The Dynamics of Hedge Fund Performance

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 performance-based ratings of funds, for a given risk-adjusted 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.

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 cross-sectional 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 one-dimensional 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:

$$\begin{aligned} Y=F(X). \end{aligned}$$

(1)

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 cross-sectional 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 cross-sectional ergodicity, the empirical cross-sectional 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:

$$\begin{aligned} Y_1=F_1(X_1), Y_2=F_2(X_2). \end{aligned}$$

(2)

$Y_1$ 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 variance-covariance matrix of levels and ranks is given by:

$$\begin{aligned} V \left( \begin{array}{c} X_1 \\ X_2 \\ Y_1 \\ Y_2 \end{array} \right) = \left( \begin{array}{cccc} V X_1 &{} Cov(X_1,X_2) &{}Cov(X_1,Y_1) &{} Cov(X_1,Y_2) \\ . &{} V X_2 &{} Cov(X_2, Y_2) &{}Cov(X_2, Y_2) \\ . &{} . &{} V Y_1 &{} Cov(Y_1, Y_2) \\ . &{} .&{} . &{} V Y_2 \end{array} \right) . \end{aligned}$$

(3)

The different elements of this variance-covariance 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:

$$\begin{aligned} Cov[G_1(X_1),G_2(X_2)] = \int \int [F_{1,2}(x_1,x_2)-F_{1}(x_1)F_{2}(x_2)]dG_1(x_1)dG_2(x_2). \end{aligned}$$

(4)

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 variance-covariance matrix (A.3). We get:

$$\begin{aligned} Cov(X_1,X_2)&= \int \int [F_{1,2}(x_1,x_2)-F_{1}(x_1)F_{2}(x_2)]dx_1dx_2, \end{aligned}$$

(5)

$$\begin{aligned} Cov[X_1,F_2(X_2)]&= \int \int [F_{1,2}(x_1,x_2)-F_{1}(x_1)F_{2}(x_2)]dx_1dF_2(x_2),\end{aligned}$$

(6)

$$\begin{aligned} Cov[F_1(X_1),X_2]&= \int \int [F_{1,2}(x_1,x_2)-F_{1}(x_1)F_{2}(x_2)] dF_1(x_1)dx_2,\end{aligned}$$

(7)

$$\begin{aligned} Cov[F_1(X_1),F_2(X_2)]&= \int \int [F_{1,2}(x_1,x_2)-F_{1}(x_1)F_{2}(x_2)]dF_1(x_1)dF_2(x_2). \end{aligned}$$

(8)

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]:

$$\begin{aligned} F_{1,2}(x_1,x_2) \le \min [F_{1}(x_1),F_{2}(x_2)], \end{aligned}$$

(9)

we get:

$$\begin{aligned} Cov[X_1,F_2(X_2)]&\le \int \int \left[ \min [F_{1}(x_1)F_{2}(x_2)]-F_{1}(x_1)F_{2}(x_2) \right] dx_1dF_2(x_2) \nonumber \\&\le Cov[X_1,F_1(X_1)]. \end{aligned}$$

(10)

Similarly, by considering the Frechet lower bound [24,  Proposition 4], we get an inequality in the other direction. Finally, we know:

$$\begin{aligned} -Cov[X_1,F_1(X_1)] \le Cov[X_1,F_2(X_2)] \le Cov[X_1,F_1(X_1)]. \end{aligned}$$

(11)

1.1.3 A.1.3 Correlations Between Levels and Ranks

Let us now focus on the correlation matrix:

$$\begin{aligned} R = Corr \left( \begin{array}{c} X_1 \\ X_2 \\ F_1(X_1) \\ F_2(X_2) \end{array} \right) = \left( \begin{array}{cccc} 1 &{} \rho _{12} &{} \lambda _{11} &{} \lambda _{12} \\ . &{} 1 &{} \lambda _{21}&{} \lambda _{22} \\ . &{} . &{} 1 &{} r_{12} \\ . &{} .&{} . &{} 1 \end{array} \right) . \end{aligned}$$

(12)

$\rho _{1,2}$ 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 L-moments of order 2 introduced by [18], $\lambda _{1,2}$ and $\lambda _{2,1}$ are the L-comoments 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 semi-definite 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:

  $0 < \mid$ $\mu$ $\mid < \lambda$,

  $\lambda ^2 + \mu ^2 - 2 \rho \lambda \mu \le 1- \rho ^2$,

  $\lambda ^2 -\mu ^2 \le \sqrt{(1-r^2)(1-\rho ^2)}$.

The first inequality in (iii) is equivalent to:

$$\begin{aligned} \frac{1}{2(1+ \rho )} (\lambda +\mu )^2 + \frac{1}{2(1-\rho )} (\lambda -\mu )^2 \le 1. \end{aligned}$$

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 risk-adjusted 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 cross-sectional 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:

$X_1 = S_{i,t}$, $X_2 = S_{i,t-h}$, $F_1(X_1)=r_{i,t}^{a}$, $F_2(X_2)=r_{i,t-h}^{a}$.

(ii) Joint analysis of levels and relatives ranks at different dates:

$X_1 = r_{i,t}^{a}$, $X_2 =r_{i,t-h}^{a}$, $F_1(X_1)=r_{i,t}$, $F_2(X_2)=r_{i,t-h}$.

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 $t$, $t-h$, the correlation matrix in (A.12) becomes the autocorrelation at lag $h$ for level and ranks (or for absolute and relative ranks):

$$\begin{aligned} R(h) = \left( \begin{array}{cccc} 1 &{} \rho _{h} &{} \lambda (0) &{} \lambda (h) \\ . &{} 1 &{} \lambda (-h)&{} \lambda (0) \\ . &{} . &{} 1 &{} r(h) \\ . &{} .&{} . &{} 1 \end{array} \right) , \end{aligned}$$

(13)

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.