MATH 366
MATHEMATICAL RISK THEORY
1. A decision maker has a utility function u(x) = x. The decision maker has to choose between the investment A and B which have returns X and Y with probability density functions
respectively.
By using the maximum utility criterion, decide in which investment the decision maker has to invest. [10 marks]
2. The aggregate claims of a property portfolio are given by
S = X1 + · · · + XN ,
where N is geometric distributed with parameter p and {Xi}i≥1 are independent and identical Exponential distributed random variables with parameter λ > 0. We assume that Xis and N are independent.
(i) For above portfolio it has been estimated that the premium P is equal to the mean plus 20% of the standard deviation of the aggregate claims. Find the premium P. [5 marks]
(ii) Find the Ee zS. What can you conclude about the distribution of S ? [5 marks]
(iii) It is estimated that the utility function for the above properties is given by u(x) = −e −ax, a > 0. What is the maximum premium, P +, that the insurered will pay buying an insurance policy for (overall) property portfolio. [5 marks]
(iv) Assume that the insurer effects an excess of loss reinsurance with retention limit m. Find the moment generating function of the reinsurer’s number of non zero claims, NR. [10 marks]
3. You are given that an insurance portfolio has aggregate claims S = X1 + · · · + XN , with
m'S (z) = aλm'X(z)mS(z),
where Xis and N are independent with mS(z) = Ee zS and mX(z) = Ee zX, and λ > 0, a > 0.
(i) Show that the claim frequency random variable, N, belongs to the Panjer’s family of distributions [10 marks]
(ii) Let X ∼ U(0, b), b > 0. Use normal approximation to calculate the prob-ability P(S > 1, 000λ) [you may express the answer in terms of a standard normal distribution function] [5 marks]