Internal Model Concepts at SCOR

1. Internal Model Concepts at SCOR Tel Aviv, November 23, 2010 Presented by Ulrich Müller, SCOR SE

2. 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

3. Agenda

4. 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) 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

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

7. 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

8. 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, …

9. 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.

10. Agenda

11. 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-AdjustedCapital (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.

12. Balance Sheet – accounting and economic view Accounting view Economic view Invested Assets Reserves Market Value of Invested Assets Discounted Reserves Other liabilities Hybrid debt Economic Capital Reinsurance assets 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

13. 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)

14. 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!

15. 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 0 20 Profit in mEUR Expected Profit

16. 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.

17. 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.

18. 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

19. 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.:

20. 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-alonextVaRdiversified = 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).

21. 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

22. 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 • Heavytails 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.

23. 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!

24. ESG based on bootstrapping • Our implementation: Economic Scenario Generator (ESG) based on bootstrapping. This is a semi-parametric method. Reviewed by FINMA • Bootstrappinghistoricalbehaviors 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

25. The bootstrapping method:data sample, innovations, simulation economic variables economic variables USD equity EUR FX rate GBP 5 year IR time time time scenarios Historic data vectors Innovation vectors Last known vector Future simulated data vectors economic variables

26. 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 σiof 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 innovationxiσj/ σi

27. 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.

28. Stochastic correction factor to obtain heavy-tailed innovation • Stochastic correction factor η to be applied to allbootstrapped innovations • Root of mean square (RMS) = 1  corrected innovations have unchanged variance • Heaviness of tail and other parameters are configurable (see paper)

29. Economic Scenario Generator Application: Functionality Reporting IglooTM Import Non-Bloomberg Time Series ALMInformationBackbone Preprocessed data EconomicRawData EnhancedTime Series Economic Scenarios IglooTM Interface Bloomberg Analysis, inter andextrapolationstatisticaltests ESG Simulation Scenario Post-processing FED

30. ESG: Simulated yield curves, example: simulation 2007Q3  end of 2008

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

32. 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 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

33. Using economic scenarios as a basis of the asset and liability models Economy Equity indices Economic Indicator (EI) Liabilities FX GDP LoB1 ... Yield curves LoB2 LoB3 Assets Cash flow LoB4 LoB5 LoB6 LoB7 Investments LoB8 LoB9 LoB10 Accounting LoB11

34. 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

35. 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

36. The credit risk model (“PL”) model predicts the credit spread derived from the default probability (EDF)

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

38. Agenda

39. 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 morbidityare a better basis for modelinglife 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

40. 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)

41. Risk factors and lines of business (LoB) in the life model Risk factors LoB • Life (EU, America, Asia, …) • Annuity • Health • Disability • Long Term Care (LTC) • Critical Illness (CI) • Personal Accident • Financing with deficit accounting • Financing without deficit accounting • Investment Treaties • Guaranteed Minimum Death Benefit • … more … • 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 Valueprocess The risk factors affect the one-year change in our view of the business, including projected future long-term cash flows

42. 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

43. 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?

44. 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

45. 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).

46. Agenda

47. 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

48. 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….

49. 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

50. 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) Plan for next UWY 2008