Unification of Stochastic Reserving Models Using Individual Claims Information

1. Introduction

When predicting the ultimate reserve amounts, general insurance actuaries use a mix of chain ladder, Bornhuetter-Ferguson (Bornhuetter and Ferguson 1972), and Cape Cod (Bühlmann and Straub 1983) methods. Most commercial reserving software systems propose these three methods. These systems will likely ask the application of chain ladder first to derive a payment or incurred pattern, which will then be used for the application of Bornhuetter-Ferguson or Cape Cod methods. In all the cases, the chain ladder method will be applied on a cumulative triangle. Below is a summary of the stochastic models underlying each of the three methods—chain ladder, Bornhuetter-Ferguson, and Cape Cod.

1.1. Chain ladder

The chain ladder method is applied on cumulative triangles.

Let $C_{i,k}$ denote the cumulative claims amount (either paid or incurred) of accident year i after k years of development, $1 \leq i,\ k \leq n$ where n denotes the most recent accident year. Then $C_{i,n + 1 - i}$ denotes the currently known claims amount of accident year i, shown in Table 1.

The basic chain ladder assumption is that there exist development factors $f_1, \ldots, f_{I-1}$ such that

$$\begin{align} E\left(C_{i, k+1} \mid C_{i, 1}, \ldots, C_{i, k}\right)&=f_k C_{i, k}, \\ 1 &\leq i \leq I, \\ 1 &\leq k \leq I-1\end{align}\tag{1}$$

where the link ratios (age-to-age factors) can be estimated as

$$\hat{f}_k=\frac{\sum_{j=1}^{I-k} C_{j, k+1}}{\sum_{j=1}^{I-k} C_{j, k}}, 1 \leq k \leq I-1, \tag{2}$$

under the assumption that $\left\{C_{i, 1}, \ldots, C_{i, I}\right\},\left\{C_{j, 1}, \ldots, C_{j, I}\right\}, i \neq j$ are independent.

In this paper, $\hat{f}_k$ will denote the estimator of the random variable $f_k$. Mack (1993) shows that the link ratios $\hat{f}_k$ are unbiased and uncorrelated.

Variance of $C_{i,k}$

In the framework of the distribution-free calculation of the standard error of the reserve estimates, several variance models exist. For the purpose of this discussion, we will focus on the Mack standard error.

As for the variance of $C_{i, k+1}$, Mack (1993) induced that $\operatorname{Var}\left(C_{i, k+1} \mid C_{i, 1}, \ldots, C_{i, k}\right)$ (where $\operatorname{Var}(A \mid B)$ denotes the conditional variance of A knowing B) should be proportional to $C_{i, k}$, i.e.:

$$\begin{align} &\operatorname{Var}\left(C_{i, k+1} \mid C_{i, 1}, \ldots, C_{i, k}\right)\\ &\quad=C_{i, k} \sigma_k^2, 1 \leq i \leq I, 1 \leq k \leq I-1 \end{align} \tag{3}$$

where

$$\begin{align} \hat{\sigma}_k^2&=\frac{1}{I-k-1} \sum_{i=1}^{I-k} C_{i, k}\left(\frac{C_{i, k+1}}{C_{i, k}}-\hat{f}_k\right)^2 \quad for \\ 1 &\leq k \leq I-2 \end{align}\tag{4}$$

It can be shown that the estimator $\hat{\sigma}_k^2$ is unbiased (Mack 1993).

1.2. Bornhuetter-Ferguson

As mentioned earlier, the Bornhuetter-Ferguson (hereinafter “BF”) is usually applied on cumulative triangles using a pattern derived from the chain ladder method. In this section, we will review the stochastic model underlying the BF method introduced in Mack (2008). In this stochastic model, the BF method should be applied on incremental triangles.

As for the chain ladder method, let $C_{i,k}$ denote the cumulative claims amount (either paid or incurred) of accident year i after k years of development, $1 \leq i,\ k \leq n$ and $\nu_{i}$ be the premium volume of accident year i where n denotes the most recent accident year. Then $C_{i,n + 1 - i}$ denotes the currently known claims amount of accident year i. Let further $S_{i,k} = C_{i,k} - C_{i,k - 1}$ denote the incremental claims amount (with $C_{i,0} = 0$) and U_i the (unknown) ultimate claims amount of accident year i. Then $R_{i} = U_{i} - C_{i,n + 1 - i}$ is the (unknown true) claims reserve for accident year i. Finally, let $S_{i,n + 1} = U_{i} - C_{i,n}$ be the incremental claims amount after development year n (tail development).

Bornhuetter and Ferguson (1972) introduced their method to estimate R_i as follows:

$${\widehat{R}}_{i}^{BF} = {\widehat{U}}_{i}\left( 1 - {\widehat{z}}_{n + 1 - i} \right)$$

where ${\widehat{U}}_{i} = \nu_{i}{\widehat{q}}_{i}$ with a prior estimate ${\widehat{q}}_{i}$ for the ultimate claims ratio $q_{i} = \frac{U_{i}}{\nu_{i}}$ of accident year i, ${\widehat{z}}_{k} \in \lbrack 0;1\rbrack$ is the estimated percentage of the ultimate claims amount which is expected to be known after development year k.

The BF stochastic model developed in Mack (2008) relies on the following assumptions related to the increments $S_{i,k}$:

• BF1: All increments $S_{i,k}$ are independent

• BF2: There are unknown parameters x_i, y_k such that:

  • $E\left( S_{i,k} \right) = x_{i}y_{k}$

  • $y_{1} + \ldots + y_{n + 1} = 1$

• BF3: There are unknown proportionality constants $s_{k}^{2}$ with $Var\left( S_{i,k} \right) = x_{i}s_{k}^{2}$

On the basis of these three assumptions, the prediction error of Bornhuetter-Ferguson can be estimated (Mack 2008). The prediction error, usually denoted as MSEP (mean squared error of prediction) consists of two components, the process error and the estimation error. Whereas the estimation error basically always can be calculated via the laws of error propagation, for the process error a stochastic model of the claims process was developed by Mack (2008).

Following Mack (2008), we have the following with x₁,… x_n known:

$$\ {\widehat{y}}_{k} = \frac{\sum_{i = 1}^{n + 1 - k}S_{i,k}}{\sum_{i = 1}^{n + 1 - k}x_{i}}$$

is a best linear unbiased estimate of y_k and

$${\widehat{s}}_{k}^{2} = \frac{1}{n - k}\sum_{i = 1}^{n + 1 - k}\frac{\left( S_{i,k}\ - \ x_{i}{\widehat{y}}_{k} \right)^{2}}{x_{i}}$$

is an unbiased estimate of s²_k.

1.3. Cape Cod

As for the chain ladder and BF methods, we denote the cumulative claims (cumulative payments or incurred losses) in accident year $i \in \left\{ 0,\ \ldots,I \right\}$ at the end of development year $j \in \left\{ 0,\ \ldots,J \right\}$ by $C_{i,j} > 0$ and we assume J ≤ I. Let $S_{i,j} = C_{i,j} - C_{i,j - 1}$ denote the incremental claims, where we set $C_{i, - 1} = 0$. The summation over an index starting from 0 is denoted with a square bracket, for example:

$$C_{\left\lfloor k \right\rfloor,j} = \sum_{i = 0}^{k}C_{i,j},\ 0 \leq k \leq I,\ 0 \leq j \leq J.$$

We assume that all claims are settled after development year J and therefore the total ultimate claim of accident year i is given by $C_{i,J}$. At time I, we have information in the upper left trapezoid/triangle:

$$D_{I} = \left\{ C_{i,j}:i + j \leq I,\ j \leq J \right\}$$

and our goal is to predict the lower right triangle:

$$D_{I}^{c} = \left\{ C_{i,j}:i + j > I,\ i \leq I,j \leq J \right\}.$$

The chain ladder prediction of the ultimate claim $C_{i,J}$ of accident year i > I − J is given by

$${\widehat{C}}_{i,J}^{CL} = C_{i,\iota(i)}\prod_{j = \iota(i)}^{J - 1}{\widehat{f}}_{j}$$

where

$$\begin{gathered} \hat{f}_j=\frac{C_{[I-j-1], j+1}}{C_{[I-j-1], j}} \\ \text { and } \iota(i)=\min (J, I-i) . \end{gathered}$$

The chain ladder development pattern is defined as

$${\widehat{\beta}}_{j}^{CL} = \prod_{k = j}^{J - 1}{\widehat{f}}_{k}^{- 1},0 \leq j \leq J - 1,\ {\widehat{\beta}}_{J}^{CL} = 1\tag{7}$$

The Cape Cod predictor (Bühlmann and Straub 1983) for the ultimate claim is given by

$${\widehat{C}}_{i,J}^{CC} = C_{i,\iota(i)} + \upsilon_{i}\widehat{q}\left( 1 - {\widehat{\beta}}_{\iota(i)} \right)$$

where

The earned premium for accident year i is denoted by $\upsilon_{i}$; and

$$\widehat{q} = \frac{\sum_{i = 0}^{I}C_{i,\iota(i)}}{\sum_{i = 0}^{I}{\upsilon_{i}{\widehat{\beta}}_{\iota(i)}}}.$$

${\widehat{\beta}}_{\iota(i)}$ is an estimate of $\beta_{\iota(i)}$ and describes the percentage of claims emerging up to development year $\iota(i)$. The incremental development pattern $\gamma_{j} = \beta_{j} - \beta_{j - 1}$ is estimated by

$${\widehat{\gamma}}_{0} = {\widehat{\beta}}_{0}$$

$${\widehat{\gamma}}_{j + 1} = {\widehat{\beta}}_{j + 1} - {\widehat{\beta}}_{j},\ \ \ 0 \leq j \leq J - 1.$$

In the original article of Bühlmann and Straub (1983), it is mentioned that the estimation of the development pattern ${\widehat{\beta}}_{j}$ is an unsolved problem. In practice, the development pattern is often estimated by the chain ladder (CL) development pattern given in (7).

Finally, we define the outstanding loss liabilities for accident year i at time I as:

$$R_{i}^{CC} = C_{i,J} - C_{i,I - i}$$

Model assumptions

Incremental claims $S_{i,j}$ are independent and there exist positive parameters $q,\ t_{k}^{3},\ 0 \leq j \leq J$ and a development pattern $\gamma_{0},\ \ldots,\ \gamma_{J}$ with $\sum_{j = 0}^{J}\gamma_{j} = 1$ such that

$$E\left\lbrack S_{i,j} \right\rbrack = \upsilon_{i}q\gamma_{j}$$

$$Var\left\lbrack S_{i,j} \right\rbrack = \left( \upsilon_{i}q \right)\ \sigma_{j}^{2}$$

where $Var\left\lbrack S_{i,j} \right\rbrack$ denotes the variance of the random variable $S_{i,j}$.

For the estimation of the variance, we need estimates for $q{\ \sigma}_{j}^{2}$. Note that

$$\begin{align} \widehat{q\ \sigma_{j}^{2}} &= \frac{1}{I - j}\sum_{i = 0}^{I - j}\frac{1}{\nu_{i}}\left( S_{i,j} - \upsilon_{i}\widehat{\gamma_{j}} \right)^{2},\\ 0 &\leq j \leq J,\ j \neq I \end{align}\tag{8}$$

is an unbiased estimator for $q{\ \sigma}_{j}^{2}$.

Note also that the above model assumptions assume that the expected loss ratio q is the same for all accident years.

2. A review of the chain ladder method

In this section, we are going to review the equation (4) of the chain ladder method:

$$\begin{align} \hat{\sigma}_k^2&=\frac{1}{I-k-1} \sum_{i=1}^{I-k} C_{i, k}\left(\frac{C_{i, k+1}}{C_{i, k}}-\hat{f}_k\right)^2 \text { for } \\ 1 &\leq k \leq I-2 . \end{align}$$

Using the definition $S_{i,k + 1} = C_{i,k + 1} - C_{i,k}$ and the chain ladder incremental pattern ${\widehat{w}}_{k} =$ ${\widehat{\beta}}_{k - 1}^{CL} - {\widehat{\beta}}_{k}^{CL}$, we can reformulate equation (4) as

$${\widehat{\sigma}}_{k}^{2} = \frac{1}{I - k - 1}\sum_{i = 1}^{I - k}{C_{i,k}\left( \frac{S_{i,k + 1}}{C_{i,k}} - \left( {\widehat{f}}_{k} - 1 \right) \right)^{2}}.$$

We can see that

$$\begin{align}{\widehat{w}}_{k} &= {\widehat{\beta}}_{k - 1}^{CL} - {\widehat{\beta}}_{k}^{CL} \\ &= \frac{1}{\prod_{l = k - 1}^{J - 1}{\widehat{f}}_{l}} - \frac{1}{\prod_{l = k}^{J - 1}{\widehat{f}}_{l}} \\ &= \frac{1}{\prod_{l = k}^{J - 1}{\widehat{f}}_{l}}\left( {\widehat{f}}_{k} - 1 \right). \end{align}$$

Hence,

$$\small{{\widehat{\sigma}}_{k}^{2} = \frac{1}{I - k - 1}\sum_{i = 1}^{I - k}{C_{i,k}\left( \frac{S_{i,k + 1}}{C_{i,k}} - {\widehat{w}}_{k}\prod_{l = k}^{J - 1}{\widehat{f}}_{l} \right)^{2}}.}$$

As a result,

$$\small{{\widehat{\sigma}}_{k}^{2} = \frac{1}{I - k - 1}\sum_{i = 1}^{I - k}{{\widehat{C}}_{i,J}^{CL}\prod_{l = k}^{J - 1}{\widehat{f}}_{l}\left( \frac{S_{i,k + 1}}{{\widehat{C}}_{i,J}^{CL}} - {\widehat{w}}_{k} \right)^{2}}.}$$

The shape of this equation is very similar to the volatility factor of the Bornhuetter-Ferguson and Cape Cod methods. Despite its unusual shape, it is the same as the usual known equation (4) used to estimate the ${\widehat{\sigma}}_{k}^{2}$ but based on incremental triangle.

As a conclusion of this section, we can mention that, in practice, actuaries usually estimate their loss ultimates on the basis of cumulative triangles. However, as we have seen, the stochastic underlying models are based on incremental claims. Incremental claims have the advantage of having independence between each triangle cell, which is not the case for cumulative claims. This is why the stochastic underlying models are based on increments. It would therefore be advisable to change the commercial reserving software and provide actuaries with analysis based on incremental claims: this would better reflect the dynamics of the claims movements.

3. Stochastic reserving methods—A unification

Equations (6), (8) and (9) are rewritten below.

Bornhuetter-Ferguson

$${\widehat{s}}_{k}^{2} = \frac{1}{n - k}\sum_{i = 1}^{n + 1 - k}{x_{i}\left( \ \frac{S_{i,k}}{x_{i}} - \ {\widehat{y}}_{k} \right)}^{2}$$

where x_i represents the a-priori ultimate of the BF method.

The stochastic model defines: $Var\left( S_{i,k} \right) = x_{i}s_{k}^{2}$.

Cape Cod

$$\widehat{q\ \sigma_{j}^{2}} = \frac{1}{I - j}\sum_{i = 0}^{I - j}\upsilon_{i}\left( \frac{S_{i,j}}{\nu_{i}} - \widehat{\gamma_{j}} \right)^{2}$$

where $\upsilon_{i}$ represents the ultimate premium of the Cape-Cod method.

The stochastic model defines: $Var\left\lbrack S_{i,j} \right\rbrack = \left( \upsilon_{i}q \right)\ \sigma_{j}^{2}$.

Chain ladder

$${\widehat{\sigma}}_{k}^{2} = \frac{1}{I - k - 1}\sum_{i = 1}^{I - k}{{\widehat{C}}_{i,J}^{CL}\prod_{l = k}^{J - 1}{\widehat{f}}_{l}\left( \frac{S_{i,k + 1}}{{\widehat{C}}_{i,J}^{CL}} - {\widehat{w}}_{k} \right)^{2}}$$

where ${\widehat{C}}_{i,J}^{CL}$ represents the prediction of the ultimate claim $C_{i,J}$.

The stochastic model defines

$$Var\left( S_{i,k + 1}|C_{i,1},\ldots,C_{i,k} \right) = C_{i,k}\sigma_{k}^{2}$$

The variance of the increment for each model is defined according to these volatility factors. These three equations have the same shape:

• The volatility factor always depends on the difference between the increments divided by the ultimate and the estimated pattern.

• The volatility factor is always a weighted average of these differences where the weights are either the ultimates (BF, Cape Cod) or derived from the ultimates (chain ladder).

In relation to the first point, in practice, the feeling for the volatility of a line of business always depends on the possibility for the actuary to have confidence in the incurred/payment pattern. When an actuary feels unsure about the incurred/payment pattern, he will say that the line of business is very volatile. On the opposite when the incurred/payment pattern is stable across the accident or underwriting years, he will say that the line of business is not volatile. The three equations reflect therefore the practice. In addition, their similarity shows that the applied method is not a determinant of the volatility of the resulting ultimates.

As for the skewness factors, the definitions are provided below.

Bornhuetter-Ferguson (Dal Moro 2021)

On the basis of the three assumptions described in 1.b (All increments $S_{i,k}$ are independent, there are unknown parameters x_i, y_k, there are unknown proportionality constants $s_{k}^{2}$ with $Var\left( S_{i,k} \right) = x_{i}s_{k}^{2}$), the prediction error of Bornhuetter-Ferguson can be estimated (Mack 2008). In order to estimate the skewness of the BF method, we need a fourth assumption:

BF4: There are unknown proportionality constants $t_{k}^{3}$ with SK$\left( S_{i,k} \right) = x_{i}^{\frac{3}{2}}t_{k}^{3}$

and

$${\widehat{t}}_{k}^{3} = \frac{1}{n - k}\sum_{i = 1}^{n + 1 - k}\frac{\left( S_{i,k}\ - \ x_{i}{\widehat{y}}_{k} \right)^{3}}{x_{i}^{\frac{3}{2}}}$$

It has to be noted that the skewness of Bornhuetter-Ferguson in the proposed model comes in a distribution-free environment. Once the best estimate, the standard deviation and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.

Cape Cod (Dal Moro 2022)

As for the skewness of Cape Cod in a distribution free environment, following on the work of Saluz (2015), we assume that there exist positive parameters $q,\ t_{j}^{3},\ 0 \leq j \leq J$ and a development pattern $\gamma_{0},\ \ldots,\ \gamma_{J}$ with $\sum_{j = 0}^{J}\gamma_{j} = 1$ such that

$$E\left\lbrack X_{i,j} \right\rbrack = \upsilon_{i}q\gamma_{j}$$

$$SK\left\lbrack X_{i,j} \right\rbrack = \left( \upsilon_{i}q \right)^{\frac{3}{2}}\ t_{j}^{3}$$

where $SK\left\lbrack X_{i,j} \right\rbrack$ denotes the third moment of the random variable $X_{i,j}$. And $\upsilon_{i}$ corresponds to the UWY premium and $q$ to the Cape Cod loss ratio.

For the estimation of the skewness, we need estimates for $q^{\frac{3}{2}}{\ t}_{j}^{3}$. Note that

$$\begin{align} \widehat{q^{\frac{3}{2}}\ t_{j}^{3}} &= \frac{1}{I - j}\sum_{i = 0}^{I - j}\frac{1}{\upsilon_{i}^{\frac{3}{2}}}\left( X_{i,j} - \upsilon_{i}\widehat{\gamma_{j}} \right)^{3},\ \\ 0 &\leq j \leq J,\ j \neq I \end{align}$$

is an unbiased estimator for $q^{\frac{3}{2}}{\ t}_{j}^{3}$.

As for Bornhuetter-Ferguson, it has to be noted that the skewness of Cape Cod in the proposed model comes in a distribution-free environment. Once the best estimate, the standard deviation and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.

Chain ladder (Dal Moro 2013)

$$\begin{align} &S K\left(C_{i, k+1} \mid C_{i, 1}, \ldots, C_{i, k}\right)\\ &\quad =C_{i, k}^{3 / 2} S k_k^3, 1 \leq i \leq I, 1 \leq k \leq I-2 \end{align}$$

where:

$$\small{ \begin{align} \hat{S} k_k^3&= \frac{1}{\left(I-k-\frac{\left(\sum_{i=1}^{I-k} C_{i, k}^{3 / 2}\right)^2}{\left(\sum_{i=1}^{I-k} C_{i, k}\right)^3}\right)} \sum_{i=1}^{I-k} C_{i, k}^{3 / 2}\left(\frac{C_{i, k+1}}{C_{i, k}}-\hat{f}_k\right)^3 \\ \text { for } 1 &\leq k \leq I-3 \end{align}\tag{10} }$$

It has to be noted that, like for Bornhuetter-Ferguson and Cape Cod, the skewness of chain ladder in the proposed model comes in a distribution-free environment. Once the best estimate, the standard deviation, and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.

Like for the volatility factor, the chain ladder skewness factor can be changed to reflect the difference between the increments divided by the ultimate and the estimated pattern.

The skewness factors show, as they do for the volatility factors, a unity in terms of the general shape of the formulae.

As a unification of these three methods, for the volatility and skewness estimators denoted respectively $v_{k}^{2}$ and ${sk}_{k}^{3}$, we could therefore consider the overall equations:

$$v_k^2=\frac{1}{m(k)} \sum_i g\left(\widehat{U}_l\right)\left(\frac{S_{i, k}}{\widehat{U}_i}-\widehat{y}_k\right)^2\tag{11}$$

where $m(k)$ represents either I-k or I-k-1, $\widehat{U_{i}}$ represents the ultimate, and $g\left( \widehat{U_{i}} \right)$ is a function of the ultimate.

$$s k_k^3=\frac{1}{o(k)} \sum_i h\left(\widehat{U}_l\right)\left(\frac{S_{i, k}}{\widehat{U}_i}-\hat{y}_k\right)^3 \tag{12}$$

with the same variables as for the volatility and o(k) is I-k or the factor before the sum in equation (10).

4. Individual claims analysis for a unified model

Based on the unified notation described above (equations 11 and 12), we see that the reserve risk distribution is defined by the relative position of the percentage of incremental claim to ultimate vs. the pattern that is defining the best estimate. As mentioned earlier, in practice, this is the work of the actuary to try and defend his choice of best estimate patterns. Nowadays, such study of the position of incremental claims vs. best estimate pattern should be looked at individual claims level.

For this purpose, let’s denote ${SIC}_{p,i,k}$ the individual incremental claim amount for claim p for accident year i at the end of development year k and ${\widehat{UIC}}_{p,i,k}$ the ultimate claim for claim p for the same accident year and development year. We have then:

$$\frac{S_{i, k}}{\widehat{U}_i}=\frac{\sum_p S I C_{p, i, k}}{\sum_p \widehat{U I C} C_{p, i, k}}=\sum_p \frac{S I C_{p, i, k}}{\widehat{U I C} C_{p, i, k}} \frac{\widehat{U I C}_{p, i, k}}{\sum_p \widehat{U I C}_{p, i, k}}$$

Let’s denote

$$a_{p, i, k}=\frac{\widehat{U I C} C_{p, i, k}}{\sum_p \widehat{U I C}_{p, i, k}}$$

Then we have the following (as $\sum_p a_{p, i, k}=1$):

$$\frac{s_{i, k}}{\widehat{U}_i}-\hat{y}_k=\sum_p a_{p, i, k}\left(\frac{S I C_{p, i, k}}{\widehat{U T C} C_{p, i, k}}-\hat{y}_k\right). \tag{13}$$

With this last equation, we can see that some information related to the unified volatility and skewness estimated in equations (11) and (12) can be derived from individual claims. In equation (13), for i and k given, the following elements are not random: $a_{p,i,k},\ {\widehat{UIC}}_{p,i,k}\ and\ {\widehat{y}}_{k}$. Therefore, the risk distribution depends on the shape (volatility and skewness) of the individual claims in year i and development year k.

In this context, let’s look at an individual claims triangle and see the consequences of looking at the reserving distribution on such a triangle, shown in Table 2.

Any projection method applied to this triangle (e.g., neural networks, chain ladder …) will just estimate the IBNER (incurred but not enough reported) and will not take into account the IBNYR (incurred but not yet reported). Therefore, in the context of a unified reserving model based on an individual claims triangle, the following procedure will have to be followed:

1. Estimate the future number of claims and the volatility and skewness of the distribution of the future number of claims. This can be done on using a chain ladder method (Mack 1993; Dal Moro 2013), shown in Table 3.

2. Estimate for each development year, the mean, volatility and skewness of individual claim incurred or payment. In this regard, we assume that the exposures remain stable across the UWYs allowing the calculation at development year level and not at development and UWY level.

3. Having the future number of claims and the individual incurred or payment per development year, it is easy to build the IBNYR risk distribution (see following sections for details).

4. As for the IBNER, we will rely on Schnieper (1991), which gives a complete description of the way in which IBNER and their volatilities can be estimated based on individual claim information. In a following section, we will extend the calculations to the skewness case.

In the next two sections, we are going to detail the four steps just described. It has to be noted that the most significant amount of reserves will come from the IBNYR as, for each new claim, the full mean ultimate payment has to be reserved.

5. IBNYR based on individual claims analysis

Based on the individual claims’ triangle of Table 2, we can estimate the following parameters:

• The mean incurred/payment per development year;

• The variance of the incurred/payment per development year;

• The skewness of the incurred/payment per development year.

Having estimated the above characteristics, we can use the law of total variance to estimate the overall variance for accident year i:

$$\small{ \begin{align} Var\left( \sum_{p = 1}^{N_{i}}{\sum_{k = I - i + 1}^{I}{SIC}_{p,i,k}} \right) &= E\left( N_{i} \right)Var\left( \sum_{k = I - i + 1}^{I}{SIC}_{p,i,k} \right) \\ &\quad + Var\left( N_{i\ } \right)E\left( \sum_{k = I - i + 1}^{I}{SIC}_{p,i,k} \right)^{2} \end{align}}$$

Due to the independence between the ${SIC}_{p,i,k}$, we have

$$\small{ \begin{align} Var\left( \sum_{p = 1}^{N_{i}}{\sum_{k = I - i + 1}^{I}{SIC}_{p,i,k}} \right) &= E\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{Var\left( {SIC}_{p,i,k} \right)} \right) \\ &\quad + Var\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{E\left( {SIC}_{p,i,k} \right)} \right)^{2} \end{align}}$$

where N_i denotes the future number of claims, which can be estimated with a standard chain ladder method from accident year i (see next section for details).

As we have

$$Var\left( {SIC}_{p,i,k} \right) = Var\left( {SIC}_{k} \right) \text{for all p and i}$$

$$E\left( {SIC}_{p,i,k} \right) = E\left( {SIC}_{k} \right) \text{for all p and i}$$

the overall variance for accident year i is

$$\scriptsize{\begin{align} Var\left( \sum_{p = 1}^{N_{i}}{\sum_{k = I - i + 1}^{I}{SIC}_{p,i,k}} \right) &= E\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{Var\left( {SIC}_{k} \right)} \right) \\ &\quad+ Var\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{E\left( {SIC}_{k} \right)} \right)^{2} \end{align} \tag{14}}$$

The same can be done for skewness with the law of total skewness and we get

$$\scriptsize{\begin{align} SK\left( \sum_{p = 1}^{N_{i}}{\sum_{k = I - i + 1}^{I}{SIC}_{p,i,k}} \right) &= E\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{SK\left( {SIC}_{k} \right)} \right) \\ &\quad+ SK\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{E\left( {SIC}_{k} \right)} \right)^{3} \\ &\quad + 3\ Var\left( N_{i} \right)\left( \sum_{k = I - i + 1}^{I}{E\left( {SIC}_{k} \right)} \right)\\ &\quad \times \left( \sum_{k = I - i + 1}^{I}{Var\left( {SIC}_{k} \right)} \right) \end{align} \tag{15}}$$

6. IBNER based on individual claims analysis

According to Schnieper (1991), the IBNER for accident year i is equal to:

$${IBNER}_{i} = C_{i,n + 1 - i}\left\{ \left\lbrack \prod_{j = n + 2 - i}^{n}\left( 1 - \delta_{j} \right) \right\rbrack - 1 \right\}\tag{16}$$

where:

$${\widehat{\delta}}_{j} = \frac{\sum_{i = 1}^{n + 1 - j}D_{i,j}}{\sum_{i = 1}^{n + 1 - j}C_{i,j - 1}}$$

and D_i,j is the decrease in total claims amount between development year j-1 and development year j with respect to claims already known in development year j-1.

In the case of the triangle shown in Table 2, we would have the following resulting triangle of D_i,j with the corresponding ${\widehat{\delta}}_{j}$, shown in Table 5.

In Schnieper (1991), we also have that:

$$Var\left( D_{i,j} \right) = C_{i,j - 1}\tau_{j}^{2}$$

where

$${\widehat{\tau}}_{j}^{2} = \frac{1}{n - j}\sum_{i = 1}^{n + 1 - j}\frac{\left( D_{i,j} - {\widehat{\delta}}_{j}C_{i,j - 1} \right)^{2}}{C_{i,j - 1}}$$

By natural extension, the skewness estimate can be derived as follows:

$$SK\left( D_{i,j} \right) = C_{i,j - 1}^{\frac{3}{2}}\zeta_{j}^{3}$$

where

$${\widehat{\zeta}}_{j}^{3} = \frac{1}{n - j}\sum_{i = 1}^{n + 1 - j}\frac{\left( D_{i,j} - {\widehat{\delta}}_{j}C_{i,j - 1} \right)^{3}}{C_{i,j - 1}^{\frac{3}{2}}}$$

As in Schnieper (1991), let’s denote ${\widehat{\theta}}_{i} = \left( \delta_{n + 2 - i},\ldots,\delta_{n} \right)$. Developing ${IBNER}_{i}\left( {\widehat{\theta}}_{i} \right)$ in a Taylor series, we obtain

$${IBNER}_{i}\left( {\widehat{\theta}}_{i} \right) = {IBNER}_{i}\left( \theta_{i} \right) + \sum_{j = n + 2 - i}^{n}\frac{\delta\ {IBNER}_{i}\left( \delta_{j} \right)}{\delta\ \delta_{j}}$$

Due to the independence of the D_i,j, we can calculate the mean standard error (hereinafter “mse”) as follows (Schnieper 1991):

$$\small{\begin{align} mse\left( {IBNER}_{i}\left( {\widehat{\theta}}_{i} \right) \right) &= E\left( {IBNER}_{i}\left( {\widehat{\theta}}_{i} \right) - {IBNER}_{i}\left( \theta_{i} \right) \right)^{2} \\ &= \sum_{j = n + 2 - i}^{n}{\left( \frac{\delta\ {IBNER}_{i}\left( \delta_{j} \right)}{\delta\ \delta_{j}}|_{\theta_{i} = {\widehat{\theta}}_{i}} \right)^{2}Var\left( \widehat{\delta_{j}} \right)} \end{align}}$$

where

$$Var\left( {\widehat{\delta}}_{j} \right) = \frac{\tau_{j}^{2}}{\sum_{i = 1}^{n + 1 - j}C_{i,j - 1}}$$

Following equation (16), we have:

$$\frac{\delta\ {IBNER}_{i}\left( \delta_{j} \right)}{\delta\ \delta_{j}}|_{\theta_{i} = {\widehat{\theta}}_{i}} = - C_{i,n + 1 - i}\prod_{\begin{array}{r} k = n + 2 - i \\ k \neq j \end{array}}^{n}\left( 1 - \delta_{k} \right)$$

which leads to:

$$\small{\begin{align} &\operatorname{mse}\left(\operatorname{IBNER}_i\left(\hat{\theta}_i\right)\right)\\ &\quad =C_{i, n+1-i}^2 \sum_{j=n+2-i}^n\left(\prod_{k=n+2-i}^n\left(1-\delta_k\right)\right)^2 \frac{\operatorname{Var}\left(\widehat{\delta}_J\right)}{\left(1-\delta_j\right)^2} \end{align}\tag{17}}$$

The same can be done for the skewness estimation which leads to the following formula:

$$\small{\begin{align} &S K\left(\operatorname{IBNER}_i\left(\hat{\theta}_i\right)\right)\\ &\quad =-C_{i, n+1-i}^3 \sum_{j=n+2-i}^n\left(\prod_{k=n+2-i}^n\left(1-\delta_k\right)\right)^3 \frac{S K\left(\widehat{\delta}_j\right)}{\left(1-\delta_j\right)^3} \end{align}\tag{18}}$$

where

$$SK\left( {\widehat{\delta}}_{j} \right) = \frac{\zeta_{j}^{3}}{\sum_{i = 1}^{n + 1 - j}C_{i,j - 1}^{\frac{3}{2}}}$$

7. Estimation of overall skewness and standard deviation

After estimating IBNYR and IBNER by UWY, based on Mack (1993), the overall standard deviation can easily be calculated as per the formula below. Let’s denote s.d.(R_i), the standard deviation of the reserve of UWY i (see 1.b for definition of (R_i) and s.d.(R) the standard deviation overall all UWYs (R=R₂ + … + R_n). Then:

$$\scriptsize{\begin{align} &s.d.\left( \widehat{R} \right)^{2} \\ &\quad = \sum_{i = 2}^{I}\left\{ s.d.\left( {\widehat{R}}_{i} \right)^{2} + {\widehat{C}}_{iI}^{CL}\left( \sum_{j = i + 1}^{I}{\widehat{C}}_{jI}^{CL} \right)\sum_{k = I + 1 - i}^{I - 1}\frac{2\ \frac{{\widehat{\sigma}}_{k}^{2}}{{\widehat{f}}_{k}^{2}}}{\sum_{n = 1}^{I - k}C_{n,k}} \right\} \end{align}\tag{19}}$$

Let’s denote the following correlations:

$$r_{i,j} = \frac{{\widehat{C}}_{iI}^{CL}{\widehat{C}}_{jI}^{CL}\sum_{k = I + 1 - i}^{I - 1}\frac{\frac{{\widehat{\sigma}}_{k}^{2}}{{\widehat{f}}_{k}^{2}}}{\sum_{n = 1}^{I - k}C_{n,k}}}{\sqrt{Var\left( {\widehat{C}}_{iI}^{CL} \right)Var\left( {\widehat{C}}_{jI}^{CL} \right)}}$$

And as $Var\left( {\widehat{C}}_{iI}^{CL} \right) = s.d.\left( {\widehat{R}}_{i} \right)^{2}$, we can write equation (19) in the following form:

$$\small{ \text { s.d. }(\widehat{R})^2=\left(\begin{array}{c} s \cdot d \cdot\left(\widehat{R_1}\right) \\ \ldots \\ \text { s.d. }\left(\widehat{R_1}\right) \end{array}\right)\left(\begin{array}{ccc} 1 & r_{12} & \ldots \\ r_{12} & \ldots & \ldots \\ \ldots & \ldots & 1 \end{array}\right)\left(\begin{array}{c} \text { s.d. }\left(\widehat{R_1}\right) \\ \ldots \\ \text { s.d. }\left(\widehat{R_1}\right) \end{array}\right)}$$

In order to aggregate the standard deviation of the proposed method based on IBNYR and IBNER by UWY, we can use the same correlation matrix on the standard deviations of each UWY to get the overall standard deviation.

As for the overall skewness, before giving a general formula, we are going to limit the estimation to the case of three accident years X1, X2, X3 as shown in the framework of lemma 1 (see appendix B). The question is the estimation of $SK\left( X_{1} + X_{2} + X_{3} \right)$. It develops as follows:

$$\small{ \begin{aligned} S K\left(X_1+X_2+X_3\right)&=E\left[\left(\left(X_1+X_2+X_3\right)-E\left(X_1+X_2+X_3\right)\right)^3\right] \\ &=E\left[\left(X_1+X_2+X_3\right)^3\right]\\ &\quad-3 E\left[\left(X_1+X_2+X_3\right)^2\right] E\left[\left(X_1+X_2+X_3\right)\right]\\ &\quad+2 E\left[\left(X_1+X_2+X_3\right)\right]^3 \end{aligned}}$$

$$\begin{aligned} S K\left(X_1+X_2+X_3\right) & =S K\left[X_1\right]+S K\left[X_2\right]+S K\left[X_3\right] \\ & +3 E\left[X_1 X_2\left(X_1+X_2\right)\right]\\ &\quad-6 E\left[X_1 X_2\right]\left(E\left[X_1\right]+E\left[X_2\right]\right) \\ & +3 E\left[X_1 X_3\left(X_1+X_3\right)\right]\\ &\quad-6 E\left[X_1 X_3\right]\left(E\left[X_1\right]+E\left[X_3\right]\right) \\ & +3 E\left[X_2 X_3\left(X_2+X_3\right)\right]\\ &\quad-6 E\left[X_2 X_3\right]\left(E\left[X_2\right]+E\left[X_3\right]\right) \\ & +3 E\left[X_1\right]\left[E\left(X_2\right)^2-\operatorname{Var}\left(X_2\right)\right] \\ & +3 E\left[X_1\right]\left[E\left(X_3\right)^2-\operatorname{Var}\left(X_3\right)\right] \\ & +3 E\left[X_2\right]\left[E\left(X_1\right)^2-\operatorname{Var}\left(X_1\right)\right] \\ & +3 E\left[X_2\right]\left[E\left(X_3\right)^2-\operatorname{Var}\left(X_3\right)\right] \\ & +3 E\left[X_3\right]\left[E\left(X_1\right)^2-\operatorname{Var}\left(X_1\right)\right] \\ & +3 E\left[X_3\right]\left[E\left(X_2\right)^2-\operatorname{Var}\left(X_2\right)\right] \\ & +6 E\left[X_1 X_2 X_3\right]+12 E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \\ & -6 E\left[X_1 X_2\right] E\left[X_3\right]\\ &\quad-6 E\left[X_1 X_3\right] E\left[X_2\right]-6 E\left[X_2 X_3\right] E\left[X_1\right] \end{aligned}$$

Following lemma 1 (see appendix B), we have

$$\scriptsize{ \begin{aligned} E\left[X_1^2 X_2\right]&=\operatorname{Cov}\left(X_1^2, X_2\right)+E\left(X_1^2\right) E\left(X_2\right) \\ &=E\left(X_1\right)\left(1+\frac{\operatorname{Var}\left(X_1\right)}{E\left(X_1\right)^2}\right) \operatorname{Cov}\left(X_1, X_2\right)\left(2+\frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left(X_1\right) E\left(X_2\right)}\right)\\ &\quad+\left(\operatorname{Var}\left(X_1\right)+E\left(X_1\right)^2\right) E\left(X_2\right) \end{aligned}}$$

For the following elements of the above equation, we therefore get

$$\scriptsize{ \begin{aligned} & E\left[X_1 X_2\left(X_1+X_2\right)\right]-2 E\left[X_1 X_2\right]\left(E\left[X_1\right]+E\left[X_2\right]\right) \\ & +E\left[X_1\right]\left[E\left(X_2\right)^2-\operatorname{Var}\left(X_2\right)\right]+E\left[X_2\right]\left[E\left(X_1\right)^2-\operatorname{Var}\left(X_1\right)\right] \\ & =E\left(X_1\right)\left(1+\frac{\operatorname{Var}\left(X_1\right)}{E\left(X_1\right)^2}\right) \operatorname{Cov}\left(X_1, X_2\right)\left(2+\frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left(X_1\right) E\left(X_2\right)}\right)\\ &\quad+\left(\operatorname{Var}\left(X_1\right)+E\left(X_1\right)^2\right) E\left(X_2\right) \\ & +E\left(X_2\right)\left(1+\frac{\operatorname{Var}\left(X_2\right)}{E\left(X_2\right)^2}\right) \operatorname{Cov}\left(X_1, X_2\right)\left(2+\frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left(X_1\right) E\left(X_2\right)}\right)\\ &\quad+\left(\operatorname{Var}\left(X_2\right)+E\left(X_2\right)^2\right) E\left(X_1\right) \\ & -2\left(E\left[X_1\right]+E\left[X_2\right]\right)\left(\operatorname{Cov}\left(X_1, X_2\right)+E\left(X_1\right) E\left(X_2\right)\right) \\ & +E\left[X_1\right]\left[E\left(X_2\right)^2-\operatorname{Var}\left(X_2\right)\right]+E\left[X_2\right]\left[E\left(X_1\right)^2-\operatorname{Var}\left(X_1\right)\right] \\ & =\operatorname{Cov}\left(X_1, X_2\right)\left(2+\frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left(X_1\right) E\left(X_2\right)}\right)\\ &\quad \times\left[E\left(X_1\right)\left(1+\frac{\operatorname{Var}\left(X_1\right)}{E\left(X_1\right)^2}\right)+E\left(X_2\right)\left(1+\frac{\operatorname{Var}\left(X_2\right)}{E\left(X_2\right)^2}\right)\right] \\ & -2\left(E\left[X_1\right]+E\left[X_2\right]\right) \operatorname{Cov}\left(X_1, X_2\right) \\ & =\operatorname{Cov}\left(X_1, X_2\right)\left(2+\frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left(X_1\right) E\left(X_2\right)}\right)\left[\frac{\operatorname{Var}\left(X_1\right)}{E\left(X_1\right)}+\frac{\operatorname{Var}\left(X_2\right)}{E\left(X_2\right)}\right]\\ &\quad+\operatorname{Cov}\left(X_1, X_2\right)^2\left[\frac{E\left[X_1\right]+E\left[X_2\right]}{E\left[X_1\right] E\left[X_2\right]}\right] \end{aligned}}$$

In addition, using lemma 2 (see appendix B), we get

$$\scriptsize{ \begin{align} & E\left[X_1 X_2 X_3\right] + 2 E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \\ &\quad - E\left[X_1 X_2\right] E\left[X_3\right] - E\left[X_1 X_3\right] E\left[X_2\right] \\ &\quad - E\left[X_2 X_3\right] E\left[X_1\right] \\ &= E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \left( 2 + \left( 1 + \frac{\operatorname{Cov}\left(X_1, X_2\right)}{E\left[X_1\right] E\left[X_2\right]} \right) \right. \\ &\quad \left. \times \left( 1 + \frac{\operatorname{Cov}\left(X_1, X_3\right)}{E\left[X_1\right] E\left[X_3\right]} \right) \times \left( 1 + \frac{\operatorname{Cov}\left(X_2, X_3\right)}{E\left[X_2\right] E\left[X_3\right]} \right) \right) \\ &\quad - E\left[X_3\right] \operatorname{Cov}\left(X_1, X_2\right) - E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \\ &\quad - E\left[X_2\right] \operatorname{Cov}\left(X_1, X_3\right) - E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \\ &\quad - E\left[X_1\right] \operatorname{Cov}\left(X_2, X_3\right) - E\left[X_1\right] E\left[X_2\right] E\left[X_3\right] \\ &= r_{12} r_{13} r_{23} \sqrt{\operatorname{Var}\left(X_1\right) \operatorname{Var}\left(X_2\right) \operatorname{Var}\left(X_3\right)} \\ &\quad \times \left( \frac{\sqrt{\operatorname{Var}\left(X_1\right) \operatorname{Var}\left(X_2\right) \operatorname{Var}\left(X_3\right)}}{E\left[X_1\right] E\left[X_2\right] E\left[X_3\right]} + \frac{\sqrt{\operatorname{Var}\left(X_1\right)}}{r_{23} E\left[X_1\right]} + \frac{\sqrt{\operatorname{Var}\left(X_2\right)}}{r_{13} E\left[X_2\right]} + \frac{\sqrt{\operatorname{Var}\left(X_3\right)}}{r_{12} E\left[X_3\right]} \right) \end{align}}$$

In the case of three accident years and under the restrictions indicated in lemmas 1 and 2, we therefore find the following aggregate skewness:

SK(X1+X2+X3)=SK[X1]+SK[X2]+SK[X3]+3r12√Var(X1)Var(X2)×[Var(X1)E(X1)+Var(X2)E(X2)][2+r12√Var(X1)Var(X2)E[X1]E[X2]]+3r212Var(X1)Var(X2)[E[X1]+E[X2]E[X1]E[X2]]+3r13√Var(X1)Var(X3)×[Var(X1)E(X1)+Var(X3)E(X3)][2+r13√Var(X1)Var(X3)E[X1]E[X3]]+3r213Var(X1)Var(X3)[E[X1]+E[X3]E[X1]E[X3]]+3r23√Var(X2)Var(X3)×[Var(X2)E(X2)+Var(X3)E(X3)][2+r23√Var(X2)Var(X3)E[X2]E[X3]]