STAT3906 RISK THEORY I
December 15, 2022
1. It is known that N has a zero-modified Poisson distribution,with P(N=1)=0.25 and P(N=2)=0.1.
(a) Find P(N=0). [5 marks]
(b) Find the variance of N. [5 marks]
[Total:10 marks]
2. Suppose that X follows the generalized Pareto distribution with cumulative distri- bution function given by
where ξ∈(0,1)and β>0.
(a) Determine the mean excess function of X,and deduce the heaviness of the tail of X. [5 marks]
(b) Compare the heaviness of the tail of X with that of Y~Exp(0).[2 marks]
(c) Find TVaRp,(X)for any p∈(0,1). [5 marks]
[Total:12 marks]
3. The number of payments N follows the zero-modified Negative Binomial distribu- tion with parameters po=0.6,r=2,β=0.5.The amounts paid per payment Y,Y2,….,are independent and identically distributed with a common cumulative distribution function given by
Assume that Yi,Y2,..are independent of N,and define the aggregate payment by S = ni=1 Yi
(a) Find P(N=n)for n=0,1,2,3. [5 marks]
(b) Find TVaRo.95(N). [4 marks]
(c) Find P(S≤10)using normal approximation.Express your answer in terms of the standard normal cumulative distribution function Φ. [5 marks]
[Total: 14 marks]
4. For an insurance coverage,claim sizes follow a distribution which is a mixture of a uniform. distribution on [0,10]with weight 0.5 and a uniform distribution on [5,1 3] with weight 0.5.For a policy with a policy limit of a,the expected payment is 6.11875.Find the value of a. [Total:10 marks]
5. You are given the following:
● In 2022,losses are exponentially distributed with mean 600.
● It is estimated that an inflation of 10%impacts all losses uniformly from 2022 to 2023.
Find the median of the portion of the 2023 loss distribution above 880,i.e.,the conditional median ofX given X>880 where X is the total loss amount in 2023. [Total: 10 marks]
6. For an insurance coverage,the ground-up losses follow a Pareto distribution with parameters α=3 andθ=5000.The coverage is subject to a deductible of 500. Calculate the deductible needed to double the loss elimination ratio. [Total: 10 marks]
7. Conditional on θ=θ,the claim size X is uniform. on the interval(θ,θ+15)for each policyholder.The parameter A varies between policyholders according to an exponential distribution with mean 10.
(a) Find the unconditional density function ofX. [6 marks]
(b) Find the mean and variance of X. [6 marks]
[Total: 12 marks]
8. For an insurance coverage,you are given:
● The number of losses follows a geometric distribution with mean 5.
● The ground-up losses follow a Poisson distribution with mean 1.
● The number of losses and loss amounts are independent.
● There is a deductible of 1 and a maximum covered loss of 3 per loss.
(a) Express the payment per loss variable as a function of the loss variable.Cal-culate the expected aggregate payment per year. [6 marks]
(b) Calculate the probability that the aggregate payment is greater than 2. [6 marks]
[Total: 12 marks]
9. In the aggregate loss model S=X₁+…+Xn,severities Xi's have a uniform distribution on [0,100]and the distribution of the claim count variable N is given by
(a) Find P(S≤50)and P(S≤150). [7 marks]
(b) Find VaR0.9(S). [3 marks]
[Total: 10 marks]