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 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

Risk factors affecting the Risk-Adjusted Capital ( ≈ Risk-Based Capital ≈ Required Capital)
What kind of risks are covered by the Risk-Adjusted Capital (RAC)? Underwriting Risk Reserving Risk (Liability Risk) Life and P&C, e.g. Natural Catastrophes Life and P&C, e.g. Reserve Strengthening Market Risks RAC e.g. Financial Crisis Operational Risks 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. 5

Internal models: evolution
Value Protection Value Sustainment Value Creation 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 Scenarios Risk Factors Financial Instruments Valuation Engine Portfolio Data IGR Management Strategy Distributional and Dependency Assumptions Balance Sheet Profit and Loss Financial Instruments Portfolio Data IGR Risk Model 1 Management Strategy Valuation Model 3 Financial Instruments Valuation Model 1 Portfolio Data Market Risk Credit Risk Insurance Risk Operational Risk Risk Model 2 Valuation Model 2 Market Risk Credit Risk Insurance Risk Distributional and Dependency Assumptions 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 (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.

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
Accounting view Economic view Invested Assets Reserves Market Value of Invested Assets Discounted Reserves Other liabilities Hybrid debt Reinsurance assets Economic Capital Discounted Reinsurance assets Other liabilities Other assets Other assets Shareholders equity Intangibles 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!

Probability distribution of year-end profits
How does a typical economic profit distribution of a modeling unit look like? Probability distribution of year-end profits Often asymmetric for insurance risks, with a heavy tail on the loss side (negative profit) -80 -60 -40 -20 20 Profit in mEUR Expected Profit

Measuring risk and capital adequacy
Different stakeholders have different views on the risk measure Different perceptions on capital adequacy: SCOR’s Group internal model, Swiss Solvency Test, Solvency II The Group Internal model interprets required capital as deviation of the economic tVaR(1%) result from the economic expected profit (= xtVaR(1%)). Consequently, available capital includes the economic expected profit The Swiss Solvency Test defines required capital as tVaR(1%) Result of the one-year change + market value margin Solvency II is based on xVaR(0.5%) The internal model should make it possible to satisfy all the requirements but should not depend on them. Different results are consistently derived from the same, common core model.

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 The following risk measures at level α, ξα, 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 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).

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 alternative formula for the Haircut principle would take dependence into account but would not realize a full allocation (that is: sum of RAC_VaR(X_i|Z) is not 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)
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 Innovation vectors Last known vector Future simulated data vectors economic variables economic variables USD equity economic variables EUR FX rate GBP 5 year IR time scenarios

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

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)

Economic Scenario Generator Application: Functionality
Reporting IglooTM Import Non-Bloomberg Time Series ALM Information Backbone Preprocessed data Economic Raw Data Enhanced Time Series Economic Scenarios IglooTM Interface Bloomberg Analysis, inter and extrapolation statistical tests ESG Simulation Scenario Post-processing FED

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 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 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

Using economic scenarios as a basis of the asset and liability models
Economic Indicator (EI) Investments GDP FX Equity indices Yield curves ... LoB1 LoB2 LoB3 Cash flow Accounting Liabilities Assets Economy LoB4 LoB5 LoB6 LoB7 LoB8 LoB9 LoB10 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)

Simulation study: simulated defaults in line with the PL model and Moody’s KMV default probability data

Modeling of Life liabilities
There are differences between P&C and Life business, such as … Life is often long-term business: cash flow projections over decades Old life business continues to generate premium, so the underwriting year and the difference between new and old business is not as relevant as for P&C Risk factors such as mortality or morbidity are a better basis for modeling life risks than the lines of business For economic life business risks, market-consistent valuation has become important: Some life business behaves like a replicating asset portfolio, typically including financial derivatives However, life reinsurers have a lot of biometric risks: mortality trends, mortality shock (pandemic), lapse risk, …. More important than economic risks! Embedded Value is a dominant valuation concept for life business. Our capital model largely relies on (side) results of the official Embedded Value computations at SCOR

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
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 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
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

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).

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) Development Years 1 2 3 4 2005 Known today Next period risk < ultimate risk 2006 Risk for end of next calendar year Risk for ultimate Underwriting Years 2007 We use currently this in the Internal Model This is what the Swiss Solvency Test requires (plus market value margin) 2008 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: 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
← 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 Windstorm ↑ Empty zone: small (attritional) losses ignored. (Some slightly larger claims also ignored, when only affecting France, not Germany).

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

Dependence modeling: Conventional correlation vs
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 Asymmetric Copula The Clayton Copula CDF is defined by:
θ = 0.1 θ = 0.5 θ = 1.0 θ = 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 1 m1 m2 1 0.3 m1 m2 1 0.6 m1 m2 1 0.9 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 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
Non-Life liability baskets of the model: hierarchical dependence structure Dependence between lines of business Dependencies between single risks within line of business X22 X31 X33 X21 X23 X34 X32

Granularity of P&C Risk Model  different risks to be aggregated
Company LOB 2 LOB 1 Reserves Unearned New Biz LOB 1 P LOB 1 NP 3 maturities: New business Unearned bus. Reserves Granularity: Lines of business Legal entities Nature Reserves Unearned New Biz Legal Entity 1 LE 2 LE 1 LE 2 LOB 1.1 LOB 1.2 Loss Model Premium Cost Stochastic Reserves Paid / incurred patterns LE 1 LE2 Loss Model Premium Cost

Comparing the number of dependency parameters: correlation matrix vs
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 = 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.

The importance of the P&C dependency calibration project
Some figures on the P&C calibration process Dependence within 19 P&C Lines of Business are calibrated via PrObEx The meetings take place between April and September, 2010 A final meeting will assess dependence between Lines of Business Around 120 experts, in 12 different locations, are taking part in the calibration process Results will have an important impact for SCOR The P&C model calibration directly aims at dependencies between concrete parts of the SCOR P&C business portfolio. Unlike the Life model, the P&C model does not separate risk factor models from exposure models. 70

How to measure dependence?
Dependence Measure Dependence Measure – What are we asking the experts? X+Y How to measure dependence? X Y 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?” 71

PrObEx, combined view Prior Information Observation Expert judgements
Final distribution via the three sources of information Prior Information Observation Expert judgements PrObEx combines the three sources to provide SCOR with the finest estimate for dependence parameters 72

Integrating all models in the approach
Cash flow Accounting Assets Investments Liabilities Lines of business (LoB) Cash & Short term investments Fixed Income Equities Real Estate Alternative Investments LoB1 Economic Indicator LoB2 LoB4 LoB4 LoB4 LoB4 LoB9 Economy Equity indices GDP Yield curves Forex

P&C risk model and its interaction with other parts of the Internal Model
P&C Plan Retrocession New Biz P&C Risk Model Projected Gross Model Net Model Unearned Reserves Losses, premiums, cost Allocated capital P&C Risk Model Full model for gross P&C Projection to the plan Retrocession Diversification Full diversification benefit calculated in capital model Allocated capital is passed back to P&C  Consistency with other business processes is ensured Life Model Capital 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 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…

Example of a result of the main aggregated model: Strategic Asset Allocation based on Efficient Frontier … 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)

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.

Internal Model Concepts at SCOR

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Internal Model Concepts at SCOR

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