USC FBE FINANCE SEMINAR presented by Jerchern LIn FRIDAY, Nov. 18, 2011 10:30 am – 12:00 pm, Room: JKP-202

Fund Convexity and Tail Risk-Taking by

Jerchern Lin*

First draft: April 6, 2011 This draft: November 13, 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 to improve performance at the expense of significant downside risk. 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 great downside risk.

* PhD student, Marshall School of Business, University of Southern California, 3670 Trousdale Parkway, Suite 308, Los Angeles, CA 90089-0804. Email: [email protected], 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 in fact 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 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 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 tournament 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 due to tournaments.

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perform well will reduce variance but show little evidence of increasing volatility for underperforming fund managers. The mixed results on risk-taking can be attributed to the longer evaluation period of fund managers (Hodder and Jackwerth (2007), Panageas and Westerfield (2009)), disincentives to liquidate funds owing to career concerns (Chevalier and Ellison (1999)) or reputation costs (Fung and Hsieh (1997)), or managerial stake 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 tail risks. Closed-end fund managers offer investors the opportunity to buy illiquid assets. Changes in discounts (market-to-book ratio) in closedend funds can be regarded as the return of the implicit option that investors sell to the management. An example of explicit convexity is high-water mark contracts 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 returns 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. It is also important to distinguish between these two types of bets, since investors can easily assess fund risk but are often unaware of tail risks. Moreover, the relation between convex incentives to skewed bets cannot be easily inferred from the mixed results in the literature.3 This paper first studies how a fund’s performance relative to its peers relates to a fund manager’s short-term tail risk-taking behavior. Fund managers have a strong incentive to improve or camouflage short-term performance because investors react to outcomes at short horizons. 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 3

Risk-seeking may imply that an outperforming fund manager tends to take more positively skewed bets because past gains allow a cushion for losses, and underperformance may encourage more skewed bets due to the protection on the downside and a strong desire to rise in relative rankings. On the contrary, risk shift may suggest that outperformance leads to fewer skewed bets because of loss and risk aversion and a strong incentive to remain at the top, and an underperforming fund manager would trade negative skewness owing to fears of fund liquidation or bet on positively skewed trades because of managerial shares in funds.

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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 gamble on a succession of solid returns to remain at the top of the rankings by taking chances that have a tiny probability of significant downside risk because negatively skewed bets have a higher probability of being winners than risky bets. High outside opportunities may encourage top managers to undertake negatively skewed bets and underestimate the magnitude of downside risk. Most interestingly, this tail risk-taking behavior is prevalent across fund types. The implications of the relation between skewed bets and relative performance are twofold. Fund managers’ incentive to bet on assets or strategies that could possibly produce a lottery reward when fund values are low conflicts with the risk-shifting behavior documented in the literature due to the managers’ longer evaluation periods, career concerns, or reputation costs. If fund values are near liquidation, fund managers can still bet on positively skewed investments, even though the risk in funds appears to be moderated. Second, the evidence of negatively skewed bets suggests that outperforming fund managers have a stronger incentive to keep the option in the money than the one predicted by symmetric risk. A successful fund manager may underestimate the probability and magnitude of an extreme downside event based on a false belief that past gains are sufficient to cushion against a downfall. The misconception encourages a successful fund manager to bet on negatively skewed investments. Negatively skewed bets further assure fund managers that the incentive contract is less likely to fall in the out-of-money zone than symmetric risk-taking. I further examine how convexity affects tail risks in different types of investment funds. Even though an explicit option contract does not exist, compensation can still be convex in performance. Both implicit and explicit convexities are examined in this study. The degree of convexity is measured as: (1) the discounts 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.

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To my knowledge, this paper is the first to examine how subsequent skewness is related to past performance and call-like features in incentives. 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 shift, 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 riskseeking 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? 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 these assets can reduce 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, market timing strategies can induce negative skewness. A market timer adjusts betas conditional on the realization of systematic factors, such as positive or negative market excess returns, and a positive market timing strategy generates negatively

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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. Mutual fund managers can write a covered call on the S&P500 index to reduce downside risk of the portfolio by limiting the upside potential of fund returns. The covered call writing also yields steady profits but incurs considerable losses when market volatility jumps.

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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 occur by the convex or concave payoff from trading strategies. Likewise, skewed trades are 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. 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 the relation of price fall and price yield has negative convexity. Interest rate products, such as swaps, offer bond managers a steady stream of interest payments but expose investors to interest rate risk and credit risk. The option-adjusted spreads in interest rate derivative products are also embedded with 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 complex fixed-income derivatives and structured products, such as pass-through securities, credit-structured products, and callable range accruals, and these products yield asymmetric payoffs. In such investments, bond fund returns appear to be skewed. Convexity in compensation structure is itself asymmetric. The convexity feature in compensation implies that fund managers value upside gains from increased compensation but are not penalized as much by downside losses. Fund managers can gain compensation by taking negatively skewed bets because the chance of earning steady profits asymmetrically outweighs the probability of losses. Fund managers are also 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. 5

For example, Engle and Mistry (2007) study negative skewness in Fama and French factors and Carhart's momentum factor.

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It is important to look at asymmetric risk beyond symmetric risk because skewness risk can help reflect the true risk of a fund. For instance, the high Sharpe ratio of hedge funds does not suggest that investors should consider them less risky investment alternatives. 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. In addition, 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 on lower-risk assets, the fund appears to have low risk and high risk-adjusted performance. If a fund manager shorts options and invests the premiums at the risk-free rate, the fund appears to have low risk, but the downside risk of this trade can mean substantial losses if the market plummets. As such, to infer the true risk hidden in the fund, one needs to look beyond volatility and measure skewness risk. Second, investors should be aware of the downside risk hidden in a fund. Fund managers can easily hide trades with significant downside risk from investors. Third, 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 risk premiums on skewness risk and that skewness risk can affect fund managers’ optimal asset allocation decisions. Investors should be compensated for higher-risk premiums if a fund’s tail risk exposures are accounted for. 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 theories and empirics show mixed results. Grinblatt and Titman (1989) and Carpenter (2000) show that a fund manager increases portfolio risk when the incentive contract is out of the money. The difference between these two papers lies in the hedgeability of the option and the class of utility function with hyperbolic absolute risk aversion (HARA) assumed by Carpenter (2000). As such, 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. In contrast, Kouwenberg and Ziemba (2007) show that loss-averse managers invest a higher proportion of risky assets in a fund in response to

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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. They also support these arguments empirically. 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 with the 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 does not necessarily lead to risk-taking due to longer evaluation periods, even for a riskneutral fund manager. In contrast to Hodder and Jackwerth, they impose the conditions that managers are evaluated instantaneously and that the shutdown barrier occurs at random in the future. Hu et al. (2011) show that a higher probability of termination leads managers to increase portfolio risk. Likewise, my results indicate that a higher probability of termination does indeed lead managers to increase skewness risk in a portfolio to try to make the option stay in the money. The reasons that factor into moderate risk-taking behavior should reject skewness risk more strongly because the probability of losing is much higher than the one under symmetric risk. Instead, I find that managers’ behavior on skewness risk exposures is supported by the data. This implies the importance of asymmetry in risk in the literature on tournaments and convex compensation. 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

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strategies are conditional on relative performance, instead of absolute performance such as a highwater 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) identify 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 indeed decrease variance, even though they face convexity in their compensation. These mixed results prompt the need to look at fund managers’ positions regarding skewness risk and to examine how convexity affects skewness risk. The reasons that underperforming funds reduce risk should encourage managers not to take skewed bets. However, 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 small. 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. 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

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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, and because investors may not be aware of the downside risk hidden in skewed trades. 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. Examples of high outside opportunities include “open-ending” for closed-end funds and switching from managing open-ended funds to handling hedge funds. On the other hand, I find that the top fund managers do take more negatively skewed bets to keep the incentive contract in the money. Although this confirms with the findings of risk reduction for outperforming funds in the literature, the inference of negatively skewed trades in the long horizon is different. When a top-performing fund manager has an incentive to reduce skewness risk, investors can suffer substantial losses in the long run, and this result is not documented by the existing models or empirics. It is also intriguing that outperforming fund managers take negative skewed bets in the short run, because any 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, but fund managers consider them safe investments in the short run and ignore the expense and possibility of large drawdowns in fund values at longer horizons. In addition, 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 differently when fund values fall in the upside and downside regions.

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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 time periods of funds. Open-ended funds are selected from the CRSP database. Hedge funds are from the Hedge Fund Research (HFR) database. Fama and French (2010) document that a selection bias due to missing returns exists in the CRSP mutual fund database before 1984. To be consistent in comparisons across fund types, both closed-end funds and open-ended funds start in January 1984. The starting period for hedge funds is January 1996. All datasets end in December 2008. The literature has identified several ex-post conditioning biases in fund returns: survivorship bias, back-fill bias, incubation bias, selection bias, and look-ahead bias. These biases may spuriously increase fund skewness and reduce fund kurtosis owing to the automatic addition of positive returns. Data vendors that I use provide survivorship-free datasets for open-ended funds, closed-end funds, and hedge funds to avoid survivorship bias. 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 closedend 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. Fund styles are classified by style codes provided in respective datasets. Closed-end fund styles are identified by Morningstar styles.6 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).7 Hedge funds are grouped by HFR

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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. 7 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

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main strategies.8 To construct returns of peer funds, I use all funds in the same style at any given months to calculate equal-weighted returns. Both bond and equity funds are included. 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 risktaking in the analysis. 5. Empirical Methods and Results 5.1 Changes in Tail Risks 5.1.1 Tail Risks on Relative Performance Elton, Gruber, and Blake (2003) assume that the fund manager reexamines his or her position at the end of 24 months and takes a position with respect to risk over the next 12 months. I follow the same assumption on the 36-month evaluation periods to study changes in fund tail risks across years.9 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 performance10 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.11 In addition, disjoint and unequal return and skewness periods avoid sorting bias.12

income open-ended funds are classified as Index, Global, Short Term, Government, Mortgage, Corporate, and High Yield. 8 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. 9 Similar to variance, the 36-month evaluation period allows a sufficient period to estimate skewness. I select 24 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. 10 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. 11 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.

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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 relative to that of peer funds in the next 12 months is calculated. In addition, I also calculate the percentage change in relative skewness by dividing the relative skewness in the next 12 months by the relative skewness in the previous 24 months. Table 1 reports the pooled distribution of individual fund skewness and kurtosis around the peer fund return in each quintile groups in the next 12 months after sorting on relative performance, 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.13 Clearly, fund skewness fluctuates over time, and the average fund skewness differs between outperforming funds and underperforming funds, regardless of fund

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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. 13 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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type. It is also evident that the skewness of low-performing funds is more positive than that of highperforming funds during most time periods. However, the source of skewness may come from the existing fund portfolios or from trades conditional on relative performance owing to overlapping data. 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 negative t-statistics across fund types suggest that outperforming funds have more negative skewness than underperforming funds conditional on past relative performance and contradict the alternative hypothesis that the differences in skewness are driven mainly by skewness in existing holdings. 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 second and the third moment of

, denoting the fund volatility and skewness relative to the peer fund.

and

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are fund volatility and skewness during months t+1 and t+12. Time fixed effects are year dummies, and standard errors are clustered at the style level.14 Time fixed effects would absorb commonality 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 across time. 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. The coefficients on relative performance are negative and statistically significant across fund types, reinforcing the findings by the sorting methodology above. A one standard deviation increase from the peer fund returns decrease 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.15 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 14

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 quantitatively unchanged. 15 The one standard deviations from the peer funds are 5.65%, 2.54%, and 4.43% for closed-end funds, open-ended funds, and hedge funds, respectively.

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positively skewed bets to improve their rankings by the evaluation date. This contradicts with the riskshifting behavior of an underperforming manager found in the literature. Risk shift predicts that losing managers will take negatively skewed bets to hedge their position or that they have no incentive to take skewed bets to avoid job loss or reputation damage. In contrast, my finding on managers’ reduction in portfolio skewness coincides with the risk-shift behavior of an 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 cushions 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 skewness

17

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. To study the possibility that the incentive to take skewed bets may differ across fund managers, I estimate equation (1) separately across groups. 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 can indicate that funds around the kink of the convexity take more skewed bets. 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 pvalues of less than 0.1. These results of the test on the differences in coefficients on relative performance show that fund managers’ positions on skewed trades respond differently to past relative performance. Fund managers around the kink take skewed bets more aggressively. In addition, outperforming fund managers have a stronger incentive to take skewed bets than underperforming fund managers. This result has two implications. First, extremely outperforming managers are worried about losing the tournament rankings by the evaluation date. Second, nonetheless, negatively skewed bets may generate solid profits over time, but the hidden risk that the fund could blow up may incur substantial losses for investors when an extreme event occurs.

18

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 ( more negative than that on the bottom fractional rank (

) is

). 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 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 show some significance to reject the hypothesis of equal coefficients on the top and

19

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 closedend 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, borrow lots of debt, and invest more income-producing assets (e.g., bonds and preferred securities) to offer high yields to investors. 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 open-ended fund managers comes from a combination of options instead of a strictly convex or concave one. Therefore, close-end fund managers have less incentive to remain at the top than openended fund or hedge fund managers. To further explore findings in Table 5, I examine the differential response of relative skewness for the next 12 months to past relative performance by the following regression. ∑

Table 6 shows the results of the 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 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 ). Hedge funds have statistically significant loadings on the coefficients and satisfy the

20

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. Closed-end funds have low statistical significance on the coefficients of the ranks, except the bottom 20% group. Unlike open-end funds and hedge funds, 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 risk-taking behavior in the closed-end fund, open-ended, and hedge fund industries in relation to discounts, the relation of tournaments and 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 will aggressively write covered calls to protect against downside losses.16 As a result, the subsequent fund returns exhibit negative skewness. 5.1.2 The Impact of Fund Characteristics on Skewed Bets—Size and Age From the previous literature, we know that the size and age of a fund influence how fund managers take risk when facing convexity in compensation (e.g., Chevalier and Ellison (1997, 1999)). 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: 16

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.

21

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. Open-ended funds and closed-end funds have positive and significant interaction terms between relative performance and size. Hedge funds display a marginally negative (−0.009) and marginally significant interaction term between relative performance and size at the 10% level. These findings indicate that: (1) managers of young funds are less likely take skewed bets, and (2) managers of small funds, except hedge funds, tend to take more skewed bets. The first finding is consistent with results in Chevalier and Ellison (1999) and others. Young fund managers have career concerns and would like 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 as high flexibility in trading strategies and broad asset classes in hedge funds are documented in the literature. 5.1.3 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

22

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 become more conservative and 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 and makes winners subjectively overweight it. Consequently, outperforming managers change trading behavior and become more risk averse. This is observed in Figure 1. Across all fund types, outperforming funds consistently take more negative skewness after 2002. In short, could we relate a fund manager’s skewness risk-taking in response to relative performance to changing economic variables? 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

23

(YieldSp t), and the lagged default spread (DefaultSp t). 17 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. 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 and liquidity. 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. 5.2 Convexity and Tail Risks 5.2.1 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. Cherkes, Sagi, and Stanton (2009) show that closed-end fund investors buy an option on liquidity because the cost of direct investments on illiquid assets is high. As such, if a fund manager generates liquidity benefits to investors, funds will be 17

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

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. The premium or discount is similar to high-water marks faced by a hedge fund manager. It measures the option added values “standardized” by net asset value—that is, how much option value is created by a manager by a one dollar increase in net asset value. It is important to use measures relative to net asset value because option values can simply increase by the investment opportunities a fund has. Therefore, the premium and discount can be used to measure the degree of convexity that a closed-end fund faces. Unlike the explicit option contracts in hedge funds, a closed-end fund manager faces implicit optionality in incentives. I calculate the closed-end fund premium and discount as follows: ⁄

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 and

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

are the fund’s stock return and net asset value return, respectively. Equation (7)

implies that when a fund’s net asset value increases, the option return increases and changes in discounts increase—that is, there are higher premiums. The manager has an implicit incentive to improve the fund’s option or stock return, and the option on the fund’s net asset value introduces the optionality in incentives. Because the compensation for a closed-end fund manager is a fraction of the fund’s net asset value, the manager has an incentive to 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. In doing so, closed-end

25

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 and assign their rankings to the next month (ex-post ranking). 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 lower the discount, the higher the future returns. The bottom 20% of funds (most discounts) 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 the finding in Thompson (1978) and Pontiff (1995) that fund premia are negatively correlated with future returns. Second, it is interesting to observe that the equity funds with most discounts display more negative skewness than those with most premia, but bond funds show the opposite patterns. For equalweighted (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

26

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. 5.2.2 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. As such, we can regard peer fund returns as the strike an open-end fund manager has to face in his or her implicit convexity in incentives. I sort funds by relative performance as the average of the difference between fund returns and peer fund returns from the beginning of a year to each quarter t within the same year into quintile groups and assign their rankings to quarter t. For instance, I calculate the average of the difference between fund returns and peer fund returns from January to February in the year 2000 and sort all funds by the average difference. Then I assign their rankings to March 2000. Similarly, for the second, third, and fourth quarter of a year, I use the average of the differences in returns between funds and peer funds up to May, August, and November in the same year to sort funds and assign their rankings to respective quarters. This sorting mechanism assumes that fund managers reevaluate their positions relative to peers quarterly and care about the relative performance from the beginning of the year up to each quarter.18 The reason to use a quarterly frequency is to match the frequency of the fund flowperformance relation below since both forms of convexity may work together to induce fund managers’ tail risk-taking behavior. 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 18

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.

27

future returns. This may be attributed to momentum effects or persistence in short-term performance. Lin (2011) reports that coskewness contributes to 75% to 90% of equity open-ended fund skewness, depending on the choice of systemic factors.19 If a manager over-weights the portfolio with assets that have positive coskewness or constantly use positively skewed bets, the regression coefficient that captures the coskewness risk must be negative. The return spread or coskewness factor, measured by selling the assets that have the most negative coskewness and buying the assets that have the most positive coskewness, is positive (see e.g., Harvey and Siddique (2000)). Therefore, when funds that perform poorly take positively skewed bets, systematic coskewness is negative. This explains why the bottom 20% equity open-ended funds show the most negative skewness in Table 10. 5.2.3 Fund Flow-Performance Relation in Open-Ended Funds The other type of convexity that an open-ended fund may face is fund flow-performance relation. Fund manager have an incentive to take trading strategies that increase assets under management since their compensation is based on them. 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.

19

Lin (2011) also reports that the percentage of coskewness in bond open-ended fund skewness is 64% and 23% when the systematic factor is equal-weighted portfolios of funds or exogenous bond factors, respectively. The difference can be attributed to missing systematic factors for bond funds. Therefore, it is less clear how coskewness risk affects bond open-ended funds.

28

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 (−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. The fund managers with the lowest sensitivity of fund flows to past performance (the bottom 20%) take more positively skewed trades than those facing the most convexity from flows. This is counterintuitive since convexity should imply positive skewness or positively skewed trades. The results show that when a fund manager faces convexity from flows, he or she will instead take more negatively skewed trades. 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. As such, I generate portfolios based on two0-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.

29

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. 5.2.4 High-Water Marks in Hedge Funds Hedge fund managers face high-water mark provisions, and their compensation structure is thus convex. The high-water mark of each fund is initially determined by the cumulative return of the first 12 months. Then the high-water mark is reset by the maximal returns achieved to date. If the cumulative returns are negative, hedge funds managers need to cover these losses first before incentive fees are paid. Following Getmansky, Lo, and Makarov (2004), the high-water mark is updated every month as follows: (

)

The gamma of an option based on the Black Scholes formula is normal probability distribution function, risk-free rate,

( ) ( √

)

√

, where N’(.) is the standard

, S is the stock price, X is strike, r is the

is stock volatility, and T is 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 others are held constant.20 The moneyness is calculated as: ( where

)

is fund i’s price and fund 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 in the top 20% group face the most convexity.

20

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

and

, yields

30

Table 13 shows the results on skewness risks with respect to log moneyness. As expected, future returns increase with log moneyness, except for the top 20% group. The middle two groups (P3 and P4) exhibit more positive skewness than the other groups. The bottom two groups exhibit more negative skewness than the top two groups. In particular, the P4 group shows a positive skewness of 0.213 (1.232) for equal-weighted (value-weighted) returns. This implies that when a hedge fund faces less (more) convexity, the fund manager will take negatively (positively) skewed bets. 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.21 One concern of the analysis in this section is that because idiosyncratic tail risks are diversified away in portfolios of funds, my findings show only that convexity affects systematic tail risks. However, empirical evidence has documented that idiosyncratic tail risks are still present in portfolios of funds. Lin (2011) shows that idiosyncratic skewness contributes to fund skewness between 31% (openended funds) to 44% (hedge funds) for investment funds at the style level. Brown, Gregoriou, and Pascalau (2012) study funds of hedge funds and document that overdiversification leads to increased tail risk exposures. In summary, results from Tables 9 to 13 imply how convexity in incentives and managerial behavior toward skewed bets affect total skewness risk at the portfolio level. Fund skewness risk comes from coskewness, idiosyncratic coskewness, and idiosyncratic skewness (see Lin(2011)). When a fund manager takes positions with respect to skewness risk, risk premiums associated with both systematic coskewness factors and the loadings that capture both coskewness risks affect the sign and magnitude of systematic skewness risk at the portfolio level. For example, when the portfolios of funds exhibit minimal idiosyncratic skewness and coskewness is the main contributor to total fund skewness, such as equity open-ended funds, negatively skewed bets imply more positive systematic (co)skewenss. Sorting techniques are based on the degree of convexity or “characteristics” in month t and assigning ranks in

21

th

k−1

The k moment of portfolios of funds is O(1/n ).

31

the next month (ex post). I sort funds by the convexity faced by the fund manager in month t and divide funds into quintile groups every month to construct equal-weighted and value-weighted time series of returns for each group. If the returns of the group with the most convexity display more positive skewness than those of the group with the least convexity, it may be inferred that fund managers who face the most (least) convexity take negatively (positively) skewed trades. However, this inference holds only if idiosyncratic skewness is diversified away at the portfolio level, and there is a systematic skewness factor consistently contributing to most fund skewness. 6. Conclusion This paper extends the literature on managerial incentives and risk-taking behavior to skewness risk. Two fundamental questions are asked in this study. First, do fund managers take positions with respect to skewness risk relative to their relative rankings in the tournaments—that is, relative to their peers? If yes, do we observe that open-ended fund and hedge fund managers take skewed positions differently from closed-end fund managers in a systematic way since the literature has well documented the convexity faced by former two types of managers—tournaments, fund flow-performance relationship, and option payoffs? Second, does convexity faced by fund managers impact tail risk-taking? Do we observe any systematic patterns across fund types? In other words, the first question asks how a fund manager takes skewed bets conditional on past relative performance, and the second question asks which skewed trades a fund manager will take conditional on convexity in incentives that he or she faces. I show that when a fund manager underperforms (outperforms) his or her peers, he or she takes more positively (negatively) skewed trades. This is quite intuitive since betting on lottery-like returns, if successful, can significantly increase fund performance and relative rankings but is more likely to yield steady losses. On the other hand, if a fund manager has been successful, he or she is more motivated 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.

32

In addition, I show that convexity can affect fund skewness. This implies that fund managers take positions with respect to skewness risk in response to the convexity that they face. The discounted closed-end fund returns are more negatively skewed than their premium counterparts. The open-ended funds with the worst relative performance have more negatively skewed returns than the outperforming funds. This coincides with our intuition that convexity increases skewness. However, the fund flow-performance relation offers counterintuitive results. The open-ended funds that face the most convexity from expected flows exhibit more negative skewness than those facing the least convexity. Double sorting on relative performance and expected fund flows does not identify a dominant effect from either source. I also find that hedge fund tail risks are related to convexity induced by high-water marks relative to a fund’s net asset values.

33 References Aragon, George O., and Vikram K. Nanda, 2011, On tournament behavior in hedge funds: high water marks, fund liquidation, and the backfilling bias, Unpublished working paper. Anson, Mark J. P., 2002, Symmetrical performance measures and asymmetric trading strategies: A cautionary example, Journal of Alternative Investments 5, 81–86. Barberis, Nicholas, and Ming Huang, 2008, Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review 98, 2066–2100. Berk, Jonathan, and Richard Stanton, 2007, Managerial ability, compensation, and closed-end fund discount, Journal of Finance 62, 529–556. Brown, Keith C., W. V. Harlow, and Laura T. Starks, 1996, Of tournaments and temptations: An analysis of managerial incentives in the mutual fund industry, Journal of Finance 51, 85–110. Brown, Stephen J., William N. Goetzmann, and James Park, 2001, Careers and survival: Competition and risk in the hedge fund and CTA industry, Journal of Finance 56, 1869–1886. Brown, Stephen J., Greg N. Gregoriou, and Razvan Pascalau, 2012, Is it possible to overdiversify? The case of funds of hedge funds, Review of Asset Pricing Studies, Forthcoming. Brunnermeier, Markus K., and Jonathan A. Parker, 2005, Optimal expectations, American Economic Review 95, 1092–1118. Busse, Jeffrey A., 2001, Another look at mutual fund tournaments, Journal of Financial and Quantitative Analysis 36, 53–73. Carpenter, Jennifer, 2000, Does option compensation increase managerial risk appetite? Journal of Finance 55, 2311–2331. Cherkes, Martin, Jacob Sagi, and Richard Stanton, 2009, A liquidity-based theory of closed-end funds, Review of Financial Studies 22, 257–297. Chevalier, Judith, and Glenn Ellison, 1997, Risk taking by mutual funds as a response to incentives, Journal of Political Economy 105, 1167–1200. Chevalier, Judith, and Glenn Ellison, 1999, Career concerns of mutual fund managers, Quarterly Journal of Economics 114, 389–432. Dass, Nishant, Massimo Massa, and Rajdeep Patgiri, 2008, Mutual funds and bubbles: The surprising role of contractual incentives, Review of Financial Studies 21, 51–99. Dittmar, Robert F., 2002, Nonlinear pricing kernels, kurtosis preference, and evidence from the cross-section of equity returns, Journal of Finance 57, 369–403.

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Elton, Edwin J., Martin J. Gruber, and Christopher R. Blake, 2003, Incentive fees and mutual funds, Journal of Finance 58, 779–804. Engle, Robert, and Abhishek Mistry, 2007, Priced risk and asymmetric volatility in the crosssection of skewness, Working paper, New York University. Fama, Eugene F., and Kenneth R. French, 2010, Luck versus skill in the cross-section of mutual fund returns, Journal of Finance 65, 1915–1948. Ferson, Wayne, and Rudi Schadt, 1996, Measuring fund strategy and performance in changing economic conditions, Journal of Finance 51, 425–462. Fung, William, and David Hsieh, 1997, Survivorship bias and investment style in the returns of CTAs: The information content of performance track records, Journal of Portfolio Management 24, 30–41. Getmansky, Mila, Andrew W. Lo, and Igor Makarov, 2004, An econometric model of serial correlation and illiquidity in hedge fund returns, Journal of Financial Economics 74, 529–609. Goetzmann, William N., Jonathan E. Ingersoll Jr., and Stephen A. Ross, 2003, High-watermarks and hedge fund management contracts, Journal of Finance 58, 1685–1718. Goetzmann, William, Jonathan Ingersoll, Matthew Spiegel, and Ivo Welch, 2007, Portfolio performance manipulation and manipulation-proof performance measures, Review of Financial Studies 20, 1503–1546. Grinblatt, Mark, and Sheridan Titman, 1989, Adverse risk incentives and the design of performance-based contracts, Management Science 35, 807–822. Harvey, Campbell R., and Akhtar Siddique, 2000, Conditional skewness in asset pricing tests, Journal of Finance 55, 1263–1295. Hodder, James E., and Jens Carsten Jackwerth, 2007, Incentive contracts and hedge fund management, Journal of Financial and Quantitative Analysis 42, 811–826. Hu, Ping, Jayant R. Kale, Marco Pagani, and Ajay Subramanian, 2011, Fund flows, performance, managerial career concerns,and risk-taking: Theory and evidence, Management Science 57, 628– 646. Huang, Jennifer, Kelsey D. Wei, and Hong Yan, 2007, Participation costs and the sensitivity of fund flows to past performance, Journal of Finance 62, 1273–1311. Kempf, Alexander, and Stefan Ruenzi, 2008, Tournaments in mutual-fund families, Review of Financial Studies 21, 1013–1036.

35 Kempf, Alexander, Stefan Ruenzi, and Tanja Thiele, 2009, Employment risk, compensation incentives, and managerial risk taking: Evidence from the mutual fund industry, Journal of Financial Economics 92, 92–108. Koski, Jennifer Lynch, and Jeffrey Pontiff, 1999, How are derivatives used? Evidence from the mutual fund industry, Journal of Finance 54, 791–816. Kouwenberg, Roy, and William T. Ziemba, 2007, Incentives and risk taking in hedge funds, Journal of Banking and Finance 31, 3291–3310. Leland, Hayne E., 1999, Beyond mean-variance: Risk and performance measurement in a nonsymmetrical world, Financial Analysts Journal, January–February, 27–36. Lin, Jerchern, 2011, Tail risks across investment funds, Unpublished working paper, Marshall School of Business, University of Southern California. Mitton, Todd, and Keith Vorkink, 2007, Equilibrium under diversification and the preference for skewness, Review of Financial Studies 20, 1255–1288. Panageas, Stavros, and Mark Westerfield, 2009, High water marks: High risk appetites? Convex compensation, long horizons, and portfolio choice, Journal of Finance 64, 1–36. Pontiff, Jeffrey, 1995, Closed-end fund premia and returns: Implications for financial market equilibrium, Journal of Financial Economics 37, 341–370. Schwarz, Christopher G., 2011, Mutual fund tournaments: The sorting bias and new evidence, Review of Financial Studies 24, 1–24. Sirri, Erik R., and Peter Tufano, 1998, Costly search and mutual fund flows, Journal of Finance 53, 1589–1622. Thompson, Rex, 1978, The information content of discounts and premiums on closed-end fund shares, Journal of Financial Economics 6, 151–186. Whitelaw, Robert F., 1994, Time variations and covariations in the expectation and volatility of stock market returns, Journal of Finance 49, 515–541.

36 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 highperforming 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 closedend funds, open-ended funds, and hedge funds, respectively. Both bond and equity funds are included in the analysis. Panel A: Closed-End Funds

37

Panel B: Open-Ended Funds

38

Panel C: Hedge Funds

39 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 Fund Type CEFs

OEFs

HFs

Bond CEFs

Bond OEFs

Equity CEFs

Group

Mean

Bottom 10%

Bottom 25%

Median

Top 25%

Top 10%

Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3

0.186 0.018 −0.046 −0.071 −0.002 0.048 0.050 0.024 −0.009 −0.050 0.140 0.072 0.036 0.028 0.006 0.073 0.000 −0.058 −0.150 −0.096 0.117 0.070 0.035 −0.026 −0.141 0.235 0.085 0.017

−0.831 −0.939 −1.026 −1.068 −1.001 −0.841 −0.859 −0.907 −0.912 −0.913 −0.750 −0.913 −0.935 −0.933 −0.970 −0.925 −0.980 −1.010 −1.176 −1.118 −0.859 −0.943 −0.942 −0.987 −1.155 −0.784 −0.852 −0.975

−0.333 −0.473 −0.525 −0.564 −0.509 −0.393 −0.405 −0.434 −0.459 −0.488 −0.307 −0.403 −0.433 −0.451 −0.490 −0.451 −0.512 −0.529 −0.621 −0.628 −0.346 −0.405 −0.431 −0.483 −0.613 −0.283 −0.406 −0.482

0.175 0.012 −0.037 −0.050 0.016 0.051 0.046 0.023 −0.003 −0.043 0.128 0.093 0.059 0.025 −0.006 0.058 −0.025 −0.038 −0.115 −0.109 0.098 0.061 0.010 −0.021 −0.102 0.219 0.087 0.044

0.671 0.506 0.436 0.427 0.486 0.497 0.506 0.486 0.452 0.386 0.588 0.574 0.535 0.510 0.477 0.592 0.500 0.408 0.329 0.404 0.604 0.557 0.507 0.461 0.377 0.713 0.540 0.510

1.235 0.994 0.910 0.888 0.941 0.935 0.976 0.968 0.885 0.814 1.057 1.042 0.988 0.997 0.976 1.128 0.999 0.877 0.794 0.883 1.128 1.103 1.074 0.948 0.826 1.360 1.038 0.976

40

Equity OEFs

P4 Top 20% Bottom 20% P2 P3 P4 Top 20%

−0.013 0.053 0.045 0.023 0.031 0.012 −0.050

−0.981 −0.918 −0.830 −0.866 −0.851 −0.858 −0.906

−0.487 −0.439 −0.384 −0.430 −0.421 −0.436 −0.485

0.019 0.069 0.055 0.026 0.029 0.014 −0.045

0.476 0.525 0.490 0.472 0.479 0.454 0.380

0.915 0.988 0.916 0.918 0.916 0.873 0.808

Mean

Bottom 10%

Bottom 25%

Median

Top 25%

Top 10%

−1.137 −1.150 −1.146 −1.135 −1.156 −1.214 −1.186 −1.201 −1.196 −1.204 −1.207 −1.161 −1.162 −1.163 −1.184 −1.136 −1.175 −1.165 −1.095 −1.109 −1.185 −1.184 −1.218 −1.235 −1.152 −1.149 −1.113 −1.173 −1.187 −1.162 −1.219 −1.191 −1.197 −1.193 −1.212

−0.682 −0.712 −0.703 −0.668 −0.714 −0.800 −0.760 −0.755 −0.762 −0.794 −0.786 −0.723 −0.719 −0.723 −0.751 −0.676 −0.727 −0.719 −0.650 −0.644 −0.712 −0.721 −0.745 −0.735 −0.690 −0.687 −0.672 −0.714 −0.758 −0.714 −0.812 −0.780 −0.773 −0.776 −0.801

0.075 0.007 0.021 0.064 −0.029 −0.172 −0.090 −0.046 −0.087 −0.173 −0.149 −0.037 −0.040 −0.037 −0.054 0.102 0.031 0.010 0.082 0.168 0.016 0.050 0.055 0.078 0.045 0.100 −0.018 0.009 −0.061 −0.074 −0.196 −0.136 −0.123 −0.139 −0.189

Panel B: Fund Kurtosis Fund Type CEFs

OEFs

HFs

Bond CEFs

Bond OEFs

Equity CEFs

Equity OEFs

Group Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20%

Panel C: Change in Skewness

0.579 0.428 0.471 0.489 0.412 0.201 0.302 0.354 0.259 0.158 0.270 0.410 0.401 0.427 0.370 0.491 0.447 0.444 0.524 0.569 0.494 0.533 0.517 0.493 0.517 0.701 0.425 0.441 0.370 0.374 0.168 0.218 0.207 0.176 0.131

1.272 1.075 1.115 1.182 1.078 0.775 0.919 0.993 0.858 0.736 0.812 0.999 1.001 1.051 1.007 1.255 1.171 1.042 1.220 1.310 1.204 1.291 1.229 1.168 1.168 1.367 0.982 1.088 1.034 0.995 0.725 0.820 0.801 0.766 0.707

3.019 2.599 2.769 2.685 2.560 2.049 2.286 2.414 2.121 1.947 2.192 2.535 2.415 2.522 2.452 2.615 2.658 2.736 2.823 2.716 2.758 2.934 2.877 2.684 2.739 3.581 2.599 2.643 2.551 2.434 1.980 2.052 2.021 1.944 1.896

41 Fund Type CEFs

OEFs

HFs

Bond CEFs

Bond OEFs

Equity CEFs

Equity OEFs

Group Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20%

Mean −79.83 −109.39 −91.44 −119.06 −94.75 −82.37 −99.57 −82.06 −87.54 −94.31 −93.30 −82.73 −93.83 −92.42 −94.15 −92.97 −112.08 −92.61 −132.91 −111.81 −80.12 −60.69 −61.24 −92.87 −72.97 −51.87 −97.28 −122.83 −109.39 −85.76 −86.83 −100.40 −94.49 −95.80 −92.52

Bottom 10% −400.52 −481.23 −472.52 −529.41 −471.87 −480.60 −493.38 −460.16 −489.21 −473.54 −460.03 −478.73 −485.30 −475.42 −453.97 −401.15 −462.57 −472.36 −558.32 −499.88 −396.61 −446.05 −432.69 −445.17 −496.83 −393.38 −477.92 −484.00 −500.79 −455.73 −495.11 −496.99 −492.58 −499.37 −465.58

Bottom 25% −177.53 −212.36 −211.95 −233.58 −218.84 −212.80 −217.76 −202.30 −218.94 −210.37 −198.55 −203.16 −211.15 −208.72 −198.45 −188.57 −218.71 −204.80 −226.91 −227.70 −171.67 −190.52 −183.93 −197.06 −209.81 −167.66 −206.38 −224.91 −221.19 −216.37 −217.78 −225.69 −218.34 −222.99 −209.52

Median −85.96 −96.94 −97.81 −94.70 −91.69 −93.46 −97.86 −93.17 −95.63 −92.55 −92.99 −91.67 −96.49 −93.02 −92.24 −96.42 −100.93 −101.26 −100.38 −94.83 −93.94 −92.35 −94.37 −96.50 −89.12 −75.95 −94.07 −91.87 −87.76 −92.07 −93.21 −98.21 −94.25 −97.12 −92.48

Top 25%

Top 10%

31.84 21.40 24.47 26.50 19.23 32.26 22.39 26.75 30.16 26.25 25.51 29.54 27.03 26.76 16.98 21.12 17.30 21.58 10.87 36.50 2.07 10.43 12.00 7.76 28.04 49.35 25.25 19.99 24.16 14.96 35.94 30.07 34.46 31.51 27.57

324.83 289.73 309.25 299.28 276.49 325.18 301.26 303.66 312.95 285.22 289.03 320.19 292.07 296.37 260.98 297.16 265.61 289.09 300.27 299.28 262.15 307.31 311.48 281.31 325.21 354.49 338.66 295.69 287.97 266.29 328.45 309.12 302.86 303.26 286.99

Panel D: Change in Kurtosis Fund Type CEFs

OEFs

Group Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3

Mean −115.90 −107.71 −88.01 −86.06 −93.47 −106.06 −104.64 −102.23

Bottom 10% −600.07 −557.27 −534.31 −556.15 −514.00 −546.92 −546.47 −535.60

Bottom 25% −198.71 −206.67 −201.06 −207.60 −204.59 −218.45 −212.47 −207.53

Median −99.95 −99.78 −94.87 −98.17 −98.00 −101.22 −100.90 −102.25

Top 25% 18.58 22.83 36.96 32.61 29.22 31.43 28.82 20.76

Top 10% 271.01 279.02 343.83 320.40 329.14 291.77 306.32 298.12

42

HFs

Bond CEFs

Bond OEFs

Equity CEFs

Equity OEFs

P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20% Bottom 20% P2 P3 P4 Top 20%

−102.04 −104.94 −102.62 −109.65 −110.60 −99.77 −103.32 −92.39 −117.41 −46.59 −122.52 −85.57 −110.12 −99.39 −107.21 −83.24 −96.24 −117.97 −98.38 −98.55 −90.37 −93.48 −102.41 −110.16 −104.54 −100.69 −105.69

−529.23 −538.40 −486.68 −537.90 −534.36 −559.20 −503.38 −547.53 −592.33 −506.75 −573.28 −544.99 −518.53 −556.17 −535.74 −470.08 −516.77 −664.22 −524.86 −538.42 −536.82 −485.56 −544.04 −559.59 −536.18 −542.39 −541.41

−214.17 −219.06 −200.84 −216.80 −213.59 −210.87 −191.07 −187.57 −214.17 −188.62 −225.18 −199.35 −183.12 −194.94 −195.35 −182.79 −193.78 −209.33 −205.21 −199.97 −202.69 −193.51 −221.28 −222.70 −222.74 −223.41 −220.00

−99.39 −101.33 −101.64 −102.34 −101.68 −98.91 −95.31 −99.99 −96.60 −91.90 −103.72 −95.33 −104.16 −106.54 −105.94 −101.05 −98.65 −99.35 −98.95 −96.28 −94.50 −99.64 −99.47 −97.91 −100.99 −98.38 −101.67

26.12 33.54 21.21 22.02 23.12 32.46 22.07 13.68 24.62 48.03 17.85 39.51 −7.70 6.45 −4.77 3.43 18.30 33.48 19.19 28.89 29.93 21.63 38.72 37.84 32.21 33.99 33.92

300.50 317.13 276.31 313.68 305.95 367.05 289.60 226.91 286.20 370.47 283.94 361.18 213.10 293.45 270.82 273.92 316.28 361.59 260.99 343.83 273.50 347.83 309.74 314.10 300.67 313.72 319.42

43 Table 2. Comparison of Differences in Average Fund Skewness This table shows the differences in average fund skewness around the peer fund return between lowand 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 CEFs OEFs HFs Bond CEFs Bond OEFs Equity CEFs Equity OEFs

High-Low Skewness

Paired T-value

Paired P-value

−0.187 −0.061 −0.098 −0.168 −0.161 −0.216 −0.093

−4.61 −1.67 −2.70 −3.55 −2.71 −3.40 −2.51

< 0.01 0.097 < 0.01 < 0.01 < 0.01 < 0.01 0.012

Signed-Rank Test P-value < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

44 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

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.067 0.031 0.020 0.041 −0.014 0.019

T-value −5.62 2.45 2.38 2.46 −1.49 1.29

Bond Funds Coeff −0.082 0.019 0.021 0.008 −0.004 0.008

T-value −4.83 1.18 1.15 0.24 −0.28 0.80

Equity Funds Coeff −0.059 0.035 0.015 0.051 −0.018 0.052

T-value −4.69 2.06 1.44 2.73 −1.39 4.88

Panel B: Open-Ended Funds All Funds

Performance t Vol t+1 Vol t Skew t Vol t−1

Coeff −0.053 0.010 −0.021 0.046 0.027

T-value −2.43 3.91 −0.91 7.57 1.07

Bond Funds Coeff −0.332 −0.193 0.102 0.023 0.048

T-value −1.63 −2.81 1.54 2.76 2.60

Equity Funds Coeff −0.042 0.013 −0.024 0.055 0.025

T-value −1.85 6.41 −0.94 4.90 1.00

45 Skew t−1

−0.004

−0.64

0.036

4.55

−0.016

−3.27

Panel C: Hedge Funds All Funds

Performance t Vol t+1 Vol t Skew t Vol t-1 Skew t-1

Coeff −0.040 0.021 −0.008 0.054 0.003 −0.014

T-value −3.21 2.34 −0.61 3.72 0.35 −1.57

Panel D: Test Differences in Coefficients on Performance t Across Fund Types Test CEFs=OEFs=HFs CEFs=OEFs CEFs=HFs OEFs=HFs Bond CEFs=Bond OEFs Equity CEFs=Equity OEFs

P-value 0.365 0.661 0.162 0.524 0.163 0.544

46 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. Tstatistics are adjusted for clustering at the style level. Panel A: Closed-End Funds All Funds Group Bottom 20%

P2

P3

Performance t-1 Vol t+1 Vol t Skew t Vol t−1 Skew t-1 Performance t-1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t

Coeff −0.018 0.051 −0.032 0.100 0.018 −0.040 −0.299 0.013 0.034 0.021 −0.030 0.011 −0.346 0.010 0.132

T-value −0.35 3.09 −2.36 2.89 0.98 −0.90 −3.50 0.87 1.28 0.61 −1.46 0.31 −3.22 0.60 3.58

Bond Funds Coeff 0.010 0.026 −0.024 0.026 0.046 −0.044 −0.131 0.019 0.012 0.005 −0.003 −0.016 −0.155 0.037 0.152

T-value 0.29 1.02 −1.46 0.28 2.88 −0.87 −1.42 0.58 0.21 0.15 −0.06 −0.41 −0.76 1.04 2.64

Equity Funds Coeff −0.032 0.053 −0.016 0.094 −0.003 0.022 −0.204 0.024 0.042 0.019 −0.048 0.041 −0.017 0.001 0.129

T-value −0.67 2.39 −0.91 2.20 −0.27 0.64 −1.64 1.53 1.92 0.71 −2.14 1.07 −0.11 0.05 6.20

47

P4

Top 20%

Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1

0.059 −0.117 −0.032 0.090 0.006 0.105 0.007 −0.082 0.061 −0.013 0.033 −0.007 0.036 0.006 0.063

1.23 −3.08 −0.82 0.68 0.46 3.44 0.23 −3.74 1.92 −0.53 3.15 −0.39 1.25 0.38 1.94

0.019 −0.160 0.006 −0.034 0.003 0.046 −0.037 −0.034 0.086 −0.055 0.017 0.022 0.063 −0.007 0.015

0.30 −3.25 0.12 −0.26 0.17 0.89 −0.67 −0.76 1.53 −1.25 0.85 1.21 2.21 −0.35 0.36

0.033 −0.115 0.016 −0.194 0.003 0.081 −0.015 −0.068 0.106 −0.042 0.048 −0.027 0.025 0.015 0.086

0.46 −4.42 0.34 −1.52 0.22 3.95 −0.33 −2.90 3.83 −3.37 4.96 −2.24 0.77 1.45 2.40

Panel B: Open-Ended Funds All Funds Group Bottom 20%

P2

P3

P4

Top 20%

Performance t-1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t

Coeff 0.014 0.016 0.003 0.045 0.015 −0.020 0.075 0.006 −0.020 0.019 0.025 0.005 −0.663 0.007 −0.056 0.046 0.076 −0.006 0.064 −0.010 −0.051 0.023 0.091 0.008 −0.065 0.012 −0.006 0.072

T-value 0.43 1.61 0.06 2.01 0.36 −0.95 0.68 0.51 −0.52 1.38 0.52 0.43 −1.96 0.86 −0.89 3.94 1.11 −0.67 0.85 −0.64 −1.51 2.23 2.27 0.63 −2.63 5.43 −0.60 4.53

Bond Funds Coeff 0.021 0.036 0.012 0.069 −0.009 −0.005 −0.719 −0.277 0.099 0.020 0.052 0.039 −1.879 −0.339 0.294 −0.005 −0.025 0.041 −0.799 −0.392 0.257 0.001 0.118 0.016 −0.188 −0.317 0.137 0.042

T-value 0.13 0.26 0.10 1.91 −0.09 −0.11 −1.67 −1.70 0.85 1.48 1.59 2.39 −2.12 −1.98 2.24 −0.39 −0.22 2.65 −1.83 −2.71 1.89 0.05 1.15 0.90 −2.69 −5.46 2.34 1.94

Equity Funds Coeff 0.021 0.019 0.008 0.055 0.013 −0.023 0.189 0.013 −0.003 0.018 0.017 −0.013 −0.085 0.013 −0.078 0.057 0.078 −0.016 −0.025 0.008 −0.060 0.022 0.066 0.004 −0.061 0.013 −0.012 0.074

T-value 0.53 1.78 0.16 2.88 0.29 −1.24 2.45 2.68 −0.07 1.31 0.31 −1.66 −2.22 1.93 −1.95 6.02 1.68 −1.54 −0.39 0.57 −2.21 1.39 2.09 0.15 −2.08 5.15 −0.83 4.10

48 Vol t−1 Skew t−1

0.013 −0.003

0.69 −0.22

0.123 0.058

2.43 1.03

0.018 −0.010

1.04 −0.53

Panel C: Hedge Funds All Funds Group Bottom 20%

P2

P3

P4

Top 20%

Performance t-1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t-1 Performance t−1 Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.025 0.029 −0.067 0.018 0.054 −0.036 −0.121 0.008 −0.018 0.089 0.016 −0.059 −0.222 0.008 0.009 0.018 0.000 0.025 0.052 0.027 0.086 0.058 −0.089 −0.007 −0.046 0.025 −0.015 0.084 0.013 0.011

T-value −0.90 7.19 −8.21 0.61 4.00 −3.04 −0.93 0.35 −0.63 4.46 0.46 −1.94 −3.73 0.29 0.25 0.75 0.00 1.20 0.65 2.00 2.68 1.64 −3.87 −0.31 −3.14 3.10 −0.49 2.10 0.47 0.66

Panel D: Test Differences in Coefficients on Performance t-1 Across Quintile Groups and Between the Low- and High-Performance Groups Test CEFs: Group 1=2=3=4=5 CEFs: Group 1=5 OEFs: Group 1=2=3=4=5 OEFs: Group 1=5 HFs: Group 1=2=3=4=5

P-value 0.000 0.126 0.054 0.097 0.006

49 HFs: Group 1=5 Bond CEFs: Group 1=2=3=4=5 Bond CEFs: Group 1=5 Bond OEFs: Group 1=2=3=4=5 Bond OEFs: Group 1=5 Equity CEFs: Group 1=2=3=4=5 Equity CEFs: Group 1=5 Equity OEFs: Group 1=2=3=4=5 Equity OEFs: Group 1=5

0.814 0.107 0.932 0.000 0.395 0.019 0.486 0.040 0.073

50 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

FracRankt,1 FracRankt,2 FracRankt,3 Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.570 −0.296 0.339 0.032 0.012 0.047 −0.013 0.016

T-value −1.25 −3.57 1.58 2.74 1.37 2.75 −1.36 1.06

Bond Funds Coeff −0.061 −0.331 −0.017 0.021 0.012 0.017 −0.003 0.005

T-value −0.25 −4.02 −0.04 1.38 0.54 0.49 −0.15 0.48

Equity Funds Coeff −0.723 −0.253 0.304 0.035 0.009 0.053 −0.017 0.050

T-value −0.98 −2.29 1.60 2.27 0.76 2.98 −1.23 4.18

Panel B: Open-Ended Funds All Funds

Bond Funds

Equity Funds

51

FracRankt,1 FracRankt,2 FracRankt,3 Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff 0.087 −0.118 −0.578 0.011 −0.016 0.046 0.029 −0.005

T-value 0.79 −2.66 −2.87 4.41 −0.69 8.77 1.21 −0.78

Coeff −0.895 −0.252 −0.269 −0.202 0.101 0.027 0.038 0.036

T-value −1.77 −2.50 −0.74 −2.98 1.44 3.01 2.29 4.31

Coeff −0.063 −0.050 −0.726 0.014 −0.019 0.054 0.026 −0.016

T-value −0.54 −1.78 −2.97 7.49 −0.68 5.51 1.07 −3.44

Panel C: Hedge Funds All Funds

FracRankt,1 FracRankt,2 FracRankt,3 Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.286 −0.124 −0.497 0.021 −0.009 0.057 0.005 −0.014

T-value −1.00 −1.21 −4.08 2.43 −0.64 3.80 0.58 −1.53

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

Test FracRankt,1: CEFs=OEFs=HFs FracRankt,1: CEFs=OEFs FracRankt,1: CEFs=HFs FracRankt,1: OEFs=HFs FracRankt,3: CEFs=OEFs=HFs FracRankt,3: CEFs=OEFs FracRankt,3: CEFs=HFs FracRankt,3: OEFs=HFs FracRankt,1: Bond CEFs= Bond OEFs FracRankt,3: Bond CEFs= Bond OEFs FracRankt,1: Equity CEFs= Equity OEFs FracRankt,3: Equity CEFs= Equity OEFs

P-Value 0.003 0.020 0.001 0.273 0.146 0.083 0.054 0.503 0.313 0.273 0.092 0.167

52 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

Bottom 20% FracPRFMt,2 FracPRFMt,3 FracPRFMt,4 Top 20% Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.101 0.106 −0.119 −0.191 −0.029 0.029 0.015 0.041 −0.014 0.019

T−value −3.09 0.97 −1.33 −2.05 −1.41 2.41 1.78 2.39 −1.37 1.28

Panel B: Open-Ended Funds

Bond Funds Coeff −0.080 0.382 0.070 −0.501 −0.069 0.018 0.018 0.006 −0.002 0.009

T−value −6.33 2.07 0.35 −2.49 −1.90 1.17 0.98 0.20 −0.17 0.95

Equity Funds Coeff −0.078 0.108 −0.044 −0.208 −0.036 0.034 0.012 0.050 −0.018 0.053

T−value −1.57 1.10 −0.39 −2.02 −1.26 2.20 1.24 2.54 −1.36 4.87

53 All Funds

Bottom 20% FracPRFMt,2 FracPRFMt,3 FracPRFMt,4 Top 20% Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.022 0.220 −0.082 −0.040 −0.098 0.011 −0.016 0.044 0.027 −0.003

T-value −1.10 2.22 −0.62 −0.96 −2.73 3.77 −0.66 7.78 1.12 −0.49

Bond Funds Coeff −0.231 0.006 −0.602 −1.729 −0.493 −0.193 0.113 0.022 0.047 0.035

T-value −0.93 0.01 −1.09 −2.45 −3.34 −2.78 1.92 2.76 2.72 4.42

Equity Funds Coeff −0.021 0.094 0.129 0.038 −0.098 0.014 −0.018 0.053 0.025 −0.014

T-value −1.04 1.05 1.11 1.44 −2.37 6.66 −0.64 5.11 1.02 −3.12

Panel C: Hedge Funds All Funds

Bottom 20% FracPRFMt,2 FracPRFMt,3 FracPRFMt,4 Top 20% Vol t+1 Vol t Skew t Vol t−1 Skew t−1

Coeff −0.038 0.069 −0.071 −0.047 −0.054 0.021 −0.008 0.053 0.003 −0.013

T-value −2.46 1.05 −1.09 −0.68 −3.20 2.36 −0.56 3.69 0.33 −1.53

Panel D: Test Differences in Coefficients on Rank (Rank t) Across Fund Types Test Bottom 20%: CEFs=OEFs=HFs Bottom 20%: CEFs=OEFs Bottom 20%: CEFs=HFs Bottom 20%: OEFs=HFs Top 20%: CEFs=OEFs=HFs Top 20%: CEFs=OEFs Top 20%: CEFs=HFs Top 20%: OEFs=HFs Bottom 20%: CEFs=OEFs Top 20%: CEFs=OEFs Bottom 20%: CEFs=OEFs Top 20%: CEFs=OEFs

P-value 0.438 0.234 0.434 0.456 0.061 0.036 0.493 0.069 0.519 0.019 0.373 0.121

54 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

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1 age t size t Performance t x age t Performance t x size t

Coeff 0.011 0.035 0.021 0.048 −0.016 0.013 −0.027 0.069 −0.075 0.053

Panel B: Open-Ended Funds

T-value 0.07 2.17 1.35 1.97 −1.43 0.72 −0.60 2.18 −1.52 2.10

Bond Funds Coeff −0.429 0.001 0.079 0.030 −0.082 −0.024 −0.063 0.098 0.215 −0.144

T-value −1.00 0.03 1.93 0.91 −2.56 −2.22 −0.60 1.48 3.65 −1.14

Equity Funds Coeff 0.124 0.038 0.007 0.046 −0.006 0.059 −0.015 0.016 −0.119 0.067

T-value 0.64 2.01 0.49 1.44 −0.49 3.83 −0.37 0.41 −2.00 1.95

55 All Funds

Performance t Vol t+1 Vol t Skew t Vol t-1 Skew t-1 age t size t Performance t x age t Performance t x size t

Coeff 0.047 0.010 −0.017 0.046 0.024 −0.005 0.000 −0.001 −0.062 0.006

T-value 1.05 3.96 −0.72 8.29 0.94 −0.76 0.01 −0.41 −2.79 2.53

Bond Funds Coeff −0.329 −0.188 0.088 0.029 0.076 0.028 0.007 −0.016 −0.037 0.023

T-value −1.28 −2.53 1.61 3.74 1.18 3.51 0.15 −1.92 −0.52 1.78

Equity Funds Coeff 0.039 0.013 −0.023 0.055 0.024 −0.016 0.003 0.002 −0.055 0.006

T-value 0.93 7.09 −0.86 4.91 0.95 −3.24 0.28 0.86 −2.38 3.15

Panel C: Hedge Funds All Funds

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1 age t size t Performance t x age t Performance t x size t

Coeff 0.251 0.021 0.003 0.048 −0.006 −0.006 −0.032 0.000 −0.131 −0.009

T-value 5.30 2.10 0.21 3.24 −0.60 −0.71 −0.51 −0.07 −5.72 −1.70

Panel D: Test Differences in Coefficients on Performance t x age t and Performance t x size t Across Fund Types Test Performance t x age t CEFs=OEFs=HFs Performance t x age t: CEFs=OEFs Performance t x age t: CEFs=HFs Performance t x age t: OEFs=HFs Performance t x size t: CEFs=OEFs=HFs Performance t x size t: CEFs=OEFs Performance t x size t: CEFs=HFs Performance t x size t: OEFs=HFs Performance t x age t: Bond CEFs= Bond OEFs Performance t x size t: Bond CEFs= Bond OEFs Performance t x age t: Equity CEFs= Equity OEFs Performance t x size t: Equity CEFs= Equity OEFs

P-value 0.171 0.548 0.236 0.132 0.092 0.052 0.038 0.254 0.058 0.067 0.011 0.000

56 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

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1 T-Bill t DivYield t YieldSp t DefaultSp t Performance t x T-Bill t Performance t x DivYield t Performance t x YieldSp t

Coeff −0.104 0.031 0.018 0.042 −0.012 0.018 0.007 0.054 0.015 0.023 0.018 −0.031 0.019

T-value −1.44 2.42 1.81 2.58 −1.21 1.28 0.78 0.98 0.99 0.24 0.90 −0.69 0.85

Bond Funds Coeff −0.009 0.018 0.028 0.013 −0.009 0.007 0.007 0.178 0.011 0.064 −0.015 −0.077 −0.027

T-value −0.04 1.23 1.67 0.38 −0.69 0.69 0.47 4.63 0.43 0.80 −0.42 −1.25 −0.59

Equity Funds Coeff −0.114 0.034 0.015 0.054 −0.017 0.050 0.001 −0.056 0.015 0.001 0.035 −0.038 0.038

T-value −1.72 1.98 1.29 2.98 −1.29 4.38 0.06 −0.63 1.06 0.01 1.16 −0.69 1.10

57 Performance t x DefaultSp t

−0.007

−0.11

0.174

1.48

−0.080

−1.87

Panel B: Open-Ended Funds All Funds

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1 T-Bill t DivYield t YieldSp t DefaultSp t Performance t x T-Bill t Performance t x DivYield t Performance t x YieldSp t Performance t x DefaultSp t

Coeff −0.132 0.011 −0.022 0.042 0.028 −0.001 −0.012 −0.039 −0.040 −0.062 0.045 −0.122 0.064 0.029

T-value −0.82 3.99 −1.03 8.67 1.15 −0.22 −1.65 −0.48 −3.36 −1.29 1.84 −5.92 2.57 0.43

Panel C: Hedge Funds All Funds

Performance t Vol t+1 Vol t Skew t Vol t−1 Skew t−1 T-Bill t DivYield t YieldSp t DefaultSp t Performance t x T-Bill t Performance t x DivYield t Performance t x YieldSp t Performance t x DefaultSp t

Coeff 0.148 0.020 −0.009 0.041 0.006 0.000 −0.014 0.060 −0.037 −0.147 0.012 −0.187 0.030 0.012

T-value 2.07 2.13 −0.60 2.89 0.70 0.03 −0.95 0.88 −1.21 −1.71 1.44 −5.39 2.43 0.81

Bond Funds Coeff −0.898 −0.205 0.113 0.026 0.047 0.036 0.037 −0.344 −0.008 0.098 0.013 −0.016 0.165 0.241

T-value −1.65 −2.97 1.39 3.57 2.15 4.13 1.74 −2.39 −0.32 1.17 0.09 −0.08 1.17 1.57

Equity Funds Coeff −0.149 0.014 −0.029 0.050 0.029 −0.012 −0.023 0.038 −0.045 −0.108 0.048 −0.100 0.066 0.008

T-value −0.98 6.55 −1.15 5.03 1.24 −2.38 −3.47 0.57 −3.36 −1.56 2.29 −5.22 2.92 0.10

58 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

Bottom 20% P2 P3 P4 Top 20%

Mean 1.343 0.886 0.719 0.393 −0.131

StdDev 2.611 2.429 2.544 3.540 3.811

Skewness 0.930 0.262 −0.510 −1.723 −1.260

Value-Weighted Kurtosis 8.746 4.874 1.672 11.152 4.983

Mean 1.343 0.964 0.752 0.379 −0.079

StdDev 2.592 2.285 2.427 3.213 3.472

Skewness 0.814 0.218 −0.598 −1.497 −1.296

Kurtosis 8.783 4.410 1.760 8.840 4.423

F-test of Differences in Skewness: Top 20% – Bottom 20% p-value