In the process of data cleansing, claims data are cleansed while the obvious mistakes are corrected. In the case of large companies who mostly have larger claims compared to the smaller entities, the cleansing of data may be done by teams that specialize in data cleansing wherein they have systematic ways of checking for the anomalies such as data in the wrong format, material misstatements, and so on. Anything that could be found dubious will be referred back to the analyst and if some fields are missing, suitable assumptions must be made about them.

The ruin theory or also known as risk theory uses mathematical models to describe an insurer’s vulnerability to insolvency/ruin. In such models, key quantities of interest are the probability of ruin, distribution of surplus immediately before ruin, and deficit at the time of ruin. This theory of the classical risk model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims.

The classical risk model is one of the basic models used in general insurance. It aims to build a time-resolving risk model that captures the evolution of the reserves of an insurance company. It also follows the principle of putting separate models together for the arrivals of the claims and the claim sizes. It is an approach that is already proven to be useful and tractable. Furthermore, it incorporates an extra ingredient in describing the inflow of premium income which would help the insured company to better understand and to have better references for the ideas.

Based on all the assumptions and processes, the classical risk model involves several simplifications of how an insurance business operates in practice. One good example of this is when the model assumes that the claim-arrival rate remains constant over time, that no interest will arise on the surplus, that there is no inflation, that the premium income is received continuously in time, that the claims are paid out immediately, and many more independence assumptions. Classical risk models form the basis of many models used in insurance mathematics.

The adjustment coefficient depends on the other parameters in the risk model such as in the situation where it depends on the relative safety loading wherein If the parameter 0 increases, therefore it can be interpreted in terms of the portfolio being “safer” so the adjustment coefficient R is expected to be larger. The adjustment coefficient is the number R appearing in the famous Lundberg upper bound wherein a compound Poisson risk process with initial capital u ≥ 0, the ruin probability, ψ(u), is bounded by e−Ru. By changing the safety loading or the distribution of the individual claims that are involved in its definition, the value of R, and with it, the ruin probability can be adjusted.

In pricing, there are several requirements that the data should meet. The first part of the actual analysis mostly starts with data and it is important to have appropriate data to undertake the risk costing exercise. There are several sources of data that are discussed which include the information from prospective clients and external information to the client such as portfolio or market information. There are also several reasons why understanding the data is important for the pricing exercise and one of the most obvious reasons is that it is necessary as this data is often requested and needed to make sure that it meets all the requirements.

The first step in understanding the data needed for the pricing exercise is the collection of the basic information on the policies that are supposed to price. In the case of commercial lines, different policies may arise. There may be a single policy such as the employers’ liability purchased by an organization. In the case of personal lines, pricing is carried out on a portfolio of policies rather than on each policy in isolation. In the case of reinsurance treaties, it normally prices a single reinsurance policy with an underlying portfolio of policies underwritten by the reinsured.

The claims data in the general insurance includes several fields but it is important to take note that there are times when the practitioners refer to ‘loss’ and ‘claim’ in two different things. According to the practitioners, a loss refers to the loss of the insured while a claim refers to the amount the insured claims based on the policy that the insured purchased. However, in normal circumstances, the two terms are often used interchangeably. There are ideal situations where one can obtain a full list of movements for all claims. However, we are often asked to price a policy simply based on a snapshot of the claims situation overall years under examination at one point in time, rather than having the full history for each claim.

The exposure is only a rough measure of risk so it is important to improve it by taking into consideration all the factors that are related to the risk. In the situation of Mr. Smith, it is important to split the total wageroll of the workers into its two different factors which is the number of clerical and manual workers the DG Corp. have so, in case of any changes in the mix of the workforce which would correspond to a change in the total risk, it would be easy to detect where the mistakes or the data came from. This would also provide a better measure of exposure than only considering the total wageroll.

The third criterion identified by the Casualty Actuarial Society is that there are historical precedents for the use of the measure of exposure. This criterion should be taken with a pinch of salt as the use of non-traditional measures of exposures may cause problems of acceptance and may also change the way that premiums are being determined. However, those problems should be an excuse for persisting in using inadequate measures otherwise, the same argument would be used for discouraging the introduction of new rating factors because they lead to changes in the premium structures.

Experience rating is a type of pricing that is based on the analysis of past experience. It is the amount of loss that the insured party experiences compared to the amount of loss that similar insured parties have in the previous years. Experience rating is most commonly associated with workers’ compensation insurance. It is used to calculate the experience modification factor.

To deal with the claims inflation appropriately, one should be aware that in the wider environment, drives an increase in claims payouts a function of time. It also needs to be able to tap into the correct information using publicly available inflation indices. The example of the relevant indices is the consumer price index (CPI), the retail price index (RPI), the average weekly earnings (AWE) index, and several specialized indices for different classes of risks.

The basket method is a method of choosing claims that have similar effects in its outcome and see the difference in how those claims with similar effects would be priced from one year to another. This method only relies on the selected samples as its basis so the data that could be collected would be small concerning the data sets used in statistical methods but it would not lessen the reliability of the result. It has different components that could be broken down into pieces but the problem is that it would have difficulty in finding claims that have similar effects which are the needed data to be chosen in a basket method.

The inflation for large losses is not necessarily the same as that for smaller losses as there are many circumstances, for example, is when pricing excess-of-loss reinsurance, where we are often specifically more interested in the behavior of large losses than the smaller losses even with the fact that we do not even have visibility on losses below a certain threshold. We need to be able to estimate large loss inflation from a cohort of losses above threshold wherein the term ‘large loss’ means ‘loss above a given threshold’.

There is a guiding principle in the revaluation process which is that you want to make an informed guess as to what the value of a claim that occurred at time t would be if it occurred at time t*. Due to the principle, it is safe to assume that all the characteristics of the claim apart from the claim amount remain unchanged, but are translated by years or days. In mathematical terms, this assumption is equivalent to the assumption that the claim payment and reserving process is stationary except for the monetary amounts and that all the monetary amounts are all scaled by the same factor.

In insurance, all claim amounts would be converted into the currency that we intend to use. In the process of pricing the claim, the same exchange rate would be used for all the claims, regardless of the date they occurred because the purpose of doing the claims revaluation and currency conversion is to determine what past claims would cost if they were happening today while the currency exchange rate at the date of occurrence of the claims are ignored because it has no bearing on the amount that would need to be paid today for a past claim.

Just like in any other risk model, the classical risk models also have their assumptions of data which includes the assumption that the claim sizes X1, X2, … are all positive random variables with distribution function Fx and a finite mean. It also has an assumption that the claims arrive in a Poisson process and that the claims sizes X1, X2, … are all independent of the claim-arrival process. Other assumptions include the premium incomes accrues linearly in time at rate c (>0), so that by time t, the total amount of premiums received is ct and that at time t =0, the insurance company has non-negative initial capital.

In the classical risk model, various important stochastic processes arise which include the Poisson process, Stochastic process, and Risk reserve process. The first one which is the Poisson process is given by {N (t): wherein t is greater than or equal to 0. The N (t) in the equation represents the number of claims that arrive up to and including the time t. The second stochastic process is given by { S (t): where t is greater than or equal to 0 where the S (t) represents the total amount claimed by time t. The third process is the Risk reserve process, or can also be called the surplus process. It is given by {U (t): wherein the U (t) is the risk reserve at time t. It also represents the amount of surplus at time t, taking account of the inflow of premiums and outflow of claim payments.

The quantity that is roughly proportional to the risk of a policyholder or a group of policyholders is called the exposure measure. There are many examples of the exposure measure in practice which include the number of vehicle years for motor, the wageroll or number of employees for employers’ liability, the turnover for public liability, and the total sum insured for property and the like. The rationale is that if the exposure doubles whilst everything else remains the same, therefore the risk would also double.

When carrying out a pricing exercise of exposure data for experience rating, we ideally need the historical exposure data over the same period for which we have claims and the estimated exposure for the renewal year. The ultimate reason why we need the historical exposure data for experience rating is because of the need to have a comparable data of the loss experience of different years on a like-for-like basis and to adjust it to current exposure levels.

Take note that the measure of exposure we need to use also depends on what loss variable we are trying to project. Specifically, the measure of exposure may be different depending on whether we are projecting the aggregate losses or the number of claims. It is important to consider, for example, the employer’s liability cover which includes the following: In the burning cost analysis, the next period’s aggregate losses based on an exposure-adjusted average of the aggregate losses of previous years would be estimated. In the frequency analysis, the next period’s number of claims based on an exposure-adjusted average of the previous years’ claim amount would be estimated. Lastly is an aviation hull where the average fleet value (AFV) is often used as a measure of exposure.

There are several examples of how a portfolio/market information can be used in practice such as in the case wherein the individual claims data sets for the same risk and different clients are available, then a portfolio severity curve can be derived which would be used to complement a client’s information or replace it altogether. The exposure curve could be used as a model for the property losses of a client in practice. In a credibility theory, the portfolio data is combined with the client’s data to produce a more reliable estimate of the expected losses and hence a better premium and the portfolio/market information could be compared to the frequency per unit of exposure of the policyholder. More examples can be found as the portfolio/market information is often used in the practice.

In insurance, each policy is important to be considered on its own merits, and sometimes a very specific index is used. In the case of looking to insure losses that derive from oil spills from ships, the portion of the claim that relates to the lost oil itself might be usefully linked to the oil price. Some things should be taken off such as the importance of taking into consideration other things such as court inflation as sometimes you add a margin to the chosen index. Either of the raw index or fit a best-fit function can be used through the index to produce a smoothed index with possibly a single number for inflation per annum. Take note that the large losses inflation may be different from normal loss inflation because different factors could affect inflation.

The most obvious way of estimating the claims inflation that one could think of is to track the average claim amount for a cohort of claims and design an index based on that or to calculate the overall trend. The method of estimating the claims inflation using Robust Statistics works as follows: first is to calculate the empirical average of the loss amount for the claims that occurred in the previous years over several months or years. The second is to build an empirical index based on this average loss amount or the third step which is to assume that the average loss amount for the year grows by an approximately constant inflation rate which can be used as an alternative to the second step.

Although the method of estimating the claims inflation using the Robust statistics is conceptually reasonable and computationally straightforward, there are still several difficulties that may arise with the use of this method. The difficulties that may arise include the use of the mean as the average might distort the estimate in a given year, typically, the mean will jump up and down with no apparent regularity. In the case of scant data, even more, robust statistics such as the median will exhibit numerical instabilities which could also lead to poor claims inflation estimates. There are circumstances where only the portion of claims below a certain limit counts which might result in a skewed inflation calculation and some cases in which there are different cover changes and claims, the claims inflation estimate will be irrelevant. More difficulties may arise aside from all the examples that have been discussed.

Due to the difficulties that may arise, precautions should be done in taking in the calculation of claims inflation from data. The precautions that one should do include the use of portfolio/market information such as large data sets of claims from many different policyholders rather than data from a single policyholder to reduce numerical instabilities as much as possible. The only thing that should be used is the empirical index approach in the cases of a massive data set because the empirical index approach tends to give more erratic results from one year to another. The Robust statistics that should be used for the average claim amounts are those that are not heavily affected by large claims and lastly, one should use possible ground up and uncapped losses as it would prevent deductibles and limits from skewing the claims inflation calculations.

A normal statistical approach for the calculation of large losses would not be valid because the effect of inflation is to push claims above the selected threshold. To overcome this problem, other statistical approaches that are not based on a monetary threshold would be used, but some problems may affect any of the statistics. One of the problems is due to the use of the small samples as the basis of the statistics, there would be a numerical instability. This could be mitigated through the use of large samples. Also one of the problems is that the average or smallest of the claims in the sample will be affected by the ground-up frequency, but it could be mitigated by taking into consideration the market information on the ground-up frequency.

With a prepared data preparation and data summarization table, several observations can already be seen. Some of the example observations that can be seen in the table include the number of claims which has increased from 2006 to 2020 from 100 to 500, the total losses could have a complicated behavior wherein the total loss amount can be high while the number of losses is low, and that the average amount could vary erratically, even on a revalued basis. These observations emphasized the importance of doing a data summarization as with the use of simple statistics, you can already see different issues with the data.

The burning cost analysis could only estimate a few percentiles of the aggregate loss distribution, so it is not possible to get the estimate of the full aggregate loss distribution with only the use of the burning cost analysis. The ultimate reason why solely relying on the burning cost analysis is inadequate because of the fact that no matter how rich the initial claims data set is, the information would be aggregated into a small set of numbers by the burning cost approach wherein representing the total losses for each year or any other model with an only small set of claims is doomed to fail to adequately predict the richness of possible customers.

Large losses have a big impact on the final outcome of a burning cost analysis. The effects of large losses include the following: the first effect is that the existence of large losses in the burning cost analysis may skew not only the burning cost but also affect the overall distribution which would give the distribution a fatter tail, the second effect is that the large loss omission could result in the underestimation of the burning cost and the tail of the distribution and lastly, removing a large loss from the data set and spreading it over the years could lead to the underestimation of the year-to-year volatility, furthermore, it is unclear what probability should be assigned to a large loss.