internal model concepts at scor tel aviv, november 23, 2010 presented by ulrich müller, scor se
TRANSCRIPT
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Presented by Ulrich Müller, SCOR SEPresented by Ulrich Müller, SCOR SE
2Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Initial remarks
The emerging European supervisory framework Solvency II not only has a Standard Model (successor of QIS5) but offers the possibility of employing an Internal Model.
Motivation: an Internal Model assesses the risks of large insurers and reinsurers more accurately than the Standard Model.
The internal modeling methods presented here reflect the requirements of the reinsurer SCOR. They are based on the work of the FinMod team and other departments at SCOR
SCOR developed its Internal Model for internal use, before Solvency II, in the sense of Own Risk and Solvency Assessment (ORSA).
Now the enhanced model is in the Solvency II pre-approval process
As a large reinsurer, SCOR has a more diversified business portfolio than most primary insurance companies of similar size
Therefore the scope of modeling challenges is huge: modeling of P&C and Life business, dependencies, retrocession, asset and credit risk etc
3Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
4Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
The Internal Model as a stochastic simulation engine
The Internal Model is comprehensive: All risks of the company are stochastically simulated (Monte-Carlo simulation)
Stress scenarios are fully contained in the normal stochastic simulation: the simulation scenarios with the most extreme outcomes behave like stress scenarios
Then there is no need to add some artificial extra stress scenarios
The main result is required Risk-Adjusted (or Risk-Based) Capital (RAC) for the whole company and for individual parts and risk types
Capital is required to cover extreme outcomes. These arise from extreme events (heavy tails of distributions) and dependencies between risks.
Therefore the modeling of distributions including realistic (often heavy) tails and dependencies is key
5Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
5
Risk factors affecting the Risk-Adjusted Capital ( ≈ Risk-Based Capital ≈ Required Capital)
RAC
Underwriting Risk
e.g. Default of Retrocessionaires
e.g. Financial Crisis
Operational Risks
e.g. Reputational, Fraud, System Failures,
Misconceived Processes
(Liability Risk)
Life and P&C, e.g. Natural Catastrophes
What kind of risks are covered by the Risk-Adjusted Capital (RAC)?
Reserving Risk Life and P&C, e.g. Reserve
Strengthening
Credit Risks
Market Risks
Correlation (more general: dependence) has a primary importance in determining the RAC.
6Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Internal models: evolution
Mar
ket
Risk
Cred
it Ri
sk
Insu
ranc
e Ri
sk
Ope
ratio
nal
Risk
Financial Instruments
Portfolio Data
IGR
Total Risk
Market Risk
Credit Risk
Insurance Risk
Financial Instruments
Portfolio Data
Scenarios
Risk Factors Financial Instruments
Valuation Engine
Portfolio Data
IGR
Management Strategy
Distributional and Dependency Assumptions
Balance Sheet
Profit and LossDistributional and Dependency
Assumptions
Valuation Model 1
Valuation Model 2
Risk Model 1
Risk Model 2
Valuation Model 3
Collection of sub models quantifying parts of the risks
Quantification of different risk types
Risk types are combined to arrive at
the company’s total risk
Modelling of underlying risk
drivers
Value Protection Value Sustainment Value Creation
Management Strategy
7Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Applications of the Internal Model: internal use, Swiss Solvency Test (SST), Solvency II
Internal use of the Group Internal Model:
Risk assessment, capital allocation, planning, basis for new business pricing, asset allocation, retrocession optimization etc.
Report on results to the Executive Committee and the Risk Committee of the Board of Directors
European regulators encourage the internal use under the heading “Own Risk and Solvency Assessment” (ORSA)
Swiss Solvency Test (SST):
SCOR Switzerland (a legal entity of the SCOR Group) produces SST reports based on the Internal Model since 3 years.
The Swiss regulator (FINMA) has reviewed the Internal Model, with a focus on some parts of special interest
Solvency II: The Internal Model (with some adaptations to Solvency II guidelines) is in the pre-approval process
8Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Methodology: Solvency II and Swiss Solvency Test (SST)
Both use the same underlying mathematical methodology:
Solvency Capital Requirement should buffer risks emanating during a 1-year time horizon
Risk is defined on the basis of the change in economic value (available capital) over a 1-year time horizon
A risk margin is assessed to cover the cost of the capital necessary to buffer non-hedgeable risks during the entire run-off of the liabilities.
There are differences between Solvency II and SST: Treatment of group solvency, standard model vs standard formula, VaR at 0.5% vs tVaR at 1% as a risk measure, treatment of operational risk, …
9Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Dependency modeling in the Internal Model and the Solvency II Standard Model (or QIS 5)
Comparing two approaches:
QIS 5 / possible Solvency II Standard Model: Loss distributions with thin tails (normal or log-normal) low capital requirement per single risk or line of business flat, uniform correlation of risk factors also in the tail. This is compensated by of high, prescribed correlation coefficients between risks low diversification benefit.
Internal Model of SCOR: Loss distributions with heavy tails wherever appropriate in realistic modeling; increased correlation of risk factors in the tails (case of stress, extreme behavior) higher capital requirement. But: The correlation of average events / risks factors is often quite moderate larger diversification effect between risks for a well-diversified company.
Main problem: QIS 5 tends to underestimating risks of single risk factors, single lines of business and “monoliners” and to overestimating risks of strongly diversified companies
Approval process: pre-approval of the Internal Model and its dependence model by national regulator(s). Essential for a globally well-diversified reinsurer such as SCOR and for any insurance business based on strong diversification between different risks.
10Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
11Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Measuring risk: Risk-Based Capital and economic profit distribution
A (re)insurance company is assessing the risk of existing or new business for several purposes: regulatory solvency tests, rating agency models, capital allocation in planning and pricing, …
The risk of a certain business is usually measured in terms of the capital required to carry it: Risk-Adjusted Capital (RAC) = Risk-Based Capital ≈ Required Capital
The RAC has to be compared to the available capital of a company in order to assess its solvency. Both capital measures rely on the economic valuation of business
Here we focus on risk-adjusted capital and its computation
Risk implies uncertainty. The economic profit (= change in economic value) is not certain; we model its distribution as a basis for RAC calculations.
12Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Balance Sheet – accounting and economic view
Reserves
Hybrid debt
Shareholdersequity
Invested Assets
Accounting view
Reinsurance assets
Other assetsIntangibles
Other liabilities
Discounted Reserves
Economic Capital
Market Value of Invested
Assets
Economic view
Discounted Reinsurance
assets
Other assets
Other liabilities
Main adjustments to the accounting view balance sheet:
• Discounting reserves and Reinsurance assets
• Considering loss value of Unearned Premium Reserves
• Hybrid debt can be considered as capital
• Intangibles has economic value of zero
13Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Profit distribution as a centerpiece of risk modeling
There are different definitions of risk and risk-based capital (Internal Model, Solvency II, Swiss Solvency Test, rating agency models, models for capital allocation in pricing and planning, …)
Some (traditional) models are simple factor models: short-cuts that directly aim at results using fixed parameters and formulas.
For large multi-line companies, factor models are of little use as they are too coarse and underestimate diversification
For state-of-the-art models, we need full profit distributions of all parts of the business
Profit distributions can be used for the stochastic simulation of the future behavior (Monte-Carlo simulation)
A set of simulated scenarios can serve as a substitute of profit distributions (e.g. in Property Cat modeling)
14Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Economic profit distributions and model granularity
Economic profit distribution = distribution of the future change in economic value. This profit is uncertain, stochastic
Time horizon: usually one year. What will be the value of the business at the end of this period?
We take economic values as best estimates at the end of the stochastically simulated period. This implies discounting of all projected cash flows, for all simulated scenarios
We want to know profit distributions not only for the whole company but also for its many parts high granularity
Granularity: different legal entities, segments and lines of business, types of risks, ….
The lowest level of granularity is a modeling unit. We model profit distributions by modeling unit. A large model has hundreds of units!
15Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
-20-40-60-80 0 20
Probability distribution of year-end profits
Often asymmetric for insurance risks, with a heavy tail on the loss side (negative profit)
Expected ProfitProfit in mEUR
How does a typical economic profit distribution of a modeling unit look like?
16Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Measuring risk and capital adequacyDifferent stakeholders have different views on the risk measure
Different perceptions on capital adequacy: SCOR’s Group internal model, Swiss Solvency Test, Solvency II
The Group Internal model interprets required capital as deviation of the economic tVaR(1%) result from the economic expected profit (= xtVaR(1%)). Consequently, available capital includes the economic expected profit
The Swiss Solvency Test defines required capital as tVaR(1%) Result of the one-year change + market value margin
Solvency II is based on xVaR(0.5%)
The internal model should make it possible to satisfy all the requirements but should not depend on them. Different results are consistently derived from the same, common core model.
17Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Economic value and profit: variations in definition
Different stakeholders and users need different definitions of economic value and profit. Model developers have to be ready to support different definitions in their stochastic simulations
Ultimate view vs one-year (or year-by-year) view:
Ultimate view: Economic value of all future cash flows until the business is totally over
Year-by-year view: Given the known starting condition at the end of a future year, the economic value at the end of the following year (relevant for computing the Market Value Margin in solvency tests)
One-year view: Economic value at the end of the first future year (relevant for required capital in solvency tests)
Value before tax or after tax (also: before or after dividend payment)
Using different interest rates for discounting future cash flows. We prefer using the risk-free yield curve at valuation time.
18Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Aggregating profit distributions
We model economic profit distributions for small pieces of business, but we often need results for larger segments – and the whole company
Many aggregate views are of interest. Example: Aggregating from the modeling unit “New Business Motor proportional, underwriting risk, Legal Entity A”. First aggregation:
– Total new business Motor, underwriting risk, Legal Entity A; or
– Total new proportional P&C business, underwriting risk, Legal Entity A; or
– Total risk new business Motor, Legal Entity A (including interest rate risk) Second aggregation:
– Total new business Motor, Legal Entity A; or
– Total new proportional P&C business, underwriting risk, all legal entities consolidated Third aggregation:
– Total new P&C business; or
– Total Legal Entity A Last aggregation:
– Total consolidated company, all risks
Different user want to see different aggregate results, based on aggregated profit distributions
For aggregating profit distributions, we need dependency models
19Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Risk measures
1)( |inf xXPRxXVaR
XVaRXXEXES |
• Value-at-Risk
• Expected Shortfall (= tVaR)
The following risk measures at level α, ξα, are commonly used:
Recall that, unlike ES, VaR is generally not coherent due to lack of subadditivity. i.e.:
20Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Risk-based capital: tVaR and xtVaR
For any stochastic economic value change ΔEV, ultimate or not, the required capital per liability (or asset) segment can be measured in terms of the Tail Value at Risk (tVaR): tVaRstand-alone = - E[ ΔEV | case of the 1% shortfall of the EV of the stand-alone segment ] tVaRdiversified = - E[ ΔEV | case of the 1% shortfall of the EV of the whole entity ] Euler principle
While tVaR is “Swiss-Solvency-Test-compatible”, our method of choice in the Group Internal Model is xtVaR, its difference from the unconditional expectation: xtVaRstand-alone = E[ ΔEV] - tVaRstand-alone
xtVaRdiversified = E[ ΔEV] - tVaRdiversified This is our standard definition of risk-based capital
We do not use VaR (but for Solvency II, we are adding this).
21Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
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Allocation of diversified Risk-Based Capital (RAC) to Partial Risks Xi
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Euler principle (our preferred choice)
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Haircut principle
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ZRAC
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- Contribution of Xi to Z (whole portfolio)
- Risk Adjusted Capital (RAC) allocated to Xi
- Percentage of RAC allocated to Xi
d
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iiVaR
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XVaRZXRAC
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22Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
The Economic Scenario Generator (ESG) of SCOR
Consistent scenarios for the future of the economy, needed for:
Modeling assets and liabilities affected by the economy
Expected returns, risks, full distributions
Business decisions (incl. asset allocation, hedging of risks) Many economic variables: yield curves, asset classes, inflation, GDP … Credit cycle level, supporting the credit risk model 6 currency zones (EUR, USD, GBP, CHF, JPY, AUD; flexible) and FX rates Correlations, dependencies between all economic variables Heavy tails of distributions Realistic behavior of autoregressive volatility clusters Realistic, arbitrage-free yield-curve behavior Short-term and long-term scenarios (month/quarter … 40 years)
Typical application: Monte-Carlo simulation of risks driven by the economy.
23Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Quarterly changes in EUR interest rates (maturities 3 months, 1 year, 5 years, 30 years)
Quarterly changes in EUR riskfree zero-coupon interest rates
-3.0
-2.0
-1.0
0.0
1.0
2.0
Time
Qu
art
erl
y in
tere
st
rate
ch
an
ge
in
%
3m
1y
5y
30y
zero change
Old rule of thumb: Interest rates move by 1% per quarter, at maximum. This rule was broken in autumn 2008 (financial crisis) by a large amount!
24Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
ESG based on bootstrapping
Our implementation: Economic Scenario Generator (ESG) based on bootstrapping. This is a semi-parametric method. Reviewed by FINMA
Bootstrapping historical behaviors for simulating the future
Bootstrapping is a method that automatically fulfills many requirements, e.g. realistic dependencies between variables
Some variables need additional modeling (“filtered bootstrap”):
Tail correction for modeling heavy tails (beyond the quantiles of historical data)
GARCH models for autoregressive clustering of volatility
Yield curve preprocessing (using forward interest rates) in order to obtain arbitrage-free, realistic behavior
Weak mean reversion of some variables (interest rates, inflation, …) in order to obtain realistic long-term behavior
25Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
The bootstrapping method:data sample, innovations, simulation
Historic data
vectors
eco
no
mic
va
riab
les
Future simulated data vectors
eco
no
mic
va
riab
les
Innovation
vectors
Last known
vector
scenariostime timetime
eco
no
mic
va
riab
les
US
D
eq
uity
EU
R F
X ra
teG
BP
5 ye
ar IR
26Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Volatility modeling in the ESG: GARCH The volatility of most variables in finance exhibits autoregressive
clusters: long periods of low volatility / long periods of high volatility.
The bootstrapping method (random sampling) disrupts those clusters.
Solution: GARCH model to re-introduce volatility clusters:
• GARCH model for the volatility σi of the time series of innovations xi , for each variable, where
• Iterative GARCH(1,1) equation:
• Robust calibration of the GARCH parameters on historical samples:
The bootstrapping method uses normalized innovations: xi / σi .
At each simulation step, the resampled innovation xi / σi is rescaled by the
current, updated GARCH volatility σj new innovation xi σj / σi
2 2 20 1 1i i ix
,,0
),0( 2ii Nx
27Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Heavy tails in the ESG
Market shocks and extreme price moves matter in economic risk assessment. Look at the tails of distributions!
Bootstrapping covers some shocks: those contained in historical data.
The size of historical samples (for many variables) is limited.
Extreme shocks (such as a “1 in 200 years” event) are probably missing in the recorded history.
Solution in the ESG: use “tail-corrected” innovations.
Corrected innovation = Historical innovation * , where is a positive random variable with a mean square of 1 and a Pareto-shaped upper tail (with a realistic tail index).
Due to this tail correction, some occasional simulation scenarios will behave like “stress scenarios”: larger shocks than in the samples.
28Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Stochastic correction factor to obtain heavy-tailed innovation
Stochastic correction factor η to be applied to all bootstrapped innovations
Root of mean square (RMS) = 1 corrected innovations have unchanged variance
Heaviness of tail and other parameters are configurable (see paper)
Density of the stochastic tail correction factor eta (with RMS value 1)
0
2
4
6
8
10
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Tail correction factor eta
Pro
ba
bili
ty d
istr
ibu
tio
n, d
en
sit
y
Density of eta,with heavyupper tail
29Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Economic Scenario Generator Application: Functionality
FED
Preprocessed data
Non-Bloomberg Time Series
Economic Raw Data
Enhanced Time Series
Economic Scenarios
IglooTM Interface
IglooTM Import
ALM Information Backbone
Analysis, inter and extrapolation
statistical tests
ESG
Simulation
Scenario
Post-processing
Reporting
Bloomberg
30Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
ESG: Simulated yield curves, example: simulation 2007Q3 end of 2008
EUR yield curve (zero coupon, risk-free), Sep 2007 and examples of simulated curves for Dec 2008
2%
3%
4%
5%
6%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Maturity [years]
Inte
res
t ra
te le
ve
l
curve of 2007Q3
simulation example 1
EUR yield curve (zero coupon, risk-free), Sep 2007 and examples of simulated curves for Dec 2008
2%
3%
4%
5%
6%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Maturity [years]
Inte
res
t ra
te le
ve
l
curve of 2007Q3
simulation example 1
simulation example 2
EUR yield curve (zero coupon, risk-free), Sep 2007 and examples of simulated curves for Dec 2008
2%
3%
4%
5%
6%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Maturity [years]
Inte
res
t ra
te le
ve
l
curve of 2007Q3
simulation example 1
simulation example 2
simulation example 3
EUR yield curve (zero coupon, risk-free), Sep 2007 and examples of simulated curves for Dec 2008
2%
3%
4%
5%
6%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Maturity [years]
Inte
res
t ra
te le
ve
l
curve of 2007Q3
simulation example 1
simulation example 2
simulation example 3
simulation example 4
31Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Backtesting the ESG distributions of USD Equity index during the crisis; case of an extreme loss
32Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Extreme scenarios are an integral
part of our ESG
SCOR ESG withstands extreme scenarios
Extreme rates of 0% or below Extreme rates of around 40%
The national banking institutions have raised the amount of money in circulation on levels not seen for decades
Expected inflation can only be fought by high interest rates
Historic examples show that extreme rates can become reality: Mexico, Argentine, Turkey or other EMEA-countries, 26% US Fed rate in the 1980’s, hyperinflation of the 1920’s in Germany
The ESG calculates scenarios with interest rates of 0% or slightly below (not below -1%)
Historic data shows examples of such occasions
Yen – rates fell slightly below Zero in the early 1990’s
Swiss national bank in the 1980’s used negative interest rates as a tool to make investments in Swiss Francs unattractive to fight the strength of the currency
33Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Using economic scenarios as a basis of the asset and liability models
Economic
Indicator (EI)
Investments
GDPFX
Equity indices
Yield curves... LoB1
LoB2
LoB3
Cash flow
Accounting
Liabilities
Assets
Economy
LoB4LoB5
LoB6LoB7
LoB8LoB9LoB10
LoB11
34Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Simulation of invested assets
All invested assets are modeled based on the ESG scenarios
Example: bond portfolios are valuated based on interest rate scenarios, with roll-overs
Asset allocation as important input to the asset model
Cash flows from liabilities are invested as well Credit risk of corporate bonds is applied Resulting asset positions after 1 year are simulated
taking into consideration ESG returns, asset allocation, cash flows from liabilities and credit risk
35Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Credit risk model based on credit spreads of corporate bonds
We are able to explain most of the credit spread seen in the market by the probability of default given by structural credit risk models. Denzler et al.: From default probabilities to credit spreads: credit risk models do explain market prices. Finance Research Letters, 3:79-95
This is possible by assuming a non-Gaussian credit migration rate for the default probability.
Simulation results show that a Pareto-like log-gamma type of distribution for the migration rate describes the process reasonably well.
The model is powerful enough to explain credit spreads from general parameters obtained from the market. Thus the model can be used to compute the price of credit risk for a corporate bond from a default probability – and the other way around.
The model reproduces default statistics (e.g. S&P) and has been calibrated with Moody’s KMV default probabilities
36Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
The credit risk model (“PL”) model predicts the credit spread derived from the default probability (EDF)
0
50
100
150
200
250
300
Nov 95 Nov 96 Nov 97 Nov 98 Nov 99 Nov 00 Nov 01 Nov 02 Nov 03 Nov 04
credit spread BM model, G = -11.58 PL model, G = 0.97
Credit Spread / EDF Implied Spread (in bp), Global Index, Maturity 5 years
37Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Simulation study: simulated defaults in line with the PL model and Moody’s KMV default probability data
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20Time to Maturity in yrs
risk-neutral def. prob. PL model log-gamma sim. model (discr. = 0.25 yrs)
Annualized Default Probability (in %), Global Index
38Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
39Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Modeling of Life liabilities
There are differences between P&C and Life business, such as … Life is often long-term business: cash flow projections over decades Old life business continues to generate premium, so the underwriting
year and the difference between new and old business is not as relevant as for P&C
Risk factors such as mortality or morbidity are a better basis for modeling life risks than the lines of business
For economic life business risks, market-consistent valuation has become important: Some life business behaves like a replicating asset portfolio, typically including financial derivativesHowever, life reinsurers have a lot of biometric risks: mortality trends, mortality shock (pandemic), lapse risk, …. More important than economic risks!Embedded Value is a dominant valuation concept for life business. Our capital model largely relies on (side) results of the official Embedded Value computations at SCOR
40Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Life business with a saving component: cash flow projections over 70 years are relevant
Examples of ESG simulations over time
Equity investments supporting a guaranteed saving performance are profitable over a long time – but there are long drawdowns (loss periods)
ESG simulation examples over 70 years (notice the loss periods)
100
1'000
10'000
100'000
1 41 81 121 161 201 241 281
Number of quarters starting on 30 June 2009 (grid: decades)
No
min
al v
alu
e in
EU
R
(lo
ga
rith
mic
sc
ale
)
Bond, simulation example 1Equity, simulation example 1Bond, simulation example 2
Equity, simulation example 2Constant growth, 2.25%
41Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Risk factors and lines of business (LoB) in the life model
Life (EU, America, Asia, …) Annuity Health Disability Long Term Care (LTC) Critical Illness (CI) Personal Accident Financing with deficit accounting Financing without deficit accounting Investment Treaties Guaranteed Minimum Death Benefit … more …
Risk factors
Random fluctuations (mixed factors) Mortality trend (EU, America, Asia, …) Longevity trend Disability trend Long term care (LTC) trend Critical illness (CI) trend Lapse Local catastrophy Pandemic (Europe, America, Asia, …) Financial risks (inflation, deflation, …) … more …
LoB
The risk factors affect the one-year change in our view of the business, including projected future long-term cash flows
The list of LoB corresponds to the list of LoB used in the Embedded Value process
42Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Profit distributions of life business based on risk factors
Simulation of changes of Present Values of Future Profit (PVFP), similar to Embedded Value
By risk factor. Some risk factors have dependencies on other risk factors
Pandemic as a main risk factor has a truncated Pareto model for excess mortality
By line of business (LoB). Each LoB has an exposure function against each risk factor (matrix)
By legal entity By currency Thus the modeling units have a 4-dimensional
granularity
43Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Dependencies between Life risks: excess mortalities in two different regions, due to pandemic risk
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6%
Excess mortality Europe
Exc
ess
mo
rtal
ity
Am
eric
a
for theta = 0
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6%Excess mortality Europe
Exc
ess
mo
rtal
ity
Am
eric
a
for theta = 1
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6%
Excess mortality Europe
Exc
ess
mo
rtal
ity
Am
eric
a
for theta = 3
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6%
Excess mortality Europe
Exc
ess
mo
rtal
ity
Am
eric
a
for theta = 8
Two regions: America, Europe
The same pandemic model for both regions: Pareto with lower and upper cut-off, 3 pandemics expected per 200 years.
The cumulative probabilities (CDFs) follow an upper-tail Clayton copula with parameter theta (θ); 2500 simulations
Exploring the following theta values: 0 (independent), 1, 3, 8
Scattergrams for resulting excess mortalities in America and Europe (not for the CDFs here)
What is the right degree of dependency, in your opinion? Which theta?
44Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Example: Hierarchical dependency of regions and sub-regions, due to the same risk type
Hierarchical tree of regions and sub-regions. Sub-regions within the same main region have stronger dependency for a certain risk factor (e.g. pandemic)
Modeling all regions cumulative probability distributions (CDFs) for all of them
At each node of the tree, there is an upper-tail Clayton copula with parameter theta (θ); 400 simulations here
Theta between sub-regions (WestAsia and EastAsia): θ = 7; theta between main regions: θ = 2
It is numerically possible to apply hierarchical dependency between risk factors without any exposure information
Resulting scattergrams for the CDFs show the desired dependency behavior
EastAsia
Asia
WestAsiaEuropeAmerica
World
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
CDF WestAsia
CD
F E
as
tAs
ia
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
CDF WestAsia
CD
F A
me
ric
a
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
CDF Europe
CD
F A
me
ric
a
45Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Example: Complete dependency tree for all risk factors of Life insurance
Hierarchical tree of all risk factors (a simple, schematic proposal)
Different copula types (including independence) are possible at each node of the tree
The risk factors “Mortality Trend” and “Longevity” refer to changes in long-term trend expectations within one simulation year (e.g. change in underlying mortality tables)
The preferred copula for “Mortality Trend” and “Longevity” is the Gauss copula (= rank correlation) because these factors are correlated throughout the distribution, not only in the tails
The preferred copula for “Pandemic” (= “Mortality Shock”) is the Clayton copula. Severe pandemics are more likely to spread over the whole world than small ones (tail dependence)
Economic risks covered by Economic Scenario Generator (ESG, also affecting P&C business and invested assets).
Other Biometric Risks
"Independent Copula" (All Biometric risks) Economic Risks (ESG)
Undetermined Copula (All Risk Factors of Life Insurance)
Longevity Longevity Europe RestOfWorld
Mortality Trend Mortality TrendEurope RestOfWorldAsia
Longevity Asia
Mortality TrendAmerica
Longevity America
PandemicEurope RestOfWorld
Mortality Trend
Gauss Copula with 8*8 correlation matrix (General Mortality Trend)Clayton Copula (Pandemic World)
Pandemic PandemicAsia America
Pandemic
46Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
47Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Overview: P&C liability modeling
Property and Casualty (P&C) reinsurance is the dominant business of SCOR. We distinguish between the following business maturities: Reserve business (insured period over, just development risk) Unearned prior-year business (still under direct insurance risk) New business to be written in the simulation year
We distinguish between further categories (high granularity): Many lines of business (LoB), grouped in categories Proportional / non-proportional treaty and facultative reinsurance
business Business in different legal entities
We model the effect of retrocession gross and net profit distributions
Hierarchical dependency tree between the many modeling units
48Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Granularity of P&C Scenarios
Legal Entities: e.g. SCOR_PC, SCOR Switzerland… Items: Premiums, Losses, Expenses Perspective: Gross, Retro Maturity: New Business, Reserves, Prior-Year Business Lines of Business: e.g. Property, Motor, Aviation, Credit & Surety… Reinsurance Type: Treaty Business, Facultative Business Cover: Proportional, Non-Proportional Programme: Retro programme names… Currencies of Programmes: e.g. EUR, USD, GBP Patterns
The input granularity is important to support output reporting flexibility!...but with this, increasing performance issues have to be carefully considered….
49Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Modeling P&C reserve risk based on the historical development of insurance losses
Loss reserves of a (re)insurance company:
Amount of reserves = Expected size all of claims to be paid in the future, given all the existing “earned” (≈ old) contracts
Reserves are best estimates.
Estimates may need correction based on new claim information
Upward correction of reserves loss, balance sheet hit
Reserve risk = risk of correction of loss reserves
Reserve risk is a dominant risk type, often exceeding the risks due to new business (e.g. future catastrophes) and invested asset risk
Reserve risks can be assessed quantitatively.
For assessing reserve risks, we use historical claim data
50Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Reserve triangles: ultimate risk vs yearly fluctuations
From historical claim data triangles, we derive a model for reserve risks (both for ultimate and one-year risk)
Known today
Plan fornext UWY
Risk fo
r end of n
ext ca
lendar year
Ris
k fo
r u
ltim
ate
This is what the Swiss Solvency Test requires (plus market value margin)
Next period risk <
ultimate risk
We use currently this in the Internal Model
Development Years
Und
erw
ritin
g Y
ears
1 2 3 4
2005
2006
2007
2008
51Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Triangle analysis of cumulative insurance claimsDevelopment year (years since the underwriting of contracts)
Under-writing year (when contracts were written)
↓
This triangle is the basis of further analysis. Here: cumulative reported claims. There are other types (claims paid, …).
52Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Measuring the stochastic behavior of historical claim reserves: Mack’s method
Chain-ladder method: computing the average development of claims over the years
Result: Typical year-to-year development factors for claims ( patterns)
Method by Mack (1993): computing local deviations from these average development factors
Variance of those local deviations estimate of reserve risk
Very sensitive to local data errors overestimation of risk
Correctness of data is very important, data cleaning needed
We developed a robust variation of the Mack method (published in the Astin Bulletin)
53Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Development of cumulative reported claims for one underwriting year of one line of business
False booking in development year 11, corrected in subsequent year 12.
All claim reports are cumulative (since underwriting of contracts).
↑ ↓
54Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Modeling the profit distributions of new and unearned prior-year P&C business
New business is subject to technical pricing at SCOR
SCOR has a sophisticated pricing tool profit distributions per
treaty
The tool NORMA aggregates treaties with proper dependency
assumptions between treaties profit distributions per modeling unit
Our risk-based capital calculation uses the resulting gross profit
distributions, for new and unearned business
NORMA models dependencies between the modeling units of P&C
business
NORMA also models retrocession treaties (for new, unearned and
reserve business stochastically simulated scenarios for
retrocession recoveries and net losses per modeling unit
55Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Dependency between risks is key
Risk Diversification reduces a company’s need for risk-based capital. This is key to both insurance and investments.
However, risks are rarely completely independent: Stock market crashes are usually not limited to one market. The
financial crisis again shows that local markets depend on each other. Certain lines of business are affected by economic cycles, such as
liability, credit & surety or life insurance. Motor insurance is correlated to motor liability insurance and both will
vary during economic cycles. Big catastrophes can produce claims in various lines of business.
Dependency between risks reduces the benefits of diversification.
The influence of dependency on the aggregated risk-based capital is thus crucial and needs to be carefully analyzed.
56Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Extreme events and dependencies
Extreme events are major risk drivers for insurers. Examples: Natural catastrophes (Non-Life insurance) Pandemic (Life insurance)
Dependencies between different risks are also major risk drivers. Risk diversification between different lines of business is limited by
dependencies.
Large or extreme events are often the cause of dependencies. A large windstorm may affect different countries whose risk exposures are
independent in case of smaller events. September 11, 2001, caused large losses in different lines of business (Life,
Property, Aviation, Business Interruption) that are usually less dependent.
The coincidence of extreme events and increase dependence is called tail dependence. Tail dependence > “everyday dependence”.
Large events should be explicitly modeled as common causes, if possible. If not possible, we need a dependence model (e.g. copula-based).
57Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Empirical evidence for tail dependence: rank scatter plot of French and German windstorm claims
Data: European windstorm event loss set French and German exposure of a reinsurer
Claims in France and Germany (plotting the ranks of the claims for each windstorm)
Small events are frequent, but their aggregate claims are comparably low. We separate them out ( “attritional model”)
Large events are not frequent, but their large claims constitute the bulk of the risk factor Windstorm
↑ Empty zone: small (attritional) losses ignored. (Some slightly larger claims also ignored, when only affecting France, not Germany).
← Condensed zone: Extreme claims in both countries are strongly correlated: Tail Dependence
58Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Empirical evidence from French and German windstorms
We observe a concentration of correlation in the upper tail: large windstorms in France are often large windstorms in Germany as well.
If we assume a uniform correlation everywhere,
we underestimate the (value at) risk due to large, common events in both countries;
and/or we overestimate the correlation of average-sized events.
In the example of windstorms, we do not have to model the dependency explicitly as long as we have event sets.
For other perils and lines of business, we have no event set We need an explicit dependency model with upper tail dependence.
Our choice: copulas rather than uniform linear correlation.
59Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Why is correlation in the upper tail often higher?The basic reason for increased correlation in the upper tail of loss distributions: Large events often have a wide range of impact and high severity at the same time
Examples for large events with wide impact and high severity:
A large European windstorm causes simultaneous, large losses in different countries (e.g. Lothar)
September 11, 2001, had simultaneous, large losses in several lines of business: Life, Property, Aviation, Business Interruption, …
A change in law simultaneously affects the settlement of different Liablility and Professional Liability treaties of certain types (in markets that were initially thought to be independent)
Examples for small (but frequent) events and lower severity:
A smaller windstorm causes notable losses only within a limited area of one market
A fire in a factory causes local damage, in only one market and line of business: Property
A specific court decision leads to a moderately higher individual loss in Motor Liability, with no consequences for other treaties or lines of business
The opposite can also happen: large localized losses and small losses with a wide range of impact. But these types of events are less typical.
60Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
A very simple model leads to tail dependence
Very simple simulation study
Two zones A and B, observing claims in both zones in a rank scatter plot
Random events, random center of impact, random severity
The width of the impact range is correlated with severity
Simulation result: Tail dependency in the upper tail, similar to the windstorm example asymmetric empirical copula found, similar to Clayton copula
Rank scatter plot (empirical copula) of yearly claims
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cdf of claims in zone A
Cd
f o
f cl
aim
s in
zo
ne
B
61Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Dependence modeling: Conventional correlation vs. copulas with tail dependence
Linear correlation as well as rank correlation are models for a unified dependency behavior, regardless of the size of losses or events.
Therefore correlation-based models tend to underestimating the tail dependence ( underestimation of capital requirement!) and overestimating dependence in case of average behavior.
We need a dependency model that supports increased tail dependency. Our choice is copulas. Which copulas?
The tail dependency is related to large losses (often due to extreme events) rather than small losses Tail dependency affects only one of the two tails asymmetric copula needed
62Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Clayton Copula
The Clayton Copula CDF is defined by:
With a Generator of the Copula:
θ = 0.1 θ = 0.5 θ = 1.0 θ = 2.0
0%
5%
10%
15%
20%
25%
30%
35%
0.1 0.5 1.0 2.0
Correlation Coefficient
Div
ersi
ficat
ion
Ben
efits
The Clayton copula is Archimedean
Asymmetric Copula
63Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
where is the inverse of the CDF N(0,1) and I is the identity matrix of size n.
The rank correlation is an elliptical copula.
Rank Correlation (= Gauss Copula)
The multivariate Normal distribution copula has a matrix as a parameter. The PDF of a Normal copula is:
m1 m2
m1 1 0
m2 0 1
m1 m2
m1 1 0.3
m2 0.3 1
m1 m2
m1 1 0.6
m2 0.6 1
m1 m2
m1 1 0.9
m2 0.9 1
Symmetric Copula
64Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Many different dependencies are modeled, some with copulas
We model the marked dependencies with copulas.
Dependencies between risk factors (e.g. trends in mortality and longevity in Life modeling)
Dependencies between different treaties within the same line of business (LoB)
Dependencies between loss developments in new and old business (reserves) within the same line of business
Dependencies between events in neighbouring regions (e.g. windstorms in France and Germany).
Dependencies between related LoB (e.g. Fire and Engineering)
Dependencies between less related LoB (e.g. Fire and Professional Liability)
Dependencies between Life and Non-Life (through Cat, terrorism etc)
Dependencies between economy and insurance liabilities (through discount rate etc)
Dependencies between economy and credit risk (credit cycle modeled in the ESG)
Dependencies between invested assets and the economy (rather obvious)
65Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Reducing the number of dependency parameters in a hierarchical dependency tree
Dependencies between
single risks within line of
business
11X
12X 13X
14X15X
1Y 3Y
Z
2YDependence between
lines of business
X21
X22
X23
Non-Life liability baskets of the model: hierarchical dependence structure
X33
X32X34
X31
66Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Granularity of P&C Risk Model different risks to be aggregated
Company
LOB 1
LOB 1 P LOB 1 NP
LOB 2
Reserves New Biz
Reserves
Legal Entity 1 LE 2
New Biz
LOB 1.1 LOB 1.2
Unearned
LE 1 LE 2
Stochastic Reserves
Paid / incurred patterns
Unearned
LE 1 LE2
Loss Model Premium Cost
Loss Model Premium Cost
3 maturities: New business Unearned bus. Reserves
Granularity: Lines of business Legal entities Nature
67Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Comparing the number of dependency parameters: correlation matrix vs. copula tree
Task: Modeling all P&C liabilities of a large company in 500 modeling baskets (different risk factors, lines of business, legal entities, markets, business maturities (reserves vs. new business), business types (proportional, non-proportional, facultative).
Alternative 1: Using a correlation matrix between all the 500 modeling baskets We need 500 * 499 / 2 = 124759 correlation coefficients. This is not a parsimonious parameter set.
Alternative 2: Using a hierarchical copula tree with (typically) 350 nodes on 7 hierarchical layers, each node with one parameter (e.g. a Clayton copula theta). We need 350 parameters. This is parsimonious and manageable in comparison.
68Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Strategy for modeling dependencies
Using the knowledge of the underlying business, develop a hierarchical model for dependencies in order to reduce the parameter space and describe more accurately the main sources of dependent behavior
Wherever we know a causal dependency, we model it explicitly
Otherwise we systematically use non-symmetric copulas: Clayton copula
Wherever there is enough data, we statistically calibrate the parameters
SCOR has a launched a new project to improve the calibration of copula parameters ProbEx
In absence of data, we use stress scenarios to estimate conditional probabilities
69Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Dependencies between Property & Casualty Risks: PrObEx
Combining three sources of information
SCOR developed a new method to calibrate P&C dependence parameters
Through a Bayesian model, three sources of information are combined:
• Prior information (regulators)• Observations (data)• Expert judgements
We invite experts to a Workshop where they are asked to assess dependencies within their LoB.
70Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
70
The importance of the P&C dependency calibration project
Dependence within 19 P&C Lines of Business are calibrated via PrObEx
The meetings take place between April and September, 2010
A final meeting will assess dependence between Lines of Business
Around 120 experts, in 12 different locations, are taking part in the calibration process
Results will have an important impact for SCOR
The P&C model calibration directly aims at dependencies between concrete parts of the SCOR P&C business portfolio. Unlike the Life model, the P&C model does not separate risk factor models from exposure models.
Some figures on the P&C calibration process
71Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
71
Dependence MeasureDependence Measure – What are we asking the experts?
X+Y
X Y
How to measure dependence?
We ask the experts:
“Suppose Y exceeds the 1-in-100 year threshold. What is the probability that also X
exceeds its 1-in-100 year threshold?”
)()( 99.099.0 YVaRYXVaRXP
72Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
72
PrObEx, combined viewFinal distribution via the three sources of information
PrObEx combines the three sources to provide SCOR with the finest estimate for dependence parameters
Prior Information Observation Expert judgements
73Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
74Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Economy
Equity indices
GDP
Yield curves
Forex
Liabilities
Lines of business (LoB)
Assets
Investments
Integrating all models in the approach
Economic
Indicator
Cash flow
Accounting
LoB1
LoB2 LoB4LoB4
LoB4LoB4
LoB9
Cash & Short terminvestmentsFixed Income
Equities
Real Estate
AlternativeInvestments
75Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
P&C risk model and its interaction with other parts of the Internal Model
P&C Risk Model
Reserves
New Biz
Unearned
P&C Plan
Projected Gross Model
Retrocession
Net Model
Capital ModelLife Model
Asset Model Economic Scenarios
Losses,
premiums,
cost
Allocated
capital
P&C Risk Model Full model for gross P&C Projection to the plan RetrocessionDiversification Full diversification benefit
calculated in capital model Allocated capital is
passed back to P&C
Consistency with other business processes is ensured
76Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Which dependencies are modeled between the main modeling blocks?
The two marked dependencies are between main model blocks and have to be modeled in the main aggregate risk calculation rather than within a partial block.
Dependencies between risk factors (e.g. trends in mortality and longevity) in Life modeling
Dependencies between different treaties within the same lines of business (LoB)
Dependencies between loss developments in new and old business (reserves) within the the same line of business
Dependencies between events in neighbouring regions (e.g. windstorms in France and Germany).
Dependencies between related LoB (e.g. Fire and Engineering)
Dependencies between less related LoB (e.g. Fire and Professional Liability)
Dependencies between Life and Non-Life (through Cat, terrorism etc)
Dependencies between economy and insurance liabilities (through discount rate, claims inflation, etc)
Dependencies between economy and credit risk (credit cycle modeled in the ESG)
Dependencies between invested assets and the economy (rather obvious)
77Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Results are per Legal Entity / Consolidated Group
All results are simulated per legal entity
Internal reinsurance, legal entity relationships, taxes etc. are considered
It is essential to have modeling flexibility regarding legal entities (but of course also for other dimensions) as those structures can change…
78Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
The investment strategy is based on:
Risk/return considerations for the entire shareholder’s equity (including liability risk)
and risk aversion as defined by top management (slope of tangent)
…
Example of a result of the main aggregated model: Strategic Asset Allocation based on Efficient Frontier
Downside risk (based on expected shortfall)
Exp
ecte
d re
turn
Scenarios of equity allocations
0% equity allocation
Optimum equity allocation
Risk versus return (efficient frontier)
79Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Agenda
1 Internal Models and regulation: SST and Solvency II
2 Economic profit distribution, risk-adjusted capital, market risk, credit risk
3 Risks in life (re)insurance
4 P&C liabilities: underwriting, reserving, dependencies, retrocession
5 Integrated company model: aggregation, additional dependencies
6 Conclusions
80Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Conclusions (I)
The Internal Model …
… is used internally for capital allocation, planning etc. (ORSA)
… is a part of regulatory solvency tests (SST, Solvency II)
… captures the risks of a large, highly diversified company better than a standard model or standard formula
Modeling many partial risks: economy, market and credit risk, invested assets, Life liabilities, P&C liabilities, …
As a basis of the risk-adjusted capital calculation, we use economic profit distributions per modeling unit
A central Economic Scenario Generator (ESG) determines the stochastic simulation of all assets and liabilities as far as they depend on the economy
81Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Conclusions (II)
For aggregating profit distributions, the modeling of the dependence between partial risks and units plays a key role
The dependence between large losses (strongly negative profits) is often stronger than for average profits tail dependence
We model tail dependence with copulas, often the Clayton copula, sometimes in hierarchical dependency trees
The life model distinguishes between primary risk factors (such as pandemic) and lines of business depending on these factors through exposure functions
Our preferred choice of the overall risk-based capital is the xtVaR at 1%, where the Euler Principle is used to allocate the total amount to the different risks and segments of the company
82Internal Model Concepts at SCORTel Aviv, November 23, 2010Ulrich Müller
Thank you …
… for your attention.
Your comments and questions are welcome.