On the impact of outliers in loss reserving

The classical loss reserving problem is interested in estimating existing (unpaid and/or unreported) claim liabilities, based on past data. Traditional reserving techniques, widely used in practice, are applied on data at a certain level of aggregation (usually quarters or years), often presented in triangles. Parameters typically rely on only a few such data points - sometimes even only one. This makes those techniques very vulnerable to the presence of outliers.

This issue is well known, however, being able to quantify the specific impact of each observation to certain statistics of interest provides greater information regarding the nature of the data at hand and will often provide insights regarding the techniques themselves. This is of particular importance when implementing and adjusting models. The objective of this paper is to provide a mathematically tractable approach to understanding how changes in each incremental claim in a loss triangle will impact certain statistics of interest. The presented impact functions are not designed to detect outliers but rather to deepen our understanding of how outliers may impact the results of a model. Impact functions coupled with statistically sound procedures to detect and treat abnormal observations will improve the robustness of reserving techniques and ultimately lead to more informed and reliable decisions.

Venter and Tampubolon (2008) calculate the impact of incremental claims on total reserve estimates under a range of models. They consider the traditional chain-ladder technique however the impacts are calculated numerically and hence no closed form equations are provided. Additionally, no consideration of individual accident years, mean squared errors (mse) or quantiles is given which constitutes a major contribution of this paper. Verdonck, Van Wouwe and Dhaene (2009) show that traditional chain-ladder reserve estimates are highly susceptible to even just one outlier and highlight that the impact on reserves may be positive or negative. See Verdonck, Van Wouwe and Dhaene (2009); Verdonck and Debruyne (2011) for recent robust reserving techniques.

In this paper we rigorously investigate the impact that incremental observations have on reserve estimates, their variability, and their quantiles. Notably, we provide closed form equations for the first derivative of these statistics of interest under Mack’s Model (Mack, 1993), which highlights numerous properties of this technique, including areas of a loss triangle where outliers are likely to have the greatest effect on results and hence where observations should be most heavily scrutinised. It appears that observations in the corners of a loss triangle have the potential to impact results most significantly. Additionally, we compare the impact of incremental observations on reserves under Mack’s Model and the Bornhuetter Ferguson technique (Bornhuetter and Ferguson, 1972) which suggests that the latter approach is more robust.

These techniques may be applied in practice to identify areas of a given loss triangle that reserves are particularly sensitive to and hence where outliers, if present, may have a significant impact on results. These impact functions may also be used to compare reserve sensitivities under different techniques as we have done for Mack’s Model and the Bornhuetter Ferguson approach. The impact that incremental observations are having in different loss triangles may be calculated using these impact functions and comparisons made between areas of sensitivity and properties of these different triangles. Through such a comparative study, trends may begin to emerge, making it easier to identify anomalous observations or even whole data sets with abnormal properties.

The paper is structured as follows. In Section 2 we define notation and briefly review the reserving techniques that will be considered. Impact functions are defined in Section 2.5. The following sections summarise the impact functions for certain statistics of interest and apply them to real data where 3D graphical representations are used to highlight their features. Section 3 focuses on central estimates, Section 4 on mean squared errors, and Section 5 on quantiles. The data used for the examples in this paper is presented and discussed in Appendix A and detailed proofs for the impact functions can be found in Appendix B. Section 6 concludes.

The loss reserving problem is concerned with using currently available data to predict future claim amounts in a reliable manner. The available data is often arranged in a loss triangle which provides a visual representation of the development of claims up to the current time as well as what is required to be predicted (see Figure 1). We denote by $X_{i,j}$ and $C_{i,j}$ the incremental and cumulative claims for accident year $i$ and development year $j$ respectively. Denote by $\mathbb{B}=\{X_{i,j}:i+j\leq I+1\}$ the past claims data. Let $R_{i}$ represent reserves for accident year $i$ and $R$ represent total reserves.

The traditional chain-ladder method is probably the most famous reserving technique. This approach hinges on the assumption that development factors $f_{1},f_{2},...,f_{I-1}$ exist, such that $E[C_{i,j+1}|C_{i,j}]=f_{j}C_{i,j}$. These development factors are unknown and estimated by

$\displaystyle\widehat{f}_{j}=\frac{\sum_{i=1}^{I-j}C_{i,j+1}}{\sum_{i=1}^{I-j}C_{i,j}},\ 1\leq j\leq I-1.$

(2.1)

Ultimate claims for accident year $i$ are then estimated by $\widehat{C}_{i,I}=C_{i,I-i+1}\widehat{f}_{I-i+1}\cdots\widehat{f}_{I-1}$. From here accident year reserves and total reserves are subsequently estimated by

$$\widehat{R}_{i}=\widehat{C}_{i,I}-C_{i,I-i+1}\quad\text{and}\quad\widehat{R}=\sum_{i=1}^{I}\widehat{R}_{i}$$

(2.2)

Mack’s Model (Mack, 1993) is able to retain much of the simplicity of the deterministic chain-ladder whilst providing a formula for the mse of reserve estimates. The assumptions underlying classical chain-ladder reserves are the same for the first moment of reserves under Mack’s Model. The mse of prediction for individual accident year reserves is given by

$\displaystyle\text{mse}(\widehat{R}_{i})=C_{i,I-i+1}\sum_{j=I-i+1}^{I-1}(f_{I-i+1}\cdots f_{j-1}\sigma^{2}_{j}f^{2}_{j+1}\cdots f^{2}_{I-1})+C_{i,I-i+1}^{2}(f_{I-i+1}\cdots f_{I-1}-\widehat{f}_{I-i+1}\cdots\widehat{f}_{I-1})^{2}$

(2.3)

The estimate for the mse of total reserves is given by

$$\widehat{\text{mse}(\widehat{R})}=\sum_{i=2}^{I}\left\{\text{mse}(\widehat{R}_{i})+\widehat{C}_{i,I}\left(\sum_{j=i+1}^{I}\widehat{C}_{j,I}\right)\sum_{k=I-i+1}^{I-1}\frac{2\widehat{\sigma}_{k}^{2}/\widehat{f}_{k}^{2}}{\sum_{n=1}^{I-k}C_{n,k}}\right\}$$

(2.4)

where

$$\widehat{\sigma}_{k}^{2}=\frac{1}{I-k-1}\sum_{i=1}^{I-k}C_{i,k}(\frac{C_{i,k+1}}{C_{i,k}}-\widehat{f_{k}})^{2},\ 1\leq k\leq I-2$$

(2.5)

This paper is not conerned with tail-fitting, however, as outlined in (Mack, 1993), this approach leaves us without an estimator for $\sigma_{I-1}$ and we follow the same approach as in that paper to estimate $\sigma_{I-1}$ by requiring that $\frac{\widehat{\sigma}_{I-3}}{\widehat{\sigma}_{I-2}}=\frac{\widehat{\sigma}_{I-2}}{\widehat{\sigma}_{I-1}}$ holds as long as $\widehat{\sigma}_{I-3}>\widehat{\sigma}_{I-2}$ leading to

$$\widehat{\sigma}_{I-1}^{2}=\min(\frac{\widehat{\sigma}_{I-2}^{4}}{\widehat{\sigma}_{I-3}^{2}},\min(\widehat{\sigma}_{I-3}^{2},\widehat{\sigma}_{I-2}^{2}))$$

(2.6)

The Bornhuetter-Ferguson (BF) reserving methodology (Bornhuetter and Ferguson, 1972) is an opposing extreme to the chain-ladder (and Mack’s Model) in that it uses prior estimates of ultimate claims and development patterns rather than inducing them from the available data to date. It can be considered a highly robust method as the presence of outliers will not influence reserve estimates. However in practice, chain-ladder development factors are often used to infer the development pattern when using the BF approach. It is this BF approach that we will consider (otherwise meaningful results will not present themselves). This means that the only difference between the two techniques is the estimate of ultimate claims for a given accident year. In particular, the BF method uses a prior (exogenously determined) estimate whereas the chain-ladder method uses the available data to estimate ultimate claims. Bornhuetter-Ferguson reserves are given by

$\displaystyle\widehat{R}_{i}^{BF}$

$\displaystyle=$

$\displaystyle\widehat{C}^{BF}_{i,I}-C_{i,I-i+1}=\widehat{\mu}_{i}-\widehat{\mu}_{i}\frac{1}{\widehat{f}_{I-i+1}\cdot...\cdot\widehat{f}_{I-1}}\quad\text{and }\quad\widehat{R}^{BF}=\sum_{i=1}^{I}\widehat{R}_{i}^{BF},$

(2.7)

where $\widehat{\mu}_{i}$ represents the prior estimate of ultimate reserves for accident year $i$.

In this section we present impact functions for numerous statistics of interest under the assumptions of Mack’s Model. An impact function is able to highlight the sensitivity of a statistic of interest to a particular observation as well as pinpoint the marginal contribution of that observation to the final value of the statistic in some instances. This is done by taking the first derivative of the statistic with respect to the given observation. In our case we are interested in how an incremental claim $X_{k,j}$ may influence a given statistic $T$ such that the impact function is given by

$$\text{IF}_{k,j}(T)=\frac{\partial T}{\partial X_{k,j}}.$$

(2.8)

Further we have that if the statistic of interest $T$ is homogeneous of order one with respect to the $X_{k,j}$’s then

$$T=\sum_{\{k+j\leq I-1\}}\frac{\partial T}{\partial X_{k,j}}X_{k,j}.$$

(2.9)

The statistic, $T$, may represent reserves, the mse of reserve estimates or quantiles. It is interesting to investigate both the sign and magnitude of the impact functions to better understand the relationship a statistic has with an incremental claim. Furthermore, for those statistics that are homogeneous of order one, we may wish to see whether $\text{IF}_{k,j}(T)\cdot X_{k,j}$ is bounded or not. Boundedness will highlight that an outlying value of $X_{k,j}$ may only have a limited effect on T and hence this is a desirable property for robust estimators. If bounded, one should investigate the maximum value that $\text{IF}_{k,j}(T)\cdot X_{k,j}$ may take.

In the following subsections, we provide closed form equations to calculate the impact that each incremental claim is having on the aforementioned statistics in Mack’s Model. We also provide the impact that incremental claims are having on reserves under the Bornhuetter-Ferguson methodology (see Section 2.4).
The impact functions will be presented with the aid of an example using real data. This data is given in Appendix C. By applying Mack’s Model to this data set we calculate reserves for accident year 8 as $226 403 952, total reserves as $1 463 388 942, rmse for accident year 8 reserves as $9 448 925 and the rmse for total reserves as $45 480 914.

When calculating the impact function for this statistic we have considered the $\sigma_{j}$ and $f_{j}$ terms as known constants such that we are calculating the sensitivity of the mean squared error to incremental claims rather than the sensitivity of the estimate of this term. To approximate this function we may then then plug in the estimated values of $\sigma_{j}$ and $f_{s}$. The impact function is given by

$\displaystyle\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))=\begin{cases}0&\text{if }k>i\\ \\ \sum_{j=I-i+1}^{I-1}(f_{I-i+1}\cdot...\cdot f_{j-1}\sigma^{2}_{j}f_{j+1}^{2}\cdot...\cdot f_{I-1}^{2})+\\ 2C_{i,I-i+1}(\widehat{f}_{I-i+1}\cdot...\cdot\widehat{f}_{I-1})^{2}\sum_{s=I-i+1}^{I-1}\frac{\widehat{\sigma}^{2}_{s}/\widehat{f}_{s}^{2}}{\sum_{i=1}^{I-s}C_{i,s}}&\text{if }k=i\\ \\ -2C_{i,I-i+1}(\widehat{f}_{I-i+1}\cdot...\cdot\widehat{f}_{I-1})\sqrt{\sum_{s=I-i+1}^{I-1}\frac{\widehat{\sigma}^{2}_{s}/\widehat{f}_{s}^{2}}{\sum_{i=1}^{I-s}C_{i,s}}}\\ \widehat{C}_{i,I}\sum_{p=k}^{i-1}\left(\left(\frac{1}{\sum_{i=1}^{p}C_{i,I-p+1}}\right)\textbf{1}_{\{j\leq I-p+1\}}-\left(\frac{1}{\sum_{i=1}^{p}C_{i,I-p}}\right)\textbf{1}_{\{j\leq I-p\}}\right)&\text{if }k\leq i-1\end{cases}$

(4.1)

See Appendix B.6 for proof. Note that as the results for $\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))$ will be in units of $² it is often desirable to look at the impact function for the root mean squared error (rmse). This is simply given by $\text{IF}_{k,j}\left(\sqrt{\text{mse}(\widehat{R}_{i})}\right)=\frac{1}{2}\cdot\frac{\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))}{\sqrt{\text{mse}(\widehat{R}_{i})}}$ and allows for the results given to be in the same units as reserves. The impact that each incremental claim is having on the rmse of the reserves for accident year 8 is given in Table 2 with a graphical representation provided in Figure 7. From Equation (4.1) we have that for $k=i$, the impact is always the sum of two positive terms and is independent of $j$. Hence for $k=i$ the impact is always positive and equal. This is represented by the row of equal height positive columns for accident year 8 in Figure 7. Additionally, we observe that the sign of the impact is the opposite of that for $\text{IF}_{k,j}(\widehat{R}_{8})$ (except when $k=8$) and we observe similar trends in terms of magnitude. In particular, note that the impact is increasing in magnitude towards the top right corner observation however these impacts are negative.

For the cases when $k\leq i-1$ note that the term $-2C_{i,I-i+1}(\widehat{f}_{I-i+1}\cdot...\cdot\widehat{f}_{I-1})\sqrt{\sum_{s=I-i+1}^{I-1}\frac{\widehat{\sigma}^{2}_{s}/\widehat{f}_{s}^{2}}{\sum_{i=1}^{I-s}C_{i,s}}}$ is always negative and is independent of $k$ and $j$ (i.e. the same value for this term is used throughout the triangle for all $k\leq i-1$). Additionally, the term
$\widehat{C}_{i,I}\sum_{\{p\in\mathbb{D}|p\geq k\}}\left(\left(\frac{1}{\sum_{i=1}^{p}C_{i,I-p+1}}\right)\textbf{1}_{\{j\leq I-p+1\}}-\left(\frac{1}{\sum_{i=1}^{p}C_{i,I-p}}\right)\textbf{1}_{\{j\leq I-p\}}\right)$ is equal to $\text{IF}_{k,j}(\widehat{R}_{i})$ provided above. Hence we see that $\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))$ will have opposite sign to $\text{IF}_{k,j}(\widehat{R}_{i})$ throughout the triangle for $k\leq i-1$.

Notably, for $k\leq i-1$ and $j\leq I-i+1$ the impact will be positive which is shown in Figure 7 for $k\leq 7$ and $j\leq 3$. We will see a change of sign for $\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))$ from positive for $j\leq I-i+1$ to negative for $j=I-i+2$ when $k=i-1$. A proof of this property is given in Appendix B.7.

Similar trends in terms of the magnitude are seen for $\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))$ as was outlined above for $\text{IF}_{k,j}(\widehat{R}_{i})$. For instance as we move towards the top right corner of the loss triangle we will tend to see the impact become increasingly negative (as opposed to increasingly positive for $\text{IF}_{k,j}(\widehat{R}_{i})$). Further investigation into the relationship between $\text{IF}_{k,j}(\widehat{R}_{i})$) and $\text{IF}_{k,j}(\text{mse}(\widehat{R}_{i}))$, particularly the change in sign when $k\leq i-1$ is a warranted extension of this work.

We have that for Mack’s Model the mean squared error of prediction for total reserves is given by

$$\text{mse}(\widehat{R})=\sum_{i=2}^{I}\left\{(\text{s.e.}(\widehat{R}_{i}))^{2}+\widehat{C}_{i,I}\left(\sum_{q=i+1}^{I}\widehat{C}_{q,I}\right)\left(\sum_{r=I-i+1}^{I-1}\frac{2\sigma_{r}^{2}/\widehat{f}_{r}^{2}}{\sum_{n=1}^{I-r}C_{n,r}}\right)\right\}$$

(4.2)

In this case, we are again considering the unknown $\sigma_{r}$ values as constants rather than taking their estimates. This again allows us to focus on the impact that incremental claims are having on the mean squared error rather than its associated estimate. The impact function is given by

$\displaystyle\begin{split}\text{IF}_{k,j}\left(\widehat{\text{mse}(\widehat{R})}\right)=\sum_{i=2}^{I}\left\{\widehat{\text{IF}}_{k,j}(\text{mse}(\widehat{R}_{i}))+\widehat{C}_{i,I}\left(\sum_{q=i+1}^{I}\widehat{C}_{q,I}\right)\sum_{r=I-i+1}^{I-1}\frac{-2\sigma_{r}^{2}\sum_{n=1}^{I-r}\widehat{f}_{r}^{2}C_{n,r}\left(\frac{\partial\ln C_{n,r}}{\partial X_{k,j}}+\frac{2\partial\ln\widehat{f}_{r}}{\partial X_{k,j}}\right)}{\left(\sum_{n=1}^{I-r}C_{n,r}\widehat{f}_{r}^{2}\right)^{2}}\right.+\\ \left.\left(\sum_{r=I-i+1}^{I-1}\frac{2\sigma_{r}^{2}/\widehat{f}_{r}^{2}}{\sum_{n=1}^{I-r}C_{n,r}}\right)\left(\widehat{C}_{i,I}\left(\sum_{q=i+1}^{I}\left(\text{IF}_{k,j}(\widehat{R}_{q})+\frac{\partial C_{q,I-q+1}}{\partial X_{k,j}}\right)\right)+\left(\sum_{q=i+1}^{I}\widehat{C}_{q,I}\right)\left(\text{IF}_{k,j}(\widehat{R}_{i})+\frac{\partial C_{i,I-i+1}}{\partial X_{k,j}}\right)\right)\right\}\end{split}$

(4.3)

Note that this formulation of the impact function still contains derivative terms which are readily calculable. Importantly, these impacts are not simply the sum of the impacts for the mse of each individual accident year. A similar point is made regarding taking the impact of the rmse for total reserves ($\sqrt{\text{mse}(\widehat{R})}$) when calculating impact functions in practice such that we are looking at the impact in the same units as reserves. The impact of individual claims on the estimated rmse of total reserves is given in Figure 8. It appears that the main result for these impacts is that for development period 1, all impacts are positive and then holding $k$ constant the impacts are decreasing with $j$ towards zero and then continuing in this pattern, are becoming increasingly negative towards the upper right corner.

We have that

$$\widehat{R}_{i}=C_{i,I-i+1}(\widehat{f}_{I-i+1}\cdots\widehat{f}_{I-1})-C_{i,I-i+1}$$

(B.1)

Write

	$\displaystyle f(X)$	$\displaystyle=$	$\displaystyle C_{i,I-i+1}(\widehat{f}_{I-i+1}\cdots\widehat{f}_{I-1})$
	$\displaystyle\ln(f(X))$	$\displaystyle=$	$\displaystyle\ln(C_{i,I-i+1})+\ln(\widehat{f}_{I-i+1})+\cdots+\ln(\widehat{f}_{I-1})$
	$\displaystyle\frac{1}{f(X)}\frac{\partial f(X)}{\partial X_{k,j}}$	$\displaystyle=$	$\displaystyle\frac{\partial\ln(C_{i,I-i+1})}{\partial X_{k,j}}+\frac{\partial\ln(\widehat{f}_{I-i+1})}{\partial X_{k,j}}+\cdots+\frac{\partial\ln(\widehat{f}_{I-1})}{\partial X_{k,j}}$
	$\displaystyle\frac{\partial f(X)}{\partial X_{k,j}}$	$\displaystyle=$	$\displaystyle f(X)\left(\frac{\partial\ln(C_{i,I-i+1})}{\partial X_{k,j}}+\frac{\partial\ln(\widehat{f}_{I-i+1})}{\partial X_{k,j}}+\cdots+\frac{\partial\ln(\widehat{f}_{I-1})}{\partial X_{k,j}}\right)$
		$\displaystyle\therefore$

$$\frac{\partial\widehat{R}_{i}}{\partial X_{k,j}}=\widehat{C}_{i,I}\left(\frac{\partial\ln(C_{i,I-i+1})}{\partial X_{k,j}}+\frac{\partial\ln(\widehat{f}_{I-i+1})}{\partial X_{k,j}}+\cdots+\frac{\partial\ln(\widehat{f}_{I-1})}{\partial X_{k,j}}\right)-\frac{\partial}{\partial X_{k,j}}C_{i,I-i+1}$$

(B.2)

We have that

$\displaystyle\frac{\partial}{\partial X_{k,j}}C_{i,I-i+1}=\begin{cases}1,&\text{if }k=i\\ 0,&\text{otherwise}\end{cases}$

(B.3)

$\displaystyle\frac{\partial\ln(C_{i,I-i+1})}{\partial X_{k,j}}$

$\displaystyle=$

$\displaystyle\frac{\partial}{\partial X_{k,j}}\ln(X_{i,1}+\cdots+X_{i,I-i+1})=\begin{cases}\frac{1}{C_{i,I-i+1}},&\text{if }k=i\\ 0,&\text{otherwise}\end{cases}$

(B.4)

Now for $s\in{I-i+1,\dots,I-1}$,

	$\displaystyle\frac{\partial}{\partial X_{k,j}}\ln(\widehat{f}_{s})$	$\displaystyle=$	$\displaystyle\frac{\partial}{\partial X_{k,j}}\ln\left(\frac{\sum_{i=1}^{I-s}C_{i,s+1}}{\sum_{i=1}^{I-s}C_{i,s}}\right)$		(B.5)
		$\displaystyle=$	$\displaystyle\frac{\partial}{\partial X_{k,j}}(\ln\left(X_{1,1}+\cdots+X_{1,s+1}+\cdots+X_{I-s,1}+\cdots X_{I-s,s+1})\right.$
		$\displaystyle-$	$\displaystyle\left.\ln(X_{1,1}+\cdots+X_{1,s}+\cdots+X_{I-s,1}+\cdots X_{I-s,s})\right)$

Note that equation (B.1) is equal to zero if $k>I-s$. The largest value $I-s$ can take is $I-I+i-1=i-1$. The first two cases of the impact function (equation (3.2)) follow from equations (B.3), (B.4) and (B.1)).
We now derive the result for $k\leq i-1$. In these cases the impact function is given by

$\displaystyle\widehat{C}_{i,I}\cdot\frac{\partial}{\partial X_{k,j}}\left(\sum_{s=I-i+1}^{I-1}\ln(\widehat{f}_{s})\right)$

(B.7)

First consider when $s=I-i+1$.

	$\displaystyle\frac{\partial}{\partial X_{k,j}}\widehat{f}_{I-i+1}$	$\displaystyle=$	$\displaystyle\frac{\partial}{\partial X_{k,j}}\left(\ln(X_{1,1}+\cdots+X_{1,I-i+2}+\right.$		(B.8)
	$\displaystyle\left.\cdots+X_{i-1,1}+\cdots X_{i-1,I-i+2}\right)$	$\displaystyle-$	$\displaystyle\ln(X_{1,1}+\cdots+X_{1,I-i+1}+\cdots+X_{i-1,1}+\cdots X_{i-1,I-i+1}))$

Notice that $X_{k,j}$ will appear once only in the first logarithmic term on the right hand side of equation (B.8) if $j=I-i+2$. However it will appear once in both terms if $j\leq I-i+1$. Hence we have the following form for the cases when $s=I-i+1$.

$\displaystyle\frac{\partial}{\partial X_{k,j}}\ln(\widehat{f}_{I-i+1})=\begin{cases}\frac{1}{\sum_{q=1}^{i-1}C_{q,I-i+2}}-\frac{1}{\sum_{q=1}^{i-1}C_{q,I-i+1}},&\text{if }j\leq I-i+1\\ \frac{1}{\sum_{q=1}^{i-1}C_{q,I-i+2}},&\text{if }j=I-i+2\end{cases}$

(B.9)

A similar result can be obtained for $s=I-i+2$ such that

$\displaystyle\frac{\partial}{\partial X_{k,j}}\ln(\widehat{f}_{I-i+2})=\begin{cases}\frac{1}{\sum_{q=1}^{i-2}C_{q,I-i+3}}-\frac{1}{\sum_{q=1}^{i-2}C_{q,I-i+2}},&\text{if }j\leq I-i+2\\ \frac{1}{\sum_{q=1}^{i-2}C_{q,I-i+3}},&\text{if }j=I-i+3\end{cases}$

(B.10)

An important point to note is that for each development factor, we are summing the cumulative payments in the denominator(s) up to $I-s$. In equation (B.9) this is to $i-1$ such that all values of $k\leq i-1$ are included. However in equation (B.10) this is only up to $i-2$. This implies that if $k=i-1$ then $X_{k,j}$ will only appear in the first development factor in equation (B.2). This pattern will continue such that for $k=i-2$, $X_{k,j}$ will only appear in the first two developments factors and so on. For $k=1$, $X_{k,j}$ will appear in all development factors. This completes the proof.