Robustness is key when assessing the performance of hedge funds. Since investment strategies are very diverse, sources of risk and corresponding exposures are hard to identify. Moreover, pension funds and rich individuals do not exhibit the same risk tolerance and will not assess performance in the same way. Traditional measures cannot offer such robustness to complex risk-factor exposures and investors’ risk aversion.

Ratios involving mean returns and volatility (Sharpe, Treynor or Information ratios) or even more sophisticated ratios accounting for downside risk (Sortino, Gain-Loss or Omega ratios), do not control for the wide variety of strategies followed by hedge funds and therefore are not sufficiently informative to rank funds. Peer benchmarking also has its limits: homogeneous peer grouping is difficult given the absence of information regarding the hedge funds’ holdings and strategies, and investors cannot assess whether or not the average manager within the peer group generates any value. Measuring performance through alpha after adjusting the returns for the risk exposures to several factors delivers a finer basis for comparing funds.

Equity, fixed income, credit, commodity or currency risks are included, together with returns on option strategies built on these primitive factors. A main drawback of the current approaches is to relate hedge fund returns to such risk factors linearly since it has been shown that hedge fund returns exhibit complex non-linear exposures to traditional asset classes.

**Non-linear exposures and risk factors**

We propose a new method that captures the complex non-linear exposures of a hedge fund strategy to several risk factors. It accommodates many non-linear functions of factor returns, hence the term nonparametric, over and above the usual option payoff patterns. In addition, it produces a risk adjustment function that weights hedge fund returns differently depending on the risk tolerance of an investor. Therefore, the computed alpha performance – the average of the risk-adjusted returns – is robust in both the non-linear exposure to risk factors and investors’ risk aversion.

Alpha performance is measured by the averaged product of a fund’s returns by a risk-adjustment function, called stochastic discount factor in asset pricing theory. Intuitively, the methodology is based on seeking to identify a non-linear function of risk factors that remains positive at all times to avoid arbitrage and that prices perfectly the basis assets selected as factors. In other words, the methodology will assign a zero alpha to any payoff that is trivially related to available factors, so that the measure of abnormal performance for hedge fund returns will only capture the fraction of the hedge fund returns that is due to the managers’ active skills. The risk adjustment function makes clear which risk factors are really important for the performance of hedge funds under analysis.

**Risk adjusting hedge fund payoffs**

The main idea is to risk-adjust hedge fund payoffs in a way that accounts for the asymmetry or tail risk exposures created by the dynamic strategies pursued by hedge funds. Indeed, the risk-adjusted performance measure will not only be based on means and volatilities of hedge fund returns, which are not sufficient statistics given the strong deviations from normality that hedge fund returns exhibit, but also on higher-order moments of the distribution of hedge fund returns.

To establish the relevance of our approach from an empirical standpoint, we evaluate the performance of various hedge fund indices, considering a set of risk factors including equities, bonds, credit, currencies and commodities, as well as several straddle strategies. We report how the measured alpha varies with the inclusion of option-based factors and with the risk tolerance of the investor. We find that, as we decrease risk tolerance, the alpha decreases significantly for some categories (emerging markets), while remaining fairly unchanged for others (equity hedge and macro). In the latter case, we conclude that the performance is robust. Yet in other cases, some funds pay well in bad times (Asian funds or short bias), offering insurance value, and their alpha increases as we decrease risk tolerance. These findings strongly suggest that what was incorrectly measured as hedge fund alpha in previous studies is actually some form of fair reward obtained by hedge fund managers from holding a set of relatively complex linear and non-linear exposures with respect to various risk factors. Often the reduction in performance comes from a small number of extreme events which are not captured well with the usual linear approach. Our findings also support the view that higher-moment equity risks capture a large part of the non-linear risk exposure of several hedge fund strategies. However, exposure to higher-moment risks for bond, interest rate or currency is essential for other strategies, in particular emerging markets. Finally, we also illustrate with individual funds how a fund manager can measure the sensitivity of his portfolio of funds to shocks affecting risk factors, that is macro shocks, or to idiosyncratic shocks impacting a particular fund.

The approach can be extended to evaluate hedge fund managers’ performance conditionally to specific macroeconomic environments such as high or low interest-rate states, large or limited economic uncertainty, boom or bear markets, liquid or illiquid markets, making the performance measurement more transparent to general economic conditions.

**Evaluating performance **

Performance evaluation of hedge funds has proceeded with two main approaches. One considers only absolute returns, while the other rests on the identification of risk factors behind hedge fund returns. It has been quickly recognised that the absolute performance measurement for hedge funds needs to go beyond the traditional Sharpe or Treynor ratios.

Indeed, hedge fund return distributions are distinctly abnormal and measures based on the mean and variance are not sufficient to capture the risk associated with hedge fund returns. Other measures have been proposed to account for the negative skewness and positive large kurtosis exhibited by hedge fund return distributions, namely the Sortino ratio, the Stutzer index or the Omega ratio. While these adapted measures are better able to capture the higher-moment risk present in hedge fund returns, they are not sufficient to rank funds. We need additional indicators to know if a given fund is doing well relative to other funds using similar strategies. The relative approach starts with the peer analysis, whereby funds in comparable groups are evaluated based on absolute return measures. Performance relative to peers may be measured during market cycles or over short or long periods. However, the groups may not be homogeneous enough to capture the underlying exposures to different risks. Therefore, measuring performance through alpha after accounting for the beta risks appears to deliver a finer basis for comparison between funds.

**Risk factors**

The alpha approach necessitates spelling out the risk factors that may affect hedge fund returns. Given the wide diversity of strategies followed by hedge funds the literature has evolved to include exposures to the main sources of risk, such as equity, fixed income, credit, commodity or currency. The main approach is to estimate linear factor models where hedge fund returns are regressed linearly on such risk factors. Such an approach captures only linear exposure to the risk factors, but several studies have shown that a large number of equity-oriented hedge fund strategies exhibit payoffs resembling a short position on a put option on the market index. These approaches to capturing non-linearities in hedge funds’ payoffs are targeted towards specific option-like strategies.

Fung and Hsieh (2001) analyse trend-following strategies and show that their payoffs are related to those of an investment in a lookback straddle. Mitchell and Pulvino (2001) show that returns to risk arbitrage are similar to those obtained from selling uncovered index put options. Agarwal and Naik (2004) extend these results and show that, in fact, a wide range of equity-oriented hedge fund strategies exhibit this non-linear payoff structure. Diez de los Rios and Garcia (2011) propose a more general approach to identify the nature of the option that best characterises the payoffs of a hedge fund. It can therefore be used to analyse any strategy. However, one needs to specify the risk factor underlying the option and the number of kinks allowed. Most of their applications have used the market index with one kink, to identify positions in put or call options or straddles. Extending the method to several risk factors and more than one kink runs into the obstacle of limited length time series of hedge fund returns. Therefore, it is empirically impossible given the amount of data available to identify spread positions on several risk factors.

**Limitations overcome**

The approach proposed in this paper overcomes these limitations. First, it allows oneto look at the non-linear exposure of a hedge fund strategy to several risk factors. Second, it is not limited to shapes resembling standard option payoff patterns. Being nonparametric, it produces a factor model that captures many non-linear functions of returns for the assets chosen as risk factors, overcoming the above-mentioned problem of limited data availability.

Moreover, it can add non-linearities to option risk factors such as the straddle strategies used in Fung and Hsieh (2001). The main idea is to risk-adjust hedge fund payoffs in a way that accounts for the asymmetry or tail risk exposures created by the dynamic strategies pursued by the hedge funds. Abnormal performance is measured by the expected product of a portfolio’s returns and a risk-adjustment function also called stochastic discount factor. The evaluation can proceed unconditionally or conditionally to a set of lagged instruments.

The methodology is based on minimising a general convex function to obtain a Minimum Discrepancy (MD) measure (Corcoran, 1998) that exactly prices the basis assets selected as factors. A well known example of such discrepancy measures is the Kullback-Leibler information criterion (KLIC). We choose a family of discrepancy functions that admits as particular cases the quadratic criterion of Hansen and Jagannathan (1991), hereafter HJ, and the KLIC, but offers other information criteria that have different implications for assessing performance.

The solutions for these risk-adjustment non-linear functions are obtained more easily by solving a portfolio problem, with the maximisation of a specific utility function in the Hyperbolic Absolute Risk Aversion (HARA) family. The first-order conditions for these HARA optimisation problems provide solutions for the risk adjustment weights that are non-linear and positive, directly generalising the linear SDF in HJ (1991) with positivity constraints and guaranteeing no arbitrage when pricing hedge fund payoffs. An additional advantage is that the approach shows how reference assets chosen as risk factors should be weighted within the risk adjustment function, thus indicating which risk factors are really important when analysing hedge fund performance.

**Using conditioning information**

Several studies have used conditioning information to evaluate the performance of managed portfolios. Performance can be evaluated conditionally to specific macroeconomic environments such as high or low interest-rate states, large or limited economic uncertainty, boom or bear markets, liquid or illiquid markets, making the performance measurement more transparent to general economic conditions. These studies usually limit themselves to conditional measures of performance that involve only conditional means and variances of portfolio returns.

We extend the literature on conditional performance measurement by producing conditional measures that take into account all conditional moments of the risk-adjustment functions. Conditional approaches have the potential advantage of having thinner tailed conditional distributions that better control the effect of extreme observations on the moments of asset returns. However, our generalised discrepancy measures, even taken unconditionally, are better able to capture the effect of these extreme observations because they account for higher moments in the unconditional distribution of returns. This is especially important when evaluating the performance of managed portfolios since private information on which fund managers condition their trades is unobservable.

In this case only the potentially fatter-tailed unconditional returns are observable. Our unconditional risk-adjustment measures will account for the effect of this unobservable information. Our implied non-linear measures are related to a number of previous studies that feature non-linear risk-adjustment or discounting functions. Bansal and Viswanathan (1993) propose a neural network approach to construct a non-linear stochastic discount factor that embeds specifications by Kraus and Litzenberger (1976) and Glosten and Jagannathan (1994).

**Family of hyperbolic functions**

Our approach provides a family of different hyperbolic functions of basis factor returns implied by the solution to portfolio problems. In Dittmar (2002), who also analyses non-linear pricing kernels, preferences restrict the definition of the pricing kernel. Under the assumption of decreasing absolute risk aversion, he finds that a cubic pricing kernel is able to best describe a cross-section of industry portfolios. Our nonparametric approach embeds such cubic non-linearities implicitly. Boyle et al. (2008) obtain robust prices for derivative securities based on discounting functions that cause minimum perturbations on prices of derivatives payoffs. Our methodology, if used to price derivatives, will provide pricing intervals based on the HARA implied risk-adjustment functions.

To establish the relevance of our approach, we evaluate the performance of various hedge fund indices, considering a set of risk factors including equities, bonds, credit, currencies and commodities, as well as several straddle strategies. We compare the measured alphas to option-based performance measures obtained by a linear model. To capture non-linearities and measure the alpha performance of the funds, Agarwal and Naik (2004) use a linear regression in which they introduce the returns on a portfolio of options along with the other usual risk factors.

Our analysis accounts explicitly for higher moments of returns induced by option-like strategies.Moreover, an important feature of our discrepancy-based approach is the possibility to capture more complex non-linearities since options portfolios can be included as factors themselves. Alpha valuations obtained with the implied non-linear risk-adjustment measures are in general lower than the performances exhibited when introducing option factors linearly.

Our study complements the analysis of Agarwal, Bakshi and Huij (2010) who investigate the relationship between the cross-section of hedge fund returns and higher-moment equity risks. We directly relate the performance of hedge funds to higher moments of all risk factors, including straddles on equity, commodity, currency, bond, and short interest rate (see Fung and Hsieh, 2001). Our findings support the view that higher-moment equity risks capture a large part of non-linear risk exposure of several hedge fund strategies. However, exposure to higher-moment risks for bond, interest rate or currency is essential for other strategies, particularly for emerging markets.

**Caio Almeida** is an assistant professor in the Graduate School of Economics of the Getulio Vargas Foundation in Rio de Janeiro. René Garcia is Professor of Finance at EDHEC Business School and the Academic Director of the EDHEC-Risk Institute PhD in Finance programme.

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## Commentary

## Issue 79

## Nonparametric Discounting

## A robust assessment of hedge fund performance

## CAIO ALMEIDA and RENE GARCIA, EDHEC-RISK INSTITUTE