Methodology of measuring performance in alternative investment Alexis BONNET, Methodology Group Isabelle NAGOT, Universit´e Paris 1 Panth´eon-Sorbonne Preliminary version The strong development of alternative investment has highlighted the limits of usual performance measures like the Sharpe ratio. Indeed, hedge-fund managers are facing a large choice of strategy and payoff types, resulting in the observation of potentially any kind of distribution, involving for example asymmetry and fat tails characterics, and even multimodality. Despite its natural appeal as a simple, intuitive tool, the Sharpe ratio is particularly misleading when the shape of the return distribution is far from normal. To remedy this, we propose a new framework in which trades, portfolios or strategies of various types can be analysed. In particular this allows a fund manager to compare and workout his optimal allocation. The class of measures is derived from natural and simple properties of the asset allocation. We give representation results which allows to describe the whole set of measures. These measures contains as particular cases the squared Sharpe ratio, the Stutzer’s rank ordering index and the Hodges measure. Any measure is shown to be proportional to the squared Sharpe ratio for gaussian distributions. For non gaussian distributions, asymmetry and fat tails are taken into account. More precisely, the risk preferences are in fact separated into a gaussian risk aversion and a non-gaussian risk aversion. We present and illustrate here the main results proved in two academic papers, [2] and [3]. Limits of Sharpe ratio Classical performance measurements and portfolio optimizations are usually based on Sharpe ratio. This ratio is the excess return over the risk-free rate divided by its standard deviation and relies on the assumption that investors choose portfolios according to a mean-variance framework. But using variance as the risk measure is not adapted anymore when investment strategies have asymmetric or leptokurtic distributions. Such distributions can derive from the asset itself or from the sale of options. It happens that option sellers score particularly higher Sharpe ratios. To illustrate this, we consider a future whose final net worth is assumed to be gaussian and we assume that we have access to all standard calls and puts on it. We derive the optimal portfolio based on these assets for achieving the maximum expected Sharpe ratio. More precisely, we compute the optimal payoff having a present value equal to 0 ; we know that this payoff can be achieved with the available assets (options and future). The volatility itself is assumed to be properly priced, i.e. no money can be made by pure gamma trading. Assuming that the underlying future is cheap, we can choose parameters giving a Sharpe ratio of 2 for it1 . We find in [2] that the Sharpe ratio is boosted from 2 to 7.32 for the optimal strategy, which has the following payoff: 1 0.5

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1 i.e. mean 2, standard deviation 1 of payoff. We note that although currently no exchange traded future is likely to offer such risk-returns characteristics, Sharpe ratios above 2 are not uncommon in the hedge fund universe, dynamic scaling on the strategies providing the equivalent to options in our example.

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Unfortunately, this optimal strategy could end up being a very dangerous choice. It is based on heavy short positions in out-of-the-money puts (see [2] or [5]). It has a truncated right tail and a very fat left tail. That means that for given expected returns, high Sharpe ratio strategies are those that generate small profits punctuated by occasional crashes. Several extensions have been proposed to remedy the limits of the mean-variance framework. The most common approach is to replace the standard deviation by another risk measure, for example the downside deviation (Sortino ratio), the value-at-risk, the expected shortfall, a low partial moment (Omega measure), or risk measures involving higher moments of the payoff distribution. Alternative investment context The need for a measure which is not restricted to gaussian distributions is particularly clear in the hedge funds context where “exotic” return distributions (skewed and fat-tailed) can be seen. Another issue is that the usual notion of return is not adapted to describe the performance of a derivative. Even for simple futures this notion is not well defined. In addition, classical methods, introducing constraints on sign and sum of weights of assets, cannot cope easily with short sales and leverage. This will be adressed by allowing negative weights for the assets and no constraint on the sum of weights.

I. Our framework We choose to work with prices (or P&L) rather than returns, since it is more appropriate in the alternative investment context. However, when limited to a unique period of time, the methodology can be directly transposed to returns. A position (or asset) is a given risk exposure, considered as a gain opportunity. One decides how much exposure to that risk one wants, by choosing the quantity λ of the position. All cash-flows linked to a position are taken into account at the end of a given trading period and properly discounted, including premia, transaction costs and payoff of the asset itself. We implicitly assume that there is no limit to borrowing at any time. The weights of the positions do not need to sum to 1 as it is the case when considering returns. The relevant variable is then the discounted net P&L at the end of the trading period. It can be either the realized or the forecast P&L (measure for comparison of funds, considering their past performances or as criterion for asset allocation), for a trade, a fund, a part of a fund... The final net worth of a position is described as a random variable X on a probability space (Ω, F, P ). We limit the analysis to the set X of random variables X such that {λ|E(e−λX ) < +∞} contains 0 in its interior (implying a proper Laplace transform)2 . We consider below a performance measure π. It is a functional on X with values in IR ∪ {+∞}. We assume law invariance for the performance measure: π(X) depends only on the law of X. Desirable properties of the optimal asset allocation We look for measures π having the following property: the optimal portfolio obtained on a set of independent assets has a measure equal to the sum of the measures of each separate independent bet, and in this optimal portfolio, the weight of an asset depends only on its own characteristics and not on the other assets, i.e. we assume that we have: P P P (A) for any X1 , ..., Xn independent, max π( i λi Xi ) = i π(Xi ) = π( i λXi Xi ) {λi }

where we have denoted by λXi the weight of the asset Xi in the optimal portfolio, depending only on Xi (the maximum is achieved at (λ1 , ..., λn ) = (λX1 , ..., λXn )). As seen above, we do not impose that the weights sum to 1, since no a priori limit is set on leverage. It is natural to expect λX to depend only on the characteristics of X: adding an independent asset to a given optimized portfolio should not modify the existing allocation. To reflect the benefits of diversification, a performance measure has to be superadditive. Here we assume further that for 2 The

study can be extended by using characteristic functions instead of log-Laplace transforms.

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independent assets, there is no term of dependence and the measure of the portfolio is just the sum of the different contributions. This allows a top down approach in the asset allocation. We remark that this property is quite a weak assumption since it applies only to independent variables. Moreover the condition is satisfied for example by the squared Sharpe ratio (computed on returns or on P&L) and is one of the properties that make the Sharpe ratio an intuitive measure. We have indeed, with Sh2 (X) denoting the squared Sharpe ratio of X: P P P i) for any X1 , ..., Xn independent, max Sh2 ( i λi Xi ) = i Sh2 (Xi ) = π( i VE(X (Xi ) Xi ), {λi }

i.e. the optimal weight for an asset X is λX =

E(X) V (X) .

II. The class of measures. Representation theorems Assumptions We add a few technical conditions on π to avoid pathological cases3 . We assume that:  ∗    ∗ (T)  ∗   ∗

T here exists a gaussian X such that π(X) 6= 0. x 7→ π(N (x, x)) is bounded on a non empty open interval. ∀X ∈ X , ∃Z gaussian such that π(Z + X) < +∞. λ1 > 0, where λ1 denotes the weight of a random variable with law N (1, 1).

The last assumption ensures that N (1, 1) is preferred to N (−1, 1), i.e. that the measure has the good sign.

First properties Theorem 1 If π satisfies the assumptions (A) and (T), then: π(0) = 0, π has values in [0, +∞], ∀α ∈ IR∗ , π(αX) = π(X), and if X1 , ..., Xn are independent and identical distributed, then π(X1 + ... + Xn ) = nπ(X1 ). Moreover π is proportional to the squared Sharpe ratio on the set of gaussian variables.

Representation result The representation involves the log-Laplace of a random variable X. We set: DX = {λ|E(e−λX ) < +∞}, and for λ ∈ DX , HX (λ) = − ln E(e−λX ), i.e. HX is the opposite of the log-Laplace transform of X. The role played by this functions comes from the following property: if X1 , ..., Xn are some independent random variables, then HX1 +...+Xn = HX1 + ... + HXn , and assumption (A) will result in an additivity of the performance measure as a function of HX , for variables in their optimal proportions. ◦

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Remark that X is now the set of random variables X such that 0 ∈DX , where DX denotes the interior of DX . Adding a natural regularity assumption (R) which corresponds to the Fr´echet-differentiability of π (as a function of the log-Laplace) at Z1 , in the direction of HX , we prove in [2] the following result: Theorem 2 Any performance measure satisfying assumptions (A), (T ) and (R) can be expressed as: π(X) = sup J (HλX ), for X ∈ X λ∈IR where J is a continuous linear function on a subset of X . 3 N (x, x) denotes the gaussian law with mean and variance equal to x. π(N (x, x)) denotes the measure π for a random variable having this law (cf law invariance).

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Comments We set J(X) = J (HX ). This quantity is in itself a performance measure for the investment in X. Then π(X) = supλ J(λX) is the best performance obtained by varying the quantity (bought or sold) of X. It refers to the potentiality of the investment in X, which is effective only if this asset is used in the right quantity: π(X) = J(λX X). π is defined up to a multiplicative constant. We choose it such that J(m) = m when m is a constant. Then from theorem 1, we have: λ1 2 for any X gaussian, π(X) = Sh (X) 2 Remark: we could choose to have π(X) = J(m) = λm1 .

1 2 2 Sh (X)

for X gaussian, by dividing J by λ1 . Then

Examples Those measures contain as particular cases: • the squared Sharpe ratio: it satisfies all considered properties and can be expressed by setting 0 J (H) = H 0 (0) + 12 H 00 (0) and π(X) = supλ J (HλX ). Indeed, we get, since HX (0) = E(X) and 2 00 HX (0) = −V (X), π(X) = supλ [λE(X) − λ2 V (X)], which corresponds to the squared Sharpe ratio. • the measure π(X) = supλ HX (λ), which can be written as π(X) = supλ J (HλX ) with J (H) = H(1). This measure is described differently in two previous works, we will refer to it as Hodges measure: − in [6], Hodges defines a measure of market opportunities based on the exponential utility function U (X) = −e−αX . It is presented as a generalization of the Sharpe ratio, reducing to it for gaussian distributions. This generalized Sharpe ratio (GSR) satisfies: 12 GSR2 = − ln(−U ∗ ), where U ∗ is the optimal expected utility obtainable when the investor chooses the best level of investment (or sale), i.e. U ∗ = supx E(e−αxX ). Then 21 GSR2 = supλ [− ln E(e−λX )], which corresponds to our definition. − in [7], Stutzer considers asset returns instead of prices. He proposes a ”rank ordering index” based on the minimization of the probability that the growth rate of invested wealth will not exceed an investor-selected target growth rate. The measure is obtained by a large deviation approach. Denoting by R the log return in excess of the benchmark, the measure can be written in the i.i.d. case (see [7]) as: supλ [− ln E(e−λR )]. Stutzer shows that when R is gaussian, this criterion is half the squared Sharpe ratio. ◦

Since the HX are C ∞ functions (HX is C ∞ on DX ), J can be viewed as a Schwartz distribution: J (HX ) =< g, HX >. Adding the following financial assumption, we prove that the order of the distribution g is at most two4 . (F): It cannot exist centered variables5 with an arbitrary large π. The idea is that there no reason to be interested in a centered P&L, then its measure cannot be high. We will find in fact that this measure is null. We call “admissible” a performance measure satisfying assumptions (A), (T), (R) and (F). Theorem 3 For any admissible performance measure, the Schwartz distribution g has order at most two and we get: 1 00 for X ∈ XK , J(X) = E(X) + < Γ, HX > (2) 2λ1 where Γ is a non negative Radon measure6 with compact support K, such that < Γ, 1 >= 1, and ◦

XK = {X | K ⊂DX }. 4 i.e.

it involves only derivatives of HX with order at most two. (F) can be weakend by considering only Bernouilli variables. This is the choice made in [2]. 6 i.e. a distribution with order 0. 5 Assumption

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J(X) splits into E(X) and a term independent of E(X) and not positive, which will be interpreted below as a term of risk. Moreover, λX and E(X) have same sign. Remark: this second representation excludes the measures based on the four first moments (i.e. including skewness and kurtosis) since they involve third and fourth derivatives of HX . Indeed they do not satisfy assumption (F) and would lead to buy unlimited amounts of centered X (and even of binary contracts with negative expected returns). Examples The squared Sharpe ratio corresponds to7 g = −δ00 + 21 δ000 , λ1 = 1 and Γ = δ0 and Hodges measure to g = δ1 , λ1 = 1 and Γ(x) = 2(1 − x)1I[0;1] (x). Application: optimal payoff according to various measures Coming back to the example used for the Sharpe ratio, we compute (see [2]) the payoff of the optimal portfolio for two other choices of admissible measure: the Hodges measure and the measure π obtained by setting8 J (HX ) = HX (1) + aHX (α), for a, α > 0, parameters choosen to ensure good properties of the measure (see [3]), in particular it is monotone. Remark that for a given set of available assets P, since no a priori limit is set on leverage, solving max π(X) = max max J(λX) is equivalent to solving max J(X). X∈P X∈P λ∈IR X∈P We draw below the payoff of the optimal portfolio (see [2]), with the same assumptions than with the Sharpe ratio in Figure 1: 1. According to Hodges measure (CARA utility): J (HX ) = HX (1): 4 3 2 1

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We observe that the investor chooses to hold the future. In this respect, this measure favors gaussian returns. 2. According to a π monotone J (HX ) = HX (1) + aHX (α): (a, α > 0)

5 4 3 2 1

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The investor chooses to partially cover the gaussian downside risk by buying a fraction of puts. This third measure gives then an optimal strategy more coherent with usual risk preferences. This proves that such measure will be much more adapted to alternative investment than the Sharpe ratio.

7 We

denote by δx the Dirac at x. that the cardinal of the distribution support K is 1 for Hodges measure and 2 for the second measure. This last choice is then the simplest after Hodges measure. 8 Remark

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III. Study of the performance measure J In this section we study the performance measure J itself. Interpretation in terms of risk aversion and CAPM In equation (2), Γ depends on λ1 . We denote by Γ1 the Radon measure associated to π if λ1 = 1. From the definition of J, we can rewrite (2) as: J(X) = E(X) +

λ1 < Γ1 , H 00X > λ1 2

and Γ1 and λ1 are now independent. This new formulation allows to reveal two kinds of risk aversion preferences. For X gaussian, we have J(X) = E(X) − 2λ1 1 V (X), then λ11 corresponds to the usual risk aversion parameter for gaussian risks. Independent of this parameter, Γ1 characterizes the risk aversion relative to non-gaussian risks. Then if two investors have same Γ1 they will choose portfolios with same relative proportions of assets. Potentially different λ11 may arise from different gaussian risk aversion or different capital. For investors with the same Γ1 , the separability property of the mean-variance context still holds. Replacing the mean-variance assumption by the existence of an aggregate Γ1 , our performance measure gives a new family of CAPM, including non-gaussian risk aversion. Preferences We study in [3] the preferences relationship associated to J, defined by: ∀X, Y ∈ X , X is preferred to Y if and only if J(X) ≥ J(Y ). We prove that the axioms allowing a representation of the preferences relationship with a utility function (i.e. the existence of U such that ∀X, Y ∈ X , J(X) ≥ J(Y ) if and only if E[U (X)] ≥ E[U (Y )]) are all satisfied except the independence axiom, which holds if and only if J corresponds to a CARA utility function (giving the Hodges measure for π). Monotone case We consider an admissible performance measure π. We prove in [3] that monotonicity (among variables with non negative mean) with respect to first order stochastic dominance of π, of J and monotonicity with respect to second order stochastic dominance of π, or of J are four equivalent properties. Any of them will be refered to as ”monotonicity”. This result holds under another financial assumption, stating that among simple variables (binary bets), only arbitrage opportunities can have an infinite measure. (F’): Bernouilli variables which are not arbitrage opportunity have a finite π. Theorem 4 Under assumption (F’), the following conditions are equivalent for an admissible measure: (i) Monotonicity (ii) g is a non negative measure with support ⊂ IR+ (iii) supp Γ ⊂ IR+ , Γ is a convex function on IR∗+ and Γ(0+ ) = 2λ1 . Remark that in order to have π monotone, the order of the distribution g has to be null. In the example of the squared Sharpe ratio, g has order two and it is indeed well known that the Sharpe ratio is not monotone. Monotone measures have good financial properties, as will be seen below with the study of the risk measure associated to J in that case. For example, application λ 7→ J(λX) is strictly concave for non constant X, and the unicity of λX is then ensured (in addition λ 7→ J(λX) and m 7→ λX+m are dual with respect to Legendre transform).

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Risk measure associated to J Setting ρ(X) = −J(X), we get a risk measure9 . When J is monotone, ρ satisfies the properties of a convex risk measure as defined in [4]: translation invariance, monotonicity and convexity, i.e. ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ), for 0 ≤ λ ≤ 1, property which reflects the benefits of diversification. To be a coherent risk measure (see [1]), in particular subadditive, ρ should in addition satisfy positive homogeneity. To explain why we do not wish this property to hold10 (and ρ will not satisfy it), we consider a position λX where X has an assymetric distribution with fat tails. For small λ, the mean and the variance of the position λX are sufficient to measure its contribution to the portfolio (effect on utility). For large λ, the investor becomes highly sensitive to the non-gaussian character of the position. Therefore J(λX) = λJ(X) cannot hold as a general rule. In other words, sensitivity to non-gaussian risk depends strongly on the size of the position.

Conclusion From a limited set of natural assumptions, we have characterized a family of good performance measures for alternative investments and derived their general expression in equation (2). These measures take into account a gaussian and a non-gaussian risk aversion and have good behavior when used on portfolios having exotic profit and loss distributions, for example including options. The sub-family of monotone measures is the one to consider since it ensures good financial properties. In particular, the associated risk measure is convex, which gives a concave efficient frontier. By choosing the appropriate non-gaussian risk aversion function Γ, the investor can tailor the monotone performance measure to his business and risk aversion.

REFERENCES [1] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9(3), 203-228. [2] Bonnet, A., Nagot, I. (2004) Methodology of measuring performance, I: Axiomatic description and representation theorems. Working paper, CERMSEM, Universit´e Paris 1 Panth´eon-Sorbonne. [3] Bonnet, A., Nagot, I. (2004) Methodology of measuring performance, II: Properties. Working paper, CERMSEM, Universit´e Paris 1 Panth´eon-Sorbonne. [4] F¨ollmer, H., Schied, A. (2002) Robust preferences and convex measures of risk. Advances in Finance and Stochastics, 39-56, Springer-Verlag. [5] Goetzmann, W., Ingersoll, J., Spiegel, M. and Welch, I. Sharpening Sharpe Ratios (February 2002). Yale ICF Working Paper No. 02-08; AFA 2003 Washington, DC Meetings. [6] Hodges, S. (1998) A generalization of the Sharpe ratio and its application to valuation bounds and risk measures. University of Warwick Financial Options Research Center, FORC Pre-Print 98/88, April 1998. [7] Stutzer, M. (2003) Portfolio Choice with Endogenous Utility: A Large Deviations Approach. Journal of Econometrics, Vol.116, 2003, pp. 365-386.

9 In [3], we consider also ρ(X) =< Γ, −H 00 >= 2λ [−J(X) + E(X)] which is a convex combination of Esscher 1 X variances. 10 In [4], F¨ ollmer and Schied remark that the axiom of positive homogeneity is debatable e.g. facing liquidity risk.

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