framework
Eva Roelse
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: Eva Roelse
Student nr.: 10180699
Email: evaroelse@live.nl
Date: July 15, 2016
Supervisor UvA: Mr. F. van Berkum MSc
Second reader: Mr. prof. dr. ir. M.H. Vellekoop
This document is written by Eva Roelse who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is 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.
Abstract
In this thesis a literature study is performed to gain insights in relevant theory about Solvency II, stochastic mortality models, several longevity risk mitigating instruments and how to model and quantify basis risk. The fictive life insurer under consideration pays her policyholders pension benefits, these are considered the only liabilities she has. As the benefits are lifelong annuities, the insurer is involved with longevity risk. Therefore, the insurer would like to know whether it is beneficial to purchase a longevity risk mitigating instrument. The Solvency II position of the insurer is clarified first. Then, the financial impact on the Solvency II framework of the indemnity and index-based swap are compared. Finally, the situation is evaluated at time 10.
Keywords Longevity risk, longevity risk mitigating instrument, longevity swap, indemnity swap, index-based swap, survivor bond, basis risk
Preface vi
1 Introduction 1
2 Literature study 3
2.1 Solvency II for a life insurer . . . 3
2.1.1 Solvency Capital Requirement . . . 4
2.1.2 Risk margin . . . 4
2.1.3 Life underwriting risk . . . 6
2.2 Stochastic mortality models . . . 8
2.3 Hedging longevity risk . . . 11
2.3.1 Q-forward . . . 12
2.3.2 Survivor bond . . . 13
2.3.3 Indemnity swap. . . 15
2.3.4 Index-based swap. . . 16
2.3.5 Comparison of longevity risk mitigating instruments . . . 17
2.4 Modelling longevity basis risk . . . 19
3 Methodology 22 3.1 Description of the data. . . 22
3.2 Mortality model . . . 23
3.3 Solvency II position of the insurer . . . 25
3.4 Longevity swaps . . . 26
3.5 Basis risk . . . 28
3.6 Evaluation of the insurer’s Solvency II position . . . 29
4 Results and analysis 31 4.1 Mortality and survival . . . 31
4.2 Liabilities of the insurer . . . 34
4.3 Indemnity and index-based swap . . . 35
4.3.1 Balance sheet . . . 36
4.3.2 Impact on SCR and risk margin . . . 37
4.3.3 Basis risk . . . 37
4.3.4 Comparison . . . 38
4.4 Evaluation at time 10 . . . 39
4.4.1 Bear the risk . . . 39
4.4.2 Hedge with an indemnity swap . . . 40
4.4.3 Hedge with an index-based swap . . . 41
4.4.4 Comparison . . . 43
5 Conclusion and further research 44 5.1 Conclusion . . . 44
5.2 Discussion . . . 45
5.3 Further research . . . 45 iv
Bibliography 47
Appendices 49
A R code 50
A.1 Main script . . . 50
This master’s thesis is written by Eva Roelse in order to obtain the degree in Actuarial Science and Mathematical Finance. With lots of enthusiasm I have worked on this thesis within the company EY. EY has offered me the opportunity to develop myself in the area of life insurance, in particular in hedging longevity risk.
I would like to thank my supervisors from the University of Amsterdam and EY, Mr. F. van Berkum, Mr. J. de Mik and Mr. M.P. Matthijssen, as well as my colleagues with whom I worked at EY Actuarial Services, for supporting me and sharing thoughts. Finally, I would like to thank EY for providing an internship to write my master’s thesis in a dynamic, companionable and professional environment.
Introduction
Over the last few decades, life expectancy has been increasing in every region in the world. This is illustrated in figure 1.1. According to De Boer et al. (2014) the life expectancy in the Netherlands increased with two years every decade over the last fifty years. Therefore, every generation lives approximately five years longer than the previous one. This has an enormous impact on the health sector, demographics, and life insurers and pension funds.
Figure 1.1: Over the last decades life expectancy has been increasing all over the world, source: UN World Population Prospects (2008)
For life insurers and pension funds it is important to have a clear view on the survival forecasts. When life expectancy is increasing, they have to pay out the contractual obligations towards the policyholders longer than expected. The problem that occurs, is that these institutions do not have the resources to cover these additional benefits. Therefore, the increasing life expectancy is a major issue for life insurers and pension funds.
The particular risk that is involved with the increasing life expectancy is called longevity risk. This risk is one of the major risks life insurers have to deal with (EIOPA, 2011). To deal with the risk, the insurer could hedge it. She could do this by means of purchasing a longevity risk mitigating instrument. If the insurer decides to bear the risk, she reserves capital in the form of the Solvency Capital Requirement (SCR) and the risk margin.
The most popular longevity risk mitigating instrument is a longevity swap. The world’s first capital market longevity swap was completed in July 2008 between JP Morgan and Canada Life. According to Blake (2011) Canada Life hedged £500 million of JP Morgan’s annuity book. The swap transaction consisted of 125,000 lives in a 40-year swap. The longevity swap was customized to her longevity exposure, but it was based on LifeMetrics Index improvements. The longevity risk was fully transferred to
Canada Life due to this swap. JP Morgan has been particularly active in trying to establish a benchmark for the longevity market (Barrieu and Veraart, 2016).
Most of the longevity swap transactions were performed in the United Kingdom, the Dutch longevity market is still immature. Only five longevity swap transactions have been completed in the Netherlands yet. These were purchased by AEGON and Delta Lloyd. The first longevity swap transaction of AEGON was in 2012, this was one of the first outside the United Kingdom and one of the largest heard of in the market at that time. It had a size of e12 billion. According to Artemis (2015), longevity swaps generally tend to cover retired policyholders. However, the latest longevity swap of AEGON includes a significant number of younger policyholders. Therefore, AEGON states that the longevity risk transfer market is still in development.
In this thesis a life insurer is considered to be the company that has insured the pension rights of her policyholders. Because the portfolio of the insurer is subject to longevity risk, the insurer is considering to purchase a longevity risk mitigating instru-ment. The two main types of the longevity swap are compared comprehensively. These are the so-called indemnity and index-based swap. The indemnity swap is a reinsurance type of contract, the index-based swap is a longevity derivative. This thesis focuses upon the financial impact of either the indemnity or the index-based swap on the Solvency II position of the insurer.
To investigate this, a literature study is performed in chapter 2. The literature study covers subjects such as Solvency II risk modules that are relevant in this thesis, stochastic mortality models to forecast mortality, and several longevity risk mitigating instruments as well as the risks that are potentially involved in these. The literature study shows that a longevity swap is the most evident product to hedge longevity risk. Therefore, the remainder of this thesis focuses on the indemnity and index-based swap. In chapter 3 the methodology to investigate the impact of each swap contract is described. The results of the research are analyzed in chapter 4. The mortality model is discussed first, since this will be necessary in the remainder of the thesis. Then, the current situation of the insurer is taken into consideration in which she does not hedge the longevity risk. Thereafter, the impact of each swap contract is investigated at time 0 and evaluated at time 10. The impact of each swap on the Solvency II position of the insurer is determined by the value of each swap, as well as the difference in SCR and risk margin. It is out of the scope of this thesis to price the risk premium of the longevity swaps. In chapter5the conclusion of the thesis is described, as well as several ideas for what we could have done differently and multiple suggestions for further research.
Literature study
Since several decades the life expectancy is increasing all over the world. Therefore, life insurers have to pay out their contractual obligations towards policyholders during a longer period than expected. This may cause major problems, since they have not reserved enough capital for this. To deal with this risk, known as longevity risk, the insurer could decide to purchase a longevity risk mitigating instrument in order to hedge the risk.
If she decides to not hedge the risk, she has to reserve capital for it in the form of the Solvency Capital Requirement (SCR) and the risk margin. The SCR and risk margin are part of the Solvency II framework. The parts of this framework that are relevant in this thesis are discussed in paragraph2.1. The insurance-specific part for the life insurer is called the life underwriting risk module. This part contains three sub-modules that relate to survival rates, i.e. longevity, mortality and catastrophe risk. Therefore, these sub-modules are explained.
A stochastic mortality model is used to forecast the survival rates of the policyhold-ers. These are necessary to calculate the best estimate of the liabilities of the insurer, the SCR and to construct the longevity risk mitigating instruments. In paragraph 2.2
three stochastic mortality models are discussed: the Lee-Carter, Cairns-Blake-Dowd and AG2014 model.
If the insurer decides to hedge her longevity risk, she transfers the risk to a counter-party for a certain price. In paragraph 2.3 three longevity risk mitigating instruments are discussed: the survivor bond, the indemnity swap and the index-based swap. At the end of the paragraph the risks that could be related to these instruments are compared. One of the risks that occur when the survivor bond or the index-based swap are used is basis risk. This risk is quantified in this thesis, therefore several models to establish this are discussed in paragraph 2.4.
2.1
Solvency II for a life insurer
Since the start of 2016, all insurers in the European Union have to comply with the Solvency II framework. This framework contains three pillars. The first pillar consists of financial requirements, the second pillar contains requirements for governance and supervision and the third focuses on reporting and disclosure. In this thesis only the first pillar is focused on. For this pillar the insurer needs to calculate its SCR and risk margin, see EIOPA (2010).
The SCR is a measure to ensure that insurers are able to meet their financial obliga-tions over the horizon of one year with a probability of 99.5%. This is equivalent to the 99.5% Value-at-Risk (VaR) of the available capital in a one-year period. Hence, the SCR is determined as the economic capital which should be held in order to prevent falling into financial ruin no more than once every two hundred years. When an insurer does not have sufficient capital available to cover the SCR, it will likely result in regulatory
intervention and require remedial action.
In this paragraph the general part of the SCR model is discussed, as well as the specific part for the life insurer. Only the relevant risk modules which are needed in this thesis are covered in this paragraph. In the general part the calculation of the SCR and the risk margin are focused on. The specific part for a life insurer is called the life underwriting risk module of which longevity, mortality and catastrophe risk are discussed.
2.1.1 Solvency Capital Requirement
The SCR can be calculated by using the standard formula or an internal model. In figure2.1the structure of the standard formula is illustrated. The SCR covers different types of risks that an insurer faces, such as underwriting risk, market risk, credit risk and operational risk. In the upper layer of figure2.1the SCR is calculated by aggregating the adjustment for loss absorbing capacity of technical provisions and deferred taxes (Adj), the capital requirement for operational risk (SCRop) and the Basic Solvency Capital
Requirement (BSCR). In the second layer the BSCR is calculated by aggregating the required capital of six underlying risk modules: market risk, health underwriting risk, counterparty default risk, life underwriting risk, non-life underwriting risk and intangible asset risk. The BSCR is calculated as:
BSCR = s
X
i,j
Corri,j× SCRi× SCRj+ SCRintangibles (2.1)
where Corri,j are the entries of correlation matrix Corr, and SCRi and SCRj are the
capital requirements for risks according to the rows and columns of Corr. SCRintangibles
is the capital requirement for intangible asset risk.
Figure 2.1: Calculation of the SCR according to the standard formula, source: EIOPA (2010)
2.1.2 Risk margin
The Solvency II guidelines require that all balance sheet items should be valued on market value. Most assets on the balance sheet of an insurer are tradeable assets which makes valuation on market value manageable. To value the items on the liability side of the balance sheet is more difficult, since these are not tradeable assets. To obtain a market-consistent value, the best estimate of the liabilities LBE is calculated as the sum of the expected, discounted cash flows. Together with the risk margin the best
estimate of the liabilities form the technical provisions of the insurer. The technical provisions are the reserves of the insurer, considered to be sufficient to fulfill all her future contractual obligations.
Pang (2014) explains that the uncertainty of the liabilities is not accounted for in the SCR, although this is the primary risk for the hypothetical buyer of the liabilities. To fulfill the requirement of a market-consistent balance sheet, the risk margin is added on top of the best estimate of the liabilities. This is illustrated in figure 2.2 below. According to EIOPA (2010) the risk margin is an increment to ensure that the value of the technical provisions is equivalent to the amount that the buyer requires, in order to take over the insurance obligations. Purcell and Mee (2012) explain that this means that if the buyer would use all its free surplus and capital, it would still have sufficient assets to safely transfer its obligations to a third party.
Figure 2.2: The risk margin is an increment on top of the best estimate, source: Zaremba (2012)
The risk margin only applies to non-hedgeable risks. According to Pang (2014) the buyer does not have to hold capital in the risk margin to fulfill the obligations of the liabilities if she could transfer the risk to the market. A risk that is generally seen as non-hedgeable is longevity risk. Hence, the insurer should hold capital for it in the risk margin. When a longevity risk mitigating instrument is used to hedge longevity risk, the SCR and risk margin decrease. The size of decrease depends on the characteristics of the instrument.
Zaremba (2012) discusses that the risk margin can be calculated by determining the cost of providing an amount of own funds equal to the SCR, necessary to support runoff of the insurance obligations. This can be done by using the cost-of-capital (CoC) which equals 6%. The risk margin is calculated as:
RM = CoC n X t=0 SCRRU,t (1 + rt+1)t+1 (2.2) where SCRRU,t is the SCR of the reference undertaking at time t and rt is the
risk-free rate at time t. The reference undertakings’ SCR is based on the underwriting risk, counterparty default risk and operational risk, hence it is not equal to the insurer’s full SCR. The reason for this is that the buyer only takes over the liabilities of the insurer, not the assets.
It is a major amount of work to calculate the SCR for all future years t = 1, . . . , n, so EIOPA (2010) proposed four simplified ways to project the SCR. The four simplified ways as well as the comprehensive way to calculate all future SCRs are listed below. The list below should be used to make a decision regarding the method for projecting the future SCRs.
1. Full calculation of the SCRs without simplifications.
2. Approximate the individual (sub-)risks within some or all (sub-)modules used for SCR calculation.
3. Approximate the whole SCR for all the future years (proportional approach). 4. Approximate all future SCRs at once (duration approach).
5. Approximate the risk margin as a percentage of the best estimate.
According to Zaremba (2012) method 3 is the most commonly used simplification to calculate the future SCRs. In this thesis this method is used to calculate the risk margin, since all required input is available. The future SCRs are calculated as is shown in equation2.3. The method projects the SCR in proportion to the best estimate of the liabilities.
SCRRU,t = SCRRU,0×
LBEt
LBE0 for t = 1, 2, . . . , n (2.3) where SCRRU,t is the SCR for the reference undertaking’s portfolio of insurance
obli-gations at time t and LBEt the best estimate of the liabilities at time t.
2.1.3 Life underwriting risk
According to EIOPA (2011) life underwriting risk is the second most material module after market risk for a life insurer. This risk module includes mortality, longevity, dis-ability, lapse, expenses, revision and catastrophe risk. In this paragraph only mortality, longevity and catastrophe risk are discussed, because these are the only risks that de-pend on mortality and survival of the policyholders. These risks are modelled in this thesis to determine the capital requirement for the life underwriting risk module.
The SCR for the life underwriting risk module is calculated as the aggregated sum of the capital requirements of the sub-modules mortality, longevity, disability, lapse, expenses, revision and catastrophe risk. The SCR for life underwriting risk is determined as follows:
SCRlif e=
s X
r,c
CorrLif er,c× Lif er× Lif ec (2.4)
where CorrLif er,c are the entries of the correlation matrix CorrLif e, Lif er and Lif ec
are capital requirements for individual life sub-risks according to the rows and columns of correlation matrix CorrLif e, see EIOPA (2010). The correlation matrix CorrLif e is defined by EIOPA and can be found in figure2.3. The matrix shows that longevity risk is negatively correlated with mortality risk. This is due to the fact that mortality and longevity risk are unlikely to occur at the same moment, since mortality risk is related to increased mortality rates and longevity risk to decreased mortality rates.
Figure 2.3: Correlation matrix for life sub-risks, source: EIOPA (2010)
According to Van Strien (2012) there are three kinds of longevity (mortality) risk: • Trend risk: the risk that the entire population will survive for a longer (shorter)
period than expected. Van Strien (2012) argues that about 80-85% of longevity (mortality) risk consists of trend risk, the rest is mainly level risk.
• Level risk: the risk that a specific group will survive for a longer (shorter) period than expected. Hence, the ratio ’portfolio survival/population survival’ is bigger (smaller) than expected. Not every group of people has the exact same average survival as the national population, so there is always the risk that portfolio mor-tality differs from population mormor-tality. Therefore, level risk is the mismatch be-tween the realized survival and the survival used for the longevity risk mitigating instrument.
• Volatility risk: the risk of random fluctuations in survival. For a large insurer or pension fund this risk is negligible, because the effect of one outlier tends to go to zero if the sample size increases. If longevity (mortality) risk is modelled by simulations, this risk can be modelled by incorporating a binomial distribution. The distribution determines whether a policyholder survives or not.
Mortality risk
An insurer who sells term assurance or endowment policies typically faces mortality risk. The insurer makes a single payment or a series of payments in the event of the policyholder’s death during the policy term. In the case that the policyholders in a portfolio die earlier than expected, the technical provisions held may be insufficient. Hence, the policyholders’ benefits upon death exceed the amount which is reserved. The risk that this will occur is called mortality risk. When mortality rates increase, the reserves need to increase which has a negative impact on the net asset value (EIOPA, 2010).
The capital requirement for the mortality risk sub-module is calculated as:
Lif emort= max(0, ∆N AV |mortality shock). (2.5)
In formula2.5∆N AV is the change in net asset value, this is the change in assets minus liabilities. According to EIOPA (2010), in the standard model the mortality shock is a 15% increase in mortality rates for each age and each policy where the payments of benefits are contingent on mortality risk.
Longevity risk
Longevity risk is the major risk sub-module in the life underwriting risk module. Ac-cording to EIOPA (2011), 44% of the life capital requirement is held for longevity risk. This risk is especially relevant for insurers who sell annuities or pure endowment policies. These insurers have to pay a recurring series of benefits until the event of the policy-holder’s death or a single payment at the end of the policy duration. If the policyholder lives longer than expected, the insurer needs to pay more benefits than the amount initially reserved in the technical provisions. The risk that this will happen is called longevity risk. When mortality rates decrease, the reserve for longevity risk increases which has a negative impact on the net asset value.
The capital requirement for longevity risk is determined as follows:
Lif elong = max(0, ∆N AV |longevity shock) (2.6)
where ∆N AV is the change in net asset value, i.e. the change in assets minus liabilities. In the standard model the longevity shock is a (permanent) decrease of 20% in mortality rates for each age and each policy where the payment of benefits is contingent on longevity risk, see EIOPA (2010).
Catastrophe risk
Catastrophe risks is one of the smallest risk sub-modules in the life underwriting risk module for a life insurer. Around 8% of the life capital requirement is held for catas-trophe risk (EIOPA, 2011). According to EIOPA (2010) the risk stems from extreme or irregular events, catastrophes, that are not sufficiently captured in other life under-writing risk sub-modules. Such events could be pandemic events, terrorist attacks or a nuclear explosion. If such an event occurs, multiple policyholders die. Therefore, the capital requirement is calculated by increasing the mortality rates for all ages and all future years by the catastrophe shock.
Lif eCAT = max(0, ∆N AV |catastrophe shock) (2.7)
where ∆N AV is the change in net asset value, the catastrophe shock is a (permanent) increase of 1.5 per mille in mortality rates.
2.2
Stochastic mortality models
Life insurers and pension funds use models to forecast mortality of the policyholders. These models are used to calculate the technical provisions and premiums. There are two types of mortality models, namely deterministic and stochastic mortality models. Deterministic mortality models forecast the best estimate of the future mortality while stochastic mortality models also give insight in deviations of the best estimate. For insurers and pension funds this is important information in order to hold a sufficient buffer in relation with the uncertainty of mortality.
De Boer et al. (2014) discuss multiple advantages of why stochastic mortality models are preferred over deterministic mortality models. The first reason is that stochastic mortality models are investigated a lot in scientific literature over the last couple of years. These models provide the possibility to a goodness-of-fit analysis which deterministic models do not provide. The second advantage is the possibility to give insight in the uncertainty of future mortality. When mortality is illustrated in a graph, the uncertainty can be seen as a cloud of possible outcomes. In this paragraph three stochastic mortality models are described, in particular the Lee-Carter, the Cairns-Blake Dowd and the AG2014 model.
A mortality model models the force of mortality µx,t or the initial mortality rate
qx,t. The force of mortality µx,t is defined as the probability of failure at time t given
survival up to time t − 1 for an individual aged x (Promislow, 2011, p. 192). The initial mortality rate qx,t represents the probability that an individual aged x dies between t
and t+1, i.e. the one-year mortality rate. The mortality rates qx,tand µx,t are connected
by the following equation:
qx,t= 1 − exp(−µx,t). (2.8)
To calculate the best estimate of the liabilities, as well as the value of the longevity risk mitigating instruments, the cumulative survival rate is needed. The one-year sur-vival rate of an individual age x at time t is determined as px,t = 1 − qx,t. The k-year
cohort survival rates can be calculated using the one-year rates:
kpx,t= k−1
Y
s=0
px+s,t+s (2.9)
A mortality model is constructed using exposures and deaths. Dx,t is the number of
deaths for people aged x during calendar year t, and Ex,t is the average population of
people aged x during calendar year t. Vellekoop (2016) explains that the parameters of the mortality model can be estimated using the method of maximum likelihood. The likelihood of surviving less than one year and dying then, equals exp(−µx,tti)·µx,twhere
ti equals the years the individual would survive. The likelihood to survive more than
one year equals exp(−µx,tti). The random variable Ti indicates whether the individual
survived that period, Di indicates whether the individual died. The likelihood then
becomes:
L =Y
i
(µx,t)Di· exp(−µx,tTi) = µDx,tx,texp(−µx,tEx,t) (2.10)
Therefore, the likelihood is the same as if Dx,t would have a Poisson distribution with
mean λx,t = µx,tEx,t, but it does not necessarily have to be Poisson distributed. The
loglikelihood of the Poisson distributed mortality rates is as follows: ln L =X
x,t
Dx,tln(µx,t) − µx,tEx,t+ C. (2.11)
where C is a constant that does not depend on µx,t. The maximum likelihood estimate
can be found by maximizing equation2.11with respect to all values µx,t. The following
expression is found: 0 = ∂ ln L ∂µx,t = Dx,t µx,t − Ex,t ˆ µx,t = Dx,t Ex,t (2.12)
The method of maximum likelihood can be used to determine the force of mortality µx,t. If this method is used, it is assumed that the deaths Dx.t are Poisson distributed.
As the mortality rates µx,tare obtained, the one-year mortality rates and the cumulative
survival rates can be calculated using equations 2.8and 2.9. The Lee-Carter model
One of the most well-known stochastic mortality models is the model of Lee and Carter (1992). It is a simple one-factor model that is described by the following equation:
ln µx,t = αx+ βxκt. (2.13)
In equation 2.13, αx is an age-specific parameter that reflects the general shape of
mortality, βx is an age-specific parameter that characterizes the sensitivity of κt. The
parameter κt is a time-specific parameter that signifies the general speed of mortality
rate changes.
The parameters αx, βx and κtcan be estimated using the method of maximum
like-lihood that is described above. It is assumed that the parameters are all deterministic, because it is hard to do a full optimization. Vellekoop (2016) explains two other methods to obtain estimates of the parameters, namely homoskedastic fitting and Singular Value Decomposition (SVD). After the parameters αx and βx are estimated, the time series
κthas to be modelled. The most popular choices for this are the random walk with drift
(RWD) and the first order autoregressive model (AR(1)). To obtain a unique solution, one can choose between three possibilities for the vector α:
• the starting values ln µx,tstart, then κstart= 0
• the final values ln µx,tend, then κend= 0
• the average values (tstart− tend)−1Ptend
t=tstartln µx,t, then
P
tκt= 0
Plat (2009) summarizes a few disadvantages of the model, one of those disadvantages is that it is a one-factor model. This leads to mortality improvements at all ages being perfectly correlated. This is called a trivial correlation structure. Another disadvantage described is that for countries where a cohort effect is observed in the past, the model gives a poor fit to historical data. Besides that, the uncertainty in future death rates for
high ages can be too low. This is due to the fact that the improvement parameter βx
is often lower at high ages. Despite these disadvantages, it is a very popular stochastic mortality model, because it is easy to fit and the parameters have a clear interpretation. According to Cairns et al. (2008) it is a robust model which has a good fit over wide age ranges.
The Cairns-Blake-Dowd model
The model of Cairns et al. (2006) is a stochastic mortality model that is designed for higher ages. Therefore, it could be used to model longevity risk. The Cairns-Blake-Dowd model has multiple factors which result in a non-trivial correlation structure. The structure of the model is quite simple. It is described by the following equation:
logit qx,t= κ(1)t + (x − ¯x)κ (2)
t (2.14)
where logit qx,t = log(1−qqx,tx,t), κ(1)t is a time-specific parameter that represents the level
of mortality and κ(2)t is a time-specific parameter that represents the slope of mortality with age. The time-varying parameters of the Cairns-Blake-Dowd model can be modelled by a bivariate random walk with drift.
Cairns et al. (2008) describe that this model is designed for higher ages, not for lower ones. Therefore, it is not an accurate model to use to forecast the mortality rates for an entire population. However, it could be used to model longevity risk in annuities for life insurers or pension funds. According to Cairns et al. (2008) the model is based upon the observation that log mortality rates are approximately linear after the age of 40. A disadvantage of the model is that the overall fit is not as good as the Lee-Carter’s. This is because the Lee-Carter model better gathers small non-linearities at lower ages. An advantage of the Cairns-Blake-Dowd model is that it is a robust model with simple age effects.
The AG2014 model
A mortality model that is widely used in the Dutch financial sector is constructed by the Royal Dutch Actuarial Association (AG). De Boer et al. (2014) describe that it is the task of the AG to provide statistical information and insights regarding the mortality tables. The AG2014 model is a stochastic mortality model which makes it possible to assess the uncertainty of the forecast.
Since 1970 mortality of multiple European countries and the Dutch converge. The life expectancies of the populations increase at almost the same speed. Therefore, the model is based on mortality data from the Netherlands and European countries with above-average GDP, i.e. West-European countries. The data can be found in the Human Mortality Database (HMD) and at the website of the Statistics Netherlands (CBS). De Boer et al. (2014) explain that the European mortality ensures that the model is more stable and less sensitive for deviations in the Dutch mortality in a certain year.
The AG2014 model is a variant of the model of Li and Lee (2005). The Li-Lee model extends the Lee-Carter model by applying it twice: first to all countries and then to the residuals. The parameters of the original Li-Lee model are estimated by SVD. In the AG2014 model the Lee-Carter model is applied first to model European mortality and then to model Dutch deviations. The observed deaths are assumed to be Poisson distributed, hence the method of maximum likelihood is used to determine these.
The model is described by the following equations: ln µx,t= ln µEUx,t + ln µNLx,t
ln µEUx,t = Ax+ BxKt
ln µNLx,t = αx+ βxκt.
where Ax, Bx, αx and βx are age-specific parameters and Kt and κt are time-specific
parameters. The time series parameter Kt measures the global trend, κt measures the
country-specific trend. To obtain a unique solution of the parameters, the parameters are standardized. The sums of the values Kt and κt are both set equal to 0, the sums
of the parameters Bx and βx are equal to 1.
The time series Kt is a random walk with drift. At each time t the series takes
a step with mean size θ. The time series κt is an AR(1) model. This means that κt
depends linearly on κt−1 by the coefficient α. The coefficient α lies within the unit
circle, otherwise the time series will explode. A time series that explodes is useless, because the values κtfor t = 2, ..., T move exponentially far from κ1. Therefore, |α| < 1
holds. This means that the values κtare mean reverting, hence these tend to go to their
mean which equals zero. Therefore, the log mortality rate ln µx,t goes to αx if years go
by. This is the reason why the AG has chosen to use an AR(1) model to model κt for
the Dutch deviations of mortality. The time series Ktand κt are defined as in equation
2.16 below: Kt κt =Kt−1+ θ aκt−1 +t δt (2.16) where (t, δt)0are i.i.d. normally distributed variables with expectation (0, 0) and
variance-covariance matrix C. The best estimate scenario of mortality is defined as the most likely development and is therefore found by setting the white noise processes t and δtequal
to zero.
The used data range covers the ages 0 up to 90 years old. De Boer et al. (2014) explain that the amount of observations declines for higher ages and can therefore lead to uncertainty in the estimates of the modelled mortality. To tackle this problem, the so-called closing of mortality is performed by using the extrapolation technique of Kannisto (1994).
Kannisto
De Boer et al. (2014) discuss the method of Kannisto to model mortality for the ages x = 91, . . . , 120. The mortality rates of these ages are based on a logistic regression for which mortality rates of the ages y = 80, . . . , 90 are used. The method is described by the following equation:
µx,t= L n X k=1 wk(x)L−1(µy(k),t) ! . (2.17)
The functions L(x) and L−1(x) are logistic and inverse logistic functions:
L(x) = 1
1 + exp(−x) L−1(x) = − ln (1/x) − 1,
(2.18)
and the weights wk(x) are determined as:
wk(x) = 1 n+ (yk− ¯y)(x − ¯y) P j(yj− ¯y)2 (2.19)
2.3
Hedging longevity risk
Longevity risk is generally seen as a non-hedgeable risk. For this reason insurers need to include it in the risk margin. However, in this thesis longevity risk is seen as a hedgeable risk. Different instruments to hedge longevity risk exist. Therefore, the risk margin decreases, due to the characteristics of the instrument. In this paragraph several
longevity risk mitigating instruments are discussed, such as the survivor bond, the indemnity swap and the index-based swap. At the end of the paragraph the risks that are potentially involved in these instruments are compared.
2.3.1 Q-forward
According to Coughlan et al. (2007) the q-forward is the simplest form of a longevity and mortality derivative. It is an important product, because more complex longevity hedges can be constructed from them. The name ’q-forward’ stems from the probability that an individual of age x dies within the next year, i.e. the mortality rate qx.
When two parties have agreed to use a q-forward, the insurer receives an amount proportional to the realized mortality rate of a portfolio from the counterparty. This is the floating leg. In return for that, it pays an amount proportional to the fixed mortality rate. The fixed mortality rate is agreed upon at the inception of the contract. The fixed cash flow is increased by a certain risk premium to compensate the counterparty for the risk he bears. The structure of a q-forward is similar to a zero-coupon swap contract that exchanges fixed and realized mortality. The structure of a q-forward is illustrated in figure 2.4.
Figure 2.4: The structure of a q-forward contract
At maturity the fixed and floating payment are exchanged. The amount the insurer receives is called the settlement or Net Payoff Amount (NPA). It is determined as:
NPAT = Notional · [qrealx,T − (1 + π)q f ixed
x,T ]
= zT · [qx,T − (1 + π)¯qx,T]
= zT · [qx,T − Kx]
(2.20)
where qx,T is the realized one-year mortality rate for an individual of age x at time T,
¯
qx,T is the forward or fixed one-year mortality rate for and individual aged x at time 0.
The fixed leg of the q-forward consists of the fixed mortality rate and the risk premium π. The fixed leg is determined as Kx= (1 + π)¯qx,T.
Usually forward contracts are traded on a market. The forward exchange rates follow from transactions made in this market. However, for q-forwards there is no liquid market where you can buy and sell the contracts. All the q-forward contracts are sold over-the-counter (OTC), which means that they are closed directly between two parties. Therefore, market prices are not available, so multiple methods to determine the price of a q-forward are discussed below.
Pricing of a q-forward
To price a q-forward, multiple approaches are possible. Barrieu and Veraart (2016) discuss the fair premium principle, the standard deviation principle and the principle of zero utility. Loeys et al. (2007) describe a method that uses the Sharpe ratio λt. These
four approaches are discussed below.
• Fair premium or net premium principle: this principle assumes that the value of the q-forward at time 0 can be determined by considering the expectation of the NPA under the physical probability measure P. When the expectation is used
as a price, the Law of Large Numbers can be used. The Law of Large Numbers states that the sample mean 1nPn
i=1Xi of i.i.d. randomly distributed variables
X1, X2, . . . , Xn with E[X] = µ converges to the mean µ as n → ∞. In the case
of q-forward contracts this would mean that there needs to be a large amount of q-forwards to make sure that the Law of Large Numbers is applicable. This is a limitation of the fair premium pricing principle, as there is no liquid market for q-forwards.
• Standard deviation principle: this principle assumes that the value of a q-forward equals the sum of the expectation under the physical probability measure P and a constant λ times the standard deviation of the NPA as risk premium. According to Barrieu and Veraart (2016) this price can be interpreted in terms of a market risk premium as λ can be related to a measure such as the Sharpe ratio of the realized mortality rate.
• Principle of zero utility: this method is based on the principle of zero-utility. The principle assumes that the counterparty is indifferent between buying the asset and not engaging in the transaction.
• According to Loeys et al. (2007) the price of a q-forward can be determined using the Sharpe ratio, the volatility of historical mortality rates, the horizon of the q-forward and the expected cumulative mortality rate. The Sharpe ratio λ is the market price of risk and is often used to indicate how well the return of an asset compensates the risk the investor has to take.
Valuation of a q-forward
The value of a q-forward on time t is equal to the discounted future cash flow. In the valuation the risk premium π is not taken into account. A q-forward has only one moment at which cash flows are exchanged, namely at maturity T .
Vt=
zT · (qx,T − ¯qx,T)
(1 + rt,T)T −t
(2.21) In the above equation zT is the notional amount at time T that is exchanged, qx,T is
the realized one-year mortality for a person aged x at time T , i.e. the probability that an individual aged x dies before maturity of the contract. The fixed mortality rate is denoted by ¯qx,T for a person aged x at time t. Discounting is done using the forward
rate rt,T for period [t, T ].
A mortality model is needed to be able to calculate the value of the q-forward. The realized mortality rate qx,t can be forecasted by the mortality model. The fixed
mortality rate ¯qx,T is agreed upon at the inception of the contract. The forward rate
can be calculated using the current term structure. As the size of the notional amount zx,k is agreed upon before the contract, the value of the q-forward can be calculated
using equation 2.21.
2.3.2 Survivor bond
The survivor bond is a longevity risk mitigating instrument that is not used much in practice due to the risks involved with it. In paragraph2.3.5the risks that are potentially involved with the survivor bond, indemnity swap and index-based swap are discussed. De Crom et al. (2011) explain that if the insurer decides to purchase a survivor bond, it pays a certain capital to the counterparty at the inception of the contract. In return for this it receives floating cash flows until maturity T of the contract in return. Therefore, the survivor bond can be seen as a bond that pays coupons every period. The coupons
are the floating cash flows. The structure of a survivor bond is illustrated below in figure
2.5.
Figure 2.5: Survivor bond
De Crom et al. (2011) explain that an insurer who purchased a survivor bond receives floating cash flows based on an index. The index is determined as the realized survival of a reference population multiplied by an experience factor. A reference population is a population that has the same characteristics as the portfolio. The reference population of a Dutch portfolio is for example the Dutch population. As an index is used, the index survival and the estimated survival could be not exactly equal. The mismatch between the realized and floating cash flows is called basis risk. Hence, the survivor bond does not fully hedge the longevity risk the insurer bears. This is one of the main disadvantages of the survivor bond. Another major disadvantage of the survivor bond is the counterparty default risk it entails. Since a capital is paid upfront of the contract, the counterparty default risk is high.
Valuation of a survivor bond
The price and floating cash flows of the survivor bond are based upon (a prediction of) the cumulative survival of the policyholders. In paragraph2.2is described how the one-year mortality rate that is predicted by the mortality model, can be used to calculate the cumulative survival rates. To determine the value of the survivor bond at time t, the payment upfront P and the future cash flows CFx,kf loating for policyholders of age x = 0, ..., ω at time k = t, ..., T are discounted to time t. In the valuation of the survivor bond, the risk premium is not taken into consideration.
Vt= −P + ω X x=0 T X k=t CFx,kf loating (1 + rk,T)T −k = −P + ω X x=0 T X k=t zx,k·k−tpx,t (1 + rk,T)T −k (2.22)
In the above formula the cash flows are based on the survivor rates, since policyholders need to be alive in order to receive the benefits. The amount that is paid to policyholders of age x at time k is denoted by zx,k. This is multiplied by the floating k −t-year survival
rate for an individual of age x at time t, i.e.k−tpx,t. The cash flows are discounted using
the interest rates rk,T for period [k, T ].
To construct the fixed and floating cash flows, a forecasts of the survival rates of the population and the portfolio are necessary. These can be calculated as described in 2.2
using a stochastic mortality model. The experience factor that is used, is calculated as the difference in predicted survival rates between the population and the portfolio for every age x and every time t. The price of the survivor bond is paid as a capital at the inception of the contract. This is based on the expected survival rates of the portfolio, increased by a risk premium to compensate the counterparty. The experience factor that
is used in the floating cash flows of the survivor bond are determined at the start of the contract as well. It is based on the difference in best estimate survival rates between the reference population and the portfolio. The floating survival rates are determined in a Monte Carlo simulation in this thesis. These are based on the forecast of the population survival multiplied by the experience factor.
2.3.3 Indemnity swap
The most popular instrument to hedge longevity risk is the longevity swap. The swap is an agreement by which two parties agree to exchange one or more future cash flows (Dowd et al., 2006). According to Melcer (2011) longevity swap contracts are in essence similar in structure to interest rate swap contracts. The structure of a longevity swap is illustrated in figure 2.6. When making use of a longevity swap, longevity risk is transferred to another (re)insurance company. The cash flows are called the fixed and floating leg of the swap contract. The fixed leg is paid by the insurer to the counterparty and is agreed upon at the inception of the contract. The fixed leg equals the best estimate of the future cash flows increased by a risk premium. The risk premium is added to the fixed payments to compensate the counterparty for the borne longevity risk. The floating cash flows are paid by the counterparty to the insurer. These cash flows are based on the realized survival rates. In practice, only the difference between the fixed and floating cash flows is settled. The reason for this is to reduce the counterparty default risk. As the insurer receives floating payments based on the realized survival in return for fixed cash flows, it does not bear the uncertainty in the survival rates anymore. Therefore, the insurer has hedged her longevity risk.
Figure 2.6: Longevity swap contract
Longevity swaps can be classified into two main types: indemnity and index-based swaps. According to De Crom et al. (2011) the indemnity swap resembles a reinsurance product with periodic payments. These payments are based on the realized mortality of the portfolio. The floating leg of this swap contract contains the realized payments the insurer has to pay her policyholders that period. The fixed cash flows are agreed upon at the start of the contract. These are based on the best estimate of the forecast of the portfolio mortality. A longevity swap contract is made for a predefined group of insured people. It usually ends at the event of the death of the last insured.
The indemnity swap is a longevity swap that provides a full hedge of the risk. This is because the floating cash flows the insurer receives from the counterparty equal the exact payments she has to pay the policyholders. This is the main advantage of the indemnity swap. This also has a downside, namely the indemnity swap is an expensive product to hedge longevity risk. Since the indemnity swap is defined at participant-level, the administration costs and therefore the price of the swap, are high. Another disadvantage of this type of swap is that the counterparty exposure is quite high if the duration of the swap is long. Mostly the duration is high, because the swap ends when the last insured of the portfolio dies.
Valuation of an indemnity swap
The value of an indemnity swap at time t is the present value of the future cash flows. Therefore, the future cash flows CFx,kf loating and CFx,kf ixed are discounted using the risk-free term structure. In the valuation of the indemnity swap, the risk premium is not taken into account.
Vt= ω X x=0 T X k=t CFx,kf loating− CFx,kf ixed (1 + rk,T)T −k = ω X x=0 T X k=t zx,k· (k−tpx,k−k−tp¯x,k) (1 + rk,T)T −k (2.23)
In equation2.23 the cash flows for time k = t, ..., T consist of the notional amount zx,k
that is assumed to be equal for all policyholders of age x = 0, ..., ω, multiplied by the difference between floating and fixed cumulative survival rates, i.e. k−tpx,k −k−tp¯x,k.
The expected cash flows are discounted using the interest rate rk,T for period [k, T ].
To determine the fixed and floating cash flows, a similar methodology as is used for the survivor bond is applied. This is described in paragraph2.3.2. The fixed cash flows are calculated at the start of the contract. They are based on the best estimate of the forecast of the portfolio survival. The floating cash flows are determined in a Monte Carlo simulation and are based on the realized survival of the portfolio.
2.3.4 Index-based swap
The other main type of the longevity swap is the index-based swap. The index-based swap can be seen as a collection of q-forwards. The floating cash flows of this product are constructed differently than the floating cash flows of the indemnity swap. When an indemnity swap is used to hedge the longevity risk, the insurer receives the exact payments she is required to pay her policyholders. The index-based swap has floating payments, that are based on an index. The index depends on the survival rates of a reference population multiplied by an experience factor. The index-based swap is therefore similar to the survivor bond, but consists of fixed cash flows that are paid periodically instead of upfront.
An advantage of the index-based swap, is that it costs less than the indemnity swap. This is due to the floating leg that is based on an index, instead of the realized payments to the policyholders. According to Coughlan et al. (2011) this ensures that there is potential to create greater liquidity. Because the hedge is based on an index, it is a standardized hedge. Therefore, it saves time, which makes it less expensive. The major disadvantage of the index-based swap is that it introduces basis risk due to the use of the index. Basis risk is the mismatch between the realized and floating payments. When basis risk occurs, some residual amount of longevity risk remains. Therefore, the index-based swap does not provide a full hedge of the risk.
Valuation of an index-based swap
The value at time t of an index-based swap can be calculated using equation 2.23. The risk premium is not taken into account in the calculation of the value. The value is the sum of discounted future cash flows:
Vt= ω X x=0 T X k=t CFx,kf loating− CFx,kf ixed (1 + rk,T)T −k = ω X x=0 T X k=t zx,k· (k−tpx,k−k−tp¯x,k) (1 + rk,T)T −k . (2.23)
In the above equation CFx,kis the cash flow at time k for policyholders of age x. The cash
flows depend of the notional amount zx,k, as well as the floating and fixed cumulative
survival rates, i.e. k−tpx,k respectivelyk−tp¯x,k. The cash flows are discounted using the
interest rate rk,T for period [k, T ].
To calculate the value of the index-based swap, fixed and floating cash flows have to be constructed. These can be constructed using a similar approach as described in paragraph 2.3.2. The fixed cash flows are based on the best estimate of the forecasted survival of the portfolio. The floating cash flows depend on the population survival rate multiplied by a predefined experience factor. The floating cash flows are constructed in a Monte Carlo simulation.
2.3.5 Comparison of longevity risk mitigating instruments
In this paragraph the risks that are likely to be involved in the survivor bond, indemnity swap and index-based swap are compared. The potential risks involved in a q-forward are not described, since the q-forward can be used to ’build’ the longevity risk mitigating instruments. At the end of the paragraph the discussed risks are illustrated to give an overview, see figure 2.7. This overview is made to give an idea of the risks, the exact sizes of the yellow parts could be somewhat different.
Basis risk
When the insurer buys an indemnity swap, he does not have to deal with basis risk, because the floating payments are exactly equal to the benefits the insurer pays the pol-icyholders. Therefore, no mismatch occurs between the floating and realized payments. When the insurer hedges her longevity risk by using an index-based swap or a survivor bond, basis risk does occur. The reason for this is that the floating payments are not necessarily equal to the payments towards the policyholders. This is because the floating payments are based upon an index. Besides that, the estimated number of policyholders who are alive in a certain period differs from the realized number of policyholders who are alive. This leads to differences in future cash flows of the instrument. Due to the mismatch between realized and floating cash flows a part of the longevity risk remains. The advantage of hedging longevity risk using an index is that it the instrument is less expensive than the indemnity swap. A disadvantage is, that the insurer will suffer a loss if the realized survival is the opposite of the index survival.
Counterparty default risk
Counterparty default risk is the risk that the counterparty is not able to pay out its contractual obligations. For the discussed longevity risk mitigating instruments this risk is the highest for a survivor bond, because a single payment is made by the insurer at the inception of the contract. In return for this the insurer receives coupons every period. According to De Crom et al. (2011) the survivor bond is therefore not a widely used instrument to hedge longevity risk. For indemnity swaps and index-based swaps, the counterparty default risk is reduced by transferring the difference between the fixed and floating leg. Another way of reducing the risk is to use a collateral. Pinsent Masons LLP (2009) argues that the party who is likely to be expected to make payments in the future, has to make collateral payments on a regular basis. This has to be done to compensate the party that is likely to suffer a loss if the swap contract is terminated. Collateral assets are typically bonds or gilts.
Liquidity risk
For longevity risk mitigating instruments no liquid market is available. All transactions that have been done are over-the-counter. Therefore, prices of the instruments are not
available for public. At the moment, there are no pricing models available for the in-struments yet. For this reason the survivor bond, indemnity and index-based swap are involved with liquidity risk. The liquidity of an index-based swap is higher than of an indemnity swap, because the index-based swap is more standardized than the indemnity swap. This makes the liquidity of the index-based swap higher than of an indemnity swap. According to De Crom et al. (2011) longevity swap contracts can contain exit clauses which enables both the insurer and the counterparty to liquidate the contract on market value. This increases the liquidity of the instruments.
Credit spread risk
According to the Risk Management Group of the Basel Committee on Banking Super-vision (2000) credit risk is the risk that a borrower fails to repay his loan or contractual obligation. This risk only occurs when a survivor bond is used to hedge longevity risk, because the insurer pays an amount upfront of the contract and receives payments every period. For longevity swaps, this risk does not occur, since every period the difference between the fixed and floating leg is transferred.
Inflation risk
The survivor bond is the only longevity risk mitigating instrument of those three that is involved with inflation risk. This is due to the single payment upfront of the contract in return for the periodic coupons. According to Biffis and Blake (2010) this is a ma-jor reason why the European Investment bank failed to introduce the survivor bond. Longevity swaps can hedge inflation risk as well instead of solely longevity risk, therefore it could be possible that this risk does not occur when the indemnity or index-based swap are used.
Rollover risk
OSFI (2014) discusses rollover risk, which occurs when longevity risk hedging contracts are closed for a shorter time period than the liabilities covered. A new contract could be entered once the contract expires. However, this could be more expensive than expected due to survival rates that have changed. The risk that this will occur is called rollover risk. This risk can only occur when the index-based swap is purchased, because the contract is for a predefined duration. The survivor bond has a duration that is not finite, therefore rollover risk does not occur. The indemnity swap usually ends at the moment of death of the last policyholder. Therefore, rollover risk will not occur. Legal risk
Legal risk is the risk of a financial loss arising from regulatory or legal action. According to OSFI (2014) longevity risk mitigating contracts are legal agreements that are not traded on exchange. The terms of the transaction should be fully understood by plan administrators, and legal advice should be acquired before entering a longevity risk mitigating contract. Legal risk only arises for an indemnity swap and an index-based swap. A survivor bond is not involved with legal risk, because it is more of a bond contract than a swap contract.
Hence, the survivor bond is involved with the most risks. Longevity swaps are there-fore more interesting for the insurer to purchase. For that reason from this point forward this thesis focuses on the indemnity and index-based swap. The indemnity swap is in-volved with the least risks, but it is a more expensive instrument than the index-based swap. The most important risk that is involved with the index-based is basis risk. This risk is modelled to give a better insight in the index-based swap. In paragraph 2.4
Figure 2.7: Longevity risk transfer structures, source: EY EAS Learning (2015)
2.4
Modelling longevity basis risk
Longevity basis risk is the risk that occurs when an index-based swap or survivor bond are used to hedge the insurer’s longevity risk. It arises because different populations or sub-populations experience different longevity outcomes. Consider e.g. a portfolio that consists of smokers while the population is diversified with smokers and non-smokers. Then the survival rates of the portfolio and population will differ. If an index-based swap is used, the floating survival rates are determined as the realized survival rates of the population multiplied by an experience factor. The experience factor is based on the difference between their survival rates. However, the index survival rates could differ from the realized survival rates. Therefore, a part of the longevity risk remains. This part is called basis risk. According to Haberman et al. (2014b) the need to quantify and reserve for any potential basis risk is receiving increasing focus, particularly under Solvency II. In this paragraph multiple methods are explained how one can model basis risk.
According to Haberman et al. (2014a) there are three different types of basis risk, i.e.:
• Structuring risk: the risk that the hedging instrument has a different payoff struc-ture than those of the hedged portfolio. This can occur for example when the hedg-ing instrument pays out annually, while the portfolio makes monthly payments or for instance when the duration of the hedge is different than the duration of the liabilities.
• Sampling risk: the risk that arises from different random outcomes of the indi-vidual’s lives within the portfolio and the index population. This means that the realized survival experienced by the two populations differs, other than by chance. • Demographic risk: the risk that there are demographic and socio-economic differ-ences in the composition of the portfolio and the index population which lead to different rates of mortality and trends.
For the first two types of basis risk there exist well-established methods to model these. Haberman et al. (2014a) explain that structuring risk can be modelled by simulating the cash flows under the portfolio and simulating the payoffs under the hedging instrument. The structures of the portfolio of the contract and the portfolio of the insurer need to be similar. Sampling risk can be assessed by simulating the outcomes for the respective populations. In a simulation this risk can be incorporated by including a binomial dis-tribution that determines how many policyholders survive every period. The last one, demographic basis risk, can be modelled on several ways, depending on the data. The direct and indirect way to model demographic basis risk are explained in this paragraph.
Figure 2.8: Choosing a method for modelling demographic basis risk, source: Haberman et al. (2014b)
Demographic basis risk can be modelled in two ways: directly and indirectly. When it is modelled directly, one can apply the M7-M5 model or the Common Age Effects (CAE)+cohorts model right away. In figure 2.8 a decision tree is illustrated. This can be used to make a decision regarding the model to model demographic basis risk. If the portfolio is not sufficiently large or it lacks sufficient back history, the demographic basis risk is modelled indirectly. The characterizing approach has to be applied to the data in order to be able to use the M7-M5 or CAE+cohorts model. The characterizing approach is illustrated in figure 2.9and explained below.
Figure 2.9: The characterizing approach for modelling demographic basis risk, source: Haberman et al. (2014b)
Haberman et al. (2014b) explain that the basic principle of modelling demographic basis risk indirectly is to divide the portfolio in a small number of characterizing groups. Every group captures key aspects of demographic risk. The groups can be projected using an alternative data source with a reliable and longer back history of mortality experience. This is called the characterizing approach. A reference population is also divided into groups, based on the same characteristics. The future mortality rates of the groups are simulated and then mapped across the characterizing groups of the portfolio. After that, it is possible to simulate mortality of the portfolio. The M7-M5 and CAE+cohorts model that are explained hereafter could be used for this.
The M7-M5 model
When the portfolio of the insurer is sufficiently large, the M7-M5 model can be used. This model is a two-population extension of the Cairns-Blake-Dowd mortality model that is discussed in paragraph 2.2. In this model the logit of mortality takes up to a quadratic form with age. The Cairns-Blake-Dowd model is a model that is designed
to model mortality of higher ages. Therefore, it could be used to model longevity risk. The M7 part of the model is a formula to calculate the mortality rates of the reference population. The M5 part of the model is a formula for the difference in mortality rates between the portfolio and the reference population.
The Common Age Effect+cohorts model
Haberman et al. (2014a) argue that if we instead allow the shape of mortality to have a non-parametric relationship with age, we obtain the extended Lee-Carter family of models. This resuls in the Common Age Effect (CAE)+cohorts model. The Lee-Carter model is discussed in paragraph 2.2. The addition of a cohort term to the Lee-Carter model performs best for the reference population. The model consists of two formu-las, just like the M7-M5 model. The first formula calculates mortality of the reference population, the second the difference in mortality of the reference population and the portfolio.
The M7-M5 model is based on the Cairns-Blake-Dowd model that is designed for higher ages. If the portfolio mainly consists of policyholders above the age of 60, the Cairns-Blake-Dowd model provides a good fit of the mortality. Therefore, the M7-M5 model can be used. The CAE+cohorts model is based on the Lee-Carter model. This model can be used to model mortality of all ages. In paragraph3.5a method is described how the amount of basis risk is quantified in this thesis. This method considers the difference in realized cash flows to the policyholders and the floating cash flows of the index-based swap as the amount of basis risk.
In this chapter a literature study was performed to gain insights in the subjects that are relevant to this thesis. In the first paragraph, the Solvency II standard formula is described. In particular mortality and longevity risk are discussed which are parts of the life underwriting risk module. The calculation of the SCR and the risk margin are explained as well. The Solvency II framework states that all balance sheet items have to be valued on market value. Assets are easily valued by market-consistent valuation, since a liquid market is available to buy and sell them. Liabilities are harder to value, therefore the expected future cash flows are discounted using the current risk-free term structure.
In order to calculate the best estimate of the liabilities, as well as the capital re-quirements, we need a stochastic mortality model. In paragraph 2.2 the Lee-Carter, Cairns-Blake-Dowd and AG2014 model are discussed. The mortality model is used to calculate the force of mortality µx,t. Using equations2.8and2.9, the one-year mortality
and survival rates, as well as the cumulative survival rates can be calculated.
In paragraph 2.3 four longevity risk mitigating instruments are described, in par-ticular the q-forward, the survivor bond, indemnity swap and index-based swap. The q-forward can be used to ’build’ the others. The risks that are potentially involved in these are compared. Based on the risks, the indemnity and index-based swap are focused on in the rest of this thesis. This is because the survivor bond is involved with too many risks. One of the risks that the index-based swap and survivor bond are involved with is basis risk. To provide a better comparison of the indemnity and index-based swap, this risk is modelled in this thesis. Therefore, the last paragraph of this chapter describes the M7-M5 and CAE+cohorts model that can be used to quantify the amount of basis risk.
Methodology
Life insurers bear longevity risk due to the risk that policyholders live longer than ex-pected. Hence, lifelong annuities to policyholders have to be paid for a longer period than expected. The insurer reserves capital to be able to pay out her contractual obli-gations, but this capital may not be sufficient if the insured live longer than expected. Therefore, the insurer could decide to hedge her longevity risk by purchasing a longevity risk mitigating instrument. In paragraph2.3 multiple instruments were discussed, such as the survivor bond, the indemnity and index-based swap. Based on the comparison of potential risks that are involved in these instruments, the indemnity and index-based swap are preferred over the survivor bond. Therefore, this thesis focuses on the indem-nity and index-based swap.
In this chapter the methodology to determine the impact of each longevity swap on the Solvency II position of the insurer is described. In paragraph 3.1 the data used in this thesis is described. This data is used to estimate the mortality model. In paragraph
3.2is discussed which mortality model is chosen to use, and how mortality is estimated and forecasted. The mortality rates are used to calculate the cumulative survival rates. These are used to determine the Solvency II position of the insurer. Paragraph 3.3
describes how the best estimate of the liabilities, the SCR and the risk margin are calculated in order to do this. Each longevity swap is then added to the portfolio of the insurer. The value of each swap, as well as the insurer’s SCR and risk margin are now calculated again. The methodology to calculate these values is described in paragraph
3.4. To provide a better comparison of the swaps, the amount of basis risk is quantified, as this risk is the major risk that arises if the index-based swap is used. The method that is used to establish this is explained in paragraph 3.5. At time 10 the situation of the insurer is evaluated, based on updated survival rates. The last paragraph of this chapter, paragraph 3.6, describes how this is established.
3.1
Description of the data
The data used in this thesis contains information to estimate the stochastic mortality model and a method to construct the liabilities of the insurer. The mortality model is used to calculate the best estimate of the liabilities and the fixed and floating legs of the longevity swaps. The liabilities are constructed as if the insurer has to pay out pension benefits to the policyholders. Therefore, the pension rights that the policyholders have accrued are insured.
Mortality model
In order to calculate the best estimate of the liabilities and the fixed and floating legs of the longevity swaps, a stochastic mortality model can be used. Three mortality models were discussed in paragraph 2.2, i.e. the Lee-Carter, Cairns-Blake-Dowd and AG2014
model. To estimate the parameters of the model, observed deaths and exposures are used. The time series are then simulated for future years. These series are used to forecast the survival of England and Wales and the portfolio. Paragraph 3.2 describes the methodology used to estimate the mortality model and forecast the survival rates.
In order to estimate the stochastic mortality model, observed exposures and deaths are needed. The exposures and deaths of European countries with above-average GDP, England and Wales and a portfolio with policyholders from England and Wales are used. The data covers the deaths and exposures of men. The used ages range from 0 to 90 years old, for the ages 90 to 120 the method of Kannisto is used which is explained in paragraph 2.2. For simplicity it is assumed that every policyholder is born on January 1st. The used years range from 1970 to 2005. The current exposures are the number of policyholders for which the swap is arranged. This means that no inflow of new policyholders is accounted for in the swap contract.
The portfolio that is used consists of 887,528 male policyholders in the year 2005. The distribution of ages is summarized in the table below. The ages range from 20 to 90 years. Age Percentage 20 - 29 <1% 30 - 39 9% 40 - 49 23% 50 - 59 36% 60 - 69 21% 70 - 79 7% 80 - 89 3% 90+ <1%
Table 3.1: Distribution of the ages in the portfolio
Pension benefits
The policyholders will start to receive pension benefits as of the age of 67. It is assumed that all policyholders in the portfolio have accrued pension rights with a defined benefit average-wage pension scheme since they are 25 years old. For policyholders of the ages 25 to 67 the salaries Sx are set to grow from 30,000 to 70,000. A franchise F of 15,000
is used and the accrual percentage ap% is set at 1.75%. The accrued pension rights OPt
at time t are then defined as:
OPx,t = OPx−1,t−1+ (Sx,t− F ) · ap%. (3.1)
The accrued pension rights are the benefits that policyholders receive as they reach the pensionable age of 67. Before the pensionable age they do not receive anything, i.e. OPx,t = 0 for x < 67. The accrued pension rights are insured at the inception of the
contract which means that future pension accrual is not accounted for in the swap.
3.2
Mortality model
To be able to calculate the best estimate of the liabilities as well as the fixed and floating cash flows of the longevity swaps, a stochastic mortality model is needed. In 2.2three mortality models are discussed, in particular the Lee-Carter, Cairns-Blake-Dowd and AG2014 model.
The mortality model that is used in this thesis is an extension of the AG2014 model. The AG2014 model estimates the mortality of West-European countries and subse-quently the deviation of Dutch mortality. As mortality of comparable countries is used,
