Insurance against the death of their animals normally arises from establishments such as horse establishments that breed horses for racing or other purposes. This is an example of property insurance. Horse insurance would be the best example of insurance in the individual risk model as the loss amount that may arise are normally fixed, and this is commonly agreed at the outset of the insurer. It normally reflects the value of the horse, whether it be a racehorse, a stallion, or a broodmare.

It is important to take note that not all of the assumptions of the individual risk model have equal appropriateness in every circumstance. One instance is in the assumption that all of the loss events are independent, there are circumstances such as many bankruptcies may all derive from a common cause like a recession. The assumption relating to the severities may be inappropriate in circumstances such as a big happening which could not only affect several companies to bankruptcy but also that their bankruptcies may be more severe, leaving a different residual value and therefore generating a different LGD. The inappropriateness of an assumption would require the correlation in frequency to be dealt with by modeling a common shock to the system that increases all probabilities at the same time.

The collective risk model is the main model that is being used in non-life insurance. The individual risk model mostly focuses on the number of risks in the book but the main objective or idea of the collective risk model is to shift the focus from the risks to the losses. In the collective risk model, the number of risks and the individual probability of loss and severity of loss in each risk are completely ignored. There is an assumption that there will always be a number N of losses in each particular year wherein the N is a discrete random variable.

In the collective risk model, there are no assumptions that could be considered to be completely valid. However, one of the assumptions is considered to be the thorniest one which is the assumption of the independence of N with the loss amounts of X as it is quite easy to differentiate the situations and find what situations are independent and which are not. For example, is how we all already know for a fact that bad weather is a factor that could affect the number of motor losses.

Familiarising yourself with the risk is a phase of pricing that has difficulties in overemphasizing the importance of familiarizing oneself with the risk one is trying to provide before diving into the technicalities of pricing. It is a phase of pricing that is least susceptible to being automated because it requires the insurer to know his client with a mind that is as open as possible. It is also important to emphasize that knowing the client will not only be useful in pricing the particular risk, but it helps the insurer to be able to identify other insurance needs of the client and would help in advising on the overall risk management strategy.

The risk profile of the client is directly related to its particular company or individual that is relevant in identifying its level of risk. In the case of an individual who buys car insurance, there are factors such as the age and type of the car would affect the individual’s risk profile while for a company buying employers’ liability, it would be determined by the factors such as the type of work its employees do and their salaries. In the case of reinsurance, the risk profile would be affected by factors such as the type of property portfolio. The most important thing to take note of in this consideration is that it would affect the degree to which the past claims experience is a good guide to future claims experience for the risk under consideration.

The purpose of risk sharing is to be able to spread the risk among other things involved. The one called reinsurer is the principal or direct insurer who is responsible for passing on some of the risks to another insurance company. It would allow the direct insurer to purchase insurance from the reinsurer. Furthermore, the direct insurer may structure the policy that the policyholder is responsible for by including a deductible or policy excess on the conditions of the cover.

In the book for the sake of discussion, it is common that only simple reinsurance arrangements are being considered which could be very different in the practice as insurance companies may have to place arrangements with very complicated structures. It is important to note that each reinsurance contract involves at least the random variables X, Y, and Z in which we use quantities with suffice to denote which of the random variables would be involved.

The model for S as a compound distribution incorporates important structural assumptions. The assumption that the sequence of claim amounts is a sequence of independent and identically distributed random variables is one of the structural assumptions. This assumption is considered to be a realistic assumption for the risks in many portfolios, but there are particular circumstances that it would be inappropriate for this assumption to be used as the independence of the variables may be questioned.

The deductibles or the policy excesses is an arrangement that is familiar to anyone in general insurance. The most obvious benefit of this arrangement would include deductibles in policies that would require the removal of the small claims from the record of the insurer. This is a benefit to the insurer as it would increase the savings in administrative efforts and would decrease the cost of managing the claims as it would result in a smaller number of claims that would be needed to be managed. The overall reduction in the number and the average amount of potential claims to be settled allows the possibility of a lowering of premiums, with consequent market advantages.

Some situations should be dealt differently and one best example of this situation is a situation in which individual losses are modeled by a Pareto random variable X with parameters chosen so that the monetary unit is the expected loss for convenience. This type of situation could be dealt with by the insurer needing to write policies that would cover such losses with an individual excess of D per loss; the policyholder would submit a claim on any loss which would exceed D.

In the optimization of the reinsurance contracts based on the variance of aggregate claims, the maximization of the expected income, profits, or wealth, or minimizing expected payout are not necessarily the only desirable outcomes insurers would want to consider as other outcomes such as the reduction of the uncertainty inherent in the situation may also be considered by the insurer. Statistically, the insurer may want to consider not only the level but also the spread of the insurer’s payout, for a shorter explanation, to consider second-order moments in addition to the first one of the payout distribution.

The reinsurance arrangements have two different types which are the proportional reinsurance and the excess of loss reinsurance. Proportional reinsurance requires the primary or ceding insurer and the reinsurer to maintain a post-transfer relationship. It also requires an assessment of all risks and uses the known risk to prorate the proportion of premiums, expenses, and losses for all parties to the agreement. Non-proportional reinsurance, or excess of loss basis, is based on loss retention. The ceding insurer agrees to accept all losses up a predetermined level. The reinsurer agrees to reimburse the ceding insurer for losses above the predetermined level and up to the reimbursement limit provided for in contact. Whether the reinsurance is proportional or non-proportional, the payments collected under the contract are assets that belong to the ceding insurance company, and the reinsurer is granted indemnity against loss.

The burning cost analysis is one of the most widespread pricing techniques despite its simplicity and its many limitations because it is considered to be the simplest yet has the most intuitive approach in figuring out costs. It works with the estimate of the expected losses to a policy based on average losses in past years, after allowing for claims inflation, exposure changes, incurred but not reported (IBNR) claims, and any other amendments that need to be made to make the past claims data relevant to today’s situation. It is mostly based on aggregate losses. It is important to take that this approach easily falls apart in the presence of deductibles and limits, as the policy might have had different levels of deductibles over the years. Furthermore, the effect of inflation is non-linear in the presence of a deductible.

Adjustments for large losses is one of the steps of the burning cost methodology. There would be an adjustment if we believe that the large losses are either overrepresented or underrepresented in the data set. In the situation of the ABC corporation, they have large losses of $20M for the 10-year data set, therefore affecting constant exposure of $2M on the burning cost. They believe that it is a 5 in 100 events so we will remove that loss and add back the amount to the burning cost by dividing 5 over 100 which would be equal to 0.05 then it would be multiplied to the $20M large losses to get the value of $1,000,000.

The calculation of the premium is the last step of the burning cost methodology. Through the use of the results of the burning cost analysis in determining the premium that the insurer would charge for a given risk. In this situation, we would assume that the insurer becomes liable for the ceded losses and charges premium for them. The targeted loss ratio is 75%, so for the simplicity of achieving this target ratio, we would use the weighted average method. The burning cost at renewal exposure amounting to $1,250,000 would just simply be divided to the targeted ratio which is 75% and it would amount to the premium of $1,666,667.

In individual risk modeling, the assumptions can be summarized as follows: First is that there is a finite number n of risks. Second is that in this type of risk modeling, only one loss is possible for every risk and this happens with a given probability. The third is that all of the loss events are independent which came from the fact that a given risk has a loss does not have an effect on whether or not another risk has a loss. Lastly, the severities are all independent however, they are not necessarily identically distributed.

In collective risk modeling, the assumptions can be summarized as follows: First is that there is an assumption that the number of losses N can be described as a discrete random variable with a distribution Fn-. The second assumption is that all of the loss amounts X are independent which are identically distributed random variables unlike in the individual risk model wherein the independent variables are not necessarily identically distributed. The last assumption is that the loss number N and the severity X of the losses are independent which can be elaborated as the severity of a given loss does not depend on how many losses there were in a given period.

The moment generating function can be interesting because of the two reasons which can be enumerated as First, as the name suggests, it allows the calculation of all moments of the aggregate loss distribution by calculating the derivative as many times as necessary. By using the equation of the moment generating function, you can then express E(S^k) as a function of the moments X and N and alternative derivations can be used through other equations. The second reason is that the moment generating function is known to be similar to the Laplace transform and the Fourier transform of the underlying distribution. These similarities make it possible to know how to calculate numerically the aggregate loss distribution and the use of the fast Fourier transform techniques which can increase the efficiency of the process.

If the details of the application of the collective risk model to motor fleet losses are seen closely, it could be seen that some additional difficulties may emerge. In motor fleet losses, every vehicle can have losses more than one, but in the situation that a vehicle crashes too badly, it will need to be written off and will therefore not be able to have further loss as the underlying fleet of the insurance will be changed and so the overall risk changes that would make the initial model invalid. The vehicles that are included in the motor insurance may be removed or added to the fleet for several reasons. In practice, If there are significant changes in the insurance, the approach would be to price the policy to a certain size and just adjust the final premium at the end of the policy. The motor loss insurance normally has several components that might require the separate modeling of each component.

Several things are important to be looked out for in commercial lines and reinsurance. It is a custom that each of the clients would be looked at as an individual, However, there are types of clients, such as smaller entities, that might be treated in a streamlined fashion. The kind of information of the client that is very important to be considered are listed as follows: It is important for the underwriter to know what does the client do and what is its business as the client’s risk profile will always be connected and related to its business. The second is to know how and where to contact the client so the main location and main number should be known. Third, it is important to know if there are any notable changes in the risk profile of the client over time. Fourth is to know if there has been any significant merger/acquisition or divestitures in the company over the years. The fifth is if the company has been implicated in any large claims that did not appear in the report. Sixth is the industry-related questions and lastly, the sources of the knowledge on the risk.

It is important to consider the sources of knowledge on the risk for the commercial lines and reinsurance. Several suggestions on where these type of information could be found and it can be enumerated as the client’s website which can give a lot of useful information even by merely looking on their web, the client’s accounts as some of the accounts of the companies are required to be posted it can be used as a piece of information about their company by the public, insurance submission as the companies normally prepare a submission for the insurer which includes the claims and exposure information, scientific publications as there are many scientific papers about the actuarial nature of the risk that could be used as a reference, and many more sources that are too many to discuss.

The purpose of buying insurance is to protect the policyholder against the effects of large losses. Particularly, it protects the direct insurer against having sole responsibility for the tails of the distribution of large claims. This has several effects on the claims of insurance: First is that there would be a reduction in the mean amount paid out by the direct insurer on claims. Second is that there would be a reduction in the variability of the amount paid out by the direct insurer on claims. Last is that there would be a reduction in the probability that the direct insurer will face a very large payout on any particular claims. Therefore, it could be concluded that the reinsurance stabilizes the direct insurer’s payouts on claims.

Under the excess of loss reinsurance, The direct insurer would set a retention level and pays in full any claim for which X is less than or equal M. The direct insurer would retain an amount M of the risk. The direct insurer would pay M and the reinsurer would pay the remaining amount for the claims for which X is greater than M. Under this type of reinsurance arrangements, the direct insurer only has limited liability which is only limited to M on each claim and no exposure to the risky tail of the claims distribution. It is one of the reasons why this type of arrangement is very attractive to direct insurers.

Under the proportional reinsurance, The direct insurer would pay a proportion of each claim while the reinsurer would be the one responsible for paying the remaining amount of claims. The proportion of the risk retained by the direct insurer would be agreed upon in advance by the involved parties. Under this type of reinsurance arrangement, both the direct insurer and the reinsurer would be involved inlaying each claim and both have unlimited liability unless there is a cap on the amount of the claim. The direct insurer is exposed to the risky tail of the distribution making this type of arrangement less attractive to the direct insurers.

The issues concerning the optimization of the reinsurance arrangements seen from the direct insurer or the reinsurer as appropriate are discussed in chapter 5. It includes the establishment of the optimal retention levels for the direct insurer based on maximizing the expected utility of the direct insurer’s assets at the end of the period of the insurance. The problem in minimizing the variance of the aggregate claims for each party involved. The consideration of the optimization of reinsurance contracts for a portfolio of independent risk, based on minimizing the uncertainty of the direct insurer’s net profit.

The result of the equation and situation in the excess of loss reinsurance that is considered to be interesting and instructive in several ways: First, the direct insurer’s optimal retention level doesn’t depend on the initial wealth of the direct insurer. This is to be expected in the presence of the exponential utility function. Second, the direct insurer’s optimal retention level does not depend on the individual claim size distribution. Third, the direct insurer’s optimal retention level is a decreasing function of the direct insurer’s utility function parameter. Increasing the parameter corresponds to setting a lower retention level and passing more of the risk to the reinsurer. Last is that the direct insurer’s optimal retention level is an increasing function of the reinsurer’s premium security loading wherein the more expensive the reinsurance, the less the direct insurer will purchase.

Burning cost analysis is the farthest one can get toward costing a risk and its volatility without resorting to stochastic modeling. The use of the burning cost analysis as a pricing technique can give several advantages which include its use to provide a different estimate of the cost of the risk alongside frequency or severity modeling analysis. Burning cost analysis can also provide a sense check on the results of frequency or severity modeling analysis because when the results for the basic statistics such as the mean and the volatility differ significantly between the two methods, this might indicate that there is a mistake in one of the methods and the difference may be used to explain the added value of frequency/severity modeling. The results that can be acquired with the use of the burning cost analysis can be used as an intermediate step between the initial summary statistics.

Burning cost analysis consists of inputs or information that is used as the basis of figuring out the costs. The inputs to the burning cost analysis include the individual claims listing, policy structure, and information, and the historical exposure information in which depending on the cases, we may also have market/benchmark information on large losses, IBNR/incurred but not enough reserved.

The burning cost methodology consists of 9 steps to get the calculation of the cost. The first step is to revalue losses and apply policy modifiers to all claims. The second is to aggregate loss by policy year. The third is to adjust changes due to exposure to bring the historical figures to current terms. Fourth, make adjustments to IBNR and IBNER based on either client information or benchmark data. The fifth step is to make adjustments for large losses if it is believed that there are over presentation or under presentation of the large losses. Sixth is to make other necessary adjustments. Seventh is to impose aggregate Deductibles/limits as necessary before calculating the burning cost in the eighth step. And the last step is to do a premium calculation based on the burning cost analysis.