L-Performance with an Application to Hedge Funds

This paper introduces a new fund performance measure, called the L-performance. It is proposed as an alternative to the Sharpe performance measure that is commonly used for fund performance valuation despite its inability to account for skewness and thick tails of fund return distributions. The L-performance improves upon the Sharpe measure in this respect. Technically, the L-performance is based on sample statistics, called L-moments, which are conceptually close to the conventional power moments, but provide more detailed information about the extremes. For this reason, the L-moments are used for prediction and assessment of extreme events, such as floods and earthquakes. In this paper, the new L-performance measure is calculated for a variety of hedge funds and is used to derive a fund ranking.


Introduction
The so-called Sharpe performance is a commonly accepted measure of fund performance.
It is defined as a ratio of the expected excess return and volatility, and used for segmentation and fund ranking [see e.g. Sharpe (1966), Lo (2002), Darolles, Gourieroux (2008)]. Its main shortcoming is that it relies on only two statistics, i.e. the sample mean and volatility, which do not always provide a sufficiently precise characterization of return distributions, especially when these distributions feature skewness and thick tails.
In particular, the sample mean may not provide a sufficiently robust approximation of the location parameter, and the standard error may not properly account for the size of tails. This is of special importance for the (unconditional) distributions of hedge fund returns, whose departures from normality are related to some liquidity and management characteristics [see e.g. Getmansky, Lo, Makarov (2003)].
The aim of this paper is to introduce a battery of alternative performance measures  Hosking (2007)]. It seems that there exists only two published applications of L-Moments to financial returns [Karvanen (2006), Gourieroux, Jasiak (2008)].
In Section 2, we recall and extend the definitions of trimmed L-moments of orders 1 and 2, and discuss their interpretations in terms of the quantile and concentration functions. The L-performances are defined in Section 3 as ratios of trimmed L-moments of order 1 and 2. We explain how the L-performances are estimated and derive the asymptotic distribution of the L-performance estimator. The performance measures are used in Section 4 to rank a set of hedge funds. Section 5 concludes. Proofs are gathered in Appendices.

Trimmed L-moments of order 1
The n-trimmed L-moment of order 1 is defined by: It is equal to the expectation of the median of conceptual sample. The trimmed L-moment of order 1 of X has the following expressions involving the cdf and the quantile function, respectively [Elamir, Seheult (2003), Hosking (2007)]: The polynomials For n = 0, the mean is obtained: λ 1,0 = 1 0 Q(u)du = EX and, when n tends to infinity, the median is obtained: λ 1,∞ = lim n→∞ λ 1,n = Q(0.5). When n increases, the trimmed L-moments of order 1 bridge the mean and the median. The existence of trimmed Lmoments requires less restrictions than the existence of conventional moments. Indeed, Hosking (2007), Theorem 1]. Thus, the trimmed L-moments for n ≥ 1 can be defined even when the expectation of returns does not exist, and they are less sensitive to outliers when n increases.
The patterns of polynomials P 1,n are displayed in Figure 1. The distortion measure associated with polynomial P 1,n is symmetric, centered at 0.5, and becomes more peaked when n increases. In contrast, it tends to a uniform distribution (resp. point mass at 0.5), when n tends to 0 (resp. infinity).

Trimmed L-moments of order 2
The (r, n) trimmed L-moment of order 2 is defined by 1 : It measures the expected range of a conceptual sample after deleting the r smallest and r largest conceptual observations. These measures increase with r. This L-moment of order 2 exists when E[|X| 1/(r+1) ] < ∞ [see Hosking (1990)]. Thus, the measures are less sensitive to outliers when r increases. Definition (2.4) extends the concept introduced in Elamir, Seheult (2003), which corresponds to the special case r = n − 1, and λ 2,n−1,n = E(X n+2:2n+1 −X n:2n+1 ). For this particular value of r, the conceptual range is determined by variables with ranks n and n+2. In financial applications, it seems preferable to choose a value of r much smaller than n − 1 in order to better capture extreme risks.
The analytical expression of the trimmed L-moment of order 2 is: The definition above assumes an odd size of the virtual sample. This definition does not include the basic L-moment of order 2 initially introduced by Hosking (1990). This L-moment is: dx the cumulated quantile function, that is the Gini (or concentration) curve [Gini (1912)], we get by integrating by part: λ2 The polynomials: can be of any sign and have mass zero. Thus the associated distortion measures are not positive. The patterns of polynomials P 2,r,n vary with n and r, as shown in The L-moment at order 2 has an integral representation equivalent to (2.5) and based on the cumulated quantile function G(u) = u 0 Q(x)dx. Function G is the concentration (or Lorentz) curve used in concentration analysis and in stochastic dominance of order 2.
In Figure 2, we see that the L-moment assigns positive weights to the extremes (i.e. the values of u close to 0 and 1), and thus focuses on extreme risks. Equivalently, the L-moment of order 2 can also be written in terms of covariance, since λ 2,r,n = cov(X, P 2,r,n [F (X)]).
It measures the link between the return and its (historical) rank [Serfling, Xiao (2006)].

L-Performance
Let us now consider a sequence x 1 , ..., x T , say, of portfolio excess returns. In this paper

Definition
By analogy to the Sharpe performance ratio, which is the ratio of the expected net return and volatility [Sharpe (1966)], the L-performance is defined as the ratio of a trimmed L-moment of order 1 and a trimmed L-moment of order 2: L r,n = λ 1,n /λ 2,r,n (3.7) 3 For the standard Sharpe performance measures, two extensions to non-iid returns are possible: i) First, one can still consider the historical performance measures, and in addition derive their asymptotic distributions in a non-iid framework [see e.g. Lo (2002) for this approach applied to Sharpe performance].
ii) Second, one can consider the conditional performance measures instead of unconditional ones [see e.e. Darolles, Gourieroux (2008) for conditional Sharpe performances and Gourieroux, Jasiak (2008) for the definition of conditional L-Moments].
These extensions are clearly out of the scope of the present paper.
This leads to a battery of L-performances that depend on the selected shrinkage parameters r and n. For example, we have: When n → ∞ , r → ∞, and r/(2n + 1) → 5%, say, we have: , (3.8) where VaR denotes the Value-at-Risk computed by historical simulation as suggested in the standard approach by Basle Committee. In fact, the L-performance measures extend the (inverse) Gini concentration index [Gini (1912)]. Indeed When the returns are such that x t = m + σu t , where the error terms have a symmetric distribution and m, σ are the location and scale parameters, respectively, we have: Q(u) = m + σG(u), where G is the quantile function of the standardized error term. By the symmetry condition, we have: G(−u) = G(1 − u). It follows that: In this framework, the L-performances are proportional to the ratio m/σ, up to a scale factor depending on r, n, and on the error term distribution. In particular, for Gaussian errors, the L-performance is equivalent to the standard Sharpe performance measure, up to a multiplicative scalar function of the shrinkage parameters only.

Estimation of L-performance
The L-performances are easily estimated from their sample counterparts. The estimator is defined by (see Appendix 1): This is a ratio of two linear combinations of order statistics 4 . These estimators are consistent, asymptotically normal under standard regularity conditions (see Appendix 1). For instance, for independent and identically distributed 5 excess returns, we have: where the asymptotic variance is given by: (3.11)

Estimation of the asymptotic variance ofL r,n,T
The asymptotic variance can be alternatively written as:  At issuing, the hedge funds can be self-declared in one or several categories, called the styles, from a given list [see e.g. Das, Das (2004) for a description of styles in various databases]. These categories describe either the type of assets in the portfolio (as the styles "Currencies", "Distressed Securities"), or the type of portfolio management (for instance the styles "Global Macro", "Merger Arbitrage"), or both of these (as "Fixed Income Arbitrage", "Equity Long/Short"). A majority of pure hedge funds belong in 9 categories, that are "Equity Long/Short", "Fixed Income", "Global Macro", "Currency", "Futures", "Equity Long Short Equally Weighted", "Fixed Income Arbitrage", "Merger Arbitrage" and "Distressed Securities".
We select 36 funds from various categories and report the information on the management company, the self-declared strategy and the assets under management. This information is displayed in Table 1, where tickers are used to abbreviate the name of each fund in the sample.

Fund Return Distributions
The unconditional return distribution of the hedge fund excess return is summarized by its conventional and L-moments of order 1 to 4.
[Insert Table 2 : L-Moments of order 1] The first and last columns in Table 2 provide the mean and the median, respectively.
Since the L-moments of order 1 are constructed from a basis of symmetric distortion polynomials, the distribution is symmetric if and only if all L-moments of order 1 are equal.
For several funds such as CSS ED or DF MF, the distribution is almost symmetric, but in general the distribution of fund returns is skewed. We also note that a simple test of [Insert Table 3 : Power Moments and L-Moments of order 2] The L-moments of order 2 for n varying provide similar classifications of risks. For instance the fund RO ED, with the largest variance, has also the largest L-moments of order 2 for any value of trimming parameters. However, the funds next in the variance ranking, in a descending order such as funds DF MF, RC EH or WI MA, have a different ranking with respect to their L-moments of order 2. In particular, fund RC EH is more risky than WI MA in variance, but less risky in all L-moments of order 2 in columns 2-6.
[Insert Table 4 : Power Moments and L-Moments of order 3 and 4] Table 4 provides the conventional skewness and excess kurtosis. It also displays the L-skewness and L-kurtosis defined by λ 3 /λ 2 , λ 4 /λ 2 where: . The L-skewness (resp. L-kurtosis) measures the difference between the slopes of the quantile function for u smaller and greater than the median, respectively, to detect asymmetries (resp. outside and within the quartile interval to detect the magnitude of the tail). The standard Gaussian L-kurtosis is equal [Hosking, Wallis (1997)]. As expected, the conventional and L-skewness of funds CSS ED and DF MF are close to zero (see the discussion of Table 2). More importantly, some of the funds which appear very risky in Table 3, with respect to their moments of order 2, can feature L-kurtosis smaller than the Gaussian L-kurtosis, which is equal to 0.0692. For example, funds WI MA or DF MF have large risks in the central part of the distribution, and rather small risks associated with the extremes.
The vertical and horizontal axes of Figure 4 measure the L-skewness and L-kurtosis, respectively. The set of admissible values of the L-skewness and L-kurtosis is the grey region above the lowest parabola [Hosking, Wallis (1997)]. The parabolic curves represent various 3-parameter families of distributions, and the points with coordinates x= sample L-skewness and y=L-kurtosis, respectively, represent the funds. Each curve represents a particular distribution family: glo (resp. gev, gpa, pe3) corresponds to the generalized logistic (resp. generalized extreme value, generalized Pareto, Pearson type III) distribution.
Among the standard 3-parameter families, the generalized logistic distributions feature the heaviest tails. We observe that almost half of the funds have fatter tails than the generalized logistic.
[Insert Table 5 : Sharpe and L-Performances] The values of different performance measures cannot be compared directly, as they do not have the same interpretation. In contrast, the rankings are comparable. We observe that the rankings based on the unconditional distributions are quite stable. For example, LES SS is always the worst performing fund, while FS EM is always the best performer. It is important to detect the hedge funds that have significantly changed their rankings. For example, the ranking of SP EH fluctuates between 3 and 15, which reveals the presence of extreme returns. This is largely due to the bias ratio observed in Table 2 with a jump in the L-moment of order 1 from 1.74% to 2.37%, which is not strictly in line with the self-declared "Equity Hedge" strategy.

Concluding Remarks
The Sharpe ratio is commonly used in the hedge fund industry and provides a reliable performance measure for retail investors. However, the Sharpe ratio can be misleading for other types of investors such as banks, or regulators. Indeed, it is very sensitive to outliers, unable to correctly assess the sizes of tails that reflect extreme risks, or to detect ii) The fitted performance is computed, for instance, when there is a portfolio of two assets, which are the fund and a market index (tracker). Then, it is necessary to disentangle the component of the hedge fund return which can be hedged by the market from its idiosyncratic component. The fitted L-performance is simply the L-performance applied to the residual plus intercept from the linear regression of hedge fund excess return on market excess return.

Appendix 1
Asymptotic Properties of Estimated L-Performance

Asymptotic Properties of L-moments
A trimmed L-moment is a linear form of the quantile fonction. It can be written as where P is a polynomial. It can consistently be estimated by the sample counterpart.
where ⇒ denotes the weak convergence (convergence in distribution), and the asymptotic behaviour of a trimmed L-moment: In particular, the estimator λ(Q T , P ) is consistent, asymptotically Gaussian with zeromean and a variance given by :
Moreover, we have : The expression of the asymptotic variance of the estimated L-performance directly follows from (A.1) applied to polynomial P = 1 λ 2,r,n (P 1,n − L r,n P 2,n ).