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.

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.

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.

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.

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.

The job of pricing actuaries would be much easier for the insurers if they could remain oblivious to everything except the portfolio of risks that they have to price. However, the price of the insurance and even the pure cost of risk is linked to the numerous factors that are out of the control of the individual or the company that wishes to transfer the risk and to the company that takes this risk.

The changes in the legal environment could either increase or decrease the frequency and severity of claims. It may increase the frequency and severity of claims by extending the scope of compensation or by introducing new counts of liability. Allowing the possibility of no-win, no-fee claims in the late 1990s became a reason for opening the floodgates to a large number of claims is an example of an increase while all of the other choices are an example of a decrease of frequency and severity of claims.

There are several other premium calculation principles available that could be used as an alternative principle, in particular, one or two based on an adjustment to the distribution function of the risk which produces a weighted version of the original density function wherein the weight decreases as x increases, producing a density with fatter tails than it had originally.

The purpose of the excess amount in the personal lines insurance is to discourage reporting of small claims which helps in decreasing the cost of managing the claims. Also one of its purposes is to discourage petty claims. One of the good purposes of the excess amount is in the point of view of the insured as it keeps both the compensation and the costs low so the premium is reduced.

Short term risk models are a representation of risk over a fixed period. Two of the popular models that have always been discussed are both examples of short term risk models. One of the short term models is the individual risk model which has a portfolio that consists of a fixed number. Another short term model is the collective risk model or also known as the aggregate risk model where successive claims from the portfolio were modeled as an independent.

It is important to look beyond the mean and the variance of the random sum because there are circumstances when other kinds of behavior would be a concern such as its tail behavior which would require the use and consideration of the whole distribution of the random sum S.

At a very high level in the pricing process, it consists of several steps: First is familiarizing the risk with the insured business. Then, with the given information on claims, exposure, and cover, the cost of taking on the risk would be estimated. The fourth step is taking into account the related cost of running the business and investment income. The fifth step is the addition of allowance for profit, thus obtaining the technical premium, and lastly, based on the technical premium and commercial considerations, the actual premium to be charged would be determined.

Despite the limitation of only focusing on the aggregate information of the claims, there are good reasons why it made sense to dwell on it. There are two reasons why it is being used; The first reason is that it is actually the basis on which many underwriters and risk managers look at risk and the second reason is that this approach provides a broad feel to the risk managers of what the risk is like and a sense check on any subsequent actuarial analysis.

The expected ceded losses of 279,920 that are computed by multiplying total losses to the percentage of available insurance will be divided by 80% to get the total expected ceded losses plus expenses of $349,900. The 20% premium for covering the expenses would now be written-off.

The formula of a conditional expectation is E [E [X │ W]]= E [X] while the formula of conditional variance is Var [X]=E [Var [X │ W]] + Var [E [X │ W]]. When adding the terms of the conditional expectation formula, it will be easy to detect that the right-hand side of the conditional variance formula is equal to the left-hand side.

Both statements are wrong because, in the first statement, the notation Fx should be described as a non-decreasing and right continuous function. The notation that is commonly used for the expected value of a random variable x is the notation Ex. The second statement is also wrong as it should be a probability density function instead of a cumulative distribution function as it is the one known as a continuous function.

Statistical software package R is a system used in carrying out the simulations, statistical analyses, and numerical approximations of random variables. It has a variety of packages that are available wherein users can find a diversity of codes, functions, and features that are designed to be equipped with a large amount of programming and analytical tasks. It is used with the assumption that the user is familiar with how R works and with its basic commands.

The properties of a normal distribution are:
The mean, median, and mode have all the same value
The curve in the center of the graph is symmetric
The area from the center to the right and from the center to the left is equal
The total area under the curve is 1

Probability is focused on analyzing the likelihood of occurrence of an event. There are two possibilities which are called a certain event and an impossible event. A certain event has a probability of 1 which implies that the likelihood that event would occur is sure. An impossible event has a probability 0 which implies that the occurrence of the event would not happen. The probabilities of all the events would just be either 0 or 1 which implies that the event would or would not occur.

Both probability generating functions and moment generating functions are an example of transforms. Transforms are used for the calculation that involves the sum of independent random variables. Probability generating function is described as a series of presentation of probability mass function of a discrete random variable while moment generating function is used as an alternative route to analytical results instead of working directly with probability density function or cumulative distribution function.