Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation

Abstract

For the multiplicative background risk model, a distortion-type risk measure is used to measure the tail risk of the portfolio under a scenario probability measure with multivariate regular variation. In this paper, we investigate the tail asymptotics of the portfolio loss $\sum _{i=1}^{d}R_iS$, where the stand-alone risk vector ${\mathbf {R}}=(R_1,\ldots ,R_d)$ follows a multivariate regular variation and is independent of the background risk factor S. An explicit asymptotic formula is established for the tail distortion risk measure, and an example is given to illustrate our obtained results.

Appendix

We discuss results and assumptions from Section 4.2 in Resnick (2002) that are used in this paper for the sake of completeness. The following results provide conditions under which the second-order regular variation of Definition 2.4 can be represented as vague convergences of measures. Assumption 6.1 gives the appropriate conditions when the limit measure $\nu (\cdot )$ as obtained in Definition2.3 has a density with respect to the Lebesgue measure; hence ${\mathbf {X}}$ is not asymptotically independent. On the other hand, Assumption 6.3 gives appropriate conditions when $\nu (\cdot )$ does not has a density, it means that asymptotic independence holds for the tail distribution of ${\mathbf {X}}$. Suppose ${\mathbf {X}}$ is a d-dimensional non-negative random vector with distribution function F and identical one-dimensional marginals $F_1$.

Assumption 6.1

We assume the following on F.

• 1. Let F have a density $F'$ such that for $b(t)\rightarrow \infty$,

  $$\begin{aligned} \lim _{t\rightarrow \infty } |b(t)^dtF'(b(t){\mathbf {x}})-p({\mathbf {x}})|=0, {\mathbf {x}}\in {\mathbb {E}}, \end{aligned}$$

  (6.1)

  where $p(\cdot )\ne 0$ is bounded on ${\mathcal {N}}$ and moreover

  $$\begin{aligned} \lim _{t\rightarrow \infty } \sup \limits _{{\mathbf {a}}\in {\mathcal {N}}}|b(t)^dtF'(b(t){\mathbf {a}})-p({\mathbf {a}})|=0,{\mathbf {x}}\in {\mathbb {E}}, \end{aligned}$$

  (6.2)

  The limit function $p({\mathbf {x}})$ necessarily satisfies $p(t{\mathbf {x}})=t^{-\alpha -d}p({\mathbf {x}})$. This implies from Resnick (2008) that there exists $V \in {RV}_{-\alpha }$ such that

  $$\begin{aligned} \lim _{t\rightarrow \infty } \frac{1-F(b(t){\mathbf {x}})}{V(b(t))}= \int \limits _{[{\mathbf {0}},{\mathbf {x}}]^c} p({\mathbf {u}})d{\mathbf {u}} = \nu ([{\mathbf {0}},{\mathbf {x}}]^c) , {\mathbf {x}}>{\mathbf {0}}. \end{aligned}$$

  (6.3)

  Thus, conditions (6.1) and (6.2) imply the multivariate regular variation. Instead of conditions (6.1) and (6.2), it is sufficient to assume ${\overline{F}}_1\in {RV}_{-\alpha }$ and

  $$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Big |\frac{F^{'}(t{\mathbf {x}})}{t^{-d}{\overline{F}}_1(t)}-p({\mathbf {x}})\Big |=0,{\mathbf {x}}\in {\mathbb {E}}, \quad and\quad \lim \limits _{t\rightarrow \infty } \sup \limits _{{\mathbf {a}}\in {\mathcal {N}}}\Bigg |\frac{F^{'}(t{\mathbf {a}})}{t^{-d}{\overline{F}}_1(t)} -p({\mathbf {a}})\Bigg |=0\nonumber \\ \end{aligned}$$

  (6.4)

  and we can take $V={\overline{F}}_1$.

• 2. Assume that the second-order condition given in (2.2) holds for ${\overline{F}}_1$ so that ${\overline{F}}_1\in {RV}_{-\alpha }$ and $A \in {RV}_\rho$, $\rho \le 0$, $A\rightarrow 0$ and for ${\mathbf {x}}\in {\mathbb {E}}$,

  $$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Bigg |\frac{\frac{F^{'}(t{\mathbf {x}})}{t^{-d}{\overline{F}}_1(t)}-p({\mathbf {x}})}{A(t)}-\chi ^{'}({\mathbf {x}})\Bigg |=0, \end{aligned}$$

  (6.5)

  where $\chi ^{'}\ne 0$ is integrable on sets bounded away from ${\mathbf {0}}$. We also assume uniform convergence on ${\mathcal {N}}$:

  $$\begin{aligned} \lim _{t\rightarrow \infty } \sup \limits _{{\mathbf {a}}\in {\mathcal {N}}}\Bigg |\frac{\frac{F^{'}(t{\mathbf {a}})}{t^{-d}{\overline{F}}_1(t)}-p({\mathbf {a}})}{A(t)}-\chi ^{'}({\mathbf {a}})\Bigg |=0, \end{aligned}$$

  (6.6)

Also assume that $\chi ^{'}$ is finite and bounded on ${\mathcal {N}}$.

Remark 6.2

For ${\mathbf {X}}\sim F$ with identical marginals $F_1$, assuming conditions (6.4) – (6.6) is sufficient for (6.1) – (6.3) to hold with $V={\overline{F}}_1$.

Using $\nu$ as defined in (6.3), we define the signed measure

$$\begin{aligned} \mu _t([{\mathbf {0}},{\mathbf {x}}]^c):=\frac{t{\mathbb {P}}\Big [\frac{{\mathbf {X}}}{b(t)\in [{\mathbf {0}},{\mathbf {x}}]^c}\Big ]-\nu ([{\mathbf {0}},{\mathbf {x}}]^c)}{A(b(t))}, \end{aligned}$$

(6.7)

which has a density given by

$$\begin{aligned} \mu _t^{'}([{\mathbf {0}},{\mathbf {x}}]^c):=\frac{tb(t)^dF^{'}\big (b(t){\mathbf {x}}\big )-p({\mathbf {x}})}{A(b(t))}, \quad x\in [0,\infty )^d. \end{aligned}$$

(6.8)

Assumption 6.3

We assume the following on F.

• 1. Suppose (2.4) holds with $\nu ([{\mathbf {0}},{\mathbf {x}}]^c)=k \sum _{i=1}^d x_i^{-\alpha }$, where k is some constant.

• 2. Moreover, the one-dimensional marginals are identical and satisfy the second-order condition as in Definition 2.2 such that we also have

  $$\begin{aligned} \mu _{t1}^\pm :=\Bigg ( \frac{t{\mathbb {P}}[\frac{X_1}{b(t)}\in \cdot ]-\nu _\alpha (\cdot )}{A(b(t))} \Bigg )^\pm {\mathop {\rightarrow }\limits ^{v}}\chi _1^\pm \end{aligned}$$

  (6.9)

on $(0,\infty ]$ where $\chi _1(x,\infty ]=cx^{-\alpha }\frac{x^\rho -1}{\rho }.$