STATS 3ST3: ACTUARIAL MODELS IN NON-LIFE INSURANCE - Winter 2024
Question I:
We assume the classical case Poisson/Gamma.
We have: St|Θ = θ ~ Poisson(θ) Θ ~ Gamma(α, λ)
Calculate the marginal distribution St (all the intermediary steps are required) using the Moment Gener- ating Function (MGF).
Question II:
The distribution of Sn+1|Si1, ..., Sin with probability density function f(x|x1 , ..., xn ) is called the predictive distribution of the random variable Sn+1 .
Show the following result (all the intermediary steps are required):
Bi;n+1 = E[μ(Θ)jSi1, ..., Sin] = E[Si;n+1 jSi1, ..., Sin].
Question III:
We assume the following mixing distributions: St|Θ ~ Poisson(Θ), with Θ ~ Gamma(α, τ ). The distribution functions are then:
(a) Find the posterior distribution of the heterogeneity parameter for the year T + 1, knowing S1 = s1 , ..., ST = sT , i.e. u (θ|S1 , ..., ST ) (all the intermediary steps are required).
(b) Find the predictive distribution of ST+1 knowing S1 = s1 , ..., ST = sT , i.e. Pr(ST+1|S1 , ..., ST ) (all the intermediary steps are required).
(c) Find the Bayesian (predictive) premium using the result in (a)
(d) Find the Bayesian (predictive) premium using the result in (b)
Question IV:
You are given:
(i) An individual insured has annual claim frequencies that follow a Poisson distribution with mean Λ .
(ii) An actuary’s prior distribution for the parameter Λ has probability density function: πΛ (λ) = 0:5 [5e-5λ + 0:2e-0:2λ]
(iii) In the first policy year, no claims were observed for the insured.
Determine the expected number of claims in the second policy year.
Question V:
You are given:
(i) The size of a claim has an exponential distribution with probability density function:
(ii) The prior distribution of Λ is an inverse gamma distribution with probability density function for x ≥ 0.
For a single insured, two claims were observed that totaled 50. Determine the expected value of the next claim from the same insured.