Fund Convexity and Tail Risk-Taking

by

Jerchern Lin*

First draft: April 6, 2011

This draft: January 2, 2011

Abstract

This paper 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 bets with tail risks.

* PhD student, Marshall School of Business, University of Southern California, 3670 Trousdale Parkway, Suite 308, Los Angeles, CA 90089-0804. Email: jercherl@marshall.usc.edu, www-scf.usc.edu/~jercherl, (213) 740-9663.

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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 Vorkink (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 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

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 Von 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 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.

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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 outperforms his or her peers,

he or she will be more likely to take a negatively skewed bet. If, however, a fund underperforms

peer funds, the fund manager will make a lottery-like bet. The underperforming 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.

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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 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.

The rest of the paper proceeds as follows. Section 2 addresses the importance of

skewness risk. Section 3 reviews the literature on risk-taking and relates it to skewness risk-

taking. Section 4 outlines the data. Section 5 describes empirical methods and results. Section 6

concludes.

2. Why Should Investors Move Beyond Volatility and Care About Skewness Risk?

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.

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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 produce 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 returns. 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 asymmetric because the bond holders have the right to convert the bond into a fixed

number of shares of 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 products, such as swaps, offer bond managers a steady

4 Hedge fund managers can engage in short-volatility trades by longing an undervalued asset and shorting an

overvalued 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 upside 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.

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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 asymmetrically 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 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 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,

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to infer the true risk hidden in the fund, one needs to look beyond volatility and measure

skewness risk.

Investors may not be sophisticated enough to understand the downside risk hidden in a

fund. Agency costs between investors and fund managers provide a strong incentive to trade

skewed bets. Fund managers may be able to hide trades with significant downside risk from

investors.

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, risk-taking will be moderated.

Panageas and Westerfield (2009) also analytically derive the same conclusion that convexity

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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,

Harlow, 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 manager’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 termination. 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).

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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 relative 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 positively 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 possible 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

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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 occurrences 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 management

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

example, 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.

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.

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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 spuriously increase fund mean returns and skewness and reduce fund variance and

kurtosis. Data vendors 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 closed-end funds and hedge funds, to remove back-fill bias, I drop returns before the

inception date. 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.

5. Empirical Methods and Results

5.1 Changes in Tail Risks

5.1.1 Tail Risks on Relative Performance

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, Intermediate 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.

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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 managerial behavior toward skewness risk and captures skewness

changes across years. Fund managers are rewarded for short-term performance11 and thus have

a stronger incentive to gamble when the evaluation 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% 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.

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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. I also test fund skewness in the subsequent 36 and 60 months as a robustness check. The results are qualitatively unchanged, but significance levels drop for longer periods. 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 autocorrelation. 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.

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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.

Table 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.

Figure 1 shows the differences in average fund skewness between low- and high-

performing quintile groups across investment funds.14 Clearly, fund skewness fluctuates over

time, and the average fund skewness differs between outperforming funds and

underperforming 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 1 may come from the existing portfolios in the past

24 months or from trades in the subsequent 12 months.

5.2 Multivariate Analysis

Table 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 hypothesis of no difference between the two groups. The tests for the top 20% and

bottom 20% groups in bond and equity funds also show a strong statistical difference. The

14

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.

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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 1 and Table 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 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:

is the average of the differences in returns between fund i and its peer fund

based on 24 monthly returns up to month t. and

are the volatility and

skewness around the peer fund return in the past 24 months. and

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.15 Time fixed effects would absorb

15

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. Results are also qualitatively unchanged when controlling for only one lagged volatility and skewness from t to t-23. Controlling for lagged volatility and skewness from t to t-11 and t-12 and t-23 yield 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.

15

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 ( ) is zero. The t-statistics for closed-end funds, open-ended funds, and hedge

funds are −5.62, −2.43, and −3.21, 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 −0.17 for closed-end funds, open-ended funds, and hedge funds,

respectively.16 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 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 0.365. 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.

The results from Table 3 suggest the importance of skewed bets over risky trades.

Underperforming fund managers wish to improve their ranking and do not wish to lose their

job; outperforming 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

16

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.

16

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 outperforming 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

cumulative 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.

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 managers to undertake 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

outperforming funds have positively skewed returns in the past 24 months, the subsequent

17

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.

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 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, except for closed-end funds. For instance, hedge

funds have a coefficient of −0.046 for the top 20% and one of −0.025 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 −0.346, −0.663, and −0.222 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 4 reports the test of differences in

the coefficients on relative performance across five groups. P-values associated with the test of

equal coefficients between the top and bottom funds are less than 0.1 for open-ended funds.

When tested for differences in coefficients across five groups, closed-end funds and hedge

funds have p-values of less than 0.01. Open-ended funds have p-values of less than 0.1. These

18

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.

To examine how extreme performing groups take positions with respect to skewness

risk, I construct the fractional rank (FracRank) of relative performance for fund i as follows:

= Min ( ,0.2), = Min (0.6, ),

, where 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:

∑

Table 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

( ) is different from that on the bottom fractional rank ( ). The

coefficients on the bottom and top fractional ranks in open-ended funds are 0.087 and −0.578,

and their associated t-values are 0.79 and −2.87. Hedge funds have −0.286 and −0.497 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.

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

19

behavior on skewed bets may differ across fund industries. Panel D of Table 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 positively 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.

As a robustness check, I examine the differential response of relative skewness for the

next 12 months to past relative performance by the following alternative piecewise linear

regression.

∑

equals to fund i’s relative performance in month t if the relative

performance is in the 20th percentile and zero otherwise. The variable q equals 2 if the relative

20

performance is in the next quintile, and so on. For q equals 2 to 5, equals the

difference between fund i’s relative performance in month t and (q-1)th quintile if the relative

performance is in the qth quintile and zero otherwise. I hypothesize that the top 20% of funds

( ) have more negative sensitivity than the bottom 20% of funds ( ).

Table 6 shows the results that hedge funds have statistically significant loadings on the

coefficients and satisfy the hypothesis. The coefficient on the top 20% of open-ended funds is

more negative than that on the bottom 20%, but the coefficient on the latter is not significant.

The bottom 20% group in closed-end funds has a more negative loading on the rank of relative

performance than the top 20% group.

Panel D of Table 6 further shows that the loadings on the top 20% ranks differ across

fund types, but not the bottom 20% ranks. Comparing to results from Table 5, extremely

underperforming fund managers do not behave differently across fund industries. This may

suggest that managers commonly have a preference for lottery-type returns in response to being

well below in the tournament rankings. The top 20% ranks across fund types behave differently

toward skewness risk. This supports the results from Table 5. One possible explanation is that

the type of convexity in compensation differs across fund types. The literature documents the

managerial variance risk-taking behavior in the closed-end, open-ended, and hedge fund

industries in relation to discounts, tournaments and the fund flow performance, and high-water

marks, respectively. In addition, unlike hedge fund managers, open-ended fund managers use

buy-and-hold trading strategies. Derivative use in open-ended funds is used mainly for

hedging (Koski and Pontiff (1999)). As such, the top open-ended fund managers might write

covered calls to protect against downside losses. As a result, the subsequent fund returns

exhibit negative skewness.

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,

21

1999)). Managers 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 the following regression:

The main interests lie in the interaction terms between relative performance and age

(size). Results are reported in Table 7. 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 7 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

22

managers can invest in broad asset classes and have high flexibility in trading strategies as

documented in the literature.

5.2.3 How Does Managerial Behavior Toward Skewed Bets Vary with Macroeconomic Conditions?

One interesting observation from Figure 1 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

periods 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, outperforming 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 become 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 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

23

observed in Figure 1. Across all fund types, outperforming funds consistently take more

negative skewness after 2002. Last, the gap in skewness between high and low performing

groups in closed-end funds between 2002 and 2004 may be related to the low interest rate

environment, and losers ride on the wave of mortgage backed securities. Could we relate a fund

manager’s skewness risk-taking in response to relative performance to changing economic

variables?17

Ferson and Schadt (1996) show that performance measures can be affected by changing

economic conditions. Whitelaw (1994) concludes that macroeconomic variables can predict

expected returns and volatility. I use public information variables in both studies to study

whether managerial behavior on skewed bets changes with economic conditions. The

macroeconomic variables include the lagged one-month Treasury Bill (T-Bill t), the lagged

dividend yield (DivYield t), the lagged term spread (YieldSp t), and the lagged default spread

(DefaultSp t).18 I interact the public information variables with relative performance, and the

regression specification is as follows:

Results are reported in Table 8, and main effects are reflected in the interaction terms

between public information variables and relative performance. The dividend yield and term

spread are two statistically significant factors for both open-ended funds and hedge funds.

17

Olivier and Tay (2009) study the manager’s incentives from the convexity of the flow-performance relationship vary with economic activity. Instead of macroeconomic variables, they use GDP growth as a proxy for economic activity. I don’t find significant results when using the interaction term with GDP growth. 18

The default spread is the yield difference between Moody’s Baa-rated and Aaa-rated corporate bonds. The term spread is the yield difference between a constant maturity ten-year Treasury bond and the three-month Treasury bill. The dividend yield is the sum of dividends paid on the S&P500 index over the past 12 months divided by the current level of the index.

24

Closed-end funds show no significant interaction terms. During periods of high dividend yield,

fund managers are more likely to take skewed bets. During periods of high term spread, fund

managers are less likely to take skewed bets. The huge spike in positive skewness for funds that

underperform in 1987 may be related to economic conditions. The underperforming managers

bet on high dividend yield stock, which exhibit positive skewness due to overvaluation before

October 1987. The occurrence of the market crash in 1987 reduces positive skewness

dramatically.

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.

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

25

implicit optionality in incentives. I calculate the closed-end fund premium and discount as

follows:

⁄

where and 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:

where

denotes the fund i’s option return and the underlying instrument is net asset

value, and and

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 9 reports the impact of premiums/discounts on fund tail risk. Returns and

skewness follow 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)

26

have higher expected returns 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 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

significantly 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.

5.3.2 Tournaments in Open-Ended Funds

The literature has documented managerial risk-taking behavior with respect to tournament

objectives. 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

27

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.

Table 10 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 10 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 portfolios of

funds into coskewness, idiosyncratic coskewness, and idiosyncratic skewness.20 Coskewness

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. 20

For details, see Lin (2011).

28

measures the covariance between fund returns and market volatility. Idiosyncratic coskewness

measures 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 10. Consistent with the

findings in the previous section, idiosyncratic skewness risk increases 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. 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. 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 10.

29

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:

[ ( )]

where is the total net assets of the fund and 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.

Table 11 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

30

(−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 12 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 12 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, which is consistent with

results from Table 11. Incentives from tournaments and fund flows should be viewed

independently.

5.3.4 High-Water Marks in Hedge Funds

31

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:

( )

where is fund i’s return at t. The gamma of an option based on the Black Scholes formula is

√ , where N’(.) is the standard normal probability distribution function,

(

) (

)

√ ,

S is the stock price, X is the strike price, r is the risk-free rate, is the stock’s volatility, and T is

the option’s time to maturity. 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 at t+1 is calculated as:

( )

where is fund i’s return at t+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.

Table 13 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

21

Sorting funds based on alternative measures, such as and , yields the same rankings of funds.

32

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 9 to 13 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 negative systematic shock will impact the entire style group. From the

sign and magnitude of systematic skewness risk at the portfolio level, we can infer fund

managers’ positions on systematic-skewed bets. For example, when the portfolios of funds

22

The kth

moment of portfolios of funds is O(1/nk−1

).

33

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 consistent with findings by Hu

et al. (2011). An interesting question for further research is the interaction between risk and

skewness.

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 returns than the outperforming funds. This indicates that equity open-

ended funds gamble on assets or trading strategies associated with negative systematic

34

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 separately. 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.

35

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38

Figure 1. Differences in Fund Skewness Between High- and Low-Performing Funds This graph shows the average fund skewness around the peer fund return of the 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 month t 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 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. 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. The black (red) line represents the average fund skewness of the low- (high-) performing groups, respectively. Panels A, B, and C are results for closed-end funds, open-ended funds, and hedge funds, respectively. Both bond and equity funds are included in the analysis. Panel A: Closed-End Funds

39

Panel B: Open-Ended Funds

40

Panel C: Hedge Funds

41

Table 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 month t. 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 month t+1. Panels C and D report the changes in relative fund skewness and kurtosis in the following 12 months from month t+1, compared to those in the previous 24 months up to month t. CEFs, OEFs, and HFs refer to closed-end funds, open-ended funds, and hedge funds, respectively. Panel A: Fund Skewness

Subsequent Skewness:

Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%

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

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

42

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

Subsequent Kurtosis:

Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%

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

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

43

Panel C: Change in Skewness

Change in Subsequent Skewness:

Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%

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

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

Change in Subsequent Kurtosis:

Fund Type Group Mean Bottom 10% Bottom 25% Median Top 25% Top 10%

CEFs Bottom 20% −115.90 −600.07 −198.71 −99.95 18.58 271.01

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

44

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

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

45

Table 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 month t 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 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 Groups

Skewness Paired

T-value Paired

P-value Signed-Rank 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

46

Table 3. The Sensitivity of Fund Skewness to Lagged Relative Performance This table shows the relation between fund skewness over the 12-month period from month t+1 and relative fund performance in the prior 24 months up to month t. The regression is as follows:

is the average of the difference between fund i’s returns and its peer fund returns

based on 24 monthly returns up to month t ,i.e., from month t−23 to month t. and

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. and

are the lagged fund volatility and skewness around the peer

fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in returns during the 24 month

interval. and

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 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.

Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t −0.067 −5.62 −0.082 −4.83 −0.059 −4.69

Vol t+1 0.031 2.45 0.019 1.18 0.035 2.06

Vol t 0.020 2.38 0.021 1.15 0.015 1.44

Skew t 0.041 2.46 0.008 0.24 0.051 2.73

Vol t−1 −0.014 −1.49 −0.004 −0.28 −0.018 −1.39

Skew t−1 0.019 1.29 0.008 0.80 0.052 4.88

Panel B: Open-Ended Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t −0.053 −2.43 −0.332 −1.63 −0.042 −1.85

Vol t+1 0.010 3.91 −0.193 −2.81 0.013 6.41

Vol t −0.021 −0.91 0.102 1.54 −0.024 −0.94

Skew t 0.046 7.57 0.023 2.76 0.055 4.90

Vol t−1 0.027 1.07 0.048 2.60 0.025 1.00

47

Skew t−1 −0.004 −0.64 0.036 4.55 −0.016 −3.27

Panel C: Hedge Funds

All Funds

Coeff T-value

Performance t −0.040 −3.21

Vol t+1 0.021 2.34

Vol t −0.008 −0.61

Skew t 0.054 3.72

Vol t-1 0.003 0.35

Skew t-1 −0.014 −1.57

Panel D: Test Differences in Coefficients on Performance t Across Fund Types

Test P-value

CEFs=OEFs=HFs 0.365

CEFs=OEFs 0.661

CEFs=HFs 0.162

OEFs=HFs 0.524

Bond CEFs=Bond OEFs 0.163

Equity CEFs=Equity OEFs 0.544

48

Table 4. Regression of Fund Skewness on 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:

is the average of the difference between fund i’s returns and its peer fund returns

based on 24 monthly returns up to month t, i.e., from month t−23 to month t. and

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. and

are the lagged fund volatility and skewness around the peer

fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in returns during the 24-month

interval. and

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. 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 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.

Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Group

Coeff T-value Coeff T-value Coeff T-value

Bottom 20% Performance t-1 −0.018 −0.35 0.010 0.29 −0.032 −0.67

Vol t+1 0.051 3.09 0.026 1.02 0.053 2.39

Vol t −0.032 −2.36 −0.024 −1.46 −0.016 −0.91

Skew t 0.100 2.89 0.026 0.28 0.094 2.20

Vol t−1 0.018 0.98 0.046 2.88 −0.003 −0.27

Skew t-1 −0.040 −0.90 −0.044 −0.87 0.022 0.64

P2 Performance t-1 −0.299 −3.50 −0.131 −1.42 −0.204 −1.64

Vol t+1 0.013 0.87 0.019 0.58 0.024 1.53

Vol t 0.034 1.28 0.012 0.21 0.042 1.92

Skew t 0.021 0.61 0.005 0.15 0.019 0.71

Vol t−1 −0.030 −1.46 −0.003 −0.06 −0.048 −2.14

Skew t−1 0.011 0.31 −0.016 −0.41 0.041 1.07

P3 Performance t−1 −0.346 −3.22 −0.155 −0.76 −0.017 −0.11

Vol t+1 0.010 0.60 0.037 1.04 0.001 0.05

Vol t 0.132 3.58 0.152 2.64 0.129 6.20

49

Skew t 0.059 1.23 0.019 0.30 0.033 0.46

Vol t−1 −0.117 −3.08 −0.160 −3.25 −0.115 −4.42

Skew t−1 −0.032 −0.82 0.006 0.12 0.016 0.34

P4 Performance t−1 0.090 0.68 −0.034 −0.26 −0.194 −1.52

Vol t+1 0.006 0.46 0.003 0.17 0.003 0.22

Vol t 0.105 3.44 0.046 0.89 0.081 3.95

Skew t 0.007 0.23 −0.037 −0.67 −0.015 −0.33

Vol t−1 −0.082 −3.74 −0.034 −0.76 −0.068 −2.90

Skew t−1 0.061 1.92 0.086 1.53 0.106 3.83

Top 20% Performance t−1 −0.013 −0.53 −0.055 −1.25 −0.042 −3.37

Vol t+1 0.033 3.15 0.017 0.85 0.048 4.96

Vol t −0.007 −0.39 0.022 1.21 −0.027 −2.24

Skew t 0.036 1.25 0.063 2.21 0.025 0.77

Vol t−1 0.006 0.38 −0.007 −0.35 0.015 1.45

Skew t−1 0.063 1.94 0.015 0.36 0.086 2.40

Panel B: Open-Ended Funds

All Funds Bond Funds Equity Funds

Group

Coeff T-value Coeff T-value Coeff T-value

Bottom 20% Performance t-1 0.014 0.43 0.021 0.13 0.021 0.53

Vol t+1 0.016 1.61 0.036 0.26 0.019 1.78

Vol t 0.003 0.06 0.012 0.10 0.008 0.16

Skew t 0.045 2.01 0.069 1.91 0.055 2.88

Vol t−1 0.015 0.36 −0.009 −0.09 0.013 0.29

Skew t−1 −0.020 −0.95 −0.005 −0.11 −0.023 −1.24

P2 Performance t−1 0.075 0.68 −0.719 −1.67 0.189 2.45

Vol t+1 0.006 0.51 −0.277 −1.70 0.013 2.68

Vol t −0.020 −0.52 0.099 0.85 −0.003 −0.07

Skew t 0.019 1.38 0.020 1.48 0.018 1.31

Vol t−1 0.025 0.52 0.052 1.59 0.017 0.31

Skew t−1 0.005 0.43 0.039 2.39 −0.013 −1.66

P3 Performance t−1 −0.663 −1.96 −1.879 −2.12 −0.085 −2.22

Vol t+1 0.007 0.86 −0.339 −1.98 0.013 1.93

Vol t −0.056 −0.89 0.294 2.24 −0.078 −1.95

Skew t 0.046 3.94 −0.005 −0.39 0.057 6.02

Vol t−1 0.076 1.11 −0.025 −0.22 0.078 1.68

Skew t−1 −0.006 −0.67 0.041 2.65 −0.016 −1.54

P4 Performance t−1 0.064 0.85 −0.799 −1.83 −0.025 −0.39

Vol t+1 −0.010 −0.64 −0.392 −2.71 0.008 0.57

Vol t −0.051 −1.51 0.257 1.89 −0.060 −2.21

Skew t 0.023 2.23 0.001 0.05 0.022 1.39

Vol t−1 0.091 2.27 0.118 1.15 0.066 2.09

Skew t−1 0.008 0.63 0.016 0.90 0.004 0.15

Top 20% Performance t−1 −0.065 −2.63 −0.188 −2.69 −0.061 −2.08

Vol t+1 0.012 5.43 −0.317 −5.46 0.013 5.15

Vol t −0.006 −0.60 0.137 2.34 −0.012 −0.83

Skew t 0.072 4.53 0.042 1.94 0.074 4.10

50

Vol t−1 0.013 0.69 0.123 2.43 0.018 1.04

Skew t−1 −0.003 −0.22 0.058 1.03 −0.010 −0.53

Panel C: Hedge Funds

All Funds

Group

Coeff T-value

Bottom 20% Performance t-1 −0.025 −0.90

Vol t+1 0.029 7.19

Vol t −0.067 −8.21

Skew t 0.018 0.61

Vol t−1 0.054 4.00

Skew t−1 −0.036 −3.04

P2 Performance t−1 −0.121 −0.93

Vol t+1 0.008 0.35

Vol t −0.018 −0.63

Skew t 0.089 4.46

Vol t−1 0.016 0.46

Skew t−1 −0.059 −1.94

P3 Performance t−1 −0.222 −3.73

Vol t+1 0.008 0.29

Vol t 0.009 0.25

Skew t 0.018 0.75

Vol t−1 0.000 0.00

Skew t−1 0.025 1.20

P4 Performance t−1 0.052 0.65

Vol t+1 0.027 2.00

Vol t 0.086 2.68

Skew t 0.058 1.64

Vol t−1 −0.089 −3.87

Skew t-1 −0.007 −0.31

Top 20% Performance t−1 −0.046 −3.14

Vol t+1 0.025 3.10

Vol t −0.015 −0.49

Skew t 0.084 2.10

Vol t−1 0.013 0.47

Skew t−1 0.011 0.66

Panel D: Test Differences in Coefficients on Performance t-1 Across Quintile Groups and Between the

Low- and High-Performance Groups

Test P-value

CEFs: Group 1=2=3=4=5 0.000

CEFs: Group 1=5 0.126

OEFs: Group 1=2=3=4=5 0.054

OEFs: Group 1=5 0.097

HFs: Group 1=2=3=4=5 0.006

51

HFs: Group 1=5 0.814

Bond CEFs: Group 1=2=3=4=5 0.107

Bond CEFs: Group 1=5 0.932

Bond OEFs: Group 1=2=3=4=5 0.000

Bond OEFs: Group 1=5 0.395

Equity CEFs: Group 1=2=3=4=5 0.019

Equity CEFs: Group 1=5 0.486

Equity OEFs: Group 1=2=3=4=5 0.040

Equity OEFs: Group 1=5 0.073

52

Table 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:

∑

The fractional rank (FracRank) for fund i is defined as follows: [Low]= Min ( ,0.2), [Mid]= Min (0.6, ), is fund i’s percentile 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 month t ,i.e., from month t−23 to month t. and

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. and

are the lagged fund volatility and skewness around the peer

fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in returns during the 24-month

interval. and

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 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. Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

FracRankt,1 −0.570 −1.25 −0.061 −0.25 −0.723 −0.98

FracRankt,2 −0.296 −3.57 −0.331 −4.02 −0.253 −2.29

FracRankt,3 0.339 1.58 −0.017 −0.04 0.304 1.60

Vol t+1 0.032 2.74 0.021 1.38 0.035 2.27

Vol t 0.012 1.37 0.012 0.54 0.009 0.76

Skew t 0.047 2.75 0.017 0.49 0.053 2.98

Vol t−1 −0.013 −1.36 −0.003 −0.15 −0.017 −1.23

Skew t−1 0.016 1.06 0.005 0.48 0.050 4.18

Panel B: Open-Ended Funds

All Funds Bond Funds Equity Funds

53

Coeff T-value Coeff T-value Coeff T-value

FracRankt,1 0.087 0.79 −0.895 −1.77 −0.063 −0.54

FracRankt,2 −0.118 −2.66 −0.252 −2.50 −0.050 −1.78

FracRankt,3 −0.578 −2.87 −0.269 −0.74 −0.726 −2.97

Vol t+1 0.011 4.41 −0.202 −2.98 0.014 7.49

Vol t −0.016 −0.69 0.101 1.44 −0.019 −0.68

Skew t 0.046 8.77 0.027 3.01 0.054 5.51

Vol t−1 0.029 1.21 0.038 2.29 0.026 1.07

Skew t−1 −0.005 −0.78 0.036 4.31 −0.016 −3.44

Panel C: Hedge Funds

All Funds

Coeff T-value

FracRankt,1 −0.286 −1.00

FracRankt,2 −0.124 −1.21

FracRankt,3 −0.497 −4.08

Vol t+1 0.021 2.43

Vol t −0.009 −0.64

Skew t 0.057 3.80

Vol t−1 0.005 0.58

Skew t−1 −0.014 −1.53

Panel D: Test Differences in Coefficients on Fractional Rank (FracRank t) Across Fund Types

Test P-Value

FracRankt,1: CEFs=OEFs=HFs 0.003

FracRankt,1: CEFs=OEFs 0.020

FracRankt,1: CEFs=HFs 0.001

FracRankt,1: OEFs=HFs 0.273

FracRankt,3: CEFs=OEFs=HFs 0.146

FracRankt,3: CEFs=OEFs 0.083

FracRankt,3: CEFs=HFs 0.054

FracRankt,3: OEFs=HFs 0.503

FracRankt,1: Bond CEFs= Bond OEFs 0.313

FracRankt,3: Bond CEFs= Bond OEFs 0.273

FracRankt,1: Equity CEFs= Equity OEFs 0.092

FracRankt,3: Equity CEFs= Equity OEFs 0.167

54

Table 6. Piecewise Regression of Fund Skewness on Relative Performance This table shows the relation between fund skewness over the 12-month period and relative fund performance in the prior 24 months. The regression is as follows:

∑

equals the difference between fund i’s relative performance in month t and (q-1)th

quintile if the relative performance lies in group q and 0 otherwise. q equals 1 (5) if the relative performance is in the bottom (top) 20%. q equals 2 if the relative performance is in the next quintile, and so on. 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 month t, i.e., from month t−23 to

month t. and

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. and

are the lagged fund volatility and

skewness around the peer fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in

returns during the 24-month interval. and

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 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.

Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Coeff T−value Coeff T−value Coeff T−value

Bottom 20% −0.101 −3.09 −0.080 −6.33 −0.078 −1.57

FracPRFMt,2 0.106 0.97 0.382 2.07 0.108 1.10

FracPRFMt,3 −0.119 −1.33 0.070 0.35 −0.044 −0.39

FracPRFMt,4 −0.191 −2.05 −0.501 −2.49 −0.208 −2.02

Top 20% −0.029 −1.41 −0.069 −1.90 −0.036 −1.26

Vol t+1 0.029 2.41 0.018 1.17 0.034 2.20

Vol t 0.015 1.78 0.018 0.98 0.012 1.24

Skew t 0.041 2.39 0.006 0.20 0.050 2.54

Vol t−1 −0.014 −1.37 −0.002 −0.17 −0.018 −1.36

Skew t−1 0.019 1.28 0.009 0.95 0.053 4.87

Panel B: Open-Ended Funds

55

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Bottom 20% −0.022 −1.10 −0.231 −0.93 −0.021 −1.04

FracPRFMt,2 0.220 2.22 0.006 0.01 0.094 1.05

FracPRFMt,3 −0.082 −0.62 −0.602 −1.09 0.129 1.11

FracPRFMt,4 −0.040 −0.96 −1.729 −2.45 0.038 1.44

Top 20% −0.098 −2.73 −0.493 −3.34 −0.098 −2.37

Vol t+1 0.011 3.77 −0.193 −2.78 0.014 6.66

Vol t −0.016 −0.66 0.113 1.92 −0.018 −0.64

Skew t 0.044 7.78 0.022 2.76 0.053 5.11

Vol t−1 0.027 1.12 0.047 2.72 0.025 1.02

Skew t−1 −0.003 −0.49 0.035 4.42 −0.014 −3.12

Panel C: Hedge Funds

All Funds

Coeff T-value

Bottom 20% −0.038 −2.46

FracPRFMt,2 0.069 1.05

FracPRFMt,3 −0.071 −1.09

FracPRFMt,4 −0.047 −0.68

Top 20% −0.054 −3.20

Vol t+1 0.021 2.36

Vol t −0.008 −0.56

Skew t 0.053 3.69

Vol t−1 0.003 0.33

Skew t−1 −0.013 −1.53

Panel D: Test Differences in Coefficients on Rank (Rank t) Across Fund Types

Test P-value

Bottom 20%: CEFs=OEFs=HFs 0.438

Bottom 20%: CEFs=OEFs 0.234

Bottom 20%: CEFs=HFs 0.434

Bottom 20%: OEFs=HFs 0.456

Top 20%: CEFs=OEFs=HFs 0.061

Top 20%: CEFs=OEFs 0.036

Top 20%: CEFs=HFs 0.493

Top 20%: OEFs=HFs 0.069

Bottom 20%: CEFs=OEFs 0.519

Top 20%: CEFs=OEFs 0.019

Bottom 20%: CEFs=OEFs 0.373

Top 20%: CEFs=OEFs 0.121

56

Table 7. Impact of Firm Characteristics on the Sensitivity of Fund Skewness to Lagged 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:

is the average of the difference between fund i’s returns and its peer fund returns

based on 24 monthly returns up to month t, i.e., from month t−23 to month t. and

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. and

are the lagged fund volatility and skewness around the peer

fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in returns during the 24-month

interval. and

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. is fund i’s age (the natural logarithm of months since inception) in

month t, and is fund i’s size (the natural logarithm of TNA) in month t. 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 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.

Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t 0.011 0.07 −0.429 −1.00 0.124 0.64

Vol t+1 0.035 2.17 0.001 0.03 0.038 2.01

Vol t 0.021 1.35 0.079 1.93 0.007 0.49

Skew t 0.048 1.97 0.030 0.91 0.046 1.44

Vol t−1 −0.016 −1.43 −0.082 −2.56 −0.006 −0.49

Skew t−1 0.013 0.72 −0.024 −2.22 0.059 3.83

age t −0.027 −0.60 −0.063 −0.60 −0.015 −0.37

size t 0.069 2.18 0.098 1.48 0.016 0.41

Performance t x age t −0.075 −1.52 0.215 3.65 −0.119 −2.00

Performance t x size t 0.053 2.10 −0.144 −1.14 0.067 1.95

Panel B: Open-Ended Funds

57

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t 0.047 1.05 −0.329 −1.28 0.039 0.93

Vol t+1 0.010 3.96 −0.188 −2.53 0.013 7.09

Vol t −0.017 −0.72 0.088 1.61 −0.023 −0.86

Skew t 0.046 8.29 0.029 3.74 0.055 4.91

Vol t-1 0.024 0.94 0.076 1.18 0.024 0.95

Skew t-1 −0.005 −0.76 0.028 3.51 −0.016 −3.24

age t 0.000 0.01 0.007 0.15 0.003 0.28

size t −0.001 −0.41 −0.016 −1.92 0.002 0.86

Performance t x age t −0.062 −2.79 −0.037 −0.52 −0.055 −2.38

Performance t x size t 0.006 2.53 0.023 1.78 0.006 3.15

Panel C: Hedge Funds

All Funds

Coeff T-value

Performance t 0.251 5.30

Vol t+1 0.021 2.10

Vol t 0.003 0.21

Skew t 0.048 3.24

Vol t−1 −0.006 −0.60

Skew t−1 −0.006 −0.71

age t −0.032 −0.51

size t 0.000 −0.07

Performance t x age t −0.131 −5.72

Performance t x size t −0.009 −1.70

Panel D: Test Differences in Coefficients on Performance t x age t and Performance t x size t Across

Fund Types

Test P-value

Performance t x age t CEFs=OEFs=HFs 0.171

Performance t x age t: CEFs=OEFs 0.548

Performance t x age t: CEFs=HFs 0.236

Performance t x age t: OEFs=HFs 0.132

Performance t x size t: CEFs=OEFs=HFs 0.092

Performance t x size t: CEFs=OEFs 0.052

Performance t x size t: CEFs=HFs 0.038

Performance t x size t: OEFs=HFs 0.254

Performance t x age t: Bond CEFs= Bond OEFs 0.058

Performance t x size t: Bond CEFs= Bond OEFs 0.067

Performance t x age t: Equity CEFs= Equity OEFs 0.011

Performance t x size t: Equity CEFs= Equity OEFs 0.000

58

Table 8. Impact of Macroeconomic Variables on the Sensitivity of Fund Skewness to Lagged 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:

is the average of the difference between fund i’s returns and its peer fund returns

based on 24 monthly returns up to month t, i.e., from month t−23 to month t. and

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. and

are the lagged fund volatility and skewness around the peer

fund return by first differencing fund i’s returns from the peer fund returns from month t−24 to month t−1 and then computing the volatility and skewness on these differences in returns during the 24-month

interval. and

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. Public information variables include the lagged one-month Treasury Bill (T-Bill t), the lagged dividend yield (DivYield t), the lagged term spread (YieldSp t), and the lagged default spread (DefaultSp t). Panels A, B, and C summarize results for closed-end funds, open-ended funds, and hedge 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.

Panel A: Closed-End Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t −0.104 −1.44 −0.009 −0.04 −0.114 −1.72

Vol t+1 0.031 2.42 0.018 1.23 0.034 1.98

Vol t 0.018 1.81 0.028 1.67 0.015 1.29

Skew t 0.042 2.58 0.013 0.38 0.054 2.98

Vol t−1 −0.012 −1.21 −0.009 −0.69 −0.017 −1.29

Skew t−1 0.018 1.28 0.007 0.69 0.050 4.38

T-Bill t 0.007 0.78 0.007 0.47 0.001 0.06

DivYield t 0.054 0.98 0.178 4.63 −0.056 −0.63

YieldSp t 0.015 0.99 0.011 0.43 0.015 1.06

DefaultSp t 0.023 0.24 0.064 0.80 0.001 0.01

Performance t x T-Bill t 0.018 0.90 −0.015 −0.42 0.035 1.16

Performance t x DivYield t −0.031 −0.69 −0.077 −1.25 −0.038 −0.69

Performance t x YieldSp t 0.019 0.85 −0.027 −0.59 0.038 1.10

59

Performance t x DefaultSp t −0.007 −0.11 0.174 1.48 −0.080 −1.87

Panel B: Open-Ended Funds

All Funds Bond Funds Equity Funds

Coeff T-value Coeff T-value Coeff T-value

Performance t −0.132 −0.82 −0.898 −1.65 −0.149 −0.98

Vol t+1 0.011 3.99 −0.205 −2.97 0.014 6.55

Vol t −0.022 −1.03 0.113 1.39 −0.029 −1.15

Skew t 0.042 8.67 0.026 3.57 0.050 5.03

Vol t−1 0.028 1.15 0.047 2.15 0.029 1.24

Skew t−1 −0.001 −0.22 0.036 4.13 −0.012 −2.38

T-Bill t −0.012 −1.65 0.037 1.74 −0.023 −3.47

DivYield t −0.039 −0.48 −0.344 −2.39 0.038 0.57

YieldSp t −0.040 −3.36 −0.008 −0.32 −0.045 −3.36

DefaultSp t −0.062 −1.29 0.098 1.17 −0.108 −1.56

Performance t x T-Bill t 0.045 1.84 0.013 0.09 0.048 2.29

Performance t x DivYield t −0.122 −5.92 −0.016 −0.08 −0.100 −5.22

Performance t x YieldSp t 0.064 2.57 0.165 1.17 0.066 2.92

Performance t x DefaultSp t 0.029 0.43 0.241 1.57 0.008 0.10

Panel C: Hedge Funds

All Funds

Coeff T-value

Performance t 0.148 2.07

Vol t+1 0.020 2.13

Vol t −0.009 −0.60

Skew t 0.041 2.89

Vol t−1 0.006 0.70

Skew t−1 0.000 0.03

T-Bill t −0.014 −0.95

DivYield t 0.060 0.88

YieldSp t −0.037 −1.21

DefaultSp t −0.147 −1.71

Performance t x T-Bill t 0.012 1.44

Performance t x DivYield t −0.187 −5.39

Performance t x YieldSp t 0.030 2.43

Performance t x DefaultSp t 0.012 0.81

60

Table 9. Convexity Impact on Tail Risks in Closed-End Funds—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 month t, 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.

⁄

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 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 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)

61

Table 10. Convexity Impact on Tail Risks in Open-Ended Funds—Tournaments 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 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.

Panel A: Bond Funds

Equal-Weighted Value-Weighted

Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

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 Equal-Weighted Value-Weighted

Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

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

F-test of Differences in Skewness: Top 20% – Bottom 20% p-value=0.032 (EW) and p-value=0.033 (VW)

62

Panel C: Skewness Decomposition of Equal-Weighted Equity Fund Returns

Skewness Coskewness

Idiosyncratic Coskewness

Idiosyncratic 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

63

Table 11. Convexity Impact on Tail Risks in Open-Ended Funds—Fund Flow-Performance Relation 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. The quarterly fund flows are calculated as follows:

[ ( )]

where is the total net assets of the fund and 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 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 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. Panel A: Bond Funds

Equal-Weighted Value-Weighted

Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

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

Equal-Weighted Value-Weighted

64

Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

Bottom 20% 0.026 0.075 −0.118 1.150 0.031 0.077 −0.052 1.394

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)

65

Table 12. Convexity Impact on Tail Risks in Open-Ended Funds –Tournament and Fund Flow-Performance Relation 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. The quarterly fund flows are calculated as follows:

[ ( )]

where is the total net assets of the fund and 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. Panel A: Bond Funds

Equal-Weighted Value-Weighted

Flow Tourn Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

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

66

P1 P5 0.028 0.027 0.751 1.070 0.028 0.029 0.788 1.491

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

Equal−Weighted Value−Weighted

Flow Tourn Mean StdDev Skewness Kurtosis Mean StdDev Skewness Kurtosis

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

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

67

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

68

Table 13. Convexity Impact on Tail Risks in Hedge Funds—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 month t, the moneyness of an option is calculated as:

( )

where is fund i’s return at t+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:

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.

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)