SOA Exam ASTAM Spring 2023, Question 6

Part a (2 points)

(i) Describe the purpose of Outstanding Claim Reserves (OCR).
(ii) Write down three reasons why there might be a delay between a loss event and the claim settlement.

Part b (2 points)

You are given the following data on cumulative claims, $C_{i,j}$ , for accident year (AY) $i$ , $i = 0, 1, \dots, 4$ , and development year (DY) $j$ , $j = 0, 1, \dots, 4$ .

Accident Year $i$	DY 0	DY 1	DY 2	DY 3	DY 4
0	100	120	135	145	158
1	135	168	190	210
2	175	210	225
3	212	265
4	260

Let $X_{i,j}$ denote the incremental claims for AY $i$ in DY $j$ .

Use the chain ladder method to calculate the projected incremental claims $\hat{X}_{2,3}$ and $\hat{X}_{2,4}$ .

Part c (3 points)

The Poisson model for outstanding claims assumes that $X_{i,j}$ are independent random variables, with

 $X_{i,j} \sim \text{Poisson}\left(\mu_i\gamma_j\right), \text{ where } \mu_i > 0,\ \gamma_j > 0, \text{ and } \sum_{j=0}^{4}\gamma_j = 1.$

(i) Show that for any given $i$ , $j$ , $E\left[C_{i,j}\right] = \mu_i\sum_{k=0}^{j}\gamma_k$ under this model.
(ii) Interpret the $\mu_i$ parameters.
(iii) Interpret the $\gamma_j$ parameters.

Part d (3 points)

You are given that the chain ladder method generates the same estimate of projected claims as the Poisson model, with maximum likelihood estimation.

(i) Use the values in (b) to show that $\hat{\gamma}_3 = 0.0776$ and $\hat{\gamma}_4 = 0.0823$ .
(ii) Calculate the estimated standard deviation of the expected outstanding claims for AY 2.
(iii) Using a normal approximation, calculate a 95% confidence interval for the expected outstanding claims for AY 2.

Part e (1 point)

Your colleague notes that:

"The chain ladder estimate gives the same outstanding claims estimate as the Poisson model, so that the chain ladder model is the same as the Poisson model."

Critique your colleague's remarks.

Answer this question and see the worked solution