SOA Exam ASTAM Spring 2025, Question 5

You are calculating outstanding claim reserves for your company's portfolio of short-term insurance contracts.

You are given the following cumulative claims run-off triangle.

Accident Year $i$	Development Year $j$ = 0	1	2	3
0	660	1155	1403	1568
1	792	1304	1650
2	825	1568
3	941

Part a (2 points)

Show that the estimated outstanding claims for Accident Year (AY) 3, using the chain ladder method, are 1370 to the nearest 10. You should calculate the value to the nearest 1.

Part b (3 points)

You are given the following notation and assumptions for the Bornhuetter-Ferguson (BF) method.

Given parameters $\mu_i,\ i = 0, 1, \cdots, I$ and $\beta_j,\ j = 0, 1, 2, \cdots, J$ , where $\beta_J = 1$ :

BF Assumption (1):

$E\left[C_{i,0}\right] = \beta_0 \mu_i$

$E\left[C_{i,j+1} \mid C_{i,0}, C_{i,1}, \cdots, C_{i,j}\right] = C_{i,j} + \left(\beta_{j+1} - \beta_j\right)\mu_i$

BF Assumption (2): $C_{i,j}$ and $C_{l,k}$ are independent for $i \ne l$ , and for all $j$ , $k$ .

(i) Show using iteration that $E\left[C_{i,J} \mid C_{i,j}\right] = C_{i,j} + \left(1 - \beta_j\right)\mu_i$ .

(ii) Show that $E\left[C_{i,j}\right] = \beta_j \mu_i$ .

(Hint: Use induction.)

Part c (3 points)

You are given the following additional information.

(i) The estimated values of the $\beta_j$ parameters are $\hat{\beta}_j = \dfrac{1}{\hat{\lambda}_j}$ .

(ii) The BF estimate of ultimate cumulative claims from AY $i$ , given $C_{i,j}$ , is $\tilde{C}_{i,J} = C_{i,j} + \big(1 - \hat{\beta}_j\big)\mu_i$ .

(iii) $\hat{C}_{i,J}$ is the chain ladder estimate of ultimate cumulative claims for AY $i$ .

(i) Derive the following formula for the estimated cumulative claims using the BF method:

$\tilde{C}_{i,J} = \hat{\beta}_j\, \hat{C}_{i,J} + \big(1 - \hat{\beta}_j\big)\mu_i$

(ii) Explain how the BF estimate of cumulative claims can be interpreted as a credibility estimate.

Part d (2 points)

You are also given:

(i) For each AY $i$ , $\mu_i$ is based on a loss ratio of 90%.

(ii) The earned premiums for AY 3 are 2400.

Calculate the BF estimate of the outstanding claims for AY 3.

Part e (1 point)

Write down one advantage and one disadvantage of using the BF method over the chain ladder method.

Answer this question and see the worked solution