SOA Exam ASTAM Sample Question 15

Study guide: Exam ASTAM

The ground-up loss severity random variable for a liability insurance risk is denoted $Y$ . Conditioning on $\Lambda = \lambda$ , $Y$ follows an exponential distribution with mean $1/\lambda$ , and $\Lambda$ is gamma distributed with $\alpha = 4$ , and $\theta = 0.002$ .

The number of losses has a negative binomial distribution with parameters $r = 2$ and $\beta = 1.5$ .

Suppose that an ordinary deductible of 100 is applied to each individual loss.

Part a

Show that the unconditional distribution function of the ground-up loss is

$F_Y(y) = 1 - \left(\frac{500}{500 + y}\right)^4, \qquad y \ge 0.$

Part b

Show that the distribution for the number of payments is negative binomial with $r = 2$ , and $\beta^* = 0.72$ to the nearest 0.01. You should calculate the value to the nearest 0.001

Part c

Show that the distribution of the amount paid per payment $Y^P$ is given by

$F_{Y^P}(y) = 1 - \left(\frac{600}{600 + y}\right)^4, \qquad y \ge 0.$

Part d

Using the method of rounding, with a span of $h = 50$ , calculate the first three values of the probability function of the discretized distribution for $Y^P$ .

Part e

Use the recursive formula, with the discretized probabilities from (d), to estimate the probability that aggregate payments are less than or equal to 100.

Answer this question and see the worked solution