Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach

1. Introduction

There has been extensive literature on loss reserving models over the past 25 years, including the Mack (1993) model. While the focus has been largely on how to tackle outstanding claims liabilities, relatively few materials have been presented for premium liabilities. Some references include Cantin and Trahan (1999), Buchanan (2002), Collins and Hu (2003), and Yan (2005), which focus on the central estimate (i.e., the mean) of premium liabilities but not on the underlying variability.

As noted in Clark et al. (2003), the International Accounting Standards Board (IASB) has proposed a new reporting regime for insurance contracts, in which both outstanding claims liabilities and premium liabilities should be assessed at their fair values. It is generally understood that this fair value includes a “margin” allowing for different types of variability for insurance liabilities. Accordingly, the Australian Prudential Regulation Authority (APRA) has prescribed an approach similar to the fair value approach. Under Prudential Standard GPS 310, a “risk margin” has to be explicitly calculated such that outstanding claims liabilities and premium liabilities are assessed at a sufficiency level of 75%, subject to a minimum of the mean plus one-half the standard deviation. Australian Accounting Standard AASB 1023 also requires inclusion of a risk margin, though there is no prescription on the adequacy level. No matter what approach one takes, it is obvious that urgency for developing proper tools to measure liability variability exists not only for outstanding claims liabilities but also for premium liabilities. In addition, according to Yan (2005), premium liabilities account for around 30% of insurance liabilities for direct insurers and 15% to 20% for reinsurers in Australia from 2002 to 2004. Premium liabilities represent a significant portion of an insurer’s liabilities, and proper assessment of the underlying variability should not be overlooked.

The definition of premium liabilities varies for different countries. Broadly speaking, premium liabilities refer to all future claim payments and associated expenses arising from future events after the valuation date which are insured under the existing unexpired policies. Buchanan (2002) notes that there are two main methods of determining the central estimate of premium liabilities. The first method is prospective in nature and involves a full actuarial assessment from first principles. Yan (2005) calls this method the claims approach and differentiates it into the loss ratio approach and historical claims approach. The loss ratio approach is the most common one for premium liability assessment in practice and is essentially an extension of the outstanding claims liability valuation. It applies a projected loss ratio to the unearned premiums or number of policies unexpired. The historical claims approach uses the number of claims and average claim size and is more suitable for short-tailed lines of business where data is sufficient. While the historical claims approach has been studied extensively under the classical risk theory, the loss ratio approach has received relatively little attention in the literature. In this paper we follow the loss ratio approach and attempt to supplement this knowledge gap.

On the other hand, the second method noted in Buchanan (2002) is retrospective in nature and involves an adjustment of the unearned premiums to take out the profit margin. As discussed in Cantin and Trahan (1999) and Yan (2005), both Canadian and Australian accounting standards require a reporting of this unearned premiums item, in which a premium deficiency reserve is added if this item is less than the premium liability estimate determined by the first method. Obviously the first method above plays a key role in premium liability assessment, and we focus on the loss ratio approach under this prospective method.

In this paper we construct a stochastic model to estimate the standard error of prediction under the loss ratio approach of assessing premium liabilities. We focus on modeling the future claims which form the largest component in premium liabilities (about 85% according to Collins and Hu 2003). We look at the weighted average ultimate loss ratio and simple average ultimate loss ratio, and derive approximation formulae to estimate the corresponding mean square error of prediction with respect to the accident year following the valuation date. As similarly reasoned in Taylor (2000), the resulting mean square error of prediction is composed of the process error component and the estimation error component, and no covariance term exists as one part is related only to the future while the other only to the past. We also illustrate the application of our model to Australian private-sector direct insurers’ public liability data and some hypothetical data simulated from the compound Poisson model.

The outline of the paper is as follows. In Section 2 we introduce the basic notation and assumptions of our model. In Section 3 we present the formulae for estimating the standard error of prediction for premium liabilities. In Section 4 we apply the model to public liability data and simulated data and analyze the results. In Section 5 we set forth our concluding remarks. Appendices A to D furnish the proofs for the formulae stated in this paper.

2. Notation and assumptions

Let \(C_{i, j}\) (for \(1 \leq i \leq n+1\) and \(1 \leq j \leq n\) ) be a random variable representing the cumulative claim amount (either paid or incurred) of accident year \(i\) and development year \(j\). Assuming all claims are settled in \(n\) years, \(C_{i, n}\) represents the ultimate claim amount of accident year \(i\). We consider the case where a run-off triangle of \(C_{i, j}\) 's is available for \(i+j \leq n+1\). In effect, the valuation date is at the end of accident year \(n, C_{i, j}\) 's for \(i+j>n+1\) and \(1 \leq i \leq n\) refer to the future claims of outstanding claims liabilities, and \(C_{n+1, j}\) 's refer to the future claims of premium liabilities. Let \(E_i\) (for \(1 \leq i \leq n+1\) ) be the premiums of accident year \(i\). The premiums are assumed to be known. The term \(C_{i, n} / E_i\) then becomes the ultimate loss ratio of accident year \(i\).

It is also assumed that exposure is evenly distributed over each year, and the exposure distribution of accident year n + 1 is the same as that of the past accident years. In reality, the future exposure relating to premium liabilities would arise more from the earlier part of accident year n + 1, while the past exposure would spread more uniformly across the whole year. Although the timing of claims development is actually different between the two cases, the way that the claims develop to ultimate remains basically the same. As our focus is on the ultimate loss ratio, this approximation is reasonable and represents a convenient simplification for the model setting.

As mentioned in the Introduction, the loss ratio approach for the premium liability valuation is basically an extension of the outstanding claims liability valuation. Hence we start with the structure of the chain ladder method, which is the most common method for assessing outstanding claims liabilities in practice and is linked to a distribution-free model in Mack (1993). Incorporating \(E_{i}\) into the three basic assumptions of the Mack (1993) model, we deduce the following for 1 ≤ i ≤ n + 1:

\[ \begin{array}{c} \mathrm{E}\left(\left.\frac{C_{i, j+1}}{E_{i}} \right\rvert\, C_{i, 1}, C_{i, 2}, \ldots, C_{i, j}\right)=\frac{C_{i, j}}{E_{i}} f_{j} ; \\ (\text { for } 1 \leq j \leq n-1) \end{array} \tag{2.1} \]

\[ \begin{array}{c} \operatorname{Var}\left(\left.\frac{C_{i, j+1}}{E_{i}} \right\rvert\, C_{i, 1}, C_{i, 2}, \ldots, C_{i, j}\right)=\frac{C_{i, j}}{E_{i}^{2}} \sigma_{j}^{2} ; \\ (\text { for } 1 \leq j \leq n-1) \end{array} \tag{2.2} \]

\[\begin{array}{c} C_{i, j} \text{ and } C_{g, h} \text{ are independent.} \\ (\text{for } i \neq g ) \end{array} \tag{2.3} \]

The parameter \(f_j\) is the development ratio and the parameter \(\sigma_j^2\) is related to the conditional variance of \(C_{i, j+1}\). These two parameters are unknown and need to be estimated from the claims data.

As of the valuation date, there is no claims data for accident year n + 1. In order to model the future claims in the first development year of accident year n + 1, we add the following two assumptions for 1 ≤ i ≤ n + 1, which are analogous to those for new claims in Schnieper (1991):

\[ \mathrm{E}\left(\frac{C_{i, 1}}{E_{i}}\right)=u ; \tag{2.4} \]

\[ \operatorname{Var}\left(\frac{C_{i, 1}}{E_{i}}\right)=\frac{v^{2}}{E_{i}} . \tag{2.5} \]

Rearranging assumptions (2.4) and (2.5) into \(\mathrm{E}\left(C_{i, 1}\right)=E_i u\) and \(\operatorname{Var}\left(C_{i, 1}\right)=E_i v^2\), we can see that the mean and variance of the claim amount of the first development year is effectively assumed to be proportional to the premiums. This is analogous to assumptions (2.1) and (2.2), in which the conditional mean and variance of the claim amount \(C_{i, j}\) for \(2 \leq j \leq n\) depends on the previous development year’s claim amount \(C_{i, j-1}\). The parameters \(u\) and \(v^2\) are unknown and can be estimated from the claims and premiums data.

Mack (1993) suggests the following unbiased estimators for \(f_j\) and \(\sigma_j^2\) and proves that \(\hat{f}_j\) and \(\hat{f}_h\) are uncorrelated for \(j \neq h\) :

\[ \begin{array}{c} \hat{f}_{j}=\frac{\sum_{r=1}^{n-j} C_{r, j+1}}{\sum_{r=1}^{n-j} C_{r, j}}=\frac{\sum_{r=1}^{n-j} C_{r, j} \frac{C_{r, j+1}}{C_{r, j}}}{\sum_{r=1}^{n-j} C_{r, j}} ; \\ \quad(\text { for } 1 \leq j \leq n-1) \end{array} \tag{2.6} \]

\[ \begin{aligned} \hat{\sigma}_{j}^{2}= & \frac{1}{n-j-1} \sum_{r=1}^{n-j} C_{r, j}\left(\frac{C_{r, j+1}}{C_{r, j}}-\hat{f}_{j}\right)^{2} ; \\ & \quad(\text { for } 1 \leq j \leq n-2) \\ \hat{\sigma}_{n-1}^{2}= & \min \left(\frac{\hat{\sigma}_{n-2}^{4}}{\hat{\sigma}_{n-3}^{2}}, \hat{\sigma}_{n-3}^{2}\right) . \end{aligned} \tag{2.7} \]

We now introduce two unbiased estimators for u and v² as follows, which are again based on Schnieper (1991):

\[ \hat{u}=\frac{\sum_{r=1}^{n} C_{r, 1}}{\sum_{r=1}^{n} E_{r}}=\frac{\sum_{r=1}^{n} E_{r} \frac{C_{r, 1}}{E_{r}}}{\sum_{r=1}^{n} E_{r}} ; \tag{2.8} \]

\[ \hat{v}^{2}=\frac{1}{n-1} \sum_{r=1}^{n} E_{r}\left(\frac{C_{r, 1}}{E_{r}}-\hat{u}\right)^{2} . \tag{2.9} \]

It can be seen that both formulae (2.6) and (2.8) are weighted averages and that both formulae (2.7) and (2.9) use weighted sums of squares. The proofs for unbiasedness of û and v̂² are given in Appendix A.

In effect, we integrate the model assumptions in Mack (1993) with those of development year one for new claims in Schnieper (1991). The overall structure is based on the chain ladder method. It then becomes possible to assess the next accident year’s ultimate loss ratio using the observed run-off triangle. As shown in the next section, the results of the outstanding claims liability valuation (i.e., projected ultimate loss ratios of the past accident years) are carried through to the premium liability valuation (regarding the expected ultimate loss ratio of the accident year following the valuation date).

3. Standard error of prediction

In practice, actuaries often examine the projected ultimate loss ratios of past accident years and compare these figures with target or budget ratios or industry ratios to obtain an estimate of the next accident year’s ultimate loss ratio. Here we assume no such prior knowledge or objective information is available and investigate the following two estimators for the next accident year’s expected ultimate loss ratio \(q=\) \(\mathrm{E}\left(C_{n+1, n} / E_{n+1}\right):\)

\[ \begin{aligned} \hat{q} & =\frac{\sum_{i=1}^{n} C_{i, n+1-i} \hat{f}_{n+1-i} \hat{f}_{n+2-i} \cdots \hat{f}_{n-1}}{\sum_{i=1}^{n} E_{i}} \\ & =\frac{\sum_{i=1}^{n} C_{i, n+1-i} S_{n+1-i, n-1}}{\sum_{i=1}^{n} E_{i}}=\frac{\sum_{i=1}^{n} \hat{C}_{i, n}}{\sum_{i=1}^{n} E_{i}} \\ & =\frac{\sum_{i=1}^{n} E_{i} \frac{\hat{C}_{i, n}}{E_{i}}}{\sum_{i=1}^{n} E_{i}} ; \end{aligned} \tag{3.1} \]

\[ \begin{aligned} \hat{q}^{*} & =\frac{1}{n} \sum_{i=1}^{n} \frac{C_{i, n+1-i} \hat{f}_{n+1-i} \hat{f}_{n+2-i} \cdots \hat{f}_{n-1}}{E_{i}} \\ & =\frac{1}{n} \sum_{i=1}^{n} \frac{C_{i, n+1-i} S_{n+1-i, n-1}}{E_{i}}=\frac{1}{n} \sum_{i=1}^{n} \frac{\hat{C}_{i, n}}{E_{i}} . \end{aligned} \tag{3.2} \]

Let \(S_{j, h}=\hat{f}_j \hat{f}_{j+1} \ldots \hat{f}_h\) (for \(j \leq h\); equal to one otherwise) and \(\hat{C}_{i, j}=C_{i, n+1-i} S_{n+1-i, j-1}\) (for \(i+\) \(j>n+1\) and \(1 \leq i \leq n) . \hat{C}_{1, n}\) is read as equal to \(C_{1, n}\). Formula (3.1) gives a weighted average, while formula (3.2) provides a simple average. Both estimators are unbiased and the proofs are set forth in Appendix B. For the expected future claims component of premium liabilities of the next accident year \(\mathrm{E}\left(C_{n+1, n}\right)\), we define its estimator as

\[ \hat{C}_{n+1, n}=E_{n+1} \hat{q} . \tag{3.3} \]

For now we deal with (3.1) and, as shown later, the results of (3.1) can readily be extended to the use of (3.2). We will also look at the effects of excluding some accident years when calculating q, as a practitioner may exclude or adjust a few years’ projected loss ratios that are regarded as inconsistent with the rest, out of date, or irrelevant. Such circumstances arise when there have been past changes in, for example, the regulations, underwriting procedures, claims management, business mix, or reinsurance arrangements.

Using the idea in Chapter 6 of Taylor (2000), we define the mean square error of prediction of the estimator q̂ as follows:

\[ \begin{array}{l} \begin{aligned} \operatorname{MSEP}(\hat{q}) & =\mathrm{E}\left(\left(\frac{C_{n+1, n}}{E_{n+1}}-\hat{q}\right)^{2}\right) \\ & =\mathrm{E}\left(\left(\frac{C_{n+1, n}}{E_{n+1}}-q+q-\hat{q}\right)^{2}\right) \\ & =\mathrm{E}\left(\left(\frac{C_{n+1, n}}{E_{n+1}}-q\right)^{2}\right)+\mathrm{E}\left((\hat{q}-q)^{2}\right) \end{aligned}\\ \text { ( } C_{n+1, n} \text { and } \hat{q} \text { are independent due to (2.3)) }\\ =\operatorname{Var}\left(\frac{C_{n+1, n}}{E_{n+1}}\right)+\operatorname{Var}(\hat{q}) . \quad(\hat{q} \text { is unbiased }) \end{array} \tag{3.4} \]

The mean square error of prediction above consists of two components: the first allows for process error and the second for estimation error. The process error component refers to future claims variability and the estimation error component refers to the uncertainty in parameter estimation due to sampling error. As similarly noted in Taylor (2000), there is no covariance term in (3.4) because at the valuation date, \(C_{n+1, n}\) is entirely related to the future while \(\hat{q}\) is completely based on the past observations.

The corresponding standard error of prediction can then be calculated as

\[ \operatorname{SEP}(\hat{q})=\sqrt{\operatorname{MSEP}(\hat{q})} .\tag{3.5} \]

For the next accident year’s expected ultimate claim amount, we compute the standard error of prediction of its estimator as

\[ \operatorname{SEP}\left(\hat{C}_{n+1, n}\right)=E_{n+1} \operatorname{SEP}(\hat{q})=E_{n+1} \sqrt{\operatorname{MSEP}(\hat{q})} .\tag{3.6} \]

We derive the process error component as follows and the proof is given in Appendix C:

\[ \begin{aligned} \operatorname{Var}\left(\frac{C_{n+1, n}}{E_{n+1}}\right)= & \frac{1}{E_{n+1}} \mathrm{E}\left(\frac{C_{n+1, n}}{E_{n+1}}\right) \\ & \times \sum_{j=1}^{n-1} \frac{\sigma_{j}^{2}}{f_{j}} f_{j+1} f_{j+2} \ldots f_{n-1} \\ & +\frac{v^{2}}{E_{n+1}} f_{1}^{2} f_{2}^{2} \ldots f_{n-1}^{2}, \end{aligned} \tag{3.7} \]

which can be estimated by

\[ \begin{aligned} \widehat{\operatorname{Var}}\left(\frac{C_{n+1, n}}{E_{n+1}}\right)= & \frac{\hat{q}}{E_{n+1}} \sum_{j=1}^{n-1} \frac{\hat{\sigma}_{j}^{2}}{\hat{f}_{j}} S_{j+1, n-1} \\ & +\frac{\hat{v}^{2}}{E_{n+1}} S_{1, n-1}^{2} . \end{aligned} \tag{3.8} \]

The estimation error component requires more computation. We derive the following approximation for this component and the proof is provided in Appendix D:

\[ \begin{aligned} \operatorname{Var}(\hat{q}) \approx & \frac{1}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{j=1}^{n-1}\left(\sum_{i=n+1-j}^{n} \frac{\mathrm{E}\left(C_{i, n}\right)}{f_{j}}\right)^{2} \operatorname{Var}\left(\hat{f}_{j}\right) \\ & +\frac{1}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{i=1}^{n} f_{n+1-i}^{2} f_{n+2-i}^{2} \ldots f_{n-1}^{2} \operatorname{Var}\left(C_{i, n+1-i}\right) \\ & +\frac{2}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{j=1}^{n-1} \sum_{i=1}^{n-j}\left(\sum_{r=n+1-j}^{n} \frac{\mathrm{E}\left(C_{r, n}\right)}{f_{j}}\right) \\ & \times\left(f_{n+1-i} f_{n+2-i} \ldots f_{n-1}\right) \operatorname{Cov}\left(\hat{f}_{j}, C_{i, n+1-i}\right). \end{aligned} \tag{3.9} \]

which can be estimated by

\[ \begin{aligned} \widehat{\operatorname{Var}}(\hat{q})= & \frac{1}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{j=1}^{n-1}\left(\sum_{i=n+1-j}^{n} \frac{\hat{C}_{i, n}}{\hat{f}_{j}}\right)^{2} \widehat{\operatorname{Var}}\left(\hat{f}_{j}\right) \\ & +\frac{1}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{i=1}^{n} S_{n+1-i, n-1}^{2} \widehat{\operatorname{Var}}\left(C_{i, n+1-i}\right) \\ & +\frac{2}{\left(\sum_{i=1}^{n} E_{i}\right)^{2}} \sum_{j=1}^{n-1} \sum_{i=1}^{n-j} \sum_{r=n+1-j}^{n} \frac{\hat{C}_{r, n}}{\hat{f}_{j}} \\ & \times S_{n+1-i, n-1} \widehat{\operatorname{Cov}}\left(\hat{f}_{j}, C_{i, n+1-i}\right), \end{aligned} \tag{3.10} \]

where the estimators of the variance and covariance terms are derived as

\[ \widehat{\operatorname{Var}}\left(\hat{f}_{j}\right)=\frac{\hat{\sigma}_{j}^{2}}{\sum_{r=1}^{n-j} C_{r, j}} ; \tag{3.11} \]

\[ \widehat{\operatorname{Var}}\left(C_{i, n+1-i}\right)=C_{i, n+1-i} \sum_{j=1}^{n-i} \frac{\hat{\sigma}_{j}^{2}}{\hat{f}_{j}} S_{j+1, n-i}+E_{i} \hat{v}^{2} S_{1, n-i}^{2} ; \tag{3.12} \]

\[ \widehat{\operatorname{Cov}}\left(\hat{f}_{j}, C_{i, n+1-i}\right)=\frac{C_{i, n+1-i}}{\sum_{r=1}^{n-j} C_{r, j}} \frac{\hat{\sigma}_{j}^{2}}{\hat{f}_{j}} . \tag{3.13} \]

By now we have shown all the formulae that are needed to calculate the standard error of prediction of (3.1). Note that the term \(f_j f_{j+1} \ldots f_{n-1}\) for \(j>n-1\) is read as equal to one in the summations. In the next section we will apply these formulae to some real claims data and simulated data.

4. Illustrative examples

We first apply the formulae shown previously to some public liability data. We use the aggregated claim payments and premiums (both gross and net of reinsurance) of the private-sector direct insurers from the “Selected Statistics on the General Insurance Industry” (APRA) for accident years 1981 to 1991 (n = 10). Adopting the approach as described in Hart, Buchanan, and Howe (1996), all the figures have been converted to constant dollar values in accordance with the average weekly ordinary time earnings (AWOTE) before the calculations. This procedure is common in practice and is based on the assumption that wage inflation is the “normal” inflation for the claims.

The inflation-adjusted claims (incremental) and premiums data are presented in Table 1 below for gross of reinsurance and in Table 2 for net of reinsurance. All the figures are in thousands.

The two run-off triangles show that it takes several years for public liability claims to develop and this line of business is generally regarded as a long-tailed line of business. We use formulae (2.6) to (2.9), (3.1), and (3.3) to estimate the parameters, accident year 1991’s expected ultimate loss ratio, and so the expected future claims of premium liabilities. We then adopt formulae (3.4) to (3.13) to compute the corresponding standard error of prediction. Table 3 below presents our results both gross and net of reinsurance.

As shown in Table 3, the estimated gross and net expected ultimate loss ratios for accident year 1991 are 49.2% and 53.6%. The standard error of prediction for the future claims of premium liabilities, expressed as a percentage of the mean, is greater for gross than for net. The gross and net percentages are 47.1% and 33.1%, respectively. This feature is in line with the general perception that gross liability variability is greater than its net counterpart. In both cases the process error component is larger than the estimation error component.

All accident years’ estimated ultimate loss ratios are fairly consistent with one another except the gross loss ratio of accident year 1983. A closer look at the claims data reveals that the gross claim payments made at accident year 1983 and development year 6 are $92,888 thousand, the amount of which is much larger than the other figures in the same development year. We find that if the amount is changed to say $18,000 thousand, then the standard error of prediction reduces significantly from 47.1% to 35.5%. Whether to allow for this extra variability or adjust the data is a matter of judgment and in practice requires further investigation into the underlying features of those claims.

As mentioned in the previous section, one can exclude some accident years’ loss ratios when calculating q if those loss ratios are considered inconsistent, out of date, or irrelevant. This computation can readily be done by setting an indicator variable for each accident year, in which the indicator is one if the loss ratio of that accident year is included and zero otherwise. Table 4 below demonstrates some results of using different numbers of accident years in computing q and SEP(q̂) with (3.1), (3.8), and (3.10).

For each case of a particular number of accident years being included, Table 4 sets out the average figures across all the possible combinations of accident years in that case. It can be seen that the estimation error component and so the standard error of prediction decreases when more accident years (i.e., more data) are used. The process error component is stable because in our analysis, the indicator adjustments are only applied to (3.1) and (3.10) but not (2.6) to (2.9).

Hitherto we have been focusing on the use of (3.1). In many situations one may prefer using a simple average of loss ratios as in (3.2). We only need to replace (3.9) and (3.10) with the following, the proof of which is analogous to Appendix D:

\[ \begin{aligned} \operatorname{Var}\left(\hat{q}^{*}\right) \approx & \frac{1}{n^{2}} \sum_{j=1}^{n-1}\left(\sum_{i=n+1-j}^{n} \frac{\mathrm{E}\left(C_{i, n}\right)}{E_{i} f_{j}}\right)^{2} \operatorname{Var}\left(\hat{f}_{j}\right) \\ & +\frac{1}{n^{2}} \sum_{i=1}^{n} \frac{f_{n+1-i}^{2} f_{n+2-i}^{2} \ldots f_{n-1}^{2}}{E_{i}^{2}} \operatorname{Var}\left(C_{i, n+1-i}\right) \\ & +\frac{2}{n^{2}} \sum_{j=1}^{n-1} \sum_{i=1}^{n-j}\left(\sum_{r=n+1-j}^{n} \frac{\mathrm{E}\left(C_{r, n}\right)}{E_{r} f_{j}}\right) \\ & \times\left(\frac{f_{n+1-i} f_{n+2-i} \ldots f_{n-1}}{E_{i}}\right) \operatorname{Cov}\left(\hat{f}_{j}, C_{i, n+1-i}\right), \end{aligned} \tag{4.1} \]

which can be estimated by

\[ \begin{aligned} \widehat{\operatorname{Var}}\left(\hat{q}^{*}\right)= & \frac{1}{n^{2}} \sum_{j=1}^{n-1}\left(\sum_{i=n+1-j}^{n} \frac{\hat{C}_{i, n}}{E_{i} \hat{f}_{j}}\right)^{2} \widehat{\operatorname{Var}}\left(\hat{f}_{j}\right) \\ & +\frac{1}{n^{2}} \sum_{i=1}^{n} \frac{S_{n+1-i, n-1}^{2}}{E_{i}^{2}} \widehat{\operatorname{Var}}\left(C_{i, n+1-i}\right) \\ & +\frac{2}{n^{2}} \sum_{j=1}^{n-1} \sum_{i=1}^{n-j} \sum_{r=n+1-j}^{n} \frac{\hat{C}_{r, n}}{E_{r} \hat{f}_{j}} \\ & \times \frac{S_{n+1-i, n-1}}{E_{i}} \widehat{\operatorname{Cov}}\left(\hat{f}_{j}, C_{i, n+1-i}\right). \end{aligned} \tag{4.2} \]

The estimated results using (3.2), (3.8), and (4.2) are shown in Table 5. The resulting ultimate loss ratios are slightly larger than previously while the standard error of prediction estimates are slightly larger in magnitude but smaller in percentage.

Finally we apply our formulae to some hypothetical data simulated from the compound Poisson model \(X_{i, j}=\sum_{k=1}^{N_{i, j}} Y_{i, j, k}\). Let \(X_{i, j}\) be a random variable representing the incremental claim amount of accident year \(i\) and development year \(j\) and so \(C_{i, j}=C_{i, j-1}+X_{i, j}\) for \(2 \leq j \leq 10\) and \(C_{i, 1}=X_{i, 1}\). Let \(N_{i, j}\) and \(Y_{i, j, k}\) be independent random variables representing the number of claims and the size of the \(k\) th claim of accident year \(i\) and development year \(j\). Let \(N_{i, j} \sim \operatorname{Pn}\left(E_i \lambda_j\right)\) and \(Y_{i, j, k} \sim \mathrm{LN}(\mu, \sigma)\) where \(\lambda_j\) 's are equal to \(9 \times 10^{-6}\), \(8 \times 10^{-6}, 7 \times 10^{-6}, 6 \times 10^{-6}, 5 \times 10^{-6}, 5 \times 10^{-6}\), \(4 \times 10^{-6}, 3 \times 10^{-6}, 2 \times 10^{-6}, 1 \times 10^{-6}\) respectively for \(1 \leq j \leq 10, \mu=8.8638\), and \(\sigma=\) \(0.8326 \quad\left(\mathrm{E}\left(Y_{i, j, k}\right)=10,000 \quad\right.\) and \(\quad \operatorname{SD}\left(Y_{i, j, k}\right)=\) \(10,000)\). We assume \(E_i\) grows from \(1,000,000\) at \(10 \%\) each year and the unearned premiums are half of \(E_{11}\). Effectively, accident year 11’s ultimate loss ratio has a mean of \(50 \%\) and a variance of 0.0077 . We then simulate a run-off triangle based on this compound Poisson model and apply our formulae (3.2), (3.8), and (4.2) to this triangle.

Under the compound Poisson model above, \(X_{i, j}\) 's are independent while under our model, \(C_{i, j+1}\) depends on \(C_{i, j}\). Hence we expect our formulae to produce a process error estimate larger than the true variance underlying the simulated data. The simulated run-off triangle and estimated results are presented in Tables 6 and 7.

As expected, the process error estimate of 0.0259 is larger than the underlying variance of 0.0077. In dealing with real claims data, one should check the underlying assumptions thoroughly regarding the conditional relationships or independence between different development years.

5. Concluding remarks

In this paper we examine the weighted and simple average loss ratio estimators and construct a stochastic model to derive some simple approximation formulae to estimate the standard error of prediction for the future claims component of premium liabilities. Based on the idea in Taylor (2000), we deduce the mean square error of prediction as comprising the process error component and the estimation error component, and no covariance term exists as the first part is associated only with the future while the second part only with the past observations. We apply these formulae to some public liability data and simulated data and the results are reasonable in general. Since the starting part of our model follows the structure of the chain ladder method, one may apply the various tests stated in Mack (1994) to check whether the model assumptions are valid for the claims data under investigation.

The formulae derived in this paper appear to serve as a good starting point for assessment of premium liability variability in practice. Nevertheless, there are other practical considerations in dealing with premium liabilities such as the insurance cycle, claims development in the tail, catastrophes, superimposed inflation, multi-year policies, policy administration and claims handling expenses, future recoveries, future reinsurance costs, retrospectively rated policies, unclosed business, refund claims, and future changes in reinsurance, claims management, and underwriting. To deal with these issues, a practitioner needs to judgmentally adjust the data or make an explicit allowance, based on managerial, internal, and industry information.

Acknowledgments

The author thanks the editors and the anonymous referees for their helpful and constructive comments. This research was partially supported by Nanyang Technological University AcRF Grant, Singapore.