Assessment of capital reservation and hedging for longevity risk

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Assessment of capital reservation and

hedging for longevity risk

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: Onurcan Tatlier

Student nr: 10682619

Email: onur.tatlier@gmail.com

Date: November 6, 2018

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Statement of Originality.

This document is written by Onurcan Tatlier who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

Various factors, like advances in science as well as more health awareness among the population, lead to a persistent increase in the life expectancy of the population. This creates challenges for insurance products that have a dependency on the life expectancy. Consequently it has become increasingly more important for insurers to assess how the trend in mortality will develop. Solvency II prescribes that insurance companies have to hold capital as a buffer for this risk type, the level of this capital can be determined using the standard model. Insurance companies can also choose to develop an internal model, which can benefit insurers by enabling them to better assess their specific busi-ness risk profile and capital requirements. Additionally insurance companies can benefit by hedging away longevity risk, for which purpose longevity swaps and caps can be used that are constructed in this thesis and valued. This thesis examines the benefits and trade-offs between both these options.

A hypothetical portfolio is constructed based on the Dutch population composition in order to examine the differences in capital requirements between the internal and stan-dard SCR model. Additionally for a single person portfolio the differences between the longevity swap and cap are examined and a scenario analysis is conducted in order to get insight into the costs of capital reservation and longevity hedging, using longevity caps and swaps. The conducted analysis shows that the internal model leads to signifi-cantly lower capital requirements for the examined portfolio. Additionally the scenario analysis shows that capital reservation is costly compared to hedging in most consid-ered scenarios. From these results it is concluded that Dutch insurers can benefit from developing and using an internal longevity model and hedging their longevity at least partly.

Keywords: Longevity risk, Solvency II, Economic capital, Longevity hedging, Longevity swaps, Longevity caps, Mortality modeling, Lee-Carter, GAPC, Costs analysis, Scenario analyss

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Acknowledgements

This thesis concludes my master’s degree in Actuarial Science and Mathematical Fi-nance at the University of Amsterdam. While writing this thesis I was working as an Actuary at Achmea.

I am grateful to my supervisor Rob Bruning for his guidance and support during this process. I also wish to thank all my other teachers at the university, whose cooperation has helped me to conduct this analysis.

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Chapter 1

Introduction

From the mortality data of the last century a slow but persistent positive trend in life expectancy of the population in developed countries can be observed. In figure 1.1 it can be seen that both for males and females the average life expectancy has increased by around 10 years, in which the trend in life expectancy between both can be seen to be strongly correlated. This trend can be attributed to various factors, the main reason being advances in science and technology having led to better living conditions and better health care. Additionally a change in lifestyle also plays an important role, through better education and more health awareness within the population.

Figure 1.1: Development in life expectancy for the Dutch population from 1950 until 2017 (based on data derived from the Human Mortality Database)[4]

While the increase in life expectancy is a very positive trend overall, it also poses risks for insurance companies and pension funds. For life annuities, pensions and other similar products in which payments are contingent on survival, this namely poses the risk that policyholders also outlive the capital that was reserved for them within the product. As the payout term for life annuities and pensions is often directly linked to the life expectancy of policy holders, it is needless to say that in pricing these products it needs to be assessed thoroughly how this trend will develop1. This has however not proven to be an easy task for actuaries. An additional complexity is that the insurance market

1_{To give a more material view, the size of the longevity risk exposure in private-sector corporations}

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has become more competitive, also driven by the low interest rates. Being too conser-vative in assessing insurance risks can result in expensive insurance products that fail to attract clients (Bravo J.M. and Diaz-Jimenez J., 2014)[3].

In the Netherlands, the professional association of actuaries, ”Actuarieel Genootschap”, presents the projected mortality rates pension funds use to compute their obligations. Having mentioned the importance of accuracy between these projections and realiza-tions, it is interesting to compare those statistics. Table 1.1 shows that for the years 2012 and 2013 life expectancy rates were overestimated both for males and females. For pension funds this results in a higher need for capital in order to meet obligations. Although an overestimation of a couple of months, as showcased in table 1.1, may not seem imposing, the effect it has on the balance sheet of an insurer should not be un-derestimated. It also needs to be noted that with longer term projections accuracy can further decrease.

Table 1.1: Period life expectancy of men(left) and women(right) compared with the realization (Royal Dutch Actuarial Association, 2015)

An important question in modeling and forecasting life expectancy is whether the cur-rent trend will continue to persist. Some demographic experts are of the opinion that this trend can impossibly persist and the human lifespan has a certain cap. Additionally they refer to factors like increasing obesity and environmental pollution, that counteract the positive trend. On the other hand it is not unthinkable that in the coming decades effective medicines get developed to slow down or even cure deadly diseases like cancer and heart diseases. As these two categories of diseases make up for almost 60% of the causes of deaths in the Netherlands in 2016, it shows what the impact of these advance-ments can be for the life expectancy of the population (volksgezondheiden.info, 2018)[6]. However as there is a high uncertainty in these expectations, this also translates itself into a high uncertainty in predictions of the future mortality rates (International Mon-etary Fund, 2012)[7].

As the insurance and pension field has a high exposure to this longevity risk, it is need-less to say that different strategies are being employed in order to provide protection against the realization of this risk. One of these strategies is to maintain capital as a protection buffer against longevity risk, in which case the question arises what the appropriate capitalization level should be. Within the insurance industry, Solvency II actually requires an appropriate level of capitalization but also provides a helping hand in determining this level.

Solvency II requires that sufficient capital is held for the next year in order to withstand a once in a 200 year shock, corresponding with a 99.5% confidence level. Solvency II however gives insurers two options for determining this buffer. The first and computa-tionally simplest option is to use the standard model within Solvency II. This provides a comprehensive framework in which one-year capital requirements can be calculated that are necessary in order to withstand one in a 200 year shock for various risk

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cate-gories. Insurers should assess what risk categories are relevant for them and calculate the corresponding capital requirement. These capital requirements are then aggregated through a correlation matrix. In this thesis, only the longevity risk under the life module is considered. Solvency II prescribes to use a 20% decrease in the morality rate structure in coming to the one-year capital requirements for longevity risk. This decrease that is applied is calibrated at the European level in order to calculate the necessary capital level to withstand a once in a 200 year longevity shock and might not be an accurate representation of what is applicable in the Dutch market (EIOPA)[8].

Solvency II additionally allows insurers to develop their own internal model which can, in case approved by the regulator, be used to come to capital requirements. As the stan-dard model is calibrated on the European industry as a whole, an internal model can benefit companies by enabling them to better assess their specific business risk profile and capital requirements. This is especially interesting if they expect this approach to lead to lower capital requirements. Insurers are in this way incentivized to build and employ an internal model as this can possibly lead to lower capital requirements than under the standard model. Constructing an internal model is however also the more complicated and demanding route as it requires establishing internal expertise to model the mortality structure in the company context.

Various mortality models have been developed in the last decades of which the Lee-Carter model is probably the most well known. Several other mortality modeling meth-ods have been developed as extensions. The Lee-Carter model will however be the base model for developing an own internal model for capital assessments.

Another option that insurers have in dealing with longevity risk is hedging this risk. Due to demand for longevity hedging products, various products for this purpose have been developed. In this thesis longevity caps and longevity swaps will be considered. Through entering in these derivatives, effectively longevity risk can be transferred to a counterparty. It needs to be noted however that although the longevity insurance mar-ket has an impressive expected deal size over £30 billion in 2018 at the global level, it is still a relatively new and developing market in which most transactions also take place in the UK. Due to immaturity in this market, illiquidity issues exist. This will mean that counterparties can demand relatively large concessions from the parties that are interested in these products, possibly resulting in high risk premiums in the swap. Another reason that the benefit of longevity hedging instruments should be assessed be-forehand, is that they expose the owner to new hedging and counterparty risks (Aon., 2018; Pigott C. and Walker M., 2016)[5][9].

The question this thesis examines is whether insurance companies can benefit from developing and using an internal model for longevity risk and hedging longevity risk us-ing derivatives, in specifically usus-ing longevity swaps and longevity caps. This question is specifically examined for the Dutch insurance market. In performing these assessments the gains and costs of these possibilities need to be specified and compared. This re-quires that an internal model is developed in order to compare with the outcomes of the standard model. Additionally longevity swaps and caps need to be valued and their characteristics need to be examined and compared for different scenarios.

This thesis is structured in the following way. Chapter 2 describes and discusses the standard model in Solvency II and how it deals with longevity risk in specific. As the first goal of this thesis is to make a comparison with an internal model, in chapter 3 the theory behind mortality modeling is discussed and used in R to build mortality models to come to a stressed mortality rate structure for a once in a 200 year old situation in

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the Dutch market. This is specifically done using Dutch mortality data from the online human mortality database. In chapter 4 the valuation of longevity swaps and caps is discussed and benefits of hedging longevity risk through entering into these derivatives is examined. Finally all this comes together in chapter 5, in which the theory is applied. A hypothetical portfolio based on the current Dutch population composition is con-structed in order to determine and compare the capital requirements for the standard and internal model. Furthermore the characteristics of the longevity cap and swap are examined on a single person portfolio and finally a scenario analysis is conducted in or-der to get insight and rank the costs for both capital reservation and longevity hedging using longevity swaps and caps.

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Chapter 2

Solvency II capital requirements

within the Standard model

At 1 January 2016 Solvency II came into effect, providing a supervisory framework for the European insurance sector. It succeeded the Solvency I framework that already dated from 1973 and was widely recognized to be flawed and outdated. The main points of critique were that the prescribed formula to determine the capital requirement was very simple and therefore unable to catch many actual risks that insurers assume in calculating their solvency capital requirement. Additionally it was widely recognized to be calibrated at a too low level of capital, the European regulators expected insurers to hold a multiple of the capital prescribed by this framework. A much more integral risk framework that addressed these flaws was therefore needed (Olleson A. and Lobregt J.H., 2006)[10]. Solvency II can be seen to be very similar to the Basel II and III frameworks for banks in that it prescribes a three-pillar approach(EIOPA)[8]:

• Pillar I: Quantitative capital requirements. This pillar is concerned with the val-uation of liabilities and assets and coming to a Solvency Capital Requirement (SCR). This thesis focuses on this pillar.

• Pillar II: Governance and Risk management requirements. This pillar prescribes the requirements for an effective governance, as well as the Own Risk and Solvency Assessment (ORSA), which is a yearly internal risk assessment that should be reported to the regulator. The second pillar aims to enforce additional capital evaluation based on internal assessment of risks and controls.

• Pillar III: Disclosure requirements. This pillar contains reporting requirements in order to enforce the disclosure of information on risk and capital levels. Both a public (SFCR) as well as a confidential report(RSB) are required.

2.1 Solvency Capital Requirement

Within the first pillar, Solvency II provides a comprehensive set of risks that are relevant for insurers. Based on these risks, organizations should assess which risks are relevant for them specifically and determine capital requirements to withstand these risks with a 99.5% certainty in the next year. More specifically this necessary capital has three building blocks:

1. Best Estimate Liabilities (BEL): This is the current value of all future liabilities, discounted using the risk-free curve.

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2. Risk margin (RM): The risk margin together with the BEL forms the Technical Provisions (TP) and is intended to ensure that in case necessary, because an insurance company has used its surplus and capital in the event of a shock, a third party can take over the obligations and meet the insurance and reinsurance obligations. The RM is intended to reflect the non-hedgeable risks to which the insurer is exposed.

3. Solvency Capital Requirement (SCR): The SCR comes on top of the TP in order to ensure that insolvency only occurs once in 99.5% years. Solvency II gives insurers the option to either use the standard model or come up with an internal model in order to determine the SCR. The standard model is briefly discussed below, while mortality modeling is discussed in the next chapter[8].

This capital framework is an example of what is more broadly known as an economic capital framework. The TP in Solvency II constitute the expected loss, while the eco-nomic capital is a buffer that needs to be held on top of this expected loss in order to guarantee that the insolvency as a result of adverse shocks only occurs once in x years. This buffer is the SCR in Solvency II, while x is chosen to be 200 years corresponding with a confidence level of 99.5%. Figure 2.1 below gives an depiction of these concepts. It is important to note based on this figure that certain extreme events can occur that can lead to losses the company can not withstand.

Figure 2.1: Concept of economic capital[13]

Using the standard model, the SCR is obtained by application of pre-calibrated Sol-vency II shocks on factors that affect the assets and liabilities, both sides of the balance sheet are shocked. More specifically within the standard model, six modules are distin-guished that consist of various sub-modules corresponding with a risk dependence, in the next section a few examples of these modules will be given. The shocks are applied at the sub-module level and eventually aggregated to a total SCR by application of the correlation matrix provided in the standard model.

As the actual shocks are applied at the sub-module level, it is good to get a better un-derstanding what happens here. Figure 2.2 provides depiction of this concept, basically the balance sheet is stressed in order to get to an economic loss per factor. Insolvency occurs when the post-shock liabilities are higher than the post-shock assets, sufficient capital should be maintained in order to have a 99.5% probability that this does not happen in the upcoming year. The shocks prescribed by the standard model were cali-brated such that this corresponding amount is obtained per factor.

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example being interest rate sensitive products. In these cases both an upward as well as a downward shock needs to be applied after which the shock should be chosen that leads to the biggest loss. Eventually all these determined losses are aggregated as was explained earlier.

Figure 2.2: Stressing the balance sheet (Morin F., 2011)[14]

2.2 Risks in life annuities and pensions

For a simple life annuity or pensions product, policyholders will often pay a certain premium over an agreed period and subsequently receive periodic payments that are contingent on their survival. For these kind of products broadly the following risk types are encountered:

• Generic risk types as regulatory and operational risks for the insurance contracts and the management of the products.

• Market risks, including equity, interest rate and credit risks: Premiums need to be invested carefully, such that sufficient (liquid) resources are available at the time of the payouts to the policyholders. Typically insurance companies will invest a large portion of the premium in fixed-income products with a high credit rating, while a smaller portion is typically invested in more risky asset classes like equities or corporate bonds. This means that the interest rate risk will often be the main market risk. Interest rates have shown a persistent negative trend in the last 10 years and some rates have even become negative, which effectively means that investors need to pay money in order to store their capital. This has heavily impacted the industry and made insurance products more expensive and therefore less appealing.

• Insurance risks: Solvency II speaks here of underwriting risks and makes a dis-tinction between the life, underwriting and health risk modules. Depending on the insurance product a shock in these submodules can have a beneficial or adverse impact: While a steep increase in the number of deaths leads to more life insur-ance payouts, the opposite holds for life annuities. Per insurinsur-ance product it should therefore be considered what the relevant submodules are.

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In this thesis the emphasis lies on the longevity shock within the life module, as can be seen in figure 2.3. By developing an own model for the mortality structure and stressing it in line with the Solvency II requirements, it can be examined how much the loss implied by this model deviates from the shock prescribed in Solvency II[8].

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Chapter 3

Mortality Modeling

The goal of this chapter is to build an internal model to determine the capital that should be held in line with the Solvency II capital requirements for longevity risk. In doing this the force of mortality will be introduced, the longevity improvement over time will be shown and different mortality models will be discussed and compared that capture this time improvement. The different models are obtained from the STMoMo package in R and are applied on the Dutch mortality data was obtained from the human mortality database.

The internal model that is developed in this chapter will be used later in this thesis in order to compare the capitalization levels obtained by the internal model and the Solvency II standard model.

3.1 Data

Dutch mortality data is used in order to fit the models that are discussed in this chapter. This data was obtained from the Human Mortality Database (HMD, 2018)[1]. The HMD provides open access to mortality and population data for various countries, including the Netherlands. The HMD has the additional advantage that there are packages in R that make use of its datasets.

More specifically in this chapter the 'Deaths' and 'Exposure to Risk' datasets obtained from this database are used for analysis and mortality modeling purposes. The data set 'Deaths' gives an overview of the number of deaths both for the female and male population, for the time span 1850 until 2016, for ages 0 until 110 years old. 'Exposure to Risk' gives an overview of the total size of these particular classes. From this data various other important mortality related measures can be obtained. It is important to note that there is little data on people that are 95 years or older. This leads to more uncertainty on model fits in this range and is something that should be considered. This range was therefore also used for plotting figure 3.1.

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Figure 3.1: Force of mortality, for years 1900 until 2014 and ages 0 to 95 (own fit using data from the Human Mortality Database and the R package STMoMo)

3.2 Mortality rates

In this section basic notation will be introduced that is relevant in mortality modeling1. By introducing the random variable Lx as the number of people in a population aged x, the number of deaths for this age class in the coming year can be further defined as:

Dx = Lx− Lx+1 (3.1)

The mortality rate qx for this age group can then be defined as: qx=

Dx Lx

(3.2) In practice actuaries often work with discrete yearly mortality rates as defined above. However in order to model the mortality rate using regression methods and longevity improvement trends, a continuous model needs to be introduced for the mortality rate, which is done in the next section[12].

3.3 Force of mortality

Rather than thinking about mortality rates as proportions of death occurrences at a certain age, it is very useful to think more abstractly about occurrences that happen once and only once at a future time T . The time T can then be seen as the ’time of failure’. Note that this type of thinking does not necessarily have to apply to the time of death, but can apply to all kinds of occurrences. This type of thinking has therefore

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been heavily utilized and researched in stochastics. In this section, it will be explained how this can be applied in order to get to the force of mortality[12].

Given the random variable Td ∈ [0, ω] representing the time of death of an individual, in which ω is set as the maximum age that a person can achieve, the survival function is given by:

sTd(x) = P {Td> x} (3.3)

In order to avoid confusion with time notations later on in this chapter, the subscript Tdwill be left out. This function gives the probability that death2 occurs after the age of x. The probability density function (pdf) f is defined as:

f (x) = −s'_(x) _(3.4)

Subsequently, using the survival function and the pdf, the force of mortality3 at time x is defined as:

µ(x) = f (x)

s(x) (3.5)

The force of mortality is the conditional probability of dying at age x, given survival until that age. Reversely if the force of mortality is known, the survival function can be determined as:

s(x) = exp(− Z x

0

µ(z)dz) (3.6)

The force of mortality µ(x) as defined in this way does not include a time dependence. It can therefore be considered as a slice for a certain time within the mortality surface as depicted in figure 3.1. The time dependence is important as longevity will be expected to improve in the future and it is important to include this expectation into the calculation. For the mortality models in the next sections the time dependence will therefore also be included, the notation µx,t will be used. These models will also have interactive terms between x and t. In order to treat the age and time variable symmetrically, both are denoted as a subscript.

3.4 Lee-Carter model

The most elementary and well-known mortality model is the Lee-Carter model, which has the following form (Haberman S. and Renshaw A., 2011; Wang J.Z., 2007)[15][16]:

ln(µx,t) = αx+ βxκt+ x,t (3.7)

Note here that a time dimension is included in order to capture the longevity trend over time. A subscript notation µx,t is used in line with academic literature, but also to be consistent in the notation for the variables x and t. In the above equation, αx is the average age-specific pattern of mortality. κt captures the time trend in mortality, while βx gives the sensitivity to this time trend at age x and time t. βx can therefore also be seen as an interaction effect between time and age. The error term is assumed to have a normal distribution and captures the variability of the log force of mortality that is not captured by the rest of the model. It is important to note that none of the terms at

2

More broadly the term time of failure is used within stochastics as stated earlier

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the right hand side of the equation is observable, so all these terms are estimated based on data that is available for the force of mortality.

This model is invariant under certain transformations, for example under the transfor-mation

(αx, βx, κt) → (αx, βx/c, cκt) (3.8) In order to make this model identifiable, additional requirements need to be set on the parameters, by demanding thatP

tκt= 0 and P

xβx= 1. This implies that

αx = 1 T − t1+ 1 T X t=t1 ln(µx,t) (3.9)

From this equation it can be seen more clearly that this term is the average age-specifc pattern of mortality.

In general the underlying time trend κtis modeled as an autoregressive process, as this turns out to fit data of many countries in practice very well:

κt= κt−1+ c + t (3.10)

This relation can then be used for forecasting the mortality rates in the future, by plugging it into equation 3.5 once all the parameters are estimated[15][16].

Although Lee-Carter is a simple and powerful model, it also has shortcomings. The main issue is that equation 3.10 implies that mortality improvements for all ages will follow the same pattern. It fails to catch certain mortality trends, like the so-called mortality rotation in which mortality development is seen to be decelerating at younger ages and accelerating at older ages in developed countries (Li N., 2015)[18].

In order to tackle these shortcomings, quite a few of extensions of Lee-Carter have been developed, some of which will be discussed in this chapter. In order to discuss these, it is first of all useful to discuss how the Lee-Carter model can be generalized and see what the common features are with the other mortality models.

3.5 Broader class of mortality models

In recent years different attempts have been made to unify the mortality models in use under one category. There are in fact different ways to generalize these models. In Appendix B for example it is explained that Lee-Carter is a factor principal component model. This section will mainly focus on the so-called Generalized Age-Period-Cohort stochastic mortality models, GAPC in short, as most the mortality models that are introduced in literature, fit into this class. GAPC models are a type of the well-known Generalized Linear Models(GLM) as the name suggests and are composed of four com-ponents (Shang H.L., 2012)[19]:

1. Random component: Dx,t∼ Binomial(Lx,t, qx,t), with E _D

x,t

Lx,t

= qx,t. Note that here the notation from section 3.2 was generalized by including a time index t, but apart from that the definitions do not change.

2. Systematic component: Similar to 3.7 and GLMs, this component captures the effects per year and time through the predictor ηt,x. Additionally note that a cohort factor γ is introduced that captures the year-of-birth effects:

ηt,x = αx+ N X

i=1

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3. Similar to GLMs there is a link function g that relates 1 to 2, through g E Dx,t Lx,t = ηx,t (3.12)

4. Set of parameter constraints: In the Lee-Carter model it was seen that a set of constraints need to be imposed in order to make the model identifiable. This also holds for the broader class of GAPC models.

3.6 Models used for Mortality modeling

Table 3.1 below gives an overview of the models that are used in this thesis in order to model the mortality surface and make projections of future longevity.

Model Formula

Lee-Carter ln(µx,t) = αx+ βxκt+ x,t

Renshaw and Haberman(RH) ln(µx,t) = αx+ β1,xκt+ β2,xγt−x+ x,t Age-Period-Cohort(APC) ln(µx,t) = αx+ κt+ γt−x+ x,t

Table 3.1: Different mortality models

Various mortality models use the Lee-Carter model as a basis. The Renshaw and Haberman model extends the Lee-Carter model by adding a term for cohort effects. In the given formula, αx is the average age-specific pattern of mortality, while the pa-rameters β1,x and β2,x measure the age interaction, γt−xis a random cohort effect as a function of the year of birth and x,t is a random period effect (StMoMo, 2018)[19].

3.7 Model fitting

This section is dedicated to showcasing outcomes of the mortality modeling analysis. Further in-depth theoretical analysis is included in the appendix A. The models de-scribed in section 3.6 are fitted on data from the Human Mortality Database. During the analysis it was observed that the time range used for fitting the models has a strong impact on the fit. Therefore a trade-off needs to be made between representativeness and quantity of the data. The choice is made to select a recent and representative time range, therefore the range 1980 until 2016 is used for fitting the models. This means 2017 is the first year for which projections are computed.

Figure 3.2 displays the Lee-Carter fit and projection outcomes for the ages 40, 60 and 80. The blue line represents the fit on the historical mortality rates, which are themselves represented by the blue dots. The striped black line shows the best estimate projec-tion, surrounded by the 95% confidence interval (black dotted). Confidence intervals are derived by creating parameter samples for the GAPC model using the semipara-metric bootstrap method (Brouhns N. et al., 2005)[19]. The red striped lines represent the same 95% interval including parameter uncertainty within the model. Based on this graph, no significant impact can be seen in including parameter uncertainty.

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Figure 3.2: Lee-Carter fit and projection on Dutch male mortality data

To display the Lee-Carter forecast for the upcoming 50 years, figure 3.3 was con-structed. In a rainbow view, red being the earliest point ending with purple, projected male mortality rates are showcased.

Figure 3.3: Dutch male Lee-Carter forecast for the years 2017 until 2066

To make a comparison in figures 3.4 and 3.5 forecasts of the ”Actuarieel Genootschap” which are based on the Li-Lee model are included. Based on the above analysis and fits, as discussed in appendix A, it can be seen that the extensions of Lee-Carter do not have a significantly better performance. Therefore the Lee-Carter model will be used for the SCR calculations in chapter 5, as it is the simplest model to work with and interpret.

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Figure 3.4: Li-Lee and Lee-Carter forecast for the year 2017 for Dutch male and female mortality

Figure 3.5: Li-Lee, Lee-Carter, Age-Period-Cohort(APC) and Renshaw and Haberman model forecast for the year 2017 for Dutch male and female mortality

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Chapter 4

Longevity Hedging

Rather than holding capital in order to withstand the realization of longevity risks, companies can also choose to reduce their exposure to this risk. Companies can choose from different options (Kaufhold K., 2013)[29]:

1. Pension buy-out: Company chooses for transferring responsibilities and liabilities completely to another party. This option will not be the first choice for most com-panies as baring certain risks while receiving premiums in return is an intentional choice. Additionally companies want to retain a contract with their clients. 2. Pension buy-in: Company chooses for insuring certain risks with another insurance

company. This will be more desired by most companies compared to option 1 as contract with the clients is retained.

3. Hedging: Company chooses for hedging longevity risk with suited instruments. Different instruments are available in the market, one popular instrument is the longevity swap.

In this chapter the last option is examined. First the longevity insurance market will be briefly discussed in order to get a better understanding of this market and its mechanics. Two derivatives are constructed for hedging the longevity risk in a life annuity, being the longevity cap and longevity swap. The valuation of both derivatives is discussed and for this purpose an own notation is introduced. In specific the longevity cap is constructed in such a way that it is comparable to the swap, i.e. reduces to the swap if the cap is put to zero. In the last section an approach is introduced to assess the gains and costs of capital reservation and hedging longevity risk, which will be used in chapter 5.

4.1 Longevity insurance market

Longevity risk is often a very material risk for insurers and pension funds and is also seen as an unrewarded risk, in the sense that it does not contribute to higher expected returns in baring, contrary to for example investment risks. Following from this consid-eration, it makes sense for affected parties to engage in longevity hedging. The Dutch sector is however seen to be hesitant, an important reason to explain this are the neg-ative sentiments that still exist within the market about (exotic) derivneg-atives. Although the longevity insurance market is quickly growing, it is still a new market with a lack of standardized products, in which contract terms need to be carefully negotiated with counterparties. Considering this, it is not surprising most of the trades in these instru-ments are currently executed in the UK. Nevertheless the recent demand by pension

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funds and insurers for instruments to hedge longevity risks is leading to further matu-rity in this market and the creation of circumstances for parties to eventually engage in trades (Bravo J.M. and Diaz-Jimenez J. 2014; Pigott C. and Walker M., 2016)[3][9]. An important condition for liquidity is that there are enough parties willing to cre-ate and offer insurance. The counterparties that are willing to engage in trades in the longevity insurance market are often the larger pension funds, (re-)insurers and invest-ment banks. Various incentives exists for them to be active in this market (van Diepen P., 2010)[27]:

• Return through favorable terms: As the market is not very liquid, parties to which risks are transferred can demand a relatively high risk premium. Similarly, par-ties can demand favorable terms on other parts of the contract that need to be negotiated.

• Information advantage: Issuers can gain from their expertise on mortality devel-opment and modeling in entering into contracts.

Additionally, longevity instruments are not only interesting for longevity hedging, but as these instruments have almost no correlation with other instruments, they can also be interesting from a portfolio diversification perspective for investors. This leads to more players in the longevity insurance market, increasing liquidity.

In the longevity insurance market, parties looking for hedging their longevity risk will often need to make the trade-off between standardized and customized instruments. Standardized instruments will be more liquid but might not fully cover the needs of a party. If therefore willing counterparties can be found, it can be the better option to negotiate and create contracts that specifically meet ones needs. In the next sections two such types of derivatives will be constructed and discussed that are intuitive in hedging longevity risks, being the longevity cap and longevity swap. The longevity cap, or mortality floor depending on the point of view, is an insurance against the mortality dropping more than a certain percentage that is agreed upon up front. The other in-strument is the more well known longevity swap1, in which two parties agree to swap a stream of fixed and variable longevity dependent payments. Valuation of these instru-ments will be discussed as well as the benefits they provide in hedging longevity risks (Coughlan G., 2014; Hull J., 2011)[30][31].

Both the mentioned derivatives have the risk that the freedom to customize the in-struments to specific needs also means that certain features within the contract turn out to be disadvantageous afterwards. It is therefore important that both sides assess what features pose risks for them and what conditions should be avoided. Examples of features that need to be specified carefully are[27]:

• Valuation curves and methodology: This includes the interest rate structure and mortality rate structure that will be used in the valuation of the derivatives, as well as the specific methodology that will be used.

• Reference group: Standardized derivatives will be based on a benchmark popu-lation, indexed-based longevity swaps are an example of such derivatives. The composition of the reference group determines the mortality rates that are used in the valuation of the derivative, this will also be seen in section 4.3 and 4.4. As a result basis risk might arise due to differences in the country, age and other attributes of the actual client base of the insurer and the reference group the

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derivatives are based on. Parties can therefore decide to base the contract on a specific and relevant target group.

• Collateral requirements: In case the counterparty credit risk is high, it makes sense to mitigate this risk by including collateralization requirements in the contract • Maturity and termination: Maturity of the contract as well as the conditions under

which the contract can be ended pre-maturily need to be specified.

4.2 General notation

In sections 4.3 and 4.4, respectively a longevity swap and cap will be constructed that can be used in order to hedge longevity risk in a life annuity portfolio. In order to construct these instruments and value them, first a suited notation needs to be introduced that will be used2:

• *: the asterisk is used to denote that the given life and morality rates are random variables. Rates that do not contain an asterisk are basically best estimates at time 0 and are known numbers that are provided in a mortality table. Rates that do contain an asterisk basically have to be simulated. In the calculations made it will hold that E(p∗) = p and E(q∗) = q regardless of the indices used3.

• _npxj,t: Probability (non-stochastic, known value) that an individual aged xj will

survive for n years, as seen on time t. If the left index n is left out, it can be assumed that n = 1.

• np∗xj,t: The n-year survival probability (stochastic) equivalent of previous line.

• q_x∗

j,t: The probability of death (stochastic) for an individual aged xj in the coming

year as seen on time t.

• bt,type: Fraction of the notional that is made as a fixed payment at the end of year t for a derivative of a given type. Note that for a swap it will also needs to be specified whether it this term refers to the floating or fixed leg. The three used types are lc (longevity cap), lsf ixed (longevity swap fixed leg payer) and lsf loating(longevity swap floating leg payer).

• dt= d0,t: The discount factor that is applied on payments at time t4. • N : Notional amount set in the contract.

• i: Yearly rate of inflation, is assumed to be constant.

• G = ((g0, x0), (g1, x1), ...) : Denotes the group of policyholders, i.e. client base, characterized by gi and xispecifying the gender and age respectively of the person indexed by i = 0, 1, ....

• A: Age at which the policyholder group members start receiving payments, can be seen as retirement age.

2

This notation is mainly an own extension of Promislow S.D.[12]

3_{In academic literature the tilde notation, for example ˜}_{T for discrete failure times, is used in line}

with theory introduced. But for more consistency the introduced notation will be used

4_{Note that d}

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• ω: Maximum age a policyholder is assumed to reach, i.e. q_ω,t= 15.

• V0,t,type(G): Present value of a time t cashflow of a derivative of a given type, for policyholder group G. Note that for a swap it also needs to be specified whether this term refers to the floating or fixed leg. The indexP t will be used to denote summation over all times.

• cmrs: Abbreviation of consensus mortality rate structure, i.e. the mrs both parties agree to base the contract on. Depending on the agreed conditions in the contract the cmrs can be the mrs that is applicable for the own portfolio of the insurer.

4.3 Longevity swap

A longevity swap is similar to a currency or interest rate swap, in that an agreement is made between two parties to swap a stream of future variable payments (floating leg of the swap) for fixed future payments (fixed leg of the swap). The longevity swap is effectively a bet on the movement in the cmrs. Insurers or pension funds can enter a longevity swap in order to hedge the longevity risk in an annuity or pension product, by agreeing on fixed payments.

Figure 4.1: Mechanics of a longevity swap

Figure 4.1 provides a depiction of the longevity swap. The fixed leg payments are based on the cmrs at time zero and are therefore known initially. The floating leg payments on the other hand will depend on the development in the cmrs and are therefore stochastic. Initially both the floating and fixed legs for all times will coincide and the value of the swap will theoretically be zero. The difference in legs will be caused by new information that arises at later times, which impacts the cmrs and thereby the floating legs. In practice the counterparty will demand a risk premium for having the longevity risk transferred over to them. This can be incorporated in various ways in the contract, for example as a simple cost addition or a shift in the cmrs incorporating the risk premium. The latter one will be used in this thesis, as it is computationally easier to work with.

The counterparty in the swap can again engage in a similar type of swap with a third party for various reasons, for example to earn a fee in facilitating this structure or to reduce exposure to longevity risks. Although this can affect the counterparty credit

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risk, it will not be considered in the valuation of the swap.

The longevity swap construction and valuation will initially be derived by assuming the policyholder group G consists of one male of exactly age x0 during initiation of the derivative, i.e. G = ((m, x0)), the policyholder group will subsequently be expanded. For convenience the time of initiation of the derivative is set to zero6, the contract has no delay and starts immediately. The swap of cashflows starts once the payments in underlying annuity start, which is at age A for all policyholders. For the above policy-holder this is on time A − x0and will last as long as the person is still alive, so payments are made for t ∈ [A − x0, ω − x0], in which it is assumed that t is an integer.

It is good to note that both btj,lsf ixedand btj,lsf loating depend on mortality rates as well

as the expected inflation rate i, the latter will be kept constant. The fraction of the notional of the fixed leg payments are given by the following expression:

bt,lsf ixed = (1 + i)t+1×(t+1)px0,0 (4.1)

For the floating legs this fraction is given by:

bt,lsf loating = (1 + i)t+1×(t+1)p∗x0,0 (4.2)

bt,lsf loatinghere is a random variable as it is a function of(t+1)p∗x0,0. At time 0, E(bt,lsf loating)

will coincide with bt,lsf ixed in case of no cost distortions like a risk premium. More gen-erally, the time 0 value of the end of time t cashflows for the fixed rate payer will be given by:

V0,t,lsf ixed((m, x0)) = N × dt+1× (E(bt,lsf loating) − bt,lsf ixed) (4.3) The above calculation needs to be done for all times. As the time of initiation is 0, payouts need to be made for the remainder of the time that the policyholder is alive which will be ω − x0, so that the total value of the cap will be given by:

V0,P t,lsf ixed((m, x0)) = N × ω−x0

X

t=A−x0

dt+1(E(bt,lsf loating) − bt,lsf ixed) (4.4)

The discounting happens per term inside the sum. Apart from cost distortions, it will hold that V_{0,P t,lsf loating} = −V_{0,P t,lsf ixed}.

As the values of the contracts are additive, the policyholder group G can be expanded by adding in more policyholders. The value of the longevity swap on such a group is basically the sum over the one-person swap from 4.4:

V_{0,P t,lsf ixed}(G) = X (gi,xi)∈G

V_{0,P t,lsf ixed}((gi, xi)) (4.5)

Theoretically a capital reservation to withstand a specific shock will equal the value of the longevity swap that is used to hedge this risk. This can be seen by looking at figure 2.2 and reasoning both from the point of the liabilities and assets. A longevity shock will lead to a liability increase, this gap can be filled with the capital that was specifically reserved for this purpose. More specifically the capital requirement is the summed present value of the differences between the post-stressed and pre-stressed liabilities. But note that the longevity swaps in the asset side counteract this movement

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exactly: The summed present value of the fixed legs theoretically should correspond with the best estimate of the liabilities while the summed present value of the floating legs on the other hand will correspond with the present value of the post-shock liabilities, the difference between the two being the value of the swap. In case of a perfect hedge the increase at the asset side, due to the longevity swaps, matches the increase at the liability side resulting from the longevity shock.

One should note however that the above conclusion holds in a fair theoretical situation, as all kind of conditions in the derivatives contract might impose costs. These can cause the swap price to deviate from this theoretical value and will often make the swap more expensive for the insurer. Examples of such costs are the inclusion of a risk premium or biases in the contract. These costs in the contract should naturally be considered before entering swap contracts.

4.4 Longevity cap

A longevity cap is similar to a currency or interest rate cap, in that the party with the long position is compensated if the probability of staying alive until a certain age exceeds a certain pre-agreed level. This level will often be based on a percentage increase in the crms. Insurers or pension funds can use the longevity cap in order to hedge longevity risks in life annuities and pensions, as they will be protected from shocks in the longevity above a pre-agreed cap.

In deriving the longevity cap, the same approach is used as for the longevity swap, so the swap is first derived for a single male of age x0 during start of the contract at which the time is set to 0. The cap applies on yearly payments that the insurer will make to this person starting from age A until time of death.

The cap rate is given by c and will lie in the interval [0, 1]. The fraction of the notional payment, bt,lc, is a function of random variables and is therefore also a random variable. In order to make the longevity cap comparable to the longevity swap, the payment needs to be constructed in a careful way, which is explained in appendix C. Here it is shown that in this case, for bt,lc it needs to hold that:

bt,lc= max (1 + i)t+1 tp∗x0,0× 1 − qx0+t,t 1 − _q x0+t,t− q ∗ x0+t,t qx0+t,t − c −_(t+1)px0,0 , 0 (4.6) Note now that choosing c = 0, results exactly in the difference between the fixed and floating legs of the corresponding longevity swap given in equations 4.1 and 4.2. This means that in case of a cap of 0, the investor is fully compensated for increases in longevity as should be expected. The compensation will be lower with an increase in the cap.

The value of the cap at time 0 for the payment at time t will then be given by:

V0,t,lc((m, x0)) = N × dt+1× E(bt,lc) (4.7) Similarly to the swap, the above calculation needs to be done for all times, so that the total value of the cap will be given by:

V_{0,P t,lc}((m, x0)) = ω−x0

X

t=A−x0

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The policyholder group G can be expanded by adding in more policyholders. The value of the longevity cap on such a group is basically the sum over the one-person caps:

V0,P t,lc(G) = X

(gi,xi)∈G

V0,P t,lc((gi, xi)) (4.9)

It can be seen that the valuation of this product is not straightforward, as the max-term is not known initially. However it can be seen that the payoff structure is similar to European call options, so option pricing theory and techniques can be utilized in order to price this derivative. The approach that will be used here is that of Monte Carlo simulations, which means that the expectation in equation 4.2 will be approximated through sufficiently simulating and averaging the max-term.

4.5 Costs comparison and risk assessment

The insurer should assess in what cases hedging should be preferred over capital reser-vation. In doing this the following points need to be considered:

• Cost comparison: The expected costs of both options should be compared. The costs between capital reservation and hedging will depend on the realization of the loss. Capital reservation will have opportunity costs, while a counterparty will demand compensation for accepting to transfer risk.

• Risk Assessment: A risk assessment will be an important extension in the decision making process as entering into derivatives exposes the insurer to new risk types. Additionally the risk appetite and preferences of the insurer should be included in the consideration. It should be carefully assessed what the impact is and whether the insurer is willing to bare these risks.

For the value comparison the costs of capital reservation, denoted by Costscr, need to be compared with the costs of hedging with derivatives, denoted by Costsder. Looking at the first term, the main downside of capital reservation is that in case losses do realize, own capital need to be utilized to cover them, which will be denoted by Costsreal. The capital that is necessary might be already present in liquid or illiquid assets or can be raised in the market by issuing equity or debt. In either case, capital reservation will have opportunity costs Costsopp as the company misses out on returns of the capital that could otherwise be invested. These costs will increase with increases in rates of re-turns on investment. Additionally costs will be made in order to obtain capital through liquidation of assets, denoted by Costsliq. So combining these costs, it will hold that Costscr = f (Costsreal, Costsopp, Costsliq, ..) for some function f .

In case of derivatives contracts a compensation will need to be paid to the counter-party for transferring the risk, which will be denoted by Costscomp. This will be the time 0 payment (premium) as well discounted value of all risk premiums. This com-pensation can also introduce significant opportunity costs Costsopp if sufficiently high, as the compensation could otherwise have been invested. Additionally in case of hedg-ing, counterparty credit risk is introduced as the counterparty might default on its debt7_{. This risk can be quantified by looking at insurance against the credit risk,}

for example in the form of credit default swaps(CDS). Note that Costsccr will also

7_{The counterparty also runs a risk as the insurer can also default on its debt, however for convenience}

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depend on Costscomp or might be even completely included in this term, as credit ratings might be considered in coming to a risk premium. Finally it is also impor-tant to note that certain derivatives, like swaps, can have downside risks, denoted by Costsdown. The total costs will therefore be a function of all mentioned contributions, i.e. Costsder = g(Costscomp, Costsopp, Costsccr, Costsdown, ...).

Based on the costs comparison, the derivative should be preferred in case that:

Costsder < Costscr (4.10)

It can be seen that as so many different factors need to be considered in making this comparison, some of them being also very difficult to quantify, this comparison should be made on a case by case basis and possibly partly qualitatively.

Some costs can be capped however. The price of a longevity cap can be used to put an upper bound on the compensation for the corresponding swap. The reason behind this is that a longevity cap with a cap equal to zero, provides an insurance against all longevity shocks. Swaps on the other hand expose the holder to downside risk. In case the risk premium on the swap is higher than the price of the corresponding cap, no rational investor would want to purchase this swap. The risk premium on the swap is therefore bounded by the corresponding cap price, including the risk premium on this cap.

Next to the above analysis it is good to perform an assessment of the options based on more than only the present values of the expected costs, as was motivated earlier in this section. Examples of other risks that will be relevant are:

• Downside risk: The longevity swaps might become (severely) negative. This can for example happen in case that catastrophe risk unfolds within the reference group. Looking only at the expected costs does not take extreme negative events sufficiently into consideration.

• Basis risk: The contract is based on a reference group that is often different from the actual client base. And even if the two perfectly coincide initially, the client base can change over time. In this way basis risk can arise, making the hedge of the longevity risk imperfect.

As so many factors play a role in making the decision, it is important to perform a scenario analysis and examine how choices in situations match up with the risk appetite and preferences of the firm.

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Chapter 5

Application of theory on a life

annuity

In this chapter the theory from previous chapters will be applied. First the standard model in Solvency II will be compared with the internal model developed in chapter 3. As the comparison is dependent on the age composition of the policyholder group, the choice is made to construct a hypothetical portfolio that is based on the population composition of the Netherlands.

In order to perform the calculations for the portfolio in an efficient manner a specific class was written in Python. In the calculations the interest rate structure for pension funds for the start of 2018 was used which is published by the DNB (DNB, 2018)[39]. The notional is taken to be e10.000 and a stable inflation of 1% is assumed for indexa-tion. It needs to be noted that the yearly benefits and indexation are actually not very relevant for most plots, as mostly ratios will be examined. The Dutch mortality data that is used, was obtained from the Human mortality database, as also explained and used in chapter 3. It is assumed that the maximum age policyholders can achieve is 99 years, because of the uncertainty in the models for higher ages due to lack of training data.

Subsequently a longevity swap and cap are valued using the theory from sections 4.3 and 4.4, in order to better understand the characteristics of these derivatives. Finally a scenario analysis is done in order to compare capital reservation with hedging through the use of longevity swaps and caps, using the approach from section 4.5. For these purposes the age composition of the portfolio is far less of an issue and a one-person portfolio is used in order to better highlight the features of the derivatives and make the comparisons more intuitive.

5.1 Construction of portfolio

A first observation in the comparison of the standard model and developed internal model is that the age composition of the underlying policyholder group matters. In or-der to make this comparison for Dutch insurers it is therefore necessary to construct a hypothetical portfolio that is representative. Assuming that the adult Dutch composi-tion is a good base for the insurance portfolio, a hypothetical portfolio was constructed using the Dutch population pyramid from the Dutch central Bureau for Statistics (CBS) for 20181 (Centraal Bureau voor Statistiek., 2018)[34].

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An issue that is encountered with a portfolio in which all ages from 18 until 99 are represented for both genders is that a large number of simulations need to be performed in the internal model for coming to the 99.5% level in the capitalization requirements. A possible solution is to work with age groups rather than with all separate ages. It was examined and observed for samples that similar SCRs are obtained using individual ages and age groups if the groups are correctly averaged and weighted. For efficiency reasons therefore, age ranges are averaged into 8 groups, mostly of blocks of 10 years apart from the first and last one, as can be seen in table 5.1. In this way the number of simulation runs is drastically reduced to 16.

age group 18-30 30-40 40-50 50-60 60-70 70-80 80-90 90-99

weight males 0.195 0.153 0.175 0.185 0.154 0.098 0.035 0.005

weight females 0.183 0.147 0.169 0.177 0.151 0.104 0.056 0.013

Table 5.1: Portfolio weights per age-gender group

5.2 Standard model vs the internal model

In figures 5.1 and 5.2 the results of the SCR calculations can be seen for the inter-nal and standard models for the hypothetical portfolio that was constructed in section 5.1. Figure 5.1 displays the present value of yearly amounts (TP+SCR), the use of age groups leads to the chainsaw structures. The differences between both models can not be clearly seen from this figure. Figure 5.2 provides the yearly SCRs as percentage of the yearly TPs, which provides a better understanding of the differences of both models for capital requirements.

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Both the standard model and the internal model give quite similar outcomes. This can also be seen from figure 5.2, especially in the middle range. The standard model however leads to slightly higher SCRs, the ratio between the SCR provided by the stan-dard model and internal model being 1.073. The total SCR for the stanstan-dard model is e10381, while the total SCR for the internal model is e9678. To give an idea of the SCR to TP ratio, the total SCR for the standard model is 0.069 times the total technical provisions.

From performing the calculations on age groups separately it was observed that for the younger age groups the internal model gives a higher total SCR compared to the stan-dard model. Looking at the mortality rate distributions, this difference is caused by the distributions for the younger policyholders being wider and having fatter tails. In this way the derived shocks for the 99.5% capitalization requirements are more severe than implied by the standard model. As this effect reverses however for the older participants, it results in a total SCR that is smaller for the internal model.

From this analysis it can be concluded that depending on the composition of the policy-holder group, mainly for a relatively old group, an internal model can lead to significantly lower capital requirements.

Figure 5.2: Yearly SCRs as percentage of the TPs for defined portfolio

5.3 Valuation longevity swap and cap

In this section the longevity swap and caps that were discussed in chapter 4 are valued. It was explained in the introduction of this chapter that the analysis is done on an one-person annuity. While it seems intuitive here to choose the average age of the adult population for making this analysis, which is 50 years, a younger person of 35 years old will be used. The reason is that while this choice does not significantly affect the

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scenario analysis, it leads to more interesting results for the longevity cap as a result of the higher volatility in the obtained simulations due to the wider underlying mortality distribution.

Looking first at the swap, in chapter 4 it was noted that in case no risk premium is demanded by the counterparty, the swap should have an initial value of 0. As in practice a risk premium will be demanded by the counterparty, most commonly incorporated in the swapped cashflows, the value of the swap at time 0 will be determined by the present value of all these risk premium payments. Assuming that the risk premium will result from a percentage increase in the mortality rates, the impact of the risk premiums can be calculated similarly to the SCR. In figure 5.3 the effect of a longevity shock on the swap value is shown in blue by displaying the ratio of the present value of the swap to the present value of the fixed legs of the swap. The linear red line is added in order to show that the increase in this ratio is not linear. The inclusion of a risk premium and other costs will cause the blue line to shift to the right, i.e. the value of the swap per shock will be lowered.

Figure 5.3: Ratio of total swap value to total fixed leg value as function of longevity shock

Looking now at the longevity cap valuation, figure 5.4 provides the value of the longevity cap for different caps using the valuation from section 4.4. Note that as this valuation uses simulations, there will be a small volatility in calculated values with new simulations. The longevity cap price for a full protection is seen to be e1792, which is 17.8% of the SCR given by the internal model assuming no risk premium. This cap is not cheap but it provides protection against all longevity shocks for the duration of the contract. The value of the derivative quickly decays with an increase in this cap, which should be expected as it is very unlikely to see big shocks. The corresponding insurance quickly becomes cheaper as a result. The longevity cap therefore has the nice feature that it is a relatively cheap option as a protection for tails of large shocks.

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Figure 5.4: Longevity cap as a function of shock in mortality

The risk premium in the cap and swap is unknown and not incorporated in the above valuations. The longevity cap can be used however, as was explained in chapter 4, to put a bound on the amount an investor is willing to pay for the value of the imperfections in the longevity swap. The value of the longevity cap was seen to bee1792 for a cap of 0, which implies a maximum margin of 404 bps on the corresponding swap if the risk margin on the longevity cap is assumed to be zero.

Looking only at the direct costs of entering into swaps and caps, it can be concluded from the above valuation that longevity caps are quite expensive for a small cap, while swaps require little to no upfront payment. In case the insurer only wants insurance for very large shocks above the cap, the longevity cap can be a cheap option however. A comparison should be made between the present value of the future risk premiums of the swap and the direct cost of entering the cap. In the next section a scenario analysis will be discussed in order to provide a better understanding of what choice is better in which situation.

5.4 Scenario analysis

In this section a few different scenarios will be examined in order to make the advantages and disadvantages of capital reservation and hedging for longevity risk more specific. Longevity caps and swaps are examined separately as they have different properties that lead to different outcomes within the scenarios.

• Scenario 1: Base case will persist. This scenario assumes that there will be no deviation from the cmrs.

• Scenario 2: Modest improvement in longevity. All mortality rates are decreased by 2%.

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• Scenario 3: Modest worsening in longevity. All mortality rates are increased by 2%.

• Scenario 4: Realization of solvency longevity shock. This is the scenario that the 1 in 200 year longevity shock of 20% realizes.

• Scenario 5: Realization of solvency mortality shock. This is the scenario that the 1 in 200 year mortality shock of 15% realizes.

Table 5.2 displays by what type of costs the insurer is affected in each scenario look-ing at capital reservation and longevity hedglook-ing. Here swaps and caps are considered separately. The same cost components are examined that were introduced in chapter 4, only costs that are relevant and significant are listed. It is good to note that while this overview is comprehensive, it is still a simplified representation of costs in practice.

Costs capital reservation Costs longevity swap Costs longevity cap sc1 Costscapopp+ Costsliq Costslscomp+ Costslsopp Costslccomp+ Costslcopp

sc2 Costs

cap

opp+ Costsliq+ Costsreal,sc2

Costsls_comp+ Costsls_opp+ Costsls_CCR,sc2

Costslc_comp+ Costslc_opp+ Costslc_CCR,sc2

sc3 Costs

cap

opp+ Costsliq+ Costsreal,sc3

Costsls_comp+ Costsls_opp+ Costsls

CCR,sc3

Costslc

comp+ CostslcCCR,sc3+ Costslc_opp

sc4 Costscapopp+ Costsliq

Costsls

comp+ Costslsopp+

Costsls_down,sc4 Costs

lc

comp+ Costslcopp

sc5 Costscapopp+ Costsliq

Costsls_comp+ Costsls_opp+

Costsls_down,sc5 Costs

lc

comp+ Costslcopp

Table 5.2: Overview of costs for capitalization, swap and cap

Some costs are unknown, while other costs are strongly time dependent. While it is therefore difficult to estimate costs accurately, it is easier to determine upper limits to costs. Additionally the aim is not to quantify costs as accurately as possible, but to get insight in how the insurer is impacted in the different cases and in this way rank the different options for total costs. It is therefore more appropriate to look at cost ranges in making the comparisons rather than single values.

The person of 35-year is expected to live another 50 years, which is important in calcu-lating the opportunity costs. The opportunity costs can be estimated assuming that the capital can be otherwise invested in risk-free Dutch government bonds with a duration of 50 years that yield the risk-free return obtained from the DNB. In this way it is obtained that:

Costscap_opp= SCR d50

− SCR

∗ d₅₀= 0.62 × SCR (5.1)

The opportunity costs for the swap and cap are unknown as the risk premium is variable and not known upfront. However using the constraint on the risk premium for the swap that was derived, a rough upper limit for the risk premiums and opportunity costs can be set. Reasoning in this way, the maximum Costsls_compat time 0 equals the longevity cap value without a risk premium, which is 0.176 × SCR as derived in section 5.3. Assuming that the same maximum risk premium would be applicable for the longevity cap, this will mean the maximum Costslc_comp will be twice its value without a risk premium. Note that both the compensation costs are written as a factor times the SCR, similarly the opportunity costs for the swap and cap can be obtained applying the same factors on

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Costscapopp.

An upper bound for the CCR costs is easy to set, as in the worst case the total exposure is lost. The liquidation costs Costsliqon the other hand will be assumed to be proportional to the SCR with factor α. This factor α is very timing dependent and difficult to estimate.

In table 5.3 the above reasoning is incorporated, only the liquidation cost factor α remains as an unknown variable. In most cases estimation intervals are given, rather than values, in order to account for the uncertainty in the estimates. Here the maximum risk premiums and correspondingly maximum opportunity costs that were estimated are used to put an upper bound on the intervals. It is good to note that the upper bounds that were put are pretty rough and therefore the considered costs will in practice not lie close to the upper bound.

All elements are expressed as factors of the SCR, i.e. the factors should be multiplied with the SCR in order to obtain the true estimates. Additionally the numbers were rounded to two decimals.

capital reservation Longevity swap Longevity cap (c = 0)

sc1 0.62 + α [0, 0.29) [0.18, 0.57)

sc2 0.71 + α [0, 0.38) [0.18, 0.66)

sc3 1.62 + α [0, 1.29) [0.18, 1.57)

sc4 0.62 + α [0.09, 0.38) [0.18, 0.57)

sc5 0.62 + α [0.62, 0.91) [0.18, 0.57)

Table 5.3: Estimates and estimation intervals of costs from table 5.2, divided by the SCR

Although table 5.3 contains pretty wide ranges for costs, it provides remarkably well insights for making comparisons. Based on table 5.3 it can be concluded that in most cases capital reservation is the most expensive option, as only in scenario 5 it might be cheaper than the longevity swap. Additionally it can be seen that the longevity swap is cheaper than the longevity cap with c = 0 in almost all cases, except for scenario 5. This seems intuitive as the longevity cap does not have a downside risk as opposed to the swap. As a result, the investor either needs to pay a higher compensation or go for a higher cap. Investors should therefore examine for what choice of c the longevity cap is interesting for them to invest in.

In the above analysis it is assumed that the hedge provided by the derivatives is perfect, while in practice this will not be the case. Pursuing a hedging strategy also requires com-mitment. Maintaining capital provides the freedom to enter into derivatives positions at a later time. On the other hand it can be costly at later times to sell the purchased longevity derivatives, due to their illiquid nature, in order to pursue other strategies. In order to get insight in these type of risks, insurers should additionally perform a risk assessment to examine risks that might arise for them as a consequence of pursuing a (partial) hedging strategy.

From this analysis it can be concluded that it is beneficial for insurers to use the con-sidered derivatives in order to hedge the longevity risk in the portfolio at least partly. The degree to which this should be done depends on many factors like risk preferences of the insurer, timing and outcomes of the risk assessment.

Assessment of capital reservation and hedging for longevity risk