Allocation of life underwriting risk to different products under Solvency II : effects of different allocation methods

(1)

Different Products under Solvency II

Effects of Different Allocation Methods

Pascal Vrolijk

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: Pascal Vrolijk Student nr: 9415300

Email: pascalvrolijk@hotmail.com Date: November 30, 2015

Supervisor: Prof. Dr. Ir. Michel Vellekoop Second reader: Prof. Dr. Roger Laeven

(2)

(3)

Abstract

This thesis answers the question whether different allocation methods lead to signifi-cantly different allocations when the life underwriting risk SCRLif e, as prescribed by

Solvency II, is allocated to the products term life insurance, funeral insurance and life an-nuity. In order to analyze the life underwriting risk and its underlying risk sub-modules, the products under consideration are analyzed with respect to the risks underlying the life underwriting risk. The allocation is performed by using the framework provided by Dhaene et al. (2012). This framework provides an overview and classification of dif-ferent optimal allocation methods. These allocation methods are optimal in the sense that for each loss, the allocated capital comes as close as possible to the loss that it concerns. From the results of the analysis performed in this thesis, it follows that almost all allocations per product are quite close together, which is exactly what one would expect. For practical purposes therefore, the pro rata SCRLif e allocation suffices. The

only allocation method which does show a significant difference in capital allocation is the Covariance allocation, which allocates an extremely low capital to funeral insurance since the life underwriting risk of this product does not move as much with the life underwriting risk of the total portfolio as that the other products do.

Keywords Optimal capital allocation, allocation methods, modelling of risk, scenarios for change in risk driver, Solvency II, life underwriting risk, SCR, mortality risk, longevity risk, expenses risk, lapse risk, catastrophe risk, term life insurance, funeral insurance, life annuity.

(4)

Preface

Foremost I would like to thank my supervisor Michel Vellekoop for all the help provided in guiding and supporting me during this enterprise called thesis, my second reader Roger Laeven for finding time to evaluate my thesis on such short notice and Rob Kaas for providing good advice during my studies. Also I want to thank my colleagues for supporting me in my flexible working times, with special thanks to Jos Noordam who found the patience and time to answer all my questions about modelling in Prophet, and to Caspar Kroon for exchanging ideas. And special thanks go out to my parents, to my girlfriend who understood I had to borrow night time in order to finish my thesis and to my sister and her boyfriend.

(7)

Introduction

The framework for capital requirement known as Solvency I was introduced in 1973 and the capital requirement was not based on all the risks that are related to an insurance company. It was a fixed percentage of the provisions and did not take into account the risk profile of the insurer. Moreover, certain important risks, like market risk and operational risk were not included in the calculation of the capital requirement. Neither were the assets and liabilities valued on the economical principle of market value. With the introduction of Solvency II the required capital is calculated by a risk based method that represents the risk profile of the insurance company and assets and liabilities are valued on the economical principle of market value. This gives a better match between the required capital and the risk of the insurance company.

The process of creating the Solvency II legislation followed the Lamfalussy approach. This approach has the purpose to simplify legislation and to come more quickly to a definite version of the legislation that is supported by the member states. Part of this process was the use of Quantitative Impact Studies, the last one being QIS5. These impact studies helped shaping the legislation and made the implementation easier for insurance companies because they had already performed part of the calculations. There-fore it is not surprising that QIS5 is very comparable to the final version of Solvency II that comes into force as of 2016.

The most important goal of Solvency II is to protect the interests of the policy holders. Other goals are an increased understanding and governance of the risks that the insur-ance company is confronted with, supervisory interventions that better match the risk profile of the insurance company, increased transparency and public confidence in the financial sector.

Solvency II prescribes that an insurance company needs to hold a certain level of capi-tal, the Basic SCR, in order to be solvent. The risk measurement in this thesis focusses on the required capital for the life underwriting risk. Solvency II prescribes the level of capital that an insurance company needs to hold as a whole, but it does not prescribe how the insurance company should allocate this capital to underlying levels like busi-ness units, busibusi-ness lines, individual products, life versus non-life insurance or selling versus non-selling portfolio. This thesis discusses and analyzes methods for the opti-mal allocation of the capital for life underwriting risk to the products of the insurance company.

(8)

Chapter 2

Measurement of Risk under

Solvency II

2.1 Solvency II and the Three Pillar Approach

Solvency II is based on three pillars that provide a framework for risk measurement, risk management and transparency towards the public and supervisor.

The first pillar describes the quantitative requirements like the calculation of the Sol-vency Capital Requirement (SCR) and the Minimum Capital Requirement (MCR). The idea behind the SCR threshold is that the capital of the insurance company should be sufficient to meet her obligations with a probability of at least 99.5% for the next 12 months. The idea behind the MCR is that this capital corresponds to meeting the obli-gations for the next 12 months with a probability of at least 85%. The level of the MCR usually lies between 25% and 45% of the SCR.

These two capital thresholds provide the regulator the means of intervention: if the insurers capital becomes lower than the SCR, the supervisor may take certain actions. The lower the capital is with respect to the SCR, the stronger will the intervention of the supervisor be. When the level of the MCR is reached, the insurance company is no longer permitted to sell insurance products

The second pillar of the Solvency II framework stimulates good risk management. The focus here is on the risk governance system and on the Own Risk and Solvency As-sessment (ORSA). For the risk governance system on one hand the focus is on the professionality and integrity of (key) employees and members of the board in order to assure a good risk management system. On the other hand it is focused on the inte-gration of the risk management system into the insurance company. In the ORSA the insurance company must on one hand look with a broad view at the future to identify possible risks that might not yet have been identified in the SCR. On the other hand in the ORSA the insurance company looks at how the SCR may develop in time to assess their Solvency position in the future.

The third pillar of the Solvency II framework provides guidelines for reporting to the supervisor and for public disclosure in order to achieve more transparency. For public disclosure the company publishes once per year a report about the financial solidity of the company. This is sort of a summary of the yearly report that is sent to the supervi-sor. Next to the yearly report, the insurance company also sends a quarterly report to the supervisor for a more recent update of the risks. The yearly report to the supervisor is broader and also includes an opinion of higher management (board of directors) about risk measurement and risk management.

(9)

2.2 Measurement of Risk, the Applicable Framework of

Solvency II

It is important for insurance companies to measure the risk they take. This risk can be measured in different ways. The risk measurement in this thesis follows the idea of Solvency II. In Solvency II the risk is quantified by the Solvency Capital Requirement, the SCR. This risk measurement is described in article 101 (3) from the Solvency II Directive (Directive 2009/138/EC [recast]), which was adopted in November 2009. The SCR is calculated according to the tree in figure 2.1. The lowest levels of risk in this tree refer to the risk sub-modules that are colored light blue, such as Mortality. The capital requirements for these lowest levels of risk correspond with a Value-at-Risk ap-proach, subject to a confidence level of 99.5% over a one-year period. These capital requirements are aggregated by the means of a correlation matrix in order to calculate the capital requirements for one higher level, the risk modules that are colored medium blue in figure 2.1, such as SCRLif e. Since for this thesis the conclusions based on the

SCRLif e calculation will also apply to the Basic SCR, the analysis will be confined to

the calculation of the SCRLif e.

In order to calculate the Basic SCR, the BSCR which is colored blue in this figure, the capital requirements for the underlying risk modules are again aggregated by means of a correlation matrix. Finally at the highest level of risk, the capital requirement for the insurance company as a whole, is calculated by the SCR, which is colored dark blue. This SCR is calculated by adding up the capital requirements for the three underlying risks.

(10)

4 Pascal Vrolijk — Allocation of Risks

2.3 Value-at-Risk

In the case of the Solvency II regulation, the Value-at-Risk approach calculates a capital that the insurance company should hold. This capital should provide policy holders with more certainty that the insurance company will be able to meet her obligations in adverse scenarios for the insurance company. It should provide a coverage for all different kinds of risk to which the insurance company is subject. Of course there is no 100% protection from all risk but the idea behind the Value-at-Risk approach is to calculate a required capital such that during a one-year period there is only a 0.5% chance that the capital is not sufficient to meet the obligations of the insurance company and that as a result there is a 0.5% chance that the insurance company becomes insolvent.

Therefore the Value-at-Risk method is used to calculate the level of capital that the insurance company should hold to cover the lower levels of risk.

An advantage is that the use of the modular approach makes it possible to compare the risks of different insurance companies, to look at the different kind of risks in a unified way.

A disadvantage is that this method only looks at the 0.5% percentile and it does not look at more severe risks that have a smaller possibility of occurring. When these more severe risks do occur, they do so with devastating effects. For this reason it gives a false sense of safety because it works good in the normal situations but as soon as the bad scenarios coincide and amplify one another, the capital that is held may not be sufficient. This can lead to extreme risk taking i.e. risks that have such a small possibil-ity of occurring that from a managements perspective everything seems to be fine. But when the risk does occur it can be disastrous and possibly lead to insolvency.

A disadvantage of the Value-at-Risk method is that it is not subadditive. This means that the at-Risk of two risks can be larger than the sum of the individual Value-at-Risks. Therefore it is possible that the total required capital to cover two risks is bigger than the sum of the required capitals to cover each risk.

2.4 Calculation of SCR

Lif e

The calculation of this SCR is divided into different risk modules. One risk module con-cerns the calculation of the capital requirement for the life underwriting risk (SCRLif e).

Since for this thesis the conclusions based on the SCRLif e calculation will also apply to

the SCR, the analysis will be confined to the calculation of the SCRLif e for 3 different

products. These products are described in chapter 4.

The SCRLif e calculation itself corresponds to a Value-at-Risk approach, subject to

a confidence level of 99.5% over a one-year period.

Solvency II prescribes the calculation of the SCRLif e as follows:

SCRLif e= v u u t m X i,j=1 CorrijSCRiSCRj, (2.1)

where SCRi denotes the submodule i and Corri,j denotes the correlation between

sub-module i and subsub-module j

In the case of the 3 considered products the submodules of SCRDisabilityand SCRRevision

are not taken into account in this thesis. From figure 2.1 it follows that 5 submodules remain. In this case Solvency II prescribes the following calculations for the submodules of SCRLif e:

(11)

1. The submodule SCRM ortality equals the loss resulting from an instantaneous and

permanent increase of 15% in the mortality rates (i.e. the one year probabilities of dying for a given age);

2. The submodule SCRLongevity equals the loss resulting from an instantaneous and

permanent decrease of 20% in the mortality rates;

3. The submodule SCRExpenses equals the loss resulting from the combination of:

• An instantaneous permanent increase of 10% in the (not initial) expenses; • An increase of 1 percentage point to the expense inflation rate;

4. The submodule SCRLapse equals the maximum loss resulting from one of the

following scenarios:

• An instantaneous permanent increase of 50% in the lapse rates; • An instantaneous permanent decrease of 50% in the lapse rates; • A discontinuance of 40% of the policies in the first year;

5. The submodule SCRCAT equals the loss resulting from an instantaneous increase

of the mortality rates of 0.0015 during the first year.

Because the SCRDisabilityand the SCRRevisionequal 0 for the 3 products that are being

considered, they are deleted from the SCRLif e calculation and the correlation matrix

is thus defined by:

i/j Mortality Longevity Expenses Lapse CAT

Mortality 1 -0.25 0.25 0 0.25

Longevity -0.25 1 0.25 0.25 0

Expenses 0.25 0.25 1 0.5 0.25

Lapse 0 0.25 0.5 1 0.25

CAT 0.25 0 0.25 0.25 1

(12)

Chapter 3

Optimal allocation of risk

For an insurance company it is important that the capital that it needs to hold at an aggregated level can be allocated to underlying levels. In this way the cost of holding this capital at a specific underlying level can be calculated and this can be used for a number of important purposes.

Firstly, it is useful for the purpose of (re)pricing: to cover the cost of capital that a company needs to hold for a specific underlying level, a surcharge can be calculated more precisely, and if necessary it can be adjusted over time.

Secondly, for the purpose of performance measurement, the cost of capital is a cost component that needs to be taken into account. For this purpose the performance at a specific underlying level not only includes the cost of holding capital but the performance at this level can also be related to (among others) the cost of holding capital.

Thirdly, for the purpose of financial reporting at underlying levels, the cost of capital needs to be known at each specific underlying level.

Literature provides many ways for the allocation of capital, which can also be applied to the allocation of the SCRLif e. This thesis will use the framework of the article of

Dhaene et al. (2012) which gives an overview and classification of different optimal allo-cation methods. These alloallo-cation methods are optimal in the sense that for each loss the allocated capital comes as close as possible to the loss that it concerns. This distance between the loss and its allocated capital is measured as a quadratic or as an absolute distance and is subsequently minimized over all the losses. Various classes of optimal capital allocations result from weighing the distances between the loss and its allocated capital in different ways (from different point of views). The following classes of optimal capital allocations result from using different weights for the weighted expected loss:

1. Business unit driven allocations arise by weighing the expected loss in such a way that the highest weights are given to the scenarios where the specific loss is the largest and smaller weights are given to the scenarios where this specific loss is smaller;

2. Aggregate portfolio driven allocations arise by weighing the expected loss in such a way that the highest weights are given to the scenarios where the sum of losses is the largest and smaller weights are given to the scenarios where this sum of losses is smaller;

3. Market driven allocations arise by weighing the expected loss in such a way that the highest weights are given to the scenarios where a comparable loss in the market is the largest and smaller weights are given to the scenarios where this comparable loss in the market is smaller;

4. Default option driven allocations arise by weighing the expected loss in such a way that the highest weights are given to the scenarios where the total loss is larger

(13)

than the total allocated capital. Scenarios where the total loss is smaller than the total allocated capital get weight 0 and are therefore excluded.

In the article of Dhaene et al. (2012) the focus is mainly on the optimal capital alloca-tions that result from a quadratic measurement of the distance. This thesis handles the allocation of loss to different products under Solvency II. Therefore the focus will be on business unit driven allocations and on aggregate portfolio driven allocations. At first glance the default driven allocations also seem to be of interest to consider but in this type of allocations the scenarios where the total loss is smaller than the total allocated capital are totally neglected. This means that default driven allocations are only based on the distribution of losses under scenarios of extreme total loss. It does not consider the distribution of losses in all the other (more normal c.q. moderate) scenarios, which is the majority of all the scenarios.

This thesis studies how the SCRLif e will be allocated to 3 products for business unit

driven and aggregate portfolio driven allocations. Section 3.1 describes how the SCRLif e

module is used to measure the loss and it gives some definitions that will be used to describe the allocation methods that this article discusses. These definitions will slightly deviate from the article because they will also include the method by which the risk is measured. This means that the measurement of risk will be based on the Value-at-Risk method by which the SCRLif e is calculated in the Solvency II context. Section 3.2

de-scribes the unifying framework of the optimal allocation formula that is provided by the article of Dhaene et al. (2012). Section 3.3. will describe the business unit driven and aggregate portfolio driven allocations in more detail, while section 3.4 will only briefly explore the market driven and default option driven allocations.

3.1 SCR

Lif e

module as measurement for loss

Consider a portfolio of 3 products, and let stochastic variable Xi be the loss associated

with product i. The loss Xidepends on the risks that are described in the 5 submodules

of the SCRLif e: mortality risk, longevity risk, expense risk, lapse risk and catastrophe

risk. The modelling of these risks is described in chapter 5. Assume that the losses (X1, X2, X3) are a random vector on the probability space (Ω,F ,P). The loss Xi is

cal-culated using the SCRLif e module as if the insurance portfolio only consists of product

i. The distribution functions will be denoted by FXi(x).

The aggregate loss is defined by:

S =

3

X

i=1

Xi (3.1)

The risk capital K that the insurance company needs to hold for this aggregate loss is calculated by the SCRLif e module for the portfolio of products:

K = SCRLif e _X3 i=1 P roducti (3.2)

The purpose of capital allocation is to find a Ki that is in some way close to the loss

Xi and that it is such that the Ki satisfy the following full allocation principle:

K =

3

X

i=1

Ki (3.3)

This formula is called the full allocation formula because it ensures that all the required capital is divided over the three products under consideration.

(14)

The SCRLif e module is not a linear operator. This is because of the correlation matrix

which accounts for diversification advantages between the 5 different risks. Therefore the difference between Kiand SCRLif e(P roducti) is that the simple sum S does not include

the diversification effects between the products, but K does include these diversification effects. This is shown in the following formula:

K = SCRLif e X3 i=1 P roducti 6= 3 X i=1 SCRLif e(P roducti) (3.4)

Because the loss Xi is not equal to Ki for all i, it is important to allocate the capital

K in an optimal way to the products. The next section discusses several methods to do this.

3.2 Optimal allocation formula

As mentioned in the introduction of chapter 3, the article of Dhaene et al. (2012) provides a unifying framework to define and classify optimal allocation methods. These allocation methods are optimal in the sense that for each risk the allocated capital comes as close as possible to the risk that it concerns. In the article, this basic idea is translated into the following formula:

minK1,...,K3 3 X j=1 vjE h ξjD X_j − K_j vj i , such that 3 X j=1 Kj = K, (3.5)

where vjare non-negative real numbers such thatPnj=1vj =1, where ξj are non-negative

random variables such that E[ξi]=1, and where D is a non-negative function.

3.2.1 Quadratic distance measurement

In the case that the distance between each loss and its allocated capital is minimized by the quadratic distance formula D(x) = x2, the solution of formula (3.5) is given by:

Ki= E[ξiXi] + vi K − 3 X j=1 E[ξjXj] , f or i = 1, ..., 3 (3.6)

To understand this formula, one first needs to get a better understanding of the random variables ξi and of the numbers vi. The non-negative and increasing variable ξi, with

expectation equal to 1, quantifies the adversity in the outcomes of loss Xi. E[ξiXi]

therefore is the weighted expected loss for product i.

When ξi is a non-negative and non-decreasing function of loss Xi (ξi=h(Xi)), the

resulting allocation will be Business Unit Driven, as defined in the introduction of this chapter. Similarly when ξi=h(S) this will result in an Aggregate Portfolio Driven

allo-cation. On the other hand when ξi is a non-negative and non-decreasing function of a

comparable loss in the markt (ξi=ξM), this will result in a Market Driven allocation.

Formula (3.6) shows that the capital allocated to product i is equal to the weighted expected loss of this product plus the fraction vi of the difference between the total

al-located capital and the sum of the weighted expected losses. This weight vi determines

how important this product is regarded to be: the more important the product is, the bigger is the part of the difference that is allocated to this product. The importance can for example be measured as a fraction of production level, profits or weighted expected losses.

(15)

3.2.2 Absolute distance measurement

In the case that the distance between each loss and its allocated capital is minimized by the absolute distance formula D = |x|, the optimal allocation formula (3.5) transforms into the following formula:

minK1,...,K3 3 X j=1 E h ξj Xj− Kj + i , such that 3 X j=1 Kj = K (3.7)

This means that the distribution of losses that are smaller than its allocated capital, are not taken into account when determining the capital allocation.

This thesis only describes results for the allocations which are based on a quadratic distance measurement. Therefore the part of the optimal allocation formula that is based on an absolute distance measurement will not be discussed in more detail.

3.3 Business Unit and Aggregate Portfolio Driven

Alloca-tions

In the case of Business Unit and Aggregate Portfolio Driven Allocations the optimal capital allocation can be captured in a proportional allocation formula, formula (3.8). The method of proportional allocation allocates a proportion of the risk capital K to each product. The proportion is defined by the fraction of the risk related to a single product versus the risk related to the portfolio of products. The risk of a certain loss Xi

is quantified by the risk measure ρ. This is shown in the following proportional allocation formula: Ki= K ρ(Xi) P3 j=1ρ(Xj) f or i = 1, 2, 3 (3.8)

The next two sections describe Business Unit and Aggregate Portfolio Driven Alloca-tions.

3.3.1 Business Unit Driven Allocations in more detail

The assumption for Business Unit Driven Allocations is that ξi depends on the specific

loss Xi.This is shown by the following formula:

ξi= hi(Xi) f or i = 1, 2, 3, (3.9)

where hi is a non-negative and non-decreasing function, such that E[hi(Xi)]=1 for each

i.

If it is assumed that the importance of a product is measured as a fraction of weighted expected losses, the value of vi is given by:

vi =

E[ξiXi]

P3

j=1E[ξjXj]

f or i = 1, 2, 3 (3.10)

Filling in these values of vi and ξi into formula (3.6), results in the following formula

for Business Unit Driven Allocations:

Ki= K

E[Xihi(Xi)]

P3

j=1E[Xjhj(Xj)]

f or i = 1, 2, 3 (3.11)

This is a special case of the proportional allocation formula (3.8) by taking ρ(Xi) equal

to the weighted expected loss E[Xihi(Xi)]:

(16)

Because the risk measure ρ depends on Xi and not on S, the allocation of risk

capi-tal is not influenced by the dependency structure within the portfolio. The allocation of capital is completely defined by the risk measurement of the specific product, it is therefore called a Business Unit Driven Allocation. At first thought one might think that this leads to a very good capital allocation because the capital that is allocated to one product does not depend on how good or bad the risk of other products are in comparison to the product under consideration. Each product therefore seems to get exactly as much risk as it deserves since its risk measurement does not depend on the risk measurement of other products. However it must be realized that there is a strong dependency structure between the losses of the products because the loss is measured by the same SCRLif e module from Solvency II and therefore there is a strong

depen-dency between the losses of products. The strong dependepen-dency comes from the fact that the SCRLif e module uses the same risk drivers for all products (change in mortality

rate, change in level of expenses and change in lapse rate). Therefore within a specific scenario that contains certain levels of risk drivers, the losses of products are correlated. Since the losses are marked by a strong dependency between the products, also the risk measurement of these losses should be marked by a strong dependency. This would sug-gest that there is dependency between the risk measurement of separate products and at the aggregate level. To explore and compare this in further detail the following Business Unit Driven Allocations without this desired dependency between risk measurement of separate products will be compared with Aggregate Portfolio Driven Allocations.

Standard deviation principle:

The capital allocation following the Standard deviation principle (B¨uhlmann, 1970) results from filling in the following risk measure ρ(Xi) into the proportional allocation

formula (3.8):

ρ(Xi) = E[Xi] + aσXi f or i = 1, 2, 3, (3.13)

where a ≥ 0.

Individual conditional tail expectation:

The capital allocation following the Conditional tail expectation (Overbeck, 2000) re-sults from filling in the following risk measure ρ(Xi) into the proportional allocation

formula (3.8):

ρ(Xi) = E[Xi|Xi > FX−1i(p)] = CT Ep[Xi] f or i = 1, 2, 3, (3.14)

where p=99.5%.

It needs to be noted that the conditional tail expection that is described here de-pends on the conditional expectation of Xi, where the condition is based on Xi. Usually,

when the CTE allocation is considered, it is the conditional expectation of Xi, where

the condition is based on S. The usual CTE allocation (which was also considered by Overbeck 2000) is therefore a portfolio aggregated allocation method and is described by formula (3.25).

Exponential principle:

The capital allocation following the Exponential principle (Gerber, 1974) results from filling in the following risk measure ρ(Xi) into the proportional allocation formula (3.8):

ρ(Xi) =

1 aln(E[e

aXi_]) _{f or i = 1, 2, 3,} _(3.15)

(17)

Esscher principle:

The capital allocation following the Esscher principle (Gerber, 1981) results from filling in the following risk measure ρ(Xi) into the proportional allocation formula (3.8):

ρ(Xi) =

E[XieaXi]

E[eaXi_] f or i = 1, 2, 3, (3.16)

for a certain a > 0.

Haircut allocation:

The Haircut allocation results from filling in the following risk measure ρ(Xi) into the

proportional allocation formula (3.8):

ρ(Xi) = F_X−1_i(p) f or i = 1, 2, 3, (3.17)

where p=99.5%.

Distortion risk measure (not calculated):

To complete the overview of the article of Dhaene et al. (2012) for business driven allocations, the capital allocation following the distortion risk measure (Wang, 1996; Acerbi, 2002) is shown but it is not calculated in this thesis. This capital allocation results from filling in the following risk measure ρ(Xi) into the proportional allocation

formula (3.8):

ρ(Xi) = E[Xi∗ g0(1 − FXi(Xi))] f or i = 1, 2, 3, (3.18)

where g:[0,1] → [0,1], g0> 0 and g00< 0.

3.3.2 Aggregate Portfolio Driven Allocations in more detail

The assumption for Aggregate Portfolio Driven Allocations is that ξi depends on the

aggregate loss S. This is shown by the following formula:

ξi = h(S) f or i = 1, 2, 3, (3.19)

where h is a non-negative and non-decreasing function, such that E[h(S)]=1.

When for Aggregate Portfolio Driven Allocations the importance of a product vi is

given by:

vi=

E[ξiXi]

E[ξjS]

f or i = 1, 2, 3, (3.20)

filling in these values of vi and ξi into the optimal allocation formula (3.6) results into

the following formula for Aggregate Portfolio Driven Allocations:

Ki = K

E[Xih(S)]

E[Sh(S)] f or i = 1, 2, 3 (3.21)

This is a special case of the proportional allocation formula (3.8) by letting ρ(Xi) depend

on the aggregate loss S as shown by the following formula:

ρ(Xi) = E[ξiXi] = E[Xih(S)] f or i = 1, 2, 3 (3.22)

This subsection describes allocation methods where the risk measure ρ depends on S, the aggregate loss, calculated by summing up all separate losses conform formula (3.1). The allocation of risk capital is therefore strongly influenced by the dependency struc-ture within the portfolio. It is for this reason that this class of optimal allocations are

(18)

called Aggregate Portfolio Driven Allocations. As discussed in the subsection of Busi-ness Driven Allocations there is a strong dependency structure between the losses of the products because the loss is measured by the same SCRLif e module from Solvency II.

Overbeck principle:

Overbeck (2000) discusses two aggregated portfolio driven capital allocation principles. The first one that is considered by Overbeck (2000) results from filling in the following risk measure ρ(Xi) in the proportional allocation formula (3.8):

ρ(Xi) = E[Xi] + a

Cov[Xi, S]

σS

f or i = 1, 2, 3, (3.23)

where a = 0. Notice the similarity between this portfolio aggregate driven allocation and the business unit driven allocation following the standard deviation principle (B¨uhlmann, 1970) conform formula (3.13).

CTE allocation:

The second aggregated portfolio driven capital allocation principle that Overbeck (2002) considers is also called the CTE allocation. It results from filling in the following risk measure ρ(Xi) in the proportional allocation formula (3.8):

ρ(Xi) = E[Xi|S > F_S−1(p)] f or i = 1, 2, 3, (3.24)

where p=99.5%.

The CTE allocation depends on the conditional expection of Xi, where the

condi-tion is based on S. In general when the CTE allocacondi-tion is considered, it follows this definition, instead of (3.14).

Quantile allocation principle:

The Quantile allocation results from filling in the following risk measure ρ(Xi) in the

ρ(Xi) = F_X−1_i(βp) f or i = 1, 2, 3, (3.25)

where p=99.5% and β is such that the full allocation formula (3.3) is satisfied.

In the case that beta equals one, this portfolio aggregate driven allocation will equal the business unit driven allocation following the haircut allocation principle conform formula (3.17). Section 6.4 will explain that this happens when there is no risk mitiga-tion, i.e. when there is no diversification benefit of the losses.

Covariance principle:

The Covariance allocation results from filling in the following risk measure ρ(Xi) in the

ρ(Xi) = Cov[Xi, S] f or i = 1, 2, 3 (3.26)

Wang principle:

The capital allocation that is considered by Wang(2007) results from filling in the fol-lowing risk measure ρ(Xi) in the proportional allocation formula (3.8):

ρ(Xi) =

E[XieaS]

E[eaS_] f or i = 1, 2, 3, (3.27)

where a > 0. Notice the similarity between this portfolio aggregate driven allocation and the business unit driven allocation following the Esscher principle (Gerber, 1981) conform formula (3.16).

(19)

Tsanakas principle, 2004 (not calculated):

The capital allocation discussed by Tsanakas (2004) results from filling in the following risk measure ρ(Xi) into the proportional allocation formula (3.8):

ρ(Xi) = E[Xig0(1 − Fs(S))] f or i = 1, 2, 3, (3.28)

where g:[0,1] → [0,1], g0> 0 and g00< 0.

Notice that this portfolio aggregate driven allocation is very similar to the business unit driven allocation following the distortion risk measure (Wang, 1996; Acerbi, 2002) conform formula (3.18).

Tsanakas principle, 2008 (not calculated):

The capital allocation discussed by Tsanakas (2008) results from filling in the following risk measure ρ(Xi) into the proportional allocation formula (3.8):

ρ(Xi) = E h Xi Z 1 0 eγaS E[eγaS_]dγ i f or i = 1, 2, 3, (3.29) where a > 0.

3.4 Market Driven and Default Option Driven Allocations

As noted before, the Market Driven and Default Option Driven Allocations are not modelled in this thesis. In order to complete the overview of the optimal allocations for the different methods of quadratic distance measurement between each loss and its allocated capital, these two methods will be discussed briefly though.

Market Driven Allocations

The assumption for Market Driven Allocations is that the weights refer to a reference (or market) portfolio. This is shown by the following formula:

ξi = ξM f or i = 1, 2, 3 (3.30)

When for Market Driven Allocations the importance of a product, which is measured by vi, is given by:

vi=

E[ξiXi]

E[ξiS]

f or i = 1, 2, 3, (3.31)

then filling these values of vi and ξi into the optimal allocation formula (3.6) results

into the following formula for Market Driven Allocations:

Ki= K

E[ξMXi]

E[ξMS]

f or i = 1, 2, 3 (3.32)

Default Option Driven Allocations

For Default Option Driven Allocations the assumption is that ξi depends on the

aggre-gate loss S as is shown by the following formula:

ξi = h(S) = I(S > K)

P(S > K) f or i = 1, 2, 3, (3.33) where h is a non-negative and non-decreasing function, such that E[h(S)]=1.

When filling this in into the optimal allocation formula (3.6), the following formula results:

E[(Xi− Ki)I(S > K)] = viE[(S − K)+] f or i = 1, 2, 3 (3.34)

This formula can be solved for all Ki, where the solution will be based on the default

option E[(S − K)+]. For this reason these allocations are called Default Option Driven

(20)

Chapter 4

Description of the insurance

portfolio

This thesis studies capital allocations for a portfolio of 3 traditional life insurance prod-ucts. This chapter describes these products, which are part of a real insurance portfolio. For the description of the products the following characteristics are described:

1. The products are described by explaining for what purpose customers use the product and when the cashflows of benefits and premiums take place.

2. The tariff components of the premiums or the lump sum are described without going into details about the premium calculation or the level of tariff parameters. 3. An overview will be given of the portfolio characteristics.

4. Based on the best estimate scenario under Solvency II, an overview will be given for the expected present value of the resulting cashflow, which is calculated by the following formula:

P V(Resulting Cashflow) = PV(Premiums) − PV(Benefits) − PV(Expenses), (4.1) where PV stands for the expected Present Value, which includes mortality and lapse.

It needs to be noted that conform Solvency II calculations, the initial costs are not part of the PV(Expenses). The PV(Resulting Cashflow) is analyzed because the SCRLif eis based on the change in the PV(Resulting Cashflow). The PV(Resulting

Cashflow) represents the value of the portfolio under the best estimate scenario of Solvency II and it will be shown how this value develops over time.

5. Since most of the calculations of the loss under stress scenarios of Solvency II result from shocking the mortality rates and lapse rates, an overview will be given of how these (portfolio weighted) rates develop into the future under the basis scenario. To better understand the drivers of how the number of policies decreases in the future under the basis scenario, the decrease in the number of policies will be analyzed by quantifying the effect of mortality, lapse and policy expiration. The policy expires if there is no more insurance because the insured period is finished. It follows that Lifelong Annuities and Funeral Insurances have no policy expiration because the insured period lasts until they die. In this portfolio of products only Term Life Insurance and Limited Annuities policies expire.

The effect of lapse rates on the number of insured people is much more severe than the effect of mortality rates, but that on the other hand the effect of mortality rates on the present value of the resulting cashflow is much more severe than the effect of lapse rates. This will be shown in chapter 6 where the results will be described.

(21)

4.1 Term life insurance (ORV)

X1 is the loss for product 1, where product 1 is a term life insurance. At the moment

this type of insurance is one of the main contributors of the production of traditional life insurance policies within the Netherlands. The portfolio under consideration is the production of term life insurance policies that are sold in 2014 by SNS-REAAL (the predecessor of VIVAT), and are still in force, namely 36,584 policies. The following can be said about this product:

1. • For the payment of a level premium during the premium period, the term life insurance (immediately) pays out an insured amount to the beneficiary when one of the insured persons dies within the insured period. The insured period is a maximum of 40 years, where the last premium payment must take place before the age of 75. In case of lapse (termination of the insurance) no payment will take place from the insurance company to the customer. The reason for this is that it is incorporated into the tariff structure as a reduction on the tariff. The insurance can be based on one or two insured persons. Before accepting the insurance, the health of the insured person(s) is checked which can lead to a higher premium in certain cases of higher expected mortality risk.

• In general the beneficiary is the bank, the partner (in marriage) or the partner in an enterprise. In the case that the bank is the beneficiary, this is usually their requirement when you close a mortgage. The bank requires this so that in case of death they will be able to get back the remainder of the mortgage when the (sales) value of the house is lower than expected. In the case that the partner in marriage is the beneficiary, the term life insurance is usually taken out to provide one partner with a benefit in the case that the other partner dies. This is especially important if the beneficiary does not have (enough) income or when the beneficiary suddenly has to take care of the children. In the case that the partner in an enterprise is the beneficiary, then when one partner dies, the other partner can buy out the next of kin of that partner and continue the enterprise.

• When the insurance is taken out in combination with a mortgage, the in-sured period typically equals the duration of the mortgage. The most com-mon duration of new mortgages is around 30 years. This is mainly due to tax regulation that is less favorable for durations longer than 30 years. The insured amount can be constant or decreasing during the insurance period. A term life insurance that is sold in connection with a mortgage usually has an insured amount that decreases to zero during the insurance period. Be-cause of recent changes in tax regulation, these term life insurances with a decreasing insured amount are gaining in popularity. In case of a decreasing insured amount, the premium period is five years shorter than the insured period.

2. The vast majority of policies are premium paying policies, the few people that take a lump sum term insurance are negligible. The tariff components of a term life insurance exist of a premium for benefit payments, a premium for profit, a premium for immediate cost and a premium for recurring cost (expenses). Since new European regulation since December 2012 it is no longer permitted to dis-criminate based on sex. Therefore the premium depends on mortality rates that do not discriminate for sex: the mortality rates are combination of the mortality rates for a male and for a female. They do discriminate for the smoker status; mortality rates for a person that smokes are higher than for a person that does not smoke. In certain cases when the health check of the insured person shows that there

(22)

is a higher expected mortality risk, the mortality rates can be increased, which results in higher premiums. The premium also depends on duration and age: in general the premium is higher for older people and longer durations. Exceptions may be contributed to a difference in the premium for profit or in some exceptional cases it can be contributed to non-smoothness in mortality rates. The premium for benefit payments is calculated from the expected present value of future benefit payments. The premium for profits is typically lower for a certain chosen target group, this is in order to be more competitive in certain market segments like in the segment of the most common durations, which is a duration of 30 years. The premiums for immediate and recurring cost are based on a constant amount of costs and are calculated from the immediate cost and from the expected present value of future costs.

3. The risk is measured by the SCRLif e, which is the deviation from the PV(Resulting

Cashflow) under the standard scenario of Solvency II. The PV(Resulting Cash-flow) is calculated by formula (4.1), where the components of the PV(Premiums) are described in a previous bullet. The portfolio of Term Life Insurance is typically characterized by a high value of PV(Premiums) because premiums will be received until the policy is terminated. The following table shows the PV(Resulting Cash-flow) and its components for this product and per policy:

Present value cashflows Total portfolio Per policy

Number of policies 36,584

PV(Premiums) + 125,855,511 + 3,440 PV (Benefits) - 77,083,824 - 2,107 PV (Expenses) - 16,526,568 - 452 PV (Resulting Cashflow) + 32,245,119 + 881

Table 4.1: PV(Resulting Cashflow) and its components for this product and per policy

The portfolio can be divided into policies with a decreasing or a constant insured amount. The following table shows that both types of policies have an almost equal insured amount, but it is important to realize that the present value of cashflows for policies with a decreasing insured amount is much smaller because the insured amount decreases to zero during the policy period. The decrease in insured amount can be a linear decrease or a percentage decrease. Since the linear decrease is much less common (5% of the total portfolio) it is not separately shown.

The table also shows that the average overall insurance period is 22 years even though the most common insurance period equals 30 years. This is the result of a number of policies with shorter insurance periods.

(23)

The following table shows the main characteristics of this portfolio for policies with a decreasing insured amount and a constant insured amount.

Portfolio characteristics Decreasing Constant Total portfolio Insured amount 164,463 164,041 164,212 Percentage of portfolio 41% 59% 100% Insurance period at start 25 20 22

Age at start 39 43 41

Percentage males 57% 59% 58%

Percentage smokers 19% 10% 14%

Percentage 1-life insurances 48% 55% 52%

Table 4.2: The main characteristics of the portfolio for policies with a decreasing insured amount and a constant insured amount

4. The following figure shows the present value of premiums, benefit and expenses versus the present value of the resulting cashflow. It also shows how these values change over time. The decrease of the PV(Resulting Cashflow) can be contributed

Figure 4.1: Present Value at time t of future cashflows

to the fact that the PV(Premiums) decreases faster in time than that it should. This is caused by the level premium, which means that compared to a fairly calculated yearly premium, the level premium in the beginning is higher and later it will be lower. When the level premium is lower than the fairly calculated yearly premium, this means that the premium income is lower than that it should be to cover the benefits, costs and profits. Therefore in figure 4.1 the PV(Resulting Cashflow) decreases strongly in the beginning until at year 11 it becomes negative. One can also see that at the average insurance period of 22 years the present value of the cashflows is indeed relatively low.

5. The SCRLif e depends strongly on the development of the number of policies.

The following figure shows how the number of policies decreases by the expected effect of mortality, lapse and policy expiration. The effects are cumulative, so the length of the arrow is the cumulative decrease in the number of policies since the beginning.

The figure shows that the effect of mortality on the decrease of policies is not significant, the biggest effect is caused by lapse and expiration of policies. Since

(24)

Figure 4.2: Decrease in number of policies in time

the most common insurance period is 30 years, the figure indeed shows that the decrease by expiration increases most at 30 years from now. The effect of lapse is so significant because in the first 10 years the lapse rates can be quite high, reaching a maximum of 10% around year 10.

The average yearly lapse rate of the policies and the lapse in numbers per year are shown in the following figure. It shows for example that 5 years from now the expected average portfolio lapse rate equals about 10% which results in an expected lapse of almost 3,000 policies. Since the most common insurance period equals 30 years, the number of expected lapse is negligible in the end.

Figure 4.3: Number of policies terminated at time t, because of Lapse

Mortality is not a significant driver in the decrease in the number of policies, but the SCRLif e results from shocking the mortality rates and lapse rates. For that

reason the average portfolio mortality per year is shown in the following figure. It shows for example that 5 years from now the expected average portfolio mortality rate equals 0.15% which would result in about 40 deaths that year. It is important to notice that even though the average portfolio morality rate increases over time, the number of deaths decreases over time because, as shown before, the number of policies decrease strongly in time because of lapse and expiration.

(25)

Figure 4.4: Number of policies terminated at time t, bacause of mortality

Another important point is that the decrease in numbers is mainly caused by lapse and not by mortality, but the effects of the Solvency II shocks for lapse and mortality on the PV(Resulting Cashflow) lie closer together because the effect of mortality is much more severe. Chapter 6 will further discuss this in further detail.

4.2 Funeral insurance (Uitvaart)

X2 is the loss for product 2, where product 2 is a funeral insurance. The portfolio

under consideration is the production of all funeral insurances without profit sharing that have been sold since the year 2000 by VIVAT or its predecessors, and are still in force, which are 19.754 policies. For reasons of programming in Prophet only the funeral policies without profit sharing have been selected. The following can be said about this product:

1. For the payment of a level premium during the premium period, the funeral insur-ance pays out a fixed insured amount to the beneficiary when the insured person dies. Therefore this insurance always pays out when the insured person dies and with this benefit the beneficiary, usually the partner, can pay for the funeral. Any money that is left can be spent as the beneficiary pleases. The insurance can only be based on one insured person.

Before accepting the insurance, the health of the insured person is checked which can lead to a higher premium in certain cases of higher expected mortality risk. Since the insured amount is a lot lower than for the term life insurance, this health check is less extensive.

The maximum period of premium payments is a maximum of 40 years, where the last premium payment must take place before the age of 75.

2. The vast majority of policies are premium paying policies, only a few insured peo-ple take a lump sum term insurance. Since it concerns an existing portfolio, the portfolio exists of several different tariffs, but in general the tariff components of a funeral insurance exist of a premium for benefit payments, a premium for immedi-ate cost and a premium for recurring cost. Since there is no component for profit, the profit is calculated implicitly from the result on mortality and from the result on costs. Older tariffs use the same mortality table for males as for females, but

(26)

the age of females is reduced in order to approximate their lower mortality. Since new European regulation since December 2012 it is no longer permitted to dis-criminate based on sex. Therefore the present tariff is based on the same mortality table for males but with an equal age reduction for males and females in order to approximate lower average mortality rates. Since the funeral tariff has not really undergone any recent changes, they do not discriminate for the smoker status. In certain cases when the health check of the insured person shows that there is a higher expected mortality risk, the mortality rates can be increased, which results in higher premiums. The premium also depends on duration and age: the premium is higher for older people and for a shorter period of premium payments. The pre-mium for benefit payments is calculated from the expected present value of future benefit payments. The premiums for immediate and recurring cost are based on a percentage of the insured amount and are calculated from the immediate cost and from the expected present value of future costs.

3. The portfolio of Funeral Insurance is typically characterized by a low percentage of premium paying policies because the majority of the policies has been paid up due to various reasons. Because for the majority of the paid-up policies the assured amount was reduced, the percentage of premium paying policies increases from 20% of the total number of policies to 57% of the total sum assured. The following figure quantifies this effect for the premium paying policies versus paid-up policies with a specification for the reason why the policies are paid up.

Figure 4.5: Premium paying vs Paid-up policies

The figure shows that the biggest reason for a policy to be paid up is because the premium was no longer being paid. In that case the insurer decreases the assured amount relative to the amount of premiums that had been paid. Two other reasons for a policy to be paid up is because the client requests the premium payment to be stopped (this includes a decrease of the assured sum) and that the premium period is finished (the assured sum is not decreased).

The PV(Resulting Cashflow) is calculated by formula (4.1), where the components of the PV(Premiums) are described in the previous bullet. Because only 20% of the portfolio exists of premium paying policies, the PV(Premiums) is relatively low, resulting in a negative PV(Resulting Cashflow).

(27)

The following table shows this PV(Resulting Cashflow) and its components for this product and per policy:

PV(Premiums) + 4,461,395 + 226 PV (Benefits) - 9,884,788 - 500 PV (Costs) - 4,302,250 - 218 PV (Resulting Cashflow) - 9,725,643 - 492

The portfolio can be divided into premium paying policies and paid up policies. As mentioned before, the premium paying policies are in the minority when looking at the number of policies, but measured by the assured sum, about half of the portfolio is made up by premium paying policies. It also shows that the assured sum of the paid-up policies is reduced, and that therefore the assured sum is much lower for paid-up policies than for premium paying policies.

Portfolio characteristics Premium paying policies Paid-up policies Total portfolio Percentage

(weighted by assured sum)

57% 43% 100% Percentage (weighted by numbers) 20% 80% 100% Percentage male 47% 50% 50% Insured amount 3,396 645 1,190

Premium paying period (from start)

29 6

Premium paying period (stil remaining)

16 3

Present age 47 42 43

Table 4.4: The main characteristics of this portfolio

The table also shows that the difference between the age at start and the present age is different for premium paying policies and for paid-up policies. This difference implies that on average the premium paying policies started 13 years ago (47-34) and paid-up policies 11 years ago (42-31). Since the first policy in this portfolio was sold in the year 2000, the oldest policies are running about 15 years. From comparing the start of the portfolio 15 years ago with the fact that on average the policies are running for 11 (paid-up policies) or 13 years (premium paying policies) it follows that the majority of the policies started in the very first years after 2000.

4. The following figure shows the present value of premiums, benefit and expenses versus the present value of the resulting cashflow. It also shows how these values change over time. The values are based on the standard scenario under Solvency II and show the development of the value of the portfolio in time.

(28)

Figure 4.6: Present value at time t of future cashflows

The PV(Benefits) is much higher than the PV(Premiums) because on one hand only half of the portfolio is made up of premium paying policies, measured in assured sum and on the other hand 13 years of the on average 29 years of premium payment have already passed and are therefore not included in the PV(Premiums). Together with the negative cashflow of expenses, this results in a negative PV(Resulting Cashflow).

5. The SCRLif e depends strongly on the development of the number of policies. The

following figure shows how the number of policies decreases by the expected effect of mortality and lapse. In contrast to term life insurance the number of policies does not decrease because of expiration. The reason for this is that for funeral insurance, the policy does not expire after the premium period but it continues until the insured person dies, upon which the assured sum is paid out. The effects are cumulative, so the length of the arrow is the cumulative decrease in the number of policies since the beginning.

(29)

From the picture it follows that the cumulative effect of lapse is constant after 30 years, which means that for this portfolio the last premium will be paid 30 years from now since paid-up policies are not assumed to lapse anymore. In contrast to term life insurance, lapse is not a significant driver in the explanation of the decrease in the number of policies. The reason for this is that on one hand only premium paying policies are assumed to lapse, which is only 20% of the number of policies and that on the other hand the lapse rate is much lower than for term life insurance.

The average yearly lapse rate of the policies and the lapse in numbers per year are shown in the following figure.

Figure 4.8: Number of policies terminated at time t, because of Lapse

The figure indeed shows the very low lapse rate in comparison to term life in-surance. The portfolio average lapse rate per year for premium paying policies is about equal to 1%. The lapse measured in numbers in the first year is equal to: lapse rate (1.2%)* number of premium paying policies (20% of 19,754) = 47 policies. After 30 years from now the premium period of the last premium paying

(30)

policy is finished and therefore the lapse in numbers equals 0.

For funeral insurance mortality is the only significant driver in the decrease in the number of policies. Because the SCRLif eresults from shocking the mortality rates

and lapse rates, the average yearly portfolio mortality is shown in the following figure. It shows for example that most policy holders are expected to die 50 years from now. Since the average policy holder is 43 years old, this is to be expected.

Figure 4.9: Number of policies terminated at time t, because of Mortality

Between now and 50 years from now the average number of policies that terminate because of mortality lies around 225 per year. When multiplied by 50 years, the expected cumulative effect on the decrease in number of policies by mortality would equal around 11,000. This number can also be seen in graph 4.7.

4.3 Life Annuity (Direct Ingaande Lijfrente)

X3 is the loss for product 3, where product 3 is a Life Annuity. The portfolio under

consideration is the production of all Life Annuities that have been sold by VIVAT or its predecessors since the introduction of the present tariff (2004) until now, and are still in force, which are 11,412 policies. The following can be said about this product:

1. For the payment of a lump sum, the Life Annuity can either pay out a fixed amount for a limited period of time or until the insured person dies. When the Life Annuity is taken out on two persons, the benefit is paid until the first insured person dies after which the benefits continue at a predefined lower (or equal) level than before. The lump sum that is used to pay for the Life Annuity usually comes from money that is saved under tax regime that makes it unfavorable or sometimes even impossible to use a banking product to pay out possibly higher benefits. The benefits that are paid out result from the given lump sum. Life Annuities that are paid out for a limited period of time are usually used as additional income in the years preceding the old age pension, but can for example also be used to increase ones income in the early years of the old age pension. The lifelong annuity on the other hand is almost only used as additional income next to the old age pension. The insured person usually prefers to receive the benefits monthly, but if he or she prefers such, the benefits can be paid quarterly, half yearly or yearly.

(31)

2. The portfolio of Life Annuities is the existing portfolio that is based on the present tariff which was introduced in 2004. Therefore this portfolio exists of only one tariff, with the exception of adjustment out of legal necessities. This means that from December 2012 the mortality rates did no longer discriminate for sex: the mortality rates are an interpolation of the mortality rates for a male and for a female. And also from 2013 there was no longer a tariff component for closing commission. The tariff consists of tariff components which are lump sums for benefit payments, for immediate cost, for recurring cost, for solvency cost and for profit. Since the Life Annuities have not really undergone any recent changes, they do not discriminate for the smoker status. The level of annuity payments depends on the age, on the period in which the benefits are paid and on the interest rates: a higher age, a shorter period in which the benefits are paid and higher interest rates all result in higher annuity payments. For Life Annuities before December 2012, men had higher annuity payments than women. Price adjustments for competitive reasons are realized through changes in the tariff component for profit. The lump sum for benefit payments is calculated from the expected present value of future benefit payments. The lump sums for immediate and recurring cost are based on a fixed amount and are calculated from the immediate cost and from the expected present value of future costs. The lump sum for solvency cost is also calculated from the expected present value of future solvency cost.

3. The PV(Resulting Cashflow) is calculated by formula (4.1), where the PV(Premiums) is replaced by the PV(Lump Sum) since Life Annuities is an insurance that is paid by lump sum and not by premium payments. The components of the lump sum are described in the previous bullet. Since the lump sum is already paid at the start of the policy it follows that the portfolio of Life Annuities is typically characterized by the zero value of PV(Lump Sum).

The following table shows the PV(Resulting Cashflow) and its components for this product and per policy:

PV(Lump Sum) 0 0

PV (Benefits) - 491,064.480 - 43,031 PV (Costs) - 7,733,663 - 678 PV (Resulting Cashflow) - 498,798,142 - 43,708

The portfolio can be divided into Limited and Lifelong Annuities. The following table shows that in numbers the Limited Annuities are a vast majority of 70%, but weighted by lump sum the Limited Annuities are only a small majority of 56% of the portfolio. This is the result of the fact that the lump sum of Limited Annuities is almost half the lump sum of Lifelong Annuities. It also shows that on average the Limited Annuities started 6 years ago and Lifelong annuities 7 years ago. This follows from comparing the present age with the age at start.

(32)

The following table shows the main characteristics of this portfolio for Limited and Lifelong Annuities.

Portfolio characteristics Limited Annuities Lifelong Annuities Total portfolio Percentage of portfolio

(weighted by number policies)

70% 30% 100%

Percentage of portfolio (weighted by lump sum)

56% 44% 100%

Percentage male 53% 52% 53%

Lump sum 41,241 73,112 50,951

Annuity period at the start 9 lifelong combined Annuity period at present

time

3 lifelong combined

Present age 66 69 67

Table 4.6: The main characteristics of this portfolio

4. The following figure shows the present value of premiums, benefits and expenses versus the present value of the resulting cashflow. It also shows how these values change over time. The values are based on the standard scenario under Solvency II.

Figure 4.10: Present Value at time t of future cashflows

As mentioned before, the lump sum has already been paid, therefore the PV(Lump Sum) equals 0. Since there is no positive cashflow, the PV(Resulting Cashflow) is negative and almost completely defined by the PV(Benefits).

5. The SCRLif e depends strongly on the development of the number of policies.

The following figure shows how the number of policies decreases by the expected effect of mortality and expiration. In contrast to term life insurance and funeral insurance the number of policies does not decrease because of lapse. The reason for this is that in the case of Life Annuities, the annuity payments can only stop upon the death of the insured person(s), there is no possibility of lapse.

(33)

From the picture it follows that the cumulative effect of expiration is constant after 15 years, which means that the longest (remaining) premium period for Limited Life Annuity equals 15 years. This effect has been visualized by drawing dotted arrows for the future period where there will not be any expiration on this portfolio. The dotted arrows are therefore in the area where the cumulative decrease by expiration remains constant.

The average portfolio mortality rate and the number of mortalities per year are shown in the following graph.

Figure 4.12: Number of policies terminated at time t, because of Mortality

The graph shows that most insured people are expected to die 25 years from now. Since the average policy holder is 67 years old, this is to be expected. Another point to be noted is that from 35 to 45 years from now the portfolio average mortality rate decreases before it increases again. This is caused by the portfolio mix: in 35 years from now the average policy holder is about 100 years old and therefore the mortality rates are very high. The even higher mortality rates of the older people will lead to really high mortality rates weighing less in time because they will die and be taken out of the portfolio. Therefore the lower mortality

(34)

rates of the younger part of the group can cause a temporary decrease in average mortality rates when people become very old.

(35)

Generation of scenarios,

modelling of risks and allocations

This chapter describes in more detail how the random loss Xi can be modelled and

it provides guidelines to model the capital allocations. To give a clear understanding of this process, section 5.1 discusses the characteristics of the underlying risk drivers, section 5.2 describes how stress scenarios are generated and finally section 5.3 shows how these generated stress scenarios lead to the modelling of random losses and capital.

5.1 Defining the distributions of the change in the

under-lying risk drivers

The measurement of risk in this thesis follows the idea of Solvency II and thereby measures the risk by the calculation of the risk module SCRLif e. This SCRLif e is the

underwriting risk for life insurance and is calculated by the aggregation of 5 risk sub-modules. The term underwriting risk refers to the loss that results when the realized cashflow deviates from the expected cashflow. This deviation is caused by a change in the underlying risk driver. The calculation of the 5 risk sub-modules is based on the change in 3 underlying risk drivers. The risk sub-modules SCRM ortality, SCRLongevity

and SCRCAT are based on three different kind of changes in the underlying risk driver

mortality rate. The risk sub-modules SCRLapseand SCRExpensesare based on a change

in the underlying risk drivers lapse rate and level of expenses. For each of these 5 risk sub-modules the distribution of the change in the underlying risk driver will be implicitly defined by the VaR-method on which the idea of Solvency II is based. The assumption that we will make that the change in the underlying risk driver is normally distributed. The expectation of this change then follows from the standard scenario and the variance follows implicitly from setting the 99.5% percentile equal to the prescribed shock as defined in Solvency II.

For the 5 risk sub-modules this leads to the following quantile formula for the change in the underlying risk driver:

F_{(Change in Risk Driver}−1

m)(P rescibed Shockm) = 0.995, (5.1)

with Change in Risk Driverm ∼ N(µm, σ2m) for m = 1,..,5.

In this equation µm equals the mean change in the risk driver in the standard

sce-nario for risk sub-module m. This change can either be multiplicative or additive. Since σm is the only unknown parameter in this equation, it can be solved for each m. Hence

the distribution of the change in the risk driver for each of the 5 risk sub-modules is defined.

Solving formula (5.1) mathematically for σm shows that σm can be calculated as

(P rescribed Shockm− µm) divided by 2.5737, since 2.5737 is the 99.5% quantile of the

Allocation of life underwriting risk to different products under Solvency II : effects of different allocation methods

Different Products under Solvency II

Pascal Vrolijk

Contents

Preface

Introduction

Chapter 2

Measurement of Risk under

Solvency II

2.1

Solvency II and the Three Pillar Approach

2.2

Measurement of Risk, the Applicable Framework of

Solvency II

2.3

Value-at-Risk

2.4

Calculation of SCR

Chapter 3

Optimal allocation of risk

3.1

SCR

module as measurement for loss

3.2

Optimal allocation formula

3.3

Business Unit and Aggregate Portfolio Driven

Alloca-tions

3.4

Market Driven and Default Option Driven Allocations

Chapter 4

Description of the insurance

portfolio

4.1

Term life insurance (ORV)

4.2

Funeral insurance (Uitvaart)

4.3

Life Annuity (Direct Ingaande Lijfrente)

Generation of scenarios,

modelling of risks and allocations

5.1

Defining the distributions of the change in the

under-lying risk drivers