This document summarizes a presentation on stochastic scenario creation for Solvency II capital requirements. It discusses risk identification, data selection and limitations, calibration of risk distributions, modeling dependencies between risks, and validation of scenarios. The presentation covers aggregating risks while accounting for dependencies, producing correlated risk drivers, and using correlated random numbers to generate scenarios from calibrated distributions with restrictions on certain risks.

1. Stochastic Scenario Creation
Infoline Zurich
13 December 2012
Servaas Houben
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2. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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3. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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4. Capital requirement under SII
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5. VaR limitations - subaddivity
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Risk 1
Probability Loss
0.03 1 mln
0.97 0
95% VaR 0
Risk 2
Probability 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

6. Quiz
Data:
• 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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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

7. 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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8. “Risk mitigation” through dividends
8
-
1.000
2.000
3.000
4.000
5.000
6.000
Valuetotalreturn
Date
S&P500 Total return
total return index
without top 10

9. 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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10. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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11. 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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12. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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13. 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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14. Nominal yield curves
Risk Management  Hedging uses assets quoted on OIS
Pricing (Guarantees)  Funding for hedging based on OIS
Provisioning  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 31
Term (Years)
EUR / 24 August 2012 / Spot / Annual
Market LIBOR
Market OIS
Solvency II Risk Free
Source: Bloomberg
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15. Liquidity and Matching premium
adjustments
15
0
50
100
150
200
250
300
350
400
450
500
LQPinbasispoints
Date
LQP development over time
USD
GBP
EUR
(100)
(50)
-
50
100
150
200
250
300
350
400
MPinbp
Date
MP development
UK
US
EUR
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Source: Itraxx

16. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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17. 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
Indexvalue
Date
S&P 500
capital return
index

18. 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
Indexvalue
Date
S&P 500 return

19. 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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20. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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21. Case study - diversification
• Measurement of strength and direction of
relationship 2 risk drivers:
21
-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%
TWDequity
UK property
Property Monthly Total Return UK vs TWD equity return, Feb 1991 - Feb 2011
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22. Produce correlated risk drivers
22
Uncorrelated risk drivers Correlated risk drivers
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23. Scenario production
• Apply correlated random numbers to calibrated
distributions
• Apply restrictions to certain risk drivers
▫ Interest rates
▫ Credit spreads
▫ Volatilities
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24. Agenda
Solvency II Capital requirement
Risk identification
Data selection and limitations
Calibration
Aggregation and dependencies
Validation
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25. 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
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26. 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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27. 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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28. 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 0.7
0.7 1 0.7
0.7 0.7 1
C =
1 0 0
0.7 0.714 0
0.7 0.294 0.651
L =
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29. Appendix - From uncorrelated to
correlated random seed
• Z = uncorrelated random number stream
• X = correlated random number stream
• X = L * Z
• =
0 0
0
• X= ∗ + ∗
∗ + ∗ + ∗
=
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