Abstract
The ratings of fund managers based on past performances of the funds and the rating dynamics are crucial information for investors. This paper proposes a stochastic migration model to investigate the dynamics of performancebased ratings of funds, for a given riskadjusted measure of performance. We distinguish the absolute and relative ratings and explain how to identify their idiosyncratic and systematically persistent (resp. amplifying cycles) components. The methodology is illustrated by the analysis of hedge fund returns extracted from the TASS database for the period 1994–2008.
Keywords
 Transition Matrix
 Relative Rating
 Hedge Fund
 Transition Matrice
 Fund Manager
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 1.
In the hedge fund literature, the term “performance” is often used for return, and thus is an expost notion. In our paper we are interested in exante performance measures, i.e., before observation. These measures are adjusted for the uncertainty on future returns.
 2.
\((K1)^2\) independent factors, if the state “inactive”, i.e. state \(K\), is an absorbing state.
 3.
A transition matrix can always be diagonalized.
 4.
\(vec\) means that all the elements of the matrix \((K,K)\) are stacked in a single vector of dimension \((K^2,1)\).
 5.
TASS can take up to nine months before transferring a fund from the Live database to the Graveyard database. This implies that some dead funds can be considered as still living in the end of the sample.
 6.
The constant number of live funds at the end of the period should be due to lags in funds reporting their performance. Indeed, we observe the performances for October 2008 at the end of November 2008, which implies that only funds with short notice of publication of Net Asset Value (NAV) are incorporated at that moment. Thus, without additional information, the drop in the number of live funds cannot be directly related to an increase in fund mortality, which seems to happen at the end of 2008.
 7.
The size 24 months of the window has been chosen for illustrating the approach. In a more complete analysis, it might be preferable to consider different sizes, then to select the most informative one for predictive purpose and/or the most relevant one for the updating frequency of the investors portfolio.
 8.
Very small migration correlations have not to be put to zero due to the possible leverage effects in portfolio management strategy. Loosely speaking, even small migration correlations can allow for gain opportunities by using appropriate correlation strategies.
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Acknowledgments
We gratefully acknowledge financial support of the chair QUANTVALLEY/Risk Foundation: “Quantitative Management Initiative”, the Global Risk Institute and the chair ACPR: “Regulation and Systemic Risk”.
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Appendix A: Properties of the Theoretical Ranks
Appendix A: Properties of the Theoretical Ranks
We discuss in this appendix different properties of the theoretical ranks, such as the link between levels and ranks, or the links between the crosssectional ranks and the historical ranks. We start by reviewing some basic results without in mind the applications to fund performances and rankings. Then we particularize the results to the ranking of funds.
1.1 A.1 Basic Properties
1.1.1 A.1.1 Definition of a Theoretical Rank
Let us consider a onedimensional continuous random variable \(X\) and denote by \(F\) its cumulative distribution function: \(F(x)=P(X<x)\). If its continuous distribution admits a strictly positive probability density function (p.d.f), then the theoretical rank of \(X\) is the random variable:
The variable \(Y\) takes values in \([0, 1]\) and follows the uniform distribution on \((0,1)\).
Example A.1
To understand the definition of the theoretical rank, let us consider a large set of funds, whose performances for period \(t\) are \(X_{i,t}\), \(i=1,\ldots ,n\), say. We can construct the empirical crosssectional c.d.f. \(\hat{F}_{n,t}= \frac{1}{n} \sum _{i=1}^{n} 1_{X_{i,t}<x}\), and the empirical ranks \(\hat{Y}_{i,t}= \hat{F}_{n,t}(X_{i,t})\). Under crosssectional ergodicity, the empirical crosssectional c.d.f. tends to a limiting one \(F_t\), say, and the empirical rank \(\hat{Y}_{i,t}\) to the theoretical rank \(Y_{i,t}= F_{t}(X_{i,t})\).
1.1.2 A.1.2 Linear Links Between Levels and Ranks
Let us now consider a pair of continuous variable \((X_1, X_2)\), with joint and marginal c.d.f. denoted by \(F_{1,2}\), \(F_{1}\) and \(F_{2}\), respectively: \(F_{1,2}(x_1,x_2) = P[X_1<x_1,X_2<x_2] \), \(F_{j}=P[X_j<x_j] \), \(j=1,2\). The associated theoretical ranks are:
\(Y_1\) and \(Y_2\) are marginally uniform on \([0, 1]\), but are dependent if \(X_1\) and \(X_2\) are. This dependence can be characterized by their joint c.d.f.: \(C(y_1,y_2) = P[Y_1<y_1,Y_2<y_2] = F_{1,2}[F^{1}_1 (y_1), F^{1}_2 (y_2)] \), called the copula of \(X_1\) and \(X_2\). Let us now discuss the links between all these variables, when we focus on linear links measured by the Pearson correlations. The joint variancecovariance matrix of levels and ranks is given by:
The different elements of this variancecovariance matrix are of the form: \(Cov[G_1(X_1),\) \(G_2(X_2)]\), where \(G_1, G_2\) are two increasing functions, either the identity function, or the c.d.f. We have the following Lemma:
Lemma A.1
If \(G_1,G_2\) are increasing functions:
In fact, a functional measure of dependence is the difference between the joint c.d.f. and what would be its expression under the independence assumption, i.e. \(F_{1,2}(x_1,x_2)F_{1}(x_1)F_{2}(x_2)\). Lemma A.1 says that any covariance of this type is a weighted average of this functional measure, with weights equal to the derivatives of functions \(G_j\). In particular, the variables \(X_1\) and \(X_2\) are independent iff \(F_{1,2}(x_1,x_2)=F_{1}(x_1)F_{2}(x_2), \forall x_1,x_2\), or equivalently iff \(Cov[G_1(X_1),G_2(X_2)] =0\), for any increasing functions \(G_1,G_2\).
The formula (A.4) in Lemma A.1. provides comparable expressions of the elements of the joint variancecovariance matrix (A.3). We get:
It is known that the covariance operator is invariant by linear affine transformations of the variables. However we pass from levels to ranks by a nonlinear transform, i.e. the c.d.f. This transformation can imply a loss of information concerning linear links. For instance, by using the Frechet upper bound [10]:
we get:
Similarly, by considering the Frechet lower bound [24, Proposition 4], we get an inequality in the other direction. Finally, we know:
1.1.3 A.1.3 Correlations Between Levels and Ranks
Let us now focus on the correlation matrix:
\(\rho _{1,2} \) is the standard Pearson correlation, \(r_{1,2}\) is the correlation of ranks defined by [26], \(\lambda _{1,1}\) and \(\lambda _{2,2}\) are the Lmoments of order 2 introduced by [18], \(\lambda _{1,2}\) and \(\lambda _{2,1}\) are the Lcomoments of order 2 [see e.g. [23, 24]].
By considering the whole matrix \(R\), we perform a joint analysis of these different dependence measures. These correlations are constrained. First, matrix \(R\) is a positive semidefinite matrix. Second, additional constraints are due to the deterministic increasing relationship between \(X_1\) and \(F_1(X_1)\) [resp. \(X_2\) and \(F_2(X_2)\)]. For instance, we know that: \(\lambda _{11} \ge 0\), \(\lambda _{22} \ge 0\), \(\mid \) \(\lambda _{12}\) \(\mid \) \(\le \min (\lambda _{11},\lambda _{22})\), \(\mid \) \(\lambda _{21}\) \(\mid \) \(\le \min (\lambda _{11},\lambda _{22})\), from inequalities (A.12).
These constraints can be explicited in special symmetric cases, where \( \lambda _{11} = \lambda _{22}= \lambda \), say, \(\lambda _{12} = \lambda _{21} = \mu \), say.
Proposition
The different correlations \(\rho = \rho _{12}\), \(r=r_{12}\), \( \lambda =\lambda _{11} = \lambda _{22}\), \( \mu = \lambda _{12} = \lambda _{21} \) are constrained as follows:

(i)
If \(\rho = 1\), then \(r = 1\), \(\mu = \lambda \) with \(0 \le \lambda \le 1\).

(ii)
If \(\rho = 1\), then \(r = 1\), \(\mu = \lambda \) with \(0 \le \lambda \le 1\).

(iii)
If \(\rho \ne \pm 1\), we get:
\(0 < \mid \) \(\mu \) \(\mid < \lambda \),
\(\lambda ^2 + \mu ^2  2 \rho \lambda \mu \le 1 \rho ^2\),
\(\lambda ^2 \mu ^2 \le \sqrt{(1r^2)(1\rho ^2)}\).
The first inequality in (iii) is equivalent to:
It corresponds to an ellipsoid with the \(45^{\circ }\) and \(45^{\circ }\) lines as it main axes. The second inequality corresponds to the interior of an hyperbola with the same axes as the ellipsoid above. The domain of admissible value for \(\lambda \), \(\mu \) for given values of \(r\), \(\rho \) is the intersection of the hyperbolic and ellipsoidal domain.
1.2 A.2 Rank Autocorrelation Functions
1.2.1 A.2.1 The Variables
The results in Appendix A.1 are useful to interpret the joint dynamics of performances, absolute ranks and relative ranks. Recall that these variables are defined as follows:
\(S_{i,t}\) denotes the riskadjusted Sharpe performance for HF \(i\) and period \(t\).
For these data, we can define the historical c.d.f. of \(S\) computed on averaging on both HF and dates, and deduce the associated absolute ranks as \(r_{i,t}^{a} = \hat{F} (S_{i,t})\).
We can also consider the crosssectional c.d.f. at date \(t\), obtained by averaging on HF for given \(t\), and deduce the associated relative rank as \(r_{i,t}= \hat{F}_t (S_{i,t})\).
Thus the results of Appendix A.1 can be used with the following [\(X_1,X_2,F_1(X_1),F_2(X_2)\)] variables:
(i) Joint analysis of levels and absolute ranks at different dates:
\(X_1 = S_{i,t}\), \(X_2 = S_{i,th}\), \(F_1(X_1)=r_{i,t}^{a}\), \(F_2(X_2)=r_{i,th}^{a}\).
(ii) Joint analysis of levels and relatives ranks at different dates:
\(X_1 = r_{i,t}^{a}\), \(X_2 =r_{i,th}^{a}\), \(F_1(X_1)=r_{i,t}\), \(F_2(X_2)=r_{i,th}\).
1.2.2 A.2.2 Autocorrelograms
When the variables \(X_1\), \(X_2\) (resp. \(F_1(X_1)\), \(F_2(X_2)\)) correspond to an observation of the same variable at different dates \(t\), \(th\), the correlation matrix in (A.12) becomes the autocorrelation at lag \(h\) for level and ranks (or for absolute and relative ranks):
where for instance:
\(\rho (h)\) is the ACF on absolute ranks,
\(r(h)\) is the ACF on relative ranks,
\(\lambda (h)\) is the cross ACF between absolute and relative ranks.
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Darolles, S., Gouriéroux, C., Teiletche, J. (2015). The Dynamics of Hedge Fund Performance. In: Huynh, VN., Kreinovich, V., Sriboonchitta, S., Suriya, K. (eds) Econometrics of Risk. Studies in Computational Intelligence, vol 583. Springer, Cham. https://doi.org/10.1007/9783319134499_7
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