Common shock approach to counterparty default risk of reinsurance

Abstract

The paper deals with the construction of required capital to cover the default risk in portfolios with a smaller number of heterogeneous counterparties. The typical application is counterparty default risk of reinsurance (e.g., in Solvency II), but other applications in finance are also possible. Since the approach by means of Vasicek portfolio model is questionable in such cases the paper addresses mainly the approach based on the so-called common shock principle. An extensive numerical study compares results of various methods which are applicable in this context. The numerical results confirm that the suggested modifications of the widely accepted common shock approach implemented within the Solvency II framework might be preferred by insurance companies when constructing the portfolio of reinsurers.

Appendix

Derivation of (17)

For instance, to derive σ_ij for i ≠ j one can write using (13):

$$\begin{aligned} \sigma_{ij} & = \text{cov} (I_{i} ,I_{j} ) = E(I_{i} ,I_{j} ) - E(I_{i} ) \cdot E(I_{j} ) = E(I_{i} ,I_{j} ) - {\text{PD}}_{i} \cdot {\text{PD}}_{j} \\ & = \int\limits_{0}^{1} {[p_{i} + (1 - p_{i} )r^{{\gamma \;/p_{i} }} ][p_{j} + (1 - p_{j} )r^{{\gamma \;/p_{j} }} ]\beta r^{\beta - 1} {\text{d}}s} - \frac{{(\gamma + \beta )p_{i} }}{{\gamma + \beta p_{i} }} \cdot \frac{{(\gamma + \beta )p_{j} }}{{\gamma + \beta p_{j} }} \\ & = \frac{{\beta (1 - p_{i} ) (1 - p_{j} )}}{{\beta + \gamma p_{i}^{ - 1} + \gamma p_{j}^{ - 1} }} - ({\text{PD}}_{i} - p_{i} ) ({\text{PD}}_{j} - p_{j} ). \\ \end{aligned}$$

The derivation for i = j is analogous.

Derivation of (24)

$$\begin{aligned} P(L = 0) & = P(I_{1} = 1 \cap I_{2} = 0) = \int\limits_{0}^{1} {\text{[}1 - {\text{PD}}_{1} (r)\text{][}1 - {\text{PD}}_{2} (r)\text{]}f(r){\text{d}}r} \\ & = \int\limits_{0}^{1} {[1 - p_{1} - (1 - p_{1} )r^{{\gamma \;/p_{1} }} ][1 - p_{2} - (1 - p_{2} )r^{{\gamma \;/p_{2} }} ]\;\beta r^{\beta - 1} {\text{d}}r} \\ & = (1 - p_{1} )(1 - p_{2} )\left[ {1 - \frac{\beta }{{\gamma /p_{1} + \beta }} - \frac{\beta }{{\gamma /p_{2} + \beta }} + \frac{\beta }{{\gamma /p_{1} + \gamma /p_{2} + \beta }}} \right]. \\ \end{aligned}$$

Derivation of (26)

$$\begin{aligned} {\text{ES}}(L\left| {R \ge C)} \right. & = \sum\limits_{i = 1}^{k} {{\text{LGD}}_{i} \cdot E(I_{i} \left| {R \ge C)} \right.} \\ & = \sum\limits_{i = 1}^{k} {\frac{{{\text{LGD}}_{i} }}{P(R \ge C)}\int\limits_{C}^{1} {{\text{PD}}_{i} (r)f(r){\text{d}}r} } \\ & = \sum\limits_{i = 1}^{k} {\frac{{{\text{LGD}}_{i} }}{P(R \ge C)}\int\limits_{C}^{1} {[p_{i} + (1 - p_{i} )r^{{\gamma \;/p_{i} }} ]\;\beta r^{\beta - 1} {\text{d}}r} } \\ & = \sum\limits_{i = 1}^{k} {\frac{{{\text{LGD}}_{i} }}{P(R \ge C)}\left\{ {p_{i} (1 - C^{\beta } ) + \frac{{\beta (1 - p_{i} )}}{{\gamma /p_{i} + \beta }}[1 - C^{{\gamma /p_{i} + \beta }} ]} \right\}} \\ & = \sum\limits_{i = 1}^{k} {\frac{{{\text{LGD}}_{i} }}{P(R \ge C)}\left\{ {\frac{{(\gamma + \beta )p_{i} }}{{\gamma + \beta p_{i} }}(1 - C^{{\gamma /p_{i} + \beta }} ) + p_{i} (C^{{\gamma /p_{i} + \beta }} - C^{\beta } )} \right\}.} \\ \end{aligned}$$

Derivation of (27)

It holds

$$P(R \ge C) = \int\limits_{C}^{1} {\beta r^{\beta - 1} {\text{d}}r} = 1 - C^{\beta } .$$

Hence

$$P(R \ge \alpha^{1/\beta } ) = 1 - \alpha$$

so that (27) is proved.