Normal Distribution and Collective Premium
AMA470 Midterm exam
March 5, 2010
Please show full working out in order to obtain full marks.
1. Suppose that:
• The number of claims per exposure period follows a Poisson distribution with mean λ = 110.
• The size of each claim follows a lognormal distribution with parameters µ and σ 2 = 4.
• The number of claims and claim sizes are independent.
(a) Give two conditions for full credibility that can be completely determined by the information above. Make sure to define all terms in your definition.
(b) Suppose that 7000 claims are needed for full credibility (with range parameter k = 0.1 and and probability level P ). Determine P .
1
2
2. A portfolio contains two types of risks: risk A and risk B:
• For risk A, the number of claims per year is independent and follows a Poisson distribution with mean 1.
• For risk B, the number of claims per year is independent and follows a Poisson distribution with mean 3.
Suppose that the portfolio contains the same number of people for each risk. Consider a random insured individual and let Xj denote his claims in year j.
(a) Calculate the probability that he makes k claims in the second year (for k = 0, 1, . . .) given that he makes no claims in the first year. (b) Find E[X2 |X1 = 0].
3
4
3. Suppose that:
• Conditioned on Θ = θ, the number of claims in each year N1 , N2 , . . . are independent and Poisson distributed with parameter 3θ and the claims amount in the jth year is Xj = 1000Nj .
• The prior distribution is gamma distributed with mean 0.2 and variance 0.04.
(a) Compute the individual premium and collective premium.
(b) Suppose that in the past 5 years, an individual has made N claims.
If his Bayes premium is twice the collective premium, compute N .
5
6
March 5, 2010
Please show full working out in order to obtain full marks.
1. Suppose that:
• The number of claims per exposure period follows a Poisson distribution with mean λ = 110.
• The size of each claim follows a lognormal distribution with parameters µ and σ 2 = 4.
• The number of claims and claim sizes are independent.
(a) Give two conditions for full credibility that can be completely determined by the information above. Make sure to define all terms in your definition.
(b) Suppose that 7000 claims are needed for full credibility (with range parameter k = 0.1 and and probability level P ). Determine P .
1
2
2. A portfolio contains two types of risks: risk A and risk B:
• For risk A, the number of claims per year is independent and follows a Poisson distribution with mean 1.
• For risk B, the number of claims per year is independent and follows a Poisson distribution with mean 3.
Suppose that the portfolio contains the same number of people for each risk. Consider a random insured individual and let Xj denote his claims in year j.
(a) Calculate the probability that he makes k claims in the second year (for k = 0, 1, . . .) given that he makes no claims in the first year. (b) Find E[X2 |X1 = 0].
3
4
3. Suppose that:
• Conditioned on Θ = θ, the number of claims in each year N1 , N2 , . . . are independent and Poisson distributed with parameter 3θ and the claims amount in the jth year is Xj = 1000Nj .
• The prior distribution is gamma distributed with mean 0.2 and variance 0.04.
(a) Compute the individual premium and collective premium.
(b) Suppose that in the past 5 years, an individual has made N claims.
If his Bayes premium is twice the collective premium, compute N .
5
6