Huijuan Liu Cass Business School Lloyd’s of London 30/05/2007
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Predictive Distributions for Reserves which
Separate True IBNR and IBNER Claims
Huijuan LiuCass Business SchoolLloyd’s of London
30/05/2007
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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
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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
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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
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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?
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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
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Prediction Variances of Overall Reserves
Prediction Variance = Process Variance + Estimation Variance
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)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
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Process / Estimation Variances of Row Total
1nX
nX1
1.2 nX nX 2
2nX nnX
Recursive approach
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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The End