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IAA Colloquium Oslo
Model Risk
Authors :
Ecaterina NISIPASU Laila ELBAHTOURI Mihaela [email protected] [email protected] [email protected]
SCOR SE : Actuarial Modelling Department – 5, Avenue Kléber 75795 Paris Cedex 16
Thomas [email protected]
AXA Liability Managers : Actuarial Department – 40, Rue du Colisée 75795 Paris Cedex 08
1
Tuesday 9th June 2015
1 What is the model risk ?
2 Applications and Results
3 Improvement for quantifying the model risk
Summary of the presentation
2
1 What is the model risk ?
2 Applications and results
3 Improvement for quantifying the model risk
Summary of the presentation
3
4
How can we define the model risk ?
No precise definition nor mathematical formulas
First fields to approach this risk : banking and financial world
General definition given in an insurance or reinsurance context
« Model risk can arise from various forms of errors or frominappropriate construction or use of the model ».
(Shaun Wang et al. – « Model Validation for Insurance Entreprise Riskand Capital Models. » 2014)
Part 1 : What is model risk ?
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Specification risk : risk that the chosen model is unadapted
Coding risk : wrong algorithm for model implementation or coding error
Data risk : wrong choice of historical data
Estimation risk : error in the calibration of model’s parameters
Application risk : model complexity
Model Risk
Specificationrisk
Codingrisk
Data risk
Estimation risk
Application risk
Part 1 : What is model risk ?
What are the potential sources ?
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Approach to quantify model risk
« Reference model » methodology
Choice of the reference model : it depends on statistical results
• Validation of each assumption of the model• Backtesting the model on the historical data
Absolute measure of model risk
𝐴𝐴𝐴𝐴 = sup𝑖𝑖
𝐴𝐴𝐴𝐴𝑖𝑖 = sup𝑖𝑖
𝜌𝜌 𝑋𝑋𝑖𝑖𝜌𝜌 𝑋𝑋0
− 1 = �𝜌𝜌 ℒ𝜌𝜌 𝑋𝑋0
− 1
with : �̅�𝜌 ℒ = 𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 𝑋𝑋𝑖𝑖 | 𝑋𝑋𝑖𝑖 ∈ ℒ , 𝑖𝑖 = 0, … , 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ℒ − 1𝜌𝜌 𝑋𝑋0 = risk measure of the reference model
(Pauline Barrieu, Giacomo Scandolo – « Assessing Financial Model Risk. » 2013)
Part 1 : What is model risk ?
1 What is the model risk ?
2 Applications and results
3 Improvement for quantifying the model risk
Summary of the presentation
7
2.1. Equity risk : Data and models
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Data : Stock index GDDLUS
Daily values from January 1999 to December 2012 Descriptive statistics
Part 2 : Applications and results
0
1000
2000
3000
4000
5000
6000
Stoc
k in
dex
valu
e
GDDLUS daily values: 1999 - 2012
2.1. Equity risk : Data and models
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Chosen model for forecasting the stock index
Black & Scholes model Merton model with Poisson « jump » GARCH model
How to choose the reference model
Testing the assumptions of each model with statistical tests Backtesting of the model : Probability Integral Transform Test
(Francis X. Diebold and al. – « Evaluating Density Forecasts. » 1998, and Peter Blum. – « On some mathematical aspects of dynamic fianancial analysis. » 1998 )
Part 2 : Applications and results
Part 2 : Applications and results
2.1. Equity risk : Application
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Reference model : GARCH model
Application for forecasting the stock index final value for 2013
VaR 99,5 %Model GARCH Black & Scholes Merton à saut𝒊𝒊 0 1 2
𝑨𝑨𝑨𝑨𝒊𝒊 0 0,0312 0,0296
AM0,0312
TVaR 99 %Model GARCH Black & Scholes Merton à saut𝒊𝒊 0 1 2
𝑨𝑨𝑨𝑨𝒊𝒊 0 -0,1000 -0,0086
𝐴𝐴𝐴𝐴 = sup𝑖𝑖
𝐴𝐴𝐴𝐴𝑖𝑖 = sup𝑖𝑖
𝜌𝜌 𝑋𝑋𝑖𝑖𝜌𝜌 𝑋𝑋0
− 1 =�̅�𝜌 ℒ𝜌𝜌 𝑋𝑋0
− 1
�̅�𝜌 ℒ = 𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 𝑋𝑋𝑖𝑖 | 𝑋𝑋𝑖𝑖 ∈ ℒ , 𝑖𝑖 = 0, … , 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ℒ − 1
AM0
2.2. Reserving risk : Data and models
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Data : Non proportional reinsurance branch
Historical data of 15 years Study on paid triangle
Part 2 : Applications and results
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cum
ulat
edPa
id
Development years
1999 2000 2001 2002 2003 2004 2005 2006
2007 2008 2009 2010 2011 2012 2013
2.2. Reserving risk : Data and models
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Chosen model for the ultimate claim estimation
Chain-Ladder model Mack Model Generalized Linear Models
• Poisson• Log-Normal• Log-Gamma
How to choose the reference model
Testing the assumption of each model with statistical test Testing Bootstrap residuals assumption of each model
Part 2 : Applications and results
Part 2 : Applications and results 13
2.2. Reserving risk : Application
Reference model : Chain-Ladder model
Application for estimating the ultimate claim amount
VaR 99,5 %
Model Chain-Ladder Mack GLM PoissonGLM Log-Gamma
GLM Log-Normal
𝒊𝒊 0 1 2 3 4𝑨𝑨𝑨𝑨𝒊𝒊 0 -0,2338 0,0765 0,6909 0,1900
TVaR 99 %
Model Chain-Ladder Mack GLM PoissonGLM Log-Gamma
GLM Log-Normal
𝒊𝒊 0 1 2 3 4𝑨𝑨𝑨𝑨𝒊𝒊 0 -0,2351 0,0786 0,7015 0,1918
AM0,6909
AM0,7015
𝐴𝐴𝐴𝐴 = sup𝑖𝑖
𝐴𝐴𝐴𝐴𝑖𝑖 = sup𝑖𝑖
𝜌𝜌 𝑋𝑋𝑖𝑖𝜌𝜌 𝑋𝑋0
− 1 =�̅�𝜌 ℒ𝜌𝜌 𝑋𝑋0
− 1
�̅�𝜌 ℒ = 𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 𝑋𝑋𝑖𝑖 | 𝑋𝑋𝑖𝑖 ∈ ℒ , 𝑖𝑖 = 0, … , 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ℒ − 1
1 What is the model risk ?
2 Applications and results
3 Improvement for quantifying the model risk
Summary of the presentation
14
Part 3 : Improvement for quantifying the model risk
How can we improve the methodology ?
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Use all individual values and not just the upper bound of the model risk measure
Taking into account factors of credibility depending on two criteria : Expert judgment Statistical results
New adjusted measure for the absolute measure of model risk:
Issue : Accurate estimation of these factors of credibility
𝐴𝐴𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒 = �𝑖𝑖=0
𝑐𝑐𝑎𝑎𝑐𝑐𝑒𝑒 ℒ −1
𝜔𝜔𝑖𝑖 𝐴𝐴𝐴𝐴𝑖𝑖 = �𝑖𝑖=0
𝑐𝑐𝑎𝑎𝑐𝑐𝑒𝑒 ℒ −1𝜔𝜔𝑖𝑖 𝜌𝜌 𝑋𝑋𝑖𝑖𝜌𝜌 𝑋𝑋0
− 1
�𝑖𝑖=0
𝑐𝑐𝑎𝑎𝑐𝑐𝑒𝑒 ℒ −1
𝜔𝜔𝑖𝑖 = 1with :
Conclusion
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Model risk depends hugely on every selected model
The application of Solvency II will speed up the study of model risk
Need of a precise approach to quantify this risk
« All models are wrong, but some are useful » - Georges Box
Thank you for your attention !
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