CAS Exam 7 Spring 2015, Question 18

Study guide: Exam 9

The International Actuarial Association's A Global Framework for Insurer Solvency Assessment outlines the following procedure for simulating underwriting losses for a multi-line insurer:

For each line of insurance i:
- a. Select a random number $\chi_i$ from a gamma distribution with mean 1 and variance $c_i$ .
- b. Select a random claim count $K_i$ from a Poisson distribution with mean $\chi_i \lambda_i$ where $\lambda_i$ is the expected claim count for line of insurance $i$ .
- c. For each $i$ and for $k = 1, \ldots, K_i$ , select a random claim size, $Z_{ik}$ , from a lognormal distribution with mean $\mu_i$ and standard deviation $\sigma_i$ .
Set $X_i = \sum_{k=1}^{K_i} Z_{ik}$ = Loss for line of insurance $i$ .
Select a random number $p$ , from a uniform (0, 1) distribution. For each line $i$ , select $\beta_i$ to be the $p^{th}$ percentile of a distribution with $E[\beta_i] = 1$ and $Var[\beta_i] = b_i$ .
Set $X = \sum_i \beta_i \cdot X_i$ = Loss for the insurer.

This results in the following formula for the variance of a line's loss ratio (losses divided by expected losses):

$Var\left[\frac{\beta_i \cdot X_i}{E[\beta_i \cdot X_i]}\right] = (1 + b_i)\left(\frac{\mu_i^2 + \sigma_i^2}{\lambda_i} + c_i\right) + b_i$

Part a (0.5 points)

Derive a formula for the minimum variance of a line's loss ratio for the model described above.

Part b (0.25 points)

Briefly explain the significance of the result from part a. above for the estimation of an insurer's underwriting risk within any single line of business.

Part c (0.25 points)

Identify the correlation affected by the parameter $c_i$ above.

Part d (0.25 points)

Identify the correlation affected by the parameter $b_i$ above.

Part e (0.75 points)

For a particular line of business, the standard deviation of the loss ratio can be no smaller than 30%. In addition, the standard deviation of inflationary effects is estimated at 4%. Determine estimates of $b$ and $c$ for this line of business.

Answer this question and see the worked solution