Projection methods for life underwriting risk.

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Underwriting Risk

C.E. de Vries - Stam

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

Faculty of Economics and Business Amsterdam School of Economics Author: C.E. de Vries - Stam Student nr: 5958008

Email: ineke@atdv.nl

Date: February 2014 Supervisor: Drs. R. Bruning

Supervisor: Drs. F. Hendriks (a.s.r.) Second reader: Prof. Dr. Ir. M. Vellekoop

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Abstract

Solvency II, the new regulatory framework for insurers, identifies different types of risk and prescribes that these risks should be covered by capital, the Solvency Capital Re-quirement (SCR). The SCR is the amount of capital to be held in order to comply with a 99.5% certainty that the insurer is able to meet his obligations on an annual basis. Assets and liabilities are calculated on market value. In order to determine the SCR shocks are applied to both sides of the balance sheet. The Market Value Margin (MVM) is a liability on top of the best estimate liability. Its aim is to have a total liability (Best Estimate and MVM) that complies to a 99.5% certainty with respect to non-hedgeable risks. The risks involved are generally underwriting risks, operational risk and counter party default risk. For determining the MVM the Cost of Capital method is prescribed. This method demands projection of the SCR for the risks involved. These calculations are time-consuming so simplifications need to be made in order to ease the process.

In general risk drivers are used to project the solvency capital requirements (SCRs). In this research a certain risk driver is tested for the applicability to groups of policies and to paid up policy status. This risk driver, see equation 1, is applied to project the capital requirements for the following life underwriting risks: mortality, longevity, lapse up and lapse down risk. These risks have in common that they apply a uniform shock to the probabilities. When the effect of options and guarantees is assumed to be linear with the best estimate of the liabilities, this risk driver is exact for one policy.

\ SCR(k) = SCR(0) · ( kpB0x g kpS0x · ^BE(k)S_{− BE(k)}B ^ BE(0)S_{− BE(0)}B ), (1) where: \

SCR(k) = solvency capital requirement using the approximate risk driver;

SCR(0) = the exact calculated Solvency capital requirement, including the effect of options and guarantees,

kpB

0

x = status probabilities in the basic scenario,

g

kpS

0

x = status probabilities shocked from time t = 0 on,

^

BE(k)S _{= best estimate of the liabilities with shocked rates applied starting from time t = 0}

on excluding the effect of options and guarantees,

BE(k)B = best estimate of the liabilities in the basic scenario, excluding the effect of options and guarantees.

SCR(0) of the portfolio is known and BE(k)B and _BE(k)^S _{can be derived from the}

best estimate cash flows of the portfolio for both the basic and shocked scenario. Using this data, the risk driver can approximate SCR(k), denoted by _{SCR(k) for time t = k,}\ with k > 0. The risk driver includes the quotient ’kpB0x

g

kpS0x

’, wherekpB

0

x is the status

prob-ability in the basic scenario and gkpS

0

x is the status probability in the shocked scenario,

where probabilities are shocked from time t = 0 on. How is this quotient determined for groups of policies? This research considers three methods for approximating this quotient in the risk driver for groups of policies and compares the results to the ex-act calculation. The examined portfolios are: whole life insurance, term insurance and immediate whole life annuity with one single premium payment at the start of the policy.

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iv C.E. de Vries - Stam — Projection Methods for Life Underwriting Risk

The best result is given by the method where the ratio of the status probabilities in the risk driver is approximated by weighting the status probabilities with the cost price based reserves. For the basic and the shocked scenario, the status probability of each policy is multiplied by the reserve of each policy and summed. The ratio of these two scenarios approximates the ratio of the status probabilities in the risk driver. Lapse down risk however needs to be treated with caution, as the summing of the cash flows makes it behave differently. A solution to overcome this contrasting behaviour is to split the portfolio into one with and one without lapse down risk. For term insurance the method where the ratio of status probabilities in the risk driver is approximated by weighting the status probabilities by their insured capital, performs best. The ratio in the risk driver is formed by the ratio of the basic and shocked scenario of the summa-tion of the status probabilities per policy multiplied by their insured capital. However the first mentioned reserve weighting method follows closely; the difference between the results of the two methods is not large.

This research describes a way to implement paid up policy status in the model of a portfolio of whole life insurance. It examines whether the risk driver responds correctly within the boundaries of the model to this implementation. Mortality and longevity risk behave conform expectations. Lapse up risk acts differently, probably because the expense cash flow is calculated with survival probabilities instead of status probabilities. When applying the risk driver to the expense cash flow, the ratio of survival probabilities is required instead of the ratio of status probabilities.

Keywords Risk driver, Groups of policies, Paid up policies, Solvency capital requirement, Market value margin, Mortality risk, Longevity risk, Lapse up risk, Lapse down risk, Cost of capital method, Whole life insurance, Term insurance, Immediate whole life annuity

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Preface vi

1 Introduction 1

1.1 Background . . . 1

1.2 Research question . . . 2

1.3 Towards a solution . . . 3

1.4 Outline of the thesis . . . 3

2 Theoretical background 4 2.1 Market value margin . . . 4

2.2 Cost-of-capital method . . . 5

2.3 Solvency capital requirement . . . 7

2.4 Life risk . . . 7

3 Methodology 10 3.1 Mathematical assumptions . . . 10

3.2 Best estimate of insurance product . . . 10

3.3 Process behind risk margin calculation . . . 11

3.4 Definition and use of risk driver . . . 12

3.5 Risk driver for one policy . . . 13

3.6 Mathematical derivation for one policy . . . 14

3.7 The risk driver and groups of policies . . . 16

4 Implementation of paid up policy status and the risk driver 19 4.1 Incorporation of paid up policy status and assumptions . . . 19

4.2 Paid up policy status and the risk driver . . . 20

5 Results 23 5.1 Model assumptions . . . 23

5.2 Whole life insurance with surrender rates processed . . . 26

5.3 Whole life insurance with paid up rates processed . . . 27

5.4 Term insurance . . . 30

5.5 Immediate whole life annuity . . . 31

5.6 Concluding results . . . 32

6 Conclusions 33

References 35

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Preface

Special thanks go to my supervisors: drs. R. Bruning and drs. F. Hen-driks. They supported me throughout the whole process of the research and writing of the thesis. They helped me with their ideas and their professional knowledge. I also thank a.s.r. for giving me the oppor-tunity of an internship. I also thank my colleagues of the actuarial department of ASR Leven NV for the good learning environment. I am thankful for my parents since they made it possible for me to do this study. Especially I want to thank my father for explaining the mathematics at the start of my study. Last but not least, I thank my husband Arnoud de Vries for his moral support.

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Introduction

1.1 Background

The credit crisis of 2007 − 2008 showed that the regulation framework for banks and insurers used at that time was inadequate. A new regulatory framework that maps the risks better was required, to prepare the banks and insurers better for a next crisis. Solvency II was born.

Solvency II is the new European regulatory framework for insurers. This framework is expected to be implemented by January 1st _{2016, as proposed by the European}

Commission. It is set up similar to Basel II, the new regulatory framework for European banks. Solvency II is structured by the so called ‘three pillars’. The first pillar states the capital requirements the insurer has to hold. Pillar two consist of rules about (internal) supervision and pillar three ensures transparency. Pillar one speaks of the minimal capital requirement (MCR) and the solvency capital requirement (SCR). The insurer should aim for the SCR. If the insurer has less than the MCR, he is not allowed to run his business and will face enhanced surveillance by EIOPA; the insurer is considered insolvent. On top of the best estimate the insurer has to hold the market value margin to be sure that theoretically another insurer is willing to take over the liabilities at any time. The market value margin is the cost of holding future capital requirements of non-hedgeable risks: insurance risks, operational risks and counter party risks. The best estimate of the liabilities together with the market value margin form the technical provision.

Under Solvency I the valuation of the technical provision is based on the principle of historical cost price. This means in most cases the use of a fixed interest rate and an old mortality table without trend. Solvency II breaks with this tradition. Solvency II values the whole balance sheet, both assets and liabilities, on ‘market value’ or ‘fair value’: an up-to-date yield curve has to be used for discounting the cash flows and an up-to-date mortality table with trend has to be used to determine the cash flows. Solvency II is more risk oriented than Solvency I, all risks are visualized and capitalized. Different from the standard model, insurers are allowed and even encouraged to develop their own internal model under Solvency II. An internal model better suits the company needs then the standard model. Several quantitative impact studies, known as QIS4 and QIS5, have already been made. They are set up in order to measure the impact of the new regulatory framework on the insurer.

Solvency II prescribes the use of the cost-of-capital method to calculate the market value margin (MVM). This method requires the projection of the future SCRs of the non-hedgeable risk. For the calculation of the MVM insurance companies use certain simplifications to project future SCRs, because the full calculation is time-consuming. These simplifications are named ‘risk drivers’. A risk driver is assumed to be linear with respect to SCR. Therefore SCR(k) divided by SCR(0) is nearly equal to driver(k) divided by driver(0). Since both SCR(0) as the values of the risk drivers in all points of

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2 C.E. de Vries - Stam — Projection Methods for Life Underwriting Risk

time are known, SCR(k) can be derived. This research has been carried out in the form of an internship at a.s.r.; the situation over there is the starting point of this research. Given are the cash flow projections of policies with equal characteristics for the basic scenario and for the shocked scenario, merged in so called model points. These cash flows are for example premiums, costs, benefits and cost-price-based reserves. The shocked scenario applies shocked rates starting at time t = 0. Applying the yield curve, the cash flow projections can be discounted to form the best estimate of the liabilities in both the basic and the shocked scenario for every point in time. Options and guarantees are ignored in this calculation. The best estimates of the liabilities are needed to determine life risk. ASR documentation (2013) describes that for life risks with every year an equal shock percentage, formula 1.1 can be derived for one policy. These risks are: mortality, longevity, lapse up and lapse down risk. By applying formula 1.1, SCR(k) can be determined for every future year k. Formula 1.1 is proven for one policy in chapter 3.6. In formula 1.1 the survival probabilities of the shocked and of the basic scenario are applied. Formula 1.1 implicitly defines the risk driver for the projection of SCR(k). This formula works for lapse, but not necessarily for paying up. Paying up or ‘pupping’ is related to lapse risk. It means that a policyholder stops paying premium, while continuing the policy, so there is no surrender. The contract is then called a ‘paid-up-policy’ (’pup’). The benefit is reduced. Lapse and ‘pupping’ can form a big part of life risk. This questions the possibility of applying the risk driver to policies with the possibility of paying up. Another question is how to apply this driver to groups of policies. In other words, how to determine kpB

0

x and gkpS

0

x in formula 1.1 for groups of

policies from the available data.

\ SCR(k) = SCR(0) · ( kpB0x g kpS0x · ^BE(k)S_{− BE(k)}B ^ BE(0)S_{− BE(0)}B ), (1.1) where: \

SCR(k) = solvency capital requirement using the approximate risk driver,

SCR(0) = the exact calculated Solvency capital requirement, including the effect of options and guarantees,

kpB

0

g

kpS

0

^

BE(k)S _{= best estimate of the liabilities with shocked rates applied starting from time t = 0}

on excluding the effect of options and guarantees,

1.2 Research question

The purpose of this thesis is to investigate how the risk drivers defined in formula 1.1 can be generalised in such a way that it also works for cash flows of groups of policies. A second purpose is to examine the applicability of the risk driver to groups of policies that have the right to stop premium paying with changing the policy into a paid-up policy. In this research the focus will be on the risk drivers for mortality risk, longevity risk and lapse risk up and down, because the shocks for the calculation of these risks are similar. The central question therefore is: ’Are the risk drivers according to formula 1.1 for the calculation of the market value margin adequate and how can they be extended

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to certain groups of products like funeral-policies and to policies facing paid-up-policy-status?’ An adequate risk driver predicts the SCR well and responds well to changes in the interest and mortality rates.

1.3 Towards a solution

To answer the research question I will calculate the capital requirements for mortality, longevity, lapse up and lapse down risk in three different ways for groups of policies, using:

• the exact method policy by policy;

• the risk driver method of formula 1.1 with several options for the quotient of survival probabilities in the risk driver.

The exact method calculates the risks according to the full method policy by policy and uses no simplifications. The risk driver method is split in three separate calculation methods. They determine the ratio of survival probabilities applied in the risk driver in different ways. The first method uses average survival probability rates. The second method is the same as the first, except that the survival probabilities are weighted by the insured capital of each policy. The third method weights the survival rates by the reserve of the policy. Different from the exact method, the risk driver methods sum the cash flows of all policies before calculating the capital requirement for life risks. Calculations are done in MS EXCEL/VBA. The results of these calculation methods are analysed.

1.4 Outline of the thesis

Chapter 2 gives the theoretical background of this research. It discusses the underlying concepts as the market value margin, the cost-of-capital method, solvency capital re-quirement and life risk. Chapter 3 continues with the methodology. It starts with the mathematical assumptions and describes the basic calculation of the best estimate of the liabilities of an insurance product. It also informs about the process behind the market value margin calculation and defines a risk driver. Further on it discusses the risk driver tested in this research, the mathematical derivation and different methods to apply the risk driver to groups of policies. Chapter 4 focuses on the phenomenon of paid up policies. It describes how it can be implemented in the model and if the risk driver responds well to this. The results and analysis of the different portfolios can be found in chapter 5. Chapter 6 presents the conclusions.

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Chapter 2

Theoretical background

Relevant information for the research question is discussed as the market value margin (MVM) in chapter 2.1, the Cost Of Capital approach in chapter 2.2, Solvency capital requirement in chapter 2.3, and the computation of life risk in chapter 2.4. This research focuses on mortality risk, longevity risk and lapse up and down risk.

2.1 Market value margin

EIOPA (2013) discusses the calculation of MVM, also called the risk margin (RM), extensively. IAA (2009) state that MVM can be seen as a reward for bearing risk and as an amount to cover deviation from the estimated liability. These two views result in similar outcomes. According to the International Actuarial Association (IAA), the International Association of Insurance Supervisors (IAIS), and the International Accounting Standards Board (IASB) MVM should contain the following properties:

• higher, if there is less information about the current estimate and its trend; • higher, if the portfolio contains risks with low frequency but high severity; • higher, if the portfolio contains contracts with a longer run-off period; • higher, if the portfolio contains risks with a wider probability distribution; • lower, if the uncertainty in financial markets reduces.

EIOPA (2013): the best estimate together with the MVM form the technical provision of an insurer. The MVM is calculated under the assumption that, if the insurer becomes insolvent, another insurer, the reference undertaking, is willing to take over the business and meet his obligations for the value of the technical provision. See figure 2.1. The MVM is the cost of holding extra funds to ensure the ability of paying out the policyholders now and in the future. The yearly rate that denotes the cost of holding extra funds is the cost of capital rate. Chapter 2.2 gives more background information about this rate. The assumptions for the MVM calculation are:

• the reference undertaking takes over the whole portfolio;

• the reference undertaking does not have any funds or (re)insurance obligations before the transfer;

• the assets opposed to the liabilities are chosen in such a way that the market risk of the reference undertaking is minimized;

• the loss absorbing capacity of the technical provision of the reference undertaking equals that of the original undertaking for each risk.

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Figure 2.1: MVM, the market value of the liabilities (MVL) and the market value of the assets (MVA).

IAA (2009) describes four groups of methods to calculate the MVM:

• quantile method: uses percentile, confidence levels or multiples of the second and higher moments of the risk distribution;

• cost of capital method: the insurer should have enough capital to be adequate;

• discount related method: discounts future expected cash flows by the risk free in-terest rate with a risk premium subtracted from the risk free inin-terest rate;

• explicit assumptions method: uses specified required inputs or simpler methods, such as: a minimum loss ratio or a fixed percentage of the MVM.

The Cost Of Capital (COC) method is seen as the most risk sensitive. It possesses the listed requirements for the MVM, posted by IAA, IAIS and IASB, as mentioned above, and meets the five characteristics of the MVM. In other industries risk is priced in a similar way. It takes more effort to implement the COC method compared to the other methods. Liebwein (2006) argues that an internal individual model has to be preferred above the standard model, because it reflects the risk landscape of the company more accurately. EIOPA (2013) writes that it is allowed to use an own internal model, but the model has to be validated.

2.2 Cost-of-capital method

EIOPA (2013) defines the COC rate as the spread above the risk free rate. The principle behind this method is that capital held in own funds to cover the liabilities, can only be invested in marketable securities. This capital ’costs’ the spread above the risk free rate. The annualized COC rate is used to determine the MVM. This calculation is described in chapter 2.3. The COC rate is determined from the perspective of the reference undertaking and not from the original enterprise. It is set on 6%, however Chief Risk Officer Forum (2008) argues that 2.5% – 4.5% is more appropriate and corresponds to a 99.5% confidence level on an annual basis. Gatumel (2008) stresses that it is difficult to assess a COC rate and one uniform risk premium for all risks and all companies. The price of risks depends on the basket of risk at which it belongs, the

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considered risk level and the time period. According toGatumel (2008) the MVM should be made dependent of these factors.

Bundesamt f¨ur Privatversicherungen (2006) writes that the COC method values the assets and liabilities on market value. This concept is tested in 2004 for large companies and in 2005 for large, mid-sized and small companies, covering 90% of the insurance market. The MVM proved to be risk sensitive. A higher duration of the liabilities and more insurance risk resulted in a bigger MVM. Bundesamt f¨ur Privatversicherungen (2006) defines SCRRU(t) as the expected shortfall (Tail Value at Risk) of the change

of risk bearing capital over one year for the reference undertaking (RU). The risk bear-ing capital is the market consistent value of assets (MVA) minus the market consistent value of liabilities (MVL). MVM is the present value of the costs of holding all fu-ture SCRRU(t) during the run-off of the portfolio. See figure 2.2. The calculation of

SCRRU(t) depends solely on the portfolio of the original insurer and is independent of

the portfolio of the reference undertaking. The reason for this is that no information is known about the second insurer. In calculating the MVM certain simplifications can be made. Financial risk is assumed to be hedged as good as possible. There can be accounted for some possible residual financial risk.

Figure 2.2: The development of the future SCRt to be set up by the reference

under-taking.

Bundesamt f¨ur Privatversicherungen (2006) considers the COC method as the best: the policy holder is well protected and the method is transparent. Parameters are observable and are set completely transparent. The capitalization of (6% COC rate) is based on a 99% chance to be able to meet all obligations for the coming year. This corresponds with a Value at Risk (VaR) of 99.6% up to 99.8% and a BBB rating. Bundesamt f¨ur Privatversicherungen (2006) estimates the COC rate of an A or AA rated company to be 3% to 4.5% and for a BBB rated company on 6%. There is no double counting of risks. It is easy for supervisors to check and replicate the calculations for MVM. The MVM can be calculated on different levels of sophistication, simplifications can be made on several levels of risks. The COC method can be applied by all types of (re)insurers: life and non life. This approach ensures that the technical premium charged to the policy holders is not bigger than the technical provision. The technical provision and technical premium are related. Waszink and Voois (2013) argues that the COC method can lead to excessive high MVMs for insurers with liabilities of a high duration. This can be explained by the fact that when the return requirement is larger than the disconto curve, the present value of the return requirement is not limited by its capital requirement itself. The MVM then will exceed the value of the capital requirement. An addition needs to be made to the Solvency II framework.

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2.3 Solvency capital requirement

EIOPA (2013) and Chief Risk Officer Forum (2008): the solvency capital requirement SCRRU(t) is the amount of capital the insurer should hold on top of his best estimate

of his liabilities. This capital ensures 99.5% certainty to meet the obligations for that specific year t. SCRRU(t) covers the following types of risk: the underwriting risk of the

transferred business, the residual market risk, operational risk and risk due to material exposure related to (re)insurance obligations. Formula 2.1 denotes the MVM calculation from SCRRU(t). M V M = COC ·X t≥0 SCRRU(t) (1 + rt+1)t+1 , _(2.1) where:

M V M = Market value margin = Risk margin,

SCRRU(t) = SCR for year t calculated for the reference undertaking,

rt= Risk free rate for maturity t,

COC = Cost of Capital rate.

SCRRU(t) t is calculated by formula 2.2:

SCRRU(t) = BSCRRU(t) + SCRRU,op(t) + AdjRU(t), (2.2)

where:

BSCRRU(t) = basic SCR,

SCRRU,op(t) = operational risk,

AdjRU(t) = adjustment for loss absorbing capacity of technical provisions.

Operational risk, denoted by SCRRU,op(t), captures failures in internal process or from

external events. It covers legal risks as well but not risks due to strategic decisions. EIOPA (2013) states that projections of cash flows used for the calculation of the best estimate should be made per policy. If this is an undue burden for the undertaking, the single policies can be treated as groups, if the following requirements are met:8

• the policies in the same group have equal nature and are subject to the same type of risks;

• the risk underlying the group of policies and the expenses are well estimated; • the best estimate of the separate policies summed should about equal the best

estimate of the group of policies.

2.4 Life risk

EIOPA (2013) describes how the capital requirements are calculated of all (sub-)risks and how they combine to the total BSCRRU(t). By using a certain correlation structure,

BSCRRU(t) can form together with SCRRU,op(t) the MVM. Figure 2.3 shows SCRRU(t)

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Figure 2.3: SCR tree of (sub)risks.

The solvency capital requirement for life risk (SCRlif e(t)) is calculated as shown in

formula 2.3:

SCRlif e(t) =

s X

r,c

CorrLif er,c· Lif er(t) · Lif ec(t), (2.3)

where:

CorrLif er,c = correlation between the sub risks,

Lif er, Lif ec= capital requirements for sub risks ’r’ and ’c’,

r, c = sub life risks.

Figure 2.4: Correlation matrix of life sub risks.

The correlation matrix of the sub risks (CorrLif er,c), is defined as in figure 2.4. The

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(BOF) when a shock is applied to the rates used for the calculation of the best estimate of the assets and the liabilities. BOF needs to be calculated in both the shocked and the basic scenario. The difference between these two forms the sub risk. The (sub) risk is the maximum of zero and the loss, the loss can not be negative. Every sub risk applies a different shock.

Lif ej(t) = max [(∆BOF (t) | shock) ; 0] , (2.4)

where:

∆BOF (t) | shock = change in the value of basic own funds,

= [(M V A(t) | shock) − (M V L(t) | shock)] − [M V A(t) − M V L(t)] , shock = instantaneous increase of all future mortality or lapse rates used.

The shock for mortality risk is a permanent instantaneous upward shock to the mortality rates of 15%. The mortality rates can not exceed the value 1. For longevity risk it is a permanent instantaneous downward shock of 20% to the mortality rates. Lapse risk (Lif elapse) is calculated in a slightly other way compared to the other sub life risks.

This calculation is described in formula 2.5. The shocks are bounded, see formula 2.6.

Lif e(t)lapse= max

h

Lapse (t)_down; Lapse (t)_up; Lapse (t)_massi, (2.5) where:

Lif e(t)lapse = total capital requirement for lapse risk,

Lapse(t)down = capital requirement due to a 50% decrease in lapse rates,

Lapse(t)up = capital requirement due to an increase of 50% in lapse rates,

Lapse(t)mass= captial requirement due to a mass lapse event.

R(t)up(R (t)) = min[150% · R ; 100%],

R(t)down(R (t)) = max[50% · R ; R − 20%],

(2.6)

where:

R(t)up= shocked lapse rate in upward scenario,

R(t)down= shocked lapse rate in downward scenario,

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Chapter 3

Methodology

This chapter discusses different methods where to apply the risk driver. In chapter 3.1 the model and mathematical assumptions are outlined, followed by an explanation of the best estimate of an insurance product in section 3.2. In chapter 3.3 the emphasis is on the process of the risk margin calculation. In chapter 3.4 the definitions of the risk driver are given, chapter 3.5 describes the proposed risk drivers and in chapter 3.6 the mathematical derivation of the risk driver for one policy is examined. In chapter 3.7 the method is extended to groups of policies.

3.1 Mathematical assumptions

All cash flows are due, except for the death benefit and surrender value, which are paid out at the end of the year. The valuation date is denoted by t = 0. Symbol x means the age of the insured at the valuation date. Starting from the valuation date, n denotes the remaining term of the policy, n, can be infinite, but for practical purposes it is limited to 100. The remaining premium time is denoted by m, starting from the valuation date. The policy is valued up to 100 years from the valuation year onwards. Suppose that m = 5, which means that the policyholder pays premiums for 5 more years. I.e. the first premium payment is on t = 0 and the last on t = 4, because all premium payments are due. Suppose n = 20 for a death benefit, the last benefit payment can be on the end of the 19th year. In the model this cash flow is set on t = 20, because it has to be dis-counted with D(20). Premium payments are due, therefore the last premium payment is on t = m − 1. Figure 3.1 illustrates the time line of one policy. The valuation date is on t = 0. The policy starts prior or on t = 0. The insurance taker pays premium starting of the commencement of the policy and the end of the premium term at t = m. The time t = k is a specified time between t = 0 and t = n.

? ? ? ? ?

begin policy t = 0: valuation date t = k t = m t = n

Figure 3.1: time line with k, m and n defined.

3.2 Best estimate of insurance product

If the time value of options and guarantees is neglected, following Bruning (2011), the best estimate of the liabilities can be written as the sum of the present value of the benefits, the surrender value and the expense minus the present value of the gross premium, as is shown in equation 3.1.

On an arbitrary future point in time t = k:

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BE(k)liabilities= BE(k)benefits+ BE(k)surrender+ BE(k)expense− BE(k)gross premium.

(3.1) The market values of the best estimates of these cash flows: the benefits, surrender values, costs and gross premiums can be calculated as in equation 3.2:

BE(k)cash flow=kp

0 x· n X j=k cash flow(j) ·j−kp 0 x+k· D(k, j) · hx+j, (3.2) where:

BE(k)cash flow= the market value on time k of the best estimate of a certain cash flow: benefits,

expense, surrender benefit or gross premium,

j−kp

0

x+k = probability of survival of the ‘status’ of the policy in time t = [k, j]: insured

does not die and does not surrender,

D(k, j) = discount rate, discounting a cash flow from t = j to t = k, with j >= k, hx+j = the one-year probability that the the insurer has to pay out the

certain cash flow to the insured for that year, n = remaining term of policy in years,

m = remaining term of premium payment in years, x = age of the insured in years on t = 0,

k >= 0, reference time.

(3.3) This research focuses on mortality, longevity, lapse up and lapse down risk. The shocks to calculate these risks only affect the liability side of the balance sheet. Therefore only the market value of the liabilities have to be considered. This gives the result, that is shown in equation 3.4:

Lif e(k)j = (∆BOF (k) | shock),

(∆BOF (k) | shock) = [M V A(k)basic− M V L(k)basic] − [M V A(k)shocked− M V L(k)shocked], Since M V A(k)shocked= M V A(k)basic, this gives:

(∆BOF | shock) = M V L(k)shocked− M V L(k)basic_,

(3.4) where:

Lif e(k)j = capital requirement for life risk j on t = k,

(∆BOF (k) | shock)) = change in value of basic own funds after applying a shock on the rates on t = k, M V A(k)basic = value of the assets in the basic scenario on t = k,

M V L(k)basic = value of the liabilities in the basic scenario on t = k, M V A(k)shocked = value of the assets in the shocked scenario on t = k, M V L(k)shocked = value of the liabilities in the shocked scenario on t = k.

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3.3 Process behind risk margin calculation

In line with Solvency II, a.s.r. applies the COC method to calculate MVM. Instead of the full calculation, a.s.r. uses ’risk drivers’, to project in time the capital requirement for the sub risks. The risk driver optimally covers the underlying risk. The advantages

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of the proposed risk drivers are that they are sensitive to changes in interest rates due to its discounting element and that they are less prudent than the conventional risk drivers. A disadvantage is that more data is needed. Besides the cash flow data from the basic scenario, data from the shocked scenario is needed as well. The newly proposed risk drivers are defined in table 3.1 in paragraph 3.4. The risk driver for mortality risk, longevity risk and lapse up and down risk is the same. It contains the ratio of the number of policies in the basic scenario and the number of policies in the shocked scenario. The addition of this term can be understood intuitively. The shock causes the number of policies to change. Therefore the risk driver needs adjusting. This works well for one policy. In chapter 3.7 this method is extended to groups of policies.

a.s.r. uses a software package to project future cash flows. The status probabilities, surrender rates, paid up rates and mortality rates are processed. Among the projected cash flows, needed for the risk driver calculations, are: gross premium, expenses, benefits, surrender values, reserves, profit and loss and the balance sheet. First the software package calculates the expected cash flow per policy and then accumulates the cash flows to a model point. Another software package discounts the output to form the market value. This software package takes care of the stochastic part of the market value calculations as well to calculate time values of options and guarantees: different economic scenarios are considered. Part of the output of these software packages are: BE(k)Band ^BE(k)S_{. BE(k)}B_{is the best estimate of the liabilities in the basic scenario.}

^

BE(k)S _{is the best estimate of the liabilities in the shocked scenario. The shock rates}

are applied starting from t = 0. To determine SCR(k) for every k, the cash flow BE(k)S is needed, the best estimate of the liabilities calculated with unshocked rates from the basic scenario up to t = k and from t = k on with the shocked rates. This takes a lot of computation time, because for every SCR(k) a new projection of cash flows is needed. This is the main reason why risk drivers are used to speed up the process. BE(k)S is not available as output from both software packages. A driver is used to transform BE(k)S from BE(k)B. The process can be improved when cash flows from the output of the software packages can be used as risk drivers, because then all the data needed is already available and programmed in the systems.

3.4 Definition and use of risk driver

A risk driver is approximately linearly related to SCR(k). It is a known entity or can be derived for all points in time. Since SCR(0) is calculated, the projection of SCR in time (SCR(k)) can be made, using the risk driver. In equation 3.6 this is illustrated. A risk driver can be found by using equation 3.7.

\

SCR(k) = (driver(k)

driver(0)) · SCR(0). (3.6)

Rewriting equation 3.6, taking into account the approximate linearity, leads to equation 3.7:

SCR(k) SCR(0) ≈

driver(k)

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where:

\

SCR(k) = the projected capital requirement, using a risk driver, SCR(k) = solvency capital requirement for year k,

SCR(0) = solvency capital requirement for year 0, driver(k) = value of the risk driver for year k, driver(0) = value of the risk driver for year 0.

3.5 Risk driver for one policy

Table 3.1 outlines the risk driver. Mortality risk, longevity risk, lapse up and lapse down risks have the same risk driver. The capital requirement of these risks are caused by a uniform shock in the status probabilities.

Table 3.1: ASR documentation (2013) Risk driver for one policy.

Lif e(k)mort= [BE(k)^mort· (kp

B0 x

g

kpS0x

)] − BE(k)basic,

Lif e(k)long= [BE(k)^long· (k pB0_x g

kpS0x

)] − BE(k)basic,

Lapse(k)up= [BE(k)^lapse up· (k pB0_x g

kpS0x

)] − BE(k)basic,

Lapse(k)down [BE(k)^lapse down· (k pB0 x g kpS0x )] − BE(k)basic, where: ^

BE(k)mort= best estimate calculated with a permanent upward shock in

mortality rates, with shocked rates applied from time t=0 on, ^

BE(k)long = best estimate calculated with a permanent downward shock in

mortality rates, with shocked rates applied from time t=0 on, ^

BE(k)lapse= best estimate calculated with a permanent increase in lapse rates,

with shocked rates applied from time t = 0 on,

BE(k)basic= best estimate calculated without any shocks: the basic scenario, kpB

0

x = status probability in the basic scenario,

g

kpS

0

x = status probability in the from time t = 0 on shocked scenario.

The derivation of the driver in table 3.1 will be done in chapter 3.6. The equations in table 3.1 are exact except for the effect of options and guarantees. So the equations of table 3.1 have to be translated to a risk driver including these effects. The model con-sidered here follows table 3.1, since options and guarantees are not taken into account. In equation 3.8 is, for illustration purposes, given the explicit formula of the risk driver.

\ SCR(k) = SCR(0) · ( kpB0x g kpS0x · ^BE(k)S_{− BE(k)}B ^ BE(0)S_{− BE(0)}B ), (3.8) where:

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\

SCR(k) = solvency capital requirement using the approximate risk driver, SCR(0) = the exact calculated Solvency capital requirement, including the effect

of options and guarantees,

kpB

0

g

kpS

0

^

BE(k)S _{= best estimate of the liabilities with shocked rates applied from time t = 0 on,}

excluding the effect of options and guarantees,

3.6 Mathematical derivation for one policy

The derivation of the risk driver for mortality, longevity, lapse up and lapse down risk is similar, because the same shock method is used. Only the calculation of the best estimate of the death benefit is shown here, because the calculation of the best estimates of the other cash flows is similar. The best estimate of the cash flows can be summed, because the terms are independent, as shown in equation 3.1. The basic calculation of SCR(k) is described in equation 3.9. The best estimate of all cash flows are summed in the basic scenario and in the shocked scenario in order to determine SCR(k).

SCR(k) = BE(k)S− BE(k)B, (3.9)

where:

BE(k)S = the best estimate of all cash flows in the shocked scenario with shocked mortality rates applied starting from time t=k. This means that

before time t = k the basic scenario is applied,

BE(k)B= the best estimate of all cash flows in the basic scenario.

In this derivation only payments upon death are considered. The term U O(j, j +1)insured denotes the actual benefit paid out upon death. The terms U O(j, j + 1)B and U O(j, j + 1)S form the expected payments upon death in the basic respectively the shocked sce-nario for every year. From the regular process the cash flows of expected payments are known in both the basic and in the shocked scenarios. Calculating mortality and longevity risk a uniform instantaneous shock is applied to the mortality rates. Lapse up and lapse down risk applies a uniform instantaneous shock to the lapse rates. All shocks result in changed status probabilities compared to the basic scenario. As described in the equations 3.9, SCR(k) is calculated by subtracting the best estimate of the liabilities in the basic scenario from the best estimate in the shocked scenario.

BE(k)B = kp B0 x D(k)· ∞ X j=k j−kpB 0 x+k· qBx+j· D(j + 1) · U O(j, j + 1)insured, = 1 D(k)· ∞ X j=k U O(j, j + 1)B· D(j + 1), (3.10)

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where:

kpB

0

x = the status probability in the basic scenario,

qB_x = mortality rate in the basic scenario, D(j) = discounting factor,

U O(j, j + 1)insured = actual benefit to be paid out upon death

between t = [j, j + 1) and paid out at time t = j + 1;

U O(j, j + 1)B = the expected not discounted insured capital or benefit in the basic scenario, in the year t = [j, j + 1).

See equation 3.10. The best estimate of the death benefit for the basic scenario can be calculated by the summation of the multiplication of the insured benefit with the probability that the benefit will be paid out. This can be seen as a discounted expectation of the benefit to be paid out from year k to the future. The discounted factor is set on time j + 1, because it is assumed that the benefit payment is made at the end of the year of death. The symbol U O(j, j + 1)insured is the actual benefit to be paid out upon death of the insured, known as the insured capital. It is a fixed amount of money, but this is not the case for paid up policies. U O(j, j + 1)B can be seen as the not discounted expected payment on death in the year t = [j, j + 1). The probabilities of the benefit to occur are processed in this term, but not the discounting element. The second equation in equation 3.10 is derived from the first by relocating the term ‘kpB

0

x ’ in the summation.

The part after the summation in the second equation gives U O(j, j + 1)B, the known cash flow in the basic scenario. For BE(k)S we find:

BE(k)S = kp B0 x D(k)· ∞ X j=k j−kpS 0 x+k· qx+jS · D(j + 1) · U O(j, j + 1)insured, (3.11) where: j−kpS 0

x+k= status probability in the shocked scenario,

q_x+jS = mortality rate in the shocked scenario.

Equation 3.11 is almost equal to equation 3.10, except that in the summation the term ‘j−kpB

0

x+k’ is replaced by ‘j−kpS

0

x+k’ and ‘qBx+k’ is replaced by ‘qSx+k’. This can be explained

by the fact that up to point ’t = k’ there is no shock in the mortality rates. The shock to these rates is uniform from time ‘t = k’ to the future. That is the reason that after the summation term the shocked probabilities and before the summation term the basic probabilities are used. First the policy status has to survive until time t = k, then the shock occurs uniform for all the future years. Formula 3.11 can be rewritten to a formula consisting only of terms from the basic scenario and from the time t = 0 on shocked scenarios, see formula 3.12. This formula makes it possible to calculate BE(k)S from the given available cash flows of a.s.r.

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16 C.E. de Vries - Stam — Projection Methods for Life Underwriting Risk BE(k)S= kp B0 x D(k) · g kpS 0 x g kpS 0 x · ∞ X j=k j−kpS 0 x+k· qx+jS · D(j + 1) · U O(j, j + 1)insured, = 1 D(k) · kpB 0 x g kpS 0 x · ∞ X j=k g jpSx · qSx+j· D(j + 1) · U O(j, j + 1)insured, = 1 D(k) · kpB 0 x g kpS 0 x · ∞ X j=k ^ U O(j, j + 1)S_{· D(j + 1),} (3.12) where: ^

U O(j, j + 1)S _{= the expected not discounted death benefit in the shocked scenario,}

between time t= [j, j + 1), calculated with shocked rates from time t = 0 on,

g

kpS

0

x = shocked status probability calculated with shocked one year,

mortality rates from time t = 0 on.

In equation 3.12 the ratio kgpS0x

g

kpS0x

is added to the equation 3.11, which is possible because the ratio equals ‘1’, so there is no change in value. In the second equation in equation 3.12 the numerator of this fraction is placed in the summation, which is possible because the numerator is a constant. The result is the expected discounted benefit in the from t = 0 on shocked scenario behind the summation with a term in front. This term in front of the summation is to correct for the fact that up to time t = k the normal mortality rates and lapse rates are to be applied. So the third equation in equation 3.12 shows that for one policy BE(k)S can be written in terms of the basic scenario and terms of the from t = 0 on shocked scenario. For one policy formula 3.13 can be applied to the ratio in the risk driver to form life risk, because it can be mathematically derived.

#policies(k)B ^ #policies(k)S = amount of policies (0) ·kp B0 x amount of policies (0) · gkpS 0 x = kp B0 x g kpS 0 x . (3.13)

The derivation discussed in this section can not be applied to catastrophe risk, lapse mass risk and expense risk, because the character of the shock is different. Catastrophe risk and lapse mass risk are characterised by the same type of shock: an instantaneous increase in the mortality rates or lapse rates for only the first year, which causes a decrease of the status probabilities in the first and coming years. The shock in the first year influences the status probabilities in the years to come. However for lapse mass risk and catastrophe risk by adding a time dimension to the status probabilities and mortality rates and by working with matrices instead of lines, probably a similar equation is useful. This research will not concentrate on this topic. For expense risk there is a change in the expense rates and thus in the cash flow of the costs, this risk driver only works for a change in the status probabilities.

3.7 The risk driver and groups of policies

How can the risk driver be applied to groups of policies? The best estimates calculated from the output of the software packages used, are sums of the single cash flows of all policies together in a portfolio and not separate. Therefore the ratio ”kpB0x

g

kpS0x

” in the risk driver must be approximated for all policies in the portfolio. It seems logical to create average survival probabilities. In this research three weighting methods are explored.

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The ‘normal weighted method’ approximates the ratio by equally weighting all status probabilities of the policies in the portfolio. The ‘capital weighted method’ weights the survival status probabilities by their insured capital. The ‘reserve weighting method’ uses the reserves to define the ratio. The reserves are cost price based and are an output of the software packages used, so it is easily applied. In all three weighting methods a policy does not put weight in the ratio if the policy is ended or the insured is older than 120 years.

Table 3.2: Different weighting methods for approximating the ratio ”kpB0x

g

kpS0x

“ in the risk driver.

Normal weighting method: kpB0x

g kpS0x ≈ 1 #policies(k)B· #policies(k)B P i=1 k pB0_x,i 1 #policies(k)S· #policies(k)S P i=1 ] kpS0x,i =#policies(k)B ^ #policies(k)S

Capital weighting method: kpB0x

g kpS0x ≈ 1 T IC· #policies(k)B P i=1 k pB0 x,i·IC(i) 1 T IC· #policies(k)S P i=1 ] kpS0x,i·IC(i)

Reserve weighting method: kpB0x

g kpS0x ≈ #policies(k)B P i=1 kpB0x,i·kViN etto #policies(k)S P i=1 ] kpS0x,i·kViN etto where:

i = policy i, sokpBx,i denotes the status probability for that specific policy i,

#policies(k)B = amount of policies in the basic scenario,

#policies(k)S = amount of policies in the from time t = 0 on shocked scenario, T IC = total insured capital of all policies together,

IC(i) = insured capital of policy i,

kViN etto= the net reserve cost price based of policy i, if the policy exist on t = k.

Life risks of the portfolios are calculated in five ways, as described in table 3.3. Method 1 calculates life risk in the exact way policy by policy and then sums the risks of all policies to accomplish the total amount of capital requirement. Method 2, 3, and 4 first add the cash flows of all the policies in the portfolio and then calculate the risks by applying the risk drivers. Method 2 uses the normal weighting method for determining the quotient in the risk driver, method 3 the capital weighting method and method 4 uses the reserve weighting method. These weighting methods are described in 3.2. Method 5 calculates life risk per policy using the risk driver and then sums up the life risk for all policies in the portfolio. Method 5 is only matters for the portfolio calculated with paid up rates instead of surrender rates. For the other portfolios method 5 gives equal results to method 1. It can be seen as a mean to exercise control on the exact method. In this research life risk of four types of portfolios are calculated: whole life insurance with surrender rates applied and with paid up rates implemented instead of surrender rates, term insurance and immediate annuities with one single premium payment at the start of the policy.

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Table 3.3: Calculation methods for mortality risk, longevity risk, lapse up risk and lapse down risk.

Per policy or

Method Type per model point Weighting method Referred to in the charts with: 1 Exact method Per policy a solid light blue line

2 Risk driver method Per model point Normal, see table 3.2 a red dashed line 3 Risk driver method Per model point Capital, see table 3.2 a green line: dash dot 4 Risk driver method Per model point Reserve, see table 3.2 a purple line: dash dot dot 5 Risk driver method Per policy an orange line: small dashes

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Implementation of paid up policy

status and the risk driver

The policyholder of a whole life insurance has the right to terminate premium payment, while continuing the policy with reduced insured capital. The policyholder pays up the policy, also ‘pupping’. The policy is called a paid up (pup) policy. Chapter 4.1 explores the ability of implementing ‘pupping’ in the model. Chapter 4.2 shows mathematically what happens in the model with the implementation of the paid up policy status and how the risk driver responds to this.

4.1 Incorporation of paid up policy status and

assump-tions

The benefit payment upon death of a paid up policy is modelled as 95% of the net provision divided by the purchase price of a funeral policy of a death benefit of 1, as described in formula 4.1. The paid up policy death benefit is cost price based. A policy can only be made paid up before the last premium payment. If a a policy is made paid up between t = [k − 1, k) and the insured dies between time t = [k − 1, k), the insured will receive the full death benefit at t = k. If the insured dies after or on t = k, the insured will receive the reduced death benefit, as described above.

So, if a policy is made paid up, starting from the next premium payment, the insured has right on the reduced benefit. If there is pupping, there is no surrender and vice versa, because a paid up policy is not likely to surrender and a surrendered policy can not be made paid up. Lapse risk is calculated then by shocked paid up rates instead of shocked surrender rates.

Pupping can be incorporated in the model as described in formulas 4.1 and 4.2. In Excel U O(j, j + 1)B,accpup can be implemented by recursion, since U O(−1, 0)B,accpup = 0 is

known. It means that at time t = 0 it is assumed that there are no paid up policies. This assumption is made to ease the modelling. Otherwise the model should account for all the policies made paid up far back in time. The term ‘U O(j, j + 1)insured_pup ’ is the reduced death benefit when a policy is pupped between time t = [j, j + 1). The term U O(j, j + 1)B,accpup consists of two groups: the group that pups in year t = [j, j + 1)

and the group that already pupped before. The group that pupped before stays in the U O(j, j + 1)B,accpup group by not dying. The group that pups that year has survived the

status of not dying and not pupping up to time t = j and pups in the year t = [j, j + 1) while not dying.

U O(j, j + 1)insured_pup = 95% ·j+1V

N et

A

x+j+1:n−j+11

, (4.1)

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U O(j, j + 1)B,acc_pup = U O(j − 1, j)B,acc_pup · (1 − q_x+jB )+ U O(j, j + 1)insured_pup ·jpB

0

x · (1 − qx+jB ) · pupBx+j,

(4.2)

where:

U O(j, j + 1)insured_pup = fixed cost price based insured capital when a policy is made paid up between t = [j, j + 1),

U O(j, j + 1)B,acc_pup = accumulated value of the insured amount of all paid up basic scenarios between time t = [j, j + 1),

U O(−1, 0)insured_pup = 0 : model starts with 0 paid up policies, U O(−1, 0)B,acc_pup = 0 : model starts with 0 paid up policies,

jpB

0

x = status probability in the basic scenario,

q_x+jB = one year mortality rate in the basic scenario,

pupB_x+j = one year probability that the policy is made paid up in the basic scenario. The best estimate of death benefit in the basic scenario including paid up policy

prob-abilities, is calculated as stated in formula 4.3 and 4.4.

U O(j, j + 1)B_pup = U O(j, j + 1)insured_normal ·jpB

0

x · qBx+j+ U O(j − 1, j)B,accpup · qBx+j, (4.3)

BE(k)B_pup= 1 D(k) · ∞ X j=k U O(j, j + 1)B_pup· D(j + 1), = 1 D(k) · ∞ X j=k

[U O(j, j + 1)insured_normal ·jpB

0

x · qBx+j+ U O(j − 1, j)B,accpup · qx+jB ] · D(j + 1),

(4.4) where:

U O(j, j + 1)B_pup = expected death benefit with paid up rates processed for time t = [j, j + 1) in the basic scenario,

BE(k)B_pup = the best estimate of death benefit calculated with ‘pupping’ instead of ‘surrender’ rates,

kpB

0

x = the status probability in the basic scenario,

D(k) = the discounting factor,

U O(j, j + 1)insured_normal = the death benefit paid out upon death if the policy is not made paid up,

U O(−1, 0)insured_normal = 0: model starts with no deaths, thus no payments at t = 0.

4.2 Paid up policy status and the risk driver

This chapter describes in formulas what happens in the model when paid up status is implemented and whether the risk driver responds adequately to this implementation. If it is possible to write Lif e(j) in terms of the totally shocked scenario, where the

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shocked probabilities are used for all points in time, and in terms of the unshocked basic scenario, this would be a great enhancement, because it makes the calculation less complex. For mortality and longevity risk the following formulas can be applied. Lapse up and down risk give similar formulas, except that the paid up rates are shocked instead of the mortality rates. In the shocked scenario in the model the term ‘U O(j, j +1)insured

pup ’

does not change due to the shocked rates, because it is fixed cost price based. It is a benefit promised to the insured. The term U O(j, j + 1)B,accpup does change, as is shown in

formula 4.5.

^

U O(j, j + 1)S,accpup =U O(j − 1, j)^ S,accpup · (1 − qSx+j)+

U O(j, j + 1)insured_pup · g_jpS0

x · (1 − qx+jS ) · pupBx+j,

(4.5)

where:

^

U O(j, j + 1)S,accpup = accumulated expected value of the insured capital in the

from time t = 0 on shocked scenario, g

jpS

0

x = the status probability in the from time t = 0 on shocked scenario,

qS_x+j = the one year mortality rate of the shocked scenario,

pupB_x+j = one year probability that the policy is made paid up in the basic scenario.

The model calculates the expected death benefit with paid up rates implemented as in formula 4.6. The best estimate of the death benefit can be determined as is shown in formula 4.7:

^ U O(j, j + 1)S

pup = U O(j, j + 1)insurednormal · gjpS

0 x · qSx+j+U O(j − 1, j)^ S,acc pup · qSx+j, (4.6) ^ BE(k)S pup= 1 D(k) · ∞ X j=k ^ U O(j, j + 1)S pup· D(j + 1), = 1 D(k) · ∞ X j=k

[U O(j, j + 1)insured_normal · gjpS

0 x · qSx+j+U O(j − 1, j)^ S,acc pup · qSx+j] · D(j + 1), = 1 D(k) · ∞ X j=k

[[U O(j, j + 1)insured_normal · gjpS

0

x · qx+jS +[U O(j − 2, j − 1)^ S,acc

pup · (1 − qSx+j−1)

+ U O(j − 1, j)insured_pup · ^_j−1pS0

x · (1 − qx+j−1S ) · pupBx+j−1] · qSx+j]] · D(j + 1).

(4.7) where:

^ BE(k)S

pup = Best estimate in the from time t=0 on shocked scenario.

The risk driver places the term ‘kpB0x

g

kpS0x

’ in front of the best estimate in the starting from time t = 0 on shocked scenario. This is shown in formula 4.8. The aim of the risk driver is to rewrite _BE(k)^S

pup to BE(k)Spup, the best estimate in the starting from time t = k

on shocked scenario. It seems however that the risk driver does not have the desired effect. The formulas 4.8, last equation, middle part shows that: ‘+U O(j − 2, j − 1)^ S,accpup ·

(1 − qS

x+j−1) · qSx+j’ does not contain the term ‘ gkpS

0

x ’. It seems as if ‘ gkpS

0

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placed in front of the summation by applying the risk driver and hereby_BE(k)^S pup can

not be rewritten to BE(k)S_pup.

kpB 0 x g kpS 0 x ·_BE(k)^S pup = 1 D(k)· kpB 0 x g kpS 0 x · ∞ X j=k

[[U O(j, j + 1)insured_normal · g_jpS0

x · qSx+j

+[U O(j − 2, j − 1)^ S,accpup · (1 − qx+j−1S )

+ U O(j − 1, j)insured_pup · ^_j−1pS0

x · (1 − qSx+j−1) · pupBx+j−1] · qx+jS ]] · D(j + 1), = 1 D(k)· kpB 0 x g kpS 0 x · ∞ X j=k

[[U O(j, j + 1)insured_normal · gkpS

0

x ·j−kpS

0

x+k· qx+jS

+ [U O(j − 2, j − 1)^ S,accpup · (1 − qx+j−1S )

+ U O(j − 1, j)insured_pup · g_kpS0

x ·j−k−1^pS 0 x+k· (1 − q S x+j−1) · pupBx+j−1] · qx+jS ]] · D(j + 1), = 1 D(k)· kpB 0 x g kpS 0 x · ∞ X j=k

U O(j, j + 1)insured_normal · g_kpS0

x ·j−kpS

0

x+k· qSx+j· D(j + 1)

+U O(j − 2, j − 1)^ S,accpup · (1 − qx+j−1S ) · qx+jS · D(j + 1)

+ U O(j − 1, j)insured_pup · g_kpS0

x ·j−k−1pS

0

x+k· (1 − qSx+j−1) · pupBx+j−1· qSx+j· D(j + 1).

(4.8)

However, because the termU O(j − 2, j − 1)^ S,accpup contains gkpS

0

x as well, see formula 4.5,vit

is possible that the risk driver works well after all. It is difficult to prove this mathe-matically, because of the recursion in formula 4.5. It is questionable whether the last equation in formula 4.8 can be rewritten in an expression consisting of the basic sce-nario and the from time t = 0 on shocked scesce-nario. See formula 4.9. The results of the calculations will show if this is the case.

kpB 0 x g kpS 0 x ·_BE(k)^S pup = 1 D(k)· kpB 0 x g kpS 0 x · ∞ X j=k

U O(j, j + 1)insured_normal · gkpS

0

x ·j−kpS

0

x+k· qSx+j· D(j + 1)

+ U O(j − 1, j)insured_pup · gkpS

0 x ·j−k−1pS 0 x+k· (1 − qSx+j−1) · pupBx+j−1· qSx+j· D(j + 1), ? = 1 D(k)· kpB 0 x g kpS 0 x · ∞ X j=k

U O(j, j + 1)insured_normal ·_j−kpS_x+k0 · qS

x+j· D(j + 1)

+ U O(j − 1, j)insured_pup ·_j−k−1pS_x+k0 · (1 − q_x+j−1S ) · pupB_x+j−1· qS_x+j· D(j + 1), = BE(k)S_pup,

(4.9) where:

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Results

This chapter analyses life risk calculations of the model points, given by a.s.r. Mortality risk, longevity risk, lapse up risk and lapse down risk of the following portfolios are calculated: a portfolio of whole life insurance with surrender rates processed and with paid up rates processed instead of the surrender rates, a portfolio of term insurance and of immediate whole life annuity.

The portfolio of immediate whole life annuity is a portfolio consisting of policies with one single premium payment at the start of the policy. The first annuity payment is the year after the premium payment, so it can be seen as a one year deferred whole life annuity due as well. The capital requirements are calculated in five ways, see table 3.3. Method 1 (referred to in the charts with a solid blue line) is the exact full calculation. Method 2 to 5 apply the risk driver, but use different weighting methods to determine the ratio of survival probabilities in the risk driver. Method 2 (referred to in the charts with a red dashed line) applies the normal weighting method, method 3 (referred to in the charts with a green line dash dot) the capital weighting method and method 4 (referred to in the charts with a purple line dash dot dot) the reserve weighting method, see table 3.2 for information about the weighting methods. Method 5 (referred to in the charts with an orange line of small dashes) does not use a weighting method, but applies the risk driver per policy. Method 5 is only relevant for the analysis of the implementation of paid up policy status. For the other portfolios method 5 gives results equal to method 1, as expected.

Chapter 5.1 gives more information about the assumptions made. Chapter 5.2 anal-yses the results of the whole life insurance portfolio with surrender rates processed and chapter 5.3 with paid up rates processed. Chapter 5.4 discusses the portfolio of the term death insurance. In this chapter ‘pup rates’ are not processed. The policyholder can not pay up the policy, because with term death insurance this is not allowed. The results of immediate whole life annuity are analysed in chapter 5.5. To pay up the policy or to lapse is not possible for life annuities. In chapter 5.6 some concluding results are given.

5.1 Model assumptions

The most up-to-date mortality table with trend from AG&AI is applied, running from 2011 to 2062. A swap rate curve, valuing from December 31th2012 onwards and

accord-ing to the principles of QIS5, is used for the discountaccord-ing of the cash flows. Premiums and surrender values are determined based on an old mortality table; AG&AI 2000 – 2005, and a fixed interest rate of 3%. The valuation date is considered to be January 1st 2013. The surrender value is set to 95% of the net supply. The graphics of mortal-ity, longevmortal-ity, lapse up and lapse down risk of the methods are compared. The average linear deviation (ALD) in percentage of the methods compared to the exact method 1, is calculated according to formula 5.1 for each risk and for all methods. In this research the ALD is preferred to the average absolute deviation (AAD), see formula 5.2, because

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ALD shows the sign (positive or negative) of the deviation. This is important, because a negative ALD means that the method underestimates that certain life risk and a pos-itive ALD means that the method overestimates that certain life risk. However AAD give similar outcomes: it prefers the same methods, but with different numbers. The minimum and maximum linear deviations (M inLD and M axLD) are calculated as in formula 5.3 for each risk and each method. The best method approaches the outcomes of the exact calculation as precisely as possible which means that ALD and AAD should be (close to) zero.

ALDi,j = 1 n · n X k=0

method(k)i,j− exact(k)i

exact(k)i

· 100%, (5.1)

where:

ALDi,j = the average linear deviation for risk i of method j from the exact

method 1 in percentage on time t = k,

method(k)i,j = capital requirement on time t = k of risk i calculated by method j,

exact(k)i = capital requirement on time t = k of risk i calculated by exact method 1,

k = point in time,

i = mortality, longevity, lapse up or lapse down risk, j = method 2, 3, 4 or 5,

n = number of points in time that have capital requirement values.

AADi,j = 1 n · n X k=0

|method(k)_i,j− exact(k)_i| exact(k)i

· 100%, (5.2)

where:

AADi,j = the average absolute deviation for risk i of method j from the

exact method 1 in percentage on time t = k.

M inLDi,j = M inimum(

exact(k)i

),

M axLDi,j = M aximum(

exact(k)i

),

(5.3)

where:

M inLDi,j = the minimum linear deviation of risk i calculated by method j for all

points in time,

M axLDi,j = the maximum linear deviation of risk i calculated by method j for all

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Figure 5.1: The projection of mortality risk according to method 1, 2, 3 and 4 of whole life insurance portfolio with surrender rates applied and with time in years on the x-axis and the amount of capital requirement on the y-axis.

Figure 5.2: The projection of lapse down risk according to method 1, 2, 3 and 4 of whole life insurance portfolio with surrender rates applied and with time in years on the x-axis and the amount of capital requirement on the y-axis.

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5.2 Whole life insurance with surrender rates processed

See figure 5.1. Method 2 and 3 deviate substantially, however after 78 years the results of both methods move closer together. The capital requirements of method 2 become even bigger than the exact method 1 after 78 years. This can be explained by the fact that the portfolio probably consists of more elderly people and only a few young. As a consequence the average death rate drops because of the few young people compared to the more elderly people and the capital requirements of mortality risk turns out to be lower in the beginning. After a while, also the mortality rate of the few young people increases, which causes after 78 years both methods to converge to the results of the exact method. Also, only the young insured are left, so there are no more big differences in age of the insured. The ALDs are: of method 2: −20.1%, of method 3: −33.0%, of method 4: −13.1%. From method 2 to 4, method 4 performs best. Method 4 uses the reserves to determine the ratio of the risk driver. The reserve captures information as the term of the policy, the death rates, term of the premium payments and the insured capital. That is why method 4 probably is a better approximation of the ratio of the risk driver in comparison to method 2 and 3. See table 5.1. However ALD AAD of method 4 are considerably large as well. The cause of this can be the difference in age of the insured and the fact that the reserve used for the weighting of the status probabilities is cost price based. The fixed interest rate that discounts the reserve is larger than the interest rate curve used for the market valuation of the liabilities. This can be a cause as well.

Table 5.1: ALDs, AADs, M inLDs and M axLDs of mortality risk calculated by method 2, 3 and 4 of whole life insurance portfolio with surrender rates applied.

Method ALD AAD M inLD M axLD 2 −20.1% 27.5% −61.2% 30.8% 3 −33.0% 33.0% −65.6% 0.0% 4 −13.1% 13.1% −30.2% 0.0%

See figure 5.2. Lapse down risk gives some small capital requirements over time according to method 1. Methods 2, 3 and 4 however give no capital requirements. This is due to the fact that the methods 2 to 4 sum the best estimate of the liabilities of all policies and then calculate lapse down risk for all these policies together. So the negative capital requirements for lapse down risk and the positive capital requirements for lapse down risk cancel each other out, resulting in no capital requirements. However method 1 determines the capital requirement for each policy separately and sums the capital requirements. The capital requirement is the maximum of zero and the amount, so the negative and the positive capital requirements do not cancel each other out. Apparently the portfolio consist of a few policies with a small capital requirement for lapse down risk, while the main part of the portfolio does not have capital requirement for lapse down risk. The portfolio needs to be split in two groups: policies with lapse down risk and policies without.

Figure 5.3 shows similar results for all methods for lapse up risk. Method 2, 3 and 4 surprisingly all have a ALD of −1, 0%, see table reftab:LUARDsurrender.

Table 5.2: ALDs and AADs, M inLDs and M axLDs of laps up risk calculated by method 2, 3 and 4 of whole life insurance portfolio with surrender rates applied.

Method ALD AAD M inLD M axLD 2 −1.0% 1.0% −8.6% 0.0% 3 −1.0% 1.0% −8.6% 0.0% 4 −1.0% 1.0% −8.6% 0.0%

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Figure 5.3: The projection of lapse up risk according to method 1, 2, 3, and 4 of whole life insurance portfolio with surrender rates applied and with time in years on the x-axis and the amount of capital requirement on the y-axis.

5.3 Whole life insurance with paid up rates processed

The plots of methods 2, 3 and 4 show the same pattern as the plots of the portfolio without ‘pupping’, so both cash flow methods underestimate mortality risk as well. See figure 5.4. The explanation of this is the same as in chapter 5.2. The ALD of method 2, 3, and 4 from the exact method 3 is: −16.0%, −30.2%, −14.0%. Method 4 approximates the exact method best. Method 5 shows no ALD from the exact method 1. It seems that the risk driver responds adequately to the implementation of paid up policy status for this life risk. Method 5 applies formula 3.8 policy by policy. In paragraph 3.6 it was proved that in case of surrender this equals the exact method (method 1). Our calculation shows that this seems also be the case for pupping, though we did not prove this.

Table 5.3: ALDs, AADs, M inLDs and M axLDs of laps up risk calculated by method 2, 3, 4 and 5 of whole life insurance portfolio with pup rates applied.

Method ALD AAD M inLD M axLD 2 −15.9% 26.1% −55.6% 38.5% 3 −30.2% 30.2% −59.7% 0.0% 4 −13.7% 13.7% −28.7% 0.0% 5 0.0% 0.0% 0.0% 0.0%

See figure 5.5. Lapse down risk shows similar pattern as lapse down risk of whole life insurance without paid up rates processed, except that method 5 equals zero and only the exact method 1 gives capital requirements. This can be a combination of two facts. The first fact is the same as mentioned in chapter 5.2 for lapse down risk: the cause is the summing of the cash flows in method 2, 3, and 4. The second cause can be that the cash flow of expenses is calculated with survival probabilities instead of status probabilities. The reason for this is that when a policy is made paid up, expenses still

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Figure 5.4: The projection of mortality risk according to method 1, 2, 3, 4 and 5 of whole life insurance portfolio with paid up rates processed with time in years on the x-axis and the amount of capital requirement on the y-axis.

Figure 5.5: The projection of lapse down risk according to method 1, 2, 3, 4 and 5 of whole life insurance portfolio with paid up rates processed with time in years on the x-axis and the amount of capital requirement on the y-axis.