SOA Exam ASTAM Fall 2024, Question 6

You are analyzing a run-off triangle of aggregate claims. The development factors are:

DY, j	0	1	2	3	4	5	6	7
$f_j$	3.0982	1.4436	1.1955	1.0874	1.0360	1.0186	1.0056	1.0000

Part a (2 points)

(i) State with reasons whether this business is long tailed or short tailed.

(ii) You are given that the most recent cumulative claims data for Accident Year (AY) 6 is $C_{6,1} = 9{,}182$ .

Show that the estimated outstanding claims for AY 6 total 9,100 to the nearest 100, using the Chain Ladder method. You should calculate the value to the nearest 1.

Part b (2 points)

(i) The Earned Premium in AY 6 is 18,000 and the expected loss ratio at issue is 0.85. Show that the estimated outstanding claims for AY6 using the Bornhuetter-Ferguson method is 7,600 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Describe one advantage and one disadvantage of the Bornhuetter-Ferguson method compared with the Chain Ladder method.

Part c (3 points)

You are given the following four assumptions for Mack's model.

(A1) $C_{i,j}$ and $C_{l,k}$ are independent for $i \ne l$ and for all $j, k$ .
(A2) For a given accident year (AY) $i$ , $\{C_{i,j}\}_{j=0,1,2\ldots}$ is a Markov Chain.
(A3) There exist $f_j$ such that $\mathrm{E}\left[C_{i,j+1} \mid C_{i,j}\right] = f_j\,C_{i,j}$ .
(A4) There exist $\sigma_j^2$ such that $\mathrm{Var}\left[C_{i,j+1} \mid C_{i,j}\right] = \sigma_j^2\,C_{i,j}$ .

(i) Explain briefly how these assumptions differ from the Chain Ladder assumptions.

(ii) Show that under Mack's model

$\mathrm{Var}\left[C_{i,j+2} \mid C_{i,j}\right] = C_{i,j}\left(\sigma_{j+1}^2 f_j + f_{j+1}^2\sigma_j^2\right)$

Part d (2 points)

Your colleague has fitted Mack's model to the runoff triangle in (a), using the R ChainLadder package. The results are summarized in the graphs below. Note that there are 26 residual values shown in each of the lower four residual plots.

(i) State with reasons whether the graphs show that the data are consistent with the Mack Model assumptions.

(ii) Explain briefly how the Mack Model estimates of outstanding claims compare with the chain ladder estimates.

Part e (2 points)

(i) You note from the R output that the sum of the Mean Square Errors of Prediction (MSEP) over all the AYs is $1428^2$ , and that the total MSEP is $1547^2$ .

Your colleague says that this must be an error, as the model assumption A1 states that each AY is independent. Critique this comment.

(ii) Another colleague notes that the Bornhuetter-Ferguson estimate of the final cumulative claims for AY6 (from (b) above) lies outside the AY6 whiskers in the top left plot. State with reasons which estimate you would recommend to the insurer.

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