Best Estimate Valuation in an. Février Insurance Stress-Test Workshop. 11 April 2017, Paris, France

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1 Best Estimate Valuation in an Construire Economical un générateur Framework: de Key scénarios Points, économiques Best Practices en and assurance Pitfalls Version Version Insurance Stress-Test Workshop Février April 2017, Paris, France Frédéric PLANCHET Quentin GUIBERT 1

2 Context Since the works of Black-Scholes-Merton, hedging and pricing techniques are largely used for financial products. The concept of risk neutral measure is based on the idea that derivatives products should be hedged. This concept is adapted for a deep and liquid market of tradable instruments. For the last 20 years, insurance industry have massively used economic scenarios generators for different valuation purposes: regulation (Solvency II), accounting (IFRS) and financial reporting (MCEV). For that, insurers use methods, originally developed for pricing financial products, to valuate their liabilities based on a risk neutral measure. 2

3 Context Many theoretical and practical issues have been raised, particularly for insurance business, as noted for e.g. by Vedani et al. [2017]: life long duration of life insurance liabilities, no liquid market for insurance portfolios, partially endogenous risk factors, volatility of the economic value which does not reflect the risks carried, Not possible to construct an hedging portfolios for insurance liabilities, No strong arguments to justify that data selected for model calibration is accurate. Solvency II imposes a specific valuation framework, based on a principle of market consistency for financial risks. 3

4 Aims The approach for valuing liabilities in insurance depends on many financial and technical assumptions. For these assumptions, Solvency II merely recommends to comply with relatively high-level principles (see e.g. EIOPA guidelines). Despite these constraints, the best estimate of liabilities can vary widely, from one set of acceptable assumptions to another. As a result, each local supervisor authorities have to develop their own guidelines and standards in order to check if the homemade valuation model of an insurer is valid or not. On the other hand, insurers have developed together a set of best practices to fill the gaps in the regulatory rules or in the scientific literature for justifying that an assumption is acceptable. These practices can be defined on both national and group levels. 4

5 Aims Two main problems appear as a consequence of that: how to check that assumptions defined by an insurer are credible, whereas they can be produced by an hidden / black box model, an expert judgement or emerged from a set of best practices? how to ensure the same treatment for different insurance across country? In this presentation, we aim to underline these issues by focusing explicitly on the example of the Economic Scenario Generators (ESG). We focus on the practical issues that are raised by French insurers. 5

6 Valuation principles Under Solvency II framework, the economic balance sheet is basically established using at the valuation date: the market value of assets, the market value of liabilities, i.e. the value of technical provisions as the sum of a best estimate and a risk margin. For basic assets (equities, bonds, etc.), trading prices are generally provided by the market, if it is sufficiently deep, liquid and transparent (IFRS 13). Trading prices of comparable instruments or prices obtained with a model can be used (IFRS 7), where there are none. Such models, though, are necessary to recalculated the derivatives 'prices depending on their underlying risk factors (equity, interest rate, credit, liquidity). 6

7 Valuation principles Replicable and non-replicable liabilities are distinguished. The best estimate of the liabilities is calcuted as the sum of probabilityweighted average of future cash-flows, taking account of the time value of money (see art. 77 directive Solvency II). P Q BEt = Et δucfu u> t This quantity is generally valued by using Monte-Carlo simulations techniques, and a annual or/and monthly discretization grid BE 1 N t δu, ncfu, n N n= 1 u> t 7

8 Valuation principles Cash-flows projections should basically consider, with respect to the boundary of the contracts: future premiums, recoverables, future benefits as defined by the contracts terms (deaths, surrenders, annuities, ), Future expenses (administrative costs, management fees, ), Futures tax payments. Best estimate calculation should include: financial options and guarantees of contracts, the policyholders behaviour, the future manangement actions, a suitable modeling for the underlying risks including dependence between them. 8

9 Valuation principles Technical assumptions should be realistic, without any degree of prudence : Mortality tables, Lapse models, These assumptions should be defined with sufficient granulary, with respect to the concept of homogeneous risk groups. Data quality criteria (completeness, appropriateness and accuracy) should be satistied and regularly reviewed. 9

10 Valuation principles For valuing financial options and guarantees included in life insurance product, insurers generally used an ESG. The calculation process is as below for a French saving product (Laurent et al., 2016). Economic Scenarios Generator (risk neutral) Iteration of a discrete date Calculation of mathematical reserves before profit bonuses and calculation of S1 financial and technical reserves Application of ALM rules Application of profit sharing rules, depending on technical and financial results Iteration of a simulation given by the ESG Revaluation of liabilities 10

11 Valuation principles Some constraints appear from the regulatory framework related to the ESG: Risk-free interest rate term structure is given by EIOPA on a monthly basis. The ESG should be market consistent, i.e. it should satisfy some tests guarantying that the results provided by the ESG is consistent with financial market data (see art. 76 directive). The calibration process uses data from financial markets that are deep, liquid and transparent. The insurer should justify that data selected is relevant given the characteristics of the insurance obligations. The random numbers generator is valid and not manipulated. 11

12 Best practices for a risk-neutral ESG The use of the ESG in insurance is a rather technical subject. As the cost for developing and maintaining such models is high, 4 main cases can basically be distinguished: large insurers, which used an outsourced ESG, but have a rather good understanding of it, large insurers, which have developed their own ESG, little insurers, which used an outsourced ESG, but have difficulty to have a clear view of the underlying models, little insurers, which have developed a basic ESG, with small resources. This situation tends to legitimize models developed by external providers to address the needs of large insurers. 12

13 Best practices for a risk-neutral ESG In general, an ESG include models for the following financial risks: Interest rates risk, Real interest rates risk or inflation risk, Equity risk, Property risk, Spread risk for corporate and sovereign bonds, Currency risk. Spread risk is rarely modeled, which is largely questionable. Currency risk is rarely modeled, but concerns more specifically large groups. The dependence between these risks is generally modeled using a Gaussian copula. 13

14 Best practices for a risk-neutral ESG Some examples of model for interest rates risks: Short rate models: Hull and White model G2++ (2 factors Hull and White model) CIR++ and CIR2++ Market models: Libor Market model (LMM) with deterministic volatilities LMM+ Several degree of complexity to take into account: the shape of the interest rate curves, negative interest rates, volatility surfaces and smiles the prices of out-the-money options (caps, swaptions). 14

15 Best practices for a risk-neutral ESG Calibration approaches for interest rate models: Tests: Date : the valuation date or on a average on a specific period, Data: the interest rate curves given by EIOPA with some adjustments (CRA, VA, extrapolation, ) ATM swaption or caps prices or with implied volatilities surfaces. Implied volatilities calculation: Black formula with lognormal distribution for the swap rates, Bachelier formula with normal distribution for the swap rates. Market consistency test, Martingale tests. 15

16 Main issues for a risk neutral ESG Theoretical limitations: Financial risks covered by insurers can not be hedged No financial market for insurance products Risk neutral measure is non unique and specific to each financial instrument Insurance industry tends to reproduce financial practises, but with different aims (giving an economic values vs. hedging and pricing financial instruments on a daily basis). Absence of large financial market with derivatives products for property risk or a mix of financial risk. Some parameters should be estimated based on historical data. 16

17 Main issues for a risk neutral ESG Practical limitations: The use of financial data available at the end of the year, The number of required simulations, The choice of financial instruments to calibrate the model is tricky to justify, The EIOPA curves are different the spot curves used by traders, Models developed for managements actions and policyholders behavior are estimated with historical data some inconsistencies may appears using risk neutral scenarios. 17

18 From Pilar One to ORSA modeling Computing the economic balance sheet at time t=0 fulfills the requirements of Pillar 1 of Solvency II. To meet the requirements of Pillar 2, an insurer should also be able to project this balance sheet in the future. An ALM model in life insurance should be able: to valuate assets and liabilities, to compute quantiles of the distribution of the net asset value, which is a distribution of economic values. The first item uses a risk neutral measure, and the second one uses an historical measure. 18

19 From Pilar One to ORSA modeling As part of a comprehensive modeling aimed at providing distributions of economic values, a two-step approach should be developed by using: 1. a functional g providing the economic value based on state variables Y, know at the calculation date, 2. a dynamic model for risk factors Y t π = ( ) g Y 0 0 We can then determine the economic value at any time by using π = t ( ) g Y t The functional g is complex and is based on the a no arbitrage assumption, which leads to construct a risk neutral measure. The dynamic of Y is a problem of econometrics. 19

20 From Pilar One to ORSA modeling For example, with the classical Vasicek model, the following dynamics are used for an unique risk factor (e.g. the short rate): Quantiles ( ) dy = dr = a b r dt + σ dw t t t t Pricing Q ( ) σ dr = a b r dt + dw t λ t t and the pricing function is: b λ λσ = b a W = W + λ t Q t t 2 with r = b σ λ 2 2a We observe here that the link between the two representations is made via the parameter λ. Note: the parameter σ is theoretically invariant. ( ) σ ( ) ( ) a( T t) a T t 1 e g ( rt ) = P( rt, T t) = exp r rt T t r e 3 a 4a ( 1 )

21 From Pilar One to ORSA modelling The previous approaches by projecting the flow of benefits under the contract and obtaining numerical results relies heavily on simulation techniques. Using these approaches within the framework of internal models is particularly difficult (cf. BAUER et al. [2010]). Cumbersome calculations make these models difficult to use, configure and maintain. In particular, these approaches are poorly suited for ORSA projections, due to the large computation time needed (but optimization is possible, see NTEUKAM et al. [2014]) and the lack of robustness (which is mainly due to over parameterization). 21

22 From Pilar One to ORSA modelling BONNIN et al. [2014] propose an alternative model, which is well suited for the ORSA purpose. This model consists in computing the economic value of complex life insurance contracts by applying a coefficient on their mathematical reserves. This approach is based on the idea that the gap between the best estimate value and the mathematical reserves is between +/- 5% empirically. This gap represents the time value of financial options of the contracts. For the Solvency Capital Requirement (SCR) calculation and projection, these authors adapt the model described in GUIBERT et al. [2012] for non-life insurance contract. In this framework, the SCR is easily computable using basic simulation techniques. 22

23 From Pilar One to ORSA modelling A similar approach can be used for pensions, see BONNIN et al. [2015]. This first analytical framework can then be expanded to capture more complex effects, such as the wealth effect of the insurer through its management of unrealized losses (cf. COMBES et al. [2016]). 23

24 Conclusion Pillar one techniques need to be carefully set-up to be efficient. Such approaches are not suited to project the balance sheet. Having a closed formula to go from the mathematical reserve to the best estimate improves significantly the model s performances. Being easily reproducible, it facilitates the process of audit and control. Such models can be built analyzing the main risks of the contracts, e.g., by observing that a (French) saving contract is mainly non-hedgeable, because of the accounting rules effect on the revalorization rate of the contract. This approach also provides a powerful tool for making projections of SCR along a «critical path». This is especially interesting when for time dependent stress scenario analysis (cf. GUIBERT et al. [2014]). 24

25 References BAUER D., BERGMANN D., REUSS A. [2010] «Solvency II and Nested Simulations a Least-Squares Monte Carlo Approach», Proceedings of the 2010 ICA congress. BONNIN F., COMBES F., PLANCHET F., TAMMAR M. [2015] «Un modèle de projection pour des contrats de retraite dans le cadre de l ORSA», Bulletin Français d Actuariat, vol. 14, n 28.. BONNIN F., JUILLARD M., PLANCHET F. [2014] «Best Estimate Calculations of Savings Contracts by Closed Formulas - Application to the ORSA», European Actuarial Journal, Vol. 4, Issue 1, Page COMBES F., PLANCHET F. TAMMAR M. [2016] «Pilotage de la participation aux bénéfices et calcul de l option de revalorisation», Bulletin Français d Actuariat, vol. 16, n 31. IFERGAN E. [2013] Mise en œuvre d un calcul de best estimate, Mémoire d actuaire, Dauphine. GUIBERT Q., JUILLARD M., NTEUKAM T. O., PLANCHET F. [2014] Solvabilité Prospective en Assurance - Méthodes quantitatives pour l'orsa, Paris : Economica. GUIBERT Q., JUILLARD M., PLANCHET F. [2012] «Measuring uncertainty of solvency coverage ratio in ORSA for Non-Life Insurance», European Actuarial Journal, 2: , doi: /s GUIBERT Q., JUILLARD M., PLANCHET F. [2010] «Un cadre de référence pour un modèle interne partiel en assurance de personnes», Bulletin Français d Actuariat, vol. 10, n 20. KAMEGA A. [2009], PLANCHET F., THÉROND P.E., Scénarios économiques en assurance - Modélisation et simulation, Paris : Economica. LAIDI Y., PLANCHET F. [2015] «Calibrating LMN Model to Compute Best Estimates in Life Insurance», Bulletin Français d Actuariat, vol. 15, n 29. LAURENT J.P., NORBERG R., PLANCHET F. (editors) [2016] Modelling in life insurance a management perspective, EAA Series, Springer. 25

26 References NTEUKAM T. O., PLANCHET F., REN J. [2014] «Internal Model in Life insurance: Application of Least Square Monte-Carlo in Risk Assessment», Les cahiers de recherche de l ISFA, n NTEUKAM O., PLANCHET F. [2012] «Stochastic Evaluation of Life Insurance Contract: Model Point on Asset Trajectories & Measurement of the Error Related to Aggregation», Insurance: Mathematics and Economics. Vol. 51, pp PLANCHET F., THÉROND P.E., JUILLARD M. [2011] Modèles financiers en assurance, seconde édition, Paris : Economica. VEDANI J., EL KAROUI N., LOISEL S., PRINGENT J.-L. [2017] «Market inconsistencies of market-consistent European life insurance economic valuations: pitfalls and practical solutions», European Actuarial Journal, pp1-28. Packages & R codes related to the ESG - ESG ( - ESGtoolkit ( - ycinterextra (

27 Contact Quentin GUIBERT Frédéric PLANCHET PRIM ACT 42 avenue de la Grande Armée Paris ISFA 50 avenue Tony Garnier F Lyon