Internal Model Concepts at SCOR
Presented by Ulrich Müller, SCOR SE
Tel Aviv, November 23, 2010
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
2
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
3
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
4
Risk factors affecting the Risk-Adjusted Capital
( ≈ Risk-Based Capital ≈ Required Capital)
What kind of risks are covered by the Risk-Adjusted Capital (RAC)?
(Liability Risk)
Underwriting Risk
Reserving Risk
Life and P&C, e.g.
Natural Catastrophes
Life and P&C,
e.g. Reserve
Strengthening
Market Risks
RAC
e.g. Financial
Crisis
Operational Risks
Credit Risks
e.g. Reputational, Fraud,
System Failures,
Misconceived Processes
e.g. Default of
Retrocessionaires
Correlation (more general: dependence) has a primary importance in
determining the RAC.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
5
Internal models: evolution
Collection of sub
models quantifying
parts of the risks
Value Protection
Value Sustainment
Value Creation
Quantification of
different risk types
Risk types are
combined to arrive at
the company’s total risk
Modelling of
underlying risk
drivers
Financial Instruments
Risk Factors
Financial
Instruments
Distributional and
Dependency
Assumptions
Portfolio Data
Portfolio Data
IGR
Risk
Model 1
Operational
Risk
Risk
Model 2
Credit
Risk
Portfolio Data
Insurance
Risk
Valuation
Model 2
Valuation
Model 3
Scenarios
Financial Instruments
Market
Risk
Valuation
Model 1
Management Strategy
IGR
Management
Strategy
Valuation Engine
Balance Sheet
Market
Risk
Credit
Risk
Insurance
Risk
Distributional and Dependency
Assumptions
Profit and Loss
Total Risk
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
6
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
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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 runoff 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, …
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
8
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
9
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
10
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) = RiskBased 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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
11
Balance Sheet – accounting and economic
view
Accounting view
Invested
Assets
Reserves
Economic view
Market Value
of Invested
Assets
Discounted
Reserves
Other liabilities
Hybrid debt
Reinsurance
assets
Other liabilities
Other assets
Intangibles
Shareholders
equity
Discounted
Reinsurance
assets
Economic
Capital
Other assets
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
12
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)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
13
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!
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
14
How does a typical economic profit distribution of a
modeling unit look like?
Probability distribution of year-end profits
Often asymmetric for insurance risks, with a
heavy tail on the loss side (negative profit)
-80
-60
-40
Profit in mEUR
-20
0
20
Expected Profit
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
15
Measuring risk and capital adequacy
Different 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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
16
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
17
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
18
Risk measures
The following risk measures at level α, ξα, are commonly used:
• Value-at-Risk
VaR  X   inf x  R | P( X  x)  1   
• Expected Shortfall (= tVaR)
ES  X   EX | X  VaR  X 
Recall that, unlike ES, VaR is generally not coherent due to
lack of subadditivity. i.e.:
            
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
19
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).
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
20
Allocation of diversified Risk-Based Capital (RAC) to
Partial Risks Xi
Euler principle (our preferred choice)
Haircut principle
- Contribution of Xi to Z (whole portfolio)
d

 d

ES ( X i , Z )  E  X i |  X i  VaR   X i 
i 1
 i 1 

VaR( X i , Z ) 
VaR X i 
d
VaR X 
VaR( Z )
i
i 1
- Risk Adjusted Capital (RAC) allocated to Xi
RAC ES ( X i , Z )  ES ( X i , Z )  E ( X i )
RACVaR ( X i , Z )  VaR( X i , Z )  E ( X i )
- Percentage of RAC allocated to Xi
RAC ES ( X i , Z )
RAC ES ( X i | Z ) 
RAC ES ( Z )
RACVaR ( X i | Z ) 
VaR X i 
RACVaR ( X i | Z ) 
 d

VaR  X i 
 i 1 
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
VaR X i 
d
VaR X 
i 1
i
21
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
22
Quarterly changes in EUR interest rates (maturities
3 months, 1 year, 5 years, 30 years)
Quarterly changes in EUR riskfree zero-coupon interest rates
1.0
0.0
-1.0
3m
1y
5y
30y
zero change
-2.0
.0
9
31
.1
2
.0
8
30
.1
2
.0
7
31
.1
2
.0
6
31
.1
2
.0
5
31
.1
2
.0
4
30
.1
2
.0
3
31
.1
2
.0
2
31
.1
2
.0
1
30
.1
2
.0
0
30
.1
2
.9
9
.1
2
31
.1
2
.9
8
-3.0
31
Quarterly interest rate change in %
2.0
Time
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!
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
23
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
24
Innovation
vectors
Last known
vector
Future simulated data vectors
economic variables
economic variables
Historic data
vectors
USD equity
economic variables
The bootstrapping method:
data sample, innovations, simulation
EUR FX rate
GBP 5 year IR
time
time
time
scenarios
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
25
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
2
xi , for each variable, where xi  N (0,  i )
•
Iterative GARCH(1,1) equation:
•
Robust calibration of the GARCH parameters on historical
samples:  0 ,  , 
 i2   0   xi21   i21
 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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
26
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 Paretoshaped 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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
27
Stochastic correction factor to obtain
heavy-tailed innovation
Density of the stochastic tail correction factor eta
(with RMS value 1)
 Stochastic
correction factor
η to be applied to
all bootstrapped
innovations
Probability distribution, density
10
8
 Root of mean
square (RMS) = 1
 corrected
innovations have
unchanged
variance
6
Density of eta,
with heavy
upper tail
4
 Heaviness of tail
and other
parameters are
configurable (see
paper)
2
0
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Tail correction factor eta
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
28
Economic Scenario Generator
Application: Functionality
Reporting
IglooTM Import
Non-Bloomberg
Time Series
ALM Information Backbone
Economic
Raw Data
Preprocessed data
Enhanced
Time Series
Economic
Scenarios
IglooTM
Interface
Bloomberg
FED
Analysis, inter and
extrapolation
statistical tests
ESG
Simulation
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
Scenario
Post-processing
29
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
6%
Interest rate level
5%
4%
curve
of 2007Q3
curve
curve of
of 2007Q3
2007Q3
curve
of 2007Q3
simulation
example 1
simulation example 1
simulation example 2
1
simulation
example
2
simulation
simulation example
example 3
1
simulation
3
simulation example
example 2
4
3%
2%
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Maturity [years]
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
30
Backtesting the ESG distributions of USD Equity
index during the crisis; case of an extreme loss
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
31
SCOR ESG withstands extreme scenarios
Extreme scenarios are an integral
part of our ESG
Extreme rates of 0% or below
Extreme rates of around 40%
 The ESG calculates
scenarios with interest rates
of 0% or slightly below (not
below -1%)
 The national banking institutions
have raised the amount of
money in circulation on levels
not seen for decades
 Historic data shows
examples of such occasions
 Expected inflation can only be
fought by high interest rates
 Yen – rates fell slightly below
Zero in the early 1990’s
 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
 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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
32
Using economic scenarios as a basis of the asset
and liability models
Economy
Equity indices
FX
...
GDP
Economic
Indicator (EI)
Liabilities
LoB1
Yield curves
LoB2
LoB3
Assets
Cash flow
Investments
Accounting
LoB4
LoB5
LoB6
LoB7
LoB8
LoB9
LoB10
LoB11
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
33
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
34
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
35
The credit risk model (“PL”) model predicts the credit
spread derived from the default probability (EDF)
Credit Spread / EDF Implied Spread (in bp), Global Index, M aturity 5 years
300
250
200
150
100
50
0
Nov 95
Nov 96
Nov 97
credit spread
Nov 98
Nov 99
Nov 00
Nov 01
Nov 02
BM model, G = -11.58
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
Nov 03
Nov 04
PL model, G = 0.97
36
Simulation study: simulated defaults in line with the
PL model and Moody’s KMV default probability data
Annualized Default Probability (in % ), Global Index
2
1.8
1.6
1.4
1.2
1
0
5
risk-neutral def. prob.
10
Time to Maturity in yrs
PL model
15
20
log-gamma sim. model (discr. = 0.25 yrs)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
37
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
38
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 derivatives
However, 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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
39
Life business with a saving component: cash flow
projections over 70 years are relevant
ESG simulation examples over 70 years
(notice the loss periods)
 Examples of
ESG
simulations
over time
Nominal value in EUR
(logarithmic scale)
100'000
Bond, simulation example 1
Equity, simulation example 1
Bond, simulation example 2
Equity, simulation example 2
Constant growth, 2.25%
10'000
1'000
100
1
41
81
121
161
201
241
 Equity
investments
supporting a
guaranteed
saving
performance
are profitable
over a long
time – but
there are
long
drawdowns
(loss
281
periods)
Number of quarters starting on 30 June 2009 (grid: decades)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
40
Risk factors and lines of business (LoB) in the life
model
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 …
The risk factors affect the one-year
change in our view of the business,
including projected future long-term
cash flows
LoB
 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 …
The list of LoB corresponds to the list
of LoB used in the Embedded Value
process
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
41
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
42
Dependencies between Life risks: excess mortalities
in two different regions, due to pandemic risk
0.6%
0.6%
for theta = 0
0.5%
Excess mortality America
Excess mortality America
0.5%
0.4%
0.3%
0.2%
0.1%
0.0%
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.2%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
Excess mortality Europe
The cumulative probabilities
(CDFs) follow an upper-tail
Clayton copula with parameter
theta (θ); 2500 simulations
0.6%
for theta = 3
Excess mortality America
Excess mortality America
The same pandemic model for
both regions: Pareto with lower
and upper cut-off, 3 pandemics
expected per 200 years.
0.3%
0.0%
0.0%
0.6%
0.6%
0.4%
0.3%
0.2%
0.1%
0.0%
0.0%
0.4%
0.1%
Excess mortality Europe
0.5%
Two regions: America, Europe
for theta = 1
0.5%
Exploring the following theta
values: 0 (independent), 1, 3, 8
for theta = 8
0.4%
Scattergrams for resulting
excess mortalities in America
and Europe (not for the CDFs
here)
0.3%
0.2%
0.1%
0.1%
0.2%
0.3%
0.4%
Excess mortality Europe
0.5%
0.6%
0.0%
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
What is the right degree of
dependency, in your opinion?
Which theta?
Excess mortality Europe
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
43
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)
World
Modeling all regions  cumulative probability distributions (CDFs) for all of them
Asia
America
WestAsia
At each node of the tree, there is an upper-tail Clayton copula with parameter
theta (θ); 400 simulations here
Europe
EastAsia
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
1.0
1.0
0.8
0.8
0.8
0.6
0.4
CDF America
1.0
CDF America
CDF EastAsia
Resulting scattergrams for the CDFs show the desired dependency behavior
0.6
0.4
0.2
0.2
0.0
0.2
0.4
0.6
CDF WestAsia
0.8
1.0
0.4
0.2
0.0
0.0
0.6
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
CDF WestAsia
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
0.2
0.4
0.6
0.8
1.0
CDF Europe
44
Example: Complete dependency tree for all risk
factors of Life insurance
Undetermined Copula (All Risk Factors of Life Insurance)
"Independent Copula" (All Biometric risks)
Economic Risks (ESG)
Other Biometric Risks
Clayton Copula (Pandemic World)
Pandemic
Asia
Pandemic
America
Pandemic
Europe
Pandemic
RestOfWorld
Gauss Copula with 8*8 correlation matrix (General Mortality Trend)
Mortality Trend
Mortality Trend
Mortality Trend
Mortality Trend
Asia
America
Europe
RestOfWorld
Longevity
Longevity
Longevity
Longevity
Asia
America
Europe
RestOfWorld
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).
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
45
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
46
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
47
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….
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
48
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
49
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)
Development Years
1
2
3
4
Underwriting Years
Known today
Next period risk
<
ultimate risk
2006
2007
We use currently this in the
Internal Model
2008
Plan for
next UWY
Risk for ultimate
2005
This is what the Swiss Solvency Test
requires (plus market value margin)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
50
Triangle analysis of cumulative insurance claims
Development year (years since the underwriting of contracts) 
Underwriting
year
(when
contracts
were
written)
↓
 This triangle is the basis of further analysis. Here:
cumulative reported claims. There are other types (claims
paid, …).
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
51
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)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
52
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).
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
53
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
54
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
55
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).
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
56
Empirical evidence for tail dependence: rank scatter
plot of French and German windstorm claims
← Condensed zone: Extreme claims in
both countries are strongly correlated:
Tail Dependence
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”)
↑ Empty zone: small (attritional) losses ignored.
(Some slightly larger claims also ignored, when
only affecting France, not Germany).
Large events are not
frequent, but their large
claims constitute the bulk of
the risk factor Windstorm
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
57
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 averagesized 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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
58
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
59
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
Rank scatter plot (empirical copula) of yearly claims
Random events, random center
of impact, random severity
1.0
0.9
0.8
The width of the impact range
is correlated with severity
Cdf of claims in zone B
0.7
0.6
Simulation result: Tail
dependency in the upper tail,
similar to the windstorm example
 asymmetric empirical copula
found, similar to Clayton copula
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Cdf of claim s in zone A
0.7
0.8
0.9
1.0
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
60
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
61
Clayton Copula
θ = 0.1
θ = 0.5
θ = 1.0
Asymmetric Copula
θ = 2.0
35%
The Clayton Copula CDF is defined by:
With a Generator of the Copula:
Diversification Benefits
30%
25%
20%
15%
10%
5%
0%
0.1
The Clayton copula is Archimedean
0.5
1.0
2.0
Correlation Coefficient
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
62
Rank Correlation (= Gauss Copula)
m1
m2
m1
m1
m2
m2
m1
m2
m1
1
0
m1
1
0.3
m1
1
0.6
m1
1
0.9
m2
0
1
m2
0.3
1
m2
0.6
1
m2
0.9
1
Symmetric Copula
The multivariate Normal distribution copula has a matrix as a parameter. The PDF of
a Normal copula is:
where
of size n.
is the inverse of the CDF N(0,1) and I is the identity matrix
The rank correlation is an elliptical copula.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
63
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)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
64
Reducing the number of dependency parameters in
a hierarchical dependency tree
Non-Life liability baskets of the model: hierarchical dependence structure
Z
Dependence between
lines of business
Dependencies between
single risks within line of
business
X 15
Y2
Y1
Y3
X22
X33
X31
X 14
X11
X 12
X13
X21
X23
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
X34
X32
65
Granularity of P&C Risk Model  different risks
to be aggregated
Company
LOB 1
LOB 2
Unearned
Reserves
3 maturities:
 New business
 Unearned bus.
 Reserves
Granularity:
 Lines of business
 Legal entities
 Nature
New Biz
Reserves
Legal Entity 1
LOB 1 NP
Unearned
New Biz
LE 2
LE 1
LE 2
Loss Model
Stochastic
Reserves
LOB 1 P
Premium
LOB 1.1
Cost
Paid / incurred
patterns
LE 1
Loss Model
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
LOB 1.2
Premium
LE2
Cost
66
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
67
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
68
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
69
The importance of the P&C dependency
calibration project
Some figures on the P&C calibration process
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.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
70
Dependence Measure
Dependence Measure – What are we asking the experts?
X+Y
How to measure
dependence?
X
We ask the experts:
Y
PX  VaR0.99 ( X ) Y  VaR0.99 (Y )
“Suppose Y exceeds the 1-in-100 year threshold.
What is the probability that also X exceeds its 1in-100 year threshold?”
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
71
PrObEx, combined view
Final distribution via the three sources of information
Prior Information
Observation
Expert judgements
PrObEx combines
the three sources
to provide SCOR
with the finest
estimate for
dependence
parameters
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
72
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
73
Integrating all models in the approach
Cash flow
Liabilities
Assets
Accounting
Investments
Cash & Short term
investments
Fixed Income
Lines of business (LoB)
LoB1
Economic
Indicator
Equities
Real Estate
LoB2
LoB4
LoB4
LoB4
LoB4
LoB9
Alternative
Investments
Economy
Equity indices
GDP
Yield curves
Forex
Internal Model Concepts at SCOR
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Ulrich Müller
74
P&C risk model and its interaction with other
parts of the Internal Model
P&C Plan
Retrocession
Projected
Gross Model
Net Model
New Biz
Unearned
Reserves
P&C
Risk Model
Losses,
Allocated
premiums,
capital
P&C Risk Model
 Full model for gross P&C
 Projection to the plan
 Retrocession
Diversification
 Full diversification benefit
calculated in capital model
 Allocated capital is
passed back to P&C
cost
Life Model
Capital Model
Asset Model
Economic
Scenarios
 Consistency with other business processes is ensured
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
75
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)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
76
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…
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
77
Example of a result of the main aggregated model:
Strategic Asset Allocation based on Efficient Frontier
…
Risk versus return (efficient frontier)
The investment strategy is
based on:
Expected return
Risk/return considerations
for the entire shareholder’s
equity (including liability
risk)
Scenarios of equity allocations
0% equity allocation
Optimum equity allocation
and risk aversion as
defined by top
management (slope of
tangent)
Downside risk (based on expected shortfall)
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
78
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
79
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
80
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
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
81
Thank you …
… for your attention.
Your comments and questions are welcome.
Internal Model Concepts at SCOR
Tel Aviv, November 23, 2010
Ulrich Müller
82