For a better understanding of the pricing process, it only makes sense that the insurer knows what a model means. There are different descriptions made by different professionals of what a model means. Mae West was an American known personality who made a description of a model in terms of its philosophical depth with a statement “I’m no model lady. A model is just an imitation of the real thing.”

One of the most popular uses of the model is that it allows us to make a prediction that generalizes loss experience. A model takes the reality, which is the data, in all its muddiness and complexity. It helps to identify regularities in the data that allow the representation of the data to be worked out in a much simpler way, but still captures the essential features of the data. One of its simplest examples would be a linear model because, in many interesting circumstances, the dependent and independent variable of a phenomenon will be related by an approximately linear relationship.

IBNER stands for the incurred but not enough reserved which is sometimes also the incurred but not enough reported amount. It is the difference between the final settled amount and the overall estimate which is paid plus outstanding. In general insurance, IBNER claims is the amount owed by an insurer to all valid claimants who have had a covered loss but have not yet reported it. IBNR is necessarily an estimate since the insurer knows neither how many of these losses have occurred, nor the severity of each loss, The sum of IBNR losses plus reported losses yields an estimate of the total eventual liabilities the insurer will cover, known as ultimate losses.

It is important to take into mind the possibility of the occurrence of systematic errors. For organized checking of error, it is ideal that The insurer has the complete history of all reserve estimates and records of how they changed over time so that it can be used as a reference and a basis for checking of errors. Normally, large losses are reviewed at regular intervals and the reserve is typically updated if any new facts emerge while in the case of small losses, they will be given a temporary amount of recording to some company protocols and then, it will be updated the moment it is already settled.

By merely looking at the two empirical distributions, we cannot see the reason for the difference that the two empirical distributions have. It would not provide a clear picture of whether the reason for the difference is that there is a systematic over reserving or that larger claims take longer to settle, so, therefore, tend to be underrepresented in the settled claim data set. This difference could only be explained with the use of in-depth statistical analysis of the way that claims are reported, reserved, and settled.

The third method in identifying and dealing with IBNER is the identification of trends and adjustments for IBNER. This method can only be used if the insurer has the history of the claim reserves over the years or at least the information on the initial estimate and the settled amount. This would allow the insurer to identify trends and regularities in the way data goes from being fully outstanding to fully paid.

The simple average of the claims is 93.67. The formula of the simple average is the summation of relevant factors divided by the number of relevant factors. In the calculation of IBNER, only claims that are relevant for the calculation of the IBNER factor are the second, fifth, sixth, and seventh claims as they are the only one with a percentage outstanding and a non-zero for both year 1 and 2. So the relevant factors that would be used are 1, 1.10 & 0.71. 1 + 1.1 + 0.71 = 2.81. Simple Average is equal to 2.81/3 = 93.67.

There are advantages and disadvantages in using the method of just using closed claims. Its advantage is that it does not use loss estimates and therefore, the input to the analysis is more certain. The use of this method has more disadvantages than its advantage such as the fact that it does not use all data, so the evidence would be concealed and lastly, it should not be used without a bias-correction mechanism to take account of the fact that large claims are the ones that are more likely to take longer to settle.

The severity analysis has the objective regarding the production of severity model for the loss amounts. A severity model is not only essential to calculate, in combination with a frequency model, the distribution of aggregate losses but it will also help us to answer questions regarding the distribution of individual loss amounts.

Several ways or methods could be used for identifying and dealing with IBNER, this method is grouped into three headings. The first method that can be used is that the insurer could ignore the IBNER, however, this method can only be used if the insurer has sufficient information to use. The second method is that the insurer could just use closed claims which can be used if the insurer has at least information on which claims are settled or almost completely settled. The last method that can be used is the identification of trends and adjustments for IBNER to predict the ultimate value of each claim.

There are different ways for you to be able to identify trends and regularities in the way data goes from being fully outstanding to fully paid. Two possible methods are mostly in insurance which is the method of identifying trends with the use of development triangles and identifying trends with the use of multivariate analysis. In a more detailed manner, the first method is that the insurer would develop individual unrealized claims triangles with chain ladder-like-techniques wherein the ultimate project will be the estimated final amount. The second method is where a generalized linear modeling style approach would be used to identify the factors that are most relevant to IBNER and predict the final amount.

The method of identifying trends with the use of multivariate analysis uses a one-size-fits-all approach, in that it puts together all claims and calculates some IBNER averages by reporting in years. However, there are different factors that the IBNER depends on such as the development year wherein the younger the claim, the more scope there is for it to deviate from the incurred estimate. The size of the claim wherein the larger claims may have more uncertainty about them. The outstanding ratio wherein the larger the outstanding amount, the more conjectural the estimate of the incurred value will be, and lastly, the type of claim wherein some types of claims will be more difficult to estimate reliably.

There are advantages and disadvantages to using the method of ignoring the IBNER. The advantages of using this method include the simplicity of its use wherein it only requires a snapshot of the loss run at a given point in time, so there is no need for the full historical development of claims. It would work just fine if there is no occurrence of systematic errors and it is also the only option available when there is not enough loss data to produce a credible statistical analysis of IBNER. Its disadvantage is that if there are systematic errors, the result will be inevitably biased.

There is a common approach used in choosing a severity model that would adequately represent the data set which is the approach of fitting the historical losses to a variety of distributions from a distribution-fitting tool and simply choose the one that fits best, however, the use of this carpet-bombing approach has several problems. One of its problems is that it mostly focuses on goodness-of-fit rather than prediction and there are too many parameters that can be optimized to give a good fit. There is often no reason behind the inclusion of some models for the problem at hand as this approach accepts any distribution as long as it fits regardless of its use and there is also no consistency amongst different risks.

Just by studying the simple experiment made with lognormal data, many lessons can already be drawn such as the fact that unconstrained distribution fitting to select the best distribution does not work in the practice. The kinds of models that are more likely to be chosen in practice are more complex models that have more parameters and absurd models such as ‘beta transformed’. It also showed that the only method that can be used in assessing goodness-of-fit that sometimes gives sensible results is AIC because it penalizes the complexity of the model and the prediction error in the calculation is very large.

In another experiment, it is demonstrated that the method that will work on particular data sets is not chosen. So one of the strategies used in this experiment is the ranking of distributions according to the AIC. It would have a result that would demonstrate that with the use of the mentioned strategy, the lognormal would be in a higher ranking than the first experiment, although not by much.

For the present purposes, we only need one theorem which is called the Pickand-Balkena-de Haan theorem. It states that no matter what the ground-up distribution looks like, its tail can always be modeled by the following distribution. It is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown, however, Unlike for the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is in the values above a threshold.

Generalized Pareto distribution is a distribution whose PDF is zero up to a certain value and then declines to zero more or less quickly depending on the value of certain parameters. It is often used to model the tails of another distribution. It is specified by three parameters: location μ\mu, scale σ\sigma, and shape ξ\xi. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter.

Just like any theory, the extreme value theory or Pickand-Balkena-de Haan theorem also has its difficulties such as its difficulty in identifying the extreme value region. The key difficulty in using this theory is that understanding when μ is large enough that it can already be applied, that is finding the so-called ‘extreme value region’. However, it may be difficult to identify this region, but there are many different ways in finding this region.

Risk costing is considered to be an essential component of pricing, however, it is not the only factor that is considered as there are factors that should be taken into consideration before the translation into the actual premium charged to policyholders. There are factors such as the inclusion of loadings for expenses, commissions, reinsurance, target profit, and the like which would allow us to produce a technical premium, that is, the premium that should be charged on a purely technical basis.

In actual practice, the insurer does not normally operate in a vacuum but participates in the market in which regulations and competitive pressure force the company to deviate from the technical premium. Ultimately, the actual premium which is the premium that is charged in the actual transaction is considered to be a strategic commercial decision wherein the company is forced to have continuous monitoring of the costs, sales, and its position in the marketplace to ensure that the company makes the optimal decisions and achieves its objectives.

According to historical transactions, the main role of the pricing actuaries is risk costing. However, since there are developments and continuous change that is occurring in the market, pricing actuaries became involved in the determination of technical premium, along with other practitioners. They are also now involved in the determination of the competitive premiums, especially in personal lines.

The component of technical premium that can be defined as the mean of the aggregate loss distribution is called the ‘Expected loss’. It guarantees a breakeven result in the long term. However, some circumstances might require the expected loss to be replaced by percentile of the distribution. There are variations as some classes of business have such low-frequency losses which might result in a much higher than average experience for 5 to 10 years with the use of expected loss.

Based on the studies learned, it is important to incorporate the experience and theoretical reasons in the selection of the severity model rather than scatter-gunning for a model that fits the data well. The modeling of large losses provided reasons why choosing a particular model can be quite compelling. One of the reasons is that in modeling large losses, one needs a threshold distribution, specifically, it is the distribution with values above a certain positive threshold. The second reason is that we would want a severity distribution whose density decreases with size.

Few simple rules should be followed in choosing the threshold, however, these rules do not determine the threshold uniquely, but could only exclude the values that do not make sense. In choosing the threshold or μ, the KS distance should be less than approximately 0.8 for a 5% significance level, second is that the graph of ξ should be flat, third is that ξ should be lower than 1 because if ξ is greater than 1, then the mean would not be defined and the distribution cannot be used for pricing. The fourth rule is that there should be at least 30 data points larger than μ and you should choose the smallest value of the threshold for which the rules 1 to 5 would apply.

It is important for us to know and understand how the theory we are using is used in the actual practice. We should know how the actuaries work with the large losses, but due to the limited information from the large companies, we can only assume or say that in the actual practice the GPD model is used by many participants in the London market, Germany, Switzerland, and in the United States where its use is in the rise. Some of the large reinsurance companies in the actual practice still use Pareto with some modifications. Many practitioners in the actual practice still use the lognormal model.

The theory we are using would inevitably have its limitations, so although the extreme value theory drastically reduces the uncertainty of the model and works fine in actual practice, it still has some limitations. Its limitations include the choice in threshold as there is no guarantee that the ‘loss’ data will not change its behavior with the chosen threshold. The uncertainty of parameter could also cause significant problems in pricing especially in the case of sparse data and there is a strong bias on the parameters as the ξ is underestimated.

The simple approach to modeling loss amounts consists of 4 steps or ways that can be enumerated; first, the default distribution should be decided which would be used for modeling losses for any risk in a given portfolio of risks. The second step is that the model of the tail should be in a separate distribution. The third step is to validate the model with the use of a holdout sample or an approximate method such as the AIC and the last step in case of failure of the default model, the empirical distribution for losses should be used.

It is important to remember that the technical premium can be subdivided into five components according to Cost plus. Cost-plus pricing is a pricing strategy in which the selling price is determined by adding a specific markup to a product’s unit cost. According to cost-plus, the components of the technical premium include expected losses, allowance for uncertainty, other costs, investment income, and lastly, loading for profit.

Other costs are one of the five components of the technical premium according to cost-plus. It is the costs that are the indirect costs that are not included in the other components. It can be enumerated as the claim expenses, commissions, and reinsurance premiums. Claims expenses refer both to expenses allocated to specific losses while commissions are the amount paid to a broker or an external agent for the procurement of business and lastly, the reinsurance premium is the amount that is needed to add to the risk premium to get reinsurance.