Sensitivity analysis of Mixed Tempered Stable parameters with implications in portfolio optimization

Abstract

This paper investigates the use, in practical financial problems, of the Mixed Tempered Stable distribution both in its univariate and multivariate formulation. In the univariate context, we study the dependence of a given coherent risk measure on the distribution parameters. The latter allows to identify the parameters that seem to have a greater influence on the given measure of risk. The multivariate Mixed Tempered Stable distribution enters in a portfolio optimization problem built considering a real market dataset of seventeen hedge fund indexes. We combine the flexibility of the multivariate Mixed Tempered Stable distribution, in capturing different tail behaviors, with the ability of the ARMA-GARCH model in capturing the time dependence observed in the data.

Appendix

1.1 Generalized method of moments

We discuss briefly the GMM proposed in Hansen (1982) and show how to apply this method to the Multivariate MixedTS. The aim of this procedure is to estimate the vector of parameters $\theta _0 \in {\mathbb {R}}^p$ from a model based on the following $q\times 1$ vector of unconditional moment conditions:

$$\begin{aligned} {\mathsf {E}}\left[ g\left( \theta _0,X\right) \right] =0 \end{aligned}$$

(12)

where $g\left( \cdot \right) :\Theta \times {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{q}$, $\Theta \subset {\mathbb {R}}^p$ is a compact space. For a given sample $X_1,X_2,\ldots ,X_T$, we replace the expectation in (12) with sample means and obtain:

$$\begin{aligned} {\bar{g}}_T\left( \theta \right) =\frac{1}{T}\sum _{t=1}^T g\left( \theta ,X_t\right) . \end{aligned}$$

The GMM estimator depends on the choice of a positive definite weight matrix $W_T\in {\mathbb {R}}^{q\times q}$ and is the solution of the following minimization problem:

$$\begin{aligned} {\hat{\theta }}\left( W_T\right) =\underset{\theta \in \Theta }{{\text {argmin}}} \ {\bar{g}}_T\left( \theta \right) ^{\top }W_T {\bar{g}}_T\left( \theta \right) . \end{aligned}$$

To find an estimator, we need at least as many moment conditions as the number of parameters. In particular we have the classical Method of Moments for $q = p$ and the GMM for $q>p$. Under mild conditions, for any positive definite weight matrix $W_T$, the GMM produces consistent estimators. Moreover the asymptotic distribution of ${\hat{\theta }}$ is:

$$\begin{aligned} \sqrt{T}\left( {\hat{\theta }}-\theta _0\right) \sim N\left( 0,V\right) \end{aligned}$$

where V is the asymptotic variance-covariance matrix defined as:

$$\begin{aligned} V = \left( D^{\top } W D\right) ^{-1} D^{\top } W S W D \left( D^{\top } W D\right) ^{-1} \end{aligned}$$

with $D \in {\mathbb {R}}^{q\times p}$:

$$\begin{aligned} D:={\mathsf {E}}\left[ \frac{\partial g\left( \theta ,X\right) }{\partial \theta ^\top }\right] \end{aligned}$$

and $S \in {\mathbb {R}}^{q\times q}$:

$$\begin{aligned} S:={\mathsf {E}}\left[ g\left( \theta ,X\right) g\left( \theta ,X\right) ^{\top }\right] , \end{aligned}$$

$W_T\overset{P}{\rightarrow } W$ as $T\rightarrow \infty$. An appropriate choice of matrix W improves the efficiency within the class of GMM type estimators. The most efficient estimator is obtained if the following condition holds:

$$\begin{aligned} W_T \overset{P}{\rightarrow } S^{-1} \end{aligned}$$

and, in that case, the variance-covariance matrix V reads:

$$\begin{aligned} V = \left( D^{\top } S^{-1} D\right) ^{-1}. \end{aligned}$$

(13)

Several algorithms have been proposed in literature in order to reach the variance-covariance matrix in (13). In this paper we use the R package gmm developed in Chaussé (2010) where the optimal matrix W is estimated using the Heteroskedastic Auto-Correlation consistent (HAC) approach proposed in Newey and West (1987). It is required to choose a certain grid $u_1,\ldots , u_j, \ldots , u_q$ where $u_j \in {\mathbb {R}}^{N}$ with $i=1,\ldots ,q$ while $g\left( \theta , X, u_j\right)$ is defined as:

$$\begin{aligned} g\left( \theta , X, u_j\right) = e^{i\left\langle u_j,X\right\rangle }-\phi _{X,\theta }\left( u_j\right) , \ \ j=1,\ldots ,q, \end{aligned}$$

$\left\langle \ , \ \right\rangle$ is the scalar product. The moment conditions require:

$$\begin{aligned} {\mathsf {E}}\left[ g\left( \theta , X, u_j\right) \right] =0 \end{aligned}$$

where the j-th element of the vector function ${\bar{g}}_T\left( \theta \right)$ is:

$$\begin{aligned} {\bar{g}}_{T,j}\left( \theta \right) =\frac{1}{T}\sum _{t=1}^T \left[ e^{i\left\langle u_j, X_t\right\rangle }-\phi _{X_t,\theta }\left( u_j\right) \right] . \end{aligned}$$

Observe that for each t, $X_t$ is multivariate i.i.d MixedTS distributed though we have:

$$\begin{aligned} {\bar{g}}_{T,j}\left( \theta \right) =\frac{1}{T}\sum _{t=1}^T \left[ e^{i\left\langle u_j, X_t\right\rangle }\right] -\phi _{X,\theta }\left( u_j\right) \end{aligned}$$

The grid is chosen from a multivariate standard normal distribution and the multivariate joint characteristic function $\phi _{X,\theta }\left( u\right)$ is defined in (5).