Download - Internal Model Concepts at SCOR

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


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


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


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.


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


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


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, …


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.


Agenda






6 Conclusions


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.


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


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)


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!


-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?


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.


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.


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


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.:


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).


)(),(

1

ZVaRXVaR

XVaRZXVaR d

ii

ii

Allocation of diversified Risk-Based Capital (RAC) to Partial Risks Xi

d

i

d

iiiii XVaRXXEZXES

1 1

|),(

Euler principle (our preferred choice)

)(),(),( iiiES XEZXESZXRAC

Haircut principle

)(),(),( iiiVaR XEZXVaRZXRAC

)(

),()|(

ZRAC

ZXRACZXRAC

ES

iESiES

d

ii

iiVaR

XVaR

XVaRZXRAC

1

)|(

- Contribution of Xi to Z (whole portfolio)

- Risk Adjusted Capital (RAC) allocated to Xi

- Percentage of RAC allocated to Xi

d

ii

iiVaR

XVaR

XVaRZXRAC

1

)|(


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.


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!


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


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


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


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.


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


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


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


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




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





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






Backtesting the ESG distributions of USD Equity index during the crisis; case of an extreme loss


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


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


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


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


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


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


Agenda






6 Conclusions


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


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%


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


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


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%


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%


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?


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


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


Agenda






6 Conclusions


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


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….


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


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


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, …).


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)


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).

↑ ↓


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


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.


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).


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


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.


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.


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


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


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


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


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)


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


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


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.


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


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.


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


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


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


Agenda






6 Conclusions


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


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


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)


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…


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)


Agenda






6 Conclusions


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


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


Thank you …

… for your attention.

Your comments and questions are welcome.

Download - Internal Model Concepts at SCOR

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