Normal distribution and collective premium

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Normal distribution and collective premium

There are an unknown number of losses that may occur and each loss is an unknown amount. This situation can be modeled using a compound distribution of and. The model is specified by: This type of structure is called a collective risk model.

Normal distribution and collective premium

An alternative way to model aggregate loss is to model each risk using a different distribution appropriate to that risk. For example, in a portfolio of risks, one may be modeled using a pareto distribution and another may be modeled with an exponential distribution.

The expected aggregate loss would be the sum of the individual expected losses. This is called an individual risk model and is given by: Both of these models are tested in the exam; however, the individual risk model is usually tested in combination with the collective risk model.

An example of a problem structure that combines the two is given below. Your company sells car insurance policies. The in-force policies are categorized into high-risk and low-risk groups. In the high-risk group, the number of claims in a year is poisson with a mean of The number of claims for the low-risk group is poisson with a mean of The amount of each claim is pareto distributed with and.

Being able to see the structure of the problem is a very important first step in being able to solve it.

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In this situation, you would model the aggregate loss as an individual risk model. There are 2 individual risks— high and low risk.

For each group, you would model the aggregate loss using a collective risk model. For the high-risk, the frequency is poisson with mean 30 and the severity is pareto with and. For the low-risk group, the frequency is poisson with mean 10 and the severity is pareto with the same parameters.

For these problems, you will need to know how to: Find the expected aggregate loss.

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Find the variance of aggregate loss. Approximate the probability that the aggregate loss will be above or below a certain amount using a normal distribution.

Determine how many risks would need to be in a portfolio for the probability of aggregate loss to reach a given level of certainty for a given amount. Determine how long your risk exposure should be for the probability of aggregate loss to reach a given level of certainty for a given amount.

Problems that require you to determine probabilities for the aggregate loss will usually state that you should use a normal approximation.

This will require the calculation of the expected aggregate loss and the variance of the aggregate loss. For the individual risk model, it is Variances under the collective risk model are conditional variances.θ is a random variable with a normal distribution with mean equal to 75 and standard deviation equal to 6.

The single benefit premium for this insurance on (41) is 16, For a collective risk model the number of losses, X, has a Poisson distribution with λ= Adjustment Set-Up, and Normal Operations.

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coefficient of variation 5 coherent risk measure 9 collective premium 45 from MATH at School of Banking and Commerce. normal distribution with equal mean and variance. b) The parameters ~¢' and ~ in formula (I) are determined so that an unbiased premium refund is obtained, i.e. the mean value.

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