1. Short introduction to the model

We will use the same notation as in Siegenthaler (2017) except for the position of some indices. In order to make the understanding easier, upper indices n and m will indicate the time of the estimation, whereas lower indices i and h refer to the origin period of the corresponding claims. The indices k refer to time (development years) between the origin year and the year of estimation, i.e., i + k = n.

The model introduced in Siegenthaler (2017) tries to analyze estimates \({\widehat{U}}_{i}^{n}\) of ultimates Ui. The aim of the original paper is to present estimators for the one-year uncertainty (solvency uncertainty) and the ultimate uncertainty of the estimated ultimates independent of the underlying claim data and the reserving methodology used to estimate \({\widehat{U}}_{i}^{n}.\) Because of Solvency II, the Swiss Solvency Test and the upcoming IFRS 17, such uncertainty estimators are becoming more and more important. Up to now, reserving models based on claim data are used to estimate such uncertainties, for instance, Merz and Wüthrich (2008), Mack (1991), Dahms (2012), or bootstrapping approaches. Often the resulting estimates for uncertainties strongly depend on the reserving method used. Since real uncertainties cannot be observed, it is impossible to decide which model is the best. Therefore, an estimation method that is independent of the underlying reserving method would be very helpful. Unfortunately, the model cannot fulfill these promises. It only corrects for some bias in the estimated ultimates (see Section 3). Therefore, we will call it bias correction model.

The bias correction model assumes that there exist constants gk and \(\sigma_{k}^{2}\) such that

\[{\widehat{U}}_{i}^{n} ≔ \widehat{E}\left\lbrack U_{i} | ℱ^{n} \right\rbrack \tag{1.1}\]

\[{\widehat{U}}_{i}^{i + J} = U_{i} \tag{1.2}\]

\[E\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k}^{i + k} \right\rbrack = g_{k}{\widehat{U}}_{i}^{i + k} \tag{1.3}\]

\[\text{Var}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k}^{i + k} \right\rbrack = \sigma_{k}^{2}{\widehat{U}}_{i}^{i + k} \tag{1.4}\]

where \({F}^{n}\) represents all at the time of estimation n available information and \({D}_{k}^{n}\) consists of all information contained in the estimated ultimates \({\widehat{U}}_{h}^{m}\) that have been or will be estimated before or at \(max(n, n − i + h)\), i.e.,

\[{D}_{k}^{n}\ : = \sigma\left( {\widehat{U}}_{h}^{m}\ :m \leq \max(n,n - i + h) \right).\]

2. Model assumption

2.1. Simple example

The definition 1.1 of the estimated ultimates \({\widehat{U}}_{i}^{n}\) can be rephrased as:

\({\widehat{U}}_{i}^{n}\) are random variables measurable with respect to \({F}^{n}\).

Therefore, it is possible that there is no relation between \({\widehat{U}}_{i}^{n}\) and the real ultimate Ui at all, except for the tail condition 1.2.

Indeed, take independent origin years with stable business, i.e., \(E[U_i]=g\) and \(Var[U_i] = \sigma^{2}\), for all origin years i, then the following estimated ultimates fulfill assumptions (1.1) to (1.4):

\[{\widehat{U}}_{i}^{n} ≔ \left\{ \begin{align}1,\text{for }n < i + J, \\ U_{i},\text{for }n = i + J, \\ \end{align} \right.\ \]

with

\[g_{k} ≔ \left\{ \begin{aligned} \ \ 1,\text{for }k < J - 1, \\ g, \text{for }k = J - 1, \\ \end{aligned} \text{ and } \sigma_{k}^{2} ≔ \left\{ \begin{aligned} \ \ 0,\text{for }k < J - 1, \\ \sigma^{2}, \text{for }k = J - 1. \\ \end{aligned} \right.\ \right.\ \]

In this simple example the solvency uncertainty equals zero, except for \({\widehat{U}}_{i}^{i + J}\) at estimation time i + J − 1. At this time it is equal to the ultimate uncertainty \(\sigma^{2}\), which can be estimated by the empirical variance

\[{\widehat{\sigma}}^{2}\ : = \frac{1}{i - 1}\sum_{h = 0}^{i - 1}\left( U_{h} - \overline{U} \right)^{2}\text{ with }\overline{U} ≔ \frac{1}{i}\sum_{h = 0}^{i - 1}U_{h}.\]

The bias correction model leads to other estimates for the uncertainties, because it considers the bias correction \({\widehat{\widehat{U}}}_{i}^{n}:={\widehat{g}}_{n - i}.....{\widehat{g}}_{J - 1}{\widehat{U}}_{i}^{n}\) instead of the estimation behind \({\widehat{U}}_{i}^{n}\) (see Section 3).

2.2. Best estimates

In the last section we have seen that it may be helpful to put some restrictions on the allowed estimated ultimates. In practice, for instance under Solvency II, the Swiss Solvency Test or IFRS 17, one would take best estimates. Unfortunately, it is not exactly clear what that means mathematically. Let us look at two interpretations. Note, almost all standard stochastic reserving methods satisfy both interpretations.

Unbiased estimates: In most cases we would at least expect that our estimates \({\widehat{U}}_{i}^{n}\) are unbiased. In this case we get

\[\begin{aligned}E\left\lbrack U_{i} \right\rbrack =&\ E\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \right\rbrack \\=&\ E\left\lbrack E\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k}^{i + k}\right\rbrack \right\rbrack \\=&\ g_{k}E\left\lbrack {\widehat{U}}_{i}^{i + k} \right\rbrack \\=&\ g_{k}E\left\lbrack U_{i} \right\rbrack,\end{aligned}\]

which leads to \(g_{k} \equiv 1\).

Claims development result is predicted by zero: This is the one-year version of unbiased estimates. We would like to have

\[E\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {F}^{i + k} \right\rbrack = {\widehat{U}}_{i}^{i + k}.\]

Unfortunately, this is too strong. But we can weaken it by using the same estimation methodology for the left hand side like we have used to estimate \({\widehat{U}}_{i}^{i + k}\), i.e., we assume

\[\widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {F}^{i + k} \right\rbrack = {\widehat{U}}_{i}^{i + k}.\]

In the case of the bias correction model the only way to predict the left hand side is

\[\widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {F}^{i + k} \right\rbrack\ : = \widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k}^{i + k} \right\rbrack\ : = {\widehat{g}}_{k}{\widehat{U}}_{i}^{i + k},\]

which leads to \({\widehat{g}}_{k} \equiv 1\). Since \({\widehat{g}}_{k}\) are unbiased, it follows that \(g_{k} \equiv 1\).

2.3. Parameter error

The solvency uncertainty can be split into

\[\begin{aligned}E\left\lbrack \left( {\widehat{U}}_{i}^{n + 1} - {\widehat{U}}_{i}^{n} \right)^{2} \middle| {F}^{n} \right\rbrack\ =&\ \underbrace{{\text{Var}\left\lbrack {\widehat{U}}^{n + 1} \middle| {F}^{n} \right\rbrack}}_\text{random error}\\+&\ \underbrace{\left(E\left[\widehat{U}_{i}^{n+1} \middle| F^{n}\right]-\widehat{U}_{i}^{n}\right)^{2}}_{\text {parameter error}}\text{.}\end{aligned}\]

In the case of best estimates, assumption (1.3) of the bias correction model means that there is no parameter error, which is very unlikely in practice. Similar arguments lead to the same statement for the parameter error of the ultimate uncertainty. Note, our reasoning stays true if we replace \({F}^{n}\) by an arbitrary σ-algebra \({D}\) with respect to which \({\widehat{U}}_{i}^{n}\) is measurable.

3. Bias correction model

If we look closely at the derivation of the estimator for the solvency uncertainty in the bias correction model by Siegenthaler (2017), we note that

  • It is enough to assume

    \(\text{i)}\ E\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k} \right\rbrack = g_{k}{\widehat{U}}_{i}^{i + k},\)

    \(\text{ii})\ \text{Var}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k} \right\rbrack = \sigma_{k}^{2}{\widehat{U}}_{i}^{i + k},\)

    where \({D}_{k}\) is an arbitrary σ-algebra containing all \({\widehat{U}}_{h}^{h + k},\ h \geq 0.\)

  • If \(g_{k} \neq 1\) then the term \({\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k}\) is biased, which should be compensated for in the derivation of the uncertainty. Usually, this is done by subtracting

    \[\widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k} \middle| {D}_{k} \right\rbrack = \left( {\widehat{g}}_{k} - 1 \right){\widehat{U}}_{i}^{i + k}.\]

    This compensation is missing in the original paper.

Indeed, we get

\[\begin{aligned}&{{E}\left\lbrack \left( {\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k} - \widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k} \middle| {D}_{k} \right\rbrack \right)^{2} \middle| {D}_{k} \right\rbrack} \\&{= E\left\lbrack \left( {\widehat{U}}_{i}^{i + k + 1} - {{\widehat{g}}_{k}\widehat{U}}_{i}^{i + k} \right)^{2} \middle| {D}_{k} \right\rbrack} \\& = {\text{Var}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {D}_{k} \right\rbrack + \left( g_{k} - {\widehat{g}}_{k} \right)^{2}\left( {\widehat{U}}_{i}^{i + k} \right)^{2}.}\end{aligned}\tag{3.5}\]

The second term is approximated by the \({D}\)k-conditional variance of \(\widehat{g}\)k. This leads to

\[\begin{aligned}&{E\left\lbrack \left( {\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k} - \widehat{E}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} - {\widehat{U}}_{i}^{i + k} \middle| {D}_{k} \right\rbrack \right)^{2} \middle| {D}_{k} \right\rbrack }\\&{ \approx \text{Var}\left\lbrack {\widehat{U}}_{i}^{i + k + 1} \middle| {\widetilde{{D}}}_{i,k} \right\rbrack + \text{Var}\left\lbrack {\widehat{g}}_{k} \middle| {D}_{k} \right\rbrack\left( {\widehat{U}}_{k}^{i + k} \right)^{2} }\\&{ \approx {\widehat{\sigma}}_{k}^{2}\left( \frac{1}{{\widehat{U}}_{i}^{i + k}} + \frac{1}{\sum_{h = 0}^{i - 1}{\widehat{U}}_{k}^{h + k}} \right)\left( {\widehat{U}}_{i}^{i + k} \right)^{2}.}\end{aligned}\]

Remark 3.1 For the estimation of the ultimate uncertainty of the bias correction model we have to correct for the bias, too. This results in the well known estimators for the chain-ladder ultimate uncertainty, introduced in Mack (1991) and Buchwalder et al. (2006).

4. Conclusion

At the second line of (3.5) we see that we have not analyzed the solvency uncertainty of the estimation behind \({\widehat{U}}_{i}^{n}\). Instead of that, we have looked at the first step of the bias correction

\[{\widehat{\widehat{U}}}_{i}^{n} ≔ {\widehat{g}}_{J - 1}....{\widehat{g}}_{n - i}{\widehat{U}}_{i}^{n},\]

which is a chain-ladder projection based on the triangle of estimated ultimates. In general, we think that a bias correction using chain-ladder on estimated ultimates is not a good idea, because most reserving methods strongly couple origin years such that assumptions (1.3) and (1.4) cannot be fulfilled.