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  What is the true cost of active management? A comparison of hedge funds and mutual funds  ∗ Jussi Keppo NUS Business School and Risk Management Institutekeppo@nus.edu.sg Antti Petajisto NYU Stern School of Businessantti.petajisto@stern.nyu.edu March 16, 2014 Abstract On the surface, hedge funds seem to have much higher fees than actively managed mutualfunds. However, the true cost of active management should be measured relative to the size of the active positions taken by a fund manager. A mutual fund combines active positions witha passive position in the benchmark index, which can make the active positions expensive.A hedge fund takes both long and short positions and uses leverage, which makes the activepositions cheaper, but this can be offset by the expected incentive fees, especially for morevolatile funds. We investigate the trade-offs from the perspective of a fund investor choosingbetween a mutual fund and a hedge fund, examining the impact of leverage, volatility, ActiveShare, nominal fees, and alpha for a realistic range of parameter estimates. Our calibrationshows that a moderately skilled active manager is almost equally attractive to investors as amutual fund manager or as a hedge fund manager, showing that both investment vehicles cancoexist as efficient alternatives to investors. Further, our model explains documented empiricalfindings on career development of successful fund managers and on hedge funds’ risk taking.Finally, we show that our findings are quite robust with respect to a jump risk in the hedgefund returns. Keywords:  Management fee, incentive fee, hedge fund, mutual fund ∗ We have benefited from comments at Aalto University, Norwegian University of Science and Technology, Uni-versity of Michigan, INFORMS Annual Meeting, and more specifically Stein-Erik Fleten, Matti Keloharju, SamuliKnupfer, Mikko Leppamaki, Peter Molnar, Romesh Saigal, Matti Suominen, Sami Torstila, and Tuomo Vuolteenaho.We are also grateful to Vinay Benny and Zhichen Zhao for research assistance. 1  1 Introduction Are hedge funds more expensive than mutual funds? Initially the answer may seem obvious: hedgefunds typically charge about 2% management fee, which is slightly higher than the usual about 1%fee charged by actively managed mutual funds, and on top of this hedge funds charge an incentivefee which typically amounts to 20% of any positive profits. Hence, they clearly charge higher feesper dollar invested, but do they also deliver more in return for those fees? Naturally a hedge fundmanager would claim that they deliver greater alpha due to their superior skills. But what if boththe hedge fund and mutual fund manager are about equally skilled? How do the mutual fund andhedge fund structures differ as vehicles to deliver alpha to investors, and which vehicle would allowthe investor to benefit more from a given manager’s skills?This question has become more important recently as new legislation and regulation in the U.S.and the European Union has been increasing the costs of opening new hedge funds, thus makingthe mutual fund structure relatively more attractive to active managers than it was in the past.To the extent that this regulatory change pushes active managers to choose to run mutual fundsinstead of hedge funds, does this mean investors are getting a better deal and paying lower fees foractive management? We point out that the answer is far from obvious.In this paper we investigate whether investors would be better off investing in hedge fundsor mutual funds, and how this trade-off varies for a range of plausible parameter values. We alsoexplain the costs and benefits of each fund structure from the investor’s point of view. Our approachis conceptual: We want to understand the mechanism of the trade-off, so rather than estimatingvarious relevant quantities from the hedge fund and mutual fund data ourselves, we use otherresearchers’ estimates to evaluate the trade-off for the investors. Further, by using the trade-off our model explains several documented empirical findings on career development of successful fundmanagers and on hedge funds’ risk taking (see e.g. Li et al. (2011) and Nohel et al. (2010)).The simplest trade-off comes from the effect of leverage: A hedge fund can use leverage to scaleup the manager’s bets for each dollar of the investor’s capital, generating a greater alpha, albeitwith greater idiosyncratic volatility, than the active mutual fund, and hence offsetting the highermanagement fee. In this sense, the hedge fund can indeed provide more active management foreach dollar invested in the fund.Another potential cost for mutual fund investors arises from the market exposure embedded intheir actively managed fund: If the investors are not able (perhaps due to institutional constraints)to hedge out the market exposure themselves, then they may end up with too much beta and toolittle alpha. In contrast, a market-neutral hedge fund gives them an easy way to optimize theirexposure to both the market and the active strategies. Other smaller effects can also affect thetrade-off: For example, if the hedge fund is able to borrow at lower rates or borrow more than the1  investor himself, that can add value to the investor in case he would like to use some leverage toboost his returns.According to our model, assuming a moderately skilled equity manager who has an annualizedinformation ratio of about 0.3 before fees, mutual funds and hedge funds deliver about the sameexpected utility to investors, so both investment vehicles are equally attractive to investors, in spiteof the fact that hedge funds may initially appear to be more expensive. The investors benefit fromhedge fund leverage since the management fee is paid on the money invested in the fund, not on thegross positions that include leverage. This leverage effect is stronger the higher the hedge fund’sinformation ratio before fees (or “skill,” loosely speaking). Further, if the fund is more levered, theinvestor can effectively get the same exposure with a smaller investment in the fund. We find thatwithout leverage typical hedge funds could not compete with mutual funds, and that these findingsare quite robust with respect to a jump risk in the hedge fund returns. We also investigate impactsfrom several other factors. For instance, our model shows that a ten percentage point increase in theincentive fee should be compensated by little over one percentage point increase in the unleveredalpha.Several papers have empirically analyzed hedge fund alphas and fees. Early studies of hedge fundperformance include, for example, Fung and Hsieh (1997a, 1997b, 1999, 2000, 2001), Ackermann etal. (1999), Brown et al. (1999, 2000, 2001), Liang (1999, 2000, 2001), Agarwal and Naik (2000a,b,c),Brown and Goetzmann (2001), Edwards and Caglayan (2001), Kao (2002), and Lochoff (2002).Getmansky, Lo, and Makarov (2004) and Khandani and Lo (2009) point out that autocorrelationin returns induced by return smoothing may distort performance measures of some hedge funds.Joenvaara (2011) and Bali et al. (2011) argue that hedge fund alphas in good times are in partcompensation for systemic risk, while Jylha and Suominen (2011) attribute part of hedge fundalphas to a simple carry trade strategy. Fung et al. (2008) find that while some hedge funds appearto generate true alphas, inflows to the best-performing funds and the hedge fund industry overallappear to have pushed these alphas down. Kritzman (2008) provides an example of fees and returnsfor a hedge fund and mutual fund, demonstrating the importance of considering the size of activebets in this comparison. In general, while most studies find a positive level of skill for hedge funds,quantifying hedge fund performance remains difficult because of the wide variety of investmentstyles across funds, time-varying strategies within funds, and lack of comprehensive data due tovoluntary reporting, survivorship bias, and backfill bias.In contrast, mutual fund performance has been studied over a long time period and usingcomprehensive data: e.g., Jensen (1968), Brown and Goetzmann (1995), Carhart (1997), Grinblattand Titman (1989, 1993), Gruber (1996), Daniel et al. (1997), Wermers (2000, 2003), Pastor andStambaugh (2002), Bollen and Busse (2004), Cohen et al. (2005), and Mamaysky et al. (2007).While the average fund has lost to its benchmark net of fees and expenses, most papers find2  positive before-fee alphas of about 1%, indicating some positive average level of skill. More recentlythe literature has focused on identifying subsets of mutual fund managers that are more likely tooutperform. For example, Cremers and Petajisto (2009) and Petajisto (2010) introduce ActiveShare as a way to quantify how active fund managers are, pointing out that the most active stockpickers on average have been able to outperform their benchmarks even after fees and transactioncosts. Such evidence is reassuring for our analysis, which starts with the premise that some investorscan indeed identify value-adding managers. Even if one were to disagree with this premise, the factremains that trillions of dollars are invested with active managers, so improving this decision cansignificantly improve the welfare of investors.The rest of the paper is organized as follows. Section 2 presents our simple model. Section 3explains the optimization procedure. The calibration results are discussed in Section 4, and themain conclusions are presented in Section 5. 2 Model We consider an investor who is able to invest in an index fund, active mutual fund, and hedgefund, as well as borrow and lend cash. The investor can borrow and lend at the same risk-free rateand use leverage up to  L  ≥  0 times his wealth. His investment strategy is static over time horizon[0 ,T  ], so at time 0 the investor selects the portfolio and then just passively waits until the payoffsare realized at time  T  . However, if at any time between 0 and  T   the portfolio value falls below acertain threshold, the institution that lent the money will ask the investor to close his entire riskyinvestment. The objective of the investor is to maximize expected utility from his wealth at time T  , given constant relative risk aversion (CRRA) preferences.The value of the risk-free investment at time  T   is given by W  r ( T  ) =  W  r (0)exp( rT  ) , (1)where  r  is the risk-free rate and  W  r (0) is the amount of money in the risk-free asset at time 0. Asdiscussed above, the investor is able to borrow and lend, i.e., he can take long and short positionsin the risk-free asset. However,  W  r (0)  ≥ − LW  (0), where  W  (0)  >  0 is the initial total wealth of theinvestor and  L  ≥  0 is the maximum leverage level, i.e., the maximum loan the investor can take is L  times the initial wealth.The investor can also invest in an index fund that holds the market portfolio. The value of theindex fund follows a geometric Brownian motion, and thus the value at time  T   is given by W  i ( T  ) =  W  i (0)exp  r  +  ησ i  −  12 σ 2 i  T   +  σ i B i ( T  )  , (2)3
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