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Stochastic Scenario CreationInfoline Zurich13 December 2012
Servaas Houben
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Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Infoline 13 December 2012 Zurich
Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
Infoline 13 December 2012 Zurich
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Capital requirement under SII
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Infoline 13 December 2012 Zurich
VaR limitations - subaddivity
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Infoline 13 December 2012 Zurich
Risk 1Probability Loss
0.03 1 mln
0.97 0
95% VaR 0
Risk 2Probability Loss
0.03 1 mln
0.97 0
95% VaR 0
Risk 1 and 2
Probability Loss
One event 0.0582 1 mln
Two events 0.0009 2 mln
95% VaR 1 mln
QuizData:• Monthly capital return index S&P 500 returns from Dec
1927-Feb 2011• Dec 1927 index value 17.66, Feb 2011 1,327.22• Total number of 998 monthly returns
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Infoline 13 December 2012 Zurich
Question:When excluding 10 highest monthly returns (setting them to 0%) what would be the index value as at Feb 2011?
Answers
<250
250-500
500-750
750-1000
>1000
Results• Set highest 10 values to 0: 172.80 (-87%)• Set lowest 10 values to 0: 15,330.78 (+1.050%)
0
200
400
600
800
1,000
1,200
1,400
1,600
1927
1934
1941
1948
1955
1962
1969
1976
1983
1990
1997
2004
All inclusive
Top 10 excluded
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
1927
1935
1943
1951
1959
1967
1975
1983
1991
1999
2007
All inclusive
Bottom 10 excluded
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Infoline 13 December 2012 Zurich
“Risk mitigation” through dividends
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-
1.000
2.000
3.000
4.000
5.000
6.000
Valu
e to
tal r
etur
n
Date
S&P500 Total return
total return index
without top 10
Overall scenario creation process
Data• Regime shifts?• Proxy data• Stale prices• Volatility
clustering
Calibration• Stationarity?• Body and tail
calibration• Sensitivity
testing measures
Dependencies• Body and tail
dependencies• PSD condition• Trade-off data
and economic rationale
Validation of scenarios• Sampling error• Flooring of risk
drivers• Rare event
distortion
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Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Risk identification
• Identification of quantifiable risks • Mapping of individual risks to homogeneous risk
groups▫ Diversification▫ Reporting
• Trade-off granularity and practical implementation
• Risk universe stores information
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Infoline 13 December 2012 Zurich
Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Infoline 13 December 2012 Zurich
Data selection• Empirical data:▫ Market risks
• Non empirical data/expert judgment:▫ Operational risks ▫ Non market risks Life and non-life risks
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Nominal yield curves
Risk Management Hedging uses assets quoted on OISPricing (Guarantees) Funding for hedging based on OISProvisioning Solvency II based on LIBOR & UFR
– One-off surplus (based on current market environment)– Hedging efficiency and provisioning risk due to LIBOR-OIS basis
EUR 24 August 2012
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
1 11 21 31Term (Years)
EUR / 24 August 2012 / Spot / Annual
Market LIBORMarket OISSolvency II Risk Free
Source: Bloomberg
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Liquidity and Matching premium adjustments
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050
100150200250300350400450500
LQP
in b
asis
poi
nts
Date
LQP development over time
USD
GBP
EUR
(100)
(50)
-
50
100
150
200
250
300
350
400
MP
in b
p
Date
MP development
UK
US
EUR
Infoline 13 December 2012 Zurich
Source: Itraxx
Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Historical data collection
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0
200
400
600
800
1.000
1.200
1.400
1.600
1.800
1928 1938 1948 1958 1968 1978 1988 1998 2008
Inde
x va
lue
Date
S&P 500
capital returnindex
Data amendments• Select indices deemed most appropriate• Apply transformation to data to check if data is
stationary
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-30%
-20%
-10%
0%
10%
20%
30%
40%
1928 1938 1948 1958 1968 1978 1988 1998 2008
Inde
x va
lue
Date
S&P 500 return
Calibration
• Determine sample stats• Distribution fitting: ▫ Fit to different distributions▫ Individual country fitting/clustering
• Fit testing:▫ Kolmogorov-Smirnov goodness of fit
test▫ Anderson Darling▫ Sense test: 0.5% and 0.05%
percentiles▫ Plot sample data and fitted
distributions
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Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Case study - diversification
• Measurement of strength and direction of relationship 2 risk drivers:
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-100,00%-80,00%-60,00%-40,00%-20,00%
0,00%20,00%40,00%60,00%80,00%
-4,000% -3,000% -2,000% -1,000% 0,000% 1,000% 2,000% 3,000% 4,000%
TWD
equ
ity
UK property
Property Monthly Total Return UK vs TWD equity return, Feb 1991 - Feb 2011
Infoline 13 December 2012 Zurich
Produce correlated risk drivers
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Uncorrelated risk drivers Correlated risk drivers
Infoline 13 December 2012 Zurich
Scenario production
• Apply correlated random numbers to calibrated distributions
• Apply restrictions to certain risk drivers▫ Interest rates▫ Credit spreads▫ Volatilities
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Agenda
Solvency II Capital requirementRisk identificationData selection and limitationsCalibrationAggregation and dependenciesValidation
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Validation
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Risk drivers
• Key statistics• Mean/median• Standard deviation• Skewness• Kurtosis
• Key percentiles• 1 in 200 capital
requirement
Dependencies
• Complications for risk drivers portraying tail behaviour
Infoline 13 December 2012 Zurich
References & contact details• CEIOPS, Task Force Report on the Liquidity premium, 1 March 2010
• Cooke, Houben, Varnell, Dependencies and aggregation, AENORM August 2012
• Shaw, Smith, Spivak, Measurement and Modelling of Dependencies in Economic
Capital, 10 May 2012
• Taleb, Fooled by Randomness – the hidden role of chance in life and in the markets,
2001
• Taleb, The Black Swan – the impact of highly improbable, 2007
• Vose, Fitting distributions to data – and why you are probably doing it wrong, 15
February 2010
• Email: servaashouben@gmail.com• Blog: http://actuaryabroad.wordpress.com
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About me
Servaas Houben heads the risk scenario
generation team at Prudential, London. He
studied econometrics in the Netherlands and
worked in life insurance for the first four
years of his career. Following this, he worked
in Dublin and London. Besides actuarial,
Servaas completed the CFA and FRM
qualifications, and regularly writes on his
blog, for CFA digest and Dutch actuarial
magazines.
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Appendix - Correlated random numbers• Start with correlation matrix C• Find lower lower triangle matrix L such that
LTL = C• C needs to be positive semi definite (positive
eigenvalues)
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1 0.7 0.70.7 1 0.70.7 0.7 1
C =
1 0 00.7 0.714 00.7 0.294 0.651
L =
Infoline 13 December 2012 Zurich
Appendix - From uncorrelated to correlated random seed
• Z = uncorrelated random number stream• X = correlated random number stream• X = L * Z
• =퐿 0 0퐿 퐿 0퐿 퐿 퐿
푍푍푍
• X=푍
퐿 ∗ 푍 + 퐿 ∗ 푍퐿 ∗ 푍 + 퐿 ∗ 푍 + 퐿 ∗ 푍
=푋푋푋
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Infoline 13 December 2012 Zurich