Predictive Distributions for Reserves which

Separate True IBNR and IBNER Claims

Huijuan LiuCass Business SchoolLloyd’s of London

30/05/2007

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

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)

New ClaimsChanges in Old Claims

Development year j Development year j

Accident year i

Accident year i

Incurred

IBNR IBNER

, 1ij i j ij ijX X D N

1, jiij XX

ijDijN

jijiij EXNE 1, jjijiij XXDE 1,1,

21, jijiij EXNVar

21,1, jjijiij XXDVar

, , , 1 , 11i j t ij i j t i j t ij j t i j t ij i j tE X X E E X X X E X X E

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?

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

),(~1,

2

1,1,

ji

jjji

ji

ij

XNormalX

XD

),(~2

1,i

jjji

i

ij

ENormalX

EN

A Stochastic Model

Prediction Variances of Overall Reserves

Prediction Variance = Process Variance + Estimation Variance

)ˆ()()(

111

n

iin

n

iin

n

iin XVarXVarXMSEP

Process Variances of Row Total

Estimation Variance of Row Total

Covariance between Estimated

Row Total

n

kti

kntn

n

iin

n

iin XXCovXVarXVar

111

)ˆ,ˆ(2)ˆ()(

Process Variances of Overall Total

Estimation Variances of Overall

Total

Process / Estimation Variances of Row Total

1nX

nX1

1.2 nX nX 2

2nX nnX

Recursive approach

Estimation Covariance between Row Totals

2ˆnX

1,3ˆ

nX

2.3ˆ

nX 1,3ˆ

nX

3ˆnX 1,

ˆnnX

nX 2ˆ

nX 3ˆ

nX 4ˆ

nnX̂

1,2 nX

nX1

nX1

nX1

2,3 nX

Recursive approach

Correlation = 0

Calculate correlation between estimates

Calculate correlation

using previous correlation

The Results

n

iniinininininin EXXEXXVar

1

21,1,

21,1,

2 )1(

n

iniininin

inininniniin

VarEXXVarVar

XXVarEVarXXE

1 21,1,

1,1,

22

1,

ˆˆˆ

ˆˆ1ˆˆ

kt

nkt

knktntnkntn

knktntnknt

knktntnkntn

VarEE

XXXXCovE

XXXX

XXXXCovVar

)ˆ(

),ˆ,ˆ()ˆ1(

,ˆˆ

),ˆ,ˆ()ˆ(

2 1,1,1,1,

2

1,1,1,1,

1,1,1,1,

)(1

n

iinXMSEP

BootstrapOriginal Data with

size mDraw randomly

with replacement, repeat n times

Simulate with mean equal to corresponding Pseudo Data

Pseudo Data with size m

Simulated Data with size m

Estimation Variance

Prediction Variance

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 102242 1.6 14.8 32.1 39.6 55 60 127523 13.8 42.4 36.3 53.3 96.5 148754 2.9 14 32.5 46.9 173655 2.9 9.8 52.7 194106 1.9 29.4 176177 19.1 18129

Example Schnieper Data

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

D Triangle 2 3 4 5 6 72 -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

  Reserves estimates Estimation errors Prediction errors 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%

Analytical & Bootstrap

Empirical Prediction Distribution

-100 0 100 200 300 400 500 600 700

0.001

0.002

0.003

0.004Density

Svar1 N(s=112)

-50 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700

25

50

75

100

125 Svar1

Fig. 1 Empirical Predictive Distribution of Overall Reserves

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.

The End