SOA Exam ASTAM Sample Question 12

Study guide: Exam ASTAM

You are the actuary with the task of fitting a loss model to a set of 533 observed claims, over a two-year period, from a portfolio of liability insurance policies with very long tails. There is no censoring or truncation in the observed claims.

You are thinking of fitting a Burr XII model with distribution function of the form

$F(x) = 1 - \left(\frac{\theta}{\theta + x^\gamma}\right)^\alpha, \quad x > 0,$

where the parameters $\theta$ , $\gamma$ , $\alpha$ are all positive. Observe that the Pareto model is a special case of this model where the parameter $\gamma$ is fixed at 1.

You fitted both the Burr XII and the Pareto models to the data and the maximum likelihood estimates together with other useful statistics are given:

Model	Parameters	Estimates (MLE)	Standard errors	Log-likelihood
Pareto	$\alpha$	1.1819	0.1115	–3515.941
	$\theta$	136.6418	21.1313
Burr XII	$\alpha$	1.1931	0.2019	–3526.253 (as printed)
	$\theta$	138.6682	36.5152
	$\gamma$	0.9956	0.0639

Part a

Briefly describe four steps you would follow to use the data to select a parametric loss model for the severity.

Part b

Explain what the standard errors are in the table above, how these are to be used, and how they have been calculated.

Part c

Conduct a hypothesis test to examine whether $\gamma = 1$ based on the model estimates in the Burr XII model.

Part d

Calculate the Schwarz Bayesian Criterion (SBC) for each model. Explain what this criterion is used for and how you would choose between the Pareto and the Burr XII model based on this criterion.

Part e

Briefly summarize your findings whether the Pareto or the Burr XII is a better model to the data.

Answer this question and see the worked solution