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MATH 366

MID TERM EXAM 2023

MATHEMATICAL RISK THEORY

1. Suppose that an insurer adopts an exponential utility of the form u(x) = −ae−ax, x ≥ 0.

(i) Show that the insurer has the non-satiation and the risk aversion property.     [2 marks]

(ii) It is estimated that an insurance-based investment product has a random return/loss X that follows a normal distribution with parameters µ > 0 and σ2 > 0. Find the minimum premium, P− that the insurer will price this contract. What is P−as a → 0? Explain your answer intuitively. [8 marks]

2. (a) Let the random variable N belongs to the Panjer family of distributions, i.e.

with pn = P(N = n). Show that the probability generating function, PN (z) = EzN satisfies the differential equation

[8 marks]

(b) Let the aggregate claims of a non-life insurance portfolio be given by

S = X1 + · · · + XN,

where the claim count r.v., N, has mass function , n = 0, 1, 2, . . ., whilst the claim size r.v. are independent and identically distributed (and independent from N) taking the values 1 and 2 with prob-abilities 1/5 and 4/5 respectively.

(i) Show that N belongs to the Panjer family of distributions.        [4 marks]

(ii) Find the probability P(S = 0).              [4 marks]

(iii) Find the probability P(S ≤ 3).             [4 marks]