Optimal risk sharing in insurance networks

Abstract

We discuss the impact of risk sharing and asset–liability management on capital requirements. Our analysis contributes to the evaluation of the merits and deficiencies of different risk measures. In particular, we highlight that the class of V@R-based risk measures allows for a substantial reduction of the total capital requirement in corporate networks that share risks between entities. We provide case studies that complement previous theoretical results and demonstrate their practical relevance. For large networks, optimal asset–liability management is often contrary to those strategies that are desirable from a regulatory point of view.

Appendices

A Risk measures

We denote by ${\mathcal {X}}$ a vector space of measurable, real-valued functions on a measurable space $(\Omega , {\mathcal {F}})$ that contains the constants. If ${\mathbb {P}}$ is a probability measure on $(\Omega , {\mathcal {F}})$, typical examples of ${\mathcal {X}}$ are $L^p$-spaces, $p\in [1,\infty ]$, where ${\mathbb {P}}$-almost sure equal functions are identified with each other.

A monetary risk measure$\rho :{\mathcal {X}}\rightarrow {\mathbb {R}}$ is an inverse monotone and cash-invariant function on ${\mathcal {X}}$:

• 1. Inverse Monotonicity:$X,Y\in {\mathcal {X}}, X\le Y$$\Rightarrow$$\rho (X)\ge \rho (Y)$

• 2. Cash-Invariance:$X\in {\mathcal {X}}, m\in {\mathbb {R}}$$\Rightarrow$$\rho (X+m)=\rho (X)-m$

Property 1 states that the risk of a position Y is smaller than the risk of a position X, if the future value of Y is at least X. Property 2 states that risk is measured on a monetary scale: If m Euro are added to X, then the risk of X is exactly reduced by this amount.

In particular, any monetary risk measure corresponds to its acceptance set, ${\mathcal {A}}=\{X\in {\mathcal {X}}:\rho (X)\le 0\}$, from which it can be recovered via

$$\begin{aligned} \rho (X)=\inf \{m\in {\mathbb {R}}:X+m\in {\mathcal {A}}\}. \end{aligned}$$

Thus, a monetary risk measure can be viewed as a capital requirement: $\rho (X)$ is the minimal capital that has to be added to the position X to make it acceptable.

The choice of a meaningful risk measure for capital regulation is subject to an ongoing discussion between academics and practitioners that began in the mid 1990s. Various desirable properties of monetary risk measures have been proposed, and corresponding classes of risk measures have been identified and characterized. A common requirement in the literature is that diversification should not increase risk. In mathematical terms, diversification corresponds to quasi-convexity of $\rho$, i. e.,

$$\begin{aligned} \rho (\lambda X+(1-\lambda )Y)\le \max \{\rho (X),\rho (Y)\} \end{aligned}$$

(11)

for $X,Y\in {\mathcal {X}}$ and $\lambda \in (0,1)$. In that case, $\rho$ is also a convex functional on ${\mathcal {X}}$. A monetary risk measure is called a convex risk measure if it satisfies condition (11) of quasi-convexity and is hence convex. A convex risk measure is called coherent if it is also positively homogeneous, i. e.,

$$\begin{aligned} \rho (\lambda X)=\lambda \rho (X) \end{aligned}$$

for $X\in {\mathcal {X}}$ and $\lambda \ge 0$. Positive homogeneity is often seen as critical, in particular since the additional concentration risk caused by scaling the financial position is not captured.

A monetary risk measure on a space of random variables ${\mathcal {X}}$ on $(\Omega ,{\mathcal {F}},{\mathbb {P}})$ is distribution-based (sometimes somewhat misleading also called law-invariant), if $\rho (X)=\rho (Y)$ whenever the distributions of X and Y under ${\mathbb {P}}$ are equal, i. e., ${\mathbb {P}}^X={\mathbb {P}}^Y$ for $X,Y\in {\mathcal {X}}$.

Distribution-based risk measures include a wide variety of examples, see, e. g., Föllmer and Schied [15] and Föllmer and Weber [16]. Throughout this paper, we focus on three prominent examples with different properties:

• (a) Value at risk: The most commonly used risk measure in practice—and in particular the prescribed risk measure for Solvency II purposes—is value at risk (${{\text {V@R}}}$). For a given level $\alpha \in (0,1)$, we denote by ${{\text {V@R}}}_\alpha$ the monetary risk measure defined by the acceptance set

  $$\begin{aligned} {\mathcal {A}}_{{{\text {V@R}}}_\alpha }=\{X\in {\mathcal {X}}\vert {\mathbb {P}}[X<0]\le \alpha \}. \end{aligned}$$

  (12)

  For a financial position X, the value ${{\text {V@R}}}_\alpha (X)$ specifies the smallest monetary amount that needs to be added to X such that the probability of a loss becomes smaller than $\alpha$:

  $$\begin{aligned} {{\text {V@R}}}_\alpha (X)=&\inf \{m\in {\mathbb {R}}\vert {\mathbb {P}}[X+m<0]\le \alpha \}\\ =&-\sup \{c\in {\mathbb {R}}\vert {\mathbb {P}}[X<c]\le \alpha \}=-q_X^+(\alpha ),\nonumber \end{aligned}$$

  where $q_X^+(\alpha )$ is the upper $\alpha$-quantile of X.

  Recall that ${{\text {V@R}}}_\alpha$ has two main deficiencies: Firstly, value at risk is not a convex risk measure and may thus penalize diversification beyond the setting of Gaussian or more generally elliptic financial positions. Secondly, ${{\text {V@R}}}_\alpha$ neglects extreme losses that occur with small probability. These deficiencies of value at risk were a major reason to develop a systematic theory of coherent and convex risk measures, as initiated by Artzner, Delbaen, Eber and Heath [3]; Föllmer and Schied [14].

• (b) Average value at risk: Another basic example is the average value at risk (${{\text {AV@R}}}$), also known as conditional value at risk, tail value at risk, or expected shortfall, which plays a prominent role in the Swiss Solvency Test. The average value at risk at level $\beta \in (0,1]$ is defined by

  $$\begin{aligned} {{\text {AV@R}}}_\beta (X):=\tfrac{1}{\beta }\int _0^\beta {{\text {V@R}}}_\alpha (X)\,d\alpha ,\quad X\in {\mathcal {X}}. \end{aligned}$$

  In contrast to value at risk, ${{\text {AV@R}}}_\beta$ accounts for extreme losses per definition, and it provides incentives for diversification. More precisely, ${{\text {AV@R}}}_\beta$ is a coherent measure of risk.

• (c) Range value at risk: Cont, Deguest and Scandolo [9] suggest an alternative to ${{\text {V@R}}}$ and ${{\text {AV@R}}}$, called range value at risk (${{\text {RV@R}}}$). Letting $\alpha , \beta >0$ with $\alpha + \beta \le 1$, they define

  $$\begin{aligned} {{\text {RV@R}}}_{\alpha ,\beta }(X)=\tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }{{\text {V@R}}}_\gamma (X)\,d\gamma , \quad X\in {\mathcal {X}}. \end{aligned}$$

  Note that the limiting cases of ${{\text {RV@R}}}_{\alpha , \beta }$ correspond to ${{\text {V@R}}}_{\alpha }$ for $\beta \rightarrow 0$ and ${{\text {AV@R}}}_\beta$ for $\alpha \rightarrow 0$. Like ${{\text {V@R}}}$, ${{\text {RV@R}}}$ is a non-convex risk measure, and it may thus penalize diversification.

B Proofs

Proof of Corollary 3.1.

Proof

By Example 2.6, we have $\Box _{i=1}^{n}{{\text {RV@R}}}_{\alpha _i,\beta _i}(E_1(\delta ))={{\text {RV@R}}}_{\alpha ,\beta }(E_1(\delta ))$ for $\alpha =\alpha _1+\ldots +\alpha _n$, $\beta =\max \{\beta _1,\ldots ,\beta _n\}$. It is thus enough to show that

$$\begin{aligned}&{{\text {RV@R}}}_{\alpha ,\beta }(E_1(\delta ))\nonumber \\&\quad =-\eta ^2(\delta )S^2_0e^\mu \tfrac{1}{\beta }\left( \Phi (\Phi ^{-1}(\alpha +\beta )-\sigma )-\Phi (\Phi ^{-1}(\alpha )-\sigma )\right) -\eta ^1(\delta )+\pi . \end{aligned}$$

(13)

To this end, note first that

$$\begin{aligned} {{\text {RV@R}}}_{\alpha ,\beta }(E_1(\delta ))= & {} {{\text {RV@R}}}_{\alpha ,\beta }(\eta ^2(\delta )S^2_1+\eta ^1(\delta )-\pi )\nonumber \\= & {} \eta ^2(\delta ){{\text {RV@R}}}_{\alpha ,\beta }(S^2_1)-\eta ^1(\delta )+\pi , \end{aligned}$$

(14)

since ${{\text {RV@R}}}_{\alpha ,\beta }$ is cash-invariant and positively homogeneous. Hence, it remains to compute

$$\begin{aligned} {{\text {RV@R}}}_{\alpha ,\beta }(S^2_1)=\tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }{{\text {V@R}}}_\gamma (S^2_1)\,d\gamma =\tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }-q_{S^2_1}(\gamma )\,d\gamma . \end{aligned}$$

Using the quantile transformation rule for $S^2_1=f(W_1)$ with the increasing function $f(x)=S^2_0\exp (\mu -\tfrac{1}{2}\sigma ^2+\sigma x)$ combined with the fact that $q_X(\gamma )={\mathbb {E}}[X]+\Phi ^{-1}(\gamma )\sigma (X)$ for any normally distributed X, we obtain

$$\begin{aligned} {{\text {RV@R}}}_{\alpha ,\beta }(S^2_1)= & {} \tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }-q_{S^2_1}(\gamma )\,d\gamma =\tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }-S^2_0e^{\mu -\tfrac{1}{2}\sigma ^2+\sigma q_{W_1}(\gamma )}\,d\gamma \\= & {} \tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }-S^2_0e^{\mu -\tfrac{1}{2}\sigma ^2+\Phi ^{-1}(\gamma )\sigma }\,d\gamma =-S^2_0e^{\mu -\tfrac{1}{2}\sigma ^2}\tfrac{1}{\beta }\int _\alpha ^{\alpha +\beta }e^{\Phi ^{-1}(\gamma )\sigma }\,d\gamma . \end{aligned}$$

Substituting $y=\Phi ^{-1}(\gamma )$ with $dy=(1/\varphi (\Phi ^{-1}(\gamma ))d\gamma$ in terms of the density $\varphi$ of the standard normal distribution leads to

$$\begin{aligned} {{\text {RV@R}}}_{\alpha ,\beta }(S^2_1)= & {} -S^2_0e^{\mu -\tfrac{1}{2}\sigma ^2}\tfrac{1}{\beta }\int _{\Phi ^{-1}(\alpha )}^{\Phi ^{-1}(\alpha +\beta )}e^{\sigma y}\varphi (y)\,dy\\= & {} -S^2_0e^{\mu -\tfrac{1}{2}\sigma ^2}\tfrac{1}{\beta }\int _{\Phi ^{-1}(\alpha )}^{\Phi ^{-1}(\alpha +\beta )}e^{\sigma y}\tfrac{1}{\sqrt{2\pi }}e^{-\tfrac{1}{2}y^2}\,dy \\= & {} -S^2_0e^{\mu }\tfrac{1}{\beta }\int _{\Phi ^{-1}(\alpha )}^{\Phi ^{-1}(\alpha +\beta )}\tfrac{1}{\sqrt{2\pi }}e^{-\tfrac{1}{2}(y-\sigma )^2}\,dy \\= & {} -S^2_0e^{\mu }\tfrac{1}{\beta }\int _{\Phi ^{-1}(\alpha )-\sigma }^{\Phi ^{-1}(\alpha +\beta )-\sigma }\varphi (y)\,dy \\= & {} -S^2_0e^{\mu }\tfrac{1}{\beta }\left( \Phi (\Phi ^{-1}(\alpha +\beta )-\sigma )-\Phi (\Phi ^{-1}(\alpha )-\sigma )\right) . \end{aligned}$$

Together with (14) this proves (13). Since

$$\begin{aligned} {\mathbb {E}}[E_1(\delta )]=\eta ^2(\delta )S^2_0e^\mu +\eta ^1(\delta )-\pi , \end{aligned}$$

the formulae for $\Box _{i=1}^{n}{{\text {SCR}}}^{i}_{\mathcal {A}}(E_1(\delta ))$ and $\Box _{i=1}^{n}{{\text {SCR}}}^{i}_{\mathrm{mean}}(E_1(\delta ))$, respectively, follow from (2) immediately. $\square$

Proof of Corollary 3.2.

Proof

Recalling that the limiting cases of ${{\text {RV@R}}}_{\alpha , \beta }$ correspond to ${{\text {V@R}}}_{\alpha }$ for $\beta \rightarrow 0$ and ${{\text {AV@R}}}_\beta$ for $\alpha \rightarrow 0$, the claim follows from Example 2.6 and Corollary 3.1. More precisely, we have

$$\begin{aligned} {{\text {V@R}}}_{\alpha }(E_1(\delta ))= & {} \lim _{\beta \rightarrow 0}{{\text {RV@R}}}_{\alpha ,\beta }(E_1(\delta ))\\= & {} \lim _{\beta \rightarrow 0}\left( -\eta ^2(\delta )S^2_0e^\mu \tfrac{1}{\beta }\left( \Phi (\Phi ^{-1}(\alpha +\beta )-\sigma )-\Phi (\Phi ^{-1}(\alpha )-\sigma )\right) \right. \\&\left. -\eta ^1(\delta )+\pi \right) \\= & {} -\eta ^2(\delta )S^2_0e^\mu \left( \tfrac{d}{d\beta }\Phi (\Phi ^{-1}(\alpha +\beta )-\sigma )\right) -\eta ^1(\delta )+\pi \\= & {} -\eta ^2(\delta )S^2_0e^\mu \tfrac{\varphi (\Phi ^{-1}(\alpha )-\sigma )}{\varphi (\Phi ^{-1}(\alpha ))}-\eta ^1(\delta )+\pi \\= & {} -\eta ^2(\delta )S^2_0e^\mu \exp (\sigma \Phi ^{-1}(\alpha )-\tfrac{\sigma ^2}{2})-\eta ^1(\delta )+\pi . \end{aligned}$$

In the same manner, we derive

$$\begin{aligned} {{\text {AV@R}}}_{\beta }(E_1(\delta ))= & {} \lim _{\alpha \rightarrow 0}{{\text {RV@R}}}_{\alpha ,\beta }(E_1(\delta ))\\= & {} \lim _{\alpha \rightarrow 0}\left( -\eta ^2(\delta )S^2_0e^\mu \tfrac{1}{\beta }\left( \Phi (\Phi ^{-1}(\alpha +\beta )-\sigma )-\Phi (\Phi ^{-1}(\alpha )-\sigma )\right) \right. \\&\left. -\eta ^1(\delta )+\pi \right) \\= & {} -\eta ^2(\delta )S^2_0e^\mu \tfrac{1}{\beta }\Phi (\Phi ^{-1}(\beta )-\sigma )-\eta ^1(\delta )+\pi , \end{aligned}$$

since $\lim _{\alpha \rightarrow 0}\Phi ^{-1}(\alpha )=-\infty$ and $\lim _{\alpha \rightarrow 0}\Phi (\Phi ^{-1}(\alpha )-\sigma )=0$. $\square$