Analysis of Solvency II SCR Longevity risk valuation : standard model and stochastic model

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Analysis of Solvency II SCR Longevity

Risk Valuation: Standard Model and

Stochastic Model

Shu Zhang

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Shu Zhang Student nr: 10389776

Email: shuzhang.sue@gmail.com Date: August 21, 2014

Supervisor: Drs.Rob Bruning

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Analysis of Solvency II SCR Longevity Risk Valuation — Shu Zhang iii

Abstract

The capital requirement under Solvency II regime is determined as the 99.5% Value-at-Risk of the Available Capital. In the standard model provided by EIOPA, the Value-at-Risk capital requirement is approximated by the result of a 20% reduction of the mortality rates in the change of Net Asset Value as a pre-specified longevity shock. We study the 99.5% Value-at-Risk approach for obtaining capital re-quired for longevity risk based on the Lee-Carter model, as while the 20% shock approach as standard model brought up by EIOPA. The analysis of the adequacy of the shock approximation is taken by com-paring the resulting capital requirement between the 20% shock ap-proach and the Value-at-Risk method. The comparison reveals the shortcomings of the 20% shock approach in the standard model to be structural shortcomings.

Keywords Solvency II,SCR for longevity risk, Lee-Carter Model, Standard shock approach,

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Preface

This thesis is written in completion of the master programme of Ac-tuarial Science and Mathematical Finance in the University of Ams-terdam.

My appreciation is to send to my supervisor Rob Bruning,for his grate-ful guidance, for his enthusiasm and the motivating technical discus-sions.

I would like to thank my parents for supporting me for my studies in many ways, and to my friends for encouraging me for going through the tough but valuable academical time.

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Chapter 1

Introduction

Within the European Union, risk-based funding requirements for insurance companies are currently revised as part of the Solvency II project, in order to built up a more real-istic model and more proper assessments for the risks that the insurance companies are exposed to. The Solvency Capital Requirement (SCR) under Solvency II is determined as the 99.5% Value-at Risk(VaR) of the Available Capital(AC) over a one-year time horizon. The European Commission with support of the European Insuance and Occu-pational Pensions Authority (EIOPA) has established a scenario based standard model for all insurance companies to approximate their capital requirements. In this standard model, the overall risk for insurance companies is split into different risk modules,for example, the market risk, operational risk, or life underwriting risk. Moreover, the risk models are divided into submodules, based on which the required solvency capitals are calculated as the SCRs for various submodule risks. The EIOPA assumes a multivariate normal distribution with pre-specified correlation matrices for the SCRs of submodule risks. Based on there matrices, the SCRs are aggregated under the effects of diver-sification. A series of Quantitative Impact Studied(QIS) are brought out to establish the calibration of the standard model, also to analyze the effects of the new capital re-quirements. The standard model certainly has some shortcomings: Doff(2008) tested the Solvency II framework against the seven criteria for the fitness in efficient and complete market, found out some inappropriate incentives of the capital requirements formula; Devineau and Loisel (2009) compared the standard formula and internal model, showed some aggregation errors. Even so, most of the small-sized and medium-sized insurance companies are suggested to rely on the standard formula, while larger companies, being able to establish some calibration for the standard model, which they can adapt to their internal model for maintaining the stability of the European financial market.

Longevity risk is an important risk for life insurance companies and annuities, which is also difficult to evaluate. There was a steady increase in the average lifetime of its inhabitants in the western world over the past centuries, which could effect the potential pension costs, and present significant challenges for governments as well as individual pension funds and life insurers. Although the past trends suggest the further changes in mortality rates to be expected, a considerable uncertainty exists regarding the future development of morality. Thus the major challenge is to account for the effects of this particular uncertainty on the costs of pensions properly. The equal importance of the accurate quantification and management of the risk in pension and insurance liabilities are aware of by the insurers, while the focus is on the risk in financial investment for long, since interventions in the pension and insurance contracts can be mitigated but not eliminated. In project Solvency II, the valuation and management of pension and insurance liabilities has been put more emphasis on. Waegenaere et al.(2010) reviewed the current state of the literature about longevity risk, the model of future mortality and the importance of longevity risk for the solvency of portfolios of pension and life insurance products were discussed, also the possibility for longevity risk management

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2 Shu Zhang — Analysis of Solvency II SCR Longevity Risk Valuation

was investigated. Plat (2009) focused on the additional longevity risk which pension funds could be exposed to by comparing fund specific mortality to general population mortality. Olivieri and Pitacco (2008a) investigated rules for assessing the capital re-quired for a life annuity portfolio, concluded that the scenario for shock approach in the standard model could be deviated far from the actual experience of the insurer, which could lead to a bias for the capitals allocation. They suggested that the standard model simplifies the scenarios of loss too strongly that an internal model could be adopted instead.

SCR for longevity risk in the Solvency II standard model is determined as the change of Net Asset Value by the shock of 20% deduction of the mortality rate level. However, the adequacy of longevity shock is still questionable. In the consultation paper (CEIOPS(2008c)), the participants of QIS 5 stated the form of the longevity stress within the SCR standard formula does not appropriately reflect the actual longevity risk, specifically it does not appropriately allow for the risk of increases in future mortality improvements. In the CEIOPS (2009a) paper, they stated that a gradual change in mortality rates could be more appropriate than a one-off shock is acknowledged and also that the longevity stress should be related to age and duration instead of sticking to the a fixed level one-off shock is illustrated. The fixed shock can lead to an unnecessarily high capitalization of insurance companies especially when the longevity risk is overestimated by the fixed 20% one-off shock.Or the consequence that the default risk of a company can be significantly higher than the acceptance level of 0.5% may occur.

Thus, an analyze on whether or not the change in liabilities due to a 20% one-off longevity shock is a good approximation of the 99.5% VaR of the available capital. In this paper, we carry out the analysis by testing the adequacy of the 20% one-off shock structure.We apply the one-off shock approach and the VaR approach to the same scenario to check the difference of the results. The remainder of this paper is organized as follows: In Chapter 2, the definitions and the importance of longevity risk for annuity providers and insurance companies are introduced. Then the standard model for capital requirements for longevity risk is studied in Chapter 3, including the relevant definitions in the standard model, the risk module and submodule structures of Solvency II standard model as explained in QIS 5, and the settings for the SCRs for both longevity risk calculation approaches. The assumptions and simplifications for the specific condition to exclude other risk than longevity risk and to ensure the possibility that the one-off shock approach can be translated into gradual change in mortality. In Chapter 4, the stochastic model needed for mortality is discussed, specifically the Lee-Carter model is introduced as the basic stochastic model for the study in this paper. The forward mortality modeling framework and the specification of the model would be illustrated. The comparison of the results of SCRs via shock approach and VaR approach is carried out in Chapter 5. As well, we analyze the results and the reasons of the differences between the SCRs calculated from the two approaches. The Chapter 6 concludes.

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Chapter 2

Longevity Risk

2.1 Basic Notations for Mortality Rates and Longevity

First, we introduce the basic definitions for longevity risk and the required notifications for mortality rates.

Mortality Rates

qt,x is the underlying probability that an individual aged exactly x at time t will die

before time t + 1. The notation year t is referred to the period t to t + 1. qt,x is defined

for integer values for age x and year t here, the qt,x is assumed to be one number for

the whole period of year t. Survival Rates

pt,xshows the probability of the individual surviving at least another year at year t. If we

assume that the probability of individual surviving is independent of time t, simplified notations qx and px can be used.

pt,x= 1 − qt,x (2.1)

Central Death Rate

We use Dt,x to denote the number of death in year t at the age of x, let Et,x denote the

average population size of age x in year t. Then the central death rate is defined as mt,x =

Dt,x

Et,x

(2.2) .

The probability of the same individual surviving at least τ more years is τpx =

Qτ −1

j=0px+j, where px+j is the one-year death probability at year j. Then the expected

number of years the individual would survive can be derived as ex =Pτ ≥1 τpx. So the

individual is expected to die in year t + ex, counting from year t, at age x + ex.

However, the facts that one-year death probabilities decrease in longevity over time is simply observed from the data, showing increase in longevity risk over long period. Thus, the assumption that the one-year death probabilities are constant does not hold. Instead, the probability that an individual belonging to group g of age x survives at least τ years at year t is expressed as

τp(g)t,x = p (g) t,x · p (g) t+1,x+1· · · p (g) x+τ −1,t+τ −1, (2.3)

using p(g)_x+j,t+j = 1 − q_x+j,t+j(g) is the probability of the individual surviving at least another year at year t + j.

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Then the expected number of years the individual would at least survive at year t, is

e(g)_x,t =X

τ ≥1

τ p(g)_x,t. (2.4)

For calculating the expected number of years the individual will at least survive, further projections of the one-year death probabilities q_x,t(g)0, for t0 ≥ t are necessary.

Without such further projected one-year death probabilities, a serious underestima-tion of the expected number of years that an individual can survive will appear, also mis-valuation of the expected discounted value of the annuity. H´ari et al. (2007) showed that the expected remaining years of life changes significantly while accounting for the future changes in mortality rate changes. The expected present value, at time t, for an annuitant aged x, which belongs to group g, is

˜

a(g)_x,t = X

τ ≥max{65−x,0}

τ p(g)_x,t · P_tτ, (2.5)

where P_t(τ ) is the market value of a zero-coupon bond maturing at t + τ , at time t.By comparing the results of expected present value of annuities based on two different assumptions: the so-called period table, where the future one-year death probability is fixed; and the forecast one-year death probabilities. H´ari et al. (2007) found out that the underestimation of the present value existed when the calculation was based on period life tables.

A more challenging issue is that the one-year death probabilities do not only decrease over time, but also decrease with different rate for various ages, even different for males and females in an unpredictable way. It is unrealistic to assume the future one-year death probabilities is to be known in a deterministic way, and it seems to be more reasonable to assume the one-year death probabilities q_x,t(g)0 are stochastic at time t, for

t0 > t. Thus, the longevity risk, the risk that the probability of an individual of age x belonging to group g surviving at least τ other years at year t being not deterministic, is confronted. To be noticed, the individual mortality risk refers to the risk that an individual’s remaining lifetime is a random variable for given death probabilities; while the longevity risk refers to the risk of the results of longer term deterministic mortality projections deviations. The expected number of years that the individual can survive at year t is a random variable, as a consequence of longevity risk.

2.2 The Importance of Longevity Risk

Furthermore, the longevity risk, unlike the individual mortality risk, cannot be diver-sified by increasing the portfolio (law of large numbers). De Waegenaere et al. (2010) demonstrated the special character by enlarging the number of annuities in the port-folio but not resulting in a risk free portport-folio payoff. As a consequence, the longevity risk always remains, and the products with payoff related to future mortality will in-clude a longevity risk premium. The fact that longevity risk cannot be diversified by pooling is an important aspect of pricing and risk management. First for pricing, the non-diversifiability implies a risk premium in the price of a longevity linked asset or lia-bility. Second, for risk management, non-diversification has the important implication, that the traditional approach of increasing the portfolio size, which is used for individual mortality risk, does not reduce the impact of longevity risk (B¨orger 2010).

Longevity risk is the particular exposure of prominent risk annuity providers and pension funds, since the risks insured are related with expected survival. The longevity risk is accounted as a part of the life underwriting risk module, and gains special focus in many studies for its unique characteristics.

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Analysis of Solvency II SCR Longevity Risk Valuation — Shu Zhang 5

• The importance of longevity risk is likely to grow,as the increasing demand for annuity and endowment products, because of the decrease benefits from pension schemes public payment in most developed countries, combined with private an-nuities’ tax incentive.

• The longevity risk constitutes a systematic risk since it cannot be diversified in a large portfolio or hedged, because of no deep market for the securities of this risk. Longevity risk in the 1-year setting of Solvency II is composed by two parts from a perspective of an annuity provider: the risk that the realized mortality for next year will be below its expectation, and the risk that a deducted expected mortality will be beyond next year. Taking an example to explain the first risk component, a mild winter can lead to less people than usual dying from flu. For the second risk component, the test of a newly discovered medication against cancer may be taken first for some time before it is available for a large enough group, as a result the mortality on a population scale would be affected. Both of these two components would lead to higher liabilities than expected, because of more insured population alive than expected at the end of the period for the first case, in the second case, the best estimate basis for alive liabilities would be larger (B¨orger 2010). Therefore, a stochastic mortality model has to be accounted for these two components for longevity risk.

The importance of longevity risk is widely acknowledged and focused, resulting in difficulty of pricing model building and longevity risk model calibrating. The major challenge regarding the considerable uncertainty about the future development of mor-tality, is to account for the effects of this particular uncertainty on the costs of pensions properly. A lot of approaches dealing with the effect of the changes in life expectancy being used in practice include regularly re-estimation of the value of the liabilities on the basis of estimated death probabilities, or determine the value of the liabilities on the basis of a trend in mortality. Risk management practices need to be adjusted, since the approaches in practice are either retrospective or not concerning the uncertainty in the future development of mortality properly.

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Chapter 3

Solvency Capital Requirement

under Solvency II

The Solvency II framework is established by the European Commission for insurance companies,in order to built up a more realistic model and assessments for the risks which the insurance companies are exposure to. The Solvency Capital Requirement (SCR) under Solvency II is determined as the 99.5% Value-at Risk(VaR) of the Available Capital(AC) over a one-year time horizon. In this chapter, we introduce the Solvency II regime and the standard model for longevity SCR.

3.1 Solvency II

The European Commission is developing a new Solvency II framework. The new frame-work has a holistic balance sheet where assets and liabilities are valued and stressed using a Market Consistent Valuation approach. The aim of Solvency II is to assure that an insurance company will only default due to market turmoil once every 200 years. Although the framework is still under development, and unlikely to be implemented before 2016, the insurance companies have analyzed their balance sheet in order to pre-pare themselves to be Solvency II compliant. The assessment includes insurance policies of a new unit linked product which was recently launched. All calculations are based on the regulations of the so called Quantitative Impact Study 5 (QIS5).

In preparation for Solvency II, EIOPA (European Insurance and Occupational Pen-sions Authority), the former CEIOPS, has played, and still does, an important role. Its involvement is mainly technical. EIOPA supervised five Quantitative Impact Studies (QIS) over the years, aiming to develop the calculation of Solvency II Capital Require-ment (SCR).

Solvency II is the most comprehensive regulation ever imposed on the insurance industry across Europe, which was originally expected to be implemented on 1 January 2014. Solvency II will be adopted by all 27 European Union (EU) Member States plus three of the European Economic Area (EEA) countries.

The Solvency I framework was introduced in 1973. Since then more elaborate risk management systems have been developed to protect policyholders’ interests more ef-fectively by making firm failure less likely, and by reducing the probability of consumer loss or market disruption. Solvency II reflects new risk management practices and aims to define the required capital and manage risk to withstand a 1 in 200 year economic event and properly manage risks.

For Solvency II internal models, it is obliged to validate the model. The European Insurance and Occupational Pensions Authority (EIOPA) has written guidelines for the validation procedure. One of the requirements is that the model is back-tested: Many assumptions are set based on an analysis of historical data. There is therefore a presumption that past performance is a good indicator of future performance.

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testing may be used to assess the validity of this underlying assumption If the model is correctly specified, one expects that the probability of finding an extreme event in history is 0.5% per year. If more events are found than expected, this can be a reason to question the validity of the model. Therefore it is important that the back-test is performed in a correct way. In practice, often more historic extreme events are found than we will expect based on the confidence level of the stress.

The European Commission has established a scenario based standard model with the support of the Committee of Insurance and Occupational Pension Supervisors (CEIOPS), since the implementation of internal models for insurance companies to assess their risks as accurately as possible is quite costly and sophisticated. The stan-dard model is for all insurance companies to use, in order to approximate their capital requirements. The standard model splits the overall risk into different modules (i.e. market risk, operational risk, life underwriting risk), and sub-modules of separating SCRs computation. The sub-models of SCRs are aggregated with pre-specified corre-lation matrices for diversification effects and the assumption of a multivariate normal distribution. A serious of Quantitative Impact Studies (QIS) established the calibration of the standard model, to analyze the new capital requirement.

3.2 Solvency Capital Requirement in the Standard Model

The market value of liabilities under Solvency II is approximated by Technical Provisions consisted by the Best Estimate Liabilities (BEL) and a Risk Margin (RM).

As stated in Solvency II TP.2.1, the best estimate should correspond to the proba-bility weighted average of future cash-flows, taking account of the time value of money. The Best Estimate Liabilities can be calculated by the value of liabilities, which depend on stochastic future payments:

BELt=

X

i≥t

d(t, i + 1)E[Xi], (3.1)

where

Xi = Value at time i + 1 of liability payments in time interval [i, i + 1];

d(t, i) = The discount rate for cash-flows at time i discounted to t.

The Risk Margin needs to be added on top of Best Estimates, which is the deviation from Best Estimates, represents the additional payments due to uncertainty. The Risk Margin should be calculated the cost of providing an amount of eligible funds, which are equal to the Solvency Capital Requirement (SCR) required for supporting the insurance. So the current value and the future values of SCRs and cost of capital which determines the risk margin are needed to reserve for eventualities. The CoC (Cost of Capital) rate is used to determine the cost for providing the eligible own funds’ amount. The SCRs correspond to the Value-at-Risk of basic own funds of insurance or reinsurance undertaking over a one year horizon subject to a confidence level of 99.5%, which must be enough or in 200 year event. The CoC rate is determined as 6% in QIS5. Hence, according to CEIOPS(2009b), the Risk Margin (RM) could be defined as:

RM = CoCX

t≥0

E[SCRi] × d(t, i + 1), (3.2)

with SCR(i) the Solvency Capital Requirement prescribed by Solvency II, at the end of period [t, t+1] under the assumption of transfer to other reference undertaking. The Technical Provisions (TP), which is required to set up in Solvency II, cor-responds to the current amount undertakings would need to pay when they were to transfer the reinsurance obligations to another undertaking. The value of Technical

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Provisions is determined as the sum of a Best Estimate and a Risk Margin:

T Pt= BELt+ RMt, (3.3)

for risk could not be priced directly. While the Best Estimate and Risk Margin should not be valued separately but be calculated as a whole Technical Provisions. Under some certain conditions, the insurance and reinsurance obligation underwriting can be replicated of the cash-flows.

So SCRtis needed to be calculated, the Solvency Capital Requirement (SCR) under

Solvency II is defined as the capital required at time t = 0 to cover all losses could occur until t = 1 with a probability of at least 99.5% as previous description. In internal model, the SCRt is defined as the formula, 99.5% VaR of the Available Capital over 1

year:

SCRt= Q0.5%t (At+1− T Pt+1), (3.4)

where

Q0.5%_t = 0.5% quantile of distribution given information at t, At+1= Value of assets at time t + 1,

T Pt+1= Value of technical provisions at time t + 1.

Therefore, the Standard model can be used for all possible shocks which define 99.5% worst case scenarios for the change in value of own surplus within one year.

The SCR is calculated as the VaR of the Available Capital (AC), which depends on the SCR via Risk Margin. The Standard Formula for SCR is elaborate

SCR = BSCR + Adj + SCRop, (3.5)

where,

Adj = Adjustment for the risk absorbing effect of technical provision and deferred taxes

SCRop= The capital requirement or operational risk

BSCR = Basic Solvency Capital Requirement

The Adjustment part in the SCR formula represents the possible effect of manage-ment actions. Basic SCR is calculated under different scenarios, which can reflect sudden changes in the market or underlying risks. Some management actions might be imple-mented, based on the different scenarios. The adjustments also concern the possibility of deferring taxes after making losses. The BSCR before adjustment is the part we are interested in, which is the capital concerns the major risk categories, including: market risk, counter-party default risk, intangible asset risk, life underwriting risks, non-life underwriting risks and health underwriting risks.

The BSCR can be presented by a formula composed by the various sub-module of risks: BSCR = s X ij Corrij× SCRi× SCRj+ SCRintangibles, (3.6) where,

SCRintangibles= the capital requirement for the intangible asset risk;

Corrij = entries of the correlation matrix;

SCRi, SCRj = the capital requirements for the individual risk categories

According to the sub-modules of risks distinguished in Solvency II, the sub-modules of the Solvency Capital Requirements are SCRmkt, SCRdef, SCRintangibles, SCRlif e,

SCRnl, and SCRhealth. The life underwriting risk attracts most of our attention, for

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The sub-module for life underwriting risk is also scenario based and consists of the following sub-risks: mortality, longevity ,lapse, expenses, disability/morbidity and revision risk. The SCR for life risk is organized by the formula following.

SCRlif e=

s X

r,c

CorrLif er,c× Lif er× Lif ec, (3.7)

where CorrLif er,c = The entries of the correlation matrix,

Lif er, Lif ec= Capital Requirements for the individual life risks. The Life

correla-tion matrix defined by QIS 5 is as Table 3.1.

Table 3.1: Life correlation matrix

Life Risk Mortality Longevity Disability Lapse Expense Revision Catastrophe Mortality 1 Longevity -0.25 1 Disability 0.25 0 1 Lapse 0 0.25 0 1 Expenses 0.25 0.25 0.5 0.5 1 Revision 0 0.25 0 0 0.5 1 Catastrophe 0.25 0 0.5 0.25 0.25 0 1

The Available Capital (AC) corresponds to the amount of available financial re-sources. For measuring AC, different market- consistent valuation approaches are intro-duced. For example, the differences between the market value of assets and the market value of liabilities are discussed. For the valuation of assets, the market-to-market ap-proach is used for the typical investment portfolio of an insurance company when the market values are readily available, or when the market value can be derived from ob-servable market inputs cash flows. For the valuation of liabilities, the direct and indirect approaches are usually used. The direct method is a direct valuation of an insurance liability, based on the associated cash flows. In contrast, the valuation is calculated by considering the cash flows of the insurance business from the shareholders’ perspective in the indirect method. Since the indirect method presents a more practically accepted approach, we will focus on the indirect method, for the choice of method does not affect the results. Furthermore, the calculation of life insurance liabilities cannot be done di-rectly because of the complex financial structure of embedded options and guarantees, the mark-to-model approach based on Monte Carlo simulation is usually followed by insurance companies.

The Market-Consistent Embedded Value (MCEV) is introduced for assessing the market-consistent present value of a life insurance company’s assets and liabilities from the shareholders’ perspective, corresponds to the value of interest of shareholders in the distributable asset earnings from life insurance business (Bauer et. al 2010). Further, the market-consistent value of insurance liabilities can be calculated indirectly as the difference between the MCEV and the market value of assets. The AC under Solvency II principles is close to MCEV, so we assume AC to coincide with MCEV.

The MCEV is set as the difference between the sum of the Adjusted Net Asset Value (ANAV) and the Present Value of Future Profits (PVFP) and Cost-of-Capital (CoC), according to the CFO Forum (2008):

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The ANAV can be derived from the Net Asset Value (NAV), including adjusted intangible assets, unrealized gains and losses in assets, consisting of free surplus and required capitals. The ANAV can be calculated based on the balance sheet of a company in most cases. The PVFP represents the present value of post-taxation shareholder cash flows and the assets from the associated liabilities, which should be determined based on stochastic models, since the derivation is quite challenging. The CoC is the cost of the sum of required capital and residual non-hedgeable risks.

For calculation of AC1, we assume the profit for the first year, X1, is included here

before being paid to shareholders, AC1 = M CEV1+ X1. Under Solvency II principles,

the SCR is the smallest amount, based on which AC at t = 1 as seen from t = 0 is adequate with at least a probability of 99.5%. The SCR is defined as (cf. Bauer et al. 2010)

SCR = argminx{P (AC0− d(0, 1) · AC1 > x) ≤ 0.005)}. (3.9)

3.3 Longevity SCR Valuation

We will focus on longevity risk for its specific importance for the risk management in pension or life insurance. Most of the western world has experienced a steady increase in the average lifetime of its inhabitants over the past centuries, which can effect the mortality on pension costs potentially, also presents significant challenges for govern-ments as well as individual pension funds and life insurers. Although the past trends suggest further changes in mortality rate to be expected, a considerable uncertainty re-garding the future development of morality exists, so the major challenge is to account for the effects of this particular uncertainty on the costs of pensions properly. The equal importance of the accurate quantification and management of the risk in pension and insurance liabilities is aware of, while the focus is on the risk in financial investment for long, since interventions in the pension and insurance contracts could be mitigated but not eliminated. In project Solvency II, the valuation and management of pension and insurance liabilities has been put more emphasis on.

The SCRlongevity is computed as the change in liabilities due to a longevity shock

under the assumption that a permanent reduction of the mortality rates for all ages is 20%. The 20% is based on the regard of the insurance companies in the United Kingdom in 2004 consistent with 99.5% VaR concept of Solvency II ( CEIOPS (2007)). In QIS5, the capital requirement for longevity risk is described as the change in the net asset value due to a permanent decrease in the mortality rates of 20%.

The sub-module of longevity risk can be computed as the formula (Bauer et. al 2010):

SCRV aR_longevity = argminx{P (N AV0− d(0, 1) × N AV1 > x) ≤ 0.005}), (3.10)

where

N AVt = The Net Asset Value at time t, d(0, 1) = The discount rate for cash-flows

at time t = 1 discounted to t = 0, the NAV corresponds to the liabilities of all contracts which are exposed to longevity risk and their associate assets (B).

However, the SCR for longevity risk, is determined as the change in NAV resulting from a longevity shock at time t = 0 (cf. CEIOPS(2008a)). Specifically, in the current Solvency II standard model, the longevity shock is defined as a permanent 20% reduction of the mortality rates in each age.

SCRshock_long = N AV0− (N AV0|20%shock). (3.11)

This shock can express the systematic changes in mortality trend well, but does not include any small sample risk. For not blurring the results of numerical comparison

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of the mortality shock and VaR valuation approaches, the small sample risk in VaR approach is also disregarded here.

3.4 Model Setup

In this section, we describe the assumptions for the analysis of longevity risk under the Solvency II model, including the assumptions of interest rate evolution, the company’s asset strategy, the contracts which we concern, and also the best estimate mortality.

We set the assumptions for the company and the annuity as B¨orger (2010). The company which offers the contract that we concern is situated in the Netherlands. We set t = 0 in 2009. We assume that the company’s asset and Technical Provisions (TP) coincide at t = 0, for the approximation definition of SCR under the deterministic shock method coinciding with the exact definition of SCR, thus there is not any Excess Capital for this company. Further, the assumption that the company only invests in risk-free assets and the change in interest rates are completely hedged, so that the company is only exposed to longevity risk.

Thus the specific risk-free term structure of this company at time t can be derived from the term structure at 2009. The annual interest rate for maturity T at time t, t < T is denoted as i(t, T ). To ensure the asset cash flow coincides with the liability cash flow, we assume that the risk-free assets are only traded when premium payments are received or when survival benefits are paid. After all, the difference between the value of Technical Provisions (TP) and Assets at t > 0 is only due to the changes in expected mortality rate. At last, in the assumptions, we disregard operational risk (B). For the contracts, we consider life annuities which can pay a fixed amount of annuity yearly at the end of the year when the insured is still alive, but without any guarantees or options for death benefits. At the same time, we disregard any surplus or charges. So if an x0-year old individual at time t0 = 0 is still alive at time t, the liabilities of this

contract paying 1 will be BELt= X T >t 1 (1 + i(t, T ))T −t · Ep[T P (T ) x0 ]. (3.12)

Let CF1to denote the cash flow at t = 1, A0and A1to denote the asset at time t = 0

and t = 1 separately, based on the assumptions on payment dates and asset evolution, A1 = A0(1 + i(0, 1)) + CF1. (3.13)

The cash flow is negative when the company pay more benefits than the amount of premium it receives, and will be positive the other way around. The equation of cash flow implies N AV0− N AV1 1 + i(0, 1) = BEL1− CF1 1 + i(0, 1) − BEL0 ,

which can lead to the SCR formula SCRV aR= argminx{P (

BEL1− CF1

1 + i(0, 1) − BEL0 > x) ≤ 0.005}. (3.14) As well, the SCR formula for shock approach, can be

SCRshock = (BEL0|20%longevityshock) − BEL0. (3.15)

From the simplified SCR formulas for both approaches, we can observe that, the SCR in the VaR approach only depends on the expected mortality at the end of the first year. So it is not a significant difference whether the changes in mortality rate

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emerging gradually over the year, or happening as a one-off shock at t = 0. Thus a deterministic shock at time t = 0 can express all possible gradual mortality change, moreover, the expected survival probabilities at t = 0 can be transformed in to the one at t = 1. After all, it is feasible to express a SCR for longevity risk in VaR approach by a one-off shock, shockV aR_(B¨_{orger 2010) then}

SCRV aR= argminx P BEL1− CF1 1 + i(0, 1) − BEL0 > 0 ≤ 0.005 = BEL1− CF1 1 + i(0, 1) |shock V aR − BEL0

= (BEL0|shockV aR) − BEL0

= SCRshock.

Therefore, if the 20% reduction longevity shock coincides with a 200-year scenario for the mortality evolution in one year, the SCRs computed in the Shock Approach and the VaR Approach should also correspond to each other. While on contrast, if the SCR computed in the Shock Approach and the VaR Approach differ significantly, the 20% deterministic shock can not be a realistic approximation of VaR.

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Chapter 4

The Mortality Model

For the computation of Solvency Capital Requirement for longevity risk by the VaR method, a forward stochastic mortality model is needed. A lot of approaches for mod-eling mortality are proposed in the past literature. The models framed in discrete time mostly focus on statistical analysis and projection of annual mortality data, the other models are framed in continuous time. In this chapter, we introduce the famous Lee Carter model for forward mortality rate estimation, including the basic building of mortality, the assumptions for the Lee Carter Model and the method for measuring the model.

4.1 Model Requirements

A stochastic mortality model is required for the computation of longevity SCR from the VaR approach. Various stochastic models are proposed on literature. Carins et al. (2008a) overviews a wide range of extrapolative stochastic mortality models over the last 20 years,including discrete time model facilitating valuation of mortality-linked contracts, continuous time mortality model for examination of the potential dynamic hedging and a range of financial instruments possible for hedging morality and risk. The class of spot model where only realized period mortality is modeled is the most common used mortality model. Furthermore, the spot models contain a mortality trend assumption, which are used for anticipated changes in mortality over time. But in most mortality models, the trend is fixed as part of the calibration and the scenarios of realized mortality are derived as random deviations from the mortality trend, which means the liabilities at t = 1 are always computed based on the same trend assumption, so the models with fixed trend assumption do not account for the second component of the longevity risk, the risk that the mortality rate will be beyond the deducted expected mortality next year.

The most common used mortality model is a kind of the spot models only modeling the realized mortality. The spot models include an assumption about mortality trend, for containing anticipated changes over time. The Lee and Carter (1992) model is the most well-known of the mortality models, in which the force of mortality is composed by an age effects and an combination of age effect and random period effect. Specifically, the random period effect in the Lee-Carter model follows a random walk process with fixed drift. While the Lee-Carter model serve a useful purpose, it still has drawbacks. For example, the Lee-Carter model is a one-factor model, leading to mortality improvements at all ages with perfect correlation; the factor for setting the level of future death trend was lower at high ages, which could be underestimated for higher ages; the basic version of the Lee-Carter model could result in lack of smoothness in the estimated age effect etc (Carins et. al 2008a).

Some observed that the drift for the random period effect in Lee-Carter model is not constant over time. This mortality trend is always fixed in most spot models, and the

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random deviation scenarios for mortality rates are derived from this trend. So these spot models only account for one side of the longevity risk, that the mortality rate would not realize as the expected level. Cox et al.(2009) improved the Lee-Carter model by allowing for trend changes, a Markov regime switching model with assumption of different trend and volatility in two regimes. While in the Markov chain model, the long-term mortality trend depends on the current mortality regime and possible regime switching in the next year, but the trend is relatively fixed because the distribution of the Markov chain is quite stationary. In Bauer (2008), a stochastic mortality model is brought out as BBRZ model, where a stochastic differential equation is applied to realized survival probability. The forward mortality model is fully specified by the volatility and the initial curve. B¨orger et al. (2013) brought up a stochastic process for trend of mortality, which explicitly models changes in the long-term mortality trend assumption over time satisfied the SST (Swiss Solvency Test) criteria. This enables to quantify mortality and longevity risk over the one-year time horizon.

While on the other side, some brought up new framework for dividing long-term longevity trend risk into one-year horizon, instead of estimating the trend of mortality risk.Richards et al. (2012) is a paper on the quantification of mortality and longevity risk under solvency II, describing different approached for quantifying longevity risk over a one-year time horizon via standard mortality models such as Lee-Carter or Carins-Blake Dowd. They argued that a number of components are contained in longevity risk, of which trend risk is just one part. While the longevity trend risk faced by insurers and pension schemes could be considered as a long-term accumulation of small changes, the adverse trend in a long-term could be added up.

To assess the quality of stochastic mortality models, Carins et al. (2008a) defined criteria for models to be evaluated:

• Mortality rates should be positive

• The model should be consistent with historical data.

• Parameter estimates should be robust relative to the period of data and range of ages employed.

• Model forecast should be robust relative to the period of data and range of ages employed.

• Forecast levels of uncertainty and central trajectories should be plausible and consistent with historical trends and variability in mortality data.

• The model should be straightforward to implement using analytical methods or fast numerical algorithms.

• The model should be relatively parsimonious. The principle of parsimony refers to the rules that among competing hypotheses, the one with the fewest assumptions should be selected (Carins et. al 2009).

• It should be possible to use the model to generate sample paths and calculate prediction intervals.

• The structure of the model should make it possible to incorporate parameter uncertainty in simulations.

• At least for some countries, the model should incorporate a stochastic cohort effect.

• The model should have a non-trivial correlation structure. The ability to produce a non-trivial correlation structure refers to the correlation between the year-on-year

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changes in mortality rates at different ages. (Carins et. al 2009 ) The changes in the mt,x at different ages are found to be imperfectly correlated. Also the existence of

a non-trivial correlation structure implies that more than one hedging instrument is needed for hedging of longevity-linked liabilities.

Another important criterion is the model is applicable for the whole age range (Plan 2009). The existing models discussed in this chapter meet most of the criteria above, however, there is no existing model meet all the criteria above.

For our analysis in the following sections, the original Lee-Carter model is used, since the Lee-Carter model turns out to be a most useful basic model. Also the reason that the most important component of the longevity risk is the bias from froward expected mortality rates is considered. As well the long-term longevity risk is the insurance com-panies, annuity providers and the pensions face to, the fluctuation of the mortality trend would be assumed as added up in the long-term mean of trend.

4.2 Lee and Carter Model

The availability of annual data in integer year and individual integer ages make is relatively straightforward to fit discrete time models to data in statistical methods. The most early and most generally used discrete time model is the Lee-Carter Model, proposed by Lee and Carter (1992). The Lee Carter model has also been used as a kind of standard model in literature on mortality forecasting. We used the Lee Carter model here to model the forward mortality rate.

In the Lee-Carter model, the central death rate mt,x for age x in time t is modeled

as

ln(mt,x) = αx+ βxκt+ t,x, (4.1)

where sets of constants {αx} and {βx} are age effects and κt is a random period

effect. αxand βxare time-independent parameters, {κt} is a one-dimensional underlying

time-dependent potential process for the evolution quantification of mortality over time. αx denotes the level of log central death rate of age x, the parameter βx denotes the

improvement rate at age x and κt is an index of the level of mortality. The variations

around the systematic trend for specific age and time is showed by error term t,x.

t,x has mean 0 and variance σt2, t,x and s,y are independent when t differs from s

and x differs from y. The t,x represents particular age-specific historical influences not

captured by the model.

The period effect, κt follows a random walk with drift, which is an ARIMA(0,1,0)

process,

κt= κt−1+ γ + εt, (4.2)

where ε is an error term with normal distribution and zero mean, and γ is a constant drift. εtis an error term, with zero mean, and εt,xand εs,yare independent when t differs

from s and x differs from y. For fitting the Lee-Carter model to historical mortality data, the maximum likelihood algorithm is used.

By assuming that αx and βx to be stable in time, the Lee-Carter model is reduced

to a one-dimensional time series. For obtaining the volatility, Equation (4.2) could be expressed as

dt= γ + εt, (4.3)

in terms of

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By reshaping Equation (4.4), we could obtain that the differentiation of ln mx,t is

also Gaussian.

ln µdt,x= βx· dt+ t,x. (4.5)

However, there are some disadvantages of the Lee-Carter model (Plan 2009, Cairns et al. 2008a):

• The Lee-Carter model is a one-factor model, resulting in mortality improvements at all ages being perfectly correlated (trivial correlation structure).

• For countries where a cohort effect is observed in the past, the model gives a poor fit to historical data.

• The uncertainty in future death rates is proportional to the average improvement rate βx. For high ages this could lead to uncertainty being too low, since historical

improvement rates have often been lower at high ages.

• The basic version of the Lee-Carter model could result in a lack of smoothness in the estimated age effect βx.

• Lee-Carter model has trivial correlation. The correlation structure is called trivial when there is perfect correlation between changes in mortality rates at different ages from one year to the next. In the Lee-Carter model, the single time series process has perfect correlation at all ages.

While there are some literature on additions or modifications of the Lee-Carter model, but none of them could cover all disadvantages above.

Historical data is available in Human Mortality Database. We set Dx,tas the number

of death, Ex,t as th exposure, so the mx,t could be modeled as in Equation 4.1. The

mortality rates in the Netherlands from 1940-2009 is shown in Figure 4.1. The best estimate T-year force of mortality at time t is sufficient to obtained by using R-code of the software package ”forecast” and ”demography”.

AGE[1:100] ANNEE

log(MUT[1:100, ])

Figure 4.1: NL mortality rate 1940-2009 (Human Mortality Database)

The best estimate for liabilities could be calculated by the Lee-Carter model, with the mortality trend following ARIMA process with fixed drift, by using the historical data in years 1940-2009. The Figure 4.2 shows the log death rates for all ages in the year 1940, 1950, 1960, 1970, 1980, 1990 and 2009. While by observation, the mortality

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rate decreased in a significant level from the year 1940 to 2009 as shown in Figure 4.2. The mortality rates for all ages become smaller significantly, especially for young ages. By using the mortality data from Human Mortality Database to establish the model, we observed that the volatility of the ARIMA(0,1,0) model is dramatically large for the years 1940-2009, which is not realistic for further simulation. It could be considered as the result of significantly improved mortality rates in the middle of 20th century (1940-1970).Figure 4.3 shows the log death rates for individuals at the age of 55, 65, 75, 85 and 95, an obvious downward trend of the mortality rates for all ages could be observed, and the mortality rates are significantly lower for the year after 1970. A drastic fluctuation of log death rates in 1945-1950 could be observed, increased first then dropped shrewdly. The male mortality rates increased gradually during a bit year 1950-1970, which are relatively obvious for age 55, 65 and 75, even the ensemble trend was gliding.

For choosing a preferable time period for the estimation and forecast, we compare the estimation outcomes of ARIMA (0,1,0) models for different period (1940-2009, 1950-2009, 1960-1950-2009, 1970-2009) as shown in Table 4.1 and Table 4.2. In these two tables, the drift in the ARIMA(0,1,0) models, the standard error, the log likelihood, the AIC, the AICc and the BIC are showed. The ARIMA(0,1,0) models with drifts are actually the random walk with drifts, where the drifts are the coefficients of the interceptions. The standard errors refers ti the square roots of the estimated error variance of the quantity, which could represent the goodness of fit, but also related to the size of estimated data. So we could observe that the standard error grows as the size of data grows. The volatility of these stochastic models are shown as σ2, which get smaller as the data size get smaller. The ARIMA models are fitted by maximize the log-likelihood of the, and the results are shown as log-likelihood the Table 4.1 and Table 4.2. The log-likelihood is getting larger as the data size decreases.

The AIC, the AICc and the BIC are three model selection criteria. AIC, Akaike information criterion is a measure of the relative quality of a statistical model, which deals with the trade-off between the goodness of fit and the complexity of the model (Akaike 1974). AIC value is AIC = 2k − 2ln(L), where k is the number of parameters in the model, and L is the maximized value of the likelihood function for the model. Thus AIC rewards goodness of fit, also includes a penalty that is an increasing function of the number of estimated parameters, which discourages over-fitting. Given the set of candidate ARIMA(0,1,0) models, the preferred model should be the one with the minimum AIC value (Akaike 1974). AICc is AIC with a correction for finite sample sized, AICc = AIC + 2k(k+1)_n−k−1, where n denotes the sample size. The AICc is AIC with larger penalty for parameters, while converges to AIC as n gets large as in these ARIMA models (Burnham and Anderson 2002). The Bayesian information criterion (BIC) is a criterion for model selection among a finite set of models (Schwarz 1978). BIC = −2 ˙ln bL + k ˙ln(n), where bL is the maximized value of the likelihood function of the model. Both BIC and AIC resolve the over-fitting problem that the likelihood was increasing by adding parameters, while BIC has a larger penalty term. Also the model with smaller BIC is preferred.

The outcomes for both male and female show that the goodness of fit of ARIMA(0,1,0) models are significantly improved for shorter period, which can be observed from all model selection criterion AIC, AICc and BIC. All three criterion have smaller result when the data size gets smaller. Also the volatility becomes evidently smaller as the data period turns to be shorter, because of smaller data size and more stable mortality rates level. So the data in years 1970-2009 would be determined as the base period of data for the Lee-Carter model estimation, forecast and further simulation.

After the estimation of the Lee-Carter model, a serious of estimated parameters αx, βx, the age effects for ages x = 0 − 110 for both male and female are obtained

(See Appendix A.). The estimated time-dependent potential process κt are obtained

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18 Shu Zhang — Analysis of Solvency II SCR Longevity Risk Valuation 0 20 40 60 80 100 −6 −4 −2 0 Age(1940)

Death Rate (log)

0 20 40 60 80 100 −8 −6 −4 −2 Age(1950)

Death Rate (log)

0 20 40 60 80 100 −8 −6 −4 −2 0 Age(1960)

Death Rate (log)

0 20 40 60 80 100 −8 −6 −4 −2 0 Age(1970)

Death Rate (log)

0 20 40 60 80 100 −8 −6 −4 −2 0 Age(1980)

Death Rate (log)

0 20 40 60 80 100 −8 −6 −4 −2 0 Age(1990)

Death Rate (log)

0 20 40 60 80 100 −10 −8 −6 −4 −2 0 Age(2000)

Death Rate (log)

0 20 40 60 80 100 −10 −8 −6 −4 −2 0 Age(2009)

Death Rate (log)

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Analysis of Solvency II SCR Longevity Risk Valuation — Shu Zhang 19 1940 1950 1960 1970 1980 1990 2000 2010 −5.6 −5.4 −5.2 −5.0 −4.8 −4.6 −4.4

Death Rate in year 1940−2009

Age 55 Year

Death Rate (log)

1940 1950 1960 1970 1980 1990 2000 2010 −4.8 −4.6 −4.4 −4.2 −4.0 −3.8 −3.6

Age 65 Year

Death Rate (log)

1940 1950 1960 1970 1980 1990 2000 2010 −3.8 −3.6 −3.4 −3.2 −3.0 −2.8 −2.6 −2.4

Age 75 Year

Death Rate (log)

1940 1950 1960 1970 1980 1990 2000 2010 −2.4 −2.2 −2.0 −1.8 −1.6 −1.4

Age 85 Year

Death Rate (log)

1940 1950 1960 1970 1980 1990 2000 2010 −1.4 −1.2 −1.0 −0.8 −0.6

Age 95 Year

Death Rate (log)

Figure 4.3: Death Rate(log) in for age 55,65,75,85,95 (Female:red; Male:blue)

Table 4.1: Outcomes of ARIMA(0,1,0) model (Male) 1940-2009 1950-2009 1960-2009 1970-2009 Drift -2.4255 -1.6813 -1.7128 -2.1265 s.e. 1.4222 0.5240 0.5406 0.5205 σ2 139.6 16.2 14.32 10.57 log likelihood -263.9 -162.57 -131.49 -98.22 AIC 531.81 329.14 266.98 200.44 AICc 531.99 329.35 267.24 200.77 BIC 536.27 333.29 270.77 203.77

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Table 4.2: Outcomes of ARIMA(0,1,0) model (Female) 1940-2009 1950-2009 1960-2009 1970-2009 Drift -2.2683 -1.6572 -1.6066 -1.8962 s.e. 0.8764 0.3647 0.4099 0.4987 σ2 52.99 7.847 8.232 9.7 log likelihood -230.98 -141.55 -118.21 -96.6 AIC 465.95 287.1 240.41 197.19 AICc 466.14 287.31 240.67 197.52 BIC 470.42 291.25 244.2 200.52 2010 2030 2050 0.000 0.004 0.008 0.012

projectd Mortality Rate (Male)

Age 65 Year Expected Mor tality Rate 2010 2030 2050 0.000 0.004 0.008

Projected Mortality Rate(Female)

Age 65 Year

Expected Mor

tality Rate

Figure 4.4: Projected Mortality Rates

Appendix A.). To get the future mortality table, the forecast of future time effects κt

for years t = 2010 − 2059 should be obtained. The expected time effects κtfor the future

50 years and the 99.5% certainty forecast of time effect could be got from the result of forecasting ARIMA(0,1,0) models for future 50 periods (See Appendix A). Then by applying the mean of forecast result of κt for years 2010-2059 (See Appendix A)and the

fixed age effects αx and βx into the Lee-Carter model framework, the mean projection

of the mortality rates for age 0-100 in the yeas 2010-2059 are obtained. Also the 99.5% certainty projection could be obtained by applying the 99.5% forecast result of time parameter κt with the age effects into the Lee-Carter model framework. Figure 4.4

shows the forecast expected mortality rates and the 99.5% certainty forecast mortality rates for individuals at the age of 65 for both male and female. The forecast mortality rates for female are much lower than those for male.

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Chapter 5

Comparison of SCR

In this chapter, we compare the numerical result of SCR for longevity risk via the VaR approach and the shock approach, and investigate whether the 20% shock on mortality rate is a good approximation for the 99.5% VaR valuation of the SCR. If not, what the deficit is for the shock approach.

We start the analysis by assuming a life annuity of 1000, for a 65-year old male and a 65-year old female. The liabilities and the SCRs for the shock approach and the VaR approach are given in the Table 6.1 and we observe that for both gender the shock approach requires much more capital than the 99.5% VaR approach. The deviation between the standard formula and the VaR approach is shown in column 3, while the deviation of that in relation to the best estimate liabilities at time t = 0 is also significant. While, the difference of SCR derived by the two approaches is smaller for female than that for male.

Table 5.1: SCRs for 65-year old male and female BEL0 SCR SCR/BEL0

Shock approach Male 13957.75 984.25 7.0% VaR approach Male 13957.75 661.51 5.0% Shock approach Female 16602.29 918.20 5.5% VaR approach Female 16602.29 932.60 6.0%

We investigate more if the certain parameters vary in our computations, what the comparison result will turn to. The parameters may influence the computation result like age, the best estimate survival probabilities, or the payment dates of the survival benefits.

For study of the sensitivity of the SCRs via these two approaches, especially the sensitivity of the Lee-Carter model on the different length of historical data. We use different time period of historical mortality rates data for the Lee-Carter Model estima-tion, forecast and future mortality table projection as the same method we described in Chapter 4. Table 5.2 shows the SCRs comparison result for Dutch male based on the data from time period 1940-2009, 1950-2009, 1960-2009, 1970-2009 and 1980-2009. Table 5.3 shows the SCRs comparison for Dutch female.

From Table 5.2, we could observe that as we extend the length of historical data period, the best estimation of liabilities do not have so dramatic change, instead, the liabilities keep in a relative stable level. While the SCRs via both approaches get smaller as the basic mortality rates data period get shorter, shows that the longer period data contains more longevity exposure. However, the SCRs via VaR approach shrink more rapidly as the basic data period get shorter. Since the shorter basic data period exclude

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the dramatically fluctuate data in the 1940-1950, become more stable as we discussed in Chapter4, result in a smaller volatility. The change in volatility would effect the calculation of VaR, so the SCR via VaR approach goes down steeply. The ratio of the difference between SCRs via two approaches to the best estimation of liabilities also grows when the length of data period get shorter. The SCR via shock approach is considered to be the result in capital of a fixed deduction to the mortality rates, which could not reflect the subtle effect of the change of volatility to the mortality rates.

For Dutch female mortality rates, the results of SCRs based on the data in different period are significantly different from those for males. From Table 5.3, we observe that the best estimate of liabilities are also relative stable as the change of data period, and the SCRs via both approaches get smaller as the basic mortality rates data period get shorter too. However, the SCRs via VaR approach are larger than those via shock approach, except for the ones with smallest data size. The SCRs difference via the two approaches suggest that the volatility for female mortality rates has more effect on the VaR evaluation. Further, the SCRs via shock approach for female data give a relative better approximation for the SCRs via VaR approach, which could be observed in the eighth column, from the fact that the ratios of the differences between the SCRs to the best estimated liabilities are getting close to zero as the data size decrease.

Besides the length of data period, the trend of mortality rates after 2009, the end time point we choose, could also effect the result of SCRs. Based on the fact that after 2009, the Dutch male mortality rates were improved more than those for Dutch female, the effects of different mortality rates trends on the calculation of SCRs attract our interests. To get a relative obvious SCRs results of the different mortality trends effects, we assume that the mortality rates of a Dutch male at the age of 65 were improved by 0.7%, a 65-year old female’s mortality rates were improved by 0.5% (AG 2012), based on the assumption of Prognosetafel 2012-2016. The mortality rates in year 2009-2014 would be added to the basic table under the previous assumption. Then in the same way the expected mortality rates and the 99.5% certainty forecast mortality rates for

Table 5.2: SCRs for different data periods(Male)

Time Period BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ _SCR∆SCR_{V aR} ∆SCR_BEL₀

1940-2009 13837.76 1107.43 8.0% 2091.33 15.1% -46.0% -7.1% 1950-2009 13695.21 1055.44 7.7% 737.88 5.4% 43.0 % 2.3% 1960-2009 13741.93 1015.57 7.4 % 697.89 5.1% 45.5% 2.3% 1970-2009 13957.75 984.25 7.1 % 661.51 5.0% 48.8% 2.0% 1980-2009 14166.95 968.75 6.8 % 656.32 4.6 % 47.6% 2.0 %

Table 5.3: SCRs for different data periods(Female)

Time Period BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ _SCR∆SCR_{V aR} ∆SCR_BEL₀

1940-2009 16540.63 957.60 5.8% 1975.81 11.9% -51.5% -6.2% 1950-2009 16579.39 931.56 5.6% 968.78 5.8% -3.8 % -0.2% 1960-2009 16570.98 929.47 5.6 % 988.42 5.9% -5.9% -0.4% 1970-2009 16602.29 918.20 5.5 % 932.60 6.0% -1.5% -0.1% 1980-2009 16423.92 898.02 5.4 % 883.94 5.3 % 1.6% 0.0%

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the future 50 years are obtained by the Lee-Carter Model, added up with the 5 years mortality rates from 2010-2014 we assumed, the SCRs via both approaches would be obtained. Since the time trend of the mortality rates are changes, so the time effect factors in the Lee-Carter model would be affected. The comparison results of SCRs could give a suggest on how the time-serious factors affect the future mortality rates projection.

Table 5.4: SCRs for 65-year Old Male with 5 Extra Years Mortality Rates Improvement Mortality Improve BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ ∆SCR_BEL₀

1970-2009 Male 13957.75 984.25 7.0% 661.51 5.0% 2.0% 1970-2014 Male 15849.02 985.19 6.2% 1040.75 7.0% 0.0% 1970-2009 Female 16602.29 918.20 5.5% 932.60 6.0% -0.0% 1970-2014 Female 16930.78 935.83 5.5% 780.26 4.6% 0.9%

Table 5.4 shows the best estimation of liabilities and the SCRs via the shock ap-proach and the VaR apap-proach with the 5 extra years mortality rates. As the mortality rates improved, the best estimation of the annuity liabilities get larger obviously, which suggest that the time trend factors take important positions in the Lee-Carter model. An obvious change during time would affect the estimation of the time serious factors significantly, as a result, the forecast projected mortality rates which are based on the time serious factors in the Lee-Carter model change evidently. The SCRs for male via the shock approach do not change significantly, while the SCRs for male via the VaR approach increased dramatically, which coincides with the fact that the male mortality rates are much improved. On the other side, the SCRs for female via shock approach increased, however, the SCRs via VaR approach for female decreased, which also ap-proves the fact that the female mortality rates are not improved much in the recent few years. Nonetheless, the SCRs for female via two approaches are still consistent. The fixed 20% deduction of the mortality rates lead to a relative stable mortality rates, with little unexpected risk exposure. However, the VaR approach focus on the 99.5% proba-bility false related to the volatility, which would be a relative large one in the obviously improved mortality rates trend. So the SCRs via the VaR approach reflect the mortality trend changes in a better way.

In order to analyze the SCRs by shock approach and VaR approach for different ages, we consider ages between 55 and 95, in specific 55, 65, 75, 85 and 95 as in Table 5.5 and Table 5.6. In the third column and the fourth column, we could observe that the SCR via shock approach increases first, then decreases with age, while the SCR relative to liabilities BEL0 is growing up for old ages. The growing of SCR relative

to liabilities BEL0 due to the structure of longevity shock, the reduction of 20% shock

increases with age as the mortality rates are larger for old age obviously. In the fifth and sixth column, the SCR via VaR approach decreases with age, while the SCR relative to the liabilities up first then goes down with age.As we use the Lee Carter model, and apply the fixed volatility to κt, the volatility for each time t is fixed. While the

mortality increased with age, so the ratio of VaR for mortality to mortality rate in the same time t would be smaller for larger mortality. For the last column, we could observe that the difference between the SCR via shock approach and VaR approach relative to best estimate liabilities increase significantly with age growing.

For Female, the SCRs for different ages are in Table 5.6. The same pattern of change of SCR could be observed for female as for male. The SCR via shock approach and SCR relative to best estimate liabilities BEL0 change in the same way as those for male,

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than those for male, due to the fact that the mortality rates for female in those ages are smaller than male, thus the reduction of 20% shock is smaller also. The ratio of the difference between SCRs via the two approaches to the best estimate liabilities increases more dramatically, since except for extremely old age (90 or 100), the female mortality rates are obviously smaller than male mortality rates for old ages (55-85).

The shortcoming of the shock approach which came up in the standard model seems to be structural, since the reductions not related to ages lead to more required capital for old ages. Thus an age-dependent longevity stress which has smaller reduction rates for old ages could be a better approximation. According to the findings that the num-ber of relevant causes of death is larger for old ages than for young ages (Tabeau et al.(2001)), means that older people are like to die for more causes of death. So dramat-ically reductions are more rarely to appear for old ages.

In the Table 5.7, the SCRs under various mortality levels for 65-year old male are showed. The different mortality levels are obtained by shifting the mortality rates in the basic mortality table for male by -20%, -10%, +10%, +20% separately, then new projected tables would be obtained for the expected forecast and the 99.5% certainty forecast for the future 50 years. Then the SCRs via the shock approach and VaR ap-proach are compared for examining the property of the 20% deduction approximation for the 99.5% VaR risk exposure for longevity risk.

As shown in the Table 5.7, the SCRs via shock approach increase significantly with the mortality rates level going up, while the SCRs via VaR are relatively stable, when the mortality rates change a lot. The ratio of the difference between the SCRs via two approaches to the best estimate liabilities fluctuate according to the mortality level. However, the ratio of the difference between the SCRs via the two approaches to the SCR via VaR approaches is decreasing. The reason for the increasing SCRs via shock approaches could also be illustrated as the structure of shock, the fixed shock rate leads to larger reduction for higher mortality rates, while smaller reduction for lower mortality rates. But the ’relative volatility’ assumption for shock approach could not compensate for the long-term risk due to longer survival for small mortality rates. While the volatility in the Lee-Carter model does not has the exact fraction feature with mortality rates,

Table 5.5: SCRs for different ages(Male)

Age BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ ∆SCR_BEL₀

55 19458.36 773.69 4.0% 826.33 4.2% -0.3% 65 13957.75 984.25 7.0% 661.51 5.0% 2.0% 75 8373.50 901.31 10.8% 299.37 3.5% 7.1% 85 4090.75 638.12 15.6% 59.4 1.5% 14.1% 95 1665.43 320.60 19.3% 7.62 0.46% 18.8%

Table 5.6: SCRs for different ages(Female)

Age BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ ∆SCR_BEL₀

55 21219.10 593.35 2.8% 724.14 3.4% -0.6% 65 16602.29 918.20 5.5% 932.60 6.0% 0.5% 75 10822.55 916.74 8.5% 650.16 6.0% 2.5% 85 5413.16 726.26 13.4 % 238.03 4.4% 9.0% 95 2059.45 402.27 19.5% 32.4 1.6% 17.9%

(31)

so the SCRs derived from VaR with the fixed volatility does not grow as the SCRs by shock approach.

Table 5.7: SCRs for different mortality levels(Male)

Mortality shift BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ _SCR∆SCR_{V aR} ∆SCR_BEL₀

-20% 14983.68 984.51 6.6% 629.20 4.2% 56.5% 2.4% -10% 14442.34 985.64 6.8% 647.07 4.5% 52.3 % 2.3% Original 13957.75 984.25 7.0 % 661.51 5.0% 48.8 % 2.0% +10% 13519.67 981.45 7.3 % 673.24 5.0% 45.8% 2.3% +20% 13120.29 977.87 7.5 % 682.77 5.2 % 43.2% 2.2 %

Table 5.8: SCRs for different mortality levels(Female)

Mortality shift BEL0 SCRshock SCR_BELshock₀ SCRV aR SCR_BELV aR₀ _SCR∆SCR_{V aR} ∆SCR_BEL₀

-20% 17549.33 883.17 5.0% 880.81 5.0% 2.6% 0.0% -10% 17053.63 903.28 5.2% 909.63 5.3% -0.6 % -0.0% Original 16602.29 918.20 5.5 % 932.60 6.0% -1.5 % -0.0% +10% 16188.49 929.46 5.7 % 951.06 5.8% -2.2% -0.1% +20% 15806.75 938.09 5.9 % 965.97 6.1 % -2.8% -0.2 %

As shown in the Table 5.8, the SCRs in different mortality rates for Dutch female are shown. The best estimated liabilities decrease significantly as the mortality rates level grows up, while the SCRs via both approaches increase for higher mortality rates level. The SCRs via two approaches coincide for all mortality rates levels. The results of SCRs for female show that the SCRs via shock approach could always give good approximations for the SCRs via VaR approach for a 65-year old Dutch female. However, trivial difference could be observed for the SCRs, that the differences between the SCRs via two approaches are smaller for lower mortality rates level than those for higher mortality rates level. The smaller difference fact also exists for Dutch male data, which could suggest that the SCR via shock approach would give better approximation for SCRs via VaR approach for lower mortality rates level.

An obvious shortcomings of the shock approach could be observed, that the shock approach derives the SCR for longevity risk with fixed deduction, which would lead to too much required capital for old ages. Also the SCRs via fixed mortality rates deduction method are not as sensitive as the SCRs via VaR approach to the length of historical data period or the time trend of mortality rates. The structural shortcoming of the standard model could be modified by adding scaling factor to the shock structure and by relating the shock structure to age of individual. For further study of the shortcomings of shock approach for longevity risk in the standard model, one could investigate in the effect of shift or change of yield curve, the effect of annuity payment date on the SCR via shock approach and the VaR approach.

Analysis of Solvency II SCR Longevity risk valuation : standard model and stochastic model