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Wishart conditional tail risk measures: An analytic approach

Jose Da Fonseca, Patrick Wong

Abstract

This study introduces a new analytical framework for quantifying multivariate risk measures. Using the Wishart process, which is a stochastic process with values in the space of positive definite matrices, we derive several conditional tail risk measures which, thanks to the remarkable analytical properties of the Wishart process, can be explicitly computed up to a one- or two-dimensional integration. These quantities can also be used to solve analytically a capital allocation problem based on conditional moments. Exploiting the stochastic differential equation property of the Wishart process, we show how an intertemporal (i.e., time-lagged) view of these risk measures can be embedded in the proposed framework. Several numerical examples show that the framework is versatile and operational, thus providing a useful tool for risk management.

Wishart conditional tail risk measures: An analytic approach

Abstract

This study introduces a new analytical framework for quantifying multivariate risk measures. Using the Wishart process, which is a stochastic process with values in the space of positive definite matrices, we derive several conditional tail risk measures which, thanks to the remarkable analytical properties of the Wishart process, can be explicitly computed up to a one- or two-dimensional integration. These quantities can also be used to solve analytically a capital allocation problem based on conditional moments. Exploiting the stochastic differential equation property of the Wishart process, we show how an intertemporal (i.e., time-lagged) view of these risk measures can be embedded in the proposed framework. Several numerical examples show that the framework is versatile and operational, thus providing a useful tool for risk management.
Paper Structure (22 sections, 16 theorems, 83 equations, 3 figures, 7 tables)

This paper contains 22 sections, 16 theorems, 83 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The mathematical framework
  3. Conditional tail risk measures
  4. Multidimensional Fourier transform representations for risk measures
  5. One-dimensional Fourier transform representations for risk measures
  6. Wishart conditional tail risk measures
  7. The two dates case
  8. A note on numerical precision
  9. Conditional higher moments tail risk measures
  10. The Tail variance (TV):
  11. The Tail covariance (TCov):
  12. The Tail skewness (TS):
  13. Numerical experiments
  14. Implementation details
  15. One date risk measures
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $(x_t)_{t\geq 0}$ be a Wishart process given by eq:Wishart, and denote the MGF of $x_t$ by where $\theta_0 \in \mathbb{S}_n$. Then with the deterministic functions $(a(t,\theta_0),b(t,\theta_0))$, where $a(t,\theta_0)\in \mathsf{M}(n)$ and $b(t,\theta_0) \in \mathbb{R}$, satisfying the system with initial conditions $a(0,\theta_0)=\theta_0$ and $b(0,\theta_0)=0$. As usual $\cdot^\prime$ de

Figures (3)

  • Figure 1: Dependence of $\mathbb{E}\left[ x_{11,t} \mid x_{11,t}>x_* \right]$ on $x_*$.
  • Figure 2: Dependence of $\mathbb{E}\left[ s_{t} \mid s_{t}>s_* \right]$ on $s_*$.
  • Figure 3: Dependence of $\mathbb{E}\left[ x_{11,t_1} \mid x_{11,t_0}>x_* \right]$ on $t_1$.

Theorems & Definitions (29)

  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Corollary 2.7
  • Proposition 3.1
  • proof
  • ...and 19 more