they should measure these components to identify the
needs and choose an effective tail risk hedging mechanism accordingly. For instance, if a portfolio
faces potential tail risks when economies skid, gold and treasuries are good hedging tools. On the
other hand, if the significant portion of tail risks in a fund comes from COSKEW or VOLCOMV ,
one should look for a volatility-based tail risk hedging mechanism, such as long-short strategies or
managed futures.
1.6.2.3 GMM Estimation for Skewness and Kurtosis Decompositions
The error terms of the time-series regression in (1.6) may suffer from heteroscedasticity, autocorre-
lation, and non-normality, and thus result in inefficientβ coefficients and biased OLS standard er-
rors. Furthermore, funds in the same group share commonalities in risk and strategies, and thereby
the error terms may be correlated across funds and subject to possible fixed effects and clustering.
Hansen’s (1982) generalized method of moments (GMM) is the most robust estimation technique to
allow for heteroscedasticity, autocorrelation, non-normality, and cross-sectional correlation in error
terms. As such, I adopt GMM methodology to estimate the components from skewness and kurtosis
decompositions.
The parameters for the skewness decomposition are β
i
, μ
i
, μ
p
, COSKEW
i
, ICOSKEW
i
,
and RESSKEW
i
, for i = 1...N. N is the number of funds in the same fund style or type. μ
p
is the expected return for the portfolio of funds. μ
i
is the expected return for fund i. Following
38
equation (1.6) and (1.7), moment conditions for skewness are the following:
r
i,t
= R
i,t
−μ
i
(1.9)
r
p,t
= R
p,t
−μ
p
(1.10)
u
i,1t
= (R
p,t
−μ
p
)(u
i,t
) (1.11)
u
i,2t
= COSKEW
i
−β
3
i
r
3
p,t
−3β
2
i
(r
2
p,t
u
i,t
) (1.12)
u
i,3t
= ICOSKEW
i
−3β
i
(r
p,t
u
2
i,t
) (1.13)
u
i,4t
= RESSKEW
i
−u
3
i,t
(1.14)
Similarly, the following moment conditions are used to estimateβ
i
,μ
i
,μ
p
,COKURT
i
,VOL−
COMV
i
, ICOKURT
i
, and RESKURT
i
in the kurtosis decomposition in equation (1.6) and
(1.8).
r
i,t
= R
i,t
−μ
i
(1.15)
r
p,t
= R
p,t
−μ
p
(1.16)
u
i,1t
= (R
p,t
−μ
p
)(u
i,t
) (1.17)
u
i,2t
= COKURT
i
−β
4
i
r
4
p,t
−4β
3
i
(r
3
p,t
u
i,t
) (1.18)
u
i,3t
= VOLCOMV
i
−6β
2
i
(r
2
p,t
u
2
i,t
) (1.19)
u
i,4t
= ICOKURT
i
−4β
i
(r
p,t
u
3
i,t
) (1.20)
u
i,5t
= RESKURT
i
−u
4
i,t
(1.21)
1.7 Empirical Results
Table 1.5 reports the skewness decomposition across fund types. The first column (EW portfolio
skewness) is the total skewness for the equal-weighted portfolios of funds. The second column
(individual skewness) is the average of total skewness across all funds in a given style. Individ-
ual funds’ coskewness (COSKEW), idiosyncratic coskewness (ICOSKEW), and residual skewness
39
(RESSKEW) are reported as the proportion of total fund skewness and I denote them as COSKEW
(%), ICOSKEW (%), and RESSKEW (%), respectively. All values at the style level are calculated
as the equal-weighted average across all funds within the same style. Style averages are reported at
the bottom of the fixed income styles, equity income styles, and all fund styles. FI Average is the
average of statistics across fixed-income fund styles. EF Average is the average of statistics across
equity fund styles. Group Average is the average of statistics across all fund styles.
Managed portfolios have negative skewness and excess kurtosis at both aggregate and individual
fund levels. Note that the equal-weighted portfolio skewness and average fund skewness can be
different, although for fixed income funds, both values are close. Equal-weighted portfolios of
funds are constructed using all observations in a given month, but the number of funds changes over
time. High attrition can make the distributions of the equal-weighted portfolios of funds negatively
skewed. HFs are one example. Likewise, fund birth rates can affect the number of funds in a given
month, and thus impact the distributions of the equal-weighted portfolios.
COSKEW is an important source of skewness across fund types. The proportions of CEF skew-
ness are almost equal in the three components of skewness. The individual COSKEW, ICOSKEW,
and RESSKEW are 40.48%, 33.32%, and 26.21%, respectively. Around 80% of ETF skewness is
from COSKEW. OEF skewness mostly comes from COSKEW (71.17%) and HFs have a percent-
age of 65.93% on COSKEW. The large fractions of COSKEW in fund skewness suggest that market
volatility has a strong impact on fund returns, and fund skewness risks are not diversified. Across
fund types, HFs display the highest percentage on RESSKEW (44.29%). This can reflect the asset
classes HFs invest in, and the leverage and dynamic strategies HFs can undertake.
Most fixed income and equity fund styles have the largest component in COSKEW. Relative to
fixed income CEFs and ETFs, fixed income OEFs have a highly negative percentage on ICOSKEW,
and a highly positive percentage on RESSKEW. The negative percentage on ICOSKEW means
that fund volatility decreases when market return drops. The hedging gains from ICOSKEW are
counteracted by negative RESSKEW. This suggests that fixed income OEF managers use trading
strategies that bear high idiosyncratic skewness risk or trade negatively skewed assets with high
turnover. Equity ETFs and OEFs consistently have the highest percentages in COSKEW. Equity
40
CEFs’ percentage on three skewness components are close. This suggests that equity fund managers
engage in trades or assets that make a big marginal contribution to the skewness of the market
portfolio.
The sign and magnitude of each skewness component can be determined by multiplying indi-
vidual COSKEW (%), ICOSKEW (%), and RESSKEW (%) by the average fund skewness. CEFs,
ETFs, OEFs, and HFs all have negative COSKEW and negative RESSKEW. This result denotes that
investment fund returns and the market volatility move in opposite directions and fund managers
add individual assets with negative skewness or fund-specific strategies generate negatively skewed
payoff. Negative skewness is associated with high risk premiums. During crises, jumps in market
volatility reduce fund skewness and negatively skewed bets can blow up. Investors can suffer from
high skewness risk hidden in managed portfolios.
The sign of ICOSKEW depends on the correlation between a fund’s idiosyncratic volatility and
market returns. The relation can be positive or negative, and thus can be used to offset COSKEW.
For example, large positive ICOSKEW means that assets’ idiosyncratic risks in the fund are posi-
tively correlated with market returns. During crises, drops in returns yield positive skewness in fund
returns and offset negative COSKEW. Empirical studies show that small growth firms have high
idiosyncratic volatility; large value firms are low idiosyncratic volatility stocks. Thus, ICOSKEW
is more negative in the former.
OEFs and HFs have a negative sign on ICOSKEW (%) (positive values of ICOSKEW), but
CEFs and ETFs have a positive sign on ICOSKEW (%) (negative values of ICOSKEW). HFs and
OEFs have posititive relations between a fund’s idiosyncratic volatility and market returns, but ETFs
and CEFs have negative relations. That combined with the magnitude of ICOSKEW can reflect the
asset characteristics a fund trades. The comparison of ICOSKEW suggests that HFs and CEFs
prefer small growth stocks and ETFs and OEFs prefer large value stocks.
Table 1.6 presents the results from the kurtosis decomposition. I report individual compo-
nents as percentages of total fund kurtosis - COKURT (%), VOLCOMV (%), ICOKURT (%), and
RESKURT (%). The average ETF and OEF fund has excess kurtosis below 3 and CEFs and HFs
exhibit large kurtosis. This result confirms the analysis on the frequencies of tail returns. Fixed
41
income funds have more kurtosis than equity funds. In particular, equity ETFs and equity OEFs
show less fat-tailedness than other fund types.
COKURT (41.4%) and VOLCOMV (35.62%) contribute the most to the kurtosis of CEFs, in-
cluding fixed income and equity CEFs. COKURT (67.46%) is the most important contributor to the
kurtosis of both fixed income and equity ETFs. Fixed income and equity OEFs have the highest per-
centage on COKURT as well. HFs depend on RESKURT (39.60%) the most, and then VOLCOMV
(33.81%). These results suggest that funds are subject to different types of systematic fat tail risks,
and an effective tail risk hedging should reduce exposures an investor faces the most. Morever,
the fractions of combined COKURT and VOLCOMV exceed more than 50% of fund kurtosis, and
it implies that too much systematic fat tail risk is not diversified away in funds as suggested by
the portfolio theory. Since HFs have the highest percentage in residual tail risks (RESSKEW and
RESKURT) across fund types, this confirms that HF managers commonly use idiosyncratic assets
to improve performance. Across all fund styles and types, ICOKURT has minimal influence on
total fund kurtosis.
Similar to skewness, a fund manager’s trading strategies are reflected in COKURT, VOLCOMV ,
ICO-KURT, and RESKURT. Results show that managed portfolios have positive COKURT, positive
VOLCOMV , and positive RESKURT, suggesting fund returns and volatility are positively correlated
with market volatility and skewness and idiosyncratic assets in funds are fat-tailed. When a fund
manager has constantly trade illiquidity or volatility based products, such as VIX options or futures,
the percentage on VOLCOMV will be high. HFs are one example. On the other hand, if a fund
manager mostly trades assets in the benchmark, COKURT can have a high percentage. ETFs are
one example. The high percentage in RESKURT can reflect a fund manager’s flexibility in stock
picking. Agency costs and compensation structure give a manager incentives to take tail risks (low
skewness and high kurtosis) to generate risk-adjusted returns over time.
The comparison of the same style across fund types exhibits differences in the skewness and kur-
tosis decomposition. For instance, equity global OEFs have the largest component in COSKEW, but
most of skewness of equity global CEFs come from RESSKEW. Although COSKEW contributes
the most to long-short strategies, long-short OEFs rely more on COKURT, but equity hedge HFs
42
face more fat tail risks from RESKURT. The inconsistency shows that different fund types rely on
trading strategies that induce different levels of systematic and idiosyncratic skewness and fat tail
risks, even their fund objective is the same.
The skewness and kurtosis decomposition help understand the trading strategies commonly used
by fund managers and priced risks across fund types. If a fund manager tends to add negatively
coskewed assets to increase expected returns, one would observe negative COSKEW in the fund.
If a fund manager often chooses assets with high idiosyncratic volatility or negative idiosyncratic
skewness, the fund will exhibit higher percentage on ICOSKEW or ICOKURT. If the skewness or
kurtosis of a fund comes mostly from the idiosyncratic component of returns, one can conclude
that the fund uses individual assets to increase fund expected returns. If a fund’s common trading
strategy is to rely on volatility comovement between the assets and the market, the source of kurtosis
of the fund will mostly come from VOLCOMV .
More importantly, the examination of each component from the skewness and kurtosis decom-
position conclude that managed funds are subject to different sources of tail risks. This has sev-
eral important implications. First, it is hard to diversify tail risks in managed portfolios. Because
COSKEW, COKURT, and VOLCOMV contribute to most tail risks and they all have the same signs
and similar magnitudes for all fund types, fund returns and volatility of all fund types will move to-
wards the same direction when market volatility jumps or market skewness declines drastically.
Heterogeneity in the percentage of components across fund styles suggests that investors can select
a specific style and fund type to match their needs to hedge tail risks. Moreover, the fund industry
claims that HFs can be used to hedge tail risks because of the flexibility in asset classes and trading
strategies. Equity hedge and macro HFs do have less negative skewness, but style averages show
that most HF styles are still subject to tail risks, especially idiosyncratic tail risks. For example,
fund of hedge funds invest in a variety of different hedge funds, but their idiosyncratic tail risks are
not well reduced (skewness of -0.459 and excess kurtosis of 0.588).
Second, the measures of these components help investors examine tail risks in their investment
portfolios. The appropriate tail risk hedging fund should match investors’ risk profiles on these
components. Like hedging beta risk, investors can look for low beta securities or industries to re-
43
duce systematic risk. For instance, if an investor’s portfolio consists of low COSKEW and high
VOLCOMV , s/he should look for a tail risk hedging fund that offers fund returns positively corre-
lated with market volatility and fund volatility negatively correlated with market volatility to reduce
systematic tail risks.
Third, a one-size-fits-all tail risk hedging mechanism does not work for all funds. A fund neg-
atively correlated to investors’ portfolios is not sufficient to hedge tail risks. The fund industry has
been launching volatility-based tail risk hedging funds, which guarantee a convex payoff to the up-
side during periods of market crisis. However, an effectively tail risk hedging mechanism should
consider how fund returns and volatility respond to extreme movements in market returns, volatility,
and skewness. These components capture different sources of tail risks, and thus policy makers and
fund managers should examine these components on any funds.
Measurement errors are associated with estimation of skewness and kurtosis. I keep all funds
with at least 12 monthly returns. This causes a trade-off between survivorship bias and measurement
errors. The components in the kurtosis decomposition have higher statistical significance than those
in the skewness decomposition. RESKURT and VOLCOMV are statistically significant at 5% for
most fund styles and types. On the other hand, three components of the skewness decomposition
yield low statistical significance.
Based on model predictions, across fund types, HFs (ETFs) should be subject to idiosyncratic
risk the most (least). The compensation structure of ETFs is tied to systematic returns with no con-
vexity. Some OEFs are subject to explicit incentive fees and their assets have been growing (Elton,
Gruber, and Blake(2003)). Moreover, the fund-flow performance relation is convex for OEFs. The
implicit convexity for CEFs may come from fund tournament or price premium/discount relative
to net asset values. The compensation structure for CEFs depends more weight on idiosyncratic
returns than ETFs, because of active management in CEFs and index-tracking in ETFs. The per-
centage of RESSKEW for HFs, OEFs, CEFs, and ETFs are 44.29%, 26.21%, 31.27%, and 5.74%,
respectly. For the kurtosis decomposition, HFs, OEFs, CEFs, and ETFs have the percentage of
RESKURT as follows: 39.60%, 23.45%, 11.91%, and 10.30%. These results conincide with the
model predictions.
44
The total fund skewness from low to high is HFs, OEFs, CEFs, and ETFs. This ranking is
predicted by the model. The total fund excess kurtosis for CEFs is the highest, but only slightly
above HFs. Figure 1.2 suggest that it is possible if theα (the return decomposition parameter) andg
(the convexity parameter) for CEFs on average is close to 0. OEFs have the lowest kurtosis, but very
close to ETFs. The model fails to predict the result of total fund kurtosis, but it can be attributed to
the assumed range ofα andg for OEFs.
The order of skewness holds across fixed-income funds, but the result for kurtosis is mixed
across equity funds. The percentages for fixed-income funds across ETFs, CEFs, and OEFs are
7.62%, 11.33%, and 73.23%, respectively, for the skewness decomposition. The kurtosis decom-
position also shows that fixed-income ETFs have the lowest weight (13.30%) on the idiosyncratic
component. Equity ETFs have the percentage on RESSKEW and RESKURT - 4.48% and 8.29%,
respectively, but equity OEFs have the lowest percentage on RESKURT.
The empirical results and model predictions are in line with Starks (1987). She concludes
that the “symmetric” contract does not necessarily eliminate agency costs, but it better aligns the
interests between investors and managers than the “bonus” contract. Since ETFs use a symmetric
contract and HFs use a bonus contract, the alignment of interests is worse for HFs but agency costs
still exist in both funds. This implication is reflected in the differences in skewness and kurtosis
between these two types of funds. ETFs are less negatively skewed and fat-tailed. HFs are more
negatively skewed and more leptokurtic. ETFs are subject to more systematic tail risks, and HFs are
subject to more idiosyncratic tail risks.
45
Table 1.5: Skewness Decomposition by Equal-weighted Portfolios
This table summarizes the skewness decomposition by using equal-weighted portfolios of funds as market portfolio. EW portfolio
skewness is the skewness for theh equal-weighted portfolios of funds formed by funds in the same styles. Individual skewness is the
cross-sectional average of skewness of individual funds in each style. Skewness is the third central moment about the mean and computed
asE[r
3
i
]/σ
3
i
. r
i
and σ
i
are the demeaned return and standard deviation of fund i. COSKEW, ICOSKEW, and RESSKEW refer to the
following components in the skewness decomposition:
E(r
3
i
) = β
2
i
cov(r
i
,r
2
p
)+2β
2
i
cov(u
i
,r
2
p
)
| {z }
COSKEW
+3β
i
cov(u
2
i
,r
p
)
| {z }
ICOSKEW
+ E(u
3
i
)
|{z}
RESSKEW
wherer
p
is the demeaned return for the market portfolio. Individual COSKEW, ICOSKEW, and RESSKEW are the average of estimated
values from the above equation by GMM across individual funds and reported as the percentage of the skewness of demeaned fund
returnsE[r
3
i
]. FI and EF stand for fixed income and equity funds, respectively. Numbers in parentheses are t-statistics associated with a
null hypothesis of zero raw coskewness, idiosyncratic coskewness, and residual skewness in the respective columns. FI Average is the
average of statistics across fixed-income fund styles. EF Average is the average of statistics across equity fund styles. Group Average is
the average of statistics across all fund styles.
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
Panel A: Closed-End Funds
FI Global −1.512 −0.602 122.89 −12.21 −10.67
(−0.65) (−0.29) (0.32)
FI Sector −0.754 −0.399 105.95 −6.07 0.12
(−0.60) (−0.41) (−0.10)
FI Long Term −0.339 −1.224 57.37 40.49 2.14
(−0.33) (−0.61) (0.19)
FI Intermediate Term 0.749 0.203 −6.31 116.74 −10.43
(0.43) (0.32) (−0.22)
FI Short Term −0.419 −0.912 27.74 63.65 8.61
(−0.73) (−1.20) (−0.03)
FI Government −0.262 −0.185 32.15 36.76 31.09
(−0.14) (−0.31) (−0.59)
Continued on next page
46
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
FI High Yield 0.296 −0.620 70.36 4.62 25.01
(−0.88) (−0.16) (−0.49)
FI Others −2.273 −1.656 43.16 12.03 44.81
(−1.16) (−0.65) (−0.10)
FI Average −0.564 −0.675 56.66 32.00 11.33
(−0.51) (−0.41) (−0.13)
EF Balanced −0.157 −0.780 72.21 25.29 2.49
(−0.99) (−0.45) (0.22)
EF Global 0.598 −0.059 16.66 70.76 12.59
(−0.74) (0.61) (0.36)
EF Sector −0.896 −0.162 53.60 19.99 26.41
(−0.99) (0.30) (−0.24)
EF Commodities 0.508 −1.136 50.18 66.73 −16.90
(−0.69) (−0.38) (0.01)
EF Large Cap 2.306 −0.698 3.07 −28.21 125.15
(−1.01) (−0.25) (−0.14)
EF Mid Cap 0.247 −0.258 −67.72 77.61 90.10
(−0.41) (−0.25) (−0.47)
EF Small Cap 0.833 −0.138 68.26 9.26 22.48
(−1.26) (1.39) (−0.89)
EF Growth 0.789 −0.480 51.55 −19.30 67.76
(−1.11) (0.26) (−0.67)
EF Value −0.834 −0.996 −13.76 97.39 16.37
(−1.11) (−0.50) (−0.47)
EF Others −1.830 −1.426 41.24 24.17 34.59
(−1.07) (−0.33) (0.18)
EF Average 0.156 −0.613 27.53 34.37 38.10
(−0.94) (0.04) (−0.21)
Group Average −0.164 −0.640 40.48 33.32 26.21
Continued on next page
47
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
(−0.75) (−0.16) (−0.17)
Panel B: ETFs
FI Global −1.016 −0.314 59.64 39.26 1.10
(−0.55) (0.93) (0.00)
FI Sector 0.924 0.826 103.91 −5.69 1.79
(1.20) (−0.47) (−0.32)
FI Long Term 1.178 0.945 122.79 −20.81 −1.98
(0.94) (−0.97) (0.21)
FI Intermediate Term 0.650 0.585 85.66 5.04 9.30
(0.75) (−0.86) (0.37)
FI Short Term 0.445 −0.252 46.24 25.57 28.20
(0.27) (−1.18) (−0.34)
FI Government 0.024 0.526 −45.66 123.15 22.51
(−0.01) (0.83) (0.20)
FI High Yield 0.531 0.654 106.86 −6.95 0.09
(0.66) (−1.15) (0.48)
FI Others −1.143 −1.078 101.51 −1.50 −0.01
(−1.33) (1.54) (0.49)
FI Average 0.199 0.236 72.62 19.76 7.62
(0.24) (−0.17) (0.14)
EF Balanced −0.041 −0.215 76.95 12.18 10.87
(−0.48) (0.52) (0.54)
EF Global −0.967 −0.733 87.30 4.00 8.70
(−1.33) (0.34) (0.38)
EF Sector −0.716 −0.726 71.30 27.25 1.45
(−1.07) (−0.50) (0.19)
EF Commodities −0.751 −0.892 81.85 18.00 0.15
(−1.61) (−0.40) (−0.08)
Continued on next page
48
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
EF Large Cap −0.743 −1.310 87.54 12.13 0.33
(−1.55) (−1.11) (0.07)
EF Mid Cap −1.071 −1.194 88.21 10.39 1.41
(−1.46) (−1.07) (0.06)
EF Small Cap −1.023 −1.327 94.84 5.52 −0.36
(−1.53) (−1.23) (−0.13)
EF Growth −0.121 −1.068 94.54 5.37 0.09
(−1.72) (−0.78) (−0.09)
EF Value −0.560 −1.413 93.09 6.51 0.40
(−1.57) (−0.77) (0.03)
EF Bear Market 0.917 0.676 62.21 17.30 20.50
(1.00) (0.88) (0.46)
EF Currency −1.362 −0.670 105.81 −11.26 5.45
(−0.60) (0.10) (−0.35)
EF Others −0.321 −0.737 75.58 19.59 4.82
(−1.16) (−0.47) (0.01)
EF Average −0.563 −0.801 84.93 10.58 4.48
(−1.09) (−0.37) (0.09)
Group Average −0.258 −0.386 80.01 14.25 5.74
(−0.56) (−0.29) (0.11)
Panel C: Open-Ended Funds
FI Index −0.167 −0.035 100.30 3.62 −3.92
(−0.16) (0.04) (−0.19)
FI Global −0.849 −0.556 85.77 −66.27 80.50
(0.06) (−1.13) (0.13)
FI Short Term −0.333 −0.890 45.00 −142.57 197.57
(−0.51) (−0.35) (−0.38)
FI Government −0.158 −0.138 69.42 8.48 22.10
Continued on next page
49
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
(−0.67) (0.24) (−0.01)
FI Mortgage −0.315 −0.420 80.90 2.53 16.57
(−0.58) (−0.29) (−0.04)
FI Corporate −0.963 −0.580 113.87 −33.19 19.32
(−0.69) (−0.11) (0.07)
FI High Yield −0.776 −1.174 −32.05 −7.75 139.80
(−1.00) (−0.51) (0.06)
FI Others −0.095 0.286 48.69 −62.56 113.86
(0.00) (0.06) (0.41)
FI Average −0.457 −0.439 63.99 −37.22 73.23
(−0.44) (−0.25) (0.01)
EF Index 5.493 −0.966 95.55 2.39 2.05
(−1.45) (0.01) (−0.33)
EF commodities 0.155 −0.516 87.41 23.22 −10.63
(−1.07) (0.07) (−0.15)
EF Sector −0.569 −0.371 83.74 3.99 12.27
(−0.73) (−0.20) (0.04)
EF Global −0.918 −0.796 83.73 9.82 6.45
(−1.24) (−0.01) (−0.00)
EF Balanced −0.472 −1.098 88.85 17.72 −6.57
(−1.23) (−0.40) (−0.13)
EF Leverage and Short 2.351 −0.287 19.62 29.08 51.30
(−0.48) (−0.06) (−0.26)
EF Long Short −1.658 −0.980 76.09 −1.42 25.33
(−1.69) (−0.72) (−0.26)
EF Mid Cap −0.494 −0.919 84.11 11.02 4.87
(−1.13) (−0.56) (0.06)
EF Small Cap −0.490 −0.741 103.62 −7.17 3.54
(−1.07) (−0.41) (0.07)
EF Aggressive Growth −0.405 −0.734 101.10 −5.18 4.08
Continued on next page
50
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
(−1.05) (0.68) (0.07)
EF Growth −0.695 −0.812 81.57 14.60 3.83
(−1.30) (−0.31) (−0.00)
EF Growth and Income −0.997 −0.883 96.89 0.93 2.18
(−1.30) (−0.31) (−0.14)
EF Equity Income −0.944 −0.782 91.69 1.25 7.07
(−0.93) (−0.79) (0.06)
EF Others −0.567 −0.498 −40.20 143.89 −3.70
(−0.85) (0.73) (0.07)
EF Average −0.015 −0.742 75.27 17.44 7.29
(−1.11) (−0.16) (−0.07)
Group Average −0.176 −0.631 71.17 −2.44 31.27
(−0.87) (−0.20) (−0.04)
Panel D: Hedge Funds
Equity Hedge −0.302 −0.299 46.67 17.42 35.91
(−0.36) (−0.25) (0.04)
Event-Driven −1.899 −0.618 53.84 21.50 24.67
(−0.63) (−0.60) (0.06)
Fund of Funds −1.000 −0.981 42.13 10.99 46.88
(−0.93) (−0.92) (−0.32)
HFRI −1.074 −1.115 157.06 −95.69 38.63
(−0.64) (−0.79) (−0.17)
HFRX −2.257 −1.709 110.83 −23.03 12.20
(−0.57) (−1.70) (−0.71)
Macro 0.378 0.012 19.74 −20.42 100.68
(0.14) (0.14) (0.04)
Relative Value −4.219 −1.208 31.24 17.85 51.04
(−0.45) (−0.62) (−0.23)
Continued on next page
51
Table 1.5 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
Group Average −1.482 −0.845 65.93 −10.20 44.29
(−0.49) (−0.68) (−0.18)
52
Table 1.6: Kurtosis Decomposition by Equal-weighted Portfolios
This table summarizes the kurtosis decomposition by using equal-weighted portfolio of funds as market portfolio. EW portfolio kurtosis
is the kurtosis for the equal-weighted portfolios of funds formed by funds in the same styles. Individual kurtosis is the cross-sectional
average of kurtosis of individual funds in each style. Kurtosis is the fourth central moment about the mean and computed asE[r
4
i
]/σ
4
i
−3.
r
i
and σ
i
are the demeaned return and standard deviation of fund i. COKURT, VOLCOMV , ICOKURT, and RESKURT refer to the
following components in the kurtosis decomposition:
E(r
4
i
) = β
3
i
cov(r
i
,r
3
p
)+3β
3
i
cov(u
i
,r
3
p
)
| {z }
COKURT
+6β
2
i
E(r
2
p
u
2
i
)
| {z }
VOLCOMV
+4β
i
cov(u
3
i
,r
p
)
| {z }
ICOKURT
+ E(u
4
i
))
| {z }
RESKURT
wherer
p
is the demeaned return for the market portfolio. Individual COKURT, VOLCOMV , ICOKURT, and RESKURT are the average
of estimated values from the above equation by GMM across individual funds and reported as the percentage of the kurtosis of demeaned
fund returnsE[r
4
i
]. FI and EF stand for fixed income and equity funds, respectively. Numbers in parentheses are t-statistics associated
with a null hypothesis of zero raw cokurtosis, idiosyncratic cokurtosis, volatility comovement, and residual kurtosis in the respective
columns. FI Average is the average of statistics across fixed-income fund styles. EF Average is the average of statistics across equity
fund styles. Group Average is the average of statistics across all fund styles.
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
Panel A: Closed-End Funds
FI Global 11.897 6.080 52.09 37.14 −3.27 14.04
(1.12) (2.10) (0.19) (2.60)
FI Sector 3.395 4.611 19.84 41.68 5.24 33.24
(0.90) (1.77) (0.63) (2.86)
FI Long Term 7.365 8.322 43.68 37.20 −2.08 21.20
(0.66) (1.88) (0.05) (2.06)
FI Intermediate Term 5.568 5.443 29.75 44.91 2.92 22.42
(1.12) (2.21) (0.76) (3.24)
FI Short Term 1.814 4.361 12.36 54.30 3.88 29.47
(0.35) (1.58) (0.39) (2.35)
FI Government 2.390 2.305 14.97 38.31 8.46 38.25
(1.20) (2.13) (0.83) (2.51)
Continued on next page
53
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
FI High Yield 5.445 3.708 47.57 36.53 −5.41 21.32
(1.56) (2.27) (−0.04) (2.59)
FI Others 11.743 6.415 67.79 22.50 −0.54 10.25
(1.36) (1.73) (0.21) (2.20)
FI Average 6.202 5.156 36.01 39.07 1.15 23.77
(1.03) (1.96) (0.38) (2.55)
EF Balanced 5.747 4.193 51.43 34.53 −1.24 15.29
(1.40) (1.96) (0.29) (2.50)
EF Commodities 5.801 2.725 30.94 40.01 3.98 25.07
(1.33) (2.21) (0.48) (2.84)
EF Global 4.882 4.424 18.35 41.89 3.87 35.89
(0.96) (2.04) (0.21) (2.40)
EF Sector 5.754 4.279 34.19 43.47 −6.02 28.36
(1.19) (1.82) (−0.41) (2.27)
EF Large Cap 27.479 5.530 45.49 39.45 −0.17 15.23
(1.30) (1.85) (0.27) (2.62)
EF Mid Cap 2.958 4.565 29.65 36.43 5.81 28.10
(0.99) (1.92) (0.52) (2.51)
EF Small Cap 5.238 3.321 5.27 76.14 −16.61 35.21
(−0.43) (1.89) (−0.47) (3.32)
EF Growth 6.635 4.707 22.20 50.27 −5.23 32.76
(0.39) (1.64) (0.01) (2.77)
EF Value 4.680 3.886 56.87 37.22 −1.57 7.48
(1.38) (1.91) (−0.13) (2.06)
EF Others 7.017 5.649 58.66 33.23 −0.44 8.55
(1.18) (1.92) (0.51) (2.24)
EF Average 7.619 4.328 35.30 43.26 −1.76 23.19
(0.97) (1.92) (0.13) (2.55)
Group Average 6.989 4.696 35.62 41.40 −0.47 23.45
Continued on next page
54
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
(1.00) (1.94) (0.24) (2.55)
Panel B: ETFs
FI Global 3.555 2.910 52.77 31.84 8.92 6.47
(0.89) (1.27) (0.15) (2.18)
FI Sector 2.002 1.841 92.65 8.34 −1.43 0.45
(1.77) (2.01) (0.03) (1.97)
FI Long Term 9.208 7.720 82.56 16.75 0.16 0.54
(1.35) (1.47) (0.25) (1.60)
FI Intermediate Term 4.283 3.537 66.89 30.30 −0.64 3.46
(1.29) (1.61) (−0.35) (2.16)
FI Short Term 0.890 2.156 36.00 34.02 −1.95 31.93
(0.80) (1.14) (0.12) (2.26)
FI Government 0.123 1.444 7.41 28.26 0.77 63.56
(0.65) (1.16) (0.35) (2.47)
FI High Yield 2.787 2.636 94.35 5.68 −0.05 0.02
(1.39) (2.51) (−0.13) (2.00)
FI Others 4.516 3.699 98.79 1.20 0.01 0.01
(1.57) (2.14) (−0.22) (1.87)
FI Average 3.421 3.243 66.43 19.55 0.72 13.30
(1.21) (1.66) (0.03) (2.06)
EF Balanced 1.408 1.988 61.75 28.67 0.07 9.50
(1.24) (2.09) (0.17) (1.73)
EF Global 2.524 2.404 69.21 22.32 1.84 6.63
(1.45) (2.27) (0.15) (2.60)
EF Sector 2.259 1.687 48.06 33.30 −0.38 19.01
(1.16) (2.02) (0.04) (2.42)
EF Commodities 3.222 1.430 69.22 23.32 1.37 6.10
(1.56) (2.03) (0.07) (2.23)
Continued on next page
55
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
EF Large Cap 1.559 2.842 79.38 17.94 −0.28 2.96
(1.50) (2.31) (−0.28) (2.36)
EF Mid Cap 3.267 2.648 79.32 15.20 −0.51 5.98
(1.35) (1.96) (0.05) (2.31)
EF Small Cap 2.240 2.731 90.59 9.41 −0.56 0.56
(1.51) (2.29) (−0.09) (2.47)
EF Growth 1.453 1.884 81.83 15.01 −0.03 3.19
(1.70) (2.28) (0.21) (2.62)
EF Value 2.537 3.579 79.72 17.24 −0.44 3.48
(1.40) (2.24) (0.16) (2.43)
EF Bear Market 0.970 1.142 48.28 38.85 −1.11 13.99
(1.26) (1.83) (−0.01) (2.24)
EF Currency 3.894 3.786 55.12 26.71 0.19 17.99
(0.94) (1.60) (0.31) (2.24)
EF Others 0.489 1.749 55.47 32.84 1.62 10.08
(1.45) (2.27) (0.04) (2.63)
EF Average 2.152 2.323 68.16 23.40 0.15 8.29
(1.38) (2.10) (0.07) (2.36)
Group Average 2.659 2.691 67.47 21.86 0.38 10.30
(1.31) (1.93) (0.05) (2.24)
Panel C: Open-Ended Funds
FI Index 0.545 1.412 75.50 20.03 −1.91 6.38
(2.17) (2.30) (−0.13) (2.42)
FI Global 5.778 3.739 35.55 52.03 −4.35 16.76
(1.11) (2.29) (0.10) (2.56)
FI Short Term 1.911 5.023 16.24 71.25 −20.75 33.20
(0.47) (1.94) (−0.33) (2.20)
FI Government 0.430 1.311 56.88 26.83 −1.82 18.10
Continued on next page
56
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
(1.97) (2.18) (0.18) (2.65)
FI Mortgage 0.966 2.018 60.27 27.10 0.48 12.15
(1.99) (2.38) (0.13) (2.17)
FI Corporate 4.220 2.578 69.02 25.31 −1.93 7.60
(1.79) (2.54) (−0.15) (2.17)
FI High Yield 4.850 5.553 72.90 21.01 −0.03 6.12
(1.58) (2.28) (0.27) (2.34)
FI Others 2.824 3.412 18.32 21.92 −0.95 60.71
(0.56) (1.36) (0.14) (2.84)
FI Average 2.690 3.131 50.59 33.18 −3.91 20.13
(1.45) (2.16) (0.03) (2.42)
EF Index 87.572 2.850 77.95 18.13 −0.77 4.69
(1.69) (2.57) (−0.14) (2.44)
EF Commodities 4.114 1.585 66.37 25.43 −0.66 8.86
(1.57) (2.62) (0.05) (2.52)
EF Sector 2.142 1.268 44.65 37.35 1.03 16.97
(1.70) (2.32) (0.25) (2.68)
EF Global 3.287 2.245 73.89 20.49 0.54 5.08
(1.69) (2.75) (0.15) (2.79)
EF Balanced 2.151 3.332 73.09 21.10 −0.83 6.64
(1.55) (2.49) (0.25) (2.56)
EF Leverage and Short 23.537 1.597 44.92 38.01 −0.47 17.54
(1.39) (2.12) (0.34) (2.33)
EF Long Short 2.791 1.439 67.59 24.49 2.43 5.48
(1.36) (2.09) (0.07) (2.49)
EF Mid Cap 1.336 2.579 76.03 21.74 −1.06 3.28
(1.65) (2.43) (0.10) (2.55)
EF Small Cap 1.097 1.930 73.92 24.35 −1.25 2.98
(1.70) (2.55) (−0.03) (2.56)
EF Aggressive Growth 0.730 2.067 71.61 19.73 0.58 8.08
Continued on next page
57
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
(1.39) (2.54) (0.04) (2.55)
EF Growth 2.198 2.114 73.00 21.60 0.11 5.30
(1.86) (2.62) (0.16) (2.61)
EF Growth and Income 2.785 2.169 82.80 14.39 −0.06 2.87
(1.80) (2.69) (0.07) (2.55)
EF Equity Income 3.010 2.041 79.60 17.44 −0.21 3.17
(1.74) (2.52) (0.09) (2.57)
EF Others 1.857 1.994 66.75 23.17 0.11 9.97
(1.41) (2.54) (0.08) (2.58)
EF Average 9.901 2.086 69.44 23.39 −0.04 7.21
(1.61) (2.49) (0.11) (2.56)
Group Average 7.279 2.466 62.59 26.95 −1.45 11.91
(1.55) (2.37) (0.08) (2.50)
Panel D: Hedge Funds
Equity Hedge 2.004 2.001 17.56 34.18 1.08 47.17
(0.64) (1.50) (0.29) (2.45)
Event-Driven 6.907 4.071 23.59 31.03 3.75 41.64
(0.73) (1.58) (0.49) (2.25)
Fund of Funds 4.186 2.951 44.53 33.21 2.35 19.91
(1.17) (1.95) (0.37) (2.31)
HFRI 4.018 6.471 33.21 42.58 −0.29 24.50
(0.98) (2.14) (0.70) (2.90)
HFRX 8.220 6.048 50.56 27.05 1.26 21.13
(0.69) (1.88) (0.65) (2.27)
Macro 0.134 2.006 7.28 22.82 2.96 66.95
(0.61) (1.21) (0.29) (2.32)
Relative Value 25.845 7.001 7.27 45.82 −8.80 55.89
(0.42) (1.07) (0.20) (1.96)
Continued on next page
58
Table 1.6 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
Group Average 7.330 4.364 26.28 33.81 0.33 39.60
(0.75) (1.62) (0.43) (2.35)
59
1.8 Robustness Analysis
1.8.1 An Application of the Model on Mutual Funds
All moments in the model in section IV are standardized. One set of parameters from mutual funds
is applied to the model. Brown, Goetzmann, Ibbotson, and Ross (1992) simulate mutual fund returns
by the following:
R
i,j
= r
f
+β
i
(R
p,j
−r
f
)+
i,j
(1.22)
where the risk free rate is 0.07 and the risk premium is assumed to be normal with mean 0.086 and
standard deviation 0.208. β
i
follows the normal distribution with mean 0.95 and standard deviation
0.25 cross-sectionally. The idiosyncratic term
i,j
is assumed to be normal with mean 0 and standard
deviationσ
i
. The relationship between nonsystematic risk andβ
i
is approximated as:
σ
2
i
= k(1−β
i
)
2
(1.23)
The value of k is 0.05349. Note that β
i
(R
p,j
−r
f
) and
i,j
are equivalent to r
p,j
and r
BB,j
in the model, representing systematic and idiosyncratic components of returns. I implement these
parameters in the model and display the model’s predictions for the relation between the return
decomposition (convexity) effect and the optimal weight on the market portfolio and the big bet in
Figure 1.4.
To summarize, the model predictions hold in a qualitatively similar manner. Convexity induces
fund managers to take idiosyncratic big bets and increased weights in compensation relative to a
benchmark cause fund managers to invest more in the benchmark and thus yield more systematic
tail risks.
1.8.2 Autocorrelation
Stale pricing or serial correlation of returns has the most significant impact on HFs among fund
types. Due to the unique characteristics of HFs, such as limited regulations and the lockup and
60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
alpha: Return decomposition
The Return Decomposition and Convexity Effect on Benchmark
g: Convexity
Benchmark weight
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
alpha: Return decomposition
The Return Decomposition and Convexity Effect on Big Bet
g: convexity
Big bet weight
Figure 1.4: The Optimal Weight of the Benchmark and Big Bet
The return decomposition parameterα and the convexity parameterg are the weight of the systematic return
and convex payoff in managerial compensation, respectively. z-axis is the optimal weight.
notice periods, HF managers have more flexibility in trading illiquid assets. Since current prices
may not be available for illiquid assets, HF managers commonly use past prices to estimate them.
As a result, the presence of illiquid assets can lead to significant serial correlation on HF returns.
This link is supported by Getmansky, Lo, and Makarov (2004), who conclude that illiquidity and
smoothed returns are the main source of serial correlation in HFs. The existence of serial correlation
in returns can affect HF performance and statistics (Lo (2002) and Jagannathan, Malakhov, and
Novikov (2010)).
Following Asness, Krail, and Lieu (2001) and Getmansky, Lo, and Makarov (2004), let the true
but unobserved demeaned return satisfy the following regression:
r
∗
i,t
=β
∗
i
r
p,t
+u
∗
i,t
, E(u
∗
i,t
) = 0, r
p,t
andu
∗
i,t
arei.i.d.
61
I use three lags to model autocorrelations of the observed demeaned returns. The observed
demeaned returnr
i,t
is thus modeled as:
r
i,t
= θ
0
r
∗
i,t
+θ
1
r
∗
i,t−1
+θ
2
r
∗
i,t−2
= β
∗
i
(θ
0
r
p,t
+θ
1
r
p,t−1
+θ
2
r
p,t−2
)+(θ
0
u
∗
i,t
+θ
1
u
∗
i,t−1
+θ
2
u
∗
i,t−2
)
= β
0,i
θ
0
r
p,t
+β
1,i
θ
1
r
p,t−1
+β
2,i
θ
2
r
p,t−2
)+η
i,t
= (β
0,i
+β
1,i
+β
2,i
)(R
p,t
−μ
p
)+ ˜ u
i,t
The last equation is used by Asness, Krail, and Lieu (2001) to compute the “summed beta”
Sharpe ratios for HFs. They estimate coefficients by the second to last equation and consider the
summation of three coefficients as the true beta. They therefore compute the “summed beta” resid-
uals as
˜ u
∗
i,t
=r
i,t
−
˜
β
∗
i
(R
p,t
−μ
p
)
where
˜
β
∗
i
is the true or “summed beta”, i.e.
˜
β
∗
i
= β
0,i
+β
1,i
+β
2,i
. I follow the same approach to
construct moment conditions. GMM moment conditions are modified as follows:
For skewness decomposition:
r
i,t
= R
i,t
−μ
i
r
p,t
= R
p,t
−μ
p
u
i,1t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t
−μ
p
)
u
i,2t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t−1
−μ
p
)
u
i,3t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t−2
−μ
p
)
u
i,4t
= COSKEW
i
−
˜
β
∗
i
3
r
3
p,t
−3
˜
β
∗
i
2
(r
2
p,t
˜ u
∗
i,t
)
u
i,5t
= ICOSKEW
i
−3
˜
β
∗
i
(r
p,t
˜ u
∗2
i,t
)
u
i,6t
= RESSKEW
i
− ˜ u
∗3
i,t
62
For kurtosis decomposition:
r
i,t
= R
i,t
−μ
i
r
p,t
= R
p,t
−μ
p
u
i,1t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t
−μ
p
)
u
i,2t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t−1
−μ
p
)
u
i,3t
= (R
i,t
−μ
i
−β
0,i
(R
p,t
−μ
p
)−β
1,i
(R
p,t−1
−μ
p
)−β
2,i
(R
p,t−2
−μ
p
))(R
p,t−2
−μ
p
)
u
i,4t
= COKURT
i
−
˜
β
∗
i
4
r
4
p,t
−4
˜
β
∗
i
3
(r
3
p,t
˜ u
∗
i,t
)
u
i,5t
= VOLCOMV
i
−6
˜
β
∗
i
2
(r
2
p,t
˜ u
∗2
i,t
)
u
i,6t
= CONSKT
i
−4
˜
β
∗
i
(r
p,t
˜ u
∗3
i,t
)
u
i,7t
= RESKURT
i
− ˜ u
∗4
i,t
The decomposition results (%) for skewness and kurtosis are reported in Table 1.7. Overall, the
tail risk decompositions are robust to autocorrelation. The weight on RESSKEW increases slightly
and the weight on RESKURT stays almost the same. COSKEW and RESSKEW are still the top
two contributors to HF skewness. The components of VOLCOMV and RESKURT occupy the most
weights in HF kurtosis. More interestingly, in contrast to the finding in Asness, Krail, and Lieu
(2001) that beta risk increases after stale prices are adjusted, idiosyncratic tail risks for HFs slightly
increase. This may suggest that stale pricing helps identify true idiosyncratic tail risks undertaken
by HF managers.
63
Table 1.7: Skewness and Kurtosis Decomposition of HFs Adjusted for Autocorrelations
This table summarizes the skewness and kurtosis decompositions by using equal-weighted portfolio of funds as market portfolio, after
being adjusted for stale prices. The 3-lag autocorrelated observed return process is identified asr
i,t
= (β
0,i
+β
1,i
+β
2,i
)r
p,t
+u
i,t
. r
i,t
andr
p,t
are demeaned return for fundi and market portfolio. Substitute the true
˜
β
i
(=β
0,i
+β
1,i
+β
2,i
) in the equation ofr
i,t
=
˜
β
i
r
p,t
to
derive and compute the skewness and kurtosis decompositions. Numbers in parentheses are t-statistics associated with a null hypothesis
of zero raw coskewness, idiosyncratic coskewness, residual skewness, cokurtosis, idiosyncratic cokurtosis, volatility comovement, and
residual kurtosis in the respective columns. Group Average is the average of statistics across all fund styles.
Systematic Idiosyncratic
—————————————– ———————
Styles Individual Individual Individual
COSKEW (%) ICOSKEW (%) RESSKEW (%)
Panel A: Skewness Decomposition
Equity Hedge 23.66 36.20 43.21
(−0.32) (−0.28) (0.04)
Event-Driven 50.84 21.25 27.54
(−0.54) (−0.59) (0.08)
Fund of Funds 44.63 21.19 35.23
(−0.95) (−0.79) (−0.30)
HFRI 141.69 −49.19 8.05
(−0.75) (−0.40) (−0.26)
HFRX 66.09 −15.20 52.80
(−0.83) (−1.36) (−0.25)
Macro 42.53 −99.38 157.81
(0.11) (0.12) (0.05)
Relative Value 32.20 22.77 45.39
(−0.37) (−0.60) (−0.18)
Group Average 57.38 −8.91 52.86
(−0.52) (−0.56) (−0.12)
Continued on next page
64
Table 1.7 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles Individual Individual Individual Individual
COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
Panel B: Kurtosis Decomposition
Equity Hedge 9.77 48.98 −11.54 52.88
(0.50) (1.20) (0.22) (2.15)
Event-Driven 10.18 51.51 −6.23 44.51
(0.50) (1.26) (0.25) (2.06)
Fund of Funds 37.69 43.03 −2.42 21.82
(1.04) (1.66) (0.33) (2.04)
HFRI 36.99 34.96 4.50 23.57
(1.07) (1.91) (0.69) (2.80)
HFRX 52.37 21.01 1.58 25.76
(0.90) (1.53) (0.25) (2.25)
Macro −9.27 60.95 −35.47 84.24
(0.27) (0.82) (0.14) (1.95)
Relative Value −5.57 67.05 −21.31 60.21
(0.27) (0.95) (0.07) (1.82)
Group Average 18.88 46.79 −10.13 44.71
(0.65) (1.33) (0.28) (2.15)
65
1.8.3 Exogenous Systematic Factors
Different fund types are subject to different exogenous systematic factors due to risk characteris-
tics. ETFs are passive and index-tracking, and therefore returns are highly correlated with market
factors. The premiums on CEFs are related to market risk, small-firm risk, and book-to-market
risk (Lee, Shleifer, and Thaler (1991), Swaminathan (1996), Pontiff (1997)). Carhart (1994) shows
that momentum plays an important role in mutual fund performance. Non-linearities in HF returns
may suggest some systematic factors representing option-like payoffs (Fung and Hsieh (2001) and
Agarwal and Naik (2004)).
Following the literature, I use Fama-French 3-factor model for equity ETFs and CEFs, Carhart
4-factor model for equity OEFs and Fung and Hsieh 7 factor model for HFs. For bond fonds, I add
two more Barclay bond indexes - the Barclay U.S. government/credit index and corporation bond
index. Fama-French 3 factors are value-weighted market excess returns, and two factor-mimicking
portfolios SMB and HML. SMB and HML measure the observed excess returns of small caps over
big caps and of value stocks over growth stocks. Carhart adds the momentum factor on top of Fama-
French 3 factors. The momentum factor is constructed by the monthly return difference between
one-year prior high over low momentum stocks. Fung and Hsieh 7 factors include the equity and
bond market factor, the size spread factor,
23
the credit spread factors,
24
and three lookback straddles
on bond futures, currency futures, and commodity futures.
For simplicity, this paper adopts the single-factor model to illustrate economic intuitions on
components of skewness and kurtosis decompositions. I construct beta-weighted time series of
aforementioned factors to decompose systematic and idiosyncratic tail risks. Table 1.8 and 1.9
show the results.
25
23
Wilshire Small Cap 1750 - Wilshire Large Cap 750 return.
24
month-end to month-end change in the difference between Moody’s Baa yield and the Federal Reserve’s 10-year
constant-maturity yield.
25
I also use equal-weighted exogenous factors, but across all fund types and styles, RESSKEW and RESKURT con-
sistently have the highest percentages among all components in both skewness and kurtosis decompositions. This result
reflects that equal-weighted exogenous factors do not appropriately capture time-variation in systematic tail risks and
implies that investors can diversify tail risks across fund types. A further analysis on the correlation between equal-
weighted portfolios of funds and equal-weighted exogenous factors shows that the decomposition of the systematic and
idiosyncratic tail risks is sensitive to the chosen benchmarks, i.e. low correlation between the endogenous and exogenous
benchmarks implies the increased percentage of RESSKEW and RESKURT. All results are available upon request.
66
Table 1.8: Skewness Decomposition by Beta-weighted Exogenous Factors
Beta-weighted factors are constructed from Fama-French 3 factors, Carhart 4 factors, Fung-Hsieh 7 factors, and 2 bond factors. Equity CEFs and ETFs
use the beta-weighted Fama-French 3 factors. Equity open-ended funds and hedge funds use the beta-weighted Carhart 4 factors, and Fung-Hsieh 7
factors, respectively. Bond CEFs, ETFs, and open-ended funds use two more bond indexes in addition to the factors used in their equity counterparts -
the Barclay U.S. government/credit index and corporation bond index. The weights to construct beta-weighted factors depend on the respective betas
on each factor. Betas are estimated by regressing fund excess returns on factor excess returns. EW portfolio skewness is the cross-sectional average
of skewness of beta-weighted factors. Individual skewness is the cross-sectional average of skewness of individual funds in each style. Skewness is
the third central moment about the mean and computed asE[r
3
i
]/σ
3
i
. ri andσi are the demeaned return and standard deviation of fundi. COSKEW,
ICOSKEW, and RESSKEW refer to the following components in the skewness decomposition:
E(r
3
i
) = β
2
i
cov(ri,r
2
p
)+2β
2
i
cov(ui,r
2
p
)
| {z }
COSKEW
+3βicov(u
2
i
,rp)
| {z }
ICOSKEW
+ E(u
3
i
)
|{z}
RESSKEW
whererp is the demeaned return for the market portfolio. Individual COSKEW, ICOSKEW, and RESSKEW are the average of estimated values from
the above equation by GMM across individual funds and reported as the percentage of the skewness of demeaned fund returnsE[r
3
i
]. FI and EF stand
for fixed income and equity funds, respectively. Numbers in parentheses are t-statistics associated with a null hypothesis of zero raw coskewness,
idiosyncratic coskewness, and residual skewness in the respective columns. FI Average is the average of statistics across fixed-income fund styles. EF
Average is the average of statistics across equity fund styles. Group Average is the average of statistics across all fund styles.
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
FI Average −0.772 −0.675 24.66 38.56 36.75
(−0.35) (−0.59) (−0.47)
EF Average −1.274 −0.613 45.51 32.98 22.14
(−0.96) (−0.64) (0.28)
Group Average −1.051 −0.640 36.24 35.46 28.63
(−0.69) (−0.62) (−0.05)
Panel B: ETFs
FI Average 0.162 0.236 59.63 15.40 22.25
(0.15) (−0.39) (0.82)
Continued on next page
67
Table 1.8 (Continued)
Systematic Idiosyncratic
—————————————– ———————
Styles EW Port Individual Individual Individual Individual
Skewness Skewness COSKEW (%) ICOSKEW (%) RESSKEW (%)
EF Average −1.170 −0.801 82.55 13.16 3.85
(−1.14) (−0.22) (0.22)
Group Average −0.637 −0.386 73.38 14.06 11.21
(−0.62) (−0.28) (0.46)
Panel C: Open-Ended Funds
FI Average −0.352 −0.439 22.56 29.67 50.24
(−0.12) (−0.38) (−0.37)
EF Average −0.919 −0.742 90.53 5.69 3.85
(−1.11) (−0.19) (0.11)
Group Average −0.712 −0.631 65.81 14.41 20.72
(−0.75) (−0.26) (−0.06)
Panel D: Hedge Funds
Group Average 0.928 −0.845 27.81 23.88 47.80
(−0.45) (−0.68) (−0.29)
68
Table 1.9: Kurtosis Decomposition by Beta-weighted Exogenous Factors
Beta-weighted factors are constructed from Fama-French 3 factors, Carhart 4 factors, Fung-Hsieh 7 factors, and 2 bond factors. Equity
CEFs and ETFs use the beta-weighted Fama-French 3 factors. Equity open-ended funds and hedge funds use the beta-weighted Carhart
4 factors, and Fung-Hsieh 7 factors, respectively. Bond CEFs, ETFs, and open-ended funds use two more bond indexes in addition to
the factors used in their equity counterparts - the Barclay U.S. government/credit index and corporation bond index. The weights to
construct beta-weighted factors depend on the respective betas on each factor. Betas are estimated by regressing fund excess returns
on factor excess returns. EW portfolio kurtosis is the cross-sectional average of kurtosis of beta-weighted factors. Individual kurtosis
is the cross-sectional average of kurtosis of individual funds in each style. Kurtosis is the fourth central moment about the mean and
computed asE[r
4
i
]/σ
4
i
−3. r
i
andσ
i
are the demeaned return and standard deviation of fundi. COKURT, VOLCOMV , ICOKURT, and
RESKURT refer to the following components in the kurtosis decomposition:
E(r
4
i
) = β
3
i
cov(r
i
,r
3
p
)+3β
3
i
cov(u
i
,r
3
p
)
| {z }
COKURT
+6β
2
i
E(r
2
p
u
2
i
)
| {z }
VOLCOMV
+4β
i
cov(u
3
i
,r
p
)
| {z }
ICOKURT
+ E(u
4
i
))
| {z }
RESKURT
where r
p
is the demeaned return for the beta-weighted factors. Individual COKURT, VOLCOMV , ICOKURT, and RESKURT are the
average of estimated values from the above equation by GMM across individual funds and reported as the percentage of the kurtosis of
demeaned fund returnsE[r
4
i
]. FI and EF stand for fixed income and equity funds, respectively. Numbers in parentheses are t-statistics
associated with a null hypothesis of zero raw cokurtosis, idiosyncratic cokurtosis, volatility comovement, and residual kurtosis in the
respective columns. FI Average is the average of statistics across fixed-income fund styles. EF Average is the average of statistics across
equity fund styles. Group Average is the average of statistics across all fund styles.
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
Panel A: Closed-End Funds
FI Average 3.603 5.156 7.83 24.45 14.74 52.48
(0.62) (1.43) (1.09) (2.48)
EF Average 4.228 4.328 23.92 36.83 7.02 29.90
(1.01) (1.75) (0.74) (2.54)
Group Average 3.950 4.696 16.77 31.33 10.45 39.94
(0.84) (1.61) (0.89) (2.52)
Continued on next page
69
Table 1.9 (Continued)
Systematic Idiosyncratic
—————————————————————– ——————–
Styles EW Port Individual Individual Individual Individual Individual
Kurtosis Kurtosis COKURT (%) VOLCOMV (%) ICOKURT (%) RESKURT (%)
Panel B: ETFs
FI Average 1.832 3.243 25.53 39.35 0.01 35.18
(0.65) (1.36) (0.15) (2.12)
EF Average 2.329 2.323 68.89 21.03 1.88 7.81
(1.40) (2.14) (0.27) (2.44)
Group Average 2.130 2.691 51.55 28.36 1.13 18.76
(1.10) (1.82) (0.22) (2.31)
Panel C: Open-Ended Funds
FI Average 2.806 3.131 14.08 28.12 2.28 55.82
(0.47) (1.53) (0.39) (2.73)
EF Average 2.153 2.086 73.20 19.80 0.36 6.23
(1.65) (2.54) (0.15) (2.70)
Group Average 2.390 2.466 51.70 22.82 1.06 24.26
(1.22) (2.17) (0.23) (2.71)
Panel D: Hedge Funds
Group Average 3.664 4.364 17.07 34.68 1.46 46.83
(0.55) (1.36) (0.47) (2.46)
70
First, COSKEW contributes the most to total fund skewness, except HFs. COKURT is the most
contributing source to total fund kurtosis for ETFs and OEFs. In addition, HFs (ETFs) have the
largest (smallest) weight on RESSKEW and RESKURT. Second, RESSKEW and RESKURT tend
to be higher for fixed income funds when beta-weighted exogenous factors are used. This spurious
result may be induced by missing bond factors, such as a high-yield index or a global bond index.
1.8.4 Year 1996-2008
The starting period of four fund types differs in this study. However, the time-variation of economic
states, such as changes in yields and business cycles, may impose differential impacts on “economy-
wide” shocks on funds. Using the same time intervals for all four fund types can ascertain that all
funds are subject to the same economic shocks at any time. If the pattern of skewness and kurtosis
decomposition holds, the percentage of each component should be robust to the same starting period.
Therefore, I restrict all investment funds to have the same starting date as HFs and perform GMM
on this subsample of data.
The main inferences remain qualitatively unchanged, when I restrict the dataset for all funds
between the period from 1996 to 2008 only. Note that this period also excludes the 1987 stock
market crash. COSKEW contributes the most to the skewness of all fund types. COKURT and
VOLCOMV are the two largest components in kurtosis decomposition for CEFs, ETFs, and OEFs.
HFs’ kurtosis comes mostly from the VOLCOMV and RESKURT. At the style level of each fund
type, few fund styles have different proportions in skewness and kurtosis decompositions. It may
imply that each component is time-varying at the style level. However, at the aggregate fund type
level, the percentage on each component stays the same. In addition, HFs (ETFs) have the largest
(least) weights on idiosyncratic tail risks.
1.9 Conclusion
Different styles and types of managed portfolios execute different strategies and objectives. Tra-
ditional fund managers can make investment decisions based on returns and volatility of different
71
individual assets. They can also adjust exposure to systematic factors or asset classes, such as size,
book-to-market, or momentum. However, many stylized facts on financial asset returns refute the
validity of the mean-variance framework, and market-timing and stock-picking strategies can induce
systematic and idiosyncratic tail risks.
This study shows that managed portfolios are subject to tail risks. The frequency of tail returns
shows that CEFs and HFs are subject to more total tail risks. ETFs show a disparity in the frequency
between the systematic and idiosyncratic tail returns. Therefore, fund managers may manage sys-
tematic and idiosyncratic tail risks through investing in assets with desired properties and tail risks.
For instance, a manager can generate abnormal returns by adding assets with negative coskewness
or positive cokurtosis or selecting negatively skewed or positively kurtosised assets. The skewness
and kurtosis decompositions show the mechanisms fund managers may use to manage tail risks.
Skewness and kurtosis decompositions introduce economically important components. These
components reflect fund returns and volatility with respect to extreme movements in market returns,
volatility, and skewness. Skewness is decomposed into coskewness, idiosyncratic coskewness, and
residual skewness. Coskewness and idiosyncratic coskewness are relatively important in the total
fund skewness, but all three components do not show statistical significance. Kurtosis can be de-
composed into four components - cokurtosis, volatility comovement, idiosyncratic cokurtosis, and
residual kurtosis. The volatility comovement and residual kurtosis contribute the most to the total
fund kurtosis at a statistically significant level. Results of the skewness and kurtosis decompositions
are robust to benchmarks used.
The fund tail risks are linked to compensation structure across fund types through a simple
model. There are two main determinants of compensation schemes - the decomposition between
the systematic and idiosyncratic returns (return decomposition effect), and the convexity or degree
of option-like payoffs (convexity effect). The model predicts that the increased weight on systematic
returns can increase market exposure, and in turn increase total skewness and decrease total kurtosis.
In addition, increased convexity can increase idiosyncratic tail risks, and thus reduce asymmetry and
raise fat-tailedness. Empirical results confirm both predictions.
72
Chapter 2
Fund Convexity and Tail Risk-Taking
2.1 Introduction
Traditional utility theory suggests that risk-averse investors prefer lottery-like returns, or positive
skewness. Many studies have tried to explain this behavior (e.g., Brunnermeier and Parker (2005),
Mitton and V orkink (2007), Barberis and Huang (2008)). It is puzzling, then, that most investors
delegate investment decisions to fund managers because the majority of managed portfolios exhibit
negative skewness and excess kurtosis. Behavioral finance scholars might argue that investors can
prefer negative skewness.
1
An alternative hypothesis, however, is that investors cannot observe
tail risks or that fund managers use trading strategies that improve fund performance in terms of
mean and variance at the expense of downside risk (Leland (1999)). This paper builds on this
hypothesis and addresses: (1) how fund managers engage in skewed bets in response to relative
fund performance, and (2) how convexity in incentives affects fund managers’ tail risk-taking.
The first type of risk-taking incentives I study is tournament behavior. Tournament incentives
can be channeled through convex incentives faced by fund managers.
2
The literature on tournaments
1
The prospect theory, proposed by Kahneman and Tversky (1979), introduces a value function based on change
in wealth relative to a reference point. Unlike the conventional V on Neuman-Morgenstern utility of wealth, the value
function of prospect theory is concave in the profit region and convex in the loss region. This type of utility leads to
loss aversion and preferences for a one-time substantial loss (negative skewness) than a succession of very small losses
(positive skewness).
2
The variance strategies related to relative performance can depend on the degree of convexity of implicit and explicit
asymmetric incentive contracts. For example, when a fund manager has an incentive to outperform his or her peers,
outperforming can attract more inflows through the convex flows to fund performance relation. Kempf, Ruenzi, and Thiele
(2009) also show that managerial risk-taking depends on the relative importance of employment and convex compensation
73
has primarily addressed managerial risk-taking with respect to relative performance. Fund managers
have a strong incentive to take idiosyncratic bets to rise in tournament rankings. Brown, Harlow,
and Starks (1996) show that midyear losers tend to increase fund volatility in the second half of the
year. Elton, Gruber, and Blake (2003) show that incentive-fee mutual funds involve more risk-taking
than nonincentive mutual funds if their performance lags behind that of their peers in the first half
of the year. Brown, Goetzmann, and Park (2001) conclude that a fund manager’s variance strategy
depends on relative rather than absolute performance evaluation. They also provide evidence that
managers who perform well will reduce variance but show little evidence of increasing volatility
for underperforming fund managers. The mixed results on risk-taking may be attributed to longer
evaluation periods for fund managers (Hodder and Jackwerth (2007), Panageas and Westerfield
(2009)), disincentives to liquidate funds owing to career concerns (Chevalier and Ellison (1999)),
reputation costs (Fung and Hsieh (1997)), or managerial stakes in funds (Kouwenberg and Ziemba
(2007)).
Different types of convexity can induce tail risks in managed portfolios. Chevalier and Ellison
(1997) and Sirri and Tufano (1998) show that the overall flow-performance relation is convex, so a
manager may improve an expected fund size by taking positive or right-skewed tail risks. Closed-
end fund managers offer investors the opportunity to buy illiquid assets. Changes in discounts
(market-to-book ratio) in closed-end funds can be regarded as the return of the implicit option that
investors sell to the management. An example of explicit convexity is the high-water mark contract
for hedge fund managers. In addition to receiving a fixed percentage of assets under management,
hedge fund managers are rewarded with a fraction of the gains above the last recorded maximum.
Given that volatility and skewness are positively correlated, a fund manager may indeed take skewed
bets on top of risky bets. The relation between convex incentives and skewed bets cannot be easily
inferred from the mixed results in the literature. This paper first studies how a fund’s performance
relative to its peers relates to short-term tail risk-taking behavior. I find that if a fund manager out-
performs his or her peers, he or she will be more likely to take a negatively skewed bet. If, however,
incentives. In addition, when fund performance is evaluated relative to a benchmark, the convex relationship between
past performance and the managerial compensation can drive the risk-taking choices attributed to tournament behavior.
74
a fund underperforms peer funds, the fund manager will make a lottery-like bet. The underperform-
ing fund manager will likely gamble on winning a jackpot with a tiny probability because positively
skewed bets satisfy the need to both climb up the rankings and prevent the liquidation of funds. In
contrast, a successful fund manager will more likely adopt a strategy characterized by a succession
of solid returns to stay on top, taking a chance on the large downside risk with a small probability.
Most interestingly, this tail risk-taking behavior is prevalent across fund types. As I discuss below,
a number of practical strategies with negative skewness fit this general description.
I further examine how convexity affects tail risks in different types of investment funds. Both
implicit and explicit convexities are examined. The degree of convexity is measured as: (1) the
discount for closed-end funds, (2) the sensitivity of fund returns to fund flows and the sensitivity of
fund returns to relative ranking in tournaments for open-ended funds, and (3) the ratio of high-water
marks to fund values for hedge funds. I find that convexity induces skewness risk-taking. Sorting
funds based on these measures shows that the differences in expected skewness between the funds
facing most convexity and the funds facing least convexity are statistically significant.
To my knowledge, this paper is the first to examine how subsequent skewness is related to
past performance and call-like features in incentives.
3
The current literature only measures risk as
variance or covariance such as beta. The evidence on skewed bets sheds new light on the literature
of tournaments and incentive contracts. Recent studies have documented the risk-shifting behavior
of fund managers facing implicit and explicit incentive contracts. This paper relates these incentive
contracts to fund managers’ behavior on skewed bets. The results of taking positively skewed bets
in response to underperformance show that the contributing factors to risk shifting, such as the
expected value of continuation or career concerns, may not deter underperforming managers from
taking bets on assets or trading strategies with lottery-like returns. Fund managers’ skewness risk-
seeking behavior is evident.
3
Koski and Pontiff (1999) find that investing in derivatives does not skew the distribution of open-ended equity fund
returns. Unlike their analysis on unconditional skewness, I study skewed bets conditional on incentives.
75
The rest of the paper proceeds as follows. Section 2.2 addresses the importance of skewness risk.
Section 2.3 reviews the literature on risk-taking and relates it to skewness risk-taking. Section 2.4
outlines the data. Section 2.5 describes empirical methods and results. Section 2.6 concludes.
2.2 Why Should Investors Move Beyond Volatility and Care About
Skewness Risk?
The fund industry commonly employs two types of skewed bets: negatively skewed and positively
skewed. Even if asset returns are normal, a dynamic trading strategy or options on the assets can pro-
duce fund skewness (Leland (1999), Anson (2002)). Examples of negatively skewed trades include
short options, leveraged trades, statistical arbitrage, convergence trades, credit-related strategies,
momentum strategies, doubling strategies, convertible bond arbitrage, structured trades, illiquid
trades, and short volatility trades.
4
Furthermore, (ineffective) market timing strategies can induce
negative skewness. A market timer adjusts betas on systematic factors such as market excess re-
turns. An ineffective market timing strategy generates negatively skewed risk because systematic
factor returns are negatively skewed.
5
In contrast, buying an option can increase the skewness of a
fund. A contrarian fund -that is, one in which the manager buys losers and sells winners-can also
increase the systematic skewness (coskewness) of the fund (Harvey and Siddique (2000)). As such,
tail risks can arise from the convex or concave payoffs from trading strategies.
Skewed trades are also prevalent in fixed income funds. The relation of yield to price exhibits
positive convexity. Callable bonds, which allow issuers to buy back the bond at fixed prices, exhibit
“negative convexity”, i.e. concavity. The payoffs of noncallable convertible bonds are asymmet-
ric because the bond holders have the right to convert the bond into a fixed number of shares of
4
Hedge fund managers can engage in short-volatility trades by longing an undervalued asset and shorting an overval-
ued asset in expectation of their prices converging to fundamental values. Examples include merger arbitrage, statistical
arbitrage, event-driven strategies, convergence trades, and risk arbitrage. These strategies look like one long and one
short equity position but can incur great losses for investors when the prices don’t converge as anticipated. Mutual fund
managers can write a covered call on the S&P500 index to reduce the downside risk of the portfolio by limiting the up-
side potential. The covered call writing also yields steady profits but can incur considerable losses when market volatility
jumps.
5
For example, Engle and Mistry (2007) study negative skewness in Fama and French factors and Carhart’s momentum
factor.
76
the issuer, and the bond value can only fall to the value of bond floor. Asset-backed securities or
mortgage-backed securities are subject to prepayment risk and reinvestment risk when interest rates
fall, and thus prices fall and the price-yield relationship has negative convexity. Interest rate prod-
ucts, such as swaps, offer bond managers a steady stream of interest payments but expose investors
to interest rate risk and credit risk. The option-adjusted spreads in interest rate derivative products
also reflect counterparty risk, credit risk, default risk, and liquidity risk. Under conditions of severe
distress in the economy, any widening of the spread brings out extremely negative returns. In short,
a bond fund manager can invest in a wide variety of products that yield asymmetric payoffs. In such
investments, bond fund returns appear to be skewed. Convexity in compensation structures is often
asymmetric. Asymmetric convexity in compensation implies that fund managers value upside gains
from increased compensation but are not penalized as much by downside losses. The previously
mentioned fund flow-performance relation is a classical example. Winning fund managers can gain
compensation by taking negatively skewed bets if the chance of earning steady profits asymmetri-
cally outweighs the probability of losses. Losing fund managers, in contrast, are inclined to take
positively skewed bets because the downside is limited. Because convexity is itself asymmetric and
induces skewness risk, it is intuitive to look beyond variance strategies and relate convexity to a
manager’s positions on skewed trades.
It should be relatively difficult for funds to increase the variance of their returns without attract-
ing attention from investors, outside board members and organizations like Morningstar that provide
fund monitoring services. For example, Sirri and Tufano (1998) among others find that fund flows
are negatively related to the trailing volatility of mutual fund returns, suggesting that investors pe-
nalize funds to some extent for high volatility. Tail risks, in contrast, should be more difficult to
detect and therefore, more feasible for fund managers to manipulate. In addition, skewness risk
may be harder to detect in historical returns.
It is also important to look risk beyond symmetric risk like variance because skewness risk can
help better reflect the true risk of a fund. Higher-moment risks are priced in the investor’s pricing
kernel (e.g., Harvey and Siddique (2000), Dittmar (2002)). This suggests that investors demand
compensation for skewness risk and that skewness risk can affect fund managers’ optimal asset
77
allocation decisions. For instance, the high left tail risk in hedge funds implies that hedge fund
investors bear a significant downside risk, which cannot be captured by the first two moments of
returns. Performance measures based on mean and variance can be manipulated through trading
strategies (Goetzmann et al. (2007)). If a fund manager frequently uses dynamic trading strategies
or writes covered calls and invests the proceeds in lower-risk assets, the fund appears to have low
risk and high risk-adjusted performance. If a fund manager shorts options and invests the premiums
at the risk-free rate, the fund appears to have low risk, but the downside risk of this trade can mean
substantial losses if the market plummets. As such, to infer the true risk hidden in the fund, one
needs to look beyond volatility and measure skewness risk.
2.3 Risky or Skewed Bets? A View from the Literature
How convexity affects a manager’s risk-taking behavior is well documented in the literature, but
both theory and the empirical results are mixed. Grinblatt and Titman (1989) and Carpenter (2000)
show that a fund manager increases portfolio risk when the incentive contract is out of the money.
In contrast, Kouwenberg and Ziemba (2007) show that loss-averse managers invest in a higher
proportion of risky assets in response to an increase in the incentive fee level. They further show
that investments of the managers’ own money in the fund can greatly reduce risk-taking.
Hodder and Jackwerth (2007) introduce an endogenous shutdown barrier and compare a fund
manager’s variance strategies from short-term and long-term perspectives. Since the liquidation
boundary looks like the strike of a knock-out call, unless the outside opportunities are high, a fund
manager will try to avoid the boundary and thus reduce risk-taking. When fund values are high, they
find a “Merton flats” region, in which the optimal volatility level of a fund is equal to the one without
an incentive fee. With a one-year horizon and a higher probability of termination, when fund values
are just below the high-water mark, the manager increases risky investments to increase the odds
of the option finishing in the money. These results confirm findings by Goetzmann, Ingersoll, and
Ross (2003). However, in a multiperiod framework, since the manager would consider potential
subsequent compensation based on fund performance and the expected value of termination is low,
78
risk-taking will be moderated. Panageas and Westerfield (2009) also analytically derive the same
conclusion that convexity does not necessarily lead to risk-taking given longer evaluation periods,
even for a risk-neutral fund manager.
Much of the empirical work supports the relation between performance and risk. Brown, Har-
low, and Starks (1996) find that midyear losers tend to increase fund risk in the latter part of the
year. Kempf and Ruenzi (2008) find that mutual funds adjust risk according to their relative ranking
in a tournament within the fund families. Brown, Goetzmann, and Park (2001) find that a fund man-
ager’s variance strategies are conditional on relative performance, instead of absolute performance
such as a high-water mark. Chevalier and Ellison (1997) conclude that mutual fund managers alter
fund risk toward the end of year owing to incentives to increase fund flows. Chevalier and Ellison
(1999) find that young funds increase systematic risk or herd owing to concerns about fund termi-
nation. Dass, Massa, and Patgiri (2008) show that high incentives induce managers to deviate from
the herd and to undertake unsystematic risk to improve short-term performance. On the other hand,
Koski and Pontiff (1999) show that funds using derivatives take less risk than nonusers. Panageas
and Westerfield (2009) and Aragon and Nanda (2011) show that high-water marks can offset the
convexity of a performance contract.
Overall, the assumption that convexity affects risk-taking is that managers are concerned about
the value of the incentive contract on the evaluation date. When the incentive contract is already in
the money before the evaluation date, managers tend to reduce variance. When the evaluation date
is near and the incentive contract is out of the money, managers have a strong incentive to trade risky
assets to improve short-term performance. However, managers may have a long-term perspective on
performance-based compensation and high expected values of continuation. In addition, managers
may have disincentives to liquidate funds owing to career concerns, reputation costs, or managerial
shares in funds. These arguments are used to explain recent findings that underperforming fund
managers can indeed decrease variance, even when convexity exists in their compensation.
6
6
See Chevalier and Ellison (1999), Fung and Hsieh (1997), Kempf, Ruenzi, and Thiele (2009), and Kouwenberg and
Ziemba (2007).
79
These mixed results prompt the need to look at fund managers’ positions regarding skewness
risk and to examine how convexity affects skewness risk. Positively skewed bets can be optimal
choices for losing funds because positively skewed bets offer managers a chance to rise in rela-
tive rankings and because incurred losses are too small to cause liquidation. Findings that top-
performing funds reduce risk suggest that fund managers are less inclined to take positively skewed
bets because the probability of forcing the value of the incentive contract out of the money is high in
the short run. Furthermore, the reduced impact of convexity around the kink proposes that convexity
and skewness risk are related.
Skewness may help reconcile the inconsistency found in the literature. Carpenter (2000) finds
that when either the fund’s returns are above the benchmark or the incentive fee level increases,
a risk-averse fund manager reduces fund volatility. Brown, Harlow, and Starks (1996) find that
losers increase risk. In contrast, Hu et al. (2011) show that a higher probability of termination and
increased convexity in compensation lead managers to increase portfolio risk at all level of prior
performance. I find that managers’ behavior on skewness risk exposures conditional on volatility
is supported by the data. This implies the importance of asymmetry in risk in the literature on
tournaments and convex compensation.
I contribute to the literature by linking the asymmetry in risk to convexity in incentives. I find
that managers with funds that perform poorly, even those facing liquidation, will take more posi-
tively skewed bets. This contradicts the finding in the literature that a manager is less willing to
gamble when the incentive option is further out of the money and has longer maturity. One possi-
ble explanation is that managers have a strong incentive to take positively skewed trades because
they are not penalized by losses, owing to the optionlike feature in compensation. Another possible
reason is that when the value of the outside opportunities is sufficiently high, managers voluntarily
choose to shut down and take positively skewed bets at the lower boundary. On the other hand, I
find that the top fund managers take more negatively skewed bets. However, when a top-performing
fund manager has an incentive to reduce skewness risk, investors can suffer substantial losses in the
long run, and this result is not documented by the existing models or empirics. It is also intriguing
that outperforming fund managers take negative skewed bets in the short run, because any occur-
80
rences of extreme downside events in the long run can jeopardize their careers. However, these top
managers are also the ones who can benefit from larger outside opportunities than their current man-
agement fees. It is also possible that outperforming fund managers underestimate the significance
of downside risk and overestimate the probability of collecting a succession of “pennies.” For ex-
ample, derivative hedging and momentum strategies are characterized as negatively skewed trades.
Fund managers might make consistent gains in the short run at the expense of large drawdowns in
fund values at longer horizons.
I also document that convexity does affect fund managers’ positions on skewness. How the
skewness risk in funds responds to an increase or decrease in fund values is an empirical question
and may be different from the relation between risk-taking and performance. The call-like feature
in incentives introduces asymmetry. When fund managers value an increase and a decrease in
compensation based on performance differently, they will engage in skewed bets.
2.4 Data
I study three types of actively managed investment funds: open-ended funds, closed-end funds,
and hedge funds. Actively managed funds are identified as funds whose names do not contain the
string “index” or whose fund objective is not indexed. I download the list of closed-end funds
from Morningstar and merge it with returns from the Center for Research in Security Prices (CRSP)
dataset by tickers and the beginning and ending dates of funds. Open-ended funds are selected from
the CRSP database. Hedge funds are from the Hedge Fund Research (HFR) database.
Fama and French (2010) document that a selection bias due to missing returns exists in the
CRSP mutual fund database before 1984. To be consistent in comparisons across fund types, both
closed-end funds and open-ended funds start in January 1984. The starting period for hedge funds
is January 1996. All datasets end in December 2008.
The literature has identified several ex-post conditioning biases in fund returns: survivorship
bias, back-fill bias, incubation bias, selection bias, and look-ahead bias. These biases may spuri-
ously increase fund mean returns and skewness and reduce fund variance and kurtosis. Data vendors
81
that I use provide survivorship bias free datasets for open-ended funds, closed-end funds, and hedge
funds. For open-ended funds, I drop both returns before the inception date and first-year returns after
the inception date to remove incubation and back-fill biases. For hedge funds, to remove incubation
and back-fill biases, I drop returns before the inception date and use the data field “date added to
database” in HFR dataset to clean out back-filled hedge fund returns. For closed-end funds, I drop
returns before the inception date to remove back-fill bias. Returns of open-ended funds are dropped
before the month that their styles are assigned to prevent look-ahead bias. However, my attempt to
limit the impact of ex-post conditioning biases may not be perfect. I drop funds with fewer than
12 monthly observations so that I can have a sufficient period to estimate fund skewness and keep
funds with aggressive tail risk-taking in the analysis. This introduces a small unavoidable survivor
selection bias.
Both bond and equity funds are included. Fund styles are classified by style codes provided in
the respective datasets. Closed-end fund styles are identified by Morningstar styles.
7
I use CRSP
style codes to group open-ended funds, and the details of classification codes used for each style are
described in Lin (2011).
8
Hedge funds are grouped by HFR main strategies.
9
To construct returns
of peer funds, I use all funds in the same style at any given months to calculate equal-weighted
returns.
7
Classification codes for equity closed-end funds are Global, Balanced, Sector, Commodities, Large/Mid/Small Cap,
Growth/Value, and Others. Classification codes for fixed-income closed-end funds are Global, Sector, Long Term, Inter-
mediate Term, Short Term, Government, High Yield, and Others.
8
Equity open-ended funds are classified as Index, Commodities, Sector, Global, Balanced, Leverage and Short, Long
Short, Mid Cap, Small Cap, Aggressive Growth, Growth, Growth and Income, Equity Income, and Others. Fixed income
open-ended funds are classified as Index, Global, Short Term, Government, Mortgage, Corporate, and High Yield.
9
HFR main strategies include Equity Hedge, Event-Driven, Fund of Funds, HFRI Index, HFRX Index, Macro, and
Relative Value. Descriptions of these investment strategies are available from HFR at http://www.hedgefundresearch.com.
82
2.5 Empirical Methods and Results
2.5.1 Changes in Tail Risks
2.5.1.1 Tail Risks on Relative Performance
Elton, Gruber, and Blake (2003) assume that the fund manager reexamines his or her position at
the end of 24 months and takes a position with respect to risk over the next 12 months. I follow
the same assumption on the 36-month evaluation periods to study changes in fund tail risks across
years.
10
Unlike annual tournaments, this assumption provides a more conservative view on manage-
rial behavior toward skewness risk and captures skewness changes across years. Fund managers are
rewarded for short-term performance
11
and thus have a stronger incentive to gamble when the eval-
uation date is continuous or short-term. The 36-month rolling window allows me to have sufficient
statistical power and reduce measurement errors to examine whether that tail risk-taking behavior of
fund managers is stable over time.
12
In addition, disjoint and unequal return and skewness periods
avoid sorting bias.
13
The rolling approach also permits a more precise examination of skewed bets
because managers are more likely to need to bet on asymmetric returns more often than to adjust
risky investments only in specific times of the year to achieve their desired outcome. The probability
of “winning” is tiny.
Funds are ranked in quintile groups based on the average of the difference between their returns
and peer fund returns in the past 24 months. The peer fund returns are constructed by averaging all
fund returns in the same style in every month. The 20% of funds that underperform their peers are
in Group 1. The group in the next quintile is Group 2, and so on. Group 5 consists of the top 20%
10
Similar to variance, the 36-month evaluation period allows a sufficient period to estimate skewness. I select 24-month
evaluation periods, and the results are qualitatively the same at the total fund level, but bond closed-end funds and equity
closed-end funds show statistical insignificance.
11
Short-term performance persistence, the increased turnover rate of fund managers, and the increased share turnover
of listed firms support the notion that fund managers tend to improve short-term performance.
12
Busse (2001) finds that estimation on monthly standard deviation can be biased upward due to daily return autocor-
relation. When funds that underperform in the first half of the year have higher autocorrelation in the second half of the
year, funds appear to be riskier in the second half of the year.
13
Schwarz (2011) addresses that sorting on return will also likely sort on risk levels when return and risk are from the
same time period. When funds that perform well in the first half of the year also have a higher first-half risk, the risk level
in the second half can decline simply due to mean reversion in volatility.
83
of funds that most outperform their peers. For each group, the average of the fund skewness around
the peer fund return in the next 12 months is calculated. In addition, I also calculate the percentage
change in relative skewness by dividing the relative skewness in the next 12 months by the relative
skewness in the previous 24 months. Note that relative skewness is standardized by the variance
around the peer fund return to adjust for the positive correlation between skewness and variance.
The advantages of using equal-weighted portfolios of funds within the same style as a bench-
mark are twofold.
14
First, portfolios of funds create a peer group of managers who pursue the same
style. In tournaments, fund managers are evaluated relative to the peer group, instead of a broad-
based benchmark. Second, portfolios of funds can capture a common component in the variation
over time and across funds within the group, and have the highest correlations with funds in the same
style and represent asset classes in that style because many fund managers in the same style make
similar bets or share similar trading strategies. As a result, systematic and idiosyncratic skewness
are properly identified with portfolios of funds as the benchmark.
Table 2.1 reports the pooled distribution of individual fund skewness and kurtosis around the
peer fund return in each quintile group in the next 12 months, as well as the changes in average
skewness and kurtosis around the peer fund return in the next 12 months relative to the previous 24
months for each quintile group. Panel A shows systematic decline in skewness from the bottom 20%
to the top 20% of funds for open-ended funds and hedge funds. The pattern is less systematic for
closed-end funds, but on average, the top 20% of funds have lower relative skewness than the bottom
20% of funds. Both the average and median funds of all fund types display the same systematic
pattern of skewness. On the other hand, kurtosis does not show any systematic tendency. I therefore
concentrate on the analysis of skewness in this study.
14
The issues between equal-weighted and value-weighted indexes come from market inefficiency, bid-ask bounce, or
autocorrelation. However, these issues are more pronounced in daily returns than monthly returns. In addition, an index
constructed by the median funds does not necessarily share high correlation with funds in the same style.
84
Table 2.1: Cross-Sectional Distribution of Fund Skewness and Kurtosis
This table reports the pooled distribution of fund skewness and kurtosis of individual funds around their peer fund returns in each
percentile group of closed-end funds, open-ended funds, and hedge funds. The data are from January 1984 through December 2008
for closed-end funds and open-ended funds. The data are from January 1996 through December 2008 for hedge funds. Each fund
type is ranked in quintile groups based on the average of the difference between fund returns and peer fund returns in the past 24
months up to montht. The bottom 20% is the group with the worst relative performance. The group in the next quintile is portfolio
P2, and so on. The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the same style in a
month. Closed-end fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by CRSP objective
classification codes. Hedge fund styles are based on HFR main strategies. In Panels A and B, fund skewness and fund kurtosis
around the peer fund return are computed based on a 12-month rolling period from montht+1. Panels C and D report the changes
in relative fund skewness and kurtosis in the following 12 months from montht+1, compared to those in the previous 24 months
up to montht. CEFs, OEFs, and HFs refer to closed-end funds, open-ended funds, and hedge funds, respectively.
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
Panel A: Fund Skewness
CEFs Bottom 20% 0.186 -0.831 -0.333 0.175 0.671 1.235
P2 0.018 -0.939 -0.473 0.012 0.506 0.994
P3 -0.046 -1.026 -0.525 -0.037 0.436 0.910
P4 -0.071 -1.068 -0.564 -0.050 0.427 0.888
Top 20% -0.002 -1.001 -0.509 0.016 0.486 0.941
OEFs Bottom 20% 0.048 -0.841 -0.393 0.051 0.497 0.935
P2 0.050 -0.859 -0.405 0.046 0.506 0.976
P3 0.024 -0.907 -0.434 0.023 0.486 0.968
P4 -0.009 -0.912 -0.459 -0.003 0.452 0.885
Top 20% -0.050 -0.913 -0.488 -0.043 0.386 0.814
HFs Bottom 20% 0.140 -0.750 -0.307 0.128 0.588 1.057
P2 0.072 -0.913 -0.403 0.093 0.574 1.042
P3 0.036 -0.935 -0.433 0.059 0.535 0.988
P4 0.028 -0.933 -0.451 0.025 0.510 0.997
Top 20% 0.006 -0.970 -0.490 -0.006 0.477 0.976
Bond CEFs Bottom 20% 0.073 -0.925 -0.451 0.058 0.592 1.128
P2 0.000 -0.980 -0.512 -0.025 0.500 0.999
P3 -0.058 -1.010 -0.529 -0.038 0.408 0.877
P4 -0.150 -1.176 -0.621 -0.115 0.329 0.794
Continued on next page
85
Table 2.1 (Continued)
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
Top 20% -0.096 -1.118 -0.628 -0.109 0.404 0.883
Bond OEFs Bottom 20% 0.117 -0.859 -0.346 0.098 0.604 1.128
P2 0.070 -0.943 -0.405 0.061 0.557 1.103
P3 0.035 -0.942 -0.431 0.010 0.507 1.074
P4 -0.026 -0.987 -0.483 -0.021 0.461 0.948
Top 20% -0.141 -1.155 -0.613 -0.102 0.377 0.826
Equity CEFs Bottom 20% 0.235 -0.784 -0.283 0.219 0.713 1.360
P2 0.085 -0.852 -0.406 0.087 0.540 1.038
P3 0.017 -0.975 -0.482 0.044 0.510 0.976
P4 -0.013 -0.981 -0.487 0.019 0.476 0.915
Top 20% 0.053 -0.918 -0.439 0.069 0.525 0.988
Equity OEFs Bottom 20% 0.045 -0.830 -0.384 0.055 0.490 0.916
P2 0.023 -0.866 -0.430 0.026 0.472 0.918
P3 0.031 -0.851 -0.421 0.029 0.479 0.916
P4 0.012 -0.858 -0.436 0.014 0.454 0.873
Top 20% -0.050 -0.906 -0.485 -0.045 0.380 0.808
Panel B: Fund Kurtosis
CEFs Bottom 20% 0.579 -1.137 -0.682 0.075 1.272 3.019
P2 0.428 -1.150 -0.712 0.007 1.075 2.599
P3 0.471 -1.146 -0.703 0.021 1.115 2.769
P4 0.489 -1.135 -0.668 0.064 1.182 2.685
Top 20% 0.412 -1.156 -0.714 -0.029 1.078 2.560
OEFs Bottom 20% 0.201 -1.214 -0.800 -0.172 0.775 2.049
P2 0.302 -1.186 -0.760 -0.090 0.919 2.286
P3 0.354 -1.201 -0.755 -0.046 0.993 2.414
P4 0.259 -1.196 -0.762 -0.087 0.858 2.121
Top 20% 0.158 -1.204 -0.794 -0.173 0.736 1.947
HFs Bottom 20% 0.270 -1.207 -0.786 -0.149 0.812 2.192
P2 0.410 -1.161 -0.723 -0.037 0.999 2.535
P3 0.401 -1.162 -0.719 -0.040 1.001 2.415
Continued on next page
86
Table 2.1 (Continued)
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
P4 0.427 -1.163 -0.723 -0.037 1.051 2.522
Top 20% 0.370 -1.184 -0.751 -0.054 1.007 2.452
Bond CEFs Bottom 20% 0.491 -1.136 -0.676 0.102 1.255 2.615
P2 0.447 -1.175 -0.727 0.031 1.171 2.658
P3 0.444 -1.165 -0.719 0.010 1.042 2.736
P4 0.524 -1.095 -0.650 0.082 1.220 2.823
Top 20% 0.569 -1.109 -0.644 0.168 1.310 2.716
Bond OEFs Bottom 20% 0.494 -1.185 -0.712 0.016 1.204 2.758
P2 0.533 -1.184 -0.721 0.050 1.291 2.934
P3 0.517 -1.218 -0.745 0.055 1.229 2.877
P4 0.493 -1.235 -0.735 0.078 1.168 2.684
Top 20% 0.517 -1.152 -0.690 0.045 1.168 2.739
Equity CEFs Bottom 20% 0.701 -1.149 -0.687 0.100 1.367 3.581
P2 0.425 -1.113 -0.672 -0.018 0.982 2.599
P3 0.441 -1.173 -0.714 0.009 1.088 2.643
P4 0.370 -1.187 -0.758 -0.061 1.034 2.551
Top 20% 0.374 -1.162 -0.714 -0.074 0.995 2.434
Equity OEFs Bottom 20% 0.168 -1.219 -0.812 -0.196 0.725 1.980
P2 0.218 -1.191 -0.780 -0.136 0.820 2.052
P3 0.207 -1.197 -0.773 -0.123 0.801 2.021
P4 0.176 -1.193 -0.776 -0.139 0.766 1.944
Top 20% 0.131 -1.212 -0.801 -0.189 0.707 1.896
Panel C: Change in Skewness
CEFs Bottom 20% -79.83 -400.52 -177.53 -85.96 31.84 324.83
P2 -109.39 -481.23 -212.36 -96.94 21.40 289.73
P3 -91.44 -472.52 -211.95 -97.81 24.47 309.25
P4 -119.06 -529.41 -233.58 -94.70 26.50 299.28
Top 20% -94.75 -471.87 -218.84 -91.69 19.23 276.49
OEFs Bottom 20% -82.37 -480.60 -212.80 -93.46 32.26 325.18
P2 -99.57 -493.38 -217.76 -97.86 22.39 301.26
Continued on next page
87
Table 2.1 (Continued)
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
P3 -82.06 -460.16 -202.30 -93.17 26.75 303.66
P4 -87.54 -489.21 -218.94 -95.63 30.16 312.95
Top 20% -94.31 -473.54 -210.37 -92.55 26.25 285.22
HFs Bottom 20% -93.30 -460.03 -198.55 -92.99 25.51 289.03
P2 -82.73 -478.73 -203.16 -91.67 29.54 320.19
P3 -93.83 -485.30 -211.15 -96.49 27.03 292.07
P4 -92.42 -475.42 -208.72 -93.02 26.76 296.37
Top 20% -94.15 -453.97 -198.45 -92.24 16.98 260.98
Bond CEFs Bottom 20% -92.97 -401.15 -188.57 -96.42 21.12 297.16
P2 -112.08 -462.57 -218.71 -100.93 17.30 265.61
P3 -92.61 -472.36 -204.80 -101.26 21.58 289.09
P4 -132.91 -558.32 -226.91 -100.38 10.87 300.27
Top 20% -111.81 -499.88 -227.70 -94.83 36.50 299.28
Bond OEFs Bottom 20% -80.12 -396.61 -171.67 -93.94 2.07 262.15
P2 -60.69 -446.05 -190.52 -92.35 10.43 307.31
P3 -61.24 -432.69 -183.93 -94.37 12.00 311.48
P4 -92.87 -445.17 -197.06 -96.50 7.76 281.31
Top 20% -72.97 -496.83 -209.81 -89.12 28.04 325.21
Equity CEFs Bottom 20% -51.87 -393.38 -167.66 -75.95 49.35 354.49
P2 -97.28 -477.92 -206.38 -94.07 25.25 338.66
P3 -122.83 -484.00 -224.91 -91.87 19.99 295.69
P4 -109.39 -500.79 -221.19 -87.76 24.16 287.97
Top 20% -85.76 -455.73 -216.37 -92.07 14.96 266.29
Equity OEFs Bottom 20% -86.83 -495.11 -217.78 -93.21 35.94 328.45
P2 -100.40 -496.99 -225.69 -98.21 30.07 309.12
P3 -94.49 -492.58 -218.34 -94.25 34.46 302.86
P4 -95.80 -499.37 -222.99 -97.12 31.51 303.26
Top 20% -92.52 -465.58 -209.52 -92.48 27.57 286.99
Panel D: Change in Kurtosis
CEFs Bottom 20% -115.90 -600.07 -198.71 -99.95 18.58 271.01
Continued on next page
88
Table 2.1 (Continued)
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
P2 -107.71 -557.27 -206.67 -99.78 22.83 279.02
P3 -88.01 -534.31 -201.06 -94.87 36.96 343.83
P4 -86.06 -556.15 -207.60 -98.17 32.61 320.40
Top 20% -93.47 -514.00 -204.59 -98.00 29.22 329.14
OEFs Bottom 20% -106.06 -546.92 -218.45 -101.22 31.43 291.77
P2 -104.64 -546.47 -212.47 -100.90 28.82 306.32
P3 -102.23 -535.60 -207.53 -102.25 20.76 298.12
P4 -102.04 -529.23 -214.17 -99.39 26.12 300.50
Top 20% -104.94 -538.40 -219.06 -101.33 33.54 317.13
HFs Bottom 20% -102.62 -486.68 -200.84 -101.64 21.21 276.31
P2 -109.65 -537.90 -216.80 -102.34 22.02 313.68
P3 -110.60 -534.36 -213.59 -101.68 23.12 305.95
P4 -99.77 -559.20 -210.87 -98.91 32.46 367.05
Top 20% -103.32 -503.38 -191.07 -95.31 22.07 289.60
Bond CEFs Bottom 20% -92.39 -547.53 -187.57 -99.99 13.68 226.91
P2 -117.41 -592.33 -214.17 -96.60 24.62 286.20
P3 -46.59 -506.75 -188.62 -91.90 48.03 370.47
P4 -122.52 -573.28 -225.18 -103.72 17.85 283.94
Top 20% -85.57 -544.99 -199.35 -95.33 39.51 361.18
Bond OEFs Bottom 20% -110.12 -518.53 -183.12 -104.16 -7.70 213.10
P2 -99.39 -556.17 -194.94 -106.54 6.45 293.45
P3 -107.21 -535.74 -195.35 -105.94 -4.77 270.82
P4 -83.24 -470.08 -182.79 -101.05 3.43 273.92
Top 20% -96.24 -516.77 -193.78 -98.65 18.30 316.28
Equity CEFs Bottom 20% -117.97 -664.22 -209.33 -99.35 33.48 361.59
P2 -98.38 -524.86 -205.21 -98.95 19.19 260.99
P3 -98.55 -538.42 -199.97 -96.28 28.89 343.83
P4 -90.37 -536.82 -202.69 -94.50 29.93 273.50
Top 20% -93.48 -485.56 -193.51 -99.64 21.63 347.83
Equity OEFs Bottom 20% -102.41 -544.04 -221.28 -99.47 38.72 309.74
P2 -110.16 -559.59 -222.70 -97.91 37.84 314.10
P3 -104.54 -536.18 -222.74 -100.99 32.21 300.67
Continued on next page
89
Table 2.1 (Continued)
Subsequent Skewness:
Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%
P4 -100.69 -542.39 -223.41 -98.38 33.99 313.72
Top 20% -105.69 -541.41 -220.00 -101.67 33.92 319.42
90
Figure 2.1, Figure 2.2, and Figure 2.3 show the differences in average fund skewness between
low- and high-performing quintile groups across investment funds.
15
Clearly, fund skewness fluc-
tuates over time, and the average fund skewness differs between outperforming funds and under-
performing funds, regardless of fund type. It is also evident that the skewness of low-performing
funds is more positive than that of high-performing funds during most time periods. However, the
changes in skewness in Figure 2.1-Figure 2.3 may come from the existing portfolios in the past 24
months or from trades in the subsequent 12 months.
One interesting observation from Figure 2.1-Figure 2.3 is that average fund skewness around the
peer fund over the 12-month periods for underperforming and outperforming groups changes sign
overtime. In particular, the gap in average fund skewness around the peer fund between the two
groups is large in the periods of 1987-1988, 1997-1999, and 2000-2001. We can relate those peri-
ods to market crashes. During the period of the technical bubble, underperforming fund managers
could ride the wave and bet on technical stocks to climb up the rankings, which exhibit positive
skewness. During the period of 2000-2001, underperforming funds show more negative skewness
and outperforming funds show more positive skewness. One possible explanation relates to the
findings in Dass, Massa, and Patgiri (2008) that high-incentive funds unload technical stocks and
invest more in fundamental stocks when the probability of a bubble burst increases. As such, outper-
forming fund managers take deviating strategies from the herd, and underperforming fund managers
continue to ride the technical bubble. Ex post, the losing funds bet on negatively skewed technical
stocks. Another possible explanation is the cumulative effect on a fund manager’s behavior. Unlike
the previous two crash periods, 2000-2001 is the postcrash period after the technical bubble burst.
After experiencing the occurrence of an extreme event, underperforming fund managers might be-
come more conservative and, for example, write calls to hedge their positions. On the other hand,
outperforming fund managers have gains from the past as a cushion against the odds of a succession
of steady losses. The undervaluation of assets in the postcrash period may offer outperforming fund
managers a great opportunity to gamble on assets, which are possibly rewarded with lottery-type
15
I only include figures for all funds. The patterns for bond funds or equity funds look very similar. All figures are
available on request. In addition, I also perform analysis on kurtosis but observe no systematic patterns.
91
Figure 2.1: Differences in Skewness Between High- and Low-Performing Groups -CEFs
This graph shows the average fund skewness around the peer fund return of the low- and high-performing
groups in closed-end funds. The data are from January 1984 through December 2008. Funds are sorted on the
average of past returns relative to their peer fund returns in the last 24 months up to montht into five quintile
groups. The bottom (top) 20% of funds are classified as low- (high-) performing groups. The average fund
skewness around the peer fund return is calculated over 12 monthly returns from month t + 1 on a rolling
basis from January 1986 to December 2008. The peer fund returns are calculated as the equal-weighted fund
returns by using all funds in the same style in a month. Close-end fund styles are identified by Morningstar
style codes. The black (red) line represents the average fund skewness of the low- (high-) performing groups,
respectively. Both bond and equity funds are included in the analysis.
returns. In addition, the possibility of an extreme downside event may be neglected ex ante. After
a series of crashes, the realization of the extreme downside event becomes possible. Consequently,
outperforming managers change trading behavior and become more risk averse. This is observed in
Figure 2.1-Figure 2.3. Across all fund types, outperforming funds consistently take more negative
skewness after 2002.
92
Figure 2.2: Differences in Skewness Between High- and Low-Performing Groups -OEFs
This graph shows the average fund skewness around the peer fund return of the low- and high-performing
groups in open-ended funds. The data are from January 1984 through December 2008. Funds are sorted
on the average of past returns relative to their peer fund returns in the last 24 months up to montht into five
quintile groups. The bottom (top) 20% of funds are classified as low- (high-) performing groups. The average
fund skewness around the peer fund return is calculated over 12 monthly returns from montht+1 on a rolling
basis from January 1986 to December 2008. The peer fund returns are calculated as the equal-weighted fund
returns by using all funds in the same style in a month. Open-ended fund styles are identified by CRSP
objective classification codes. The black (red) line represents the average fund skewness of the low- (high-)
performing groups, respectively. Both bond and equity funds are included in the analysis.
2.5.2 Multivariate Analysis
Table 2.2 reports the t-statistics on the differences in average fund skewness around the peer fund
returns for the next 12 months between the outperforming and underperforming groups. Paired t
values are adjusted for 11-lag autocorrelations due to overlapping data. The t-tests for closed-end
funds, hedge funds, and open-ended funds (-4.61, -1.67, -2.70, respectively) reject the null hypoth-
esis of no difference between the two groups. The tests for the top 20% and bottom 20% groups
93
Figure 2.3: Differences in Skewness Between High- and Low-Performing Groups -HFs
This graph shows the average fund skewness around the peer fund return of the low- and high-performing
groups in hedge funds. The data are from January 1996 through December 2008. Funds are sorted on the
average of past returns relative to their peer fund returns in the last 24 months up to montht into five quintile
groups. The bottom (top) 20% of funds are classified as low- (high-) performing groups. The average fund
skewness around the peer fund return is calculated over 12 monthly returns from month t + 1 on a rolling
basis from January 1998 to December 2008. The peer fund returns are calculated as the equal-weighted fund
returns by using all funds in the same style in a month. Hedge fund styles are based on HFR main strategies.
The black (red) line represents the average fund skewness of the low- (high-) performing groups, respectively.
in bond and equity funds also show a strong statistical difference. The negative t-statistics across
fund types suggest that outperforming funds have more negative skewness than underperforming
funds conditional on past relative performance. This finding suggests that fund managers execute
negatively (positively) skewed trades when their funds outperform (underperform) peer funds.
Figure 2.1-Figure 2.3 and Table 2.2 provide preliminary evidence that positions with respect
to skewness risk differ between the top and bottom extreme performing groups. However, one
concern is that the results are driven by volatility due to the possible correlation between volatility
94
Table 2.2: Comparison of Differences in Average Fund Skewness
This table shows the differences in average fund skewness around the peer fund
return between low- and high-performing groups in closed-end funds, open-ended
funds, and hedge funds. The data are from January 1984 through December 2008 for
closed-end funds and open-ended funds. The data are from January 1996 through
December 2008 for hedge funds. Funds are sorted on the average of past returns
relative to their peer fund returns in the last 24 months up to montht into five quin-
tile groups. The bottom (top) 20% of funds are classified as low- (high-) performing
groups. The average fund skewness around the peer fund return is calculated over
12 monthly returns from montht+1 on a rolling basis from January 1986 to De-
cember 2008 for closed-end funds and open-ended funds, and from January 1998 to
December 2008 for hedge funds. The peer fund returns are calculated as the equal-
weighted fund returns by using all funds in the same style in a month. Closed-end
fund styles are identified by Morningstar style codes. Open-ended fund styles are
identified by CRSP objective classification codes. Hedge fund styles are based on
HFR main strategies. CEFs, OEFs, and HFs refer to closed-end funds, open-ended
funds, and hedge funds, respectively. Paired t-values and p-values are adjusted for
11-lag autocorrelations.
Fund Type High - Low Performing Paired T-value Paired P-value Signed-Rank
Groups Skewness Test P-value
CEFs -0.187 -4.61 < 0.01 < 0.001
OEFs -0.061 -1.67 0.097 < 0.001
HFs -0.098 -2.70 < 0.01 < 0.001
Bond CEFs -0.168 -3.55 < 0.01 < 0.001
Bond OEFs -0.161 -2.71 < 0.01 < 0.001
Equity CEFs -0.216 -3.40 < 0.01 < 0.001
Equity OEFs -0.093 -2.51 0.012 < 0.001
and skewness or by persistence in skewness due to rolling estimates using overlapped periods. I
next use a regression approach to remove effects from contemporaneous volatility, lagged volatility,
and lagged skewness. To further examine the negative relation between average fund skewness in
the next 12 months and past relative performance, I run the following regression to study the change
in fund skewness around the peer fund relative to past fund performance:
Skew
t+12
i,t+1
= a
1
+b
1
1
Performance
t−23
i,t
+b
1
2
Vol
t+12
i,t+1
+b
1
3
Vol
t−23
i,t
(2.1)
+ b
1
4
Skew
t−23
i,t
+TimeFixedEffects
Performance
t−23
i,t
is the average of the differences in returns between fundi and its peer fund
based on 24 monthly returns up to montht. Vol
t−23
i,t
andSkew
t−23
i,t
are the volatility and skewness
95
around the peer fund return in the past 24 months. Vol
t+12
i,t+1
andSkew
t+12
i,t+1
are fund volatility and
skewness around the peer fund return during months t + 1 and t + 12. Note that my measure of
skewness is a close proxy to idiosyncratic skewness. Time fixed effects are year dummies, and
standard errors are clustered at the style level.
16
Time fixed effects would absorb common related
time factors in fund skewness around the peer fund returns, and clustering at the style level would
control for any seasonal patterns and autocorrelations in the subsequent fund skewness within the
style.
Panels A, B, and C show the regression results across closed-end funds, open-ended funds,
and hedge funds, respectively. The null hypothesis is that the coefficient on relatively performance
(b
1
1
) is zero. The t-statistics for closed-end funds, open-ended funds, and hedge funds are -5.73,
-2.69, and -3.31, respectively. A one standard deviation increase in performance relative to peer
benchmarks decreases average fund skewness in the next 12 months by -0.38, -0.13, and -0.19 for
closed-end funds, open-ended funds, and hedge funds, respectively.
17
In comparison, Lin (2011)
reports the skewness of the average fund as -0.610, -0.715, and -0.566 for closed-end funds, open-
ended funds, and hedge funds. The economic size is approximately 20-60% of fund skewness. Panel
D of Table 2.3 compares the coefficients on past relative performance. The p-value associated with
the test of differences in the coefficients on past relative performance across fund types is 0.427.
The p-values associated with test of differences in the coefficients on past relative performance
between any two fund types are larger than 0.1. These results imply that the incentive generated by
tournament rankings to take skewed bets is prevalent across fund industries.
16
For all regression results, I also cluster standard errors at the fund level and t-statistics are much more significant.
Results are available on request. Using monthly dummies yields results that are qualitatively unchanged. Controlling
for lagged volatility and skewness from t to t− 11 and t− 12 and t− 23 yield stronger results. Removing volatility
in the subsequent 12 months also yields stronger results. Applying the same regression on index open-ended funds
and clustering standard errors at the fund level show no significant results. Including the fourth quarter dummy yields
qualitatively unchanged results. The coefficients on the fourth quarter dummy are negative and significant for open-ended
funds and hedge funds. Fund managers might take trades hidden with left tail risks to improve Sharpe ratios toward year-
end because they have a strong incentive to attract fund flows and win performance evaluation at the end of the year. All
results are available upon request.
17
The one standard deviations of the difference in returns between individual funds and peer benchmarks are 5.65%,
2.54%, and 4.43% for closed-end funds, open-ended funds, and hedge funds, respectively.
96
Table 2.3: The Sensitivity of Fund Skewness to Lagged Relative Performance
This table shows the relation between fund skewness over the 12-month period from montht+1 and relative fund performance in
the prior 24 months up to montht. The regression is as follows:
Skew
t+12
i,t+1
= a
1
+b
1
1
Performance
t−23
i,t
+b
1
2
Vol
t+12
i,t+1
+b
1
3
Vol
t−23
i,t
+ b
1
4
Skew
t−23
i,t
+TimeFixedEffects
Performance
t−23
i,t
is the average of the difference between fundi’s returns and its peer fund returns based on 24 monthly returns
up to montht ,i.e., from montht−23 to montht. Vol
t−23
i,t
andSkew
t−23
i,t
are the second and the third moment of fundi’s returns
in excess of its peer fund returns in the past 24 months up to month t, denoting the fund volatility and skewness around the peer
fund return in month t. Vol
t+12
i,t+1
and Skew
t+12
i,t+1
are the fund volatility and skewness around the peer fund return from months
t+1 andt+12. The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the same style in a
month. Closed-end fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by CRSP objective
classification codes. Hedge fund styles are based on HFR main strategies. Panels A, B, and C summarize results for closed-end
funds, open-ended funds, and hedge funds. Panel D shows the test on the differences in coefficients on Performancei,t across
investment funds. CEFs, OEFs, and HFs refer to closed-end funds, open-ended funds, and hedge funds, respectively. T-statistics
are adjusted for clustering at the style level.
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
Panel A: Closed-End Funds
Performance
t−23
i,t
-0.067 -5.73 -0.082 -4.80 -0.058 -4.75
Vol
t+12
i,t+1
0.031 2.46 0.019 1.18 0.035 2.09
Vol
t−23
i,t
0.007 1.11 0.017 1.75 -0.002 -0.40
Skew
t−23
i,t
0.057 2.75 0.015 0.50 0.099 5.15
Panel B: Open-Ended Funds
Performance
t−23
i,t
-0.054 -2.69 -0.347 -1.87 -0.041 -1.85
Vol
t+12
i,t+1
0.009 2.61 -0.199 -3.18 0.013 6.28
Vol
t−23
i,t
0.004 0.60 0.127 1.79 0.001 0.10
Skew
t−23
i,t
0.047 5.29 0.066 5.45 0.041 3.31
Panel C: Hedge Funds
Continued on next page
97
Table 2.3 (Continued)
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
Performance
t−23
i,t
-0.042 -3.31
Vol
t+12
i,t+1
0.019 1.98
Vol
t−23
i,t
-0.004 -0.43
Skew
t−23
i,t
0.042 4.11
Panel D: Test Differences in Coefficients onPerformance
t−23
i,t
Across Fund Types
Test P-value
CEFs=OEFs=HFs 0.427
CEFs=OEFs 0.727
CEFs=HFs 0.200
OEFs=HFs 0.608
Bond CEFs=Bond OEFs 0.118
Equity CEFs=Equity OEFs 0.556
98
The results from Table 2.3 suggest the importance of skewed bets over risky trades. Underper-
forming fund managers wish to improve their ranking and do not wish to lose their job; outper-
forming fund managers undertake trades that sustain their rankings and avoid trades that can lower
them. The result of a manager’s bets on lottery-like returns as a result of underperformance to peers
implies that fund managers are more aggressive than can be measured by symmetric risk. Fund
managers are overly concerned about being out of the money and are inclined take unsystematic
and positively skewed bets to improve their rankings by the evaluation date. This contradicts with
the risk-shifting behavior of an underperforming manager found in the literature. Risk shift predicts
that losing managers will take negatively skewed bets to hedge their position or that they have no
incentive to take skewed bets to avoid job loss or reputation damage. In contrast, my finding on
managers’ reduction in portfolio skewness coincides with the risk-shift behavior of an outperform-
ing manager when fund performance is deep in the money. However, it requires a new economic
interpretation. The asymmetry in probabilities of winning versus losing implies that fund managers
are more reluctant to take trades that might drop their rankings by the evaluation date. This further
discounts the conjecture that successful fund managers take positively skewed bets against cumula-
tive gains from the past. The findings on skewed positions relative to peer funds shed light on the
importance of examining skewness risk in managed portfolios.
Table 2.2 and Table 2.3 show a similar pattern in statistical significance. It is not surprising
to observe that open-ended funds have the lowest significance among three fund types because
they face more strict mandates and regulations. Closed-end funds and hedge funds have stronger
significance because they use leverage and have high flexibility in what and how they trade. The
higher significance for closed-end funds than hedge funds may be explained by the fundamental
differences in clientele they serve. Hedge fund investors are more sophisticated and it should be
harder to hide skewed bets from such investors.
It is important to understand the implications of the above results in the long run. Positively
skewed bets imply a succession of losses along with a tiny chance of winning lottery-like returns.
Therefore, if a fund manager continues to take positively skewed bets, he or she may end up at the
top of the rankings in a year. The incentive to rise in the relative rankings pushes underperforming
99
managers to undertake positively skewed bets. However, investors who invest in underperforming
funds may suffer losses for a long period of time because a turnaround of fund performance is less
likely due to low probability of winning in positively skewed bets. Negatively skewed bets induce
significant downside risk to investors over the long haul, and the occurrence of an extreme downside
event may force managers to liquidate funds. However, negatively skewed trades offer a succession
of steady gains, and the chance of staying put is higher than the one predicted by variance strategies.
In addition, the incentive for outperforming fund managers to take negatively skewed bets may come
from high outside opportunity values even though they may end up blowing up the funds. They may
think they have enough cushion to bet against possible large drawdowns, overestimate the chance
of remaining at the top of the relative rankings, and underestimate the magnitude of the downside
risk. These implications are not addressed by the current literature on tournaments and managerial
risk-taking.
One notion that funds that perform poorly take positively skewed bets in the subsequent 12
months is due to mean reversion of fund skewness. For instance, one may argue that when out-
performing funds have positively skewed returns in the past 24 months, the subsequent skewness
should decline due to mean reversion. However, the coefficients on the lagged skewness are all
positive and significant. Moreover, skewness is calculated around the peer fund returns, instead of
the mean of fund returns. It is less likely that fund managers tend to mean revert to the average fund.
The issue on the measurement errors on skewness should be noted as well.
18
Generally speak-
ing, measurement errors on the dependent variables should yield insignificant regression results.
The results are surprisingly strong even only 12 observations are used to calculate skewness. An-
other issue is the effect of peer group misclassification. The motivation to use style classifications
provided by data vendors because they are publicly available and it is more likely that investors rely
on them to compare across funds. If fund managers have an incentive to outperform their peers,
they should follow which funds investors compare them with. Note that misclassification will im-
pact both performance and skewness relative to the peer benchmark and the average and skewness
18
I test fund skewness in the subsequent 36 and 60 months as a robustness check. The results are qualitatively un-
changed, but significance levels drop for 60 month periods.
100
of the peer group returns. For example, adding an outperforming fund shifts the average and skew-
ness of the peer benchmark to the right. According, relative skewness in the next 12 months will
shift further left, which further strengthens my results, unless funds are misclassified in a systematic
way.
2.5.2.1 Tail Risks on Relative Performance across Groups
To study the possibility that the incentive to take skewed bets may differ across fund managers, I
estimate equation (1) separately across groups sorted on the average of the differences in returns
between funds and peer benchmarks. Table 2.4 shows the relation between average fund skewness
over the 12-month period and relative fund performance in the prior 24 months across five groups.
Although the coefficients on relative performance across quintiles do not systematically increase
or decrease along quintiles, the top 20% group has a more negative slope than the bottom 20%
group. For instance, hedge funds have a coefficient of -0.045 for the top 20% and one of -0.030
for the bottom 20% group, but both groups have almost equal coefficients in closed-end funds. The
result for closed-end funds may be attributed to the combination of options held by the managers,
and its functional form may not be definitely convex or concave. Note that the coefficients on
relative performance in the middle group (P3) are -0.327, -0.632, and -0.258 for closed-end funds,
open-ended funds, and hedge funds, respectively. This indicates that funds around the kink of the
convexity take more skewed bets. Hu et al. (2011) document that compensation structure and
employment risk leads to an approximately U-shaped relation between fund managers’ risk choices
and their prior relative performance. It implies that fund managers might reduce risk and take
skewed bets simultaneously. Panel D of Table 2.4 reports the test of differences in the coefficients
on relative performance across five groups. When tested for differences in coefficients across five
groups, closed-end funds and hedge funds have p-values of less than 0.01 and 0.05, respectively.
Open-ended funds have p-values of less than 0.1. These results of the test on the differences in
coefficients on relative performance show that fund managers’ positions on skewed trades respond
differently to past relative performance.
101
Table 2.4: Fund Skewness to Lagged Relative Performance across Groups
This table shows the relation between fund skewness around the peer fund return over the 12-month period and relative fund
performance in the prior 24 months. I apply the following regression to each quintile group separately:
Skew
t+12
i,t+1
= a
1
+b
1
1
Performance
t−23
i,t
+b
1
2
Vol
t+12
i,t+1
+b
1
3
Vol
t−23
i,t
+ b
1
4
Skew
t−23
i,t
+TimeFixedEffects
Performance
t−23
i,t
is the average of the difference between fundi’s returns and its peer fund returns based on 24 monthly returns
up to montht ,i.e., from montht−23 to montht. Vol
t−23
i,t
andSkew
t−23
i,t
are the second and the third moment of fundi’s returns
in excess of its peer fund returns in the past 24 months up to month t, denoting the fund volatility and skewness around the peer
fund return in month t. Vol
t+12
i,t+1
and Skew
t+12
i,t+1
are the fund volatility and skewness around the peer fund return from months
t+1 andt+12. The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the same style in a
month. Closed-end fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by CRSP objective
classification codes. Hedge fund styles are based on HFR main strategies. Funds are sorted in quintile groups. The bottom 20%
is the group with the worst relative performance. The group in the next quintile is portfolio P2, and so on. Panels A, B, and C
summarize results of five quintile groups for closed-end funds, open-ended funds, and hedge funds. Panel D shows the test on
the differences in coefficients on Performance
t−23
i,t
across investment funds. CEFs, OEFs, and HFs refer to closed-end funds,
open-ended funds, and hedge funds, respectively. T-statistics are adjusted for clustering at the style level.
All Funds Bond Funds Equity Funds
Group Coeff T-value Coeff T-value Coeff T-value
Panel A: Closed-End Funds
Bottom 20% Performance
t−23
i,t
-0.016 -0.34 0.011 0.35 -0.025 -0.56
Vol
t+12
i,t+1
0.051 3.12 0.026 1.02 0.054 2.46
Vol
t−23
i,t
-0.014 -1.13 0.022 1.65 -0.018 -1.63
Skew
t−23
i,t
0.064 2.57 -0.010 -0.22 0.116 4.56
P2 Performance
t−23
i,t
-0.285 -3.42 -0.130 -1.41 -0.200 -1.60
Vol
t+12
i,t+1
0.012 0.80 0.019 0.58 0.024 1.50
Vol
t−23
i,t
0.006 0.41 0.009 0.34 -0.005 -0.38
Skew
t−23
i,t
0.032 1.13 -0.010 -0.28 0.061 1.74
P3 Performance
t−23
i,t
-0.327 -3.19 -0.148 -0.68 -0.015 -0.10
Vol
t+12
i,t+1
0.010 0.60 0.036 0.99 0.003 0.19
Continued on next page
102
Table 2.4 (Continued)
All Funds Bond Funds Equity Funds
Group Coeff T-value Coeff T-value Coeff T-value
Vol
t−23
i,t
0.016 1.03 -0.006 -0.26 0.012 1.03
Skew
t−23
i,t
0.027 0.75 0.023 0.42 0.050 0.94
P4 Performance
t−23
i,t
0.085 0.66 -0.029 -0.22 -0.193 -1.50
Vol
t+12
i,t+1
0.007 0.55 0.004 0.22 0.004 0.22
Vol
t−23
i,t
0.024 1.74 0.013 0.49 0.015 1.14
Skew
t−23
i,t
0.063 2.35 0.041 0.96 0.084 2.16
Top 20% Performance
t−23
i,t
-0.018 -0.73 -0.055 -1.24 -0.042 -3.54
Vol
t+12
i,t+1
0.033 3.15 0.017 0.85 0.046 4.85
Vol
t−23
i,t
0.000 -0.05 0.015 1.67 -0.011 -1.65
Skew
t−23
i,t
0.092 4.95 0.076 2.36 0.104 3.09
Panel B: Open-Ended Funds
Bottom 20% Performance
t−23
i,t
0.015 0.51 0.022 0.14 0.022 0.56
Vol
t+12
i,t+1
0.015 1.59 0.029 0.23 0.019 1.79
Vol
t−23
i,t
0.020 1.33 -0.003 -0.03 0.023 1.85
Skew
t−23
i,t
0.029 2.22 0.065 3.42 0.035 2.28
P2 Performance
t−23
i,t
0.097 0.85 -0.834 -2.12 0.193 2.56
Vol
t+12
i,t+1
0.003 0.27 -0.229 -1.66 0.013 2.45
Vol
t−23
i,t
0.000 -0.02 0.101 1.09 0.014 0.80
Skew
t−23
i,t
0.031 2.16 0.067 3.19 0.005 0.41
P3 Performance
t−23
i,t
-0.632 -2.07 -1.713 -2.21 -0.082 -2.22
Vol
t+12
i,t+1
0.006 0.59 -0.283 -1.98 0.012 1.84
Vol
t−23
i,t
0.012 0.72 0.208 1.70 0.001 0.05
Skew
t−23
i,t
0.051 3.82 0.051 2.47 0.042 3.72
P4 Performance
t−23
i,t
0.010 0.11 -0.870 -2.29 -0.021 -0.36
Vol
t+12
i,t+1
-0.014 -0.76 -0.389 -3.28 0.007 0.49
Continued on next page
103
Table 2.4 (Continued)
All Funds Bond Funds Equity Funds
Group Coeff T-value Coeff T-value Coeff T-value
Vol
t−23
i,t
0.041 2.21 0.324 2.34 0.007 0.61
Skew
t−23
i,t
0.035 2.50 0.037 1.62 0.025 1.30
Top 20% Performance
t−23
i,t
-0.062 -2.34 -0.149 -2.01 -0.058 -1.90
Vol
t+12
i,t+1
0.012 5.08 -0.333 -5.28 0.012 4.85
Vol
t−23
i,t
0.008 0.49 0.256 2.60 0.005 0.25
Skew
t−23
i,t
0.070 5.55 0.089 2.60 0.063 3.98
Panel C: Hedge Funds
Bottom 20% Performance
t−23
i,t
-0.030 -1.13
Vol
t+12
i,t+1
0.028 7.77
Vol
t−23
i,t
-0.014 -2.14
Skew
t−23
i,t
-0.010 -0.34
P2 Performance
t−23
i,t
-0.119 -0.85
Vol
t+12
i,t+1
0.001 0.04
Vol
t−23
i,t
0.004 0.21
Skew
t−23
i,t
0.033 2.00
P3 Performance
t−23
i,t
-0.258 -3.75
Vol
t+12
i,t+1
0.004 0.15
Vol
t−23
i,t
0.010 0.39
Skew
t−23
i,t
0.045 1.77
P4 Performance
t−23
i,t
0.051 0.66
Vol
t+12
i,t+1
0.024 1.60
Vol
t−23
i,t
0.003 0.18
Skew
t−23
i,t
0.050 1.83
Top 20% Performance
t−23
i,t
-0.045 -3.24
Vol
t+12
i,t+1
0.024 3.30
Continued on next page
104
Table 2.4 (Continued)
All Funds Bond Funds Equity Funds
Group Coeff T-value Coeff T-value Coeff T-value
Vol
t−23
i,t
-0.003 -0.53
Skew
t−23
i,t
0.093 2.36
Panel D: Test Differences in Coefficients on Performance
t−23
i,t
Across Quintile Groups and Between the Low-
and High-Performance Groups
Fund Type Test P-value
CEFs: Group 1=2=3=4=5 0.000
CEFs: Group 1=5 0.135
OEFs: Group 1=2=3=4=5 0.086
OEFs: Group 1=5 0.137
HFs: Group 1=2=3=4=5 0.016
HFs: Group 1=5 0.850
Bond CEFs: Group 1=2=3=4=5 0.107
Bond CEFs: Group 1=5 0.929
Bond OEFs: Group 1=2=3=4=5 0.000
Bond OEFs: Group 1=5 0.274
Equity CEFs: Group 1=2=3=4=5 0.023
Equity CEFs: Group 1=5 0.467
Equity OEFs: Group 1=2=3=4=5 0.037
Equity OEFs: Group 1=5 0.085
105
To examine how extreme performing groups take positions with respect to skewness risk, I con-
struct the fractional rank (FracRank) of relative performance for fundi as follows: FracRank
i,t,1
= Min(Rank
i,t
,0.2), FracRank
i,t,2
= Min(0.6,Rank
i,t
− FracRank
i,t,1
), FracRank
i,t,3
=
Rank
i,t
- FracRank
i,t,2
- FracRank
i,t,1
, where Rank
i,t
is fund i’s performance percentile in
month t. Sirri and Tufano 1998 and Huang, Wei, and Yan 2007 use the fractional rank of alphas
to impose the continuous piecewise linear relationship on fund flow performance sensitivities. The
regression specification is as follows:
Skew
t+12
i,t+1
= a
3
+b
3
1,q
3
X
q=1
FracRank
i,t,q
+b
3
2
Vol
t+12
i,t+1
+b
3
3
Vol
t−23
i,t
(2.2)
+ b
3
4
Skew
t−23
i,t
+TimeFixedEffects
Table 2.5 shows the sensitivity of fund skewness for the next 12 months on the fractional
rank of past relative performance. I hypothesize that the coefficient on the top fractional rank
(FracRank
i,t,3
) is different from that on the bottom fractional rank (FracRank
i,t,1
). The coef-
ficients on the bottom and top fractional ranks in open-ended funds are 0.133 and -0.527, and their
associated t-values are 1.00 and -2.10. Hedge funds have -0.306 and -0.530 for the coefficients on
the bottom and top fractional ranks, and the latter is statistically different from zero. Closed-end
funds do not show statistical significance on the coefficients of fractional ranks for the bottom and
top fractional ranks, but the middle fractional rank displays a negative and significant coefficient.
These results further reinforce the findings that an increase in relative performance will induce fund
managers to take negatively skewed trades and that managerial behavior on skewed bets can differ
among managers in the same industry.
106
Table 2.5: Regression of Fund Skewness on the Fractional Rank of Relative Performance
This table shows the relation between fund skewness around the peer fund return over the 12-month period and relative fund
performance in the prior 24 months. The regression is as follows:
Skew
t+12
i,t+1
= a
3
+b
3
1,q
3
X
q=1
FracRanki,t,q +b
3
2
Vol
t+12
i,t+1
+b
3
3
Vol
t−23
i,t
+ b
3
4
Skew
t−23
i,t
+TimeFixedEffects
The fractional rank (FracRank) for fundi is defined as follows: FracRanki,t,1 [Low]=Min(Ranki,t,0.2),FracRanki,t,2 [Mid]=
Min(0.6,Ranki,t−FracRanki,t,1), FracRanki,t,3 [High]= Ranki,t- FracRanki,t,2- FracRanki,t,1. Ranki,t is fund i’s per-
centile on relative performance in month t. The relative performance is measured as the average of the difference between fund
i’s returns and its peer fund returns based on 24 monthly returns up to montht ,i.e., from montht−23 to montht. Vol
t−23
i,t
and
Skew
t−23
i,t
are the second and the third moment of fund i’s returns in excess of its peer fund returns in the past 24 months up to
month t, denoting the fund volatility and skewness around the peer fund return in month t. Vol
t+12
i,t+1
and Skew
t+12
i,t+1
are the fund
volatility and skewness around the peer fund return from months t + 1 and t + 12. The peer fund returns are calculated as the
equal-weighted fund returns by using all funds in the same style in a month. Closed-end fund styles are identified by Morningstar
style codes. Open-ended fund styles are identified by CRSP objective classification codes. Hedge fund styles are based on HFR
main strategies. Panels A, B, and C summarize results for closed-end funds, open-ended funds, and hedge funds. Panel D shows the
test on the differences in coefficients on FracRanki,t,q across investment funds. CEFs, OEFs, and HFs refer to closed-end funds,
open-ended funds, and hedge funds, respectively. T-statistics are adjusted for clustering at the style level.
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
Panel A: Closed-End Funds
FracRanki,t,1 -0.563 -1.32 -0.061 -0.25 -0.675 -0.96
FracRanki,t,2 -0.292 -3.63 -0.331 -4.02 -0.256 -2.36
FracRanki,t,3 0.336 1.54 -0.017 -0.04 0.314 1.68
Vol
t+12
i,t+1
0.032 2.77 0.022 1.38 0.035 2.33
Vol
t−23
i,t
-0.001 -0.15 0.010 0.95 -0.008 -1.55
Skew
t−23
i,t
0.062 3.00 0.022 0.69 0.100 5.04
Panel B: Open-Ended Funds
FracRanki,t,1 0.133 1.00 -0.620 -1.25 -0.073 -0.63
Continued on next page
107
Table 2.5 (Continued)
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
FracRanki,t,2 -0.135 -2.39 -0.281 -3.32 -0.046 -1.72
FracRanki,t,3 -0.527 -2.10 -0.439 -1.19 -0.715 -2.93
Vol
t+12
i,t+1
0.010 2.98 -0.205 -3.32 0.014 7.28
Vol
t−23
i,t
0.011 1.24 0.127 1.82 0.007 0.64
Skew
t−23
i,t
0.047 5.40 0.070 5.63 0.039 3.52
Panel C: Hedge Funds
FracRanki,t,1 -0.306 -1.11
FracRanki,t,2 -0.122 -1.24
FracRanki,t,3 -0.530 -4.12
Vol
t+12
i,t+1
0.019 2.04
Vol
t−23
i,t
-0.003 -0.31
Skew
t−23
i,t
0.044 4.28
Panel D: Test Differences in Coefficients on Fractional Rank (FracRanki,t,q) Across Fund Types
Test P-value
FracRanki,t,1: CEFs=OEFs=HFs 0.001
FracRanki,t,1: CEFs=OEFs 0.010
FracRanki,t,1: CEFs=HFs 0.000
FracRanki,t,1: OEFs=HFs 0.322
FracRanki,t,3: CEFs=OEFs=HFs 0.121
FracRanki,t,3: CEFs=OEFs 0.096
FracRanki,t,3: CEFs=HFs 0.044
FracRanki,t,3: OEFs=HFs 0.582
FracRanki,t,1: Bond CEFs= Bond OEFs 0.293
FracRanki,t,3: Bond CEFs= Bond OEFs 0.164
FracRanki,t,1: Equity CEFs= Equity OEFs 0.068
FracRanki,t,3: Equity CEFs= Equity OEFs 0.102
108
Different types of fund managers face varying types of convexity in compensation, confront
different regulations, carry unique fund characteristics, and have different degrees of flexibility in
terms of what assets to trade and what strategies to execute. As such, managerial behavior on skewed
bets may differ across fund industries. Panel D of Table 2.5 compares the coefficients on fractional
ranks across fund types. P-values reject the hypothesis of equal coefficients on the top and bottom
fractional ranks across fund types. It shows that the response to take skewed bets in relation to past
fund performance relative to peer funds for extreme performers differs across fund types.
In particular, the pairwise comparisons of the bottom and top fractional ranks show that closed-
end fund managers’ tail risk-taking behavior is different from that of managers of open-ended funds
or hedge funds. Closed-end funds are actively managed, leveraged, and income oriented. Closed-
end fund managers do not worry about redemptions or cash positions in funds, trade illiquid assets,
use leverage, and invest more income-producing assets (e.g., bonds and preferred securities). Since
managerial skills are reflected in prices, an underperforming closed-end fund manager trades posi-
tively skewed assets aggressively to rise in the relative rankings. For example, he or she shifts from
high-yield assets to individual assets with lottery-like returns. If the manager wins the lottery, the
gap between share price and net asset value will narrow. On the other hand, the convexity faced
by closed-end fund managers comes from a combination of options instead of a strictly convex or
concave one. Therefore, closed-end fund managers have less incentive to remain at the top than
open-ended fund or hedge fund managers.
2.5.2.2 The Impact of Fund Characteristics on Skewed Bets - Size and Age
From the previous literature, we know that the size and age of a fund influence how fund managers
take risk when facing convexity in compensation (e.g., Chevalier and Ellison (1997, 1999)). Man-
agers of young funds are less likely to take unsystematic risk and deviate from the herd owing to
career concerns. Managers of larger funds have fewer incentives to take risk since the relation of
flow to fund performance is less convex. For these reasons, a natural extension of this study is to
examine how fund characteristics affect a manager’s behavior toward skewed positions. I perform
109
the following regression:
Skew
t+12
i,t+1
= a
4
+b
4
1
Performance
t−23
i,t
+b
4
2
Vol
t+12
i,t+1
+b
4
3
Vol
t−23
i,t
+b
4
4
Skew
t−23
i,t
(2.3)
+ b
4
5
Age
i,t
+b
4
6
Size
i,t
+b
4
7
Age
i,t
∗Performance
t−23
i,t
+ b
4
8
Size
i,t
∗Performance
t−23
i,t
+TimeFixedEffects
The main interests lie in the interaction terms between relative performance and age (size).
Results are reported in Table 2.6. All three fund types have negative interaction terms between
relative performance and age, but the significance is weaker for closed-end funds. Thus, managers of
young funds are less likely take skewed bets. Open-ended funds and closed-end funds have positive
and significant interaction terms between relative performance and size. Small funds, except hedge
funds, tend to take more skewed bets. Hedge funds display a marginally negative (-0.009) and
marginally significant interaction term between relative performance and size at the 10% level.
The first finding is consistent with results in Chevalier and Ellison (1999) and others. Young fund
managers have career concerns that create incentives to herd with other fund managers. Small fund
managers have a stronger incentive to take skewed bets because they face fewer restrictions and
trades undertaken have a smaller market price impact. However, the disparity in the impact of
size on taking skewness risk with respect to relative performance between hedge funds and other
fund types is more intriguing. Panel D of Table 2.6 further reports that the pairwise difference in
the interaction term between relative performance and size is not significant between open-ended
funds and hedge funds, but the differences across three fund types are significant at the 1% level.
This implies that hedge fund managers have more investment opportunities than other types of fund
managers. Hedge fund managers can invest in broad asset classes and have high flexibility in trading
strategies as documented in the literature.
110
Table 2.6: Impact of Firm Characteristics on Skewed Bets
This table shows the relation between fund skewness around the peer fund return over the 12-month period and relative fund
performance in the prior 24 months. The regression is as follows:
Skew
t+12
i,t+1
= a
4
+b
4
1
Performance
t−23
i,t
+b
4
2
Vol
t+12
i,t+1
+b
4
3
Vol
t−23
i,t
+b
4
4
Skew
t−23
i,t
+ b
4
5
Agei,t +b
4
6
Sizei,t +b
4
7
Agei,t∗Performance
t−23
i,t
+ b
4
8
Sizei,t∗Performance
t−23
i,t
+TimeFixedEffects
Performance
t−23
i,t
is the average of the difference between fundi’s returns and its peer fund returns based on 24 monthly returns
up to montht ,i.e., from montht−23 to montht. Vol
t−23
i,t
andSkew
t−23
i,t
are the second and the third moment of fundi’s returns
in excess of its peer fund returns in the past 24 months up to month t, denoting the fund volatility and skewness around the peer
fund return in month t. Vol
t+12
i,t+1
and Skew
t+12
i,t+1
are the fund volatility and skewness around the peer fund return from months
t+1 andt+12. The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the same style in a
month. Closed-end fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by CRSP objective
classification codes. Hedge fund styles are based on HFR main strategies. Agei,t is fundi’s age (the natural logarithm of months
since inception) in montht, andSizei,t is fundi’s size (the natural logarithm of TNA) in montht. Panels A, B, and C summarize
results for closed-end funds, open-ended funds, and hedge funds. Panel D shows the test on the differences in coefficients on
Performancei,txAgei,t and Performancei,txSizei,t across investment funds. CEFs, OEFs, and HFs refer to closed-end funds,
open-ended funds, and hedge funds, respectively. T-statistics are adjusted for clustering at the style level.
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
Panel A: Closed-End Funds
Performance
t−23
i,t
0.006 0.04 -0.431 -1.01 0.114 0.57
Vol
t+12
i,t+1
0.035 2.18 0.001 0.04 0.038 2.01
Vol
t−23
i,t
0.006 0.68 -0.002 -0.08 0.001 0.19
Skew
t−23
i,t
0.060 2.04 0.007 0.25 0.099 3.34
Agei,t -0.026 -0.58 -0.062 -0.59 -0.012 -0.30
Sizei,t 0.070 2.21 0.100 1.52 0.016 0.39
Performancei,txAgei,t -0.074 -1.49 0.216 3.72 -0.115 -1.90
Performancei,txSizei,t 0.054 2.16 -0.144 -1.14 0.067 1.97
Panel B: Open-Ended Funds
Performance
t−23
i,t
0.042 0.98 -0.428 -1.57 0.039 0.93
Vol
t+12
i,t+1
0.009 2.71 -0.198 -2.89 0.013 7.01
Continued on next page
111
Table 2.6 (Continued)
All Funds Bond Funds Equity Funds
Coeff T-value Coeff T-value Coeff T-value
Vol
t−23
i,t
0.005 0.63 0.139 1.91 0.001 0.12
Skew
t−23
i,t
0.047 5.22 0.067 4.65 0.041 3.43
Agei,t 0.001 0.11 0.010 0.24 0.003 0.27
Sizei,t -0.002 -0.82 -0.016 -2.55 0.002 0.87
Performancei,txAgei,t -0.061 -2.87 -0.009 -0.13 -0.054 -2.36
Performancei,txSizei,t 0.006 2.63 0.027 2.10 0.006 3.16
Panel C: Hedge Funds
Performance
t−23
i,t
0.248 5.84
Vol
t+12
i,t+1
0.020 1.92
Vol
t−23
i,t
-0.002 -0.21
Skew
t−23
i,t
0.044 2.94
Agei,t -0.049 -0.84
Sizei,t 0.000 -0.03
Performancei,txAgei,t -0.130 -6.18
Performancei,txSizei,t -0.009 -1.77
Panel D: Test Differences in Coefficients onPerformancei,txAgei,t andPerformancei,txSizei,t Across Fund Types
Test P-value
Performancei,txAgei,t : CEFs=OEFs=HFs 0.143
Performancei,txAgei,t : CEFs=OEFs 0.453
Performancei,txAgei,t : CEFs=HFs 0.149
Performancei,txAgei,t : OEFs=HFs 0.168
Performancei,txSizei,t : CEFs=OEFs=HFs 0.076
Performancei,txSizei,t : CEFs=OEFs 0.043
Performancei,txSizei,t : CEFs=HFs 0.032
Performancei,txSizei,t : OEFs=HFs 0.220
Performancei,txAgei,t : Bond CEFs= Bond OEFs 0.072
Performancei,txSizei,t : Bond CEFs= Bond OEFs 0.127
Performancei,txAgei,t : Equity CEFs= Equity OEFs 0.025
Performancei,txSizei,t : Equity CEFs= Equity OEFs 0.000
112
2.5.3 Convexity and Tail Risks
How fund managers respond to incentives is not only limited to idiosyncratic skewness risk, but
also systematic skewness risk. I sort funds into five portfolios based on various convexity measures
below. Forming portfolios allows me to construct continuous time series of returns for each portfolio
and avoid 36 survivorship bias.
2.5.3.1 Premium and Discount in Closed-End Funds
Closed-end funds can be traded at a premium or a discount because closed-end funds have a finite
number of shares traded on the exchange and do not allow redemptions. Discounts reflect a series of
option values that fund managers create to investors relative to net asset value. Cherkes, Sagi, and
Stanton (2009) show that closed-end fund investors buy an option on liquidity because the cost of
direct investments in illiquid assets is high. If a fund manager generates sufficient liquidity benefits
for investors, the fund will be traded at a premium. In addition, closed-end fund managers have
an option to signal their ability (see Berk and Stanton (2007)). Funds with a premium reflect high
skills in the manager or high future performance, and the relation between discounts and future net
asset value returns is nonlinear. On the other hand, closed-end fund investors hold another option to
liquidate (or open-end) their funds if fund market values are deep out of the money.
I use the premium and discount to measure the degree of convexity that a closed-end fund
faces. Unlike the explicit option contracts in hedge funds, a closed-end fund manager faces implicit
optionality in incentives. I calculate the closed-end fund premium and discount as follows:
Discount
i,t
= ((Price
i,t
−NAV
i,t
))/NAV
i,t
(2.4)
where Price
i,t
and NAV
i,t
are the closing price and net asset value of fund i in month t.
According to Pontiff (1995) and Cherkes, Sagi, and Stanton (2009), I can rewrite equation (6) as:
ΔDiscount
i,t
≈ R
O(NAV
i,t
)
i,t
=R
Price
i,t
−R
NAV
i,t
(2.5)
113
whereR
O(NAV
(
i,t))
i,t
denotes the fundi’s option return and the underlying instrument is net asset
value, and R
Price
i,t
and R
NAV
i,t
are the fund’s stock return and net asset value return, respectively.
Because the compensation for a closed-end fund manager is a fraction of the fund’s net asset value,
the manager has an implicit incentive to improve the fund’s option or stock return, i.e. reduce
discounts to avoid funds being arbitraged or liquidated (or open-end). Likewise, lowering discounts
can signal managerial skills or high net asset value return in the future since managerial skills
are priced in closed-end funds (e.g., Gruber (1996)). In doing so, closed-end fund managers can
receive high compensation, fringe benefits, an enhanced reputation, and outside opportunities. Fund
managers can take strategic actions to elevate the option value by leveraging and trading illiquid
assets or to improve the fund’s stock return by distributing dividends or repurchasing outstanding
shares to signal managerial ability, indicate future performance, and reduce asymmetric information
on traded assets.
Every month I sort funds by discounts into quintile groups for the next month. Then I construct
equal-weighted and value-weighted time series of returns for each group and calculate moments
on these portfolios of funds. To test the differences in skewness between two groups, I use the
Generalized Method of Moments estimation.
Table 2.7 reports the impact of premiums/discounts on fund tail risk. Returns and skewness fol-
low systematic patterns across both bond and equity closed-end funds. First, the larger the discount,
the higher the future returns. The bottom 20% of funds (most discounts) have higher expected re-
turns than the top 20% of funds (most premia). The differences are between 1.4% and 1.8% per
month. This coincides with findings in Thompson (1978) and Pontiff (1995).
Second, it is interesting to observe that the equity funds with largest discounts display more
negative skewness than those with premia, but bond funds show the opposite patterns. For equal-
weighted (value-weighted) returns, the bottom 20% of bond funds have a positive skewness of 0.93
(0.814), and the top 20% of funds have negative skewness of -1.26 (-1.296). On the other hand, for
equal-weighted (value-weighted) returns, the bottom 20% of equity funds have a negative skewness
of -1.131 (-1.109), and the top 20% of equity funds have a negative skewness of -0.485 (-0.689). The
114
Table 2.7: Convexity Impact on Tail Risks in CEFs -Premiums/Discounts
This table tabulates the mean, standard deviation, skewness, and kurtosis of equal-weighted (EW) and value-
weighted (VW) portfolios of funds in each quintile group. For every montht, I compute the premiums/discounts of
individual funds as follows and rank funds by discounts into quintile groups and assign their rankings to next month
t+1.
Discount
i,t
= ((Price
i,t
−NAV
i,t
))/NAV
i,t
(2.6)
Then monthly returns on individual funds in the same quintile group are averaged across funds to obtain the monthly
returns on an equal-weighted portfolio (EW). The monthly returns on a value-weighted portfolio (VW) are con-
structed by weighting individual fund returns in the same quintile group by assets. The EW (VW) portfolio of funds
in the bottom 20% is the group with the most discounts. The EW (VW) portfolio of funds in the next quintile is
portfolio P2, and so on. The moments are calculated based on the returns of each quintile group. I use GMM to test
the differences in skewness between the bottom 20% and top 20% groups. Panels A and B show results for bond
funds and equity funds, respectively.
Panel A: Bond Funds
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Bottom 20% 1.343 2.611 0.930 8.746 1.343 2.592 0.814 8.783
P2 0.886 2.429 0.262 4.874 0.964 2.285 0.218 4.410
P3 0.719 2.544 -0.510 1.672 0.752 2.427 -0.598 1.760
P4 0.393 3.540 -1.723 11.152 0.379 3.213 -1.497 8.840
Top 20% -0.131 3.811 -1.260 4.983 -0.079 3.472 -1.296 4.423
F-test of Differences in Skewness:
Top 20% - Bottom 20%: p-value<0.01 (EW) and p-value<0.01 (VW)
Panel B: Equity Funds
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Bottom 20% 1.569 5.636 -1.311 6.933 1.801 5.743 -1.019 5.348
2 1.376 5.213 -1.069 5.836 1.525 5.067 -1.165 5.325
3 0.649 4.262 -0.792 1.868 0.804 4.336 -0.747 1.479
4 0.292 4.403 -0.581 0.803 0.466 4.626 -0.417 0.856
Top 20% -0.078 5.077 -0.485 2.228 0.169 4.870 -0.689 1.192
F-test of Differences in Skewness:
Top 20% - Bottom 20%: p-value= 0.052 (EW) and p-value= 0.434 (VW)
115
opposite patterns in skewness across quintile groups may be attributed to the difference in skewed
distribution between bond and equity returns.
The F-test of differences in skewness indicates that the top and bottom 20% of funds are sig-
nificantly different for bond and equity closed-end funds when equal-weighted returns are used.
However, when group returns are weighted by net asset values, the difference in skewness between
the top 20% and bottom 20% of equity closed-end funds is not statistically different, but the top
20% of fund returns are still more positively skewed than the bottom 20% of funds.
2.5.3.2 Tournaments in Open-Ended Funds
The literature has documented managerial risk-taking behavior with respect to tournament objec-
tives. This induces implicit convexity that an open-fund manager can face. I sort funds by relative
performance as the average of the difference between fund returns and peer fund returns from the
beginning of a year to each quarter t within the same year into quintile groups and assign their
rankings to quarter t. For instance, I calculate the average of the difference between fund returns
and peer fund returns from January to February in the year 2000 and sort all funds by the average
difference. Then I assign their rankings to March 2000. Similarly, for the second, third, and fourth
quarter of a year, I use the average of the differences in returns between funds and peer funds up to
May, August, and November in the same year to sort funds and assign their rankings to respective
quarters. This sorting mechanism assumes that fund managers reevaluate their positions relative
to peers quarterly and care about the relative performance from the beginning of the year up to
each quarter.
19
The reason to use a quarterly frequency is to match the frequency of the fund flow-
performance relation commonly used in recent studies. For instance, Huang, Wei, and Yan (2007)
use quarterly data in CRSP to estimate the convex fund flow-performance relation curve, and point
out the issue of missing monthly total net asset values in CRSP before 1991. If fund managers face
the convex fund-flow performance relation on a quarterly basis, tournaments are not necessarily
restricted to be an end-of-year event.
19
I also run results by assuming that fund managers reevaluate their positions each month. The systematic patterns of
skewness are qualitatively unchanged. However, the difference in skewness between the top and bottom 20% groups is
not significant at the 10% level.
116
Table 2.8: Convexity Impact on Tail Risks in OEFs -Tournaments
This table tabulates the mean, standard deviation, skewness, and kurtosis of equal-weighted (EW) and value-weighted (VW) port-
folios of funds in each quintile group ranked by relative performance. In every quarter of a year, I calculate relative performance
as the average of the difference between monthly fund returns and monthly peer fund returns during the year before a quarter t and
assign the ranking to the quarter t. The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the
same style in a month. Closed-end fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by
CRSP objective classification codes. Hedge fund styles are based on HFR main strategies.
I sort funds into quintile groups. The bottom 20% is the group with the lowest relative performance. The portfolio of funds in the
next quintile is portfolio P2, and so on. Quarterly returns on individual funds in the same quintile group are averaged across funds
to obtain the quarterly returns on an equal-weighted portfolio (EW). The quarterly returns on a value-weighted portfolio (VW) are
constructed by weighting individual fund returns in the same quintile group by assets. The moments are calculated based on the
returns of each quintile group. I use GMM to test the differences in skewness between the bottom 20% and top 20% groups. Panels
A and B show results for bond funds and equity funds, respectively. Panel C show results for skewness decomposition of EW equity
fund returns across groups. I use beta-weighted exogenous factors constructed by the Carhart four factors as the benchmark for the
decomposition.
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Panel A: Bond Funds
Bottom 20% 0.402 1.372 -0.686 2.155 0.439 1.489 -0.639 4.828
P2 0.531 0.970 -0.482 2.239 0.542 0.949 -0.310 0.544
P3 0.509 0.919 -0.661 3.148 0.518 0.828 -0.306 1.474
P4 0.518 1.017 -0.694 2.323 0.484 0.952 -0.888 2.644
Top 20% 0.670 1.345 -0.603 1.031 0.647 1.268 -0.722 1.068
F-test of Differences in Skewness:
Top 20% - Bottom 20%: p-value=0.818 (EW) and p-value=0.884 (VW)
Panel B: Equity Funds
Bottom 20% 0.460 4.323 -0.989 2.289 0.424 4.321 -1.020 2.125
P2 0.585 3.799 -0.923 1.250 0.582 3.805 -0.826 1.299
P3 0.645 3.698 -0.830 0.952 0.641 3.739 -0.739 0.828
P4 0.791 3.746 -0.651 1.065 0.811 3.778 -0.486 0.994
Top 20% 1.049 4.157 0.283 2.613 1.014 4.140 0.315 2.902
Continued on next page
117
Table 2.8 (Continued)
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
F-test of Differences in Skewness:
Top 20% - Bottom 20%: p-value=0.032 (EW) and p-value=0.033 (VW)
Panel C: Skewness Decomposition of Equal-Weighted Equity Fund Returns
Systematic Idiosyncratic
———————————- —————–
Idiosyncratic Idiosyncratic
Skewness Coskewness Coskewness Skewness
Bottom 20% -0.989 -1.107 0.068 0.050
P2 -0.923 -0.947 0.021 0.003
P3 -0.830 -0.851 0.019 0.002
P4 -0.651 -0.683 0.032 0.000
Top 20% 0.283 -0.072 0.368 -0.013
118
Table 2.8 shows how skewness risk in funds is related to tournaments. Only equity funds show
statistically significant differences in skewness between the top 20% and bottom 20% of funds. The
skewness increases with past performance relative to peer funds, and this relationship is statistically
significant. In addition, expected returns increase across quintile groups for both bond and equity
funds (the top 20% of funds have 0.58% and 0.27% a higher average monthly return than the bottom
20% of funds for equal-weighted equity and bond funds, respectively), and the top 20% of funds
show higher future returns. This may be attributed to momentum effects or persistence in short-term
performance. It is intriguing that results in Table 2.8 display the opposite pattern to the findings in
the previous sections. The bottom 20% of funds exhibit the most negative total skewness, but they
undertake positively idiosyncratic-skewed bet. It implies that underperforming fund managers bet
on assets or trading strategies with negative systematic skewness. I use the Carhart four factors to
construct beta-weighted factors and apply skewness decomposition on the equal-weighted portfo-
lios of funds into coskewness, idiosyncratic coskewness, and idiosyncratic skewness.
20
Coskewness
measures the covariance between fund returns and market volatility. Idiosyncratic coskewness mea-
sures the covariance between idiosyncratic fund volatility and market returns. Both components are
systematic because they relate a fund’s skewness to the market portfolio’s skewness.
The decomposition results are reported in Panel C of Table 2.8. The skewness decomposition
introduces two types of systematically skewed bets - coskewness and idiosyncratic coskewness.
Coskewness measures the covariance between fund idiosyncratic returns and market volatility. Dur-
ing a state of high market volatility, adding an asset with negative coskewness will further reduce the
skewness of the portfolio. Investors dislike negative skewness and compensate it with high expected
returns (Harvey and Siddique (2000)). Idiosyncratic coskewness measures the covariance between
fund idiosyncratic volatility and market returns. Chabi-Yo (2009) shows that idiosyncratic volatil-
ity premium is directly related to idiosyncratic coskewness. When investors’ consumption is low,
adding an asset with high volatility in a bad economic state (i.e. negative idiosyncratic coskewness)
has a severe impact on the portfolio value. As such, an asset with positive idiosyncratic coskewness,
i.e. idiosyncratic volatility is low when market returns are low, idiosyncratic volatility premium is
20
For details, see Lin (2011).
119
positive. In other words, for stocks with positive idiosyncratic coskewness, the lower the volatility,
the higher the expected returns.
Consistent with the findings in the previous section, idiosyncratic skewness risk decreases from
the bottom 20% to top 20% groups. Across all groups, systematic skewness outweighs idiosyncratic
skewness. Except for the top 20% group, the rest of the groups have large negative coskewness.
The bottom 20% group exhibits the most negative coskewness. A manager can over-weight the
portfolio with assets that have negative coskewness or constantly use negatively coskewed bets to
reduce coskewness in funds. Examples include small stocks, value stocks, and momentum strategies
(see Harvey and Siddique (2000)). When a fund underperforms its peer group, the manager tilts
portfolios toward small or value stocks or undertakes momentum strategies. Because the trades may
yield higher expected returns due to size and value premiums or the momentum effect, losing funds
can take negatively coskewed bet to climb up the rankings. However, losers are protected from
betting on negative systematic skewness because the systematic shock will impact all funds if an
extreme event occurs. In contrast, the top 20% of funds take systematic bets on assets with large
positive idiosyncratic coskewness. This suggests two possible trading strategies. First, Chabi-Yo
(2009) derives idiosyncratic coskewss as a function of individual security call (put) option betas. As
such, when fund managers outperform their peers, they bet on options written on stocks with lottery-
like returns. Examples include options on small or value stocks. The bet on options with positive
idiosyncratic coskewness gives outperforming managers a chance to further improve performance,
and its downside is protected. The larger positive idiosyncratic skewness in the top 20% of funds
than other groups implies further out-of-money options, which exhibit more positive skewness and
thus, higher expected returns. Second, funds that outperform move into low volatility assets because
assets with positive idiosyncratic skewness and low volatility have high expected returns. This is
consistent with the extant literature that winning fund managers reduce fund volatility. In addition,
because betting on idiosyncratic coskewness is systematic, any loss on the bet will not change
relative rankings. This explains why the top 20% equity open-ended funds show the most positive
skewness in Table 2.8. Overall, the results show that fund managers take systematically skewed
bets to improve fund returns to climb up the rankings. Idiosyncratically skewed bets contribute to a
120
small fraction of fund skewness, and taking either positively or negatively idiosyncratically skewed
bets depend on fund performance relative to the peers.
2.5.3.3 Fund Flow-Performance Relation in Open-Ended Funds
Another type of convexity that an open-ended fund may face is the fund flow-performance relation.
Fund manager have an incentive to take trading strategies that increase assets under management
since their compensation is based on assets under management.
Following Chevalier and Ellison (1997), I perform kernel regression to estimate the expected
fund flows conditional on several control variables used in the literature. The quarterly fund flow is
calculated as follows:
Flow
i,t
= [TNA
i,t
−TNA
i,t−1
(1+R
i,t
)]/TNA
i,t−1
(2.7)
whereTNA
i,t
is the total net assets of the fund andR
i,t
is the reported return.
The control variables include fund age (the natural logarithm of the number of months since
fund inception), size (the natural logarithm of TNA), the expense ratio plus one-seventh of any
front-end load charges, the lagged fund flow, the performance measure at several lags, the lagged
fund total return volatility, and multiplicative terms in lagged fund age and lagged performance, and
time fixed effects. Following these earlier studies, the performance measures and the total return
volatility are estimated from the 36 months before quarter t.
For each month, I sort funds independently on conditional expected flows into five portfolios
and construct equal-weighted and value-weighted portfolios of funds accordingly. Note that Group
3 and 4 are funds that face the most convexity instead of the top quintile of funds in terms of expected
fund flows as Chevalier and Ellison (1997) document that funds around the kink will take more risk.
Therefore, I conduct the test of differences in skewness between the 60th percentile group and the
bottom 20% of funds.
121
Table 2.9: Convexity Impact on Tail Risks in OEFs -Flow-Performance
This table tabulates the mean, standard deviation, skewness, and kurtosis of equal-weighted (EW) and value-weighted (VW) port-
folios of funds in each quintile group. The quarterly fund flows are calculated as follows:
Flowi,t = [TNAi,t−TNAi,t−1(1+Ri,t)]/TNAi,t−1
whereTNAi,t is the total net assets of the fund andRi,t is the reported return.
I perform kernel regression to estimate the expected fund flows conditional on several control variables used in the literature. The
control variables include fund age (the natural logarithm of the number of months since fund inception), size (the natural logarithm
of TNA), the expense ratio plus one-seventh of any front-end load charges, the lagged fund flow, the traditional performance measure
at several lags, the lagged fund total return volatility, and multiplicative terms in lagged fund age and lagged performance, and time
fixed effects. The performance measures and the total return volatility are estimated from the 36 months before quartert. Equity
funds use Fama French three factors, and bond funds use three factors plus two bond factors-the Barclay U.S. government/credit
index and corporation bond index-to measure alphas.
I sort funds by conditional expected fund flows. Quarterly returns on individual funds in the same quintile group are averaged across
funds to obtain the quarterly returns on an equal-weighted portfolio (EW). The quarterly returns on a value-weighted portfolio (VW)
are constructed by weighting individual fund returns in the same quintile group by assets. The EW (VW) portfolio of funds in the
bottom 20% is the group with the least expected fund flows. The EW (VW) portfolio of funds in the next quintile is portfolio P2,
and so on. Funds in P3 and P4 face the most convexity in view of expected fund flows. The moments are calculated based on the
returns of each quintile group. I use GMM to test the differences in skewness between the bottom 20% and P4 groups. Then Panels
A and B show results for bond funds and equity funds, respectively.
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Panel A: Bond Funds
Bottom 20% 0.016 0.021 0.069 -0.414 0.017 0.020 -0.136 -0.495
P2 0.019 0.022 0.498 1.256 0.019 0.022 0.403 0.990
P3 0.016 0.022 -0.722 3.780 0.016 0.022 -0.742 3.668
P4 0.016 0.018 0.108 -0.186 0.017 0.018 0.034 0.068
Top 20% 0.018 0.020 0.676 2.462 0.018 0.020 0.758 2.510
F-test of Differences in Skewness:
P4 - Bottom 20%: p-value=0.940 (EW) and p-value=0.666 (VW)
Panel B: Equity Funds
Bottom 20% 0.026 0.075 -0.118 1.150 0.031 0.077 -0.052 1.394
Continued on next page
122
Table 2.9 (Continued)
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
P2 0.026 0.068 -0.464 0.928 0.029 0.071 -0.145 1.526
P3 0.026 0.067 -0.609 0.805 0.029 0.067 -0.470 0.709
P4 0.026 0.072 -0.835 1.395 0.027 0.069 -0.782 1.471
Top 20% 0.028 0.086 -0.399 1.003 0.029 0.083 -0.360 1.006
F-test of Differences in Skewness:
P4 - Bottom 20%: p-value=0.064 (EW) and p-value=0.080 (VW)
123
Table 2.10: Convexity Impact on Tail Risks in OEFs -Tournament and Flow-Performance
This table tabulates the mean, standard deviation, skewness, and kurtosis of equal-weighted (EW) and value-weighted (VW) port-
folios of funds in each quintile group. The quarterly fund flows are calculated as follows:
Flowi,t = [TNAi,t−TNAi,t−1(1+Ri,t)]/TNAi,t−1
whereTNAi,t is the total net assets of the fund andRi,t is the reported return.
I perform kernel regression to estimate the expected fund flows conditional on several control variables used in the literature. The
control variables include fund age (the natural logarithm of the number of months since fund inception), size (the natural logarithm
of TNA), the expense ratio plus one-seventh of any front-end load charges, the lagged fund flow, the traditional performance
measure at several lags, the lagged fund total return volatility, and multiplicative terms in lagged fund age and lagged performance,
and time fixed effects. Following these earlier studies, the performance measures and the total return volatility are estimated from
the 36 months before quarter t. Equity funds use Fama French three factors, and bond funds use three factors plus two bond
factors-the Barclay U.S. government/credit index and corporation bond index-to measure alphas.
I sort first by expected fund flows and then by tournament ranking. Tournament ranking is determined by the average of the
difference between fund returns and the peer fund returns during the year before the quarter t and assign the ranking to the quarter t.
The peer fund returns are calculated as the equal-weighted fund returns by using all funds in the same style in a month. Close-end
fund styles are identified by Morningstar style codes. Open-ended fund styles are identified by CRSP objective classification codes.
Hedge fund styles are based on HFR main strategies.
I generate portfolios based on two-way sorts first by fund flows and then by tournament. At the end of each quarter, all funds are
first sorted into quintiles by expected fund flows predicted from the previous quarter. Funds in each quintile are then assigned to
one of five equal-sized portfolios based on the cumulative average returns in excess of peer fund returns. I form intersections of
the above two variables to form 25 portfolios. The first and second columns show the ranking by double sorting. The first (second)
column shows the ranking for expected flows (tournament). P1P1 (P5P5) represents the portfolio of funds that fall into the lowest
(highest) expected flow quintile each month and the lowest (highest) tournament quintile. Then I use all fund returns in the same
group to create time series of EW and VW returns. The moments are calculated based on the returns of each group. Panels A and
B show results for bond funds and equity funds, respectively.
Equal-Weighted Value-Weighted
Flow Tourn Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Panel A: Bond Funds
P1 P1 -0.003 0.034 -0.670 1.377 0.000 0.035 -0.613 2.661
P1 P2 0.012 0.020 -0.135 0.408 0.012 0.023 -0.030 0.274
P1 P3 0.017 0.019 0.063 -0.215 0.017 0.019 -0.052 -0.396
P1 P4 0.017 0.019 0.181 -0.029 0.016 0.020 -0.174 0.687
P1 P5 0.028 0.027 0.751 1.070 0.028 0.029 0.788 1.491
Continued on next page
124
Table 2.10 (Continued)
Equal-Weighted Value-Weighted
Flow Tourn Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
P2 P1 0.005 0.018 -0.424 0.295 0.005 0.023 -0.297 -0.401
P2 P2 0.012 0.017 -0.164 0.589 0.012 0.019 -0.041 -0.109
P2 P3 0.019 0.021 0.905 1.644 0.019 0.021 0.818 1.397
P2 P4 0.016 0.018 -0.137 1.135 0.018 0.018 -0.200 1.113
P2 P5 0.022 0.018 0.594 0.678 0.021 0.020 0.287 0.586
P3 P1 0.006 0.017 -0.209 -0.731 0.004 0.022 -0.651 0.438
P3 P2 0.012 0.014 0.099 -0.239 0.011 0.016 -0.275 -0.172
P3 P3 0.016 0.022 -0.726 3.650 0.017 0.022 -0.793 3.910
P3 P4 0.015 0.016 0.270 -0.280 0.015 0.018 0.132 -0.142
P3 P5 0.022 0.017 0.368 0.429 0.021 0.018 0.407 -0.284
P4 P1 0.007 0.020 -0.101 -0.580 0.008 0.022 0.244 -0.526
P4 P2 0.012 0.018 0.419 0.258 0.013 0.020 0.156 -0.208
P4 P3 0.016 0.017 0.170 -0.357 0.017 0.018 0.174 -0.353
P4 P4 0.014 0.017 0.166 1.348 0.014 0.017 0.225 1.455
P4 P5 0.021 0.020 1.196 2.587 0.021 0.020 0.709 1.689
P5 P1 0.009 0.018 -0.060 -0.620 0.009 0.020 -0.284 0.702
P5 P2 0.014 0.015 0.151 -0.382 0.015 0.015 0.463 -0.116
P5 P3 0.017 0.020 0.772 2.785 0.017 0.020 0.831 2.825
P5 P4 0.018 0.014 0.569 -0.601 0.017 0.014 0.777 -0.258
P5 P5 0.024 0.018 0.846 -0.010 0.024 0.018 1.127 0.811
Panel B: Equity Funds
P1 P1 -0.016 0.092 -0.517 -0.240 -0.009 0.098 -0.399 -0.104
P1 P2 0.016 0.077 -0.388 0.601 0.015 0.078 -0.440 0.611
P1 P3 0.026 0.077 0.034 1.128 0.028 0.078 -0.159 0.943
P1 P4 0.036 0.079 0.275 1.785 0.037 0.079 0.419 2.633
P1 P5 0.054 0.089 1.010 3.705 0.057 0.095 1.873 8.087
P2 P1 -0.004 0.087 -0.728 0.366 -0.004 0.089 -0.672 0.416
P2 P2 0.020 0.067 -0.569 0.728 0.022 0.069 -0.500 0.654
Continued on next page
125
Table 2.10 (Continued)
Equal-Weighted Value-Weighted
Flow Tourn Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
P2 P3 0.023 0.065 -0.266 0.514 0.023 0.064 -0.538 0.859
P2 P4 0.035 0.065 -0.421 1.582 0.037 0.066 -0.155 1.626
P2 P5 0.057 0.084 0.830 4.186 0.055 0.085 0.586 3.393
P3 P1 -0.003 0.086 -0.692 0.322 0.000 0.088 -0.918 0.698
P3 P2 0.018 0.068 -0.836 0.967 0.019 0.069 -0.836 0.940
P3 P3 0.022 0.064 -0.470 0.666 0.021 0.064 -0.511 0.819
P3 P4 0.032 0.065 -0.509 0.772 0.035 0.065 -0.277 0.886
P3 P5 0.050 0.078 0.419 2.067 0.049 0.079 0.549 2.505
P4 P1 -0.004 0.088 -0.770 0.573 -0.002 0.087 -0.861 0.861
P4 P2 0.019 0.073 -0.940 1.246 0.019 0.073 -0.887 1.144
P4 P3 0.028 0.068 -0.576 0.722 0.029 0.068 -0.596 0.682
P4 P4 0.032 0.068 -0.740 1.796 0.033 0.069 -0.693 1.834
P4 P5 0.052 0.076 0.405 1.898 0.051 0.077 1.101 4.250
P5 P1 -0.015 0.115 -0.748 0.915 -0.009 0.111 -0.848 0.919
P5 P2 0.015 0.088 -0.732 1.004 0.016 0.087 -0.779 1.104
P5 P3 0.026 0.084 -0.513 0.894 0.025 0.082 -0.507 1.104
P5 P4 0.038 0.080 -0.043 0.903 0.037 0.079 -0.072 1.013
P5 P5 0.068 0.102 1.144 4.381 0.062 0.101 1.422 5.237
126
Table 2.9 reports how the fund-flow relation influences fund skewness risks. Interestingly, the
ex-post ranking on expected fund flows shows that funds exhibit flat average returns across five
groups. However, the equity funds that face the most convexity (P3 and P4) show high negative
skewness. When using equal-weighted (value-weighted) returns, P3 and P4 show a skewness of -
0.609 (-0.47) and -0.835 (-0.782), compared to a skewness of -0.118 (-0.052) for the bottom 20% of
funds. The spread in skewness between P4 and the bottom 20% of funds is 72 basis points a month,
which is significant. One possible explanation is that the fund managers with the highest sensitivity
of fund flows to past performance (P4) take more negatively coskewed trades than those facing
the least convexity from flows. A fund manager who faces most convexity from flows will take
negatively coskewed trades because positive risk premiums are associated with negative coskewness
risk and systematic skewness risk affects the entire fund style. The manager has a strong incentive
to improve fund performance. On the other hand, I do not find any systematic patterns for bond
open-ended funds.
Since the literature has documented both tournament and fund flows as incentives for fund
managers to take risk, the impact on convexity from both effects may be amplified to induce fund
managers’ tail risk-taking behavior. As such, I generate portfolios based on two-way sorts-first by
fund flows and then by tournament. At the end of each quarter, all funds are first sorted into quintiles
by expected fund flows predicted from the previous quarter. Funds in each quintile are then assigned
to one of five equal-sized portfolios based on the cumulative average returns in excess to peer fund
returns. I form intersections of the above two variables to form 25 portfolios. For instance, the
upper left entry in Table 2.10 represents the portfolio of funds that fall into the lowest tournament
quintile and the lowest expected flow quintile each month. All funds are equal-weighted or value-
weighted in a portfolio. Table 2.10 shows that no dominant effects from one of the two sources
as the patterns hold the same as the one-dimensional sorting. After controlling for expected fund
flows, I observe a systematic pattern in expected skewness-that is, for each group sorted by expected
flows, I find a monotonic increase in returns and skewness from the bottom 20% to the top 20% of
funds. After controlling for tournaments, funds around the kink exhibit more negative skewness,
127
which is consistent with results from Table 2.9. Incentives from tournaments and fund flows should
be viewed independently.
2.5.3.4 High-Water Marks in Hedge Funds
Hedge fund managers face high-water mark provisions, and their compensation structure is thus
convex. The high-water mark of each fund is initially determined by the cumulative return of the
first 12 months. Then the high-water mark is reset by the maximal returns achieved to date. If
the cumulative returns are negative, hedge funds managers need to cover these losses first before
incentive fees are paid. Following Getmansky, Lo, and Makarov (2004), my measure of the high-
water mark is updated every month as follows:
HWM
i,t
= Min(0,HWM
i,t−1
+R
i,t
) (2.8)
where R
t
is fund i’s return at t. The gamma of an option based on the Black Scholes formula
is
N
0
(d1)
Sσ
√
T
, whereN
0
(.) is the standard normal probability distribution function,S is the stock price,
X is the strike price, r is the risk-free rate, σ is the stock’s volatility, T is the option’s time to
maturity, andd1 =
(log(S/X)+(r+1/2σ
2
)T
σ
√
T
. Note that the moneyness of an option is mapped to gamma
through a concave function and thus can represent the degree of convexity for each fund when other
parameters are held constant.
21
The moneyness att+1 is calculated as:
ln(R
i,t+1
/HWM
i,t
) (2.9)
whereR
t+1
is fundi’s return att+1. I divide funds into the bottom 20% group, three medium
groups, and top 20% group based on the log moneyness. Fund managers around the kink have the
highest gammas, and thus, face the most convexity. Accordingly, they have the strongest incentive
to increase the odds of the option finishing in the money.
21
Sorting funds based on alternative measures, such as (Rt+1−HWMt)/HWMt and (Rt+1−HWMt)/Rt+1,
yields the same rankings of funds.
128
Table 2.11: Convexity Impact on Tail Risks in HFs -High-Water Marks
This table tabulates the mean, standard deviation, skewness, and kurtosis of equal-weighted (EW) and value-
weighted (VW) portfolios of funds in each quintile group ranked by high-water mark percentiles. I sort funds
by the log moneyness. For every montht, the moneyness of an option is calculated as:
ln(R
i,t+1
/HWM
i,t
)
whereR
t+1
is fundi’s return att+1.
The high-water mark of each fund is initially determined by the cumulative returns. Following Getmansky, Lo, and
Makarov (2004), the high-water mark is updated every month as follows:
HWM
i,t
= Min(0,HWM
i,t−1
+R
i,t
)
The bottom 20% is the group with the lowest moneyness. The portfolio of funds in the next quintile is portfolio P2,
and so on. Monthly returns on individual funds in the same quintile group are averaged across funds to obtain the
monthly returns on an equal-weighted portfolio (EW). The monthly returns on a value-weighted portfolio (VW) are
constructed by weighting individual fund returns in the same quintile group by assets. The moments are calculated
based on the returns of each quintile group. I use GMM to test the differences in skewness between the bottom 20%
and top 20% groups.
Panel A: Bond Funds
Equal-Weighted Value-Weighted
Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis
Bottom 20% 0.629 1.481 -0.124 4.667 0.719 1.649 0.500 5.639
P2 0.816 1.510 -0.390 3.006 0.869 1.792 0.297 8.670
P3 0.906 1.770 0.031 1.675 1.049 1.962 0.799 4.000
P4 0.917 2.371 0.213 5.739 1.096 2.395 1.232 4.805
Top 20% 0.776 2.468 -0.244 2.982 0.951 2.468 0.992 5.709
F-test of Differences in Skewness:
P4 - Bottom 20%: p-value=0.705 (EW) and p-value=0.308 (VW)
129
Table 2.11 shows the results on skewness risks with respect to log moneyness. Future returns
increase with log moneyness, except for the top 20% group. The returns of the top 20% group
are more sensitive to the weighting scheme. This might suggest that small and value size funds in
the top 20% group have lower expected returns. The middle two groups (P3 and P4) exhibit more
positive skewness than the other groups. The bottom two groups exhibit more negative skewness
than the top two groups. In particular, the P4 group shows a positive skewness of 0.213 (1.232)
for equal-weighted (value-weighted) returns. This might imply that when a hedge fund faces most
convexity, the fund manager will take positively skewed bets - both systematic and idiosyncratic
ones. Both systematic and idiosyncratic skewed bets on assets with lottery-like returns improve the
fund’s chance to be in the money, but losses due to idiosyncratic skewness might be too small to
liquidate the fund, and systematic skewness risk applies to all funds in the same style. However,
sorting on log moneyness does not produce statistically strong skewness differentials among equal-
weighted and value-weighted portfolios.
The standard portfolio theory suggests that tail risks are diversified away at a faster rate than
volatility.
22
Brown, Gregoriou, and Pascalau (2012) study funds of hedge funds and document that
overdiversification leads to increased systematic tail risk exposures. One potential interpretation of
the analysis in this section is that because idiosyncratic tail risks are diversified away in portfolios
of funds, my findings describe the convexity affects systematic tail risks only. However, empirical
evidence has documented that idiosyncratic tail risks are still present in portfolios of funds. Lin
(2011) shows that idiosyncratic skewness contributes large portions of fund skewness, between
31% (open-ended funds) to 44% (hedge funds) for investment bond and equity funds at the style
level.
In summary, results from Tables 2.7 to 2.11 imply convexity in incentives affects how a fund
manager takes a position with respect to skewness risk. More importantly, the results imply that a
fund manager takes a systematic-skewed bet when he or she faces more convexity, and the direction
of the skewed trade depends on the type of convexity. Risk premiums associated with systematic
skewness risk offer fund managers an incentive to improve the option in compensation, but any
22
The kth moment of portfolios of funds isO(1/n
k−1
).
130
negative systematic shock will impact the entire style group. From the sign and magnitude of sys-
tematic skewness risk at the portfolio level, we can infer fund managers’ positions on systematic-
skewed bets. For example, when the portfolios of funds exhibit minimal idiosyncratic skewness
and coskewness is the main contributor to total fund skewness, such as equity open-ended funds,
negative systematic (co)skewness implies negatively systematic-skewed bets. Combined with the
findings in section 5.2, I find that fund managers engage both systematic and idiosyncratic skewed
bets. In view of variance strategies, sorting on various convexity measures shows an approximately
U-Shaped relation between fund managers’ risk choices and their prior performance. This is consis-
tent with findings by Hu et al. (2011). An interesting question for further research is the interaction
between risk and skewness.
2.6 Conclusion
This paper extends the literature on managerial incentives and risk-taking behavior to skewness risk.
Two fundamental questions are addressed. First, do fund managers take positions with respect to
skewness risk as a function of rankings in past performance relative to their peers? Do open-ended,
hedge, and closed-end fund managers behave differently?
I show that when a fund manager underperforms his or her peers, he or she is more likely to take
positively skewed trades. This is quite intuitive since betting on lottery-like returns can significantly
increase fund performance and relative rankings if successful, but are more likely to yield steady
losses. On the other hand, if a fund manager has been successful, he or she is more willing to
take negatively skewed bets since the probability of true volatility and downside risk is tiny and the
probability of steady profits is higher. However, the underestimated downside risk can wipe out past
gains and blow up the fund if an extreme event occurs.
In addition, I show that fund managers take positions with respect to skewness risk in response to
the convexity that they face. More discounted closed-end fund returns are more negatively skewed.
The equity open-ended funds with the worst relative performance have more negatively skewed re-
turns than the outperforming funds. This indicates that equity open-ended funds gamble on assets or
131
trading strategies associated with negative systematic skewness in addition to idiosyncratic skewed
bets. The open-ended funds that face most convexity from expected flows exhibit more negative
skewness than those facing least convexity. This is also attributed to negatively systematic-skewed
bets. Double sorting on relative performance and expected fund flows does not identify a dominant
effect from either source. Incentives from tournaments and fund flows should be examined sepa-
rately. I also find that hedge fund tail risks are related to convexity induced by high-water marks
relative to a fund’s returns. Hedge fund managers around the kink have a strong incentive to increase
the option in compensation, and might take both positively systematic and idiosyncratic skewed bets
to achieve the goal.
132
Bibliography
Abadie, A. and G. W. Imbens. 2007. Bias Corrected Matching Estimators for Average Treatment
Effects. Mimeo.
Abel, A. B. 1990. Asset Prices under Habit Formation and Catching up with the Joneses. American
Economic Review 80: 38–42.
Aboody, D. and B. Lev. 2000. Information Asymmetry. R&D. and Insider Gains. Journal of Fi-
nance 55: 2747–2766.
Aas, Kjersti, and Ingrid Hobæk Haff, 2006, The Generalized Hyperbolic Skew Students t-
Distribution, Journal of Financial Econometrics 4, 275-309.
Ackermann, Carl, Richard McEnally, David Ravenscraft, The performance of hedge funds: Risk,
return, and incentives, Journal of Finance 54, 833-874.
Agarwal, Vikas, and Narayan Y . Naik, 2004, Risks and portfolio decisions involving hedge funds,
Review of Financial Studies 17, 63-98.
Aggarwal,Rajesh, and Philippe Jorion, 2010, The performance of emerging hedge funds and man-
agers, Journal of Financial Economics, forthcoming.
A¨ ıt-Sahalia, 2004, Disentangling diffusion from jumps, Journal of Financial Economics 74, 487-
528.
Ang, Andrew, and Joe Chen, 2002, Asymmetric correlations of equity portfolios, Journal of Fi-
nancial Economics 63, 443-494.
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang, 2006, The cross-section of
volatility and expected returns, Journal of Finance 51, 259-299.
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang, 2009, High idiosyncratic
volatility and low returns: International and further U.S. evidence, Journal of Financial Economics
91, 1-23.
Anson, Mark J. P., 2002, Symmetrical performance measures and asymmetric trading strategies:
A cautionary example, Journal of Alternative Investments 5, 81-86.
133
Aragon, George O., 2007, Share restrictions and asset pricing: Evidence from the hedge fund
industry, Journal of Financial Economics 83, 33-58.
Aragon, George O., and Vikram K. Nanda, 2011, On tournament behavior in hedge funds: high
water marks, fund liquidation, and the backfilling bias, Unpublished working paper.
Asness, Clifford, Robert Krail, and John Liew, 2001, Do hedge funds hedge? The Journal of
Portfolio Management 28, 6-19.
Baquero, Guillermo, Jenke R. ter Horst, and Marno Verbeek, 2005, Survival, look-ahead bias,
and persistence in hedge fund performance, Journal of Financial and Quantitative Analysis 40,
493-517.
Barberis, Nicholas, and Ming Huang, 2008, Stocks as lotteries: The implications of probability
weighting for security prices, American Economic Review 98, 2066-2100.
Bekaert, Geert, and Guojun Wu, 2000, Asymmetric volatility and risk in equity markets, Review
of Financial Studies 13, 1-42.
Berk, Jonathan, and Richard Stanton, 2007, Managerial ability, compensation, and closed-end fund
discount, Journal of Finance 62, 529-556.
Blake, David, and Allan Timmerman, 1998, The birth and death processes of mutual funds, Euro-
pean Finance Review 2, 57-77.
Boguth, Oliver, 2009, Stochastic idiosyncratic volatility, portfolio constraints, and the cross-
section of stock returns, Working paper, University of British Columbia.
Brennan, Machel J., Agency and asset pricing, 1993, Working paper, University of California at
Los Angeles.
Brown, Stephen J., and William N. Goetzmann, 1995, Performance persistence, Journal of Finance
50, 679-698.
Brown, Stephen J., William N. Goetzmann, Roger G. Ibbotson, 1999, Offshore hedge funds: Sur-
vival and performance, 198995, Journal of Business 72, 91-117.
Brown, Stephen J., William N. Goetzmann, and James Park, 2001, Careers and survival: Compe-
tition and risk in the hedge fund and CTA industry, Journal of Finance 56, 1869-1886.
Brown, Stephen J., Greg N. Gregoriou, and Razvan Pascalau, 2012, Is it possible to overdiversify?
The case of funds of hedge funds, Review of Asset Pricing Studies, Forthcoming.
Brown, Keith C., W. V . Harlow, Laura T. Starks, 1996, Of tournaments and temptations: An
analysis of managerial incentives in the mutual fund industry, Journal of Finance 51, 85-110.
Brunnermeier, Markus K., and Jonathan A. Parker, 2005, Optimal expectations, American Eco-
nomic Review 95, 1092-1118.
Busse, Jeffrey A., 1999, V olatility timing in mutual funds: Evidence from daily returns, Review of
Financial Studies 12, 1009-1041.
134
Busse, Jeffrey A., 2001, Another look at mutual fund tournaments, Journal of Financial and Quan-
titative Analysis 36, 53-73.
Carhart, Mark M., 1997, On persistence in mutual fund performance, Journal of Finance 52, 57-
82.
Carpenter, Jennifer, 2000, Does option compensation increase managerial risk appetite? Journal
of Finance 55, 2311-2331.
Chabi-yo, Fousseni, 2009, Explaining the idiosyncratic volatility puzzle using stochastic discount
factors, Journal of Banking and Finance, forthcoming.
Chabi-Yo, Fousseni, and Jun Yang, 2009, Default risk, idiosyncratic coskewness and equity re-
turns, Working paper, Ohio State University.
Cherkes, Martin, Jacob Sagi, and Richard Stanton, 2009, A liquidity-based theory of closed-end
funds, Review of Financial Studies 22, 257-297.
Chevalier, Judith, and Glenn Ellison, 1997, Risk taking by mutual funds as a response to incentives,
Journal of Political Economy 105, 1167-1200.
Chevalier, Judith, and Glenn Ellison, 1999, Career concerns of mutual fund managers, Quarterly
Journal of Economics 114, 389-432.
Cvitani´ c, Jakˇ sa, Vassilis Polimenis, and Fernando Zapatero, 2008, Optimal portfolio allocation
with higher moments, Annals of Finance 4, 1-28.
Daal, Elton, and Jung-Suk Yu, 2007, Daily asset return movements and value-at-risk under alter-
native classes of model specifications, Working paper, Florida Institute of Technology.
Daniel, Gilles, Didier Sornette, and Peter Wohrmann, 2009, Look-ahead benchmark bias in port-
folio performance evaluation, Journal of Portfolio Management 36, 121-130.
Dass, Nishant, Massimo Massa, and Rajdeep Patgiri, 2008, Mutual funds and bubbles: The sur-
prising role of contractual incentives, Review of Financial Studies 21, 51-99.
Dittmar, Robert F., 2002, Nonlinear pricing kernels, kurtosis preference, and evidence from the
cross section of equity returns, Journal of Finance 57, 369-403.
Dokov, Steftcho, Stoyan V . Stoyanov, and Svetlozar T. Rachev, 2007, Computing VaR and A VaR
of skewed-t distribution, Working paper, University of California Santa Barbara.
Elton, Edwin J., Martin J. Gruber, and Christopher R. Blake, 2003, Incentive fees and mutual
funds, Journal of Finance 58, 779-804.
Engle, Robert, and Abhishek Mistry, 2007, Priced risk and asymmetric volatility in the cross-
section of skewness, Working paper, New York University.
Evans, Richard B., 2010, Mutual fund incubation, Journal of Finance, forthcoming.
Fama, Eugene F., 1965, The behavior of stock-market prices, Journal of Business 38, 34-105.
135
Fama, Eugene F., and Kenneth R. French,1993, Common risk factors in the returns on stocks and
bonds, Journal of Financial Economics 33, 3-56.
Fama, Eugene F., and Kenneth R. French, 2010, Luck versus skill in the cross-section of mutual
fund returns, Journal of Finance 65, 1915-1948.
Ferson, Wayne, and Rudi Schadt, 1996, Measuring fund strategy and performance in changing
economic conditions, Journal of Finance 51, 425-462.
Frey, R¨ udiger, Alexander J. McNeil, and Mark A. Nyfeler, 2001, Copulas and credit models, RISK,
October, 111-113.
Fung, William, and David A. Hsieh, 1997, Empirical characteristics of dynamic trading strategies:
The case of hedge funds, Review of Financial Studies 10, 275-302.
Fung, William, and David Hsieh, 1997, Survivorship bias and investment style in the returns of
CTAs: The information content of performance track records, Journal of Portfolio Management
24, 30-41.
Fung, William, and David A. Hsieh, 2001, The risk in hedge fund strategies: Theory and evidence
from trend followers, Review of Financial Studies 14, 313-341.
Getmansky, Mila, Andrew W. Lo, and Igor Makarov, 2004, An econometric model of serial corre-
lation and illiquidity in hedge fund returns, Journal of Financial Economics 74, 529-609.
Goetzmann, William N., Jonathan E. Ingersoll Jr., and Stephen A. Ross, 2003, High-watermarks
and hedge fund management contracts, Journal of Finance 58, 1685-1718.
Goetzmann, William, Jonathan Ingersoll, Matthew Spiegel, and Ivo Welch, 2007, Portfolio perfor-
mance manipulation and manipulation-proof performance measures, Review of Financial Studies
20, 1503-1546.
Grinblatt, Mark, and Sheridan Titman, 1989, Adverse risk incentives and the design of perfor-
mance-based contracts, Management Science 35, 807-822.
Grinblatt, Mark, and Sheridan Titman, A study of monthly mutual fund returns and performance
evaluation techniques, Journal of Financial and Quantitative Analysis 29, 419-444.
Gruber, Martin J., 1996, Another puzzle: The growth in actively managed mutual funds, Journal
of Finance 51, 783-810.
Hamao, Yasushi, Ronald W. Masulis, and Victor Ng, 1990, Correlations in price changes and
volatility across international stock markets, Review of Financial Studies 3, 281-307.
Hansen, Lars P., 1982, Large sample properties of the generalized method of moments estimators,
Econometrica 50, 1029-1054.
Hansen, Bruce E., 1994, Autoregressive conditional density estimation, International Economic
Review 35, 705-730.
136
Harvey, Campbell R., Merrill Liechty, John Liechty, and Peter M¨ uller, 2010, Portfolio selection
with higher moments, Quantitative Finance 10, 469-485.
Harvey, Campbell R., and Akhtar Siddique, 1999, Autoregressive conditional skewness, Journal
of Financial and Quantitative Analysis 34, 465-487.
Harvey, Campbell R., and Akhtar Siddique, 2000, Conditional skewness in asset pricing tests,
Journal of Finance 55, 1263-1295.
Hemmer, Thomas, Oliver Kim,Robert E. Verrecchia, 2000, Introducing convexity in to optimal
compensation contracts, Journal of Accounting and Economics 28, 307-327.
Hodder, James E., and Jens Carsten Jackwerth, 2007, Incentive contracts and hedge fund manage-
ment, Journal of Financial and Quantitative Analysis 42, 811-826.
Holmstrom, Bengt, 1979, Moral hazard and observability, Bell Journal of Economics 10, 74-91.
Hu, Ping, Jayant R. Kale, Marco Pagani, and Ajay Subramanian, 2011, Fund flows, performance,
managerial career concerns,and risk-taking: Theory and evidence, Management Science 57, 628-
646.
Huang, Jennifer, Kelsey D. Wei, and Hong Yan, 2007, Participation costs and the sensitivity of
fund flows to past performance, Journal of Finance 62, 1273-1311.
Hunter, David, Eugene Kandel, Shmuel Kandel, and Russell Wermers, 2010, Endogenous bench-
marks, Working paper, University of Maryland.
Jagannathan, Ravi, Alexey Malakhov, and Dmitry Novikov, 2010, Do hot hands exist among hedge
fund managers? An empirical evaluation, Journal of Finance 65, 217-255.
Jondeau, Eric, and Michael Rockinger, 2003, Conditional volatility, skewness, and kurtosis: exis-
tence, persistence, and comovements, Journal of Economic Dynamics and Control 27, 1699-1737.
Kane, Alex, 1982, Skewness preference and portfolio choice, Journal of Financial and Quantita-
tive Analysis 17, 15-25.
Kempf, Alexander, and Stefan Ruenzi, 2008, Tournaments in mutual-fund families, Review of
Financial Studies 21, 1013-1036.
Kempf, Alexander, Stefan Ruenzi, and Tanja Thiele, 2009, Employment risk, compensation incen-
tives, and managerial risk taking: Evidence from the mutual fund industry, Journal of Financial
Economics 92, 92-108.
Kim, Min S., Changes in Mutual Fund Flows and Managerial Incentives, Working paper, Univer-
sity of New South Wales.
Koski, Jennifer Lynch, and Jeffrey Pontiff, 1999, How are derivatives used? Evidence from the
mutual fund industry, Journal of Finance 54, 791-816.
Kouwenberg, Roy, and William T. Ziemba, 2007, Incentives and risk taking in hedge funds, Jour-
nal of Banking and Finance 31, 3291-3310.
137
Kraus, Alan, and Robert H. Litzenberger, 1976, Skewness preference and the valuation of risk
assets, Journal of Finance 31, 1085-1100.
Lee, Charles M. C., Andrei Shleifer, Richard H. Thaler, 1991, Investor sentiment and the closed-
end fund puzzle, Journal of Finance 46, 75-109.
Leland, Hayne E., Beyond mean-variance: Risk and performance measurement in a nonsymmet-
rical world, Financial Analysts Journal, January-February 1999, 27-36.
Lin, Jerchern, 2011, Tail risks across investment funds, Unpublished working paper, Marshall
School of Business, University of Southern California.
Lo, Andrw W., 2002, The statistics of Sharpe ratios, Financial Analysts Journa l58, 36-50.
Longin, Franc ¸ois, and Bruno Solnik, 2001, Extreme correlation of international equity markets,
Journal of Finance 56, 649-676.
Mandelbrot, B., 1963, The variation of certain speculative prices, Journal of Business 36, 394-419.
McNeil, Alexander J., R¨ udiger Frey, and Paul Embrechts, 2005, Quantitative Risk Management:
Concepts, Techniques and Tools, Princeton University Press.
Merton, Robert C., and Roy D. Henriksson, 1981, On market timing and investment performance
II: Statistical procedures for evaluating forecasting skills, Journal of Business 54, 513-534.
Mitton, Todd, and Keith V orkink, 2007, Equilibrium under diversitication and the preference for
skewness, Review of Financial Studies 20, 1255-1288.
Moreno, David, and Rosa Rodr´ ıguez, 2009, The value of coskewness in mutual fund performance
evaluation, Journal of Banking and Finance 33, 1664-1676.
Murphy, Kevin J., 1999, Executive Compensation, Handbook of Labor Economics, V ol. III, Orley
Ashenfelter and David Card, editors. North Holland.
Olivier, Jacques, and Anthony Tay, 2009, Time-varying incentives in the mutual fund industry,
Unpublished working paper, HEC Paris.
Panageas, Stavros, and Mark Westerfield, 2009, High water marks: High risk appetites? Convex
compensation, long horizons, and portfolio choice, Journal of Finance 64, 1-36.
Patton, Andrew J., 2004, On the out-of-sample importance of skewness and asymmetric depen-
dence for asset allocation, Journal of Financial Econometrics 2(1), 130-168.
Pontiff, Jeffrey, 1995, Closed-end fund premia and returns: Implications for financial market equi-
librium, Journal of Financial Economics 37, 341-370.
Pontiff, Jeffrey, 1997, Excess volatility and closed-end funds, American Economic Review 87,
155-169.
Ramakrishnan, Ram T. S., and Anjan V . Thakor, 1984, The valuation of assets under moral hazard,
Journal of Finance 39, 229-238.
138
Samuelson, Paul A., 1970, The fundamental approximation theorem of portfolio analysis in terms
of means, variances and higher moments, The Review of Economic Studies 37, 537-542.
Schwarz, Christopher G., 2011, Mutual fund tournaments: The sorting bias and new evidence,
Review of Financial Studies 24, 1-24.
Shanken, Jay, 1992, On the estimation of beta-pricing models, Review of Financial Studies 5, 1-33.
Sharpe, William F., 1992, Asset allocation: Management style and performance measurement,
Journal of Portfolio Management 18, 7-19.
Sirri, Erik R., and Peter Tufano, 1998, Costly search and mutual fund flows, Journal of Finance
53, 1589-1622.
Starks, Laura T., 1987, Performance incentive fees: An agency theoretic approach, Journal of
Financial and Quantitative Analysis 22, 17-32.
Susmel, Raul, and Robert F. Engle, 1994, Hourly volatility spillovers between international equity
markets, Journal of International Money and Finance 13, 3-25.
Swaminathan, Bhaskaran, 1996, Time-varying expected small firm returns and closed-end fund
discounts, Review of Financial Studies 9, 845-887.
Taleb, Nassim N., 2004, Bleed or blowup: what does empirical psychology tell us about the pref-
erence for negative skewness?, Journal of Behavioral Finance 5.
Taleb, Nassim N., 2007, Fooled by randomness: the hidden role of chance in life and in the
markets, 2nd ed (Random House Inc., New York, NY).
Ter Horst, Jenke R., and Marno Verbeek, 2007, Fund liquidation, self-selection, and look-ahead
bias in the hedge fund industry, Review of Finance 11, 605-632.
Ter Horst, Jenke R., Theo E. Nijman, and Marno Verbeek, 2001, Eliminating look-ahead bias in
evaluating persistence in mutual fund performance, Journal of Empirical Finance 8, 345-373.
Theodossiou, Panayiotis, 1998, Financial data and the skewed generalized T distribution, Manage-
ment Science 44, 1650-1661.
Thompson, Rex, 1978, The information content of discounts and premiums on closed-end fund
shares, Journal of Financial Economics 6, 151-186.
Titman, Sheridan, and Cristian Tiu, 2010, Do the best hedge funds hedge?, Review of Financial
Studies 24, 123-168.
Treynor, Jack, and Kay Mazuy, 1966, Can mutual funds outguess the market? Harvard Business
Review 44, 131-136.
Whitelaw, Robert F., 1994, Time variations and covariations in the expectation and volatility of
stock market returns, Journal of Finance 49, 515-541.
139
Appendix A: The Numerical Procedure
for the Optimization Problem
A fund manager solves for the optimal unconditional weight based on returns up to timet. Steps
are the following:
(a) Generate 10,000 jointly independent random variables (U,V) from the T-Copula.
(b) Use the inverse method to generate time-series of returns for the benchmark and the big bet,
i.e. F
−1
p
(U) and F
−1
BB
(V). F
−1
p
and F
−1
BB
are inverse CDFs for the normal and skewed
t-distribution, respectively.
(c) Solve for the optimal weight:
w ≡ argmax
1
t
t
X
j=1
U(W
j
) (2.10)
(d) Simulate step (a) to (c) 1000 times.
140
Appendix B: Conditioning Biases and
Benchmarks
The literature has documented the following biases in fund datasets and they might differ across
fund types and bias results on tail risks.
Incubation bias is referred to as fund families start several new funds, but only open funds that
succeed in the evaluation period to the public. Evans (2007) shows that incubated mutual funds
outperform non-incubated funds. Incubation creates upward bias on fund returns and thus increase
skewness and reduce kurtosis. In addition, when a fund enters to the database, its past return history
is automatically added to the database. The addition of past returns causes backfilling bias and it
can bias fund skewness upwards and kurtosis downwards.
For OEFs, I delete returns before the fund inception date to avoid incubation bias. This step
follows from Evans’ (2007) initial approach since I have no access to the complete list of mutual
fund tickers and their creation dates from NASD. I also delete fund returns for the first year to
remove backfill bias. For HFs, I drop returns before the inception date to remove incubation bias.
Aggarwal and Jorion (2010) use the data field “date added to database” in TASS dataset and find the
median backfill period is 480 days. I adopt the same approach to clean out back-filled HF returns.
Stale prices mean that reported asset prices do not reflect correct true prices, possibly due to
illiquidity, non-synchronous trading, or bid-ask bounce. These characteristics can cause serial-
correlation in returns. HFs suffer from this bias the most , and are adjusted for stale prices in the
robustness analysis.
If a study includes only funds that survive until the end of the sample period, survivorship bias
occurs. The survival probability of funds depends on past performance (Brown and Goetzmann
(1995)). Managers who take significant risk and win will survive. Therefore, the database is left
with high risk and high return surviving funds. The survivorship bias imparts a downward bias to
risk, and an upward bias to alpha (e.g. Carhart (1997), Blake and Timmermann (1998)). It also
induces more positive skewness and less fat-tailedness.
CEFs may suffer also from survivorship bias, due to the commonly observed discounts on
traded prices. The discounts may lead to liquidation or reorganization (“open-ending”) and leave
the dataset with surviving funds. Although the exit rate for ETFs is low, survivorship bias might still
affect their tail risks. To avoid survivorship bias, I download the lists of ETFs and CEFs from the
Morningstar survivorship free database. OEFs are taken from the CRSP survivorship free database.
The survivorship bias is more complex for HFs. HFs may decide to stop reporting because of
liquidation or self-selection (Ter Horst and Verbeek (2007), Jagannathan, Malakhov, and Novikov
(2010)). Liquidation refers to underperforming funds exiting the database. Self-selection is associ-
141
ated with a fund’s decision to be included in the database. For instance, outperforming HFs have
less incentives to report performance to attract new investors and fund managers may switch to an-
other data vendor for marketing purposes. I combine both live and dead hedge fund returns from
HFR to eliminate survivorship bias.
The look-ahead bias arises when funds are required to survive some minimum length of time
after a reference date. One type of look-ahead bias applicable to this study is the look-ahead bench-
mark bias (Daniel, Sornett, and Wohrmann (2009)). Since the time series of styles are not kept
in the database, funds that change styles over time may suffer from look-ahead benchmark bias.
This omission can bias risk-adjusted returns and tail risks. The portfolios of funds for OEFs are
constructed look-ahead bias free. Monthly returns are used only after the beginning of the assigned
style. No ex-post style returns are used.
ETFs and CEFs are subject to look-ahead bias as well since no data vendors keep the history of
their classification codes. However, it is unlikely these funds will change investment styles through
time, given their fund characteristics
23
.
Investment funds with less than twelve months of returns are excluded and all investment funds
maintain the same investment strategy for at least twelve months. Fund managers are usually eval-
uated at the end of year and the minimum of 12 observations offer sufficient degrees of freedom for
GMM estimation
24
.
Nevertheless, my attempts to control these ex-post conditional biases may be imperfect. By
construction, HFs might still suffer limited look-ahead benchmark bias and I assume no change of
styles in ETFs and CEFs. Lack of NASD data might leave backfill bias in the mutual fund sample.
In addition, it is known that the coverage of HFs has little overlap across different data vendors.
Relying on only HFR data may not represent the whole HF industry.
HFR provides main and sub strategy classification codes for HFs. I use main strategy classifi-
cation codes. Style classification codes for ETFs and CEFs are from Morningstar. The Morningstar
classification codes for ETFs and CEFs are commonly used on many financial websites and easily
accessible to investors. For OEFs, I use style classification codes in the CRSP mutual fund database.
The database uses five different classification codes to cover disjoint time periods. POLICY codes
are used before 1990. CRSP uses WIESENBERGER (WB OBJ) codes between 1990 to the end
of 1992. Strategic Insight Objective (SI OBJ) codes cover from 1993 to September, 1998. Lipper
Objective (Lipper OBJ) codes are used up to 2008. Most recent funds are classified by Thomson
Reuters Objective (TR OBJ) codes.
Benchmark data are from the following sources. Market excess returns, SMB and HML fac-
tors are obtained from Ken French’s website
25
. The momentum factor is downloaded from CRSP.
23
ETFs are index funds and CEFs do not allow the redemption of shares after IPO.
24
I remove two mutual funds (CRSP Fund ID 031241 in fixed income index and 01108 in fixed income government)
and two HFs (HFR Fund ID 17393 and 21981 in relative value) from this study manually because the percentages on the
components in skewness and kurtosis decompositions by GMM estimation are so large that the average weights across
individual funds are heavily skewed. All four funds have no monthly returns outside 3 standard deviation from the mean.
Removing these four funds has minimal effects on the univariate statistics of the style that they belong to.
25
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html.
142
The seven HF factors
26
are downloaded from David Hsieh’s website
27
. The Barclay U.S. gov-
ernment/credit index (LHGVCRP) and corporate bond index (LHCCORP) are downloaded from
Datastream.
26
The equity and bond market factor, the size spread factor, two credit spread factors, and three lookback straddles on
bond futures, currency futures, and commodity futures.
27
http://faculty.fuqua.duke.edu/
˜
dah7/DataLibrary/TF-FAC.xls.
143
Appendix C: Open-ended Fund Styles
I consider funds with the following style codes are fixed income funds - POLICY in B&P,
Bonds, Flex,GS, or I-S; WB OBJ in I, S, I-S, S-I, I-G-S, I-S-G, S-G-I, CBD, CHY , GOV , IFL,
MTG, BQ, BY , GM, or GS; SI OBJ in BGG, BGN, BGS, CGN, CHQ, CHY , CIM, CMQ, CPR, CSI,
CSM, GBS, GGN, GIM, GMA, GMB, GSM, or IMX; Lipper Class in ’TX’ or ’MB’; Lipper OBJ
in EMD, GLI, INI, SID, SUS, SUT, USO, GNM, GUS, GUT, IUG, IUS, ARM, USM, A, BBB, or
HY ; and TR OBJ in AAG, BAG, GLI, BDS, GV A, GVL, GVS, UST, MTG, CIG, or CHY . I further
screen out funds with holdings in bonds and cash less than 70% at the end of the previous year.
Fixed income funds (FI) are classified as Index, Global, Short Term, Government, Mortgage,
Corporate, and High Yield. Index funds (FI Index) are selected by matching the string “index” with
the fund name. Global funds are coded as SI OBJ in BGG or BGN, Lipper OBJ in EMD, GLI, or
INI, or TR OBJ in AAG, BAG, or GLI.
Short term funds are coded as SI OBJ in CSM, CPR, BGS, GMA, GBS, or GSM, Lipper OBJ
in SID, SUS, SUT, USO, or TR OBJ in BDS. Government funds are codes as POLICY in GS,
WB OBJ in GOV or GS, SI OBJ in GIM or GGN, or Lipper OBJ in GNM, GUS, GUT, IUG, or
IUS, or TR OBJ in GV A, GVL, GVS, or UST. Mortgage funds are coded as POLICY WB OBJ
in MTG, GM, SI OBJ in GMB, Lipper OBJ in ARM or USM, or TR OBJ in MTG. Corporate
funds are coded as POLICY in B&P, WB OBJ in CBD,BQ, SI OBJ in CHQ, CIM, CGN, CMQ,
Lipper OBJ in A, BBB, or TR OBJ in CIG. High Yield funds are coded as POLICY in Bonds,
WB OBJ in I-G-S, I-S-G, S-G-I, BY , CHY , SI OBJ in CHY , Lipper OBJ in HY , TR OBJ in CHY .
Other funds are funds that I classify as bond funds but do not meet the criteria above.
Similarly, I use the following codes to screen out equity funds - POLICY in Bal, C & I, CS,
Hedge, or Spec; WB OBJ in G, G-I, I-G, AAL, BAL, ENR, FIN, GCI, GPM, HLT, IEQ, INT, LTG,
MCG, SCG, TCH, UTL, AG, AGG, BL, GE, GI, IE, LG, OI, PM, SF, or UT; SI OBJ AGG, BAL,
CVR, ECH, ECN, EGG, EGS, EGT, EGX, EID, EIG, EIS, EIT, EJP, ELT, EPC, EPR, EPX, ERP,
FIN, FLG, FLX, GLD, GLE, GMC, GRI, GRO, HLT, ING, JPN, OPI, PAC, SCG, SEC, TEC, or
UTI; Lipper Class in EQ; Lipper OBJ in SP, SPSP, AU, BM, CMD, NR, FS, H, ID, S, TK, TL, UT,
CH, CN, CV , DM, EM, EU, FLX, GFS, GH, GL, GLCC, GLCG, GLCV , GMLC, GMLG, GMLV ,
GS, GSMC, GSME, GSMG, GSMV , GNR, GTK, IF, ILCC, ILCG, ILCV , IMLC, IMLG, IMLV ,
IS, ISMC, ISMG, ISMV , JA, LT, PC, XJ, B, BT, CA, DL, DSB, ELCC, LSE, SESE, MC, MCCE,
MCGE, MCVE, MR, SCCE, SCGE, SCVE, SG, G, GI, EI, EIEI; and TR OBJ in AAD, AAG,
AGG, BAD, BAG, CVT, EME, ENR, EQI, FIN, FOR, GCI, GLE, GPM, GRD, HLT, MID, OTH,
SMC, SPI, TCH, UTL. I further screen out funds with holdings in bonds and cash less than 70% at
the end of the previous year.
144
Equity funds (EF) are classified as Index, commodities, Sector, Global, Balanced, Leverage and
Short, Long Short, Mid Cap, Small Cap, Aggressive Growth, Growth, Growth and Income, Equity
Income, and Others. Index funds (EF Index) are identified by finding the match of the string “index”
within the fund name or funds with Lipper OBJ in SP or SPSP, or TR OBJ in SPI.
Commodities funds are coded as WB OBJ in ENR, GPM, PM, SI OBJ in GLD Lipper OBJ
in AU, BM, CMD, NR, or TR OBJ in ENR, GPM. Sector funds are codes as POLICY in Spec,
WB OBJ in FIN, HLT, TCH, UTL, SF, UT, SI OBJ in FIN,HLT, Lipper OBJ in FS, H, ID, S, TK,
TL, UT, or TR OBJ in FIN, HLT, OTH, TCH, UTL. Global funds are coded as POLICY in C & I,
WB OBJ in INT, GE, IE, SI OBJ in ECH, ECN, EGG, EGS, EGT, EGX, EID, EIG, EIS, EIT, EJP,
ELT, EPC, EPX, ERP, FLG, GLE, JPN, PAC, Lipper OBJ CH, CN, DM, EM, EU, GFS, GH, GL,
GLCC, GLCG, GLCV , GMLC, GMLG, GMLV , GS, GSMC, GSME, GSMG, GSMV , GNR, GTK,
IF, ILCC, ILCG, ILCV , IMLC, IMLG, IMLV , IS, ISMC, ISMG, ISMV , JA, LT, PC, XJ, TR OBJ
in EME, FOR, GLE. Balanced funds are coded as POLICY in Bal, WB OBJ in AAL, BAL, BL,
SI OBJ in BAL, CVR, FLX, Lipper OBJ in B, BT, CV , FLX, or TR OBJ in AAD, BAD, AAG,
BAG, CVT. Leverage and short funds are coded as POLICY in Hedge, WB OBJ in OI, SI OBJ in
OPI, or Lipper OBJ in CA, DL, DSB, ELCC, SESE. Long short funds are coded as Lipper OBJ in
LSE. Mid cap funds are coded as WB OBJ in GMC, Lipper OBJ in MC, MCCE, MCGE, MCVE,
TR OBJ in MID. Small cap funds are coded as WB OBJ in SCG, Lipper OBJ in MR, SCCE,
SCGE, SCVE, SG, or TR OBJ in SMC. Aggressive growth funds are coded as WB OBJ in GI,
GCI, SI OBJ in AGG, or TR OBJ in AGG. Growth funds are coded as WB OBJ in G,LG, SI OBJ
in GRO, Lipper OBJ in G, or TR OBJ in GRD. Growth and income funds are coded as WB OBJ
in GI, GCI, SI OBJ in GRI, Lipper OBJ in GI, or TR OBJ in GCI. Equity income funds are coded
as WB OBJ in EI, IEQ, Lipper OBJ in EI, EIEI, or TR OBJ in EQI. Other funds are funds that I
classify as equity funds but do not meet the criteria above.

Abstract (if available)

Abstract The first essay is titled ""Tail Risks across Investment Funds."" Managed portfolios are subject to tail risks, which can be either index level (systematic) or fund-specific. Examples of fund-specific extreme events include those due to big bets or fraud. This paper studies the two components in relation to compensation structure in managed portfolios. A simple model generates fund-specific tail risk and its asymmetric dependence on the market, and makes predictions for where such risks should be concentrated. The model predicts that systematic tail risks increase with an increased weight on systematic returns in compensation and idiosyncratic tail risks increase with the degree of convexity in contracts. The model predictions are supported with empirical results. Hedge funds are subject to higher idiosyncratic tail risks and Exchange Traded Funds exhibit higher systematic tail risks. In skewness and kurtosis decompositions, I find that coskewness is an important source for fund skewness, but fund kurtosis is driven by cokurtosis, as well as volatility comovement and residual kurtosis, with the importance of these components varying across fund types. Investors are subject to different sources of skewness and fat tail risks through delegated investments. Volatility based tail risk hedging is not effective for all fund styles and types. ❧ The second essay, titled ""Fund Convexity and Tail Risk-Taking,"" studies how a fund manager takes skewed bets in two dimensions. First, the fund manager constantly reexamines fund performance relative to his or her peers and takes a position with respect to skewness risk. I show that when a fund manager underperforms peers, he or she will gamble on trades with lottery-like returns. On the other hand, when a fund outperforms peer funds, the fund manager will take negatively skewed trades. The results are robust to different econometric specifications. Second, I examine how convexity in incentives affects tail risks across and within different types of investment funds. The literature has documented different forms of convexity that a fund manager faces: discounts in closed-end funds, tournaments and fund flow-performance relation in open-ended funds, and high-water mark provisions in hedge funds. Sorting funds by the degree of convexity and comparing skewness between the group with the most convexity and the group with the least convexity, I conclude that convexity affects fund tail risks. This result suggests that both implicit and explicit convexities provide incentives for fund managers to take systematic and idiosyncratic bets with tail risks.

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Asset Metadata

Creator Lin, Jerchern (author)

Core Title Essays in tail risks

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 07/30/2012

Defense Date 06/15/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag closed-end fund discounts,cokurtosis,coonvex incentives,coskewness,flow and performance relationship,hedge funds,high water marks,investment funds,kurtosis,managerial compensation,mutual funds,OAI-PMH Harvest,risk taking,skewness,tournament

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ferson, Wayne (committee chair), Jones, Christopher S. (committee member), Zapatero, Fernando (committee member)

Creator Email jercherl@usc.edu,jerchern@buffalo.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-74758

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