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Internal Model Concepts at SCOR


Internal Model Concepts at SCOR Presented by Ulrich M ller, SCOR SE Tel Aviv, November 23, 2010 Initial remarks The emerging European supervisory framework Solvency ... – PowerPoint PPT presentation

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Title: Internal Model Concepts at SCOR

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 PC and Life business,
    dependencies, retrocession, asset and credit risk

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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
6 Conclusions
The Internal Model as a stochastic simulation
  • 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

Risk factors affecting the Risk-Adjusted Capital
( Risk-Based Capital Required
What kind of risks are covered by the
Risk-Adjusted Capital (RAC)?
Underwriting Risk
Reserving Risk
(Liability Risk) Life and PC, e.g. Natural
Life and PC, e.g. Reserve Strengthening
Market Risks
e.g. Financial Crisis
Operational Risks
e.g. Reputational, Fraud, System Failures,
Misconceived Processes
Credit Risks
e.g. Default of Retrocessionaires
Correlation (more general dependence) has a
primary importance in determining the RAC.
Internal models evolution
Value Protection
Value Sustainment
Value Creation
Collection of sub models quantifying parts of the
Quantification of different risk types
Risk types are combined to arrive at the
companys total risk
Modelling of underlying risk drivers
Financial Instruments
Portfolio Data
Risk Model 1
Management Strategy
Financial Instruments
Valuation Model 3
Valuation Model 1
Insurance Risk
Market Risk
Credit Risk
Operational Risk
Portfolio Data
Risk Model 2
Valuation Model 2
Distributional and Dependency Assumptions
Market Risk
Credit Risk
Insurance Risk
Total Risk
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
  • Both use the same underlying mathematical
  • 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
  • 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.

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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
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
  • Risk implies uncertainty. The economic profit (
    change in economic value) is not certain we
    model its distribution as a basis for RAC

Balance Sheet accounting and economic view
Accounting view
Economic view
Invested Assets
Discounted Reserves
Market Value of Invested Assets
Other liabilities
Hybrid debt
Economic Capital
Reinsurance assets
Discounted Reinsurance assets
Other liabilities
Other assets
Shareholders equity
Other assets
  • Main adjustments to the accounting view balance
  • Discounting reserves and Reinsurance assets
  • Considering loss value of Unearned Premium
  • Hybrid debt can be considered as capital
  • Intangibles has economic value of zero

Profit distribution as a centerpiece of risk
  • 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
  • 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!

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 in mEUR
Expected Profit
Measuring risk and capital adequacy
  • Different stakeholders have different views on
    the risk measure
  • Different perceptions on capital adequacy SCORs
    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

Economic value and profit variations in
  • 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 PC 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 PC business, underwriting
    risk, all legal entities consolidated
  • Third aggregation
  • Total new PC 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
The following risk measures at level a, ?a, are
commonly used
  • Value-at-Risk
  • Expected Shortfall ( tVaR)

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 ?EV,
    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 ?EV     case of the 1
    shortfall of the EV of the stand-alone segment

    tVaRdiversified - E ?EV     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

    xtVaRstand-alone E ?EV -
    xtVaRdiversified E ?EV
    - tVaRdiversified
    This is our standard definition
    of risk-based capital
  • We do not use VaR (but for Solvency II, we are
    adding this).

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)
- Risk Adjusted Capital (RAC) allocated to Xi
- Percentage of RAC allocated to Xi
The Economic Scenario Generator (ESG) of SCOR
  • Consistent scenarios for the future of the
    economy, needed for
  • Modeling assets and liabilities affected by the
  • 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
  • 6 currency zones (EUR, USD, GBP, CHF, JPY, AUD
    flexible) and FX rates
  • Correlations, dependencies between all economic
  • Heavy tails of distributions
  • Realistic behavior of autoregressive volatility
  • 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)
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
  • 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 methoddata sample,
innovations, simulation
Historic data vectors
Innovation vectors
Last known vector
Future simulated data vectors
economic variables
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
  • GARCH model for the volatility si 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 / si .
  • At each simulation step, the resampled innovation
    xi / si is rescaled by the current, updated GARCH
    volatility sj ? new innovation xi sj / si

Heavy tails in the ESG
  • Market shocks and extreme price moves matter in
    economic risk assessment. Look at the tails of
  • 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
  • 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 ? 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)

Economic Scenario Generator Application
IglooTM Import
Non-Bloomberg Time Series
ALM Information Backbone
Preprocessed data
Economic Raw Data
Enhanced Time Series
Economic Scenarios
IglooTM Interface
Analysis, inter and extrapolation statistical
ESG Simulation
Scenario Post-processing
ESG Simulated yield curves,
example simulation 2007Q3 ? end of 2008
Backtesting the ESG distributions of USD Equity
index during the crisis case of an extreme loss

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 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
    1980s, hyperinflation of the 1920s 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
  • Swiss national bank in the 1980s 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
Simulation of invested assets
  • All invested assets are modeled based on the ESG
  • Example bond portfolios are valuated based on
    interest rate scenarios, with roll-overs
  • Asset allocation as important input to the asset
  • 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, 379-95
  • This is possible by assuming a non-Gaussian
    credit migration rate for the default
  • 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.
    SP) and has been calibrated with Moodys KMV
    default probabilities

The credit risk model (PL) model predicts the
credit spread derived from the default
probability (EDF)
Simulation study simulated defaults in line with
the PL model and Moodys KMV default probability
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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
6 Conclusions
Modeling of Life liabilities
  • There are differences between PC 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 PC
  • 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
  • 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

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)

Risk factors and lines of business (LoB) in the
life model
Risk factors
  • 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
  • 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 list of LoB corresponds to the list of LoB
used in the Embedded Value process
The risk factors affect the one-year change in
our view of the business, including projected
future long-term cash flows
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
  • By legal entity
  • By currency
  • Thus the modeling units have a 4-dimensional

Dependencies between Life risks excess
mortalities in two different regions, due to
pandemic risk
  • 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
    (?) 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 (?) 400
    simulations here
  • Theta between sub-regions (WestAsia and
    EastAsia) ? 7
    theta between main regions ? 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

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
  • 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 PC business and
    invested assets).

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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
6 Conclusions
Overview PC liability modeling
  • Property and Casualty (PC) 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
  • Many lines of business (LoB), grouped in
  • 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 PC Scenarios
  • Legal Entities e.g. SCOR_PC, SCOR Switzerland
  • Items Premiums, Losses, Expenses
  • Perspective Gross, Retro
  • Maturity New Business, Reserves, Prior-Year
  • Lines of Business e.g. Property, Motor,
    Aviation, Credit Surety
  • Reinsurance Type Treaty Business, Facultative
  • 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 PC 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
  • Upward correction of reserves ? loss, balance
    sheet hit
  • Reserve risk risk of correction of loss
  • 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
  • From historical claim data triangles, we derive a
    model for reserve risks (both for ultimate and
    one-year risk)

Development Years
Known today
Next period risk lt ultimate risk
Risk for end of next calendar year
Risk for ultimate
Underwriting Years
We use currently this in the Internal Model
This is what the Swiss Solvency Test requires
(plus market value margin)
Plan for next UWY
Triangle analysis of cumulative insurance claims
  • Development 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 Macks 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 PC business
  • New business is subject to technical pricing at
  • 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 PC 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 companys 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
  • Big catastrophes can produce claims in various
    lines of business.
  • Dependency between risks reduces the benefits of
  • 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
  • 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 gt 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
? 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)
  • Large events are not frequent, but their large
    claims constitute the bulk of the risk factor

? Empty zone small (attritional) losses ignored.
(Some slightly larger claims also ignored, when
only affecting France, not Germany).
Empirical evidence from French and German
  • 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

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
  • 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
  • The opposite can also happen large localized
    losses and small losses with a wide range of
    impact. But these types of events are less

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
  • The width of the impact range is correlated with
  • Simulation result Tail dependency in the upper
    tail, similar to the windstorm example ?
    asymmetric empirical copula found, similar to
    Clayton copula

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
  • 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
? 0.1
? 0.5
? 1.0
? 2.0
Asymmetric Copula
The Clayton Copula CDF is defined by
With a Generator of the Copula
The Clayton copula is Archimedean
Rank Correlation ( Gauss Copula)
m1 m2
m1 1 0.6
m2 0.6 1
m1 m2
m1 1 0.9
m2 0.9 1
m1 m2
m1 1 0
m2 0 1
m1 m2
m1 1 0.3
m2 0.3 1
Symmetric Copula
The multivariate Normal distribution copula has a
matrix as a parameter. The PDF of a Normal
copula is
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.
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
  • Dependencies between events in neighbouring
    regions (e.g. windstorms in France and Germany).
  • Dependencies between related LoB (e.g. Fire and
  • 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
Non-Life liability baskets of the model
hierarchical dependence structure
Granularity of PC Risk Model ? different risks
to be aggregated
New Biz
  • 3 maturities
  • New business
  • Unearned bus.
  • Reserves
  • Granularity
  • Lines of business
  • Legal entities
  • Nature

New Biz
LE 2
LE 1
LE 2
LOB 1.2
Legal Entity 1
LOB 1.1
Loss Model
Stochastic Reserves
Paid / incurred patterns
LE 1
Loss Model
Comparing the number of dependency parameters
correlation matrix vs. copula tree
  • Task Modeling all PC 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
  • Wherever we know a causal dependency, we model it
  • 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
Combining three sources of information
  • SCOR developed a new method to calibrate PC
    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.
The importance of the PC dependency calibration
Some figures on the PC calibration process
Dependence within 19 PC 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 PC model calibration directly
aims at dependencies between concrete parts of
the SCOR PC business portfolio. Unlike the Life
model, the PC model does not separate risk
factor models from exposure models.
Dependence Measure
Dependence Measure What are we asking the
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?
PrObEx, combined view
Final distribution via the three sources of
Prior Information
Expert judgements
PrObEx combines the three sources to provide SCOR
with the finest estimate for dependence parameters
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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
6 Conclusions
Integrating all models in the approach
Liabilities Lines of business (LoB)
Assets Investments
Economic Indicator
Economy Equity indices GDP Yield curves Forex
PC risk model and its interaction with other
parts of the Internal Model
PC Plan
New Biz
PC Risk Model
Projected Gross Model
Net Model
Losses, premiums, cost
  • PC Risk Model
  • Full model for gross PC
  • Projection to the plan
  • Retrocession
  • Diversification
  • Full diversification benefit calculated in
    capital model
  • Allocated capital is passed back to PC
  • ? Consistency with other business processes is

Capital Model
Life Model
Asset Model
Economic Scenarios
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
  • 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

Example of a result of the main aggregated model
Strategic Asset Allocation based on Efficient
  • The investment strategy is based on
  • Risk/return considerations for the entire
    shareholders equity (including liability risk)
  • and risk aversion as defined by top management
    (slope of tangent)

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 PC liabilities underwriting, reserving, dependencies, retrocession
5 Integrated company model aggregation, additional dependencies
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,
    PC 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
  • 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.