On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk

Abstract

In this paper, we address the aggregation of dependent stop loss reinsurance risks where the dependence among the ceding insurer(s) risks is governed by the Sarmanov distribution and each individual risk belongs to the class of Erlang mixtures. We investigate the effects of the ceding insurer(s) risk dependencies on the reinsurer risk profile by deriving a closed formula for the distribution function of the aggregated stop loss reinsurance risk. Furthermore, diversification effects from aggregating reinsurance risks are examined by deriving a closed expression for the risk capital needed for the whole portfolio of the reinsurer and also the allocated risk capital for each business unit under the TVaR capital allocation principle. Moreover, given the risk capital that the reinsurer holds, we express the default probability of the reinsurer analytically. In case the reinsurer is in default, we determine analytical expressions for the amount of the aggregate reinsured unpaid losses and the unpaid losses of each reinsured line of business of the ceding insurer(s). These results are illustrated by numerical examples.

Appendices

Appendix 1: Properties of mixed Erlang distribution

Lemma 6.1

For a deductible $d >0$, if $X \sim ME(\beta , \underline{V})$ then the df of $Y:=(X-d)_+$ is given by

$$\begin{aligned} F_Y(y) = \left\{ \begin{array}{lcl} F_X(d) &{} \text{ for } &{} y=0, \\ F_X(d+y) &{}\text{ for } &{} y>0, \end{array} \right. \end{aligned}$$

(6.1)

where

$$\begin{aligned} F_X(d+y) = \sum _{k=0}^\infty \Delta _{k}(d,\beta ,\underline{V} ) W_{k+1}(y,\beta ), \end{aligned}$$

with

$$\begin{aligned} \Delta _{k} (d,\beta ,\underline{V}) = \frac{1}{\beta } \sum _{j=0}^\infty q_{j+k+1}w_{j+1}(d,\beta ). \end{aligned}$$

(6.2)

Lemma 6.2

Let $X_1$ and $X_2$ be two independent risks such that $X_i\sim ME(\beta ,\underline{Q}_i), i=1,2$. If $Y_i=(X_i-d_i)_+$ with $d_i>0, i=1,2$ then $R_2= Y_1+ Y_2$ has a df

$$\begin{aligned} F_{R_2}(s) = \left\{ \begin{array}{lll} F_{X_1}(d_1) F_{X_2}(d_2) &{} \text{ for } \, s=0 , \\ F_{X_1}(d_1) F_{X_2}(s+d_2) + F_{X_2}(d_2) F_{X_1}(s+d_1) + F_{X1+X_2}(d_1,d_2,s) &{}\text{ for } \, s>0, \end{array} \right. \end{aligned}$$

(6.3)

where

$$\begin{aligned} F_{X1+X_2}(d_1,d_2,s)= \sum _ {k=0}^{\infty } \sum _ {j=0}^{\infty } \Delta _k(d_1,\beta ,\underline{Q}_1) \Delta _j(d_2,\beta ,\underline{Q}_2) W_{k+j+2}(s,\beta ). \end{aligned}$$

Remark 6.3

Given the tractable expression of the df of $R_2$, its VaR at a confidence level $p \in (0,1)$ is the solution of

$$\begin{aligned} F_{X_1}(d_1) F_{X_2}(d_2)+ F_{X_1}(d_1) F_{X_2}(VaR_{R_2} (p) +d_2) + F_{X_2}(d_2) F_{X_1}(VaR_{R_2} (p)+d_1) + F_{{T_{2,k}}}(VaR_{R_2} ( p)) = p , \end{aligned}$$

which can be solved numerically. In addition, the TVaR of $R_2$ at a confidence level $p \in (0, 1)$ is given by (set $x_p:=VaR_{R_2} (p)$)

$$\begin{aligned} TVaR_{R_2}(p)= & {} \frac{1}{1-p} \biggl ( F_{X_2}(d_2) \overline{U}_{X_1}(x_p,d_1,\beta ) + F_{X_1}(d_1) \overline{U} _{X_2}(x_p,d_2,\beta ) + \overline{U}_{X_1+X_2}(x_p,d_1,d_2,\beta ) \biggr ), \end{aligned}$$

where

$$\begin{aligned} \overline{U} _{X_i}(x_p,d_i,\beta )&= \frac{1}{\beta } \sum _ {k=0}^{\infty } (k+1) \Delta _k(d_i,\beta ,\underline{Q}_i) \overline{W}_{k+2}(x_p,\beta ),\\ \overline{U}_{X_1+X_2}(x_p,d_1,d_2,\beta )&= \frac{1}{\beta } \sum _ {k=0}^{\infty } \sum _ {j=0}^{\infty } (k+j+2) \Delta _k(d_1,\beta ,\underline{Q}_1) \Delta _j(d_2,\beta ,\underline{Q}_2) \overline{W}_{k+j+3}(x_p,\beta ). \end{aligned}$$

Proof

Since $Y_1$ and $Y_2$ are independent risks which have mixed distribution, the df of $R_2$ can also be expressed as a df of a mixed distribution which depends on the value of s as follows:

• the discrete part of $F_{R_2}$ is obtained for $s=0$, specifically we have

  $$\begin{aligned} F_{R_2}(0) = \mathbb {P}(Y_1 + Y_2 \leqslant 0) = \mathbb {P}(Y_1 + Y_2 = 0) = \mathbb {P}(Y_1=0, Y_2 = 0) = F_{X_1}(d_1) F_{X_2}(d_2), \end{aligned}$$

  (6.4)

• for $s>0$ the continious part of $F_{R_2}$ is given by

  $$\begin{aligned} F_{R_2}(s) &= \mathbb {P}(Y_1 + Y_2 \leqslant s ) \nonumber \\ &= \mathbb {P}( Y_1 + Y_2 \leqslant s, Y_1 = 0, 0< Y_2 \leqslant s) + \mathbb {P}( Y_1 + Y_2 \leqslant s, 0< Y_1 \leqslant s, Y_2 = 0) \nonumber \\&\quad+ \mathbb {P}(Y_1 + Y_2 \leqslant s , 0< Y_1 \leqslant s, 0< Y_2 \leqslant s) \nonumber \\&= \mathbb {P}( Y_1 = 0, 0 <Y_2 \leqslant s) + \mathbb {P}(0 < Y_1 \leqslant s, Y_2 = 0) + \mathbb {P}( Y_1 + Y_2 \leqslant s , 0< Y_1 \leqslant s, 0< Y_2 \leqslant s) \nonumber \\ & = F_{X_1}(d_1) F_{X_2}(s+d_2) + F_{X_2}(d_2) F_{X_1}(s+d_1) + \int _0^s F_{X_1}( s- u + d_1) f_{X_2}(u+ d_2) du. \end{aligned}$$

  (6.5)

  Let $F_{{T_{2,k}}}(s) : = \int _0^s F_{X_1}( s- u + d_1) f_{X_2}(u+ d_2) du$, by Lemma 6.1 this can be written as

  $$\begin{aligned} F_{{T_{2,k}}}(s) = \sum _ {k=0}^{\infty } \sum _ {j=0}^{\infty } \Delta _k(d_1,\beta _1,\underline{Q}_1) \Delta _j(d_2,\beta _2,\underline{Q}_2) \int _0^s W_{k+1}(s- u,\beta ) w_{j+1}(u,\beta ) du. \end{aligned}$$

  (6.6)

It can be seen that $\int _0^s W_{k+1}(s- u,\beta ) w_{j+1}(u,\beta ) du$ is a convolution of two independent Erlang risks with a common scale parameter $\beta$, which is again an Erlang risk with shape parameter $k+j+2$ and scale parameter $\beta$. Thus combining (6.4), (6.5) and (6.6) the claim follows easily. $\square$

Lemma 6.4

Let $X \sim ME(\beta , \underline{V})$ with pdf $f(x,\beta ,\underline{V})$, if g is some positive function such that $\mathbb {E}\left\{ g(X)\right\} <\infty$, then $c(x,\beta ,\underline{V})=\frac{g(x)f(x,\beta ,\underline{V})}{\mathbb {E}\left\{ g(X)\right\} }$ is again a pdf of mixed Erlang distribution with scale parameter $Z(\beta )$ and mixing weights $\underline{\Theta }(\underline{V})=(\theta _1,\theta _2,\ldots )$, with

$$\begin{aligned} c(x,\beta ,\underline{V})=\sum _{k=1}^\infty \theta _k w_k(x,Z(\beta )), \end{aligned}$$

where

• $Z(\beta )=2 \beta$ and $\theta _k = \frac{1}{2^{k-1}} \sum _{j=1}^k \begin{pmatrix} k-1\\ j-1 \end{pmatrix} q_j \sum _{l=k-j+1}^\infty q_l ,$ for $g(x)=2\overline{F}(x)$,

• $Z(\beta )= \beta$ and

  $$\begin{aligned} \theta _k= \left\{ \begin{array}{lll} 0 &{} \text{ for } \,\, k \leqslant t , \\ \frac{q_{k-t}\frac{\Gamma (k)}{\Gamma (k-t)}}{\sum _{j=1}^\infty q_j\frac{\Gamma (j+t)}{\Gamma (j)} } &{}\text{ for } \,\, k > t, \end{array} \right. \end{aligned}$$

  for $g(x)=x^t$ with $t \in \mathbb {R}$,

• $Z(\beta )= \beta + t$ and $\theta _k = \frac{q_k \overline{\beta }^k}{\sum _{j=1}^\infty q_j\overline{\beta }^j }$ with $\overline{\beta } =\frac{\beta }{\beta + t} ,$ for $g(x)=e^{-tx}$ with $t \in \mathbb {N}$.

Proof

We have

$$\begin{aligned} c(x,\beta ,\underline{V})=\frac{g(x)f(x,\beta ,\underline{V})}{\mathbb {E}\left\{ g(X)\right\} } = \frac{1}{\mathbb {E}\left\{ g(X)\right\} } \sum _{k=1}^{\infty } q_k \frac{\beta ^k}{(k-1)!} g(x) x^{k-1} e^{-\beta x}. \end{aligned}$$

(6.7)

For $g(x)= x^t$ one can write (6.7) as follows

$$\begin{aligned} c(x,\beta ,\underline{V})= & {} \frac{1}{\mathbb {E}\left\{ X^t\right\} } \sum _{k=1}^{\infty } q_k \frac{\beta ^k}{(k-1)!} x^{t+k-1} e^{-\beta x} \\= & {} \sum _{k=1}^{\infty } \biggl (\frac{q_{k}\frac{\Gamma (k+t)}{\Gamma (k)}}{\sum _{j=1}^\infty q_j\frac{\Gamma (j+t)}{\Gamma (j)} }\biggl ) w_{k+t}(x,\beta )\\= & {} \sum _{s=t+1}^{\infty } \biggl (\frac{q_{s-t}\frac{\Gamma (s)}{\Gamma (s-t)}}{\sum _{j=1}^\infty q_j\frac{\Gamma (j+t)}{\Gamma (j)} }\biggl ) w_{s}(x,\beta )\\= & {} \sum _{s=1}^{\infty } \theta _s w_{s}(x,\beta ), \end{aligned}$$

with

$$\begin{aligned} \theta _s= & {} \left\{ \begin{array}{lll} 0 &{} \text{ for } \,\, s \leqslant t, \\ \frac{q_{s-t}\frac{\Gamma (s)}{\Gamma (s-t)}}{\sum _{j=1}^\infty q_j\frac{\Gamma (j+t)}{\Gamma (j)} } &{}\text{ for } \,\, s > t. \end{array} \right. \\ \end{aligned}$$

For $g(x)= e^{-tx}$, (6.7) can be expressed as follows (set $\overline{\beta } := \frac{\beta }{\beta +t}$)

$$\begin{aligned} c(x,\beta ,\underline{V})= & {} \frac{1}{\mathbb {E}\left\{ e^{-tX}\right\} } \sum _{k=1}^{\infty } q_k \frac{\beta ^k}{(k-1)!} x^{k-1} e^{-(\beta + t) x} \\= & {} \sum _{k=1}^{\infty } \Biggl ( \frac{q_k \overline{\beta }^k}{\sum _{j=1}^\infty q_j\overline{\beta }^j } \Biggr ) w_k(x,\beta +t) \\= & {} \sum _{k=1}^{\infty } \theta _k w_k(x,\beta +t). \end{aligned}$$

For $g(x)=2\overline{F}(x)$, see [3] for the proof. The results presented in the next two lemmas can be found in Section 2.2 of [28] and Section 7.2 of [14], respectively. $\square$

Lemma 6.5

If $X \sim ME(\beta _1,\underline{V})$, then for any positive constant $\beta _2 \ge \beta _1$ we have

$$\begin{aligned} X \sim ME(\beta _2, \ \underline{\Psi }(\underline{V})), \quad \underline{\Psi }(\underline{V})=(\psi _1, \psi _2, \ldots ), \end{aligned}$$

where

$$\begin{aligned} \psi _k=\sum _ {i=1}^{k} q_i \begin{pmatrix} k-1\\ i-1 \end{pmatrix} \left( \frac{\beta _1}{\beta _2}\right) ^i \left( 1-\frac{\beta _1}{\beta _2} \right) ^{k-i}, \quad k\ge 1. \end{aligned}$$

Lemma 6.6

Let $X_1,X_2$ be two independent rv such that $X_i\sim ME(\beta , \underline{Q}_i) ,i=1,2$, then $S_2=X_1+X_2\sim ME(\beta , \underline{\Pi }\{\underline{Q}_1, \underline{Q}_2\})$ with

$$\begin{aligned} \pi _l\{\underline{Q}_1, \underline{Q}_2\}= \left\{ \begin{array}{rll} 0 &{} \text{ for } \,\, l=1, \\ \sum _ {j=1}^{l-1} q_{1,j} \ q_{2,l-j}&{} \text{ for } \,\,l>1. \end{array} \right. \end{aligned}$$

Remark 6.7

According to Cossette et al. [4] (Remark 2.1), the results in Lemma 6.6 can be extended to $S_n= \sum _{i=1}^n X_i$, provided that $X_1,\ldots ,X_n$ are independent, $X_i\sim ME(\beta , \underline{Q}_i)$ for $i=1,\ldots ,n$. Specifically, $S_n \sim ME(\beta , \underline{\Pi }\{\underline{Q}_1,\ldots , \underline{Q}_n\})$ where the individual mixing probabilities can be evaluated iteratively as follows

$$\begin{aligned} \pi _l \{\underline{Q}_1,\ldots , \underline{Q}_{n+1}\}= \left\{ \begin{array}{rll} 0 &{} \text{ for } \,\, l=1,\ldots ,n, \\ \sum _ {j=n}^{l-1} \pi _j\{\underline{Q}_1,\ldots , \underline{Q}_{n}\} \ q_{n+1,l-j}&{} \text{ for } \,\, l=n+1,n+2,\ldots . \end{array} \right. \end{aligned}$$

Appendix 2: Pearson’s coefficient of a bivariate Sarmanov mixed Erlang risk

Pearson’s coefficient $\rho _{1,2}$ is one of the most commonly used dependence measures between two risks $X_1$ and $X_2$. In this regards, $X_1$ and $X_2$ are assumed to be linearly correlated. Following (2.1), we show next that when $(X_1,X_2) \sim SME(\varvec{\beta }, \underline{Q})$, the closed expressions for $\rho _{1,2}$ depend on the dependence parameter $\alpha _{1,2}$ and the choice of kernel functions as follows:

• for $\phi _i(x_i)=x_i^t - \mathbb {E}\left\{ X_i^t \right\} , t>0$

  $$\begin{aligned} \rho _{1,2}(X_1,X_2)=\frac{\alpha _{1,2} (m_{1} ^{t+1} - m_{1} ^{t} m_{1} ^{1}) (m_{2} ^{t+1} - m_{2} ^{t} m_{2} ^{1})}{\sigma _1 \sigma _2}, \end{aligned}$$

  where $\sigma _i$ is the standard deviation of $X_i$ and

  $$\begin{aligned} m_i^{s} = \mathbb {E}\left\{ X_i^s \right\} = \frac{1}{\beta _i^s}\sum _{j=1}^\infty q_{i,j} \frac{(s+j-1)!}{(j-1)!} , \quad i=1,2, \ \text {with } s>0, \end{aligned}$$

  in particular for $t=1$

  $$\begin{aligned} \rho _{1,2}(X_1,X_2)= \alpha _{1,2} \sigma _1 \sigma _2 , \end{aligned}$$

• for $\phi _i(x_i)=e^{-t x_i}- \mathbb {E}\left\{ e^{-t X_i} \right\} , t>0$

  $$\begin{aligned} \rho _{1,2}(X_1,X_2)= \frac{\alpha _{1,2} ( \eta _{1,t}-\Gamma _{1,t} ) ( \eta _{2,t}-\Gamma _{2,t} )}{\sigma _1 \sigma _2}, \end{aligned}$$

  where $\eta _{i,t}=\frac{1}{\beta _i+t}\sum _ {j=1}^{\infty }j q_{i,j} \biggl ( \frac{ \beta _i }{\beta _i+t} \biggr ) ^j$ and $\Gamma _{i,t} = m^1_i \sum _ {j=1}^{\infty } q_{i,j} \biggl ( \frac{ \beta _i }{\beta _i+t} \biggr ) ^j , i=1,2$,

• for $\phi _i(x_i)= 2\overline{F}_i(x_i)-1$, see [3]

  $$\begin{aligned} \rho _{1,2}(X_1,X_2)= \frac{\alpha _{1,2}}{\beta _1 \beta _2 \sigma _1 \sigma _2 } \sum _{j=1}^\infty \sum _{k=1}^\infty jk (\theta _{1,j} - q_{1,j} ) (\theta _{2,k} - q_{2,k} ), \end{aligned}$$

  where $\theta _{1,j}$ and $\theta _{2,k}$ are defined in (3.1).