2. Preliminaries

Let $Y_1, Y_2, \cdots , Y_N $ be a sequence of i.i.d. random claim sizes which have a common cumulative distribution function $F({\cdot})$ and density function $f({\cdot}).$ Moreover, suppose N is a random number of claims which takes values on $\{1, 2,\cdots\}.$ Now suppose that $Y_{(1)}, Y_{(2)} , \cdots ,Y_{(N)} $ stands for the order statistics, where, in particular, the smallest and the largest claim sizes are defined by $Y_{(1)} =min(Y_1, Y_2, \cdots ,Y_N) $ and $ Y_{(N)}=\max(Y_1, Y_2, \cdots ,Y_N),$ respectively.

The $k{\rm th}$ moment of the $m{\rm th}$ largest claim sizes and the expectation of the cross product of the $ m{\rm th} $ and $ z{\rm th}$ largest claim sizes $(0<m<z),$ respectively, are given by

(1) \begin{equation} E\!\left(\left(Y_{(N-m+1)}\right)^{k}\right) = \frac{1}{\Gamma(m)} \int_{0}^{1} F^{-1}(v)^{k} \left[ 1-v \right] ^{m-1} \psi_{N} ^{(m)} (v) dv\end{equation}

(2) \begin{align} E\!\left(Y_{(N-m+1)}Y_{(N-z+1)}\right) & = \frac{1}{\Gamma(m)\Gamma(z-m)} \int_{0}^{1} F^{-1}(v) (1-v)^{z-1} \psi_{N} ^{(z)} (v)\\ & \quad \times \int_{0}^{1} F^{-1}(1-u(1-v))u^{m-1} (1-u)^{z-m-1} dv du\nonumber \end{align}

(3) \begin{equation} E\!\left(Y_{1}Y_{(N-m+1)}\right) = \sum_{n\geq m} P(N=n) \sum_{h=1}^{3}T_{h}(n;\,m),\end{equation}

where

\begin{eqnarray*}T_{1}(n;\,m)&=&\frac{(n-1)!}{(n-m-1)! (m-1)!}\int_{0}^{1} F^{-1}(v) v^{n-m-1} (1-v)^{m-1}H(F^{-1}(v))dv,\\[4pt]T_{2}(n;\,m)&=& \frac{(n-1)!}{(n-m)! (m-2)!} \int_{0}^{1} F^{-1}(v)v^{n-m} (1-v)^{m-2}(\mu -H(F^{-1}(v)))dv,\\[4pt]T_{3}(n;\,m)&=&\frac{(n-1)!}{(n-m)! (m-1)!} \int_{0}^{1} F^{-1}(v)v^{n-m} (1-v)^{m-1}dv,\end{eqnarray*}

$ v=F(Y_{(z)}),$ $u=F(Y_{(m)}),$ $\psi_{N}^{(m)}(v) $ stands for the $m{\rm th}$ derivative of $\psi_N(v)= \sum_{n=0}^{\infty} P(N =n)v^{n},$ with respect to v, and $H(z)=\int_{0}^{z} tf(t)dt, \forall z\geq0,$ which satisfies $\lim_{z\rightarrow\infty}H(z)=E(Y)=\mu.$

It should be noted that in the case of two-order statistics belonging to two different data sets, with unequal lengths, one cannot employ Equation (3). This situation may be studied for a specific case as follows.

Remark 1. Suppose that O and N are two independent random counts and write $M=O+N.$ Then, the expectation of the cross product of two $m{\rm th}$ and $z{\rm th}$ order claim sizes can be calculated as follows:

\begin{align*}E\!\left(Y_{(M-m+1)} Y_{(O-z+1)}\right) & = \sum_{n=0}^{\infty} E\!\left(Y_{(O+n-m+1)} Y_{(O-z+1)|N=n}\right)P(N=n)\nonumber\\& = \sum_{n=0}^{\infty}E\!\left(Y_{(O+n-m+1)}Y_{(O-z+1)}\right)P(N=n), \end{align*}

where $E\!\left(Y_{(O+n-m+1)}Y_{(O-z+1)}\right)$ can be calculated using the Equation (2).

For more information on order statistics, we refer interested readers to Kremer (Reference Kremer1982), Berglund (Reference Berglund1998), and David & Nagaraja (Reference David and Nagaraja2003).

In general insurance, insurance companies seek appropriate (in some sense) reinsurance protection to reduce and homogenise their risks. A reinsurance treaty is a form of an insurance contract, in which the reinsurer accepts to pay a portion of an insurer’s risk by receiving a reinsurance premium (Payandeh & Panahi, Reference Payandeh Najafabadi and Panahi Bazaz2018). Besides regulatory obligations, there are several reasons which motivate an insurer to buy a reinsurance contract, see Albrecher et al. (Reference Albrecher, Beirlant and Teugels2017), among others, for more details. In the reinsurance literature, the insurer is known as the first line insurer or ceding company. Suppose random claim Z can be decomposed as a sum of an insurance portion, $Z^{In}$ , and a reinsurance portion, $Z^{Re}$ , i.e., $Z=Z^{In}+ Z^{Re},$ where $0 \leq Z^{In}$ and $Z^{Re}\leq Z.$

The largest claims reinsurance, say LCR, and the ECOMOR treaties are two reinsurance contracts that just cover some of the largest claims. Therefore, there are appropriate treaties for a LoB that may have the potential to receive some considerable large claims.

The following recall definition of an LCR treaty, and we refer interested readers to Ladoucette & Teugels (Reference Ladoucette and Teugels2006), Jiang & Tang (Reference Jiang and Tang2008), and Fan et al. (Reference Fan, Griffin, Szimayer and Wang2017), among others.

Definition 1. Let $Y_1, Y_2,\cdots ,Y_N $ be a sequence of independent and identical random claim sizes that have a common cumulative distribution function $F({\cdot})$ and a density function $f({\cdot}).$ Moreover, suppose that $Y_{(1)} , Y_{(2)} , \cdots ,Y_{(N)} $ stand for their corresponding order statistics. An LCR treaty that covers the r largest claims is a reinsurance treaty without any priority that its insurer’s portion and cedent’s portion from random claims, respectively, are

\begin{align*} X^{Re}({{\tiny\rm LCR(r)}})= \sum_{m=1}^{r} Y_{(N -m+1)}\,\,and\,\,X^{In}({{\tiny\rm LCR(r)}})= \sum_{m=1}^{N-r}Y_{(m)}^{*}. \end{align*}

The ECOMOR treaty was introduced by Thepaut (Reference Thepaut1950), who extended the regular excess of loss treaty by letting: (1) its retention level be random and (2) just covering only that part of the r largest claims, see Kremer (Reference Kremer1982) and Ladoucette & Teugels (Reference Ladoucette and Teugels2006), among others, for more details.

The following provides the general concept of the ECOMOR treaty.

Definition 2. Let $Y_1, Y_2,\cdots ,Y_N $ be a sequence of independent and identical random claim sizes that have a common cumulative distribution function $F({\cdot})$ and a density function $f({\cdot}).$ Moreover, suppose that $Y_{(1)} , Y_{(2)} , \cdots ,Y_{(N)} $ stand for their corresponding order statistics. The ECOMOR treaty covers only that part of the r largest claims that exceed the random retention $Y_{(N-r)}.$ Therefore, its insurer’s portion and cedent’s portion from random claims, respectively, are

\begin{eqnarray*} X^{Re}({{\tiny\rm ECOMOR(r)}}) = \sum_{m=1}^{r} Y_{(N-m+1)} - r Y_{(N-r+1)} \,\, and\,\,X^{In} ({{\tiny\rm ECOMOR(r)}}) =\sum_{m=1}^{N-r} Y_{(m)} + r Y_{(N -r+1)}. \end{eqnarray*}

Consider an IBNR table that contains both observed developed claims (appeared in the upper triangle/trapezoid) and unobserved developed claims, say outstanding claims, (appeared in the lower triangle, say runoff triangle).

Table 1.

Standard Notations for an IBNR table, from Hindley (Reference Hindley2017).

To make clear, Table 1, from Hindley (Reference Hindley2017), represents all notations that will be used hereafter. Using notations represented in the Table 1, one may observe that

\begin{eqnarray*}N_{i,j}^{paid}= \sum_{l=0}^{\min(j,d)} N_{i,j-l,l}^{paid}; \,\,\,\,X_{i,j}=\sum_{k=1}^{N_{i,j}^{paid}} Y_{i,j}^{(k)}\,\, {\rm and}\,\,C_{i,j}=\sum_{k=1}^{j} X_{i,k}. \end{eqnarray*}

3. Main Results

This section employs a model introduced by Verrall et al. (Reference Verrall, Nielsen and Jessen2010) and Martinez-Miranda et al. (Reference Martinez-Miranda, Nielsen and Verrall2012, Reference Martinez-Miranda, Nielsen, Verrall and Wüthrich2015) to predict the net loss reserve under the LCR and the ECOMOR treaties.

To begin, the LCR and the ECOMOR treaties are carefully recalled given by Definitions 1 and 2. Both the LCR and the ECOMOR treaties for a LoB that some of its claims do not settle immediately after its occurrence and develop j years later have to be formulated with concern. To derive the cedent’s risk portion under these treaties for a loss developing environment, consider individual random claims $Y_{i,j}^{(k)},$ for $j=0,1,\cdots, I-1,$ and $k=1,2,\cdots, N_{i,j}^{paid}.$ Moreover, suppose that $Y_{i(1)} ,Y_{i(2)} , \cdots , Y_{i\left(N_{i,j}^{paid}\right)} $ stands for the order statistics for $Y_{i,j}^{(k)},$ then

1. (1) the cedent’s risk portion under an LCR treaty which covers the r largest claims is

   (4) \begin{eqnarray}X_{i,0}^{In}({{\tiny\rm LCR(r)}})&=&\sum_{k=1}^{N_{i,0}^{paid}} Y_{i,0}^{(k)} - \sum_{m=1}^{r}Y_{i(\xi_{i,0} -m+1)} \nonumber \\X_{i,1}^{In}({{\tiny\rm LCR(r)}})&=&\sum_{k=1}^{N_{i,1}^{paid}} Y_{i,1}^{(k)} -\left[ \sum_{m=1}^{r}Y_{i(\xi_{i,1} -m+1)} -\sum_{m=1}^{r} Y_{i( \xi_{i,0} -m+1)}\right] \\ &\vdots& \nonumber\\ X_{i,J}^{In}({{\tiny\rm LCR(r)}})&=&\sum_{k=1}^{N_{i,J}^{paid}} Y_{i,J}^{(k)} -\left[ \sum_{m=1}^{r}Y_{i(\xi_{i,J} -m+1)} -\sum_{m=1}^{r} Y_{i(\xi_{i,J-1} -m+1)}\right] \nonumber \end{eqnarray}

2. (2) the cedent’s risk portion under an ECOMOR treaty which covers the r largest claims is

   (5) \begin{align}X_{i,0}^{In}({{\tiny\rm ECOMOR(r)}})& = \sum_{k=1}^{N_{i,0}^{paid}}Y_{i,j}^{(k)}-\left(\sum_{m=1}^{r}Y_{i(\xi_{i,0}-m+1)} - rY_{i(\xi_{i,0}-r+1)}\right) \nonumber\\X_{i,1}^{In}({{\tiny\rm ECOMOR(r)}}) & = \sum_{k=1}^{N_{i,1}^{paid}}Y_{i,j}^{(k)}-\left(\sum_{m=1}^{r}\left(Y_{i(\xi_{i,1}-m+1)} -Y_{i( \xi_{i,0}-m+1)}\right)\right.\nonumber\\& \qquad\qquad\qquad -r\!\left(Y_{i(\xi_{i,1}-r+1)}-Y_{i( \xi_{i,0}-r+1)}\right)\Bigg)\\ &\vdots \nonumber\\ X_{i,J}^{In}({{\tiny\rm ECOMOR(r)}}) & = \sum_{k=1}^{N_{i,J}^{paid}}Y_{i,j}^{(k)}-\left(\sum_{m=1}^{r}\!\left(Y_{i(\xi_{i,J}-m+1)} -Y_{i( \xi_{i,J-1}-m+1)}\right)\right.\nonumber\\& \qquad\qquad\qquad -r\!\left(Y_{i(\xi_{i,J}-r+1)}-Y_{i( \xi_{i,J-1}-r+1)}\right)\Bigg)\nonumber\end{align}

where $\xi_{i,j}=\sum_{h=0}^{j}N_{i,h}^{paid}$ and $\xi_{i,0}=N_{i,0}.$

Hereafter, the discussion is based on the following model assumption.

Under distributional assumptions, given by Model Assumption 1, one may observe that $E\!\left(N_{i,j}^{paid}|\mathcal{D}_{I}\right)=\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}$ and $Var\!\left(N_{i,j}^{paid}|\mathcal{D}_{I}\right)=\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} \left(1-p_{l}^{*}\right).$ Therefore:

1. (1) The conditional expectation $E\!\left(X_{i,j}|\mathcal{D}_{I}\right)$ is

   (6) \begin{eqnarray} \nonumber E\!\left(X_{i,j}|\mathcal{D}_{I}\right)&=&E\!\left(\sum_{k=1}^{N_{i,j}^{paid}}Y_{i,j}^{(k)}\bigg| \mathcal{D}_{I}\right) = E\!\left(E\!\left(\sum_{k=1}^{N_{i,j}^{paid}}Y_{i,j}^{(k)}\big|N_{i,j}^{paid}\right)\bigg|\mathcal{D}_{I}\right) = E\!\left( N_{i,j}^{paid}\big|\mathcal{D}_{I}\right) E\!\left(Y_{i,j}^{(k)}\right)\\ &=&\gamma_{i} \mu\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report} p_{l}^{*}.\end{eqnarray}

2. (2) The conditional variance $Var\!\left(X_{i,j}|\mathcal{D}_{I}\right)$ is

   (7) \begin{eqnarray}Var\!\left(X_{i,j}|\mathcal{D}_{I}\right) &=&E\!\left(Var\!\left(X_{i,j}\big|N_{i,j}^{paid}\right)\bigg|\mathcal{D}_{I}\right)+Var\!\left(E\!\left(X_{i,j}\big|N_{i,j}^{paid}\right)\bigg|\mathcal{D}_{I}\right) \nonumber\\ \nonumber &=&E\!\left(Var\!\left(\sum_{k=1}^{N_{i,j}^{paid}}Y_{i,j}^{(k)}\big|N_{i,j}^{paid}\right)\bigg|\mathcal{D}_{I}\right)+Var\!\left(E\!\left(\sum_{k=1}^{N_{i,j}^{paid}}Y_{i,j}^{(k)}\big|N_{i,j}^{paid}\right)\bigg|\mathcal{D}_{I}\right) \\ &=&E\!\left(N_{i,j}^{paid}\big|\mathcal{D}_{I}\right)Var\!\left(Y_{i,j}^{(1)}\right)+Var\!\left(N_{i,j}^{paid}\big|\mathcal{D}_{I}\right) \left[ E\!\left(Y_{i,j}^{(1)}\right)\right] ^{2}\\ \nonumber &=&\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report} p_{l}^{*}\gamma_{i}^{2} \sigma^{2} + \sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} \left(1-p_{l}^{*}\right) \gamma_{i}^2 \mu^{2}\\ \nonumber &=&\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\gamma_{i}^2\mu^{2}\left[\eta^2+\left(1-p_{l}^{*}\right) \right],\end{eqnarray}

where $\eta$ stands for coefficient of variation for random variable $Y_{i,j}^{(k)}/\gamma_{i}.$

Under an LCR reinsurance treaty and Model Assumption 1, the following theorem develops the best prediction for the cedent’s (and reinsurer’s) portion for random claim $ X_{i,j}.$

Proof. For the first section of Part (1) observe that $\left.E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\right)=E\!\left(X_{i,j}|\mathcal{D}_{I}\right)-E\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right).$ The first expectation has been given by Equation (6). The second expectation may be restated as

\begin{eqnarray*}E\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right) &=&E\!\left(\sum_{m=1}^{r} Y_{i(\xi_{i,j} -m+1)}|\mathcal{D}_{I}\right)-E\!\left(\sum_{m=1}^{r} Y_{i(\xi_{i,j-1}-m+1)}\right)|\mathcal{D}_{I}\Bigg)\\ &=&\sum_{m=1}^{r}\frac{1}{\Gamma(m)}\int_{0}^{1} F^{-1} (v) \left[ 1-v \right] ^{m-1}\psi_{\xi_{i,j}}^{(m)} (v) dv\\&& \quad - \sum_{m=1}^{r} \frac{1}{\Gamma(m)}\int_{0}^{1} F^{-1} (v) \left[ 1-v \right] ^{m-1}\psi_{\xi_{i,j-1}} ^{(m)} (v) dv, \end{eqnarray*}

where the second equation arrives from the application of Equation (1).

For the second section of Part (1), one may decompose the conditional MSEP as

\begin{eqnarray*}MSEP_{\mathcal{D}_{I}}\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})\right)&=&Var\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\\&& +E\!\left[\left(\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})-E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\right)^{2}|\mathcal{D}_{I}\right]\\&=&Var\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)+\left(\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})\right.\\&& \qquad\qquad\qquad\qquad\qquad \left.-E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\right)^{2}\\&=&Var\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right) \end{eqnarray*}

where the second expression obtained from the fact that both $\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})$ and $E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)$ are $\mathcal{D}_{I}$ -measurable and the third expression obtained from $\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})=E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right).$

Now, observe that

\begin{eqnarray*}Var\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=&Var\!\left(X_{i,j}|\mathcal{D}_{I}\right)+Var\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)-2Cov\!\left(X_{i,j},X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right).\end{eqnarray*}

The conditional variance $Var\!\left(X_{i,j}|\mathcal{D}_{I}\right)$ has been given by Equation (7), and other terms are given by Lemma 1, in the Appendix.

Proof of Part (2) is similar.

The following theorem develops the best prediction for the cedent’s (and reinsurer’s) portion for random claim $ X_{i,j},$ under an ECOMOR reinsurance treaty and Model Assumption 1.

Proof. For the first section of Part (1) observe that $E\!\left.\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)\right)=E\!\left(X_{i,j}|\mathcal{D}_{I}\right)-E\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right).$ The first expectation has been given by Equation (6). A similar argument as provided in proof of Part (1) (Theorem, 1) leads to

\begin{align*}E\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)& = E\!\left(\sum_{m=1}^{r} Y_{i(\xi_{i,j}-m+1)} -rY_{i\left(\xi_{i,j}-r+1\right)}\right.\\& \quad \left. -\sum_{m=1}^{r}Y_{i(\xi_{i,j-1}-m+1)} +rY_{i\!\left(\xi_{i,j-1}-r+1\right)}|\mathcal{D}_{I}\right)\\ & = \sum_{m=1}^{r} E\!\left(Y_{i(\xi_{i,j}-m+1)}-rY_{i\!\left(\xi_{i,j}-r+1\right)}\right)\\& \quad -\sum_{m=1}^{r} E\!\left(Y_{i(\xi_{i,j-1}-m+1)}-rY_{i\left(\xi_{i,j-1}-r+1\right)}\right)\end{align*}

\begin{align*} & = \sum_{m=1}^{r}\frac{1}{\Gamma(m)}\int_{0}^{1} F^{-1} (v) \left[ 1-v \right] ^{m-1}\psi_{\xi_{i,j}}^{(m)} (v) dv\\& \quad -\frac{r}{\Gamma(r)} \int_{0}^{1}F^{-1} (v) \left[ 1-v \right] ^{r-1} \psi_{\xi_{i,j}}^{(r)} (v)dv\\ & \quad - \sum_{m=1}^{r} \frac{1}{\Gamma(m)} \int_{0}^{1} F^{-1}(v) \left[ 1-v \right] ^{m-1} \psi_{\xi_{i,j-1}} ^{(m)} (v)dv\\& \quad +\frac{r}{\Gamma(r)} \int_{0}^{1} F^{-1} (v) \left[ 1-v \right]^{r-1} \psi_{\xi_{i,j-1}} ^{(r)} (v) dv \end{align*}

For Part (2), similar proof of Part (2), in Theorem 2, the conditional MSEP can be restated as

\begin{align*}MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})\right)& = Var\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)\\& \quad +\left(\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})\right.\\& \quad \left.-E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)\right)^{2}\\& = Var\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right) \end{align*}

where the second expression obtained from the fact that both $\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})$ and $E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)$ are $\mathcal{D}_{I}$ -measurable and the third expression obtained from $\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})=E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right).$ Now, observe that

\begin{align*}Var\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)& = Var\!\left(X_{i,j}|\mathcal{D}_{I}\right)+Var\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)\\& \quad -2Cov\!\left(X_{i,j},X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right).\end{align*}

The conditional variance $Var\!\left(X_{i,j}|\mathcal{D}_{I}\right)$ has been given by Equation (7), and other terms are given by Lemma 2, in the Appendix.

4. Simulation Study

This section develops a simulation study to (1) show practical application of the findings and (2) make a comparison between two reinsurance treaties.

Several authors developed some simulation algorithms to simulate a full IBNR table, see Stanard (Reference Stanard1985), Bühlmann et al. (Reference Bühlmann, Straub and Schnieper1980), Vaughan (Reference Vaughan1998), Narayan & Warthen (Reference Narayan and Warthen2000), Schiegl (Reference Schiegl2002), Stelljes (Reference Stelljes2006), among others, for more details. For the simulation study, Schiegl’s (Reference Schiegl2002) simulation algorithm was adjusted, and the following numerical procedure was employed.

Algorithm 1 shows how $N_{i,j}^{report},$ $Y_{i,j}^{(k)}$ and consequently $X_{i,j}$ are simulated regardless of the cedent and reinsurer portions, in an IBNR table.

The Cedent’s portion for outstanding claims under the LCR(r) treaty has been given by Algorithm 2. The Cedent’s portion for outstanding claims under the ECOMOR(r) treaty has been given by Algorithm 2.

Before providing some examples, the following definition is recalled.

It is known that the regular gamma function can be concluded by $\Gamma(a)=\Gamma(a, \infty)$ , the gamma function. Moreover,

(8) \begin{eqnarray}\sum_{m=1}^{r}\frac{\Gamma(m+b,z)}{\Gamma(m)}&=&\frac{r\Gamma(b+r+1,z)}{\Gamma(r+1)}. \end{eqnarray}

The (Type I) Pareto distribution has a considerable application in a wide range of sciences, including social, actuarial, and financial sciences. The Pareto distribution is characterised by its scale parameter $\tau$ and tail index $\theta.$

In the following, the Numerical Procedure 1 is employed against the Pareto distribution.

Example 1. Suppose for all $i=1,2,\cdots,I,$ $j=0,1,\cdots,I-1$ and $k=1,2,\cdots,N_{i,j}^{paid},$ the individual discounted payments, $Y_{i,j}^{(k)}/\gamma_{i}$ , are mutually independent with common Pareto distribution (with parameters $ \theta$ and $\tau$ ), where $\gamma_{i}$ stands for an inflation index in accident year i. In other words, $P\!\left(T_{i,j}\leq t_{ij}\right)=1-(t_{ij}/\tau)^{-\theta},$ for $t_{ij}>\tau,$ $E\!\left(Y_{i,j}^{(k)}\right)=\gamma_{i}\theta\tau/(\theta-1)$ (for $\theta>1$ ) and $Var\!\left(Y_{i,j}^{(k)}\right)=\theta\tau^{2}\gamma_{i}^{2}/\left((\theta-1)^{2} (\theta -2)\right)$ (for $\theta>2$ ), where $T_{i,j}=Y_{i,j}^{(k)}/\gamma_{i}.$

Under this distributional assumption, given by this example, the results of the Theorems 1 and (2) may be simplified as follows.

\begin{align*} E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=\frac{\theta \tau\gamma_{i} }{\theta -1} \sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*} -\left[\sum_{m=1}^{r} M_{1}\left(\phi_{i,j},m\right) -\sum_{m=1}^{r}M_{1}\left(\phi_{i,j-1},m\right) \right]\\ E\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r} M_{1}\left(\phi_{i,j-1},m\right)\\MSEP_{\mathcal{D}_{I}}\!\left(X_{i,j}^{In} ({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})\right)&=\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}\!\left(\frac{\theta \tau \gamma_{i} }{\theta-1}\right)^{2} \left[ \frac{1}{\theta (\theta -2)}+1-p_{l}^{*} \right] +\sigma_{{\tiny\rm LCR(r)}}^{2}\\& \quad -2\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\sum_{m=1}^{r} \sum_{k=m+1}^{N_{i}} \left[P\!\left(\xi_{i,j}=k\right) -P\!\left(\xi_{i,j-1}=k\right)\right] \sum_{h=1}^{3}T_{h}(m;\,k)\\ & \quad +2\frac{\theta \tau \gamma_{i} }{\theta -1}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}\left[\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r} M_{1}\left(\phi_{i,j-1},m\right) \right]\end{align*}

\begin{align*}MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{Re} ({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{Re} ({{\tiny\rm LCR(r)}})\right)&= \sum_{m=1}^{r}M_{2}\left(\phi_{i,j},m\right) -\left(\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)\right)^{2}+\sum_{m=1}^{r} M_{2}\left(\phi_{i,j-1},m\right)\\& \quad -\left(\sum_{m=1}^{r}M_{1}\left(\phi_{i,j-1},m\right)\right)^{2}-2r\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}P\!\left(\xi_{i,j}=k\right)\sum_{h=1}^{3}T_{h}(m;\,k)\\ & \quad +2\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)\sum_{m=1}^{r}M_{1}\left(\phi_{i,j-1},m\right)\,=\!:\,\sigma_{{\tiny\rm LCR(r)}}^{2}\end{align*}

and

\begin{align*}E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)& =\frac{\theta\tau \gamma_{i} }{\theta -1} \sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} -\left[\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r} U_{1}\left(\phi_{i,j},m\right) \right] \\& \quad +\left[\sum_{m=1}^{r} M_{1}\left(\phi_{i,j-1},m\right)-\sum_{m=1}^{r}U_{1}\left(\phi_{i,j-1},m\right) \right]\end{align*}

\begin{align*}E\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)& =\left[\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r} U_{1}\left(\phi_{i,j},m\right) \right]\\& \quad -\left[\sum_{m=1}^{r} M_{1}\left(\phi_{i,j-1},m\right)-\sum_{m=1}^{r}U_{1}\left(\phi_{i,j-1},m\right) \right]\\MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})\right)& =\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}\left(\frac{\theta \tau \gamma_{i} }{\theta-1}\right)^{2} \left[ \frac{1}{\theta (\theta -2)}+1-p_{l}^{*} \right]\\& \quad +\sigma_{{{\tiny\rm ECOMOR(r)}}}^{2}\\&\quad -2\sum_{l=0}^{\min(j,d)}\! N_{i,j-l}^{report}p_{l}^{*}\sum_{m=1}^{r} \sum_{k_{1}=m+1}^{N_{i}}\!P(\xi_{i,j}=k_{1})\!\sum_{h=1}^{3}T_{h}(k_{1};\,m)\\&\quad +2\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\sum_{k_{2}=r+1}^{N_{i}}P(\xi_{i,j-1}=k_{2})\sum_{h=1}^{3}T_{h}(k_{2};\,r)\\&\quad +2\frac{\theta \tau \gamma_{i} }{\theta -1}\left[\sum_{m=1}^{r}M_{1}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r} U_{1}\left(\phi_{i,j},m\right) \right] \\&\quad +2\frac{\theta \tau \gamma_{i} }{\theta -1}\!\left[\sum_{m=1}^{r}M_{1}\left(\phi_{i,j-1},m\right)-\!\sum_{m=1}^{r} U_{1}\left(\phi_{i,j-1},m\right)\right],\\ MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})\right)& =\sigma_{{{\tiny\rm LCR(r)}}}^{2}+r^{2}L_{2}\left(\phi_{i,j},r\right)-r^{2}\left(L_{1}\left(\phi_{i,j},r\right)\right)^{2}\\&\quad +r^{2}L_{2}\left(\phi_{i,j-1},r\right)-r^{2}\left(L_{1}\left(\phi_{i,j-1},r\right)\right)^{2}\\&\quad +2r^{2}L_{1}\left(\phi_{i,j},r\right)L_{1}\left(\phi_{i,j-1},r\right)-2rL_{2}\left(\phi_{i,j},r\right)\\& \qquad -2rL_{2}\left(\phi_{i,j-1},r\right)\\&\quad +2r\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}P\!\left(\xi_{i,j}=k\right)\sum_{h=1}^{3} T_{h}(k;\,m)\\ &\quad +2rE\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)L_{1}\left(\phi_{i,j},r\right)\\& \qquad -2rE\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)L_{1}\left(\phi_{i,j-1},r\right)\\&\,=\!:\, \sigma_{{{\tiny\rm ECOMOR(r)}}}^{2}\end{align*}

where $\phi_{i,j}=E(\xi_{i,j})$ and

\begin{eqnarray*} \sum_{m=1}^{r}M_{\nu}(a,m)&=&(\tau \gamma_{i})^{\nu} (a)^{\frac{\nu}{\theta}}\frac{r\Gamma(r-\frac{\nu}{\theta}+1,a)}{\Gamma(r+1)}\\\sum_{m=1}^{r} U_{\nu}(a,m)&=&(\tau \gamma_{i})^{\nu}(a)^{\frac{\nu}{\theta}}\frac{r\Gamma(r-\frac{\nu}{\theta},a)}{\Gamma(r)}\\L_{\nu}(a,b)&=&\frac{a^{b}}{\Gamma(b)}\int_{y} y^{\nu}f(y)(1-F(y))^{b-1} e^{-a(1-F(y))} dy.\end{eqnarray*}

Now, the Numerical Procedure 1 is employed with parameters given by Table 2, to (1) predict outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ ECOMOR(r)}})$ ) and (2) compare these two reinsurance treaties. Tables 3 and 4, respectively, present the prediction of outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ ECOMOR(r)}})$ ) and the MSEP for the LCR(r) and the ECOMOR(r) treaties for different r.

As Tables 3 and 4 show (1) for a fixed shape parameter $\tau,$ the reserve and the MSEP increase as the scale parameter increases, (2) for a fixed scale parameter $\theta,$ the reserve and the MSEP increase as the shape parameter increases, (3) the cedent’s MSEP under the LCR(r) treaty is smaller than such amount under the ECOMOR(r), and (4) for both treaties, the amount of cedent’s MSEP increases as the number of claims covered by the reinsurer increases.

The Fréchet distribution is an appropriate distribution to model “heavy tail” (or “fat tail”) phenomena. The probability density function, for Fréchet distribution with the location parameter, $\mu^{*},$ the scale parameter, $\sigma^{*}$ and the shape parameter, $\alpha^{*}$ is

\begin{eqnarray*}f(s)=\frac{\alpha^{*}}{\sigma^{*}}\left(\frac{s-\mu^{*}}{\sigma^{*}}\right)^{-1-\alpha^{*}}e^{\left(\frac{s-\mu^{*}}{\sigma^{*}}\right)^{-\alpha^{*}}}\end{eqnarray*}

Mean and variance are

(9) \begin{eqnarray}\mu_{Frechet}&=&\mu^{*}+\sigma^{*}\Gamma\!\left(1-\frac{1}{\alpha^{*}}\right)\\ \nonumber \sigma_{Frechet}^{2}&=&\sigma^{*2}\left[\Gamma\left(1-\frac{2}{\alpha^{*}}\right)-\left(\Gamma\!\left(1-\frac{1}{\alpha^{*}}\right)\right)^{2}\right]\end{eqnarray}

Table 2.

Simulation parameters.

Table 3.

Mean of loss reserve net of LCR(r) treaty for Pareto distribution with parameters $\theta$ and $\tau$ .

Table 4.

Mean of loss reserve net of ECOMOR(r) treaty for Pareto distribution with parameters $\theta$ and $\tau$ .

Application of the Numerical Procedure 1 whenever individual discounted payments are sampled from Fréchet distribution.

Example 2. Suppose for all $i=1,2,\cdots,I,$ $j=0, 1, \cdots, I-1$ and $k=1, 2, \cdots,N_{i,j}^{paid},$ the individual discounted payments, $\frac{Y_{i,j}^{(k)}}{\gamma_{i}},$ are mutually independent with common Fréchet distribution (with parameter $\mu^{*},$ $\sigma^{*}$ and $\alpha^{*}$ ), where $\gamma_{i}$ stands for an inflation index in accident year i. In other words, $E\!\left(Y_{i,j}^{(k)}\right)=\gamma_{i}\mu_{Frechet}$ and $Var\!\left(Y_{i,j}^{(k)}\right)=\gamma_{i}^{2} \sigma_{Frechet}^{2}$ where $\mu_{Frechet}$ and $\sigma_{Frechet}^{2}$ are given by Equation (9).

Under the above distributional assumption, the results of Theorems 1 and 2 can be simplified as

\begin{eqnarray*}E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=&\gamma_{i}\mu_{Frechet}\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report} p_{l}^{*}-\sum_{m=1}^{r} H_{Q_{1}}\left(\phi_{i,j},m\right)\\[4pt]&& +\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\\[4pt]E\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=&\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right)-\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\\[4pt]MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})\right)&=&\sum_{l=}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} \gamma_{i}^{2}\mu_{Frechet}^{2}\left(\eta^{2}+\left(1-p_{l}^{*}\right)\right)+\sigma_{{{\tiny\rm LCR(r)}}}^{2}\\[4pt]&&-2\sum_{l=}^{\min(j,d)} N_{i,j-l}^{report} p_{l}^{*}\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}\left[P\!\left(\xi_{i,j}=k\right)\right.\\[4pt]&& \left. -P\!\left(\xi_{i,j-1}=k\right)\right] \sum_{h=1}^{3} T_{h}(k;\,m)\\[4pt]&&+2\!\left[ \gamma_{i} \mu_{Frechet}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}\right] \left[ \sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right)\right.\\[4pt]&& \qquad\left. -\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\right]\\[4pt] MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{Re}({{\tiny\rm LCR(r)}})\right)&=&\sum_{m=1}^{r}H_{Q_{2}}\left(\phi_{i,j},m\right)-\left(\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right)\right)^{2}\\[4pt] &&+\sum_{m=1}^{r}H_{Q_{2}}\left(\phi_{i,j-1},m\right)-\left(\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\right)^{2}\\[4pt] &&-2r\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}P\!\left(\xi_{i,j}=k\right)\sum_{h=1}^{3}T_{h}(k;\,m)\\[4pt] &&+2\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right)\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\,=\!:\,\sigma_{{\tiny\rm LCR(r)}}^{2}\end{eqnarray*}

and

\begin{eqnarray*}E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)& = &\gamma_{i} \mu_{Frechet}\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}- \sum_{m=1}^{r} H_{Q_{1}}\left(\phi_{i,j},m\right)\\[7pt]&&+\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)+rH_{Q_{1}}\left(\phi_{i,j},r\right)\\&& -rH_{Q_{1}}\left(\phi_{i,j-1},r\right)\\[7pt]E\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)&=&\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right) -\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\\&& -rH_{Q_{1}}\left(\phi_{i,j},r\right)+rH_{Q_{1}}\left(\phi_{i,j-1},r\right)\\[7pt]MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})\right)&=&\sum_{l=}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} \gamma_{i}^{2}\mu_{Frechet}^{2}\left(\eta^{2}+\left(1-p_{l}^{*}\right)\right)\\&& +\sigma_{{{\tiny\rm ECOMOR(r)}}}^{2}\\[7pt]&&-2\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\sum_{m=1}^{r}\sum_{k_{1}=m+1}^{N_{i}}P(\xi_{i,j}=k_{1})\\&& \times \sum_{h=1}^{3} T_{h}(k_{1};\,m)\\[7pt]&&+2\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\sum_{k_{2}=r+1}^{N_{i}}P(\xi_{i,j-1}=k_{2})\\&& \times \sum_{h=1}^{3}T_{h}(k_{2};\,r)\\[4pt]&&+2\gamma_{i} \mu_{Frechet}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*} \left[ \sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j},m\right)\right.\\&&\left. -\sum_{m=1}^{r}H_{Q_{1}}\left(\phi_{i,j-1},m\right)\right] \\[4pt] &&-2r\gamma_{i}\mu_{Frechet}\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report} p_{l}^{*}\left[ H_{Q_{1}}\left(\phi_{i,j},r\right)\right.\\&& \left. - H_{Q_{1}}\left(\phi_{i,j-1},r\right)\right]\\[4pt]MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})\right)&=&\sigma_{{{\tiny\rm LCR(r)}}}^{2}+r^{2}H_{Q_{2}}\left(\phi_{i,j},r\right)-r^{2}H_{Q_{1}}^{2}\left(\phi_{i,j},r\right)\\[4pt]&&+r^{2}H_{Q_{2}}\left(\phi_{i,j-1},r\right)-r^{2}H_{Q_{1}}^{2}\left(\phi_{i,j-1},r\right)\end{eqnarray*}

\begin{eqnarray*}&&+2r^{2}H_{Q_{1}}\left(\phi_{i,j},r\right)H_{Q_{1}}\left(\phi_{i,j-1},r\right)-2rH_{Q_{2}}\left(\phi_{i,j},r\right)-2rH_{Q_{2}}\left(\phi_{i,j-1},r\right)\\[4pt]&&+2r\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}P\!\left(\xi_{i,j}=k\right)\sum_{h=1}^{3}T_{h}(k;\,m)\\[4pt]&&+2rE\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\left(H_{Q_{1}}\left(\phi_{i,j},r\right)-H_{Q_{1}}\left(\phi_{i,j-1},r\right)\right)\,=\!:\,\sigma_{{\tiny\rm ECOMOR(r)}}^{2}\end{eqnarray*}

where $H_{Q_{\nu}}(a,b)=\frac{a^{b}}{(b-1)!}Q_{\nu}(a,b)$ and $Q_{\nu}(a,b)=\int_{y} y^{\nu} f(y)(1-F(y))^{m-1} e^{-a(1-F(y))}dy.$

Now, the Numerical Procedure 1 is employed to (1) predict outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ECOMOR(r)}})$ ) and (2) compare these two reinsurance treaties. Tables 5 and 6, respectively, present prediction of outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ECOMOR(r)}})$ ) and the MSEP for the LCR(r) and the ECOMOR(r) treaties for different r. As Tables 4 and 5 show (1) for a fixed shape parameter, the reserve and the MSEP increase as the scale parameter increases, (2) for a fixed scale parameter, the reserve and the MSEP increase as the shape parameter increases, (3) the cedent’s MSEP under the LCR(r) treaty is smaller than such amount under the ECOMOR(r), and (4) for both treaties, the amount of cedent’s MSEP increases as the number of claims covered by the reinsurer increases.

The Weibull distribution is a continuous probability distribution that is an excellent candidate whenever large claims in the portfolio are addressed. The probability density function for a Weibull distribution with the scale parameter, $\lambda^{*,}$ and the shape parameter, $\theta^{*},$ is

\begin{eqnarray*} f(s)&=&\frac{\theta^{*}}{\lambda^{*}}\left(\frac{s}{\lambda^{*}}\right)^{\theta^{*}-1}e^{-\left(\frac{s}{\lambda^{*}}\right)^{\theta^{*}}},\,\,\forall\,s\geq0.\end{eqnarray*}

Table 5.

Mean of loss reserve net of LCR(r) treaty for Frechet distribution with parameters $\mu{*}$ , $\sigma^{*}$ and $\alpha^{*}$ .

Table 6.

Mean of loss reserve net of ECOMOR(r) treaty for Frechet distribution with parameters $\mu{*}$ , $\sigma^{*}$ and $\alpha^{*}$ .

Moreover, mean and variance for such a Weibull distribution, respectively, are

(10) \begin{eqnarray}\mu_W&=&\lambda^{*}\Gamma\!\left(1+\frac{1}{\theta^{*}}\right)\\ \nonumber \sigma^2_W&=&\lambda^{*2}\left[\Gamma\!\left(1+\frac{2}{\theta^{*}}\right)-\Gamma^{2}\left(1+\frac{1}{\theta^{*}}\right)\right].\end{eqnarray}

In the following, the Numerical Procedure 1 is employed against the Weibull distribution.

Example 3. Suppose for all $i=1,2,\cdots,I,$ $j=0,1,\cdots,I-1$ and $k=1,2,\cdots,N_{i,j}^{paid},$ the individual discounted payments, $Y_{i,j}^{(k)}/\gamma_{i}$ , are mutually independent with common Weibull distribution (with parameter $\lambda^{*}$ and $\theta^{*}$ ), where $\gamma_{i}$ stands for an inflation index in accident year i.

Under the above distributional assumption,the results of Theorems 1 and 2 can be simplified as

\begin{eqnarray*}E\!\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=&\gamma_{i}\mu_{W}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}-\sum_{m=1}^{r}\left(G_{1}\left(\phi_{i,j},m\right)-G_{1}\left(\phi_{i,j-1},m\right)\right)\\E\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)&=&\sum_{m=1}^{r}\left(G_{1}\left(\phi_{i,j},m\right)-G_{1}\left(\phi_{i,j-1},m\right)\right)\\MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm LCR(r)}})\right)&=&\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}^{*}\gamma_{i}^{2}\mu_{W}^{2}\left(\eta^{2}+\left(1-p_{l}^{*}\right)\right)+\sigma_{{\tiny\rm LCR(r)}}^{2}\\&&-2\sum_{l=0}^{\min(j,d)} N_{i,j-l}^{report}p_{l}^{*}\sum_{m=1}^{r}\sum_{k=m+1}^{N_{i}}\left[(P\!\left(\xi_{i,j}=k\right)\right.\\&&\left. -P\!\left(\xi_{i,j-1}=k\right)\right] \sum_{h=1}^{3}T_{h}(k;\,m)\\&&+2\!\left[ \gamma_{i}\mu_{W}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report} p_{l}^{*}\right] \left[\sum_{m=1}^{r}(G_{1}\left(\phi_{i,j},m\right)\right.\\&&\left. -G_{1}\left(\phi_{i,j-1},m\right)\right) \bigg]\\MSEP_{\mathcal{D}_{I}} \left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}}),\hat{X}_{i,j}^{Re}({{\tiny\rm LCR(r)}})\right)&=&\sum_{m=1}^{r}G_{2}\left(\phi_{i,j},m\right)-\left(\sum_{m=1}^{r} G_{1}\left(\phi_{i,j},m\right)\right)^{2}\\&&+\sum_{m=1}^{r} G_{2}\left(\phi_{i,j-1},m\right)-\left(\sum_{m=1}^{r}G_{1}\left(\phi_{i,j-1},m\right)\right)^{2}\\&&-2r\sum_{m_{1}=1}^{r}\sum_{k=m+1}^{N_{i}}P\!\left(\xi_{i,j}=k\right)\sum_{h=1}^{3}T_{h}(k;\,m)\\&&+2\left[\sum_{m=1}^{r}(G_{1}\left(\phi_{i,j},m\right) \right] \left[\sum_{m=1}^{r}(G_{1}\left(\phi_{i,j-1},m\right)\right]\,=\!:\,\sigma_{{\tiny\rm LCR(r)}}^{2},\end{eqnarray*}

and

\begin{eqnarray*}E\!\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)&=&\gamma_{i}\mu_{W}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}^{*}-\sum_{m=1}^{r}\left(G_{1}\left(\phi_{i,j},m\right)\right.\\[2pt]&& \left. \qquad -G_{1}\left(\phi_{i,j-1},m\right)\right)\\[2pt]&&+r\left(\left(G_{1}\left(\phi_{i,j},r\right)-G_{1}\left(\phi_{i,j-1},r\right)\right)\right.\\[2pt]E\!\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})|\mathcal{D}_{I}\right)&=&\sum_{m=1}^{r}\left(G_{1}\left(\phi_{i,j},m\right)-G_{1}\left(\phi_{i,j-1},m\right)\right)\\[2pt]&& \qquad -r\!\left(\left(G_{1}\left(\phi_{i,j},r\right)-G_{1}\left(\phi_{i,j-1},r\right)\right)\right.\\[2pt]MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{In}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{In}({{\tiny\rm ECOMOR(r)}})\right)&=&\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}^{*}\gamma_{i}^{2}\mu_{W}^{2}\left(\eta^{2}+\left(1-p_{l}^{*}\right)\right)\\[2pt]&& +\sigma_{{{\tiny\rm ECOMOR(r)}}}^{2}\\[2pt]&&-2\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}^{*}\left(\sum_{m=1}^{r}\sum_{k_{1}=m+1}^{\infty}P\!\left(\xi_{i,j}=k_{1}\right)\right.\\[2pt]&& \times \sum_{h=1}^{3}(T_{h}(k_{1},m)\\[2pt]&&-\sum_{k_{2}=r+1}^{N_{i}}P(\xi_{i,j-1}=k_{2})\sum_{h=1}^{3}(T_{h}(k_{2},r)))\\[2pt] &&+2\gamma_{i}\mu_{W}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}^{*}\sum_{m=1}^{r}\left(G_{1}\left(\phi_{i,j},m\right)\right.\\[2pt] && \left.-G_{1}\left(\phi_{i,j-1},m\right)\right)\\[2pt] &&-2r\gamma_{i}\mu_{W}\sum_{l=0}^{\min(j,d)}N_{i,j-l}^{report}p_{l}\left(G_{1}\left(\phi_{i,j},r\right)\right.\\[2pt] && \left. -G_{1}\left(\phi_{i,j-1},r\right)\right)\\[2pt] MSEP_{\mathcal{D}_{I}}\left(X_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}}),\hat{X}_{i,j}^{Re}({{\tiny\rm ECOMOR(r)}})\right)&=&\sigma_{{{\tiny\rm LCR(r)}}}^{2}+r^{2}G_{2}\left(\phi_{i,j},r\right)-r^{2}G_{1}^{2}\left(\phi_{i,j},r\right)\\[2pt] && +r^{2}G_{2}\left(\phi_{i,j-1},r\right)\\[2pt] &&-r^{2}G_{1}^{2}\left(\phi_{i,j-1},r\right)+2r^{2}G_{1}\left(\phi_{i,j},r\right)G_{1}\left(\phi_{i,j-1},r\right)\\[2pt] &&+2r\sum_{m_{1}=1}^{r}\sum_{k_{1}=m_{1}+1}^{N_{i}}P(\xi_{i,j}=k_{1}) \sum_{h=1}^{3}T_{h}(k_{1};\,m)\\[2pt] &&-2rG_{2}\left(\phi_{i,j},r\right)-2rG_{2}\left(\phi_{i,j-1},r\right)\\[2pt] &&+2rE\!\left(X_{i,j}^{Re}({{\tiny\rm LCR(r)}})|\mathcal{D}_{I}\right)\left(G_{1}\left(\phi_{i,j},r\right)\right.\\[2pt]& & \left. -G_{1}\left(\phi_{i,j-1},r\right)\right)\,=\!:\,\sigma_{{\tiny\rm ECOMOR(r)}}^{2},\end{eqnarray*}

where $G_{\nu}(a,b)=\frac{a^{b}}{(b-1)!}\frac{\theta^{*}}{(\lambda^{*})^{\theta^{*}}}\int_{0}^{\infty} y^{\nu+\theta^{*}-1}e^{-b(\frac{y}{\lambda^{*}})^{\theta^{*}}-ae^{-\left(\frac{y}{\lambda^{*}}\right)^{\theta^{*}}}}dy$ and $\mu_W$ and $\sigma^2_W$ are given by Equation (10).

Now, the Numerical Procedure 1 is employed to (1) predict outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ECOMOR(r)}})$ ) and (2) compare these two reinsurance treaties. Tables 7 and 8, respectively, present prediction of outstanding claims $X_{i,j}^{In}({{\tiny\rm LCR(r)}})$ (resp. $X_{i,j}^{In}({\text{ECOMOR(r)}})$ ) and the MSEP for the LCR(r) and the ECOMOR(r) treaties for different r. As Tables 7 and 8 show (1) for a fixed shape parameter, the reserve and the MSEP increase as the scale parameter increases, (2) for a fixed scale parameter, the reserve and the MSEP increase as the shape parameter increases, (3) the cedent’s MSEP under the LCR(r) treaty is smaller than such amount under the ECOMOR(r), and (4) for both treaties, the amount of cedent’s MSEP increases as the number of claims covered by the reinsurer increases.

Table 7.

Mean of loss reserve net of LCR(r) treaty for Weibull distribution with parameters $\lambda^{*}$ and $\theta^{*}$ .

Table 8.

Mean of loss reserve net of ECOMOR(r) treaty for Weibull distribution with parameters $\lambda^{*}$ and $\theta^{*}$ .