Predictive Distributions for Reserves which Separate True IBNR and IBNER Claims

1
Predictive Distributions for Reserves which
Separate True IBNR and IBNER Claims

• Huijuan Liu
• Cass Business School
• Lloyds of London
• 30/05/2007

2
Introduction

• The Schniepers 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 Schniepers Model
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
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
1 7.5 28.9 52.6 84.5 80.1 76.9 79.5 10224
2 1.6 14.8 32.1 39.6 55 60 12752
3 13.8 42.4 36.3 53.3 96.5 14875
4 2.9 14 32.5 46.9 17365
5 2.9 9.8 52.7 19410
6 1.9 29.4 17617
7 19.1 18129
Schnieper Data
14
N triangle 1 2 3 4 5 6 7
1 7.5 18.3 28.5 23.4 18.6 0.7 5.1
2 1.6 12.6 18.2 16.1 14 10.6
3 13.8 22.7 4 12.4 12.1
4 2.9 9.7 16.4 11.6
5 2.9 6.9 37.1
6 1.9 27.5
7 19.1
15
D Triangle 2 3 4 5 6 7
2 -3.1 4.8 -8.5 23 3.9 2.5
3 -0.6 0.9 8.6 -1.4 5.6
4 -5.9 10.1 -4.6 -31.1
5 -1.4 -2.1 -2.8
6 0 -5.8
7 0
16
Analytical Bootstrap
  Reserves estimates Reserves estimates Estimation errors Estimation errors Prediction errors Prediction errors prediction error prediction error
  Analytical Bootstrap Analytical Bootstrap Analytical Bootstrap Analytical Bootstrap
2 4.4 4.4          
3 4.8 5.2 6.0 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