CAS Exam 7 Spring 2012, Question 8

Study guide: Exam 7

Given the following loss development triangle and selections:

Accident Year	Incremental Incurred Losses ($000) 0-12 Months	12-24 Months	24-36 Months	36-48 Months	48-60 Months
2007	5,800	1,754	875	310	54
2008	6,000	1,675	780	285
2009	5,700	1,850	888
2010	5,850	1,500
2011	6,300

Selected Age-to-Age Link Ratios	12-24 Months	24-36 Months	36-48 Months	48-60 Months	60 Months to Ultimate
Selected	1.290	1.112	1.035	1.006	1.000

Incremental losses, $C_{ij}$ , follow an over-dispersed Poisson distribution with mean $x_i y_j$ and variance $\phi x_i y_j$ .
The variable $x_i$ represents the expected ultimate losses for accident year $i$ .
The variable $y_j$ represents the proportion of ultimate losses that emerge in development year $j$ .
The prior distribution for $x_i$ is gamma with mean $\alpha_i / \beta_i$ and variance $\alpha_i / (\beta_i)^2$ .
The dispersion parameter, $\phi$ , for the over-dispersed Poisson distribution is 8.429.
The accident year 2010 estimates for $\alpha$ and $\beta$ are 100 and 0.01143 respectively.

The mean of $C_{ij}$ for this Bayesian model is:

$Z_{ij} \times (\lambda_j - 1) \times D_{i,j-1} + (1 - Z_{ij}) \times (\lambda_j - 1) \times x_i \times \left(\frac{1}{\lambda_j}\right)$

where:

$Z_{ij} = \frac{\sum_{k=1}^{i-1} y_k}{\beta_i \times \varphi + \sum_{k=1}^{i-1} y_k}$

$\lambda_j$ is the incremental chain ladder loss development factor for development year $j$ .
$D_{ij}$ is the cumulative losses for accident year $i$ as of development year $j$ .

Part a (3 points)

Use this model to calculate the incremental losses for accident year 2010 expected to emerge between 24 and 36 months of development.

Explain how this model can be interpreted as a trade-off between the standard chain ladder method and the Bornhuetter-Ferguson method.