## Abstract

Many innovative hedge fund fee structures have been introduced in recent years in response to concerns about both the level of hedge fund fees and the incentives they may provoke. A traditional fee structure consists of a flat fee charged as a percentage of the assets under management, together with a performance fee consisting of a percentage of the profits earned. A fee structure that has become more popular recently is the first-loss structure, in which the manager receives a higher performance fee in return for providing some downside protection to investors by insuring some of their losses. Combinations of these fee structures have also been proposed, with the possibility that investors may benefit from some diversification among the fee structures. By considering the investors’ risk-reward tradeoff, the authors show that there is in fact very little benefit from such fee diversification.

**TOPICS:** Real assets/alternative investments/private equity, performance measurement

When considering the fee structure of any investment management relationship, both sides must be concerned with the incentives the contract provides to their counterparty. Managers often want to entice further investment in order to increase assets under management, and thus must ensure that the fee structure is appealing to investors (while still being lucrative for themselves). Investors face a classical principal-agent problem, and must ensure that managers are properly motivated to run the portfolio in the best interests of the investors for whom they work.

Typically, the fee structure of a hedge fund consists of two parts: a management fee and a performance fee (perhaps restricted by a high-water mark provision). The management fee is charged as a percentage of assets under management and the performance fee is charged as a percentage of the investor’s profits. For instance, a traditional fee structure consists of a management fee of 2% of assets under management and a performance fee of 20% of net profits.

Hedge funds have faced intense scrutiny since the financial crisis, as the fees they charge investors have been outsized compared to the returns they have been posting as a group.^{1} Furthermore, it is unclear whether the option-like structure of the manager’s payoff under traditional schemes might lead managers to undertake riskier strategies that may be at odds with the risk aversion of investors.^{2} In recent years, innovative fee structures have emerged aiming at better alignment between investors’ interests and the hedge fund business objectives. An example is the class of first-loss fee structures (see Banzaca [2012] for a description). In these structures, in return for a higher performance fee, the hedge fund manager provides some downside protection to investors on their losses.

He and Kou (2018) presented a description, a mathematical analysis of the incentives that such a fee structure provokes, and studied the impact on the utility of both the hedge fund manager and investor, while Djerroud et al. (2016) analyzed the first-loss fee structure using an option-pricing perspective, leading to the identification of fair levels of performance fees. Fee innovation has led to discussions and negotiations between investors and managers on the optimal fee to be used in certain situations, amplifying the universe of available fee structures by mixing different structures together. One example is the shared-loss structure, which can be considered as a mixture of the classical structure and the first-loss structure. Under a shared-loss agreement, rather than covering *all* investor losses up to a certain limit, the manager provides compensation for a proportion of the investor’s losses (again subject to a ceiling). To draw an (imperfect) analogy with insurance, in the first loss structure, the manager provides full insurance on losses (up to a preset limit), while in the shared loss structure the manager provides partial insurance.

In analogy with the concept of diversification from portfolio theory, it may be posited that the optimal fee structure from the fund investor’s point of view should be a shared-loss structure, i.e., a combination of the extremes.^{3} Were such a diversification effect to exist, it would show up not in the (risk-neutral) valuation of the fee structures, but rather when examining the risk-return tradeoff among fee structures.

Djerroud et al. (2016) investigated the fairness of the prevailing levels of hedge fund fees in terms of risk-neutral valuation. In this article we put the hypothesis of fee diversification to the test, by examining the problem of an investor who may choose any combination of the first loss and classical fee structures, and seeks to maximize either the Sharpe ratio or the Sortino ratio of the final payoff, evaluated using real-world probabilities. We demonstrate that for the vast majority of fund mean returns and volatilities, there is no fee diversification effect: either the first-loss structure or the classical structure is optimal for the investor. We also identify the regions of the parameter space in which fee diversification prevails, and demonstrate that even in these regions the effect of fee diversification is not large.

The remainder of the article is organized as follows. The next section describes the payouts to hedge fund investors under mixed fee structures. The third section summarizes analytical results regarding the Sharpe ratio (and its maximization) and the Sortino ratio for mixed fee structures. The fourth section presents numerical examples under the assumption of a geometric-Brownian motion for the hedge fund assets.^{4} The final section concludes the article.^{5}

## PORTFOLIO OF FEE STRUCTURES

Let be the value of the hedge fund’s assets at time *t*, assuming a value of *x* at the initial date. Further, let *m*_{1}, and α_{1} denote the management fee and performance fee for the traditional fee structure, and *m*_{2}, and α_{2} denote the management fee and performance fee for the first-loss fee structure. We assume the management fees are proportional to the initial investment *x* and performance fees are proportional to the final hedge fund value . Thus, the payoff function for the investor in the traditional fee structure at maturity is:

while the payoff function under the first-loss fee structure at maturity is:

2where *c* is the deposit amount, giving the fraction of the initial investment that is insured (see Djerroud et al. (2016)). Now, let ω_{1} ≥ 0 and ω_{2} ≥ 0 denote the proportion of the fund invested in the traditional fee structure and first-loss structure respectively, where ω_{1} + ω_{2} = 1. Then, the investor’s final payoff becomes

where the second equation follows since

## PERFORMANCE RATIO MAXIMIZATION WITH PORTFOLIOS OF FEE STRUCTURES

Assuming no early withdrawals, and an investment horizon of *T*, the investor’s Sharpe ratio becomes:

where is the fund investor’s return, *r*_{f,T} is the risk-free interest rate, and ω = (ω_{1}, ω_{2}).

We assume that the investor seeks to maximize the Sharpe Ratio *SR*(ω). Let *Y*_{1} be the excess profit (above an investment at the risk-free rate) of an investment in the classical fund scheme (ω_{1} = 1), and *Y*_{2} be the excess profit of an investment in the first-loss structure (ω_{2} = 1). Let be the optimal portfolio. If , *i* = 1, 2, and is the covariance matrix of (*Y*_{1}, *Y*_{2}), then the following can be shown.^{6}

1. If 5

then 6

where .

2. Otherwise, 7

By the above results, one can distinguish between the cases for which there exists a nontrivial asset allocation for the traditional fee structure and the first-loss fee structure ( for *i* = 1, 2), and when full investment in one fee structure is optimal in terms of the Sharpe ratio. The results can be understood in terms of the Sharpe ratios , *i* = 1, 2 of the two payoffs, and their correlation ρ = corr(*Y*_{1}, *Y*_{2}). Assuming that , if the correlation is less than the threshold , then there is enough potential for diversification that the optimal portfolio will have a positive investment in each fee structure. Otherwise, all the wealth will be invested in the fee scheme with the higher Sharpe ratio. The greater the difference between the Sharpe ratios of the two stand-alone fee structures *Y*_{1} and *Y*_{2}, the lower the threshold *H*, and consequently the lower the correlation required to prompt investors to put some money in the fee structure with the lower Sharpe ratio. Since *Y*_{1} and *Y*_{2} are both increasing piecewise linear functions of the same random variable , they will tend to be highly correlated, and for most parameter sets we would expect ρ > *H*, and full investment in the fee structure with the highest Sharpe ratio.

The above discussion indicates a Catch-22 of seeking diversification in fee structures, due to the typically high correlation between *Y*_{1} and *Y*_{2}. Either the threshold *H* is low, in which case it is likely that investing all wealth in one or other fee structure is optimal, or *H* is high, in which case the Sharpe ratios of the two fee structures are very close, and the improvement due to diversification (if it exists) is small.

Real-world hedge fund return distributions are often asymmetric, with fat tails. The Sharpe-ratio, which is based on the use of variance as a measure of risk, may be misleading when used for such distributions. Consequently, we also consider the maximization of the investor’s Sortino ratio, which instead uses downside variance as the measure of risk:

8where *l* is the minimal acceptable return and is called the lower partial moment of *r*(*T*).

## NUMERICAL RESULTS

Throughout this section, we assume the hedge fund’s dynamics under the real-world measure follow:

9where μ > 0 is the annual growth rate, σ > 0 is the annual volatility, and *W*_{t} is a standard Brownian Motion.

It should be noted that in all the examples in this article we assume that the parameters of the processes are known with certainty. This is obviously not the case in the real world (indeed, in practice we do not even know the process driving the returns). In practical applications, the added uncertainty due to the need to estimate the parameters (and the structure of the process driving returns) must be taken into consideration. In particular, the drift parameter μ, which has a significant impact on the preferred fee structure, is notoriously difficult to estimate in practice (see, e.g., Merton [1980]).

In Exhibits 1 and 2, we fix the six parameters μ, σ, *T*, *m*, *c*, *r* and let the remaining parameters α_{1} and α_{2} (the performance fee rates for the traditional and first-loss fee structures respectively) vary in their reasonable ranges. For each valid point (α_{1}, α_{2}), we check condition (3.2) and color the point black if the condition is satisfied. The black regions indicate that there exists a portfolio with non-zero allocation to both fee structures ω which achieves the optimal Sharpe Ratio (an “interior maximum”). In the white regions, there is no interior maximum, and the optimal allocation is 100% in one fee structure or the other (in the area above the black region the traditional fee structure is preferred, while below the black region the first-loss fee structure is preferred).

We see that the black regions for Exhibits 1 and 2 are very narrow, implying that in most cases there is no benefit due to “fee diversification”; either the traditional fee structure or the first-loss fee structure makes the investor achieve the optimal Sharpe Ratio when α_{1} and α_{2} are changing. The first-loss fee structure appears to be preferred for the bulk of the performance fee combinations that we consider (i.e., the area below the black region is larger than the area above the black region). The black region follows a nearly linear path for all combinations of (μ, σ) that we considered. Looking at the different graphs, we see that the traditional fee structure is preferable (requires a lower performance fee to be the structure selected) when the underlying hedge fund assets have higher expected returns, and lower volatilities, with the impact of low volatilities being particularly pronounced. This is intuitive, as with higher means and lower volatilities, the put-option-like downside protection provided to the investor in the first-loss structure is less likely to be exercised (and will tend to pay off less when it is exercised).

Next, we fix α_{1} = 0.2 and α_{2} = 0.5 (typical market values) and consider all possible portfolios (ω_{1}, ω_{2}) for different values of the parameters μ, σ, and *c*. In particular, we consider funds with high expected return and low volatility, low expected return and high volatility, and different deposit amounts, for which results are given in Exhibits 3, 4, and 5 respectively. The results in Exhibits 3, 4, and 5 are rather intuitive, and reinforce the observations made above. As high expected return and low volatility indicate the hedge fund is more likely to make a profit, and the downside protection afforded by the first-loss structure will not be relevant, the investor will favor the fee structure with the lower performance fee (the classical fee structure). However, we notice that the Sharpe Ratio curves in Exhibit 3, corresponding to a low volatility regime, are rather flat; while the classical fee structure is preferred, there is not much significant difference between the two fee structures in terms of Sharpe Ratio. On the other hand, low expected return and high volatility make the fund have a higher chance of suffering a loss, leading the investor to prefer the first-loss fee structure. Although Exhibit 4 presents steeper curves, the Sharpe Ratios are very close to zero. Finally, Exhibit 5 indicates that the first-loss fee structure is more heavily favored by investors when the deposit amount increases. This is entirely intuitive; the greater the protection against losses offered, the more appealing the first-loss fee structure appears.

Exhibits 6 and 7 present detailed results on the nature of the investor’s optimal fee structure for different values of μ and σ. In Exhibit 6, α_{2} = 0.4, while in Exhibit 7, α_{2} = 0.5. The results demonstrate that the investor will always choose the traditional fee structure given the hedge fund expected return is high enough. The significance of the downside protection effect of the first-loss fee structure diminishes as the expected return increases. Essentially, the fee structures with lower performance fees result in higher Sharpe Ratios. Furthermore, we see that, in contrast to what we observed earlier, the classical fee structure is preferred for extremely high levels of volatility. As noted before, when volatility increases, both the probability that the guarantee will be triggered increases, and the expected payoff given a loss is larger. However, when volatility is very high, the possibility of an extremely high payoff in the classical structure (under which the investor keeps a greater share of the profits compared to the first-loss structure) outweighs the higher expected payoff of the insurance. Very low expected returns also can produce negative Sharpe ratios, for which our Sharpe ratio maximization framework is not appropriate. Comparing the two figures, we observe the unsurprising result that the classical fee structure is preferred more often when α_{2} (the manager’s participation rate in the first-loss structure) is higher.

Finally, we let *S*_{max} denote the optimal Sharpe Ratio and *S*_{min} denote the lowest possible Sharpe Ratio for the investor. In Exhibits 8 and 9, we plot the ratios of *S*_{max} to *S*_{min} for some parameter values for which an interior optimal point exists. Note that the lowest Sharpe Ratio is always attained on one of the boundary points, i.e., either the traditional fee structure or the first-loss fee structure admits the lowest Sharpe Ratio. Generally, we find that the optimal Sharpe Ratio is not significantly larger than the worst Sharpe Ratio of these two fee structures. In fact, the largest ratio of *S*_{max} to *S*_{min} is only about 1.006 when μ ≈ 0.15 and σ = 0.1, indicating very little benefit from “fee diversification.” This makes sense in light of our theoretical results. Since ρ = corr(*Y*_{1}, *Y*_{2}) is high, in order to meet the condition for an interior maximum , we must have *SR*_{1} ≈ *SR*_{2}, i.e., the Sharpe ratios of the two fee structures must be nearly identical, and diversification has little impact.

Next, we consider the results of maximizing the Sortino ratio. Exhibits 10 and 11 present detailed results on the nature of the investor’s optimal fee structure for different values of μ and σ, with α_{2} = 0.4 and α_{2} = 0.5 respectively. The results here again admit an intuitive interpretation, fixing a level of return μ > 0, considering beginning at σ = 0, and then increasing the volatility. For zero or very low volatility, the positive mean return dominate, and the insurance component of the first-loss structure is almost never triggered. As a result, the classical fee structure will be preferred, as the one in which investors keep a larger percentage of the gains, since α_{1} < α_{2}.

As volatility increases, so do both the probability that the insurance portion of the first-loss structure will come into effect, and the expected size of the protected losses given that losses occur. Hence, the first-loss fee structure becomes more appealing. There is a brief transition period of limited “fee diversification” before investing all the funds in the first-loss fee structure becomes optimal. As the volatility increases still further, the expected benefit of the insurance component is limited (the downside protection is capped), and the potential for very large upside gains (due to extremely high volatility) becomes increasingly important. Since investors in the classical fee structure keep a higher percentage of these extreme gains, for very high levels of the volatility, the classical fee structure is preferred. Again, there is a small black transition region, in which limited “fee diversification” exists.

To assess the potential for diversification benefits when performance is measured using the Sortino ratio rather than the Sharpe ratio, we plot the ratios of the maximal Sortino ratio (*SOR _{max}*) to the minimal Sortino ratio (

*SOR*) in Exhibits 12 and 13 for cases in which an interior optimal point exists. The benefits from diversification for the Sortino ratio, while still confined to a subset of the possible parameter values, are more significant than in the case of the Sharpe ratio, and this effect is larger at higher levels of volatility. Again, considering a fixed μ, and increasing σ, we see the case when volatilities are high and diversification is possible corresponds to the black region transitioning between exclusive investment in the first-loss fee structure (the dark grey region in Exhibits 10 and 11) and the high volatility region when exclusive investment in the classical fee structure is optimal (the upper light grey region). The scope for diversification here is more pronounced because as we move into the classical structure, we are only being penalized in the Sortino ratio for its downside performance (rather than its variance, which is affected by the potential for extremely good returns, as well as extremely bad ones).

_{min}## CONCLUSION

In this article, we study and compare the Sharpe Ratios and Sortino Ratios of hedge fund investors facing combinations of the first-loss fee structure and traditional fee structure. In particular, we maximize the Sharpe Ratio or the Sortino Ratio of a portfolio that combines payoffs of the two fee structures at maturity *T*. A criterion (3.2) is presented to distinguish whether or not an interior optimal weight allocation exists between the two fee structures for the Sharpe Ratio. Numerical examples are presented for both the Sharpe and Sortino Ratios, assuming a geometric Brownian motion process for the hedge fund assets. In most cases we find that the optimal weights are on the boundary points, indicating that one extreme fee structure is preferred compared to the other, and all possible mixtures. Typically, the classical fee structure is preferred at very low volatilities, where the insurance in the first-loss contract is unlikely to be triggered, and at extremely high volatilities, for which the share of a very large potential upside is important. At intermediate volatilities (often covering most of the range typically seen in practice), the first-loss structure is preferred, owing to the importance of its insurance component. We find that there is negligible benefit to the investor due to “fee diversification,” even in the case when a combination of fee structures is optimal.

## ADDITIONAL READING

**A Fund of Hedge Funds Under Regime Switching**

David Saunders, Luis Seco, Christofer Vogt, and Rudi Zagst

*The Journal of Alternative Investments*

**https://jai.pm-research.com/content/15/4/8**

**ABSTRACT:** *This article investigates the use of a regime-switching model of returns for the asset allocation decision of a fund of hedge funds. In each time period, returns follow a multi-variate normal distribution from one of two possible regimes, corresponding to periods of “normal” and “distressed” markets. The prevailing regime in any given period is determined by the value of a two-state Markov chain. The case where serial correlation is absent and returns in different time periods are i.i.d. Gaussian mixture variables is also considered. The models are tested on empirical data and compared to a benchmark, assuming i.i.d. normally distributed returns. The results show that in a mean–variance framework, the use of regime switching can improve risk and performance measures. The importance of the sensitivity of optimal portfolio weights to the estimate of the probability of the distressed regime is discussed, and methods for calculating sensitivities are presented and illustrated on market data.*

**Portfolio Optimization in a Multidimensional Structural-Default Model with a Focus on Private Equity**

Marcos Escobar, Peter Hieber, Matthias Scherer, and Luis Seco

*The Journal of Private Equity*

**https://jpe.pm-research.com/content/15/1/26**

**ABSTRACT:** *Investments in various asset classes, such as private equity or hedge funds, are prone to default risk, which needs to be accounted for when calculating individual investment opportunities and optimal portfolio selection. The correspondent literature on portfolio optimization, however, mostly disregards default risk and accordingly skewed return distributions. This article presents a realistic and tractable framework for a portfolio optimization, including default risk, with a specific focus on private equity investments. Default events are modeled by means of a Merton- or Black–Cox structural model. On a portfolio level, the mean and covariance of the resulting return distribution can be derived analytically, allowing for a classical mean-variance optimization. To include tail risk, we additionally present a Monte-Carlo simulation for a mean conditional value-at-risk optimization. The article concludes with an application to unlisted private equity and compares the results with a model proposed by Hamada [1972], which does not explicitly consider default risk.*

**Correlation Breakdown in the Valuation of Collateralized Fund Obligation**

Unai Ansejo, Marcos Escobar, Aitor Bergara, and Luis Seco

*The Journal of Alternative Investments*

**https://jai.pm-research.com/content/9/3/77**

**ABSTRACT:** *In a collateralized fund obligation (CFO) several hedge funds are pooled in a fund that is in turn, securitized. Since the first securitization of a fund of hedge funds, launched in June 2002, these products haven’t had much impact among investors. This is not surprising considering that it is very difficult to predict the credit solvency of each tranche of the securitization and therefore its valuation. In this article a particular CFO is analyzed in an analytical pricing framework, capturing the correlation breakdown and the leptokurtic phenomena characteristics of hedge funds. The results show that due to a lack of transparency which is common in hedge funds, confidence intervals in probabilities of default and credit spreads, are high enough to burden CFO proliferation.*

## ENDNOTES

↵

^{1}https://www.bloomberg.com/news/articles/2017-06-07/new-york-illinois-pension-funds-sayhedge-funds-fees-too-high.↵

^{2}For further discussion on this point, see He and Kou (2018), Kouwenberg and Ziemba (2007), and Hodder and Jackwerth (2007).↵

^{3}The payouts with different fee structures on the same fund are comonotonic, but not perfectly correlated.↵

^{4}Results assuming a regime-switching process for the hedge fund assets are contained in the forthcoming University of Waterloo PhD thesis of F. Meng.↵

^{5}Technical details and derivations of results are available from the authors upon request.↵

^{6}The derivation is given in a technical appendix available at http://www.math.uwaterloo.ca/dsaunder/Publications.html.

- © 2019 Pageant Media Ltd