Risk concentration under second order regular variation

Abstract

Measures of risk concentration and their asymptotic behavior for portfolios with heavy-tailed risk factors is of interest in risk management. Second order regular variation is a structural assumption often imposed on such risk factors to study their convergence rates. In this paper, we provide the asymptotic rate of convergence of the measure of risk concentration for a portfolio of heavy-tailed risk factors, when the portfolio admits the so-called second order regular variation property. Moreover, we explore the relationship between multivariate second order regular variation for a vector (e.g., risk factors) and the second order regular variation property for the sum of its components (e.g., the portfolio of risk factors). Results are illustrated with a variety of examples.

Appendix

The diversification property relating the marginal risks to the aggregate risk in Theorem 3.4(2.a), can be easily extended to tail equivalent risks. We provide the result in the following.

Theorem 6.1

Let $X\in 2\mathcal {RV}_{-\alpha ,\rho _{X}}(b_{X},A_{X},H_{X})$ and $Y\in 2\mathcal {RV}_{-\alpha ,\rho _{Y}}(b_{Y},A_{Y},H_{Y})$ with α > 0,ρ_X < 0,ρ_Y < 0 and

$$H_{X} (x):= \frac{c_{X}}{\rho_{X}}x^{-\alpha}(x^{\rho_{X}}-1), x>0, \quad\quad H_{Y}(x):= \frac{c_{Y }}{\rho_{ Y}}x^{-\alpha}(x^{\rho_{Y}}-1), x>0.$$

Assume that X and Y are tail equivalent risks, meaning that $\displaystyle 0<\lim _{t\to \infty } b_{X}(t)/b_{Y}(t)<\infty$, and define $\displaystyle \tilde {K}_{d}:=\lim _{t\to \infty } \frac {b_{Y}(t)}{d b_{ X}(t)}$. Then the following hold.

• i. If $\displaystyle {\lim _{t\to \infty }A_{X}(b_{X}(t))/A_{Y}(b_{Y}(t)) =\kappa \in \mathbb {R}}$, then, for any x > 0, we have

  $$ \begin{array}{@{}rcl@{}} \lim_{t \to \infty} \frac1{ A_{Y}(b_{Y}(t))}\times\left( \frac{\text{VaR}_{1-x/t}(Y)}{d \text{VaR}_{1-x/t}(X)} - \tilde{K}_{d}\right) = \frac{(c_{Y}-\kappa c_{ X})\tilde{K}_{d}}{\alpha\rho_{Y}} (x^{{-\rho_{Y}/\alpha}}-1). \end{array} $$

• ii. If $\displaystyle {\lim _{t\to \infty }A_{Y}(b_{Y}(t))/A_{X}(b_{X}(t)) =0}$, then, for any x > 0, we have

  $$ \begin{array}{@{}rcl@{}} \lim_{t \to \infty} \frac1{ A_{X}(b_{X}(t))}\times\left( \frac{\text{VaR}_{1-x/t}(Y)}{d \text{VaR}_{1-x/t}(X)} - \tilde{K}_{d}\right) = -\frac{c_{X}\tilde{K}_{d}}{\alpha\rho_{X}} (x^{{-\rho_{X}/\alpha}}-1). \end{array} $$

Proof

The proof of Theorem 6.1 is the same as that of Theorem 3.4(2.a) and can be obtained by replacing X₁ by X, S_d by Y (with the corresponding parameters for the $2\mathcal {RV}$ property), D_β by $\displaystyle \frac {\text {VaR}_{\beta }(Y)}{d \text {VaR}_{\beta }(X)}$, and K_d by $\displaystyle \tilde {K}_{d}$ in the proof of Theorem 3.4(2.a). □