Best Estimate Valuation in an Construire un générateur de … · 2019. 6. 26. · Best Estimate...

1Best Estimate in an Economical framework

Février 2012

Construire un générateur descénarios économiques enassuranceVersion 1.2

Best Estimate Valuation in anEconomical Framework: KeyPoints, Best Practices andPitfallsVersion 1.0

2017 Insurance Stress-Test Workshop

11 April 2017, Paris, France

Frédéric [email protected]

Quentin [email protected]


� Since the works of Black-Scholes-Merton, hedging and pricing techniques arelargely used for financial products.

� The concept of risk neutral measure is based on the idea that derivativesproducts should be hedged. This concept is adapted for a deep and liquidmarket of tradable instruments.

� For the last 20 years, insurance industry have massively used economicscenarios generators for different valuation purposes: regulation (Solvency II),accounting (IFRS) and financial reporting (MCEV).

� For that, insurers use methods, originally developed for pricing financialproducts, to valuate their liabilities based on a risk neutral measure.

Context


� Many theoretical and practical issues have been raised, particularly for lifeinsurance business, as noted for e.g. by Vedani et al. [2017]:

� 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 ofmarket consistency for financial risks.

Context


� The approach for valuing liabilities in insurance depends on many financial andtechnical assumptions. For these assumptions, Solvency II merely recommendsto comply with relatively high-level principles (see e.g. EIOPA guidelines).

� Despite these constraints, the best estimate of liabilities can vary widely, fromone set of acceptable assumptions to another. As a result, each local supervisorauthorities have to develop their own guidelines and standards in order to checkif the “homemade” valuation model of an insurer is valid or not.

� On the other hand, insurers have developed together a set of “best practices” tofill the gaps in the regulatory rules or in the scientific literature for justifying thatan assumption is acceptable. These practices can be defined on both nationaland group levels.

Aims


� Two main problems appear as a consequence of that:

� how to check that assumptions defined by an insurer are credible, whereasthey can be produced by an hidden / black box model, an expertjudgement 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 onthe example of the Economic Scenario Generators (ESG).

� We focus on the practical issues that are raised by French insurers.

Aims


� Under Solvency II framework, the economic balance sheet is basicallyestablished 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 bestestimate and a risk margin.

� For basic assets (equities, bonds, etc.), trading prices are generally providedby the market, if it is sufficiently deep, liquid and transparent (IFRS 13).

� Trading prices of comparable instruments or prices obtained with a model canbe used (IFRS 7), where there are none.

� Such models, though, are necessary to recalculated the derivatives 'pricesdepending on their underlying risk factors (equity, interest rate, credit,liquidity).

Valuation principles


� Replicable and non-replicable liabilities are distinguished.

� The best estimate of the liabilities is calcuted as the sum of probability-weighted average of future cash-flows, taking account of the time value ofmoney (see art. 77 directive Solvency II).

� This quantity is generally valued by using Monte-Carlo simulations techniques,and a annual or/and monthly discretization grid


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� Cash-flows projections should basically consider, with respect to the boundaryof 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.



� Technical assumptions should be realistic, without any degree of prudence :

� Mortality tables,

� Lapse models,

� …

� These assumptions should be defined with sufficient granulary, with respect tothe concept of homogeneous risk groups.

� Data quality criteria (completeness, appropriateness and accuracy) should besatistied and regularly reviewed.



� For valuing financial options and guarantees included in life insuranceproduct, 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)

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

Revaluation of liabilities

Iteration of a discrete date

Iteration of a simulation given by the ESG



� 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 theresults 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 andtransparent.

� The insurer should justify that data selected is relevant given the characteristics of theinsurance obligations.

� The random numbers generator is valid and not manipulated.



� The use of the ESG in insurance is a rather technical subject. As the cost fordeveloping and maintaining such models is high, 4 main cases can basicallybe 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 theunderlying models,

� little insurers, which have developed a basic ESG, with small resources.

� This situation tends to legitimize models developed by external providers toaddress the needs of large insurers.

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 Gaussiancopula.



� 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).



� Calibration approaches for interest rate models:

� 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.

� Tests:

� Market consistency test,

� Martingale tests.



� 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 dailybasis).

� Absence of large financial market with derivatives products for property risk or a mix offinancial risk.

� Some parameters should be estimated based on historical data.

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 withhistorical data � some inconsistencies may appears using risk neutral scenarios.

Main issues for a risk neutral ESG


� Computing the economic balance sheet at time t=0 fulfills the requirementsof Pillar 1 of Solvency II. To meet the requirements of Pillar 2, an insurershould 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 ofeconomic values.

� The first item uses a risk neutral measure, and the second one uses anhistorical measure.

From Pilar One to ORSA modeling


� As part of a comprehensive modeling aimed at providing distributions ofeconomic 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 thecalculation date,

2. a dynamic model for risk factors

� We can then determine the economic value at any time by using

� 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.

( )0 0g Yπ =

tY

( )t tg Yπ =



� For example, with the classical Vasicek model, the following dynamics are usedfor an unique risk factor (e.g. the short rate):

and the pricing function is:

with

We observe here that the link between the two representations is made via theparameter λ.

Note: the parameter σ is theoretically invariant.

( ) ( )( )

( ) ( ) ( )( )2 2

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t t tdr a b r dt dWλ σ= − +

Quantiles Pricing

( )t t t tdY dr a b r dt dWσ= = − +

Q

t tW W tλ= + ×



� The previous approaches by projecting the flow of benefits under the contractand obtaining numerical results relies heavily on simulation techniques.

� Using these approaches within the framework of internal models is particularlydifficult (cf. BAUER et al. [2010]). Cumbersome calculations make these modelsdifficult to use, configure and maintain.

� In particular, these approaches are poorly suited for ORSA projections, due tothe large computation time needed (but optimization is possible, see NTEUKAM

et al. [2014]) and the lack of robustness (which is mainly due to overparameterization).

From Pilar One to ORSA modelling


� BONNIN et al. [2014] propose an alternative model, which is well suited for theORSA purpose.

� This model consists in computing the economic value of complex life insurancecontracts by applying a coefficient on their mathematical reserves.

� This approach is based on the idea that the gap between the best estimatevalue and the mathematical reserves is between +/- 5% empirically. This gaprepresents the time value of financial options of the contracts.

� For the Solvency Capital Requirement (SCR) calculation and projection, theseauthors adapt the model described in GUIBERT et al. [2012] for non-lifeinsurance contract. In this framework, the SCR is easily computable using basicsimulation techniques.



� A similar approach can be used for pensions, see BONNIN et al. [2015].

� This first analytical framework can then be expanded to capture more complexeffects, such as the wealth effect of the insurer through its management ofunrealized losses (cf. COMBES et al. [2016]).



� Pillar one techniques need to be carefully set-up to be efficient.

� Such approaches are not suited to project the balance sheet. Having aclosed formula to go from the mathematical reserve to the best estimateimproves significantly the model’s performances. Being easily reproducible, itfacilitates the process of audit and control.

� Such models can be built analyzing the main risks of the contracts, e.g., byobserving that a (French) saving contract is mainly non-hedgeable, becauseof the accounting rules effect on the revalorization rate of the contract.

� This approach also provides a powerful tool for making projections of SCRalong a « critical path ». This is especially interesting when for timedependent stress scenario analysis (cf. GUIBERT et al. [2014]).

Conclusion


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 del’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 theORSA », European Actuarial Journal, Vol. 4, Issue 1, Page 181-196. http://dx.doi.org/10.1007/s13385-014-0086-z.

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 pourl'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, 2:205-226, doi: 10.1007/s13385-012-0051-7..

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.

References


NTEUKAM T. O., PLANCHET F., REN J. [2014] « Internal Model in Life insurance: Application of Least Square Monte-Carlo in RiskAssessment », Les cahiers de recherche de l’ISFA, n°2014.12.

NTEUKAM O., PLANCHET F. [2012] « Stochastic Evaluation of Life Insurance Contract: Model Point on Asset Trajectories & Measurementof the Error Related to Aggregation », Insurance: Mathematics and Economics. Vol. 51, pp. 624-631.

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 insuranceeconomic valuations: pitfalls and practical solutions », European Actuarial Journal, pp1-28.

Packages & R codes related to the ESG

- ESG (https://cran.r-project.org/web/packages/ESG/index.html)

- ESGtoolkit (https://cran.r-project.org/web/packages/ESGtoolkit/index.html)

- ycinterextra (https://cran.r-project.org/web/packages/ycinterextra/index.html)

- http://www.ressources-actuarielles.net/C1256F13006585B2/0/A5E99E9ABF5D3674C125772F00600F6C

References


Contact

Quentin GUIBERT Frédéric PLANCHET

[email protected] [email protected]

http://www.primact.frhttp://www.ressources-actuarielles.nethttp://blog.ressources-actuarielles.net

PRIM’ACT

42 avenue de la Grande Armée75017 Paris

+33-1-42-22-11-00

ISFA

50 avenue Tony GarnierF - 69007 Lyon

+33-4-37-38-74-37

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