8. The power of risk pooling. There are 20,000 policy holders in an auto insuran

Question

8. The power of risk pooling. There are 20,000 policy holders in an auto insurance pool, which charges an annual premium of $850 each. Policy holder i has Ni accidents each year, and we model Ni as a Poisson random variable with parameter 1.2. The cost (to the insurance company) of each accident (the non-deductible component) is modeled as an exponential random variable with mean $700.

(a) What is the mean and standard deviation of the (non-deductible) cost of accidents for one policy holder each year? Is the premium of $850 reasonable?

(b) What is the (approximate) probability that the insurance company will lose money in a single year?

(c) What if the premium is raised to $900, is it still reasonable for the consumer? What is the new (approximate) probability that the insurance company will lose money in a year?

Explanation / Answer

Mean and standard deviation for exponential random variable is 1/K

If X has the exponential distribution function f(x) = Ke(-Kx)

So standard deviation= 1/700

Standard deviation=$0.0014

So premium of $850 is reasonable

b) Probability of loosing money is given by =e(-850/700)

Probability of loosing money=0.296=0.3

C) Probability of loosing money=e(-900/700)

Probability of loosing money=0.28

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