1. Huijuan Liu Cass Business School Lloyd’s of London 30/05/2007
Predictive Distributions for Reserves which Separate True IBNR and IBNER Claims
Huijuan Liu
Cass Business School
Lloyd’s of London
30/05/2007

2. Introduction The Schnieper’s Model (1991) Extended Stochastic Models
Analytical Prediction Errors of the Reserves
Straightforward Bootstrapping Procedure for Estimating the Prediction Errors
The full Predictive Distribution of Reserves

3. The Schnieper’s Model + Incremental Incurred IBNR IBNER
According to when the claim occurs, we can separate Incremental Incurred into Incurred But Not Reported (IBNR) and Incurred But Not Enough Reported (IBNER)
Incremental Incurred
Development year j
Development year j
IBNR
IBNER
Accident year i +
Accident year i
Changes in Old Claims
New Claims

4. Incurred
IBNR IBNER

5. Questions from the Schnieper Model
Since the expected ultimate loss can be produced analytically,
what about the prediction variance?
Can the analytical result of the prediction variance be tested?
Is there a possibility to extend the limits of the model, which
is the model can not be applied to the data without exposure
and the claims details?

6. A Stochastic Model
To derive a prediction distribution variance and test it, a stochastic model is necessary. A normal process distribution is the ideal candidate, i.e.

7. Prediction Variances of Overall Reserves
Prediction Variance = Process Variance + Estimation Variance

8. Process Variances of Overall Total
Estimation Variances of Overall Total
Process Variances of Row Total
Estimation Variance of Row Total
Covariance between Estimated Row Total

9. Process / Estimation Variances of Row Total
Recursive approach

10. Estimation Covariance between Row Totals
Recursive approach
Calculate correlation between estimates
Correlation = 0
Calculate correlation using previous correlation

11. The Results

12. Bootstrap Bootstrap Prediction Variances Original Data with size m
Draw randomly with replacement, repeat n times
Estimation Variance
Pseudo Data with size m
Bootstrap Prediction Variances
Simulate with mean equal to corresponding Pseudo Data
Original Data with size m
Draw randomly with replacement, repeat n times
Prediction Variance
Simulated Data with size m
Pseudo Data with size m
Simulate with mean equal to corresponding Pseudo Data

13. Example
X triangle 1 2 3 4 5 6 7
exposure
7.5
28.9
52.6
84.5
80.1
76.9
79.5
10224
1.6
14.8
32.1
39.6 55 60
12752
13.8
42.4
36.3
53.3
96.5
14875
2.9 14
32.5
46.9
17365
9.8
52.7
19410
1.9
29.4
17617
19.1
18129
Schnieper Data

14. N triangle 1 2 3 4 5 6 7
7.5
18.3
28.5
23.4
18.6
0.7
5.1
1.6
12.6
18.2
16.1 14
10.6
13.8
22.7
12.4
12.1
2.9
9.7
16.4
11.6
6.9
37.1
1.9
27.5
19.1

15. D Triangle 2 3 4 5 6 7
-3.1
4.8
-8.5 23
3.9
2.5
-0.6
0.9
8.6
-1.4
5.6
-5.9
10.1
-4.6
-31.1
-2.1
-2.8
-5.8

16. Analytical & Bootstrap
Reserves estimates
Estimation errors
Prediction errors
prediction error %
Analytical
Bootstrap 2
4.4 3
4.8
5.2
6.0
9.5
9.8
196%
187% 4
32.5
32.1
13.6
13.2
27.2
30.3
84%
95% 5
61.6
60.0
21.8
20.9
39.0
41.5
63%
69% 6
78.6
77.2
22.3
21.3
41.7
45.8
53%
59% 7
105.4
104.4
26.7
25.5
47.6
50.3
45%
48%
Total
287.3
283.3
77.1
80.3
110.9
112.4
39%
40%

17. Empirical Prediction Distribution
Fig. 1 Empirical Predictive Distribution of Overall Reserves
Fig. 1 Empirical Predictive Distribution of Overall Reserves

18. Further Work
Apply the idea of mixture modelling to other situation, such as paid and incurred data, which may have some practical appeal.
Bayesian approach can be extended from here.
To drop the exposure requirement, we can change the Bornheutter-Ferguson model for new claims to a chain-ladder model type.

19. The End