Health SLT risks for income protection insurance under Solvency II

Transcription

1 riss for income protection insurance under Solvency II Nataliya Boyo Mei, 2011 Master thesis Drs. R. Bruning AAG Faculty of Economics and Business University of Amsterdam

2 Content 1 Introduction 2 2 riss for income protection insurance under Solvency II From Solvency I to Solvency II Quantitative Impact Studies SCR under QIS5 9 3 Quantitative determination of the capital requirement Theoretical bacground of the model Description of the model Model assumptions Input data Transition probabilities and transition matrices Best estimate Present value of liabilities Present value of premiums Present value of expenses Capital requirement mortality 30 longevity 30 disability expense revision 33 lapse Analysis of the results 39 5 Conclusions 55 References 58 Appendix 1 Input data 60 2

3 1 Introduction Currently the term Solvency II is well nown in the insurance branch. Solvency II is the latest solvency regime which has been developed by the European Insurance and Occupational Pensions Authority (EIOPA) under supervision of the European Commission (EC). The purpose of this new framewor is to design a European standard approach which will reflect an insurance company s full ris spectrum to limit the ris of insolvency. The EC proposed date of the Solvency II implementation is January The main reason for developing this new framewor is that the current regime Solvency I is too simple and doesn t allocate capital adequately to where the riss are. Capital requirements, defined as a simple blanet solvency margins, were intended as buffer to absorb potential riss, to act as a policyholder protection measure, but they were quite simplistic in design and didn t always reflect the actual riss in a given portfolio of insurance business (CEA, 2007, p.3). Solvency II is based on the economical principles for the measurement of the assets and liabilities. It means that the assets and liabilities must be held at maret consistent value. If there is no liquid maret for some components of assets and liabilities, then the appropriate models and maret consistent valuation techniques must be applied. Moreover, Solvency II introduces a ris oriented level of required capital nown as Solvency Capital Requirement (SCR) that enables an insurance company to cover significant unforeseen losses. Introducing Minimum Capital Requirement (MCR) the new solvency regime provides strict supervisory measures. The MCR is a threshold at which an insurer is not allowed to sell policies. Being below the SCR a company may need to discuss a recovery plan with its supervisor. Under Solvency II the insurers are given an option to use either a standard approach, nown as European Standard Formula, or Internal Model to calculate the SCR. The standard approach to the SCR can be applied by any insurer irrespective of size, portfolio mix and geographical location. To apply an internal model companies must have an approval from their regulators first. Five Quantitative Impact Studies (QIS) were carried out in order to investigate the effect on the proposals of the standard approach to calculate the 3

4 SCR on the company and the industry level. The results and reactions of the participants were directed into the development of the new solutions and calibrations. The recent fifth study, QIS5, has been conducted in July-November After each study the European Standard Formula was calibrated and tested again in the next one. For example the recent financial crisis forced adjustments in QIS5 about an illiquidity premium within maret ris module. Further, in QIS5 for the first time the health ris module has been split into three sub-modules to cover business calculated similar to life techniques ( riss), business calculated similar to non-life techniques ( non- riss) and catastrophe ris. This paper is devoted to the sub-module for the riss in the European Standard Formula. The objective of the paper is to quantify and to compare the riss for a long term income protection insurance contracts financed either with standard or ris premium. What riss is income protection insurance exposed to? What ris is most significant? Which financing system does require more solvency capital: standard tariff or ris tariff? The main goal of the research is to give the answers on these questions. Under income protection insurance one considers a health or life insurance policy that provides a single or periodic payment to replace lost income when the insured is unable to wor because of illness or injury (CEA, 2007, p. 22). There have been a lot of actuarial models for disability insurance used in the international insurance maret. The first explicit applications of the mathematics of multistate models (Marov and semi-marov process) to disability benefits have appeared in the scientific literature in the late sixties of the last century (Starr (1965), Janssen (1966)). The majority of the Dutch insurers use a model nown as a Marovian model for select probabilities. In this model there is a set of disability states which are split according to duration (Pitacco, 1995). In this paper a multi-state model for individual long-term disability insurance is represented, based on a discrete stochastic Marov process. The present values of the future liabilities and future premiums are determined according to actuarial model proposed by Promislow (Promislow (2006)).The majority of the assumptions and the input data such as disability and recovery rates are based on the traditional model for the disability insurance developed by the Actuarial Commission AOV 1. The insurance policy design used for the 1 AOV (arbeidsongeschitheidsverzeering) is a Dutch acronym for disability insurance 4

5 quantitative research is based on the disability insurance contract of one of the leaders on the Dutch income protection insurance maret, ING Movir, which sells the disability insurance contracts to high educated self-employed professionals. The model determines the capital requirements according to the standard approach of Solvency II and allows observing their changes depending on the financing system (standard premium or ris premium), disability rates, interest rates, profitability of the contracts etc. This paper is structured as follow. The theoretical bacground of the new solvency framewor is given in the second chapter. The third section describes the model used for the quantification of the riss. The analysis of the results is given in the fourth chapter. The last chapter gives a short conclusion of the research. 5

6 2 riss for income protection insurance under Solvency II This chapter gives a theoretical bacground of the Solvency II regime. The first subsection describes the main differences and the advantages of the new framewor compared to the old one, Solvency I. It also gives a short description of the basic principles of Solvency II and their implementation. The second part of the chapter gives brief information about the way the SCR standard approach was developed and tested in the quantitative impact studies. The last section is devoted to the crucial point of Solvency II, SCR itself and one of its elements the Underwriting Ris Module. 2.1 From Solvency I to Solvency II Solvency II is a new regulation on capital requirements, valuation principles and ris management. The main purpose of Solvency II is to introduce a more ris sensitive solvency regime, encouraging adequate ris management by insurers. The ey concepts are ris sensitivity, harmonisation and transparency. Under Solvency I there is room for national variations which has caused differences in local standards. The new regime provides more detailed requirements that will provide policyholder s protection across the European countries. The basic principles of Solvency II are maret valuation and ris sensitivity. The balance sheet is based on maret value and provides a realistic estimate of the insurer s solvency position. Where maret valuation is not possible it must be approximated on the basis of models. Figure 1 depicts the crucial differences between the old and new regime (Delloitte (a), 2010, p.7). The National Dutch Ban (DNB) 2 summarizes the main distinctions as follows (DNB, 2009, p. 14). First of all, the assets and liabilities under Solvency II regime must be consistent with maret value, while Solvency I recognises boo value. The current framewor does not include a ris margin in the technical provision. Under Solvency II the technical provisions consist of the best estimates of the liabilities plus an explicit ris margin. Under Solvency II the best estimate corresponds to the probability-weighted average of future cash flows, taing into account the time value of money (EIOPA, 2010, p.25). The ris 2 De Nederlandse Ban 6

7 margin in its turn represents the amount that theoretically would have to be paid to another insurer to compensate them for taing on the risy insurance liabilities. It is based on the cost of holding capital to support riss which cannot be really hedged (Towers Watson, 2010, p.7). The second crucial difference is the calculation of the Solvency Capital Requirement. Within Solvency I it is nown as a solvency margin and is quantified on the basis of insurance technical variables only. There is no confidence level specified. The new framewor presents SCR which covers quantifiable insurance, maret, counterparty, credit and operational ris. It is determined so as to provide sufficient ris coverage for an entire year with confidence level of 99.5 percent. Solvency II determines MCR as to provide sufficient ris coverage for an entire year with 85 percent confidence (DNB, 2009, p.14). Calculating the MCR is simpler than the SCR. The MCR could amount to values between 25 percent and 45 percent of the SCR. Figure 1. Differences between Solvency I and Solvency II Solvency II is based on the three pillars which are created to support each other. Each pillar is focusing on a specific regulatory component, such as quantification, governance and disclosure. All three pillars are cut across maret, credit, liquidity, operational and insurance ris. The structure of the new system is depicted in figure 2 (Delloitte (a), 2010, p.4). 7

8 Figure 2. Structure of Solvency II Pillar I imposes qualitative and quantitative requirements insurance companies have to meet. It introduces two capital requirements: a Minimum Capital Requirement and a ris-sensitive Solvency Capital Requirement. Solvency II provides the possibility to calculate the SCR either with a standard formula or an internal model. The internal model must be approved by the supervisor and must meet strict quantitative and qualitative conditions. Pillar II sets out the governance requirements. The insurer must implement ris management and an actuarial function. Every company has to carry out an Own Ris and Solvency Assessment (ORSA), which includes shortterm and long-term solvency needs of an insurer. The other component of the pillar II is the Supervisory Review Process (SPR): an instrument which ensures the supervisor that the calculation of the SCR accurately corresponds to the insurer s ris profile. Pillar III sets out the reporting requirements. There will be two separate reporting streams one for the public and one for the supervisor. Reporting to the public will promote transparency and maret discipline in the insurance maret. The MCR must be reported to the supervisor at least four times a year. The SCR must be reported at least once a year. If the own funds of an insurer are less to 8

9 satisfy the SCR requirement but yet sufficient to fulfil the MCR the insurer is placed then under intensive supervision and must tae actions to restore its financial position to fulfil the SCR. In this paper attention will be concentrated on the first pillar. The SCR calculation approach was tested by running the simulations nown as quantitative impact studies (QIS). These QIS s are the topic of the following subchapter. 2.2 Quantitative Impact Studies As mentioned in the previous section QIS stands for quantitative impact study. These studies are the simulations, performed by the insurers on a voluntary basis. The purpose of these studies was twofold. On the one hand the standard formula itself was developed while on the other side the studies enabled the participating insurers to prepare for Solvency II. The studies have been run by the European Insurance and Occupational Pensions Authority on the request of the European Commission. Till now five studies QIS1, QIS2, QIS3, QIS4 and QIS5 have been conducted. The first study QIS1 was run in A relatively small number of 312 insurance companies participated in this study. The prudence level of the technical provisions was the main topic of QIS1. Despite the fact that QIS1 was the first important step in the development of the new system, the results of the study were not taen into account because of the small number of participants. While QIS1 was considered a learning process, QIS2, which was held in 2006, was the first real step to the final European Standard Formula (EIOPA, 2010). The calculation of the SCR and MCR was introduced as a solid model for the first time. The number of participating insurers was significantly higher than that of QIS1, 514 companies from 23 EEA 3 member states (EIOPA, 2010). This gave a better general view about the shortcomings of the European Standard Formula. For example, based on the obtained experience it was discovered that the developed calculation approach was not applicable for the small-sized insurance companies. The results and the conclusions of the QIS2 were taen into account for adjustments which were introduced in the third exercise, QIS3 in In QIS3 the calculation method for insurance-groups was introduced insurers from 3 European Economic Area 9

10 28 EEA states too part in QIS3. Results of this study have shown that the European Standard Formula requires further calibration. For example, the experience of the participating insurers showed that the MCR was often higher then SCR, which contradicts with the concepts of the SCR and MCR. In 2008 QIS4 was carried out insurance companies from all 30 EEA member countries joined the exercise. Liedorp and Welie (2008, pp ) summarise the most important adjustments as follow. Based on the conclusions of the QIS3 the calculation of the MCR was calibrated as approximately 35 percent of the SCR. Further, simplification approaches were introduced for the small-sized insurers. Great attention was given to the tests of the internal models. The fifth study, QIS5, has been conducted in July-November in Recent events on the financial maret showed that the formula of the SCR does still require some adjustments. For example, one of the hot issues was the illiquidity premium sub-module, which was added the maret ris module in QIS5. There were also essential changes in the structure of the SCR. It is very liely that QIS5 will not be the last study and QIS6 is very liely to be conducted before introduction of Solvency II as a mandatory regime. The next section will focus attention on the structure of the SCR and one of its sub-modules, the Underwriting Riss module under QIS SCR under QIS5 The SCR covers all material and quantifiable riss. These riss are maret ris, counterparty ris, insurance technical ris (life, non-life and health) and operational ris. Those riss are subdivided into individual riss (see figure 3). The SCR has been determined as to be sufficient to avoid insolvency during one year with 99,5 percent certainty. The SCR can be calculated at different levels of complexity. The standard formula is a general way of calculating this. An insurer may also use simplified calculations if allowed by the supervisor. Comparing to the structure under QIS4, the SCR under QIS5 was subject to a few changes. Towers Watson (2010, p.8) summarised the main adjustments. The first adjustment, an illiquidity premium module has been added within maret ris. The second adjustment is a lapse ris stress within the non-life underwriting ris module. Furthermore, the health ris module has been split into three submodules to cover business calculated on a similar basis to life, business 10

11 calculated on a similar to non-life and catastrophe ris. The structure of the SCR under QIS5 is shown in figure 3 (EIOPA, 2010, p. 90). Figure3. Structure of the SCR calculation under QIS5 The calculation of the SCR for the (non-hedgeble) insurance-riss of the income protection insurance is represented in the Module. The SCR calculated according to specifications of the Module is denoted as SCR. Because of its nature the insurance products of the Module could bear the features of both life and non-life insurances. That is why the health ris sub-module has been redesigned to be consistent with the segregation of health business into business conducted on life () or non-life (non-) technical basis. The techniques used for these two types of businesses are mostly similar to their equivalent ris modules within the life and non-life modules. For non- business there is a specific calibration for premium and reserve ris factors for medical expenses, income protection, worers compensation and non-proportional health insurance. For business, there is a higher calibration for revision ris (stress of four percent as opposed to three percent) and a lower calibration for lapse ris (stress of 20 percent instead of 30 11

12 percent). There is also greater differentiation between medical and income protection insurance. The main purpose of this study is to quantify riss of the income protection insurance. In the QIS5 income protection insurance obligations are defined as obligations which cover financial compensation as consequence of illness, accident, disability or infirmity other than obligation considered as medical expenses insurance obligations (EIOPA, 2010, p.170). The structure of the SCR is depicted in figure 4 (EIOPA, 2010, p.165). SCR Figure 4. Structure of the under QIS5 - is a capital charge for health insurance obligations pursed on a similar technical basis to that of life insurance. Non is a capital charge for health insurance obligations not pursued on a similar technical basis to that of life insurance. CAT - is a capital charge for health insurance obligations catastrophe ris. 12

13 As can be seen in figure 4 the is a part of the SCR which is designated to cover lapse ris, expense ris, revision ris and the biometric riss, such as mortality, longevity and disability/morbidity riss. In Solvency II Glossary these riss are defined as follow. Mortality ris is defined as a change in net asset value (maret value of assets less maret value of liabilities) when the mortality rates are higher than the ones underlying the best estimate. An increase in the frequency of death of insured persons may for example result in higher claim patterns than charged for in the premiums. Mortality ris affects all life insurance contracts and the health insurance contracts where claims depend upon death of the insured (CEA, 2007, p. 42). Longevity ris is a change in value caused by the actual mortality rate being lower than the one expected. Longevity ris affects contracts where benefits are based upon the lielihood of survival, i.e. annuities, pensions, pure endowments, and specific types of health contracts (CEA, 2007, p. 37). Expense ris is the ris of a change in value caused by the fact that timing and/or the amount of expenses incurred differs from those expected, e.g. assumed for pricing basis (EIOPA, 2010, p. 219). Disability ris is a change of value caused by a deviation of the actual randomness in the rate of insured persons that are incapable to perform one or more duties of their occupation due to a physical or mental condition, compared to the expected randomness. Disability ris only relates to cover against loss of income. Such a disability may be partial (a disabled person can perform a material part of their occupation), total (a disability which is insufficient to prevent the person from performing any of the duties of their occupation), permanent (the disability is expected to be for the life of the person), or temporary (a disability from which the person is expected to recover) (CEA, 2007, p. 22). Morbidity ris is a change of value caused by the actual disability and illness rates of the persons insured deviating from the ones expected. Morbidity ris is generally considered as to only relate to insurance cover for losses other than loss of income, for example medical expenses, contrary to disability ris. An increase in the frequency of an insured becoming disabled or ill may for example result in higher claim patterns than charged for in the premiums (CEA,2007,p.41). 13

14 Revision ris covers the ris of loss, or adverse change in the value of annuity (re)insurance liabilities resulting from fluctuation in the level, trend, or volatility of revision rates applied to benefits due to changes in legal environment or state of health of the person insured (EIOPA, 2010, p.173). Lapse ris is the riss of a change in value caused by deviations from the actual rate of policy lapses from their expected rates. Lapse is defined as the expiration of all rights and obligations under an insurance contract if the policyholder fails to comply with certain obligations required upholding those, e.g. premium payment (CEA, 2007, p.36). Lapse ris is new for the health module. In QIS4 it was not taen into account for calculations of the capital requirement for the health underwriting module. The goal of the paper is to quantify the riss for a long term income protection insurance contract financed either with standard or ris premium. The research is based on the multi-state model for individual long-term disability insurance and is represented in the next chapter followed by the description of the calculation approach of the riss. 14

15 3 Quantitative determination of the capital requirement The main objective of this chapter is to show a computational approach used to determine the capital requirements of the sub-module. First of all a theoretical bacground of the actuarial models for disability insurance is given in the first sub-chapter. The second and third part describe the actuarial model used in this paper to calculate the present values of future benefits and future premiums and also best estimate of the provisions. The last sub-chapter gives the methodology for the calculations of the capital requirement according to the standard approach of Solvency II scheme. 3.1 Theoretical bacground of the model The modeling of disability insurance contracts is not a new topic in the actuarial world and goes bac to the middle of the last century. For example, one of the first publications, proposing the design of disability insurance contacts, has appeared in the scientific literature in the late sixties (Starr (1965), Janssen (1966)). These were the first explicit applications of the mathematics of multistate models (Marov and semi-marov process) to disability benefits. Since then the use of Marov chains in life contingencies and their extensions has been discussed in many papers (Hoem (1969), Mattsson (1977), Segerer (1993), Gregorious (1993)) and boos (Bowers et al (1986), Haberman and Pitacco (1999), Wolthuis (2003), Promislow (2006), Waters et al (2009)). There have been a lot of actuarial models used in the international insurance maret. A first step towards a unifying approach was proposed by Pitacco (1995). Using a multi-state Marovian structure he represents a timecontinuous general model which allows taing into consideration a wide range of different policy conditions. He stresses that the general formula can be expressed with two basic principles: the probability of being disabled and the probability of becoming and remaining disabled (Pitacco, p. 45). In the light of the general model, Pitacco illustrates some calculation techniques that have been implemented in actuarial practice. So, he explains that the methods used in Norway and United Kingdom (Norwegian model and Manchester-Unity model) are similar to the general formula based on the 15

16 probability of being disabled whilst the Swedish approach is based on the probability of becoming sic and remaining sic. Further he suggests that the inception-annuity model widely used in Germany, Austria and Switzerland could be interpreted as a time-discrete general formula based on the probabilities of becoming sic ( inception ) and on the expected present value of an annuity payable while the insured remains sic (Pitacco, p.47-51). Finally, Pitacco discusses a Marovian model for select probabilities, which is used by Dutch insurers. In this model there is a set of disability states which are split according to duration. The rationale of this splitting of the disability state is that for different durations since disability inception observed recovery and mortality rates differ. The splitting allows considering select rates and probabilities without introducing a semi-marov process, which leads to the major difficulties (Pitacco, p ). In the Dutch insurance maret the income protection insurances are divided into two categories: A and B. A short term insurance against loss of income due to illness refers to category A (or A-ris). A-ris relates to the first year of sicness and is sometimes called the first year ris. The category B (or B-ris) contains insurances against loss of income due to long term disability (AG&AI, 2007, p. 59). In this paper attention will be focused on the B-ris insurances. Dutch insurers use different actuarial models to describe the individual B- ris insurances. In the nineties of the last century the Contact Commission for Income, Medical and Accident Insurers (KAZO 4 according to its Dutch acronym) has developed the multi-state Marovian model for select probabilities (KAZO model) that was widely used by the Dutch insurance companies. After restructuring of the Covenant of the Insurers in 1992 the KAZO commission was resigned and the Actuarial Commission AOV was established instead as a supervisor for the engaged insurance companies (AG&AI, 2007, p.60). Since then the KAZO model has been modernized by the Actuarial Commission AOV. The modern version of this model is nown as the AOV model. In this paper a multi-state model is proposed, based on a discrete stochastic Marov process to represent an individual long-term disability insurance contract. The present values of the future liabilities and future premiums are determined according to approach proposed by Promislow 4 KAZO=Kontactcommissie voor Arbeidsongeschitheids-, Ziete- en Ongevallenverzeeraars 16

17 (Promislow (2006)).The actuarial model used in this paper bears a lot of features of the traditional AOV model. For example, the majority of the assumptions and the input data such as disability and recovery rates of the traditional approach are used in the research model. Nevertheless the model differs from the traditional one. First of all the lapse rates were added to the model. Because of new assumptions made especially for this research the transition probabilities and the transition matrixes were adjusted accordingly. 3.2 Description of the model In the first two paragraphs the assumptions and the basic input data of the model will be discussed. The last paragraph describes the transition probabilities and the transition matrices of the Marov chain used for the modeling of the given disability insurance Model assumptions The actuarial model used for the research is an adjusted version of the traditional AOV model. That is why some assumptions of the model are similar to the general approach. There are also few assumptions which differ from that of the traditional model. Just as in the AOV model the following assumption have been made for the research (AG&AI, 2007, p. 74): 1. At the moment t=0 the person insured is healthy. 2. There is no partial disability. In the AOV model a policyholder that is disabled is fully unable to wor (100 percent disability). 3. The age of an insured is indicated by x. It is assumed that the birthdays of the policyholders are distributed equally over the whole year. At the moment of premium payment the average age of insured is x Time to maturity of the contract ( n ) is measured in years, where n. The contract expires when an insured reaches termination age, defined by contract conditions ( end. age ). 5. The benefits are paid continuously. In the model it is simplified that half of the yearly payments fall at the beginning of the year and the other half at the end of the year. 17

18 6. There are eight states in the model: A, IS ( s 1,...6), R. The states are described in the next chapter. 7. The contract starts to pay benefits exactly after one year from the moment of the disability. This deferring period is called waiting period. The additional assumptions which are not included in the AOV model but have been made for this actuarial model: 8. After recovery from illness a policyholder doesn t return to the state active again. After recovery he or she leaves the model. 9. The premium is paid annually at the beginning of the year during the period of five years if a policyholder is healthy. In the traditional model an individual continues to pay premium till termination age. 10. There is a possibility that a policyholder could fail to pay premium. In this case the contract is terminated and the policyholder leaves the model. There are no surrender value- or paid-up-policy options included in the insurance policy. 11. The only riss that exist in the model are the underwriting riss Input data In the research four age dependent probabilities were used to model an individual B-ris insurance: 1. Mortality rate ( qx ) is the probability that an insured person who is x years old will die within a period of exactly one year. 2. Disability rate ( ix ) is the probability that an insured person who is x years old and is healthy will be disabled after exactly one year. 3. Recovery rate r ( I ) is the probability that an individual of age x x S who is disabled for s years will not be disabled any more after exactly one year. lapse 4. Lapse rate ( q ) is the probability that a policyholder who is x years x old will fail to pay the premium after exactly one year and his or her insurance contract will be terminated. 18

19 The probabilities defined above are have still unnown real values. The basis input data for the model consist of the estimated values for those probabilities. In the Netherlands the values for disability rates and recovery rates have been estimated by KAZO and later by Actuarial Commission AOV based on the statistical data of the Dutch disability insurances. The latest values of the disability and the recovery probabilities were estimated by Actuarial Commission AOV in 1999 (the estimated values AOV-2000) (AG&AI, 2007, p.66). These values are the same both for male and female individual. AOV-2000 values were used as the basis input data for the actuarial model of the research and are shown in Table 1, Appendix 1. For the mortality rates the estimated values of the mortality table GBM (AG, 2007) were used in the model. The estimated values for the lapse rates were defined as a ratio between the amount of the lapsed policies at the end of the year and the amount of the lapsed policies at the beginning of the year. The same values are applied both for a male and female participants. The estimated lapse probabilities are age dependent. These data are given in Table 2, Appendix 1 5. The cash flows were discounted with interest rates derived from the DNB zero-coupon yield curve dated by (DNB, 2010) Transition probabilities and transition matrices An actuarial model for income protection insurance is more complex than an actuarial model for life insurance. Traditionally, there are two possibilities, so called states, in the life insurance actuarial model: an individual is alive or he/she is dead. An actuarial model for disability contains one state more: event of disability. In reality an insured could be disabled partially or totally. The observations have shown that probability of recovery after illness becomes lower as disability lasts longer (AG&AI, 2007, p.60). That is why in the AOV model there are few additional states defined depending on the disability duration. The traditional AOV model distinguishes the following states in which a policyholder could occur (AG&AI, 2007, p.61). The same states are included in the actuarial model of the research: A - active or able to wor; 5 The estimated values are based on the statistical data of 2005 of the Dutch insurance company ING Movir. 19

20 I1 - disabled less than one year; I2 - disabled more than one year but less than two years; I3 - disabled more than two years but less than three years; I4 - disabled more than three years but less than four years; I5 - disabled more than four years but less than five years; I6 - disabled more than five years; R - removed from the model due to death or lapse or recovery after illness. The possible transitions between states are depicted in the table 1. For example an active participant could occur one year later in three possible states: active ( A A), disable less than one year ( A I 1) or resigned from the model due to death or lapse ( A R). One year later a participant who is disabled less than one year could still be disabled ( I 1 I 2 ) or leave the model due to death or recovery from the illness ( I 1 R). After being disabled for more than five years a participant cannot recover any more and either stays disabled or dies during the current year ( I 6 I 6 or I6 R). Table 1 Possible transition between states from / to A I I I I I I R A I I I I A A A I A R I I I R I I I R I I 0 0 I R I I 0 I R I I I I I R I I I R R R R The process in table 1 can be represented as a Marov chain with eight states. In order to model long term income protection insurance in this paper the modeling approach of Promislow was used, where he applies the Marov chain process to model life contingent insurance riss (Promislow, 2006). Promislow introduces 20

21 the transition matrix P as a convenient way of describing the stochastic process of the insurance ris. In the transition matrix P the entry in the ith row and jth column is the probability of moving from state i to state j (Promislow, 2006, p.281). Taing into account the assumption of the model the transition matrix P at time consists of the transition probabilities p ( i, j) over one period (one year) and loos lie this: P p ( A, A) p ( A, I ) p ( A, R ) p ( I, I ) p ( I, R ) p ( I, I ) p ( I, R ) p ( I, I ) 0 0 p ( I, R ) p ( I, I ) 0 p ( I, R ) p ( I, I ) p ( I, R ) p ( I, I ) p ( I, R ) p ( RR, ) (1) For an individual of age x the transition probabilities are: p ( A, A) - the probability that the individual will stay active within a period of exactly one year (from time till time 1). lapse p ( A, A) 1 q i q (2) x x x p ( A, I ) - the probability that the individual is healthy at the time and will be 1 disabled after exactly one year (at time 1). p ( A, I ) i (3) 1 x p ( I, I ) s N :[1,5] and p ( I6, I 6) are the probabilities that the individual S S 1 who is disabled for s (or 6) years stays disabled exactly one year longer (from time till time 1). p ( I, I ) 1 q r ( I ) (4) S S 1 x x S p ( I, I ) 1 q r ( I ) (5) 6 6 x x 6 21

22 p ( A, R ) - the probability that a healthy individual will leave the model after exactly one year. The possible reasons of the transition from the state active to the state resigned are death or lapse. lapse p ( A, R) q q (6) x x p ( I, R ) - the probability that the disabled individual will leave the model after S exactly one year. The possible reasons of the transition from the state disabled to the state resigned are death or recovery from the illness. p ( IS, R) qx rx ( I S ) (7) The matrix P gives the transition over a single period of one year. For an individual of age x years the matrix P for the first five periods can be represented as follows: For [0, 4] P lapse 1 i q q i q q lapse x x x x x x r ( I ) q q r ( I ) x 1 x x x r ( I ) q q r ( I ) x 2 x x x r ( I ) q 0 0 q r ( I ) x 3 x x x r ( I ) q 0 q r ( I ) x 4 x x x 4 r I x 5 q q r ( I5) x x x ( ) q q x x (8) After five years the contract expires and the healthy participants leave the model. It means that all transition probabilities from state A to any other state are equal to zero. The transition matrix P 5 then loos as: P r ( I ) q q r ( I ) x 5 1 x 5 x 5 x r ( I ) q q r ( I ) x 5 2 x 5 x 5 x r ( I ) q 0 0 q r ( I ) x 5 3 x 5 x 5 x r ( I ) q 0 q r ( I ) x 5 4 x 5 x 5 x r ( I ) q q r ( I ) x 5 5 x 5 x 5 x 5 5 x 5 x q q (9) 22

23 Just as in the matrix above the transition probabilities from state A in the matrix P 6 are also equal to zero. Also at time 6 there are no participants in the model who are disabled less than one year because there were no active participants in the previous period 5. So, the transition probabilities from state I 1 to any other state are equal to zero. The matrix P 6 could be depicted as shown below P r ( I ) q q r ( I ) x 6 2 x 6 x 6 x r ( I ) q 0 0 q r ( I ) x 6 3 x 6 x 6 x r ( I ) q 0 q r ( I ) x 6 4 x 6 x 6 x r ( I ) q q r ( I ) x 6 5 x 6 x 6 x 6 5 x 6 x q q (10) Due to the same reason the entries on the first three rows of the matrix P 7 are equal to zero P r ( I ) q 0 0 q r ( I ) x 7 3 x 7 x 7 x r ( I ) q 0 q r ( I ) x 7 4 x 7 x 7 x r ( I ) q q r ( I ) x 7 5 x 7 x 7 x 7 5 x 7 x q q (11) The matrixes P8 and P9 are then as follow: P r ( I ) q 0 q r ( I ) x 8 4 x 8 x 8 x 8 4 P (12) r ( I ) q q r ( I ) x 8 5 x 8 x 8 x r ( I ) q q r ( I ) x 9 5 x 9 x 9 x q q x 8 x q q x 9 x

24 Finally, for [10, end. age] the matrix P loos lie: P x x q q (13) Note that the only necessary condition required for the transition matrix of Marov chain is satisfied: all entries are nonnegative and each row sums to one (Promislow, 2006, p.282). The probability ( n p ) ( i, j ) is the probability of transferring from state i to state j after n periods. By the ordinary rules of matrix multiplication the probabilities ( n p ) ( i, j ) are the entries of the matrix ( n) P (Promislow, 2006, p. 287): ( n) P P0 P1... P n 1 (14) In order to model the expected value of the cash flows of the liabilities Promislow a introduces another quantity, q ( i, j ). This is the probability that when starting at time 0 in state a one will occur in state i at time and then in state j at time a 1. The probability q ( i, j) is calculated as a product of probabilities, first going from state a to state i in steps, and then moving to state j at the next transition (Promislow, 2006, p.288). q i j P a i P i j P P P a i P i j for 0 (15) a ( ) (, ) (, ) (, ) [ ](, ) (, ) a (, ) 0 a q0 i j if a iand q0( i, j) P0( i, j) if a i. The transition probabilities derived in this section are used for the calculation of the best estimates of the provisions, which is the topic of the next sub-chapter. 24

25 3.3 Best Estimates of the provision Under Solvency II the best estimate should correspond to the probability weighted average of future cash-flows taing into account the time value of money. The projection horizon used in the calculation of best estimate should cover the full lifetime of all the cash in- and out-flows required to settle the obligations related to existing insurance and reinsurance contracts on the date of the valuation (EIOPA, 2010, p. 28). The best estimate of the provisions from the insurance should be given as the expected present value of future in- and outgoing cash-flows: cash-flows resulting from future claims events; cash-flows from future premiums; cash-flows arising from claims administration expenses. The sub-chapter is organized as follows. The first section describes the calculation method of the present values of the liabilities resulting from future claim events. Then the section about present value of the future premiums follows. The third part is about present value of expenses Present value of liabilities Promislow models the general insurance contract by taing an N -state Marov chain. In his model an individual begins in state a at time 0. For any two states i, jit was considered that a contract pays a benefit of b at time 1provided that there is transfer from state i to state j at time. It means that the process was in state i at time and state j at time 1. The actuarial present value of the liabilities ( PVL ) at time 0 is calculated by multiplying the amount of the benefit by the discount factor and by the probability of receiving the benefit, and then ( i, j ) summing those values over all time periods (Promislow, 2006, p. 289). a PVL b v( 1) q ( i, j ) (16) 0 0 ( i, j) 25

26 1 where v ( 1) 1 (1 r ) 1 zero-coupon yield curve. is a discount factor and r 1is the interest rate of the The previous formula gives the actuarial present value of the liabilities for the period 0. For any random time d the actuarial present value of the liabilities can be determined as follow: 1 PVL b v( d 1) q ( i, j) ( i, j) a d d d vd ( ) d The general formula 12 can be applied for the income protection insurance contract modelled in the research. According to the model assumptions an individual starts in state A. Time to maturity of the contract is the time till the final age is reached as defined in the contract ( 0,..., end. age x ). (17) Because of the assumptions of the model (2, 4 and 6) a possible scheme of the benefits could loo lie that in figure 5. At time an individual is x 0.5 years old (assumption 2). If at the age x 1 the policyholder becomes disabled and doesn t recover after one year he or she will get the first benefit at time 2 (assumptions 4 and 6). If he or she doesn t recover after two years the next benefit will be paid at time 3. The contract continues to pay benefits till the recovery, death or the age defined in the contract. Figure 5 Possible scheme of benefit payments As it could be seen from figure 5 the benefits are not paid when the insured ( A, j) ( i, R) occurs in the states A, I1 or R ( b b 0 :[0, end. age 1 x ] ). At any time the same amount of benefits is paid to any participant who is disabled ( I, I ) ( I2, I3) ( I3, I4 ) 1 2 longer than one year ( b b b ( I4, I5) ( I5, I6 ) ( I6, I6 ) b b b b ). 26

27 In the model benefits are indexed with two percent per year (DNB, 2010): b 1 (1 0.02) b :[0, end. age 1 x ] (18) below. The actuarial present value of liabilities is determined by the formulas for s 1,2,3,4 : (19) 1 b b PVL ( v ( 1) q ( I, I ) v ( 1) q ( I, I )) 4 end. age 1 x end. age 1 x 1 S 4 A 1 A d S S 1 1 S 1 S 2 S 1 vd ( ) d 2 d 2 for s 5: 1 b b PVL ( v ( 1) q ( I, I ) v ( 1) q ( I, I )) end. age 1 x end. age 1 x S 5 A 1 A d vd ( ) d 2 d 2 (20) for s 6 : 1 b b PVL ( v ( 1) q ( I, I ) v ( 1) q ( I, I )) end. age 1 x end. age 1 x S 6 A 1 A d vd ( ) d 2 d 2 (21) Finally, d 6 PVL PVL (22) S 1 S d Present value of premiums In the traditional AOV model the premium could be determined according to two financial systems (AG&AI, 2007, p.81): a. Ris tariff or age depended tariff (the premium increases with the age). b. Standard tariff or actuarial tariff (the premium remains unchanged during the whole contract s time to maturity). In the income protection insurance business the combination of the two tariff systems (combined tariff or for short combi-tariff) has been used often. In this paper the premium will be defined according to the principals of the ris tariff and standard tariff financial systems. Under the ris tariff system the premium that is paid in a certain year must be enough to finance the expected liabilities that follow after disability events that happened in the same year. According to the standard AOV model the net ris 27

28 premium for an individual of the age of x years is defined by formula 23 (AG&AI, 2007, p.82). NP i v( 1) PVL ( I ) (23) ris. tariff x 1 1 where, PVL 1( I1) is the present value of the liabilities that follow after disability event which is occurred during the period of one year (from time till time 1). The main feature of the standard tariff (or actuarial tariff) is that the premium is constant during whole time to maturity of the contract. The standard net premium is determined according to actuarial equivalence principal and is shown in the formula below. NP PVL std. tariff 0.. a where, a xm : (24).. a a xm : is the single premium for an annuity on ( x ), paying one unit annually during m years when the person insured is alive and not disabled, beginning at time 0. For this research the insurance policy was modelled with m 5. In figure 6 it is shown how both the net standard premium and the ris premium develop during the maturity of the contract for a 30 years old individual. Figure 6 Net standard premium and net ris premium paid by an individual of 30 years old for the annual insured sum of euro standard premium ris premium For the calculation of the net premium the following assumptions have been made: 1. The same mortality rates (male) are used for the calculation of the net premium for both male and female participants. 28

29 2. The cash flows are discounted with the constant yield curve with annual interest rate of three percent. 3. There are no lapse rates taen in the calculation of the net premium. The net premium reflects the net price of the insurance product. The servicing and handling of the insurance policy incur expenses. The provision for the expected expenses is taen into account when calculating the gross premium ( GP ). Very often a profit loading is taen as well. The structure of the gross premium is similar for the both financing systems and is shown in formula 25. GP where, (1 ) NP (1 a b) (1 c) NP - net premium ( NP std. tariff or NP ) ris. tariff (25) - profit loading. In the model profit loading is equal to 20 percent of the gross premium; a - loading for the administration costs. These costs are incurred during whole time to maturity of the premium payments and are equal to ten percent of the net premium; b - loading for the expected claims expenses which will be incurred during claims handling. This loading is determined as three percent of the expected liabilities. c - commissions loading to an intermediary (it is equal to fifteen percent of the gross premium). The formula of the present value of premiums ( PVP ) is similar to that of the general annuity contract given by Promislow (Promislow, 2006, p. 290). The individual begins in state A at time 0. The annual premium is payable at time if the person is healthy ( GP ) during the period of m years. So, the formula for the PVP is, m 1 1 PVPd GP v( ) P ( A, A) d ( ) d (26) Present value of expenses In determining the best estimate, the undertaing should tae into account all cash-flows arising from expenses that will be incurred in servicing all obligations related to existing insurance and reinsurance contracts over the lifetime thereof. 29

30 The following expenses were taen into account for the calculation of the best estimate: 1. Claims handling expenses ( b actual ), caused by handling of the incurred claims. The claim costs are determined as a percentage of the expected insured benefits. 2. Administration costs ( a actual ) used for eeping of the insurance contract administration. These expenditures are defined as a percentage of the net premium. 3. Commissions paid to an intermediary ( c In the model commission depends on the gross premium. actual b actual ). The size of the was taen as three percent of the expected liabilities. actual a was defined as ten percent of the expected net premiums and percent of the expected gross premiums. The present value of the expenses is shown in the formula 27. actual c as fifteen net gross PVC 3% PVL 10% PVP 15% PVP (27) d d d d Finally, the best estimates of the provisions at time d are calculated as: BEd PVL d PVC d PVP d ; (28) 3.4 Capital requirement underwriting ris arises from the underwriting of health (re)insurance obligations, based on similar technical basis to life insurance, and is associated with both the riss covered and processes used in the conduct of the business (EIOPA, 2010, p.166). The lay-out of this section is as follow. The computation methods of capital requirements for the health underwriting riss sub-modules are given in the first six sub-sections. The seventh part represents methodology of the determination of and MVM. 30

31 3.4.1 mortality The mortality ris represents the ris of loss or adverse change in the value of insurance liabilities, resulting from changes in the level, trend or volatility of mortality rates, where an increase in the mortality rates leads to an increase in the value of insurance liabilities. The mortality sub-module aims at capturing the increase in general mortality that negatively affects the obligations of the undertaing. For the health products concerned by this ris, mortality ris relates to the general mortality probabilities used in the calculation of the technical provisions. Even if the health product doesn t insure death ris, they may be significant mortality ris because the valuation includes the profit at inception: if the policyholder dies early he/she will not pay future premiums and for the profit health insurance this can be a relevant effect. The capital charge for mortality ris is defined in the same way as in the mortality ris sub-module of the life underwriting ris module: NAV mortalityshoc ( PV BE) ( PV BE ) where, mortality assets assets mortalityshoc BE BE mortalitys hoc (29) NAV - is a change in the net value of assets minus liabilities. mortalitys hoc - is a permanent 15 percent increase in mortality rates for each age and each policy where the payment of benefits is contingent on the mortality rates longevity In the technical specification of the QIS5, longevity ris is defined as loss of adverse change in the value of insurance liabilities, resulting from the change in the level, trend or volatility of mortality rates, where a decrease in the mortality rate leads to an increase in the value of insurance liabilities (EIOPA, 2010, p. 169). 31

32 The capital requirement for longevity ris is calculated in the same way as in the longevity sub-module of the life underwriting module. Similar to the life underwriting module the capital charge for longevity ris is defined as result of a longevity scenario as follows: NAV longevityshoc ( PV BE) ( PV BE ) where, longevity assets assets longevityshoc BE BE longevitys hoc (30) NAV - is a change in the net value of assets minus liabilities. longevitys hoc - is a permanent 20 percent decrease in mortality rates for each age each policy where the payment of benefits is dependent on the mortality rates disability QIS5 defines the disability ris as the ris of loss or adverse change in the value of insurance liabilities, resulting from changes in the level, trend or volatility of the frequency or the initial severity of the claims due to changes in the disability, sicness and morbidity rates and also in medical inflation (EIOPA, 2010, p. 169). Because the disability ris sub-module is based on a distinction between medical expense insurance and income protection insurance the capital requirement for disability ris is determined as: disability medical income (31) The modelled insurance policy doesn t cover medical expense. So, the formula 31 can be simplified as: (32) disability income The determination of the capital requirement for disability ris for income protection insurance is based on disability probabilities. Because the ris in income protection insurance depends on the health status of the insured person, the disability ris for income protection insurance must be treated as disability ris of the Life underwriting module (EIOPA, 2010, p. 172). 32

33 The capital requirement for disability ris for income protection insurance is defined as result of disability scenario as follows: income ( NAV disabilityshoc ) ( PV BE) ( PV BE ) BE assets assets disabilityshoc disabilityshoc BE where, NAV - is a change in the net value of assets minus liabilities. disabilityshoc is a combination of changes applied to each policy where the payment of benefits is depends on disability rates: - an increase of 35 percent in disability rates for the next year, together with a permanent 25 percent increase in disability rates at each age in following years; - a permanent decrease of 20 percent in disability recovery rates. (33) exp ense The expense ris covers the ris of loss, or adverse change in the value of (re)insurance liabilities, resulting from changes in the level, trend, or volatility of the expenses incurred in servicing insurance or reinsurance contracts. Expense ris arises if the expenses anticipated when pricing a guarantee are insufficient to cover the actual costs accruing in the following year. All expenses incurred have to be taen into account (EIOPA, 2010, p.173). Capital requirement for expense ris ( expense ) is computed as in the life expense ris sub-module of the life underwriting ris module. NAV exp shoc ( PV BE) ( PV BE ) where: exp assets assets exp shoc BEexpshoc BE (34) NAV - change in the net value of assets minus liabilities. exp shoc - increase of ten percent in future expenses compared to best estimate anticipations, and increase by one percent per annum of the expense inflation rate compared to anticipation. 33

34 3.4.5 revision The revision ris covers the ris of loss, or adverse change in the value of annuity (re)insurance liabilities resulting from fluctuation in the level, trend, or volatility of revision rates applied to benefits due to changes in (EIOPA, 2010, p.173): the legal environment (or court decision); only future changes approved or strongly foreseeable at the calculation date under the principle of constant legal environment, or the state of health of the person insured (sic to sicer, partially disabled to fully disabled, temporarily disabled to permanently disabled). The calculation of is made in the same way as in the revision ris revision sub-module of the life underwriting ris module, but with a stress of four percent instead of three percent. NAV revshoc ( PV BE) ( PV BE ) where, rev assets assets revshoc BErevshoc BE (35) NAV - change in the net value of assets minus liabilities revshoc - increase of four percent in the annual amount payable for annuities exposed to revision ris. The impact should be assessed considering the remaining run-off period of the annuities. Because of the assumptions of the model that the person insured could only be permanently and fully disabled, it assumed that that the revision ris doesn t exists in the model. That is why in this study. revision is not calculated lapse The Lapse ris ( Lapse ) covers the ris of loss, or adverse change in the value of insurance liabilities, resulting from the changes in the level or volatility of the rate of policy lapses, terminations, renewals and surrenders (EIOPA, 2010, p.174). 34

35 The calculation of the capital requirement for Lapse ( ) is done in the same way as in the lapse ris sub-module of the life underwriting ris module, but with the following change: for Lapse and for Lapse up down decrease in lapse rates is 20 percent instead of 30 percent. lapse is calculated as follows: lapse the increase and the max( Lapse ; Lapse ; Lapse ), (36) where, lapse up down mass lapse - is a capital requirement for lapse ris; Lapse down - is a capital requirement for the ris of a permanent decrease of the rates of lapsation Lapseup - is a capital requirement for the ris of a permanent increase of the rates of lapsation Lapsemass - is a capital requirement for the ris of a mass lapse event. In their turn Lapse, Lapseup and Lapse to the formulas below: down mass should be computed according Lapse NAV lapseshoc ( PV BE) ( PV BE ) where, down down assets assets lapseshoc _ down BE BE lspeshoc _ down (37) NAV - is a change in the net value of assets minus liabilities lapseshoc down - reduction of 20 percent in the assumed option tae-up rates in all future years for all policies without a positive surrender strain or otherwise adversely affected by such ris. Affected by reduction are options to fully or partly terminate, decrease, restrict of suspend the insurance cover. Where an option allows the full or partial establishment, renewal, increase, extension or resumption of insurance cover, the 20 percent reduction should be applied to the rate that the option is not taen up. The shoc should not change the rate to which the reduction is applied to by more than 20 percent in absolute terms. Lapse NAV lapseshoc ( PV BE) ( PV BE ) up up assets assets lapseshoc _ up BE BE lspeshoc _ up (38) 35

36 where, NAV - is a change in the net value of assets minus liabilities lapseshoc up - is an increase of 20 percent in the assumed option tae-up rates in all future years for all policies without a positive surrender strain or otherwise adversely affected by such ris. Affected by reduction are options to fully or partly terminate, decrease, restrict of suspend the insurance cover. Where an option allows the full or partial establishment, renewal, increase, extension or resumption of insurance cover, the 20 percent reduction should be applied to the rate that the option is not taen up. The shoced rate should not exceed 100 percent. The shoced tae-up rate is restricted as follows: Rup ( R) min(150% R ;100%) ; (39) Rdown ( R) min(50% R; R 20%) ; (40) where, Rup - is shoced tae-up rate in lapseshoc up, Rdown - is shoced tae-up rate in lapseshoc down, R - is tae-up rate before shoc. Lapsemass is computed as 30 percent of the sum of surrender strains over the policies where the surrender strain is positive. Surrender value option is a possibility when the policyholder has the right to fully or partially surrender the policy and receive a pre-defined lump sum amount. Paid-up policy option is an option when the policyholder has the right to stop paying premiums and change the policy to a paid-up status (EIOPA, 2010, p.35). Because of the assumption (10) the surrender value or paid-up sum of the insurance contract are equal to zero. That is why Lapse mass is calculated as: Lapse max(30% (0 BE );0) (41) mass Lapsemass is positive when an insurance contract is profitable ( the best estimates of the provisions are negative). Loss of potential profit due to policy lapsation is considered as an insurance ris. 36

37 3.4.7 In the Standard Formula is calculated using a correlation matrix approach. According to this approach the required capital (SCR) is first calculated on a standalone basis for each ris and then aggregated using a correlation matrix. The correlation coefficients are used to measure the strength of linear dependency between two random variables. By using a correlation matrix the riss are diversified, which prevents holding higher capital requirements than necessary. is derived by combining the capital requirements for the sub-modules using a correlation matrix (EIOPA, 2010, p. 168): Figure 7 Correlation matrix Corr Mortality Longevity Disability Lapse Expense Revision Mortality 1 Longevity -0,25 1 Disability 0, Lapse 0 0, Expense 0,25 0,25 0,5 0,5 1 Revision 0 0, ,5 1 Corr (42) r c r c r c where: Corrr c - entries of the matrix Corr,, - the capital requirements for individual health underwriting r c sub-modules according to the rows and columns of the correlation matrix Corr. Because the main objective of this paper is to investigate the riss it was assumed that only those riss exist in the model (see assumption (11)). That is why the MVM can be calculated with only. The ris margin is a part of technical provisions in order to ensure that the value of technical provisions is equivalent to the amount that insurance and 37

38 reinsurance undertaings would be expected to require in order to tae over and meet the insurance and reinsurance obligations. The ris margin should be calculated by determining the cost of providing an amount of eligible own funds equal to the SCR necessary to support the insurance and reinsurance obligations over the lifetime thereof. The rate used in the determination of the cost of providing that amount of eligible own funds is called Cost-of-Capital rate (EIOPA, 2010, p.55). SCR( d) MVM CoC CoC where MVM CoC ( d) CoC d 1 d 1 d 0(1 rd 1) d 0 (1 rd 1) - the ris margin, (43) ( ) d - the SCR for year d as calculated for the reference undertaing, r - the ris-free rate for maturity d ; d CoC - the Cost-of-Capital rate The results of the calculations and their analysis will be presented to the reader in the next chapter. 38

39 4 Analysis of the results The main objective of this paper was to analyze the exposure to the riss of the individual long term income protection insurance contract financed according to both standard and ris tariffs. The ris exposure is reflected in the capital requirements which are needed to be held by an insurer. The structure and the calculation method of the capital requirement for the sub-module were presented in the previous chapters. This chapter gives an analysis of the calculations results. The riss were determined for both financing systems. The calculations were made for disability insurance policy with annually indexed insured benefits with initial insured sum of euro. The insurance was taen for 30 years old male. The profit loading of 20 percent was embedded in the premium. Calculations show that the riss differ from each other by level and the ris exposure period. In figure 7 the riss are shown for the disability insurance policy calculated according to the both financing systems. The capital requirements projection is given for the first eleven years. It can be seen that the disability ris is most significant ris in the sub-module. The disability ris exists for ten years: during the first five years if the number of disability inceptions grows faster than expected or during the first ten years if the recovery ability decreases comparing to the expected value. The period the disability ris could last depends on the contract duration, when the transition from the state active to the state disabled is possible. According to the model assumption (9) defined in the third chapter the contract lasts for five years. The duration of the disability ris also depends on the chosen actuarial model. The model used for the research assumes that the disabled participant could recover within first five years after disability event. So the total period the unexpected changes in disability could lead to the losses, consists of the first ten years of the insurance policy life. After ten years it is expected that insured person either already left the model or he is disabled without any chance to recover. The second essential ris that the insurance contract is exposed to is the lapse ris. This ris could exist for the period of future premium payments, in this case for the first five years. The more profitable the insurance contract, the higher 39

40 the lapse ris. The capital requirement for the expense ris must be hold for the whole time to maturity of the policy (till termination age defined in the contract conditions). Because the administration of an insurance contract and handling of the incurred claims requires expenses, the expense ris is higher in the first five years when the contract is running. After five years only the claims handling expenses remain resulting in a lower expense ris. Figure 7 Capital requirements for sub-module during whole life of the insurance policy standard tariff age : 30 years gender :male SCR health SCR disability SCR lapse SCR longevity SCR mortality SCR expense diversification ris tariff age : 30 years gender: male SCR health SCR disability SCR lapse SCR longevity SCR mortality SCR expense diversification

41 The longevity ris is low compared to the previous mentioned riss. If mortality occurs to be lower than expected both the average premium and liability payments would increase. In this case the cash flows of the liabilities are more sensitive to the negative change in mortality than that of premiums. The present value of liabilities would grow faster than the present value of the future premiums resulting in the longevity ris. The longevity ris remains during the whole life of the insurance policy. The mortality ris doesn t exist in this disability insurance contract. The aggregated capital for ris sub-module must be ept till any of those riss exist. Comparing the capital requirements for riss from figure 7 it is evident that on the total level the disability insurance contract with ris premium is less risy than the contract with standard premium. Although not significantly for the ris tariff is less than that of standard tariff. The insurance with ris premium is less exposed to the majority of ris but is more open to danger of the unexpected changes in disability. The variance in capital requirements between financing systems is caused by expected cash flows of the future premium. After five years all riss of standard tariff are equal to that of ris tariff financing system. As mentioned above the capital requirements for riss were calculated for the profitable contracts with embedded profit loading of 20 percent. The cash flows were discounted with interest rates derived from the DNB zerocoupon yield curve dated by (DNB, 2011). Do the same conclusions hold for the less (more) profitable contracts or if the cash flows are discounted with another zero-coupon yield curve? To investigate this problem calculations have been made for contracts with different profitability and different zero-coupon yield curves. Figure 8 depicts the zero-coupon yield curves of different shapes used for the calculations. Zero-coupon yield curve dated by is taen as basic curve used in all calculations in this study. The yield curved dated by is shifted upward comparing to the basic curve. The shape of the curve of is flatter than that of The yield curve dated by has more humped shape: it is more increasing in the beginning and then more decreasing than basic yield curve. Finally the flat yield curve with 41

42 constant interest rate of three percent used for the calculations of the premium tariffs was taen into consideration as well. Figure 8 Zero-coupon yield curves. 4 3,5 3 % 2,5 2 1, DNB DNB DNB constant 3 % DNB In order to discover how the required capitals depend on the profitability of the insurance contract calculations have been run for policies with different premium profit loadings both positive and negative. From table 2 one can see that for all taen yield curves the same situation holds: although not significantly the ris tariff insurance is more exposed to the disability ris than that of standard tariff. For the contract with ris premium the best estimate under disability shoc deviates more from the initial best estimate. The small difference between the capital requirements for the disability ris could be explained by the following. The disability shocs have great effect on the cash flows of the liabilities, which are the same for both financing systems. The cash flows of the future premium payments differ between the systems, but effect of the disability shocs is far less dramatic than that of liabilities. Due to assumption (8) an insured person leaves the model after recovery that is why the cash flows of the future premiums are not influenced by recovery rates. Only the changes in disability rates could lead to the changes in the expected premium payments. The second reason of the small difference between disability of both tariffs is quite short contract duration (five years). The standard premium exceeds 42

43 the ris premium for the first three years only (see figure 6, chapter 3) resulting in smaller difference between present values of the future premiums comparing to the contracts with longer duration. For example for a contract which lasts for one year there is no difference between the present values for both financing systems because the ris premium and standard premium are equal to each other. With longer contract duration the difference between the present values of the future premiums becomes higher, so the effects resulting from the changes in disability rates. For both financing systems the mutual tendency was observed: more profitable insurance contracts are exposed to higher disability ris. In all simulations the higher lapse ris was observed for the contracts with standard premium. For both financing systems the lapse ris is the lowest for the policies with profit loading equal to zero. As the profit loading deviates from zero the lapse ris is growing. For the profitable policies it could be explained by negative best estimates which result in positive mass lapse ris. The loss-maing contracts are exposed to the lapse down ris: an insurance company has to carry responsibility for higher quantity of unprofitable contracts than it was expected. Just as lapse ris the expense ris is higher for the standard premium contracts in all simulation. The administration expenses depend on the present value of premium payments which is higher for the standard tariff. For less profitable policies this ris is smaller. Compared to disability the income protection insurance contract is far less sensitive to the unexpected negative changes of the mortality rates. That is why the longevity ris is quite low comparing to the rest of the riss and almost doesn t differ for both financing systems. is higher for the standard tariff for very profitable policies only. For the rest of the cases both contracts require almost the same Comparing the calculation results from table 2 according to the yield curves it is evident that the capital requirements are higher for the flatter yield curve (dated by ) and are lower for the upwards shifted yield curve (dated by ). The yield curve of gives higher required capitals comparing to the basic yield curve dated by The interest rates are slightly higher in the beginning, but they are much lower at the long end of the curve. Despite the dependency on the chosen yield curve there were no structural changes observed within the aggregated capital requirement. The disability ris remains the highest ris for every yield curve, followed by the lapse ris, the expense ris and the longevity ris. 43

44 DNB DNB DNB DNB % yield curve Table 2 riss as function of contract profitability and yield curve. Profit loading premium ris -50% -20% -10% 0% 10% 20% 50% tariff Disability Lapse Longevity Expense Disability Lapse Longevity Expense Disability Lapse Longevity Expense Disability Lapse Longevity Expense Disability Lapse Longevity Expense standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris standard ris 44

45 In this paper it is assumed that the insurance contract is exposed to the riss only (see assumption (11)). That is why the technical provisions which consist of the best estimates of the provisions and the ris margin could be calculated with only. The solvency capital requirement is equal than to. In figure 9 the required solvency position at the moment of contract inception is shown. A solvency position is determined as a sum of the technical provisions and the solvency capital requirement. From figure 9 it is evident that in the first year for the disability insurance contract financed according to the standard tariff system an insurer must hold less capital than for the same policy but with ris tariff financing system. The main reason is the difference in the best estimates of the provisions. It could be explained by the fact that the present value of the expected premiums is higher for the standard tariff. In the beginning the standard premium exceeds the ris premium (see figure 6, chapter 3). The present value of the standard premium that exceeds the ris premium in the first three years is higher than the present value of the ris premium that exceeds the standard tariff in the last two years. As it was mentioned above the capital requirement is slightly higher for the contract with standard premium system. The MVM s don t differ significantly for both systems. Although it could be stressed that at the moment t=0 the contract financed according to the standard tariff system requires less capital that must be held in the first year. If the contract requires less mandatory capital during whole time to maturity of insurance policy, it is more favorable for the insurance company. Moreover such contract could be made more attractive for the customers. The needed solvency loading which is usually embedded in the gross premium is lower for the contract with less required solvency position. That maes the price of the insurance policy more favorable for the customers. Situation changes depending on the age of the insurance contract. In figure 10 it is shown how the solvency position is changing during a period of the future premium payments for both financing systems. During five years the policy insurance of both systems are exposed to the riss almost in the same extent. For each period there is little difference in the capital 45

46 techncal provisions technical provisions Figure 9 Required solvency position at the moment of contract inception standard tariff Solvency position BE MVM SCR Solvency position ris tariff Solvency position BE MVM SCR Solvency position requirements for both tariff systems. Because of the almost similar the MVM s don t differ from each other during whole life of the contract as well. The only significant difference appears in the best estimate, which causes the difference in the total solvency positions within two financing systems. At the moment of the contract inception the insurance policy with standard premium requires less capital to be held to stay solvable. During the following four years the situation is more favorable for the insurance with the ris premium, which increases during each following year, exceeding the standard premium. 46

47 Figure 10 Required solvency positions across contract duration BE MVM SCR health technical provisions standard tariff BE MVM SCR health technical provisions ris tariff ris premium standard premium solvency position 47

48 So in the first year the solvency position is lower for the contract with the standard premium. Starting from the second year till the sixth year the contract with ris premium requires less technical provision and so lower solvency position. After five years the technical provisions and required capital requirements don t differ from each other for both contracts. Calculation of the technical reserves and capital requirement with other yield curves gives the results which are shown in the table 3. Just as in the situation described above the standard premium contract requires less capital only in the first year. In the following four years the solvency position is higher than that of the ris premium contract. The required solvency position is subject to changes if the interest rates change. Comparing to the initial calculations with yield curve dated by the solvency position declines if it is calculated with steeper yield curve (DNB ). It could be explained by the fact that the cash flows of the premiums and liabilities are discounted with higher interest rates, resulting in the lower present values. If discounting with flatter yield curve (DNB ) Table 3 Solvency positions with different zero-coupon yield curves (Euro) 3% DNB DNB DNB DNB premium tariff standard ris standard ris standard ris standard ris standard ris one gets higher value of the required solvency position because the cash flows were discounted with lower interest rates. The yield curve dated by gives higher solvency position as well. Comparing to the basic curve the interest rates of this yield curve are slightly higher on the short end and significantly lower at the long end of the curve. That is why the present values and consequently the solvency position for both tariffs are higher. As mentioned in the previous chapter the capital requirement aggregated by mean of square-root formula using a correlation is 48

49 Correlation matrix Figure 11 Structure of for contract with standard premium 169 standard tariff separate SCR's SCR SCR mortality SCR lapse SCR longevity SCR disability SCR expense SCR 6% -19% 16% 1% 0% SCR standard tariff 97% SCR lapse SCR longevity SCR mortality SCR disability SCR expense correlation effect independencies between random variables (the capital requirements for the riss). Figures 11 and 12 show how capital requirements for separate riss are aggregated into. From these figures it becomes clear that the correlation matrix mitigates about nineteen percent of the riss for the contract with standard premium and eighteen for the ris premium contract, preventing unnecessary overcapitalization of 554 and 513 euro respectively. 49

50 Correlation matrix Figure 12 Structure of for contract with ris premium ris tariff separate SCR's SCR SCR mortality SCR lapse SCR longevity SCR disability SCR expense SCR 6% -18% 14% 1% 0% SCR ris tariff 97% SCR lapse SCR longevity SCR mortality SCR disability SCR expense correlation effect In the previous calculations of the best estimates of the provisions the best guess about disability was set equal to the disability rates used in the calculations of the premium tariffs. If the best guess differs from the premium basic input then the best estimates of the provisions will differ as well and so capital requirements (see table 4). With higher disability frequencies, chosen as the best guess, the contracts financed according to both systems are exposure to the higher disability ris and lower lapse ris. With higher disability rates the insurance contracts have to cover higher liabilities, which are resulting in the higher best estimate of the 50

51 provisions. The lapse ris (lapse mass ris) which depends on the ris capital estimate will decrease then. The disability shocs implemented on the higher disability rates are causing higher disability ris. With higher disability ris and lower lapse ris the diversification effect becomes less (456 (408) euro instead of 554 (513) euro). Table 4 capital requirements with different BE guess about disability frequency (Euro) BE guess about disability SCR lapse SCR longevity SCR mortality SCR disability SCR expense SCR diversification effect premium tariff 100% premium (19%) standard basic input (18%) ris 110% premium (15%) standard basic input (13%) ris 90% premium (25%) standard basic input (23%) ris If the chosen best guess about disability frequency is lower than that calculated in the premium, the situation will become the opposite. The contracts appear to be more profitable than expected at the moment of the premium determination, resulting in lower best estimate of the provision. The unexpected loss of more profitable insurance contract is leading to the higher lapse ris. Disability shocs performed on the lower disability rates are resulting in the lower disability ris. The correlation matrix mitigates more ris than in the initial situation (646 (609) euro). Summarizing the analysis of the results presented in this chapter the following conclusions can be drown. The insurance contracts financed according to both standard tariff and ris tariff are exposed to the disability ris at most. Despite the slight difference in disability it can be stressed that the contract with ris premium suffers more from unexpected changes in the disability frequencies than that of standard tariff. Contracts with standard premium are more exposed to the lapse and expense riss. The aggregated capital requirement for riss is slightly higher for the standard tariff for the profitable contracts. The capital requirements depend on the interest rates taing for discounting of the expected cash flows. It was observed that for the flatter yield curve the higher capital is required. The opposite occurs when using steeper 51

52 yield curve: the capital requirements become lower. The given humped shaped yield curve also causes higher capital requirements. The solvency position is higher for the contract with standard premium at the moment of contract inception. Over the whole time to maturity the contact with ris premium requires less solvency capital. If the solvency loading is embedded in the premium than the contract with ris premium is cheaper and that is why more attractive for the potential customers. If the best guess about the disability rates differ from the disability rates of basic input used for the premium calculation the capital requirements will differ as well. If the best guess is higher than the basic input than higher required capitals are needed for the disability ris, longevity ris and expense ris. The requirement for the lapse ris becomes lower. The lower best guess about disability rates causes less required capital for the disability ris, longevity ris and expense ris. The requirement for the lapse ris becomes higher. The diversification effect produced by the correlation matrix is growing when disability ris becomes lower and the lapse ris is growing. The calibration of dependencies of riss is among the most difficult tass when setting up a capital model. According to EIOPA, there is no appropriate database for the calibration of the life underwriting ris correlation factors. For the time being, the choice of these factors needs to be based on expert opinion (EIOPA (b), 2010, p. 351). The lac of empirical data is also the reason for lots of discussions about the expert opinion values. Since QIS4 the correlation matrix for life underwriting riss was subject to several changes. In November 2009 EIOPA published its advice to change the correlation factors (EIOPA, 2009). According to its consultation paper... a medium correlation factor of 0.5 for lapse, disability and revision ris in relation to expense ris seems to be appropriate... For all other pairs of riss, there is liely to be a low dependence or independence. A suitable correlation factor in this situation is 0.25 (EIOPA, 2009, p.14). The reason to change the correlation coefficients for independent riss from zero to 0.25 was the argument that the life underwriting riss are not multivariate normal or elliptically distributed. That is why the independent ris factors don t have zero correlation. The quic reaction followed from the private sector in December CRO Forum stressed its disagreement on EIOPA position to use non-zero correlations for independent pairs and proposed its calibration recommendations 52

53 (CRO, 2009). It argued not the fact that the riss don t follow elliptical distributions but a justification for maing an arbitrary adjustment as proposed. It emphasized that the assumption on the distributions should be made transparent for each individual ris distribution. The adjustments of the correlation coefficients must be derived from a transparent set of assumption. Otherwise this arbitrary adjustment is even more inaccurate than using the matrix without adjustment. Therefore CRO Forum recommended that independent pairs should be treated as having zero correlation (CRO, 2009, p. 38). EIOPA follows recommendation of the CRO Forum in its calibration paper for QIS5 (EIOPA (b), 2010). Since no general assumptions on the shape and type of the distributions of the sub-riss have been made, the correlation factors for independent riss were set at zero. For the riss which can be assumed to be independent but such uncertainties exist, a low correlation parameter 0.25 was considered to be appropriate (EIOPA (b), 2010, p. 352). Since for the time being a consensus about correlation parameters seemed to be achieved the correlation matrixes for the life underwriting riss and for riss were presented in the technical specifications for QIS5 (EIOPA (a), 2010, pp.148, 168). It seems that a discussion about calibration of the correlation matrix is still far from being over. The first paper questioning QIS5 correlation matrix has appeared in December 2010 (Christiansen, 2010). Christiansen implies that the standard formula models dependencies between riss not at the roots (mortality or disability rated etc) but on the level of the ris sub-modules. Because of the fact that the same correlation matrix is used for any ind of insurance portfolio, the standard formula doesn t cover the diversity of ris structure that insurance portfolios can have (Christiansen, 2010, p.2). Based on the research performed on the basis of the German data Christiansen showed that the QIS5 correlation matrix was not appropriate for simple disability insurances. Moreover, by modeling more complicated alternatives of disability insurance, lie disability and endowment insurance or disability and temporary life insurance, he stressed the fact that the correlations greatly varied from one product to another, while the QIS5 correlation matrix is applicable for all types of products (Christiansen, 2010, p.16). Although EIOPA conducted a huge amount of analysis to obtain the results, it is liely that the further test for standard formula, QIS6, may be wanted. Findings from QIS5 will contribute to the further development of the new 53

54 regulations and help to shape the final Solvency II landscape. It will also form a vital part of the preparations by both firms and regulators for the introduction of Solvency II. 54

55 5 Conclusion Because the current solvency regime is too simple and doesn t allocate capital adequately to the actual riss, the new regulatory regime Solvency II has been developed by EIOPA and is scheduled to become mandatory for EU countries by January The new framewor Solvency II is based on the economical principles for the measurement of the assets and liabilities. It means that the assets and liabilities must be held at maret consistent value. Solvency II introduces a ris oriented level of required capital nown as the SCR that enables an insurance company to cover significant unforeseen losses. To calculate the SCR an insurer is given option to use either a standard approach, nown as European Standard Formula, or Internal Model. The standard approach to the SCR can be applied by any insurer. To apply an internal model companies must have an approval from their supervisors first. Five Quantitative Impact Studies were carried out in order to calibrate the standard approach to calculate the SCR. In QIS5 for the first time the health ris module has been split into three sub-modules to cover business calculated similar to life techniques ( riss), business calculated similar to non-life techniques ( non- riss) and catastrophe ris. The objective of this paper was to quantify and to compare the capital requirements for a long term disability insurance contract financed according to both standard and ris tariffs. It was investigated how the riss for both financing systems react on the changes in disability rates, interest rates and profitability of the contract. To fulfill the research a multi-state model for individual long-term disability insurance was represented based on a discrete stochastic Marov process. The present values of the future liabilities and future premiums were determined according to actuarial model proposed by Promislow (Promislow (2006)). The majority of the assumptions and the input data were based the AOV model. The individual disability insurance was designed according to the contract conditions of Dutch insurance company ING Movir. The model determined the capital requirements according to the standard approach of Solvency II and allowed observing their changes depending on the 55

56 financing system (standard premium and ris premium) combining with the variation in disability rates, interest rates and profitability of the contract. Summarizing the analysis of the results of this study, the following conclusions were drown. The insurance contracts financed according to both standard tariff and ris tariff are exposed to the disability ris at most. Despite the insignificant difference in disability it was discovered that the less profitable contracts with ris premium are more exposed to the disability ris than the similar contracts with standard premium. Not significant difference could be explained by the relatively short period of the premium payments and independency of the premium cash flows from recovery rates. The disability ris exists for ten years. The period the disability ris lasts depends on the contract duration and maximal recovery period. The lapse ris is significant for profitable contracts and exists for five years till the last premium payment is made. The expense and longevity riss exist for whole time to maturity of the insurance policy. The longevity ris is the smallest ris that the contracts are exposed to. Contracts with standard premium are more exposed to the lapse and expense riss. The longevity ris is equal to each other for both contracts. The aggregated capital requirement for riss is slightly higher for the standard tariff for the profitable contracts. The capital requirements depend on the interest rates taing for discounting of the expected cash flows. The flatter yield curve produces higher capital requirement. The calculations with the steeper yield curve give the lower capital requirements. The humped shaped yield curve in this study causes also higher capital requirements. The solvency position is higher for the contract with standard premium at the moment of contract inception. If considering the whole time to maturity the contact with ris premium requires less solvency capital. If the solvency loading is embedded in the premium than the contract with ris premium could is cheaper. If the best guess about the disability rates differ from the disability rates of basic input used in the premium calculation, the capital requirements will differ as well. If the best guess is higher than the basic input than higher required capitals are needed for the disability ris, longevity ris and expense ris. The requirement for the lapse ris becomes smaller. The lower best guess about disability rates causes less required capital for the disability ris, longevity ris 56

57 and expense ris. The requirement for the lapse ris becomes higher. The diversification effect produced by the correlation matrix is growing when disability ris becomes lower and the lapse ris is growing. Review of the Solvency II consultation documents showed that the determination of the correlation coefficients is a hot issue when designing the standard formula. Due to the lac of empirical data the correlation coefficients are based on the expert opinion. There is no complete agreement between EIOPA and private sector concerning dependencies between independent riss. Some recent scientific publications raise questions about QIS correlation matrix which is applicable on the level of the SCR but not on the level of the transition probabilities. Christiansen has suggested that the QIS correlation matrix was not appropriate for the German data and that the correlation greatly varied from one life insurance contract to another (Christiansen (2010)). Although EIOPA conducted a huge amount of analysis to obtain the results, it is liely that the further test for standard formula, QIS6, may be required. Findings from QIS5 will contribute to the further development of the new regulations and help to shape the final Solvency II landscape. It will also form a vital part of the preparations by both firms and regulators for the introduction of Solvency II. 57

58 References Actuarieel Genootschap (2007) AG-tafels , Woerden Actuarieel Genootschap & Actuarieel Instituut (AG&AI) (2007) Syllabus RE 7 Arbeidsongeschitheid en Zieteosten, Utrecht Bowers, N,L., Gerber, H.U., Hicman, J.C., Jones, D.A., Nesbitt, C.J. (1986). Actuarial Mathematics. The Society of Actuaries. CEA (2006). Introductory guide. Homepage ( 12 November. CEA (March, 2007). Solvency II Glossary. Homepage ( 12 November. CEA (June, 2007). Solvency II. Understanding the process. Homepage ( 12 November. EIOPA (a) (2010). QIS5 Technical Specifications. Homepage ( 3 January. EIOPA (b) (2010). QIS5 Calibration Paper. Homepage ( 20 January. EIOPA (2009). Consultation paper No.74. Homepage ( 20 January Christiansen, M.C., Denuit, M.M., Lazar, D. (2010). The Solvency Square Root Formula for Systematic Biometric Ris. Homepage ( 20 January. CRO (2009). Calibration recommendation for the correlations in the Solvency II standard formula.homepage ( 15 January. Deloitte (a) (2010). Solvency II. Quantitative Impact Study 5. (Final technical specifications). Homepage ( 6 November. Deloitte (b) (2010). Solvency II technical provisions. Homepage ( 6 November De Nederlandse Ban (2009). Consultation document Solvency II Implementation. Homepage ( 8 November. De Nederlandse Ban (2010). Nominale rentetermijnstructuur (zero coupon). Homepage ( August. De Nederlandse Ban (2010). Rente en inflatie. Homepage ( 20 December. Dicson, D.C.M., Hardy, M.R., Waters, H. (2009) Actuarial Mathematics For Life Contingent Riss. Cambridge University Press, New Yor. European Commission (2010). 'SOLVENCY II': Frequently Ased Questions (FAQs). Homepage ( ), 10 November. Haberman, S. and E. Pitacco (1999). Actuarial Models for Disability Insurance. Chapman and Hall. Hoem, J.M. (1969) Marov chain models in life insurance, Bliitter der deutschen Gesellschaft fiir Versicherungsmathematier, IX(2), Jansen, J. (1966) Application des processus semi-maroviens à un problème d invalidité, Bulletin de lássociation Royale des Actuaries Belges, 63,

59 Leidorp, F., van Welie, D. (2008) QIS4 Laatste Halte voor Solvency II? De Actuaris, November, Munich RE (2010). QIS5:European commission publishes proposal. Homepage ( ), 9 November. Pitacco, E. (1995) Actuarial models for pricing disability benefits: Towards a unifying approach. Insurance: Mathematics and Economics, 16, Promislow S.D. (2006) Fundamentals of Actuarial Mathematics. London:Whiley Starr, J.P. (1965) Plan Design in Group Long Term Disability Insurance, Journal of Ris and Insurance, 32(4), Towers Watson (2010). Insights Solvency II. Getting to grips with QIS5. Homepage ( ), 10 November. Wolthuis H. (2003) Life Insurance Mathematics (The Marovian Model) IAE. Universiteit van Amsterdam, II edition, Amsterdam. 59

60 Appendix 1 Input data Table 1 Estimated parameters for the long-term disability rates Parameter Value a b factor prof. class 1 factor prof. class 2 factor prof. class 3 factor prof. class 4 factor prof. class 5 factor prof. class 6 factor prof. class Table 2 Estimated parameters for the recovery rates State 1 1,10040 b 0, , , , , , , , , a Table 3 Lapse rates Age Lapse rate <25 3.5% % % % % % % % % 60