The mean and the standard deviation take a while to converge to the correct theoretical value due to the effect of the tail of the lognormal distribution because some large losses will throw off the estimates of both mean and standard deviation, the latter being more strongly affected. This problem can be partially overcome through the use of more sophisticated Monte Carlo simulation methods, such as stratified sampling.

In terms of the accuracy, the difference between the Fourier transform and the Pajer recursion is that in the Fourier transform, the arbitrary degree of accuracy could be achieved by calibrating the discretization step and the number of points wherein the discretization step may need an adjustment while in the Pajer recursion, the arbitrary degree of accuracy could be achieved by only calibrating the discretization step.

One of the disadvantages of the Gaussian or parametric approximations is that it has limited flexibility. It is because when aggregate deductibles are present and when losses to a high layer need to be calculated. Also one of its main features is that its implementation is still trivial.

The Monte Carlo simulation is considered to be the most popular way of calculating the aggregate loss distribution. Its popularity is because the method used is very flexible and there is an ease in the implementation. The idea is very simple and it also consists of simulating what could happen in one particular year by randomly choosing several losses, and then, for each loss, choosing a severity based on the severity distribution.

Although the Monte Carlo simulation is considered to be the most flexible method for calculating the aggregate loss distribution, it would inevitably have some limitations. It includes the fact that the methodology of the simulation can be slow because a good precision would require a large number of simulations. The number of simulations that are needed to achieve the desired level of accuracy cannot be predicted in advance and the calculation of quantiles does require sorting of all simulated scenarios which can be challenging because of the large number of simulations.

Several methods are available to use to varying degrees to analyze not only the full losses but also the losses retained below a certain deductible or the losses ceded above a certain deductible or in general ceded to an insurance or reinsurance structure. The different methods can be enumerated as Gaussian approximation, Fast Fourier transform, Panjer Recursion, and Monte Carlo simulation.

Translated Gamma approximations is another approximate method that is considered to be similar to another method called Gaussian approximation. This method is used to model the aggregate loss distribution as a shifted gamma distribution. One of the main advantages of the use of this method with respect to the Gaussian approximation is that it is not symmetric but the other problems, such as the difficulties in modeling the tail and quantifying the errors, remain.

Numerical Quasi-Exact methods is one of the methods used in calculating the aggregate loss distribution. This method simply attempts to calculate the convolution integrals with the use of discrete mathematics, utilizing the fact that integrals can be approximated as a finite sum. Several different ways can be used for calculating this sum wherein one option would be to do a straightforward calculation which is an option that is normally done. However, this process could be computationally demanding.

The method used in calculating the multidimensional integral can only be considered simple and easy if there is only one dimension. This method, in principle, be generalized to integrals over an arbitrary number of dimensions. However, when the number of dimensions is large, it could be a difficult method because of what we call the ‘dimensionality curse’, by which the number of intervals necessary to achieve a certain degree of accuracy increases exponentially in the number of dimensions.

An aggregate annual deductible is a maximum amount that the policyholders need to pay within a policy period before their insurer would pay for covered losses based on the policy structure. In other words, if a policyholder files several claims or one large claim, they are required to pay out of pocket up to a certain dollar sum before the insurer would cover the claim. Subsequently, coverage kicks in, and the insurer starts making payments.

In the actuarial practice, it is important to keep in mind that there is a difference between the pricing framework and pricing decisions. The difference between the two is that the framework of pricing is set in the context of the company’s strategic plan which would include parameters designed to promote coherent and consistent decision making while a pricing decision is made after the framework of pricing is set in place and then monitored to track variance to plan.

There are figures in the studies that question whether the pricing control cycle is a cycle or not. Some figures suggest that the pricing control cycle is more complex than a sequential cycle as one might imagine it to be. Many things happen at the same time in the different parts of the cycle and normally, in no particular order, although the general idea is indeed that of a continually improving process so the pricing control cycle should be called a network rather than a sequential cycle.

It is important for the insurer to take into consideration the several types of uncertainty that are present in the actuarial practice because it is important for the different uncertainties to be known so that its effects on the pricing would be calculated. The types of uncertainties in the actuarial practice can be enumerated as the Process uncertainty, Parameter uncertainty, Model uncertainty, Assumption or data uncertainty, and lastly, the approximation errors in calculations.

In the process of defining the various types of uncertainty, several ideas and examples are useful to keep in mind. These examples can be enumerated as the yearly number of claims related to risk should be considered, the distribution of loss severities for a given risk should be considered, and lastly, the distribution of aggregate losses for a given risk should be considered which is calculated by combining the Poisson frequency model and the lognormal severity model.

Pricing is a cycle or process that goes through continual improvement as more information keeps evolving and the innovation of things keeps developing. This process is normally referred to as the ‘actuarial control cycle’ and it can be distinguished into three phases in the actuarial approach to the problems. The first phase is Specifying the problems, the second phase is developing a solution, and the last phase is the monitoring of the consequences.

Monitoring is considered to be the key feature of the pricing control cycle because it is the feature that keeps the cycle going and allows the pricing process to keep improving while remaining adequate. Several reasons why the insurer needs to monitor its business can be enumerated as; to be able to assess its performance against its targets, for the pricing and reserving assumptions to be validated, to report to management so they can make effective decisions, so they can report to the regulators if it requires them to do so, to see how well it is doing against its competitors, and to gather information on the underwriting cycle.

In the actual practice, insurers operate in a very competitive market which is why it is important to know the position of their competitors and what they are doing so they can make effective decisions and revise their business plans. However, some difficulties present in monitoring the competition and one of the main problems is that the businesses or competitors do not have the duty to provide detailed information on their operations, so the insurer can only rely on the unrestricted information.

The portfolio composition, which can also be called a ‘business mix’, can be defined as the percentage of policyholders that fall into each of the categories. Changes in the portfolio composition are needed to be tracked as they will result in changes in the risk profile of the portfolio and ultimately in the profitability of the portfolio.

Rates and the rating factor system have an effect on the sales just like a premium. Several variables related to sales that are typically monitored include the renewal rate. Renewal rate is the percentage throughout expiring policies that are renewed wherein a low renewal rate may be an indication that the premium rate of the business is higher than that of the competitors.

Several legitimate ways can be done by the insurer to track the competition which includes the engagement to mystery shopping where the insurer would ask for quotes on few standard policies to assess prices, terms, and conditions of the competitor. The insurer could also ask the customers the reasons why they are leaving you and for whom. They could also ask the past customers of the competitor why they left and if the business comes through intermediaries, the insurer could try to ask them about the competitors.

The monitoring system should have the following characteristics for it to be considered as a good monitoring system. These characteristics include the conciseness and relevance of the output to the strategic goals of the insurer, should also be accurate both in terms of data and results, it should be simple or easy to use, it should be well documented for references and evidence purposes, and it should be able to produce an analysis quickly from data as-at date to results.

It is important to take note that the Proportional insurance has no significant difference from the direct insurance as it would just cede a proportional amount of each risk to the reinsurer, therefore, elucidating the impression that the proportional reinsurance would not be required to have a separate treatment from the direct insurance.

In pricing non-proportional reinsurance wherein only losses above a certain threshold point or a deductible trigger a payment, so it is important to set a reporting threshold or a reporting level as it would filter the claims data wherein only claims data above the reporting threshold would be communicated to the reinsurer. However, there are some instances when it would not matter whether the claim is below the reporting threshold or a zero, it would still be communicated by the reinsured to the reinsurer.

The exposure and the contract basis are connected in direct insurance as regardless of whether the exposure is the original premium or the original exposure that is used by the direct insurance, it needs to be properly matched to the claim experience which depends on the contract basis. For the RAD policies, the exposure used is premium while for the LOD policies, the exposure used is the premium earned during the policy period.

The performance of the frequency analysis in non-proportional reinsurance is just the same way as for direct insurance. However, several differences are also present between the two such as in non-proportional reinsurance, the focus is the number of claims above the reporting threshold while vice versa to direct insurance. Another difference between the two is that the treatment of exposure in non-proportional reinsurance is more complicated than direct insurance, especially when original premiums are involved.

The exposure method is used in pricing an insurance policy when the data for a specific client is not sufficient to produce a reliable severity model. This method used the so-called ‘exposure curves’ as a basis, which are re-engineered severity curves and usually based on losses from a large number of clients in which it is collected by large institutions such as Lloyd’s of London or Swiss Re. However, the origins of many of the exposure curves used by underwriters and actuaries in the London market remain mysterious. Some actuaries would not consider these curves despite its widespread use.

Depending on the losses that are covered by the policy, several types of contracts in reinsurance are made. There are many types of contracts, however, there are three types of contracts that are considered to be the most commonly known and widely used by the insurers in actuarial practice. These include losses occurring during or LOD, risk attaching during or ROD, and lastly, the claims made.

The input of the reinsurance costing process is very similar to direct insurance. However, there are still some differences in the format of these data sets such as the format in the policy formation. In the policy formation, there are important data for the treaty reinsurance policies to be known which include the policy period, type of reinsurance, basis of cover, treaty limits and excess points, details on the indexation clause if it is applicable, and the details on hours clause and sunset clause which is the catastrophe cover.

For the exposure to be used meaningfully in comparing the number of claims coming from different policy/accident years, adjustments are needed to be made beforehand. Several adjustments are needed to be done such as ensuring that the exposure used is adequate for the contract basis and in case of monetary amount exposure, it should reflect the current environment. Other adjustments involve the IBNR analysis and the selection of a frequency model.

Several ways are used in pricing property risk for a portfolio of property which includes the experience rating: the collection of all losses over a certain period, producing a frequency model and a severity model and combining them with a Monte Carlo simulation or a similar method. However, this method would only be accurate if there is enough data which is why it requires additional care and consideration which include the case when producing a severity model, the severity distribution should be capped, producing a realistic severity model can be difficult, and lastly, the method used in scaling up a loss model in case of exposure changes seems to be adequate when it comes to property risk.