The Solvency II Square-Root Formula for Systematic Biometric Risk

The Solvency II Square-Root Formula for Systematic Biometric Risk Marcus Christiansen, Michel Denuit und Dorina Lazar Prerint Series: 2010-14 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM

The Solvency II square-root formula for systematic biometric risk Marcus C. Christiansen Institut für Versicherungswissenschaften Universität Ulm D-89069 Ulm, Germany Michel M. Denuit Institut de Statistique, Biostatistique et Sciences Actuarielles Université Catholique de Louvain B-1348 Louvain-la-Neuve, Belgium Dorina Lazar Faculty of Economics and Business Administration Babes-Bolyai University Cluj-Naoca, Romania

Abstract In this aer, we develo a model suorting the so-called square-root formula used in Solvency II to aggregate the modular life SCR. Describing the insurance olicy by a Markov jum rocess, we can obtain exressions similar to the square-root formula in Solvency II by means of limited exansions around the best estimate. Numerical illustrations are given, based on German oulation data. Even if the square-root formula can be suorted by theoretical considerations, it is shown that the QIS correlation matrix is highly questionable. Key Words: Solvency caital requirement SCR, Markov jum rocess MJP, generalized life insurance, disability insurance, systematic risk, correlation.

1 Introduction and motivation In line with the Basel II requirements for banks, the Euroean Commission has established the Solvency II Directive for insurance comanies. Solvency II is based on a three-illar framework similar to Basel II. Pillar I focuses on caital requirements, such as marketconsistent valuation of the balance sheet, including insurance liabilities and assets. Under Solvency II, two caital levels are determined: the minimum caital requirement, a threshold at which comanies are no longer ermitted to sell olicies, and a solvency caital requirement SCR below which comanies may need to discuss remedies with their regulator. The SCR is comuted by means of a 99.5% Value-at-Risk. The standard aroach to the SCR can be alied by all insurers, irresective of size, ortfolio mix and geograhical location. The standard formula given by the directive for the SCR calculation uses a modular aroach with modules for non-life underwriting risk life underwriting risk health underwriting risk market risk counterarty default risk oerational risk intangible assets risk. The biometric risks considered in this aer are contained in the life underwriting risk module. The risk modules are in turn built on submodules. The life underwriting risk module consists of 7 submodules for 1. mortality risk risk of increasing mortality 2. longevity risk risk of falling mortality 3. disability or morbidity risk 4. lase risk 5. life exense risk 6. revision risk 7. life catastrohe risk. The first 6 categories are first aggregated and the result is then aggregated with the last life catastrohe risk. Modules 1-6 are aggregated to SCR by means of the so-called square-root formula SCR = Corr ij SCR i SCR j, 1.1 i,j 1

where SCR i and SCR j refer to the 6 sub-modules listed above and the airwise correlation coefficients Corr ij come from a given correlation matrix. For examle, the 5th Quantitative Imact Study QIS5 uses the following correlation matrix: SCR mort SCR long SCR dis SCR lase SCR ex SCR rev SCR mort 1 SCR long -0.25 1 SCR dis 0.25 0 1 SCR lase 0 0.25 0 1 SCR ex 0.25 0.25 0.5 0.5 1 SCR rev 0 0.25 0 0 0.5 1 Finally, SCR and SCR cat are aggregated by using again the square-root formula with correlation 0.25. In this aer, we only consider the first ste, and more secifically, the aggregation of the three first categories SCR mort, SCR long, and SCR dis using 1.1. The technical secifications of QIS5 define the life submodules as the change in the net value of assets minus liabilities due to shock scenarios. For the submodules SCR mort, SCR long, and SCR dis the following scenarios are used: 1. ermanent 15% increase in mortality rates for each age for SCR mort 2. ermanent 25% decrease in mortality rates for each age for SCR long, and 3. 50% increase in disability rates for the next year, together with a ermanent 25% increase over best estimate in disability rates at each age in following years lus, when alicable, a ermanent 20% decrease in morbidity/disability recovery rates. The standard formulas suffer from the two following shortcomings: The Solvency II directive defines SCR by means of the 99.5% Value-at-Risk but the standard formula for the life risk module SCR life neither uses stochastic modeling nor are we aware of an equivalence result to a stochastic background model. The standard formula models deendencies between mortality, disability, lasation, etc. not at the roots but on the level of the submodules SCR mort, SCR long, SCR dis, etc. Because of this fact and the fact that the same correlation matrix is used for any kind of insurance ortfolio, the standard formula does not cover the diversity of risk structures that insurance ortfolios can have. The standard formula develoed by exerienced actuaries with a lot of exertise may yield in ractice a good aroximation to the Value-at-Risk. However, we are not aware of any sound study suorting the standard formula. In the resent aer, we roose a stochastic model that allows to theoretically substantiate 1.1. Moreover, we introduce an alternative concet for the calculation of SCR life that models deendencies between mortality, disability, etc. directly on the level of the transition rates. The aer roceeds as follows. In Section 2, we describe the multistate model for general insurance olicies used in the resent work. This allows us to quantify the effect of a dearture of future transition rates mortality rates, disability rates, and recovery rates, 2

for instance from best estimates. In Section 3, we fit a multivariate Lee-Carter model to German mortality, disability and recovery rates. This model serves as the basis to exlore the imact of systematic biometric risk to determine the numerical values for the correlations between the different sources for this risk. Section 4 rooses a formal justification to the square-root formula 1.1 by means of exansion formulas derived in Section 2. Sections 5 and 6 discuss alternatives to the square-root formula, based on asymtotic aroximations and uer bounds. The final Section 7 concludes. 2 Basic model of life insurance olicies Consider an insurance olicy that is driven by a Markovian jum rocess X t t with finite state sace S and transition sace J {j, k S 2 j k}. We write x for the age of the olicyholder at time zero usually the beginning of the contract eriod and ω = ω x where ω stands for the ultimate age. The cash-flows of the contract are described by the following functions: 1. The lum sum b jk x is ayable uon a transition of olicyholder from state j to state k at age x. We assume that the functions b jk, j, k J, are well-behaved i.e., have bounded variation on [0, ω ]. 2. The function B j x gives the accumulated annuity benefits minus accumulated remiums for a sojourn of olicy in state j during [0, x]. We assume that the functions B j, j S, are well-behaved i.e., have bounded variation on [0, ω ] and are rightcontinuous. We write vs, t for the value at time s of a unit ayable at time t > s and assume that it has a reresentation of the form vs, t = e s,t] ϕu du with ϕ being the interest intensity or short interest rate. We assume that there exists a general transition intensity matrix µx; t = µ jk x; t j,k S 2, where the first and second argument of µ are the age and the calendar time at which a transition takes lace, such that the Markovian jum rocess X t t has the transition intensity matrix t µx + t; t. We write x ; s, t = P X t = k Xs = j, 0 s t ω, j,k S 2 for the corresonding transition robability matrix, which is uniquely determined by the Kolmogorov forward equation d dt x ; s, t = x ; s, t µx + t; t, s t, and starting values x ; s, s = I for all s, where I denotes the identity matrix. In the resent aer, we build our model on log transition intensities, which means that the µ jk are exonentials of auxiliary γ jk, that is, µ jk x; t = ex γ jk x; t, j, k J. 2.1 3

Examle 2.1 Disability model. In the resent aer we focus on disability insurance for the numerical illustrations. The disability model has state sace S = {a, i, d}, where a=active, i=invalid/disabled, d=dead, and transition sace J = {a, i, a, d, i, a, i, d}. We consider the following insurance roducts: a Disability insurance. A yearly disability annuity of 1000 is aid as long as the olicyholder is in state disabled. A constant remium is aid yearly in advance as long as the insured is in state active. b Disability & ure endowment insurance. Additionally to the disability benefits in a, an endowment benefit of 10 000 is aid if the olicyholder is still alive at termination of the contract. c Disability & temorary life insurance. Additionally to the disability benefits in a, a lum sum benefit of 10 000 is aid if the olicyholder dies before termination of the contract. d Disability & endowment insurance. Additionally to the disability benefits in a, a lum sum benefit of 10 000 is aid in case of olicyholder s death or termination of the contract, whichever occurs first. e Pure endowment insurance with disability waiver. An endowment benefit of 10 000 is aid if the olicyholder is still alive at termination of the contract. A constant remium is aid yearly in advance as long as the insured is in state active. f Temorary life insurance with disability waiver. A lum sum benefit of 10 000 is aid if the olicyholder dies before termination of the contract. A constant remium is aid yearly in advance as long as the insured is in state active. g Endowment insurance with disability waiver. A lum sum benefit of 10 000 is aid in case of olicyholder s death or termination of the contract, whichever occurs first. A constant remium is aid yearly in advance as long as the insured is in state active. The rosective reserve of olicy at time s in state i is obtained as the exected resent value of future benefits minus the exected resent value of future remiums given that the olicyholder is in state i at time s, i s = vs, t ij x ; s, t B j x + dt j S s,ω ] + 2.2 vs, t b jk x + t ij x ; s, t µ jk x + t; t dt. j,k J s,ω ] The family of rosective reserves i s, i S, s [0, ω ], can be obtained as the unique solution of Thiele s integral equation system i s = B i x + ω B i x + s s,ω ] i t ϕt dt + j:j i 4 s,ω ] R ij t µ ijx + t, t dt, 2.3

with starting values i ω = 0 for all i S, where R ij t := b ij x + t + j t i t is the so-called sum-at-risk associated with a ossible transition of olicy from state i to state j at time t. According to Christiansen 2008 the rosective reserve i s seen as a maing of the general transition intensity matrix µ has a first-order Taylor exansion of the form i s, µ = Vi s, µ + vs, t ij x ; s, t R jk t µ jk x + t; t µ jk x + t; t dt j,k J s,ω ] + o µx + ; µx + ;, 2.4 where means the L 1 -norm of the maximum row sum. We refer the reader to Christiansen 2008 for more details. If the transition intensities µ jk are modeled as exonentials of the form 2.1, it can be more convenient to regard the rosective reserve i s not as a maing of µ but as a maing of γ = γ jk j,k S 2. Using 2.4, the roerty e γ jkx +t;t = e γ jkx +t;t + γ jk x + t; t γ jk x + t; t e γ jkx +t;t + o γ jk x + ; γ jk x + ;, and the chain rule of Fréchet differentiation, we get a first-order Taylor exansion in γ, s, ex γ i = i + s, exγ j,k J s,ω ] vs, t ij x ; s, t R jk t µ jkx + t; t γ jk x + t; t γ jk x + t; t dt + o γx + ; γx + ;. 2.5 The next section exlains how γ jk can be estimated from yearly transition data. Between integer ages and integer times, we assume that the log intensities γ jk x; t are constant. More recisely, where [ξ] denotes the integer art of the real ξ. γ jk x; t = γ jk [x]; [t], j, k J, 2.6 3 Stochastic modeling of the transition rates In the numerical art of the resent aer, we consider the classical 3-state Markov model for disability insurance olicies described in Examle 2.1. 5

Very few studies investigated time trends in transition rates for multistate actuarial models. A noticeable excetion is Renshaw and Haberman 2000 who considered the sickness recovery and incetion transition rates, together with the mortality rates when sick, which form the basis of the UK continuous mortality investigation Bureau s multistate model. These authors identified the underlying time trends from the observation eriod 1975-1994 using searate Poisson GLM regression models for each transition. In this aer, we allow for ossible correlations by means of a multivariate Lee-Carter tye model fitted as described in Hyndman and Ullah 2007, that is, by means of a functional data aroach. The methodology develoed by Hyndman and Ullah 2007 combines functional data analysis, nonarametric smoothing and robust statistics. It is a convenient generalization of the Lee-Carter model for mortality rates: comared with Lee and Carter 1992, Hyndman and Ullah 2007 use smooth rather than crude mortality rates, relace conventional rincial comonent analysis with its functional counterart and allow for more than one rincial comonent. This methodology is used here to reduce the dimensionality of the roblem. For each of the four transitions a, i, a, d, i, a, and i, d, the Hyndman-Ullah aroach is used as a way to summarize the information contained in the age-secific transition rates multivariate time series. Secifically, the first ste consists in smoothing the observed log-transition rates γ jk x; t across age x for each calendar year t and given transition j, k J, using a nonarametric smoothing method. In a second ste, the smoothed set of transition rates is decomosed into γ jk x; t = m jk x + φ jk x β jk t 3.1 where the m jk and φ jk are deterministic functions of age and the vectors β a,i t βt = β a,d t β i,a t β i,d t form a 4-dimensional times series. This exression for γ jk x; t is similar to the model introduced for mortality rates by Lee and Carter 1992. A two-ste rocedure for robust functional rincial comonent analysis is alied to estimate the age functions m jk and φ jk, to avoid roblems with outlying years. To erform numerical illustrations, the model is fitted to annual German age-secific transition data. The sources of the data used in this aer are as follows. First, there is the data set of the Statistisches Bundesamt Deutschland German Federal Statistical Office, www.destatis.de that contains mortality rates for the general oulation in Germany. In order to reduce inconsistency in the time series due to reunification of Eastern and Western Germany in 1990, we use only the data for Western Germany. Second, there is the data set of the Deutsche Rentenversicherung German statutory ension insurance, www.deutsche-rentenversicherung.de that gathers data about mortality, disability, and reactivation rates for all ersons emloyed in the rivate or ublic sector about 90% of the entire oulation. For consistency reasons, we use only data for Western Germany, and we sto the time series at the end of 2000, because since 2001 the definition 6

of disability is reduced to unemloyability. Based on data rovided by all its 26 member institutes, the Federation of German Pension Insurance Institutes comiles data and ublishes several statistical reorts annually. In this aer, we used the following statistical reorts: Statistikband Rentenzugang/Rentenwegfall for the years 1984 2000 reort no. 66, 71, 75, 81, 86, 91, 96, 99, 104, 109, 113, 117, 121, 125, 129, 133, 137 number of transitions from active to disabled: table 40 RV Z from 1984 to 1991, table 201.10 Z RV from 1992 to 2000 number of transitions from disabled to active: table 23 RV W from 1984 to 1991, table 204.10 W RV from 1992 to 2000 number of transitions from disabled to dead: 23 RV W from 1984 to 1991, table 204.10 W RV from 1992 to 2000 in fact, the number of transitions is given jointly for disability and old-age ensions, but before age 60 the minimum age for oldage ensions all transitions come from disability ensions Statistikband Rentenbestand for the years 1985 2000 reort no. 65, 70, 74, 80, 85, 90, 95, 100, 105, 110, 111, 116, 120, 124, 128, 132 exosure to risk in state disabled: table 13 G RV from 01.01.1985 to 01.01.1991, table 201.10 G RV from 31.12.1991 to 31.12.1999 Statistikband Versicherte for the years 1984 2000 reort no. 73, 78, 83, 88, 93, 98, 103, 106, 108, 115, 119, 123, 127, 131, 135, 139 exosure to risk in state active: table 181 RV from 1984 to 1990, table 081 RV for 1991, table 6.11 V RV from 1992 to 2000. The Deutsche Rentenversicherung also rovides data about the number of transitions from active to dead. However, for the years 1992 2000 these data are only available in age grous of five years. Therefore, we estimate the mortality in state active from the data of the Statistisches Bundesamt Deutschland. Thus, statistics are available for eriod 1984-2000 for a, d and a, i transitions and for eriod 1985-2000 for i, a and i, d transitions. Transitions within the calendar year and exosure to risk were available for each age from 21 to 64, excet for the i, d transition where data cover each age from 21 to 59. The estimation is carried on with the hel of the function fdm from the demograhy ackage, imlementing the Hyndman-Ullah methodology in the R software. The model exlains 82.1% of the variance for transition a, d, 65.9% of the variance for transition a, i, 67.3% of the variance for transition i, a, and 49.5% of the variance for transition i, d. We acknowledge the relatively low ercentage of exlained variance for the transition i, d but we nevertheless kee the simle first comonent model for the numerical illustrations roosed in this aer 1. 1 Incororating the second rincial comonents allows to exlain 90.7% of the variance for transition a, d, 90.4% of the variance for transition a, i, 83.3% of the variance for transition i, a, and 70.4% of the variance for transition i, d. Incororating the third comonent increases the exlained variance for transition i, a u to 88.6% and for transition i, d u to 95.7%. 7

The coefficients time series βt for each of the four transitions are considered as the comonents of a 4-dimensional time series. A multivariate random walk with drift aears to aroriately describe the dynamic of this time series. Indeed, the differences βt βt 1 aear to be non correlated and multivariate Normally distributed. Several multivariate residual tests have been erformed: - the null hyothesis of no serial correlation u to 4 lags is not rejected by the multivariate Ljung-Box Q-statistics the Portmanteau autocorrelation test statistic is Q = 54.8 with a -value equal to 0.78; - the autocorrelation LM test does not detect serial correlation in residuals the LM test statistic is LM = 21.54, with a -value 0.15. The multivariate Jarque-Bera Normality test statistic is JB = 2.36, with corresonding -value 0.78 so that Normality is not rejected. Hence, we consider that the differences βt βt 1 are indeendent and identically distributed, obeying the Normal law with mean vector 0.072526 0.037238 0.012203 0.039883 and variance-covariance matrix Σ = 0.008808 0.000770 0.014397 0.010440 0.000770 0.001392 0.000844 0.003532 0.014397 0.000844 0.053968 0.008771 0.010440 0.003532 0.008771 0.029803 4 Solvency II standard formula 4.1 Value-at-Risk rincile The Solvency II directive defines the solvency caital requirement by means of a Value-at- Risk at robability level 99.5%. However, the Solvency II standard formula for the life risk module SCR life does not use robabilistic arguments, and it is not clear if the formula really corresonds to the intended Value-at-Risk. In this section, we resent a stochastic model in which the standard formula for SCR life is asymtotically equivalent to the intended Value-at-Risk. According to the Solvency II framework, the life risk module SCR life is roughly defined by. SCR life = V ar 0.995 change in net value of assets minus liabilities due to changes of transition rates where change in net value means deviations from best estimates. We assume here that changes in mortality rates, disability rates, etc. have no effect on assets., 8

4.2 Single olicy For a single olicy, the change in the net value of liabilities is given by the change in its rosective reserve. Therefore, given that the olicyholder is at resent time s in state i, we rewrite the definition of SCR life to SCR life = V ar 0.995 i s, µbe i s, µ, 4.1 where µ BE is a deterministic best estimate transition intensity matrix, and where µ is the actual transition intensity matrix alying to this olicy. Since these intensities relate to the future, they are only forecasts and, thus, stochastic. A ossible exlanation for 1.1 being equal to 4.1 is a model where i s, µbe i s, µ has a decomosition into a sum with terms corresonding to the different life risk submodules being Normally distributed with zero mean. Usually we do not have such a linear decomosition because the rosective reserve i s, µ = Vi s, exγ is nonlinear in the argument µ or γ. But with the hel of 2.5 we can at least aroximate i s, exγbe i s, exγ by the sum j,k J g jk t γ BE jk x + t; t γ jk x + t; t dt, 4.2 where g jk t := 1 s,ω ]t vs, t BE ij x ; s, t R,BE jk t µ BE jk x + t; t. With the best estimate γ BE being deterministic and only γ being stochastic, each term involved in the sum 4.2 uniquely corresonds to the fluctuation risk of one transition rate. If furthermore the γ jk are Gaussian with E[γ jk x; t] = γjk BE x; t, then 4.2 is Normally distributed with zero mean and its Value-at-Risk can in fact be calculated by a square root formula similar to 1.1: V ar 0.995 i s, µbe i s, µ V ar 0.995 g jk t γjk BE x + t; t γ jk x + t; t dt 4.3 j,k J = Corr j,k,l,m SCR j,k SCR l,m j,k,l,m with Corr j,k,l,m giving the airwise correlations and SCR j,k being defined by SCR j,k = V ar 0.995 g jk t γjk BE x + t; t γ jk x + t; t dt. Solvency II ignores the fact that mortality can evolve differently in different states mortality in state active, mortality in state disabled, etc.. In order to take resect of that, we aggregate all mortality terms into a single quantity and define SCR,d = V ar 0.995 g jd t γjd BE x + t; t γ jd x + t; t dt. j:j d 9

Recall that d means state dead. Furthermore, Solvency II does not see the risk of changing mortality rates as a single risk factor, but slits the risk u into an uward risk SCR mort and a downward risk SCR long. However, usually one of the two quantities SCR mort and SCR long is zero and can be ignored. In such a case we can identify SCR,d with the one of these two quantities that is not zero. The standard formula of Solvency II combines the risk of changing disability rates and the risk of changing reactivation rates into the single submodule SCR dis. In order to relicate this submodule we define SCR a,i&i,a = V ar 0.995 j,k {a,i,i,a} 4.3 Alication to disability insurance g jk t γjk BE x + t; t γ jk x + t; t dt. For the disability insurance with state sace S = {a, i, d} and no death benefits, we have SCR mort = 0, and thus we can identify SCR,d and SCR a,i&i,a with the Solvency II submodules SCR long and SCR dis. The Solvency II correlation Corr long,dis should here be equal to Corr,d,a,i&i,a = Corr g ad t γad BE x + t; t γ ad x + t; t + g id t γid BE x + t; t γ id x + t; t dt, g ai t γai BE x + t; t γ ai x + t; t + g ia t γia BE x + t; t γ ia x + t; t dt. In order to calculate Corr,d,a,i&i,a = Cov,d,a,i&i,a / Cov,d,,d Cov a,i&i,a,a,i&i,a, we just need to know the covariances between all γ jk x; t. The covariance Cov,d,a,i&i,a can be rewritten to Cov,d,a,i&i,a = g ad t g ai u Covγ adx + t; t, γ ai x + u; u + g id t g ai u Covγ idx + t; t, γ ai x + u; u + g ad t g ia u Covγ adx + t; t, γ ia x + u; u + g id t g ia u Covγ idx + t; t, γ ia x + u; u dt ds 4.4 where we assumed E[γ jk x; t] = γjk BEx; t. Analogous formulas can be obtained for Cov,d,,d and Cov a,i&i,a,a,i&i,a. Alying the results of Section 3, the covariances between every 10

air of γ jk x; t can be obtained as follows: Cov γ ij x; t, γ kl y; s = Cov φ ij x = φ ij x φ kl y t βij τ β ij τ 1, φ kl y τ=1 t τ=1 σ=1 τ=1 s βkl σ β kl σ 1 σ=1 s βij Cov τ β ij τ 1, β kl σ β kl σ 1 min{t,s} βij = φ ij x φ kl y Cov τ β ij τ 1, β kl τ β kl τ 1 = φ ij x φ kl y min{t, s} Σ i,j,k,l. 4.5 Consider a 30-year-old male who contracts insurance olicies according to Examle 2.1 a-g. All olicies start at the beginning of year 2001 and terminate at age 60. The yearly constant remiums are calculated on the basis of the forecasts best estimates of Section 3 and an interest intensity of ln1.0225. We then get the following results: a Disability insurance. The equivalence rincile leads to a constant yearly remium of 27.58. The olicy has to a large extent a survival character, and, hence, we can identify Corr,d,a,i&i,a 0.658 with Corr long,dis. b Disability & ure endowment insurance. The equivalence rincile leads to a constant yearly remium of 247.28. The olicy has a ure survival character, and, hence, we can identify Corr,d,a,i&i,a 0.563 with Corr long,dis. c Disability & temorary life insurance. The equivalence rincile leads to a constant yearly remium of 57.93. The olicy has to a large extent an occurrence character, and, hence, we can identify Corr,d,a,i&i,a 0.375 with Corr mort,dis. d Disability & endowment insurance. The equivalence rincile leads to a constant yearly remium of 275.58. The olicy has a mixed character, and Corr,d,a,i&i,a 0.354 corresonds to a combination of Corr mort,dis and Corr long,dis. e Pure endowment insurance with disability waiver. The equivalence rincile leads to a constant yearly remium of 219.70. The olicy has a ure survival character, and, hence, we can identify Corr,d,a,i&i,a 0.407 with Corr long,dis. f Temorary life insurance with disability waiver. The equivalence rincile leads to a constant yearly remium of 30.35. The olicy has a ure occurrence character, and, hence, we can identify Corr,d,a,i&i,a 0.485 with Corr mort,dis. g Endowment insurance with disability waiver. The equivalence rincile leads to a constant yearly remium of 250.05. The olicy has a ure occurrence character, and, hence, we can identify Corr,d,a,i&i,a 0.447 with Corr mort,dis. 11

The examles a-g show that the correlation Corr,d,a,i&i,a varies significantly with the tye of insurance contract. In e we even obtain a sign that is oosite to the one contained in the QIS5 correlation matrix. This indicates that Solvency II resumably oversimlifies the risk structures of life insurance ortfolios by using the same correlation factors Corr long,dis and Corr mort,dis for all kinds of roducts. 4.4 Extension to ortfolios If we have not only a single olicy but a ortfolio of olicies N that are at resent in state i, we naturally extend definition 4.1 to SCR life = V ar 0.995 N i s, µ BE i s, µ. 4.6 Here all olicies use the same calendar time. We take the earliest start of a contract as time zero. For olicies that start at a later calendar time, the ayments functions b jk x + t and B j x + t are constantly zero at the beginning. Analogously to 4.3, we can derive the decomosition i s, µ BE i s, µ V ar 0.995 N V ar 0.995 = j,k,l,m j,k J N g jk t γjk BE x + t; t γ jk x + t; t dt Corr j,k,l,m SCR j,k SCR l,m 4.7 with Corr j,k,l,m giving the correlations between the addends of the sum j,k J SCR j,k being defined by SCR j,k = V ar 0.995 N g jk t γjk BE x + t; t γ jk x + t; t dt. and the Also comletely analogous to the single olicy case, we can define a general mortality risk quantity SCR,d = V ar 0.995 g jd t γjd BE x + t; t γ jd x + t; t dt j:j d N and identify it with either SCR mort or SCR long given that one of the two risk modules is negligible. In the same way, we generalize SCR a,i&i,a by SCR a,i&i,a = V ar 0.995 j,k {a,i,i,a} N g jk t γjk BE x + t; t γ jk x + t; t dt. 12

5 Alternative calculation of SCR life 5.1 Asymtotic method The Solvency II directive says that each risk module shall be calibrated using a Value-at-Risk measure with a confidence level of 99.5%. In articular this holds for the life risk module SCR life. We mathematically formalized this aroach in 4.1 and 4.6. Unfortunately, it is usually very difficult to calculate the Value-at-Risks in 4.1 or 4.6 analytically. In this section, we assume that the decomosition 3.1 alies and that the βn vectors obey a random walk with drift. One idea to obtain at least an aroximation of 4.1 and 4.6 is to use the first-order Taylor aroximation concet of 4.3. But instead of seeing the rosective reserve as a maing of the log intensities γ jk as done in 4.2, we now see the rosective reserve as a maing of the differences β jk n := β jk n β jk n 1 with β jk 1 := 0. Here, we assume that each β jk evolves according to 3.1. Using 2.6 and the linearity of the γ jk with resect to the β jk, we can transform 2.5 into [ω] i s, exe[γ] Vi s, exγ + o = j,k J = j,k J s,ω ] [ω ] n=0 E[ βn] βn [t] g jk t φ jk[x + t] E[ βjk n] β jk n dt n=0 E[ βjk n] β jk n g jk t φ jk[x + t] dt, n=0 s n,ω ] 5.1 where the g jk t are defined as in 4.2. Under our assumtions, the random vectors E[ βn] βn are mutually indeendent and admit the reresentation E[ βn] βn = Σ 1/2 Xn, where the Xn are indeendent column random vectors with a multivariate standard Normal distribution and Σ 1/2 is the Cholesky decomosition of Σ. With defining the row vectors Gn = g jk t φ jk[x + t] dt, we may write s n,ω ] [ω] i s, exe[γ] Vi s, exγ + o n=0 j,k J E[ βn] βn = [ω ] GnΣ 1/2 Xn. n=0 5.2 The right-hand side is a sum of indeendent standard Normal random variables. Thus, it is Normally distributed with zero mean and variance [ω ] Gn Σ Gn T. n=0 13

Given that µ BE = exe[γ] and given that o [ω ] n=0 E[ βn] βn is negligible, we can aroximate 4.1 by SCR life = V ar 0.995 i s, exe[γ] Vi s, exγ [ω ] V ar 0.995 n=0 GnΣ 1/2 Xn [ω ] 1/2 = u 0.995 Gn Σ Gn T n=0 5.3 with u 0.995 being the 0.995-quantile of the standard Normal distribution. So far we only studied a single olicy. In the ortfolio case we get the same formula but the vectors Gn are of the form Gn = g jk t φ jk[x + t] dt. 5.4 j,k J N s n,ω ] Comaring 5.3 with the Solvency II concet see the revious section, we see that both aroaches are based on the same first-order Taylor aroximation, but 5.3 has the advantage that it much better reflects the various risk structures of life insurance ortfolios when it comes to deendencies. Unfortunately, the aroximation error in formula 5.3 is difficult to control. Therefore, in the next section we give a concet that allows to calculate uer bounds of SCR life. 5.2 Uer bound Using the same model setting as in the revious section, we now see the rosective reserve i s = i s, µ β as a maing of the βn, n = 0,..., [ω ]. If M 0.995 is an arbitrary but fixed set with P β M 0.995 0.995, then we have SCR life = V ar 0.995 i s, µe[ β] Vi s, µ β inf i s, µe[ β] Vi s, µ β 5.5 in case of a single olicy and β M 0.995 = su i s, µ β i s, µe[ β] β M 0.995 SCR life = V ar 0.995 N su µ M 0.995 N i s, µe[ β] V i s, µ β i s, µ β 5.6 i s, µe[ β] N 14

for an inhomogeneous ortfolio of olicies N that are at resent in state i. In order to comute the uer bounds for SCR life in 5.5 and 5.6, we need to find suitable scenario sets M 0.995 with P β M 0.995 0.995, solve the otimization roblems su β M0.995 i s, µ β or su µ M0.995 N i s, µ β There are lots of ways to choose the scenario set M 0.995. For a single olicy, the otimal choice is the scenario set for which the uer bound 5.5 is shar: { } β i s, µe[ β] Vi s, µ β V ar 0.995 i s, µe[ β] Vi s, µ β. 5.7 However, we have no method to find this otimal set. Let us now show how to find a scenario set that leads to an uer bound 5.5 that is at least close to the true Value-at-Risk. By using the first-order Taylor aroximation 5.1 on both hand sides of the inequality in 5.7, we obtain M 0.995 = { β [ω ] [ω ] Gn E[ βn] βn u 0.995 n=0 n=0 Gn Σ Gn T 1/2}. 5.8 By following the arguments of the revious section, we see that this scenario set indeed satisfies P β M 0.995 0.995. In the ortfolio case we get the same scenario set but with the ortfolio version 5.4 of Gn. The examles given later on will show that 5.8 is indeed a good choice for M 0.995. The second ste is now to calculate the surema in 5.5 and 5.6. A solution can be found by using a gradient ascent method. From 5.1 we get N V i s, µ β = G jk n β jk n with G as defined in 5.4. The artial derivatives form the gradient vector of i s, µ β with resect to the argument β. The gradient gives locally the direction with the steeest incline. The gradient ascent method follows that direction in small stes: Algorithm 5.1 calculation of the suremum in 5.5 or 5.6. 1. Choose as starting value β 0 = E[ β] and calculate the corresonding G = G 0. 2. Calculate a new β m+1 by using the iteration β m+1 n = β m n + K G m n T, n = 0,..., [ω ], 5.9 where G m n is defined by 5.4 with the g jk t on the right hand side being calculated on the basis of µ β m. The constant K > 0 controls the stesize. 3. Reeat ste 2 as long as β m+1 is an element of 5.8, that is, as long as β m+1 satisfies the inequality in 5.8 for G = G 0. 15

5.3 Numerical illustration We continue with Examles 2.1 a-g and calculate the corresonding asymtotic Value-at-Risks according to 5.3, uer bound Value-at-Risks according to 5.5 by using Algorithm 5.1. The results are dislayed in the next table V ar 0.995 calculated by... asymtotic uer bound formula formula Disability ins. 231.29 97.27 Disability & ure endowment ins. 267.09 89.32 Disability & temorary life ins. 500.05 204.22 Disability & endowment ins. 277.00 119.50 Pure endowment ins. with disability waiver 238.22 67.30 Temorary life ins. with disability waiver 347.73 146.79 Endowment ins. with disability waiver 118.43 57.09 We also conducted 10 000 Monte Carlo simulations to evaluate the true Value-at-Risk for each a-g roduct. This shows that the uer bound 5.5 is retty accurate if we use the scenario set 5.8, overestimating the true value by less than 5% in all the cases considered in this aer. We also see that in all the examles the asymtotic formula 5.3 overestimates the true Value-at-Risk to a large extent. That means that we are always on the safe side, but the aroximation error is quite large. 6 Conclusions Solvency II is the new regulatory regime for EU countries to be mandatory for insurance comanies by 2012. Its three illar structure is similar to Basel II excet for the additional insurance risk. Under the standard aroach, SCR is aroximated with the hel of the square-root formula. In this aer, we have rovided a theoretical foundation for this formula using limited exansion around the best estimate. Numerical illustrations erformed on the basis of German data have suggested that the QIS correlation matrix was not aroriate and that the correlations greatly varied from one roduct to another. Therefore, the roblem is not so much with the square-root formula itself but well in using the same correlation values for all tyes of roducts. The numerical illustrations rovided in the resent aer constitute only a first ste. First, they are based on a single German data set, only. The correlations may vary among EU member countries and the data set is not necessarily reresentative of the German insurance market. Second, it could be interesting to relax the Markovian assumtion and to extend the results derived in this aer to semi-markov rocesses: bivariate transition rates could be secified for states where duration effects are significant tyically, disability states. Desite these limitations, this aer is a first attemt to legitimate the standard formula under Solvency II, questioning the QIS correlation matrix. 16

Acknowledgements This work has been erformed while Marcus Christiansen visited the UCLouvain suorted by a DFG research grant. The warm hositality of the Institute of Statistics is gratefully acknowledged. Michel Denuit acknowledges the financial suort of the Onderzoeksfonds K.U. Leuven GOA/07: Risk Modeling and Valuation of Insurance and Financial Cash Flows, with Alications to Pricing, Provisioning and Solvency. Dorina Lazar acknowledges the financial suort of the CNCSIS UEFISCSU, roject number PNII IDEI 2366/2008. References Christiansen, M.C., 2008. A sensitivity analysis concet for life insurance with resect to a valuation basis of infinite dimension. Insurance: Mathematics and Economics 42, 680-690. Hyndman, R.J. and Ullah, Md.S., 2007. Robust forecasting of mortality and fertility rates: a functional data aroach. Comutational Statistics and Data Analysis 51, 4942-4956. Lee, R.D. and Carter, L., 1992. Modelling and forecasting the time series of US mortality. Journal of the American Statistical Association 87, 659-671. Renshaw, A. and Haberman, S., 2000. Modelling the recent time trends in UK ermanent health insurance recovery, mortality and claim incetion transition intensities. Insurance: Mathematics and Economics 27, 365-396. 17