Risk mitigation using a q-forward contract
Cindy Brijs
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: Cindy Brijs Student nr: 10246827
Email: chpbrijs@gmail.com Date: July 15, 2016
Supervisor: Drs. Rob Bruning AAGab Drs. Mark A. Hesse AAGb
Second reader: Prof. Dr. Ir. Michel H. Vellekoopa
a
University of Amsterdam
b
This document is written by student Cindy Brijs 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
Systematic longevity risk causes problems with pension insurers as uncertainty in future mortality rates implies uncertainty in the pension liabilities. Since the expected present value of future pension payments is calculated by a mortality table containing future mortality rates, this expected value is stochastic. Longevity linked securities can take away part of this risk. A hedger of longevity risk can therefore transfer longevity risk to another party in exchange for a risk premium. In this thesis we take the position of a longevity hedger who has pension liabilities and we want to know the effectiveness of a q-forward hedge when the counterparty is an investor. In order to answer this we take the following approach. We analyse the AG 2014 model for Dutch mortality and then we add a stochastic experience factor to this model such that the expected portfolio losses are accurately estimated. Then we construct a q-forward contract tailored to our longevity risk and basis risk and assess the effectiveness. Finally, we conclude that in our case the q-forward contract is an effective hedge, even if basis risk is involved. The risk in our portfolio is reduced by 0.2 percent and higher quantiles have decreased as well. We note that parameter uncertainty is not taken into account in our study and the effectiveness may be the result of the low risk-free interest rates, so further research needs to be done.
Keywords
Longevity risk, basis risk, stochastic experience factor, pension liabilities, longevity hedge, wang transform, stochastic mortality model, AG 2014, compound Poisson, risk mitigation, rel-ative risk reduction, stochastic time process, seemingly unrelated regression.
Preface v
1 Introduction 1
2 Modeling human mortality 3
2.1 Introduction to human mortality modeling. . . 3
2.2 Mortality models for a single population.. . . 7
2.3 Mortality models for subpopulations . . . 9
3 Stochastic experience factor 13 3.1 Calculating expected portfolio loss . . . 13
3.2 Models for the experience factor . . . 15
3.2.1 Model for the portfolio specific mortality rate . . . 16
3.2.2 Model for the conditional relief . . . 17
3.3 The complete model . . . 18
3.4 Application in risk mitigation . . . 20
4 Longevity linked securities 22 4.1 Types of contracts and counterparties . . . 22
4.2 Pricing of longevity linked securities . . . 24
4.3 Construction of the hedge . . . 27
5 Results 29 5.1 Data description . . . 29
5.2 Human mortality model . . . 32
5.3 Stochastic experience factor . . . 35
5.3.1 Obtaining the stochastic experience factor . . . 36
5.3.2 Simultaneous forecasting of time index processes . . . 37
5.4 Hedge calibration and effectiveness . . . 39
6 Conclusions and further research. 42
Appendix A: calibration of the time index for Dutch mortality 44
Appendix B: code 46
References 49
This thesis is the final part of the master Actuarial Science and Math-ematical Finance at the University of Amsterdam. This research took place at the Dutch insurance company a.s.r. from February 2016 to July 2016 in Utrecht, the Netherlands.
In the writing process I received help from Rob Bruning, who is also my supervisor at the university, Mark Hesse and Menno van Wijk. Hereby I want to thank Rob, Mark and Menno for reviewing my work and giving advice. I also want to thank Kathelijn Loos of Solid Pro-fessionals for her support. I thank my co-workers for helping me to get acquainted with a.s.r. and I am looking forward to working here as a junior actuary. And of course, I thank my parents and Gabri¨el for their motivational words during this stressful time.
Cindy Brijs, BSc. July 15, 2016
Introduction
Pension liabilities depend on human survival rates of future years. There is however large uncertainty concerning these survival rates of policyholders and when the survival rates are higher than anticipated, then pension payments need to be paid for a longer time than the pension insurer expected. This may lead to problems, since the mortality improvement causes the obligations to the policyholders to increase. The uncertainty in the mortality rates is the systematic longevity risk, which cannot be diversified away by having a large portfolio, because it applies to the whole population.
There are several ways how to respond to this risk. Traditionally longevity risk is handled by applying scales to a base mortality table, but accelerations of mortality improvements have exceeded these scales in the past Li and Hardy (2011). Hence, an insurer may be interested in a longevity hedge or a reinsurance contract in order to mitigate this risk.
A longevity hedge is possible between annuity providers and life insurers, because when the mortality rates for a given portfolio are exceptionally low, this results in high annuity liabilities and low life insurance liabilities and vice versa. Since insurance companies often offer both annuities and life insurance, this results in a natural internal hedge. But also general investors may be interested in taking over longevity risk in exchange for a risk premium. This risk transfer is realized in the form of a longevity linked security, which is a derivative where the underlying is a longevity metric like a mortality rate.
The reinsurance company Swiss Re for example issued a mortality bond in 2003. This bond had fixed quarterly coupons, but the principal was dependent on a mortality rate of a large reference population (Blake et al.,2006). Other types of longevity linked securities include swaps, options, forwards, futures, swaptions and so on. Unfortunately, a liquid market of longevity linked securities does not (yet) exist, although the venture Life & Longevity Markets Association (LLMA) aims to promote this. JP Morgan, a member of the LLMA, developed a ‘q-forward’ contract which is includes one cash flow. On maturity date a swap takes place between two parties where one pays a notional value times a mortality index and the other party pays the notional value times a pre-agreed fixed rate. The mortality index is often a country mortality rate for a specific age and gender.
A large disadvantage of such a hedge is the basis risk. Basis risk is present since a country mortality rate does not perfectly correlate with the expected present value of the pension payments. Suppose there is a group of policyholders with an aggregate insured amount. After one year, some have died and the sum of their insured amounts is relieved. We therefore define this relieved proportion of insured amounts as the mortality rate per insured amount and the ratio of this rate and the country mortality rate is the experience factor. In our analysis we suppose this experience factor is also a random variable and that this denotes the basis risk.
In this study we aim to obtain an estimate of the effectiveness of a q-forward hedge 1
given the systematic longevity risk and the basis risk in our pension portfolio. This analysis consists of three parts: we need to model the country mortality rate, we model the stochastic experience factor and we construct the q-forward hedge, because we assume that besides the underlying country, age, gender also the notional value and time to maturity of the contract can be chosen freely.
A lot of research has already been done when it comes to human mortality modeling. In order to assess longevity risk, we study models where mortality rates depend on one or more stochastic time processes. Lee and Carter (1992) were the first to propose a stochastic mortality model and many extensions have been made and compared since, for example by Cairns et al. (2007) and Plat (2009). Mortality rates can also be modeled by models for multiple related populations, like the model described by Li and Lee
(2005) where the two populations share a common factor. The Dutch Royal Actuarial Association (AG1) uses this Li-Lee model with some adjustments to model the Dutch national population mortality (Boer et al., 2014). In this thesis we use this model as well since our portfolio contains Dutch lives.
For the modeling of the stochastic experience factor we rely on the method of Plat
(2009). While there is not much literature on the modeling of stochastic experience factors, there are methods to estimate the portfolio specific mortality rate, because the subpopulation mortality models are applicable here. Since we have limited experience data we prefer to model the experience factor rather simple. We suppose the relief of insured benefits show similarities with claim amount modeling, so the risk theory described byKaas et al. (2008) is also relevant.
In order to construct the q-forward hedge optimally and to assess the effectiveness of the hedge, we need an appropriate pricing method and risk measure. Pricing methods of mortality linked securities are compared by Barrieu and Veraart (2014) and Bauer et al. (2010). Instead of variance minimization we use a Wang transform described by
Wang(2001) to measure the longevity risk and the residual risk when the hedge is used. The three subquestions in our research occupy chapters 2,3 and 4. Chapter 2 de-scribes how stochastic human mortality can be modeled and we mention some criteria that such a model should satisfy. Then we zoom in on the AG 2014 model as proposed in
Boer et al.(2014). In chapter 3 we propose our method for finding the stochastic expe-rience factor of a specific portfolio and for combining this factor with the model for the country mortality rate. Chapter 4 outlines some possibilities for longevity hedging and describes the decision making process in constructing the hedge. Subsequently chapter 5 shows the results of all parts of the analysis and finally the hedge effectiveness of the q-forward contract. Eventually, chapter 6 concludes and suggests topics for further research.
Modeling human mortality
In order to assess longevity risk, we need a model that can forecast mortality rates and estimate the uncertainty in these rates. First we outline the basic theory of human mortality modeling and we supply criteria that a model should satisfy. Second we outline some well-known mortality models which apply to a single population of lives and third, we focus on the multi-population models where one is a subpopulation of the other. Here we analyse the AG 2014 model thoroughly, so we are able to reproduce this model and use it in the longevity hedging framework.
2.1
Introduction to human mortality modeling
When modeling human mortality, an estimation is made for the probability that an x-year-old person dies during calendar year t. All persons of exact age between x and x + 1 in the calendar year t are in the square Qx,t of the Lexis diagram, through which
they walk diagonally either until they become x + 1 years old, or until the year t + 1 begins or until death. See figure 2.1for an illustration.
Qx,t = [x, x + 1) × [t, t + 1). (2.1)
Define the time spent in Qx,t by person i as Ti. The binary random variable Di
is then equal to 1 of person i dies in Qx,t and 0 otherwise. Hence, summing over all
persons i in a group, the total number of deaths in Qx,t is Dx,t and the total number
of person years spent in Qx,t is the exposure Ex,t, according to 2.2 and 2.3(Vellekoop,
2016). Therefore Ex,t denotes the average population size in Qx,t.
Ex,t= X i Ti (2.2) Dx,t= X i Di (2.3)
The ratio Dx,t/Ex,t is the crude death rate for age x and year t. The instantaneous
death rate is given by the force of mortality µx,t in equation 2.4, where TD is the time
of death. This is the limit of h → 0 of the probability of dying before t + h, given that the person has survived to time t, scaled to the time interval h (Promislow,2014). From here, we can derive the probability of surviving k years by taking the integral, which leads to2.5. µx,t = lim h→0 1 hP(t ≤ TD ≤ t + h | TD > t). (2.4) kpx,t = exp − Z k s=0 µx+s,t+s ds . (2.5) 3
Figure 2.1: Diagonal walk through the Lexis diagram. Here the number of deaths is 3 and the exposure equals 3.3 years (Vellekoop,2016).
For simplicity, we assume that the force of mortality is constant over the Lexis square. Using the notation ofPromislow (2014), this assumption leads to the following mortality rate and survival rates for a person of exactly age x at the beginning of year t.
px,t= exp(−µx,t) = probability of surviving 1 year. (2.6)
qx,t= 1 − px,t = probability of dying within the year. (2.7) kpx,t=
k−1
Y
s=0
px+s,t+s = probability of surviving k years. (2.8)
Hence, knowing the force of mortality enables us to calculate life expectancies, pen-sion liabilities, life insurance premiums and so on. Accurate estimates for µx,t become
therefore essential in calculating these quantities.
It would be no surprise that for a data set of historic values Ex,t and Dx,t, the
maximum likelihood estimation for µx,tis the observed crude death rate. We now shortly
outline how the maximum likelihood function for µx,t is set up, according to Vellekoop
(2016).
Since we know that person i spent ti years in Qx,t, we define the likelihood for this
by 2.9since person i either has died or survived.
P (Ti = ti) =
(
µ exp(−µti) Di = 1
exp(−µti) Di = 0
. (2.9)
Assuming the mortality of all persons i in Qx,t are independent, the log-likelihood
L(µx,t) = Y i (µx,t)Diexp(−µx,tTi) (2.10) = µDx,t x,t exp (−µx,tEx,t) (2.11) log L(µx,t) = Dx,tlog(µx,t) − µx,tEx,t (2.12)
The log-likelihood function in2.12is similar to the Poisson likelihood, because a Poisson density satisfies:
f (x) = λ
ke−λ
k! .
log f (x) = k log(λ) − λ − log(k!).
In the estimation of µx,t, the number of deaths can therefore be considered a Poisson
distributed random variable with parameter µx,t Ex,t, because maximizing the Poisson
log-likelihood with respect to µx,twould lead to the same result. And indeed, maximizing
2.12 with respect to µx,t leads to the crude death rate.
In order to forecast mortality rates qx,t, the force of mortality should be a function
of x, t and the set of parameters Θ, so we can extrapolate the time parameters in this set. We obtain these parameters by setting all partial derivatives of 2.13 with respect to each of the parameters equal to 0.
log L(µbx,t) =
X
x,t
(Dx,tlogµbx,t−bµx,t Ex,t) (2.13) whereµbx,t= f (x, t, Θ). (2.14)
Now there are several models that estimate the force of mortality for a population. In this thesis we use a stochastic dynamic approach in estimating µx,t, so we assume that
the mortality rates are driven by some stochastic process. Model selection and criteria
A model should have some desirable properties to be considered a ‘good’ model. Some of these properties are described byCairns et al.(2007) as follows.
• Parsimony
Having too many parameters makes a model unreliable, since it may find non-existing structures in the randomness of the historical data. Therefore, if two models have approximately the same likelihood om the historical data, the model with fewer parameters is always preferred.
• Consistency with historical data
A good model should fit the historical data well. To access the goodness-of-fit, we can calculate the log-likelihood of the model or we can inspect the standardized residuals according to 2.16. A model with many parameters always leads to a high likelihood, so the consistency with the historical data and the parsimony of the model should be balanced. Therefore we regard the Bayesian Information Criterion (BIC), where a penalty is added to the negative log-likelihood for the number of parameters p, as in 2.15. A low BIC is favorable, because that means that the likelihood L(µbx,t) is relatively high and the number of parameters p is
relatively low, given the number of observations nobs.
If the death counts Dx,t are truly independent and identically distributed (i.i.d.)
Poisson random variables with parameter Ex,tµbx,t, then the standardised residuals x,t should be approximately i.i.d. standard normal distributed random variables.
Hence, if the model is accurate, the residuals should show a random pattern across ages, years and cohorts.
x,t=
Dx,t− Ex,t µbx,t pEx,t µbx,t
. (2.16)
• Transparency
The model should be understandable to the user. If the output is treated as a ‘black box’, the model might be used inappropriately, so it should be avoided. • Ability to generate sample paths
The random samples play an important role in the decision making for the hedging strategy. Therefore, it should be possible to generate samples from the model estimates.
• Incorporation of cohort effects
According toCairns et al.(2007), there are large differences in mortality improve-ments for the cohorts 1920 to 1950, so they state that cohort effects should be incorporated in the model.
• Non-trivial correlation structure
If the model includes more than one period risk factor, the correlation structure of µx,t is non-trivial. A trivial correlation structure would imply that an index hedge
of multiple underlying ages is not an improvement to an index hedge of only one underlying age. Therefore, a model that provides a non-trivial correlation structure is preferred.
• Robustness of parameter estimates
The parameter estimates should not change much if different ranges of data were used. Hence, the parameter uncertainty should be kept small.
There are multiple ways of how to deal with parameter uncertainty. For example
Plat(2009) suggests three approaches, namely a formal Bayesian framework, sim-ulation of the parameters using standard errors obtained in the estimation and applying a bootstrapping framework.
An example of a bootstrapping procedure is given byBarrieu and Veraart(2014). They use the following steps in order to find the price of a q-forward contract.
1. Obtain the estimates bµx,t of the force of mortality.
2. Fit the corresponding time series κt of the model.
3. Generate paths of κt in order to evaluate the Monte Carlo outcomes later.
4. Assume that the deaths are Poisson distributed with parameter Ex,t µbx,tand start the parametric bootstrap. For nboot times, do:
(a) Given these Poisson parameters, generate new deaths Dx,t.
(b) Treat the deaths as new observations and fit all parameters again. (c) Calculate the aimed value (for example the price of a mortality linked
security) using this model.
In order to access the robustness of the parameter estimates, we can use this procedure by generating nboot estimates for every parameter in the model.
Thus, the stochastic mortality model must be tested on each of these criteria. This framework can also help if models are to be compared.
2.2
Mortality models for a single population.
In 1825 Benjamin Gompertz introduced the Gompertz distribution for mortality, given by equation 2.17 (Promislow, 2014). He assumes that the force of mortality is a de-terministic value that only depends on age, where the logarithm of µx is assumed to
be linear in x. This leads to too high values at extremely old ages, so instead of the logarithm, for example Thatcher (1999) states that a logistic form like 2.18 is more appropriate when high ages are involved.
µx= βcx. (2.17)
µx=
α exp(βx)
1 − α exp(βx)
+ γ. (2.18)
In a deterministic dynamic approach of mortality modeling the force of mortality depends not only on age, but also on time albeit in a deterministic way. The stochastic dynamic approach however, includes stochastic time variables in the model such that µx,t is a function of some random variables (Vellekoop,2016).
A well known example of this is the Lee-Carter model proposed by Lee and Carter
(1992). They assume that µx,t depends on the age-dependent parameters αx and βxand
the stochastic process κt.
log(µx,t) = αx+ βxκt+ εx,t, where εx,t∼ Normal(0, σ). (2.19)
They achieve this by fitting2.19on historical data by least squares, where αx is the
average log death rate for age x. The parameter vectors containing βx and κt are then
found by singular value decomposition (SVD) of the matrix M containing the deviations from the mean according to 2.20. Here are u1 and v1 the first left and right singular
vectors respectively and λ1 is the first singular value of M , so λ1u1 contains βx and v1
contains κt. In order to ensure uniqueness of the parameters, they restrict βx and κtto
have a sum of respectively 1 and 0.
M ≈ λ1u1v1T. (2.20) where Mx,t = log(µx,t) − αx. (2.21) X x βx = 1. X t κt= 0. (2.22)
This second restriction in 2.22 implies that the age-specific parameter αx denotes
the mean log of µx,t for age x, because the term βxκt vanishes when taking the sum
over t. So for a given year t, the term kβx is added to this mean αx, where k can be
positive or negative and βx denotes the sensitivity to shocks in κt. This implies that the
model has a trivial correlation structure, which means that mortality improvement of two different ages are perfectly correlated.
Forecasts are made by modeling time series κt by autoregressive integrated moving
average (ARIMA) models, for instanceLee and Carter(1992) use the random walk with drift (RWD), which is ARIMA(0,1,0). In this case the parameters ˆφ and ˆσ are easily found, since the differences κt− κt−1 are then assumed to be i.i.d. so estimates for φ
and σ2 are given by the sample mean and standard deviation respectively.
κt= κt−1+ φ + σεt, where εt∼ N (0, 1). (2.23)
Instead of the approach of least squares, the parameters can also be found using the Poisson maximum likelihood method we discussed earlier. We prefer the latter, since
Koissi et al.(2006) showed that this approach leads to smaller errors overall, compared to least squares or weighted least squares (WLS).
Other types of stochastic mortality models
Meanwhile, many stochastic mortality models have been proposed. For instance, Ren-shaw and Haberman (2006) added factors γt−x to the Lee-Carter model, such that
cohort effects are accounted for, so in this model the force of mortality is given by2.24. This implies that the Lee-Carter model is a special case of Renshaw-Haberman, because the models are equivalent when βx(3)γt−x(3) is set to 0. Also the age-cohort-period model
proposed by Currie et al. (2004) is a special case of the Renshaw-Haberman model in which βx(2) = 1 and β(3)x = 1 for all ages x.
log µx,t = βx(1)+ βx(2)κ (2) t + β (3) x γ (3) t−x (2.24)
Currie(2006) has a different approach to estimating µx,t, because he uses the method
of penalized basis splines. Here log µx,tis estimated by linear regression, but instead of a
basis {1, x, x2, ...xk}, a basis of spline functions {B1(x), ..., BK(x)} is used as regressors.
A spline function is a continuous function that is piecewise polynomial inside a certain range and is equal to 0 outside it, as illustrated in figure 2.2. In order to prevent overfitting, a penalty that depends on the magnitude of the coefficients is added to the log-likelihood function, such that the penalized likelihood is maximized.
Figure 2.2: Example of a basis spline (Currie et al.,2004).
Table 2.1: Stochastic mortality models compared byCairns et al.(2007). Model Formula M1 log µx,t = βx(1)+ βx(2)κ(2)t M2 log µx,t = βx(1)+ βx(2)κ(2)t + β (3) x γt−x(3) M3 log µx,t = βx(1)+ κ(2)t + γ (3) t−x M4 log µx,t =Pi,jθijBij(x, t) M5 logit qx,t = κ(1)t + κ (2) t (x − x) M6 logit qx,t = κ(1)t + κ (2) t (x − x) + γ (3) t−x M7 logit qx,t = κ(1)t + κ (2) t (x − x) + κ (3) t ((x − x)2−σb 2 x) + γ (4) t−x M8 logit qx,t = κ(1)t + κ (2) t (x − x) + γ (3) t−x(xc− x)
Cairns et al. (2007) then compared eight stochastic mortality models given by table
2.1that either estimate log µx,t or logit qx,t of the population of England and Wales and
the population of the United States. The logit function is of the form 2.25 so that the inverse of the logit function forces the outcome to be in between 0 and 1. The models M1 to M4 are respectively Lee-Carter, Renshaw-Haberman and the models of Currie et al..
logit (x) = log x
1 − x (2.25)
logit−1(x) = exp(x) 1 + exp(x).
The model M5 is the original CBD-model, named after Cairns, Blake and Dowd, and the models M6 to M8 are extensions of this. Here are x and bσx2 respectively the mean and variance of the ages in the sample range. The age xcis a parameter that needs to
be estimated, so the third term in M8 represents the cohort effect that diminishes when the cohort ages. The models M5 to M8 assume smoothness in the ages, where M1 to M3 use one separate parameter per age.
In 2009, Plat states that his mortality model leads to a better fit in terms of BIC than M1, M2, M3, M5 and M7 when it comes to Dutch mortality data. This Plat model estimates the force of mortality by2.26.
log µx,t= βx(1)+ κ (2) t + κ (3) t (x − x) + κ (4) t (x − x)++ γ (5) t−x. (2.26)
Here is (x − x)+ = max(x − x, 0), so this term captures the trend of mortality improve-ment for lower-than-average ages.
These are examples of models that estimate the force of mortality or the mortality rate of a single human population. Now, models have been proposed that estimate these values for multiple populations simultaneously. Since using more data can increase the robustness of the model, we are interested in the application of multi-population models where one population is a small subpopulation of the other. That way, we estimate the force of mortality of the Dutch national population as a subpopulation of a large ‘European’ population.
2.3
Mortality models for subpopulations
As we aim to estimate the Dutch mortality trend, we assume that the trends show much similarities with neighboring countries, because the populations are similar in terms of health. Therefore we estimate a ‘reference’ force of mortality based on data of a ‘reference population’, which includes populations of multiple countries. Then we add country-specific parameters to this general trend to make it fit the Dutch data accurately. Examples of models that have this form are given by Li and Lee (2005),
Jarner and Kryger(2008) andDowd et al.(2011). We now discuss those models shortly and thereafter we zoom in on the AG 2014 model proposed by Boer et al.(2014).
Li and Lee(2005) extend the Lee-Carter model by modeling the reference population with a Lee-Carter model using SVD and also the country-specific mortality is estimated by a Lee-Carter model. This leads to model2.27, where BxKtis the common factor and
βxκtis the country-specific factor.Li and Lee (2005) assume a random walk with drift
(RWD) for Kt and a mean-reverting process or a random walk without drift for κt, so
that mortality will not diverge from the common trend in the long run. The processes Kt and κ are assumed to be independent. A Bayesian framework for this model is
given by Antonio et al.(2015), but we do not apply this framework here, because then the stochastic experience factor should be estimated simultaneously in our case and deriving prior and posterior distributions for the whole ourselves is beyond the scope of this thesis.
log µsubx,t = αx+ BxKt+ βxκt+ εx,t. (2.27)
Jarner and Kryger (2008) estimate the general mortality trend by using a frailty model. This means that they assume that the population consists of a heterogeneous
group of individuals where some are more ‘frail’ than others and tend to die first of the group. This again affects the composition of frail and robust individuals in the group, resulting in a concentration of robust individuals at high ages. According to the frailty theory, mortality is thus not only influenced by the level of health, but also by the average frailty. Then the force of mortality is estimated by2.28, where µrefx,t is the force of mortality of the reference population given by the frailty model, rxcontains functions
of x and wt are the so called ‘spread parameters’ that needs to be fitted.
log µsubx,t = log µrefx,t + w0trx. (2.28)
Dowd et al.(2011) propose a two-population version of the age-period-cohort model M3, also known as the M3B-model. In this model the reference mortality exerts a ‘gravitational pull’ on the subpopulation mortality, but not vice versa, similar to the gravitational relationship of a star and a planet. In this case, the force of mortality for both the reference population mortality and subpopulation mortality is given by the M3 model in table 2.1. Then the gravitational pull is processed through κt and γt−x,
where the effects {κt, γt−x} of the subpopulation depends on the values {κt, γt−x} of the
reference population, but not the other way around.
These models are examples of how the mortality trends can be estimated using not only the mortality data of the population of interest, but also the larger reference population that includes this. As the AG used features of the Li-Lee model in their prognosis for Dutch mortality improvements in2014, we now focus on this model that we define as ‘AG 2014’.
The AG 2014 model for Dutch mortality
The AG models the Dutch population mortality from 1970 until 2013 using the Li-Lee model (Li and Lee, 2005) for two populations: the Dutch national population and a ‘European’ population consisting of the countries with a gross domestic product higher than the European mean, including the Netherlands. These populations are all assumed to have a similar mortality trend, since income level is assumed to affect mortality rates (Boer et al.,2014).
log µx,t = (Ax+ BxKt) + (αx+ βxκt), (2.29)
µx,t = µEUx,t exp(αx+ βxκt). (2.30)
The force of mortality is defined as in 2.29. We find the maximum likelihood es-timates for the parameters using the two-step frequentist approach as described by
Antonio et al.(2015), which means that we fit the common parameters first and subse-quently we fit the country-specific parameters conditional on the fitted µx,tfor European
mortality. Therefore we assume that the European deaths Dx,tEU are independent Pois-son distributed random variables with parameters Ex,tEUµEUx,t , such that we maximize
2.31 with respect to Ax, Bx and Kt.
log L =X
x,t
DEU
x,t (Ax+ BxKt) − Ex,tEUexp(Ax+ BxKt) . (2.31)
To find the parameter value Bx for a certain age x, the first derivative of log L to Bx
should be set to zero. It is not possible to find a closed form solution for Bx, so this zero
can be approximated using the Newton-Raphson algorithm (Vellekoop,2016). Assume that for every iteration j, the force of mortality is:
µEU,jx,t = exp(Ajx+ BxjKtj). (2.32) Then for all parameters, the algorithm becomes (Vellekoop,2016):
Aj+1x = Ajx− η P t h DEUx,t − EEU x,t µ EU,j x,t i P t h −EEU x,t µ EU,j x,t i . (2.33) Bxj+1= Bxj − η P tK j t h Dx,tEU− EEU x,t µ EU,j x,t i P t(K j t)2 h −EEU x,t µ EU,j x,t i . (2.34) Ktj+1= Ktj− η P xB j x h DEUx,t − EEU x,t µ EU,j x,t i P x(B j x)2 h −EEU x,t µ EU,j x,t i . (2.35)
So for every iteration j, a new value of µEUx,t and then Ax, Bx and Ktis calculated. The
η in the algorithm is a shrinkage parameter, which prevents the parameter estimations to ‘jump over’ the optimum.
Subsequently we find the country specific parameters αx, βx, κt in a similar way.
Given the Dutch exposures EN L
x,t and deaths DN Lx,t , the deaths are again assumed
inde-pendent and Poisson distributed random variables with parameters EN L
x,t µx,t where µx,t
is defined as in 2.30. When maximizing the log likelihood of these Poisson deaths with respect to αx, βxand κt, the force of mortality for European deaths µEUx,t is considered to
be a given constant, so the optimization works as if the exposure is EN Lx,t µbEUx,t. Therefore we define the new exposure as in 2.37, so that now2.36 needs to be maximized. Hence, we can use the same approach as in2.32 to2.35.
log L =X x,t h DN Lx,t (αx+ βxκt) − ˜Ex,tN Lexp(αx+ βxκt) i . (2.36) where ˜Ex,tN L= Ex,tN L µbEUx,t. (2.37) When convergence of the total has occurred, that is when the log likelihood does not increase any further, the parameters will be adjusted such that the constraints2.38
and 2.39are satisfied. X x Bx = 1. X t Kt= 0. (2.38) X x βx = 1. X t κt= 0. (2.39)
Given these parameters, we can make forecasts by extrapolating Kt and κt. The
European data contains years 1970 to 2009, whereas the Dutch data also contains the years 2010 to 2013. Therefore Kt is extrapolated for years 2010 to 2013 using 2.40 to
make sure the time index vectors have the same length.
K2009+s= K2009+ s(K2009− K1970)/(2013 − 2009), for s = 1, 2, 3, 4. (2.40)
For the European time index Kt, a random walk with drift (RWD) is assumed and
for the Dutch time index a first order auto regressive model (AR(1)) is assumed. Here Kt and κt are not independent, but they are correlated normal random variables.
Kt+1= Kt+ θ + εt+1. (2.41) κt+1= φκt+ δt+1. (2.42) whereεt+1 δt+1 ∼ Normal(0, C).
We estimate the parameters θ, φ and C using the fitted values for Kt and κt. If
there are n observations, the log likelihood estimators are given by 2.43 to 2.45. The derivation can be seen in Appendix A.
ˆ θ = 1 n n−1 X t=0 (Kt+1− Kt) − C12 C22 1 n n−1 X t=0 (κt+1− φκt). (2.43) ˆ φ = Pn−11 t=0 κ2t n−1 X t=0 κtκt+1− C12 C11 n−1 X t=0 κt(Kt+1− Kt− θ) ! . (2.44) ˆ C = 1 n n−1 X t=0 (Kt+1− Kt− θ)2 (Kt+1− Kt− θ)(κt+1− φκt) (Kt+1− Kt− θ)(κt+1− φκt) (κt+1− φκt)2 (2.45)
For very high ages there is little exposure, so it would be difficult to find an accurate estimator for µx,t. Therefore the force of mortality is estimated by the algorithm only
for the ages 0 to 90. We fit the higher ages 91 to 120 by a logistic regression, based on the estimations for the ages yk= 79 + k, where k = 1, ..., 11, according to the method
used in Boer et al.(2014).
µx,t= L 11 X k=1 wk(x)L−1(µyk,t) ! , (2.46) where L(x) = 1 1 + exp(−x), wk(x) = 1 11 + (yk− y)(x − y) P j(yj− y)2 .
In this study we only consider the AG 2014 model, becauseBoer et al.(2014) suppose the model is robust and appropriate for Dutch mortality modeling. In the results, we find the parameter estimates and we test the fit on the model criteria discussed in section 2.1.
Stochastic experience factor
An insurer of pensions is interested in monetary amounts rather than actual deaths, hence we aim to extend the model for national human mortality by adding portfolio specific factors, so that it estimates the expected portfolio loss. It means that when a group of annuitants with the same one-year death probability qt, according to the
national mortality model, has an aggregated amount of insured benefits ct at time t,
then the expected cash flow at time t + 1 is equal to ct(1 − qtat). We propose a model in
which this experience factor at is stochastic and interacts with the time parameters in
the AG 2014 model. The uncertainty in expected portfolio losses is thus driven by time parameters on European, national and portfolio level. The longevity hedge should then be designed in such a way, that it takes away part of the total risk of the liabilities.
3.1
Calculating expected portfolio loss
The portfolio considered in this thesis consists of men and women who are participants of a pension plan and therefore have either a due life annuity or a deferred life annuity, corresponding to the definition of Promislow (2014). For simplicity, the pension pay-ments are assumed to be paid at the beginning of each year and are of equal height until death of the policyholder. The value of a contract for (x) is given by formula3.1, where ck is the payment at the beginning of year k, v(k) is the discount rate,kpx is the
probability that (x) survives k years and ω is the maximum age. In the case that the annuity is deferred n years, then the first n values of the vector c are 0.
¨ ax(c) = ω−x−1 X k=0 ckv(k)kpx. (3.1)
Because of the multiplication rule, the k-year survival rate is the product of one year survival rates, since it is the probability that a person survives each of the k interior years. When the survival rate depends not only on age but also on time, then the k-year survival rate is redefined by:
kpx,t= k−1
Y
j=0
px+j,t+j. (3.2)
The value of the liabilities at t = t0is the sum of the annuities and deferred annuities.
This represents the expected value of the future cash flows to policyholders, given the set of mortality rates. There are four different statuses for policyholders, namely:
1. Active participant(deferred annuity) 2. Inactive participant (deferred annuity) 3. Receiver of old age pension (annuity due)
4. Receiver of survivor’s pension (annuity due)
The aggregated insured amounts for policyholders of age x in year t and status s ∈ {1, 2, 3, 4} is given by cx,t,s, so the value of the liabilities in year t0 is equal to:
t0V =t0V1+t0V2+t0V3+t0V4 (3.3) = xp X x=0 (cx,t0,1+ cx,t0,2) xp−x|¨ax,t0+ ω−1 X x=0 (cx,t0,3+ cx,t0,4) ¨ax,t0 (3.4)
For the active and inactive participants of the pension plan the assumption is made that the pension age xp is certain and the same for all participants.
The difficulty in the liability calculation lies in the estimation of the future survivor rates px,t when t > t0. The AG provides tables containing the survival rates px,t where
px,t = exp(−µx,t). (3.5)
The force of mortality µx,t is estimated by the AG 2014 model using national data. This
is why we need an experience factor, because when using probabilities3.5in the calcula-tions3.4, the liabilities get underestimated in two ways. First, the insured population is not a representative cross section of the national population, but it is a specific subset. According to Barrieu et al.(2012) policyholders with higher socio-economic status (in-dicated by occupation, income or education) tend to have lower mortality rates. Owners of old age pension rights have been occupied, so that leads to a difference in mortality compared to the national population and therefore the hedger is subject to population basis risk. Second, a group of insured with the same age, gender and status may still not be homogeneous, because of differences in insured amount. Since pension benefits are linked to occupation and income, the policyholders with higher amounts live longer on average.
Therefore a model of spread between the Dutch national population and the portfolio should be constructed in such a way that the total model estimates the ‘mortality rate’ of one unit of money, instead of one person. Given that the aggregated insured amounts at time t is equal to cx,t,s for age x and status s, the aim is now to estimate the height
of the expected portfolio loss cx+1,t+1,s one year later. That is, to estimate the value of
the one-year mortality rate of 1 unit of insured amount qx,t,sb as in equation3.6.
cx+1,t+1,s = (1 − qbx,t,s)cx,t,s (3.6)
Traditionally, the probability qbx,t,s is estimated by applying a deterministic experi-ence factor ax, that only depends on age and gender, to the country mortality rate qx,tc ,
such that:
qx,t,sb = ax qcx,t, ∀s (3.7)
In this research we derive a stochastic experience factor ax,t,s, corresponding to
the method of Plat (2008). But where Plat (2008) estimates this ratio of probabilities directly, we imitate the approach of the AG (2012) in the sense that the experience factor is the product of two effects: the difference in mortality between the national population and the insured population (ξx,t,s) and the difference between the insured
population and the insured amounts (θx,t,s). Therefore we define the following rates and
µx,t = force of mortality of the national population. (3.8)
qcx,t= 1 − exp(−µx,t) = country mortality rate. (3.9)
qx,t,sP = ξx,t,s qx,tc = portfolio mortality rate. (3.10)
qx,t,sb = θx,t,s qx,t,sP = mortality rate per insured amount. (3.11)
ax,t,s= θx,t,s ξx,t,s = experience factor. (3.12)
The factors ξx,t,sand θx,t,sform the experience factor and both parts are typically less
than 1. AsPlat(2009) states, the experience factor should become 1 for the highest ages, because the population of extremely old persons is very small and selection effects are therefore no longer present. Besides, all types of 1-year mortality rates should approach 1 for high ages, so their ratios at these ages are also equal to 1.
3.2
Models for the experience factor
While there is a lot of literature on stochastic mortality models, there is not much research done on the modeling of stochastic experience factors for pension portfolios.
Plat (2008) models this factor directly by assuming that ax,t is normally distributed,
hence he fits the observed fraction qx,tb /qcx,t using Weighted Least Squares (WLS). Then
Berkum et al. (2015) state that this fraction is highly volatile and that it is more accurate to fit the model on the observed death counts instead. They propose a Baysian framework to model the force of mortality of the portfolio and the national population simultaneously and this has the advantage that the parameter uncertainty is taken into account. Although this is a more accurate model, we cannot apply it here, since we use a different model for the national population. Also Olivieri (2011) proposes a Baysian framework for modeling of portfolio death counts, but our data set is not suitable for calibrating this model, because we do not have cohort data for many consecutive years. Besides, we are interested in insured amounts rather than number of deaths.
Our data set contains the following information:
nx,t,s= Number of policyholders at the beginning of year t.
Dx,t,s= Number of deaths during year t.
cx,t,s= Insured amount at the beginning of year t.
Rx,t,s= Relief of insured amount due to deaths in year t.
If more information of the insured amounts were available, we could estimate the relief more accurately by creating homogeneous groups based on the insured amounts of the policyholders, so that the mortality rate in the group is equal for all policyholders. Then we use Panjer recursion (Kaas et al., 2008) where the probability distribution of one insured amount must be known, or the backwards recursion described byDe Pril(1986), that is tailored to annuity payments.
Restricted to the available information, we choose to assume that all policyholders in a [x, t, s]-group have the same mortality rate and we estimate the relief of insured amounts in two steps. Firstly we model the death counts Dx,t,s and secondly, we model
the relief given the death counts Rx,t,s | Dx,t,ssimilarly to the way that aggregate claim
amounts are modeled in Kaas et al. (2008). Then the expected height of the insured amounts for future years is simply the expected remainder, that is:
cx+1,t+1,s= cx,t,s− E(Rx,t,s) (3.13)
This represents the portfolio loss, since amounts cx,t,s need to be paid to policyholders
when the pension date is reached. In order to find this expectation, we obtain an estimate for ξx,t,s in the first model and θx,t,s in the second, which occupy respectively the first
and second subsection within this section.
3.2.1 Model for the portfolio specific mortality rate
We extend the AG 2014 model in order to predict death counts in our portfolio, since the mortality rates of the national population and this subset may differ. Mortality models for multiple populations like the ones discussed byLi and Hardy (2011) can be used to model portfolio deaths, but they may not be robust in our case, because we have limited data and these models use one estimator per age. Instead, the type of model proposed inJarner and Kryger (2008) for small subpopulations may be more appropriate. Hence, we model the mortality rate as follows:
qx,t,sP = qx,tc ξx,t,s = qx,tc exp(w + wt0rx), (3.15) where rx= rx(0) .. . r(m)x , wt= w(0)t .. . wt(m) .
Here are the values rx fixed regressors, so only w and the m elements of vector wt
need to be estimated. For example, Jarner and Kryger (2008) choose to let rx be a
scaled version of (1, x, x2), so that the spread parameters (w(0)t , wt(1), w(2)t ) estimate the level, slope and curvature of the spread. In our case, we model the spread qx,t,sP /qx,tc , where w and wt may also depend on status s.
Finding maximum likelihood estimators for w and wt.
Given the number of policies nx,t,s of age x and calendar year t, the number of deaths
Dx,t,sfollows a binomial distribution, since all policyholders with the same [x, t, s]-group
are assumed to be independent and they all have the same probability of death.
Dx,t,s∼ Binomial(nx,t,s , qPx,t,s) (3.16)
We estimate the factor ξx,t,s in equation3.15 using a Poisson regression since qx,t,sP
is usually low and nx,t,s is high and this is almost equivalent to a Poisson distribution
with parameter λx,t,s as in3.17.
Dx,t,s∼ Poisson(λx,t,s= nx,t,s qx,t,sP ) (3.17)
This model for Dx,t,sis a Generalized Linear Model (GLM), which is a generalization
of the ordinary linear model, as introduced by Nelder and Wedderburn in 1972. This makes it possible to fit the Poisson parameter according to 3.18, where the first term is an offset term. Because of the log link, the model is multiplicative and we obtain the portfolio specific mortality rate as in 3.15, using the glm function in R. Note that the model 3.18is equivalent to3.10.
A correction for delayed information
In one calendar year the reported deaths may only be a fraction of the actual occurred deaths, since some have occurred, but are not yet reported (IBNR, Kaas et al.(2008)). Given an IBNR triangle with calendar years t and lag years k, the maximum likelihood estimates for the deaths occurred in year t and reported in year t + k are found by a GLM regression where the the deaths are Poisson distributed with parameter λt,t+k in
the following model.
λt,t+k = ut πk. (3.19)
When the πk’s are scaled to have sum 1, then πk denotes the percentage of deaths
reported with a lag of k years. In this case ut is the maximum likelihood estimator for
the total deaths in calendar year t. This information is used as a deterministic constant in the model for the experience factor in order to decrease the bias. Therefore we add the reported percentage as an extra offset to the Poisson parameter in3.18, which leads to the updated model:
Dx,t,s∼ Poisson (gtλx,t,s) , (3.20) where gt= P x,sdx,t,s ut .
log λx,t,s= log(gt) + log(nx,t,s qcx,t,s) + log ξx,t,s. (3.21)
As the most recent year has a reported percentage of less than 70 percent, we remove this from the data set, since we think the data is not reliable to estimate ξx,t,swith.
Of course, there are many other possible ways to estimate the reported percentage. We choose this form, because it is equivalent to the well-known chain ladder method for claim modeling. Another option would be to replace ut by the known exposure nt, the
number of policyholders, but this would force the proportion of deaths to be constant over the year. For more information on IBNR methods, we refer to Kaas et al.(2008).
3.2.2 Model for the conditional relief
In the historic data there is little known about the height of the insured amount of the policyholders. We know that the average amount is equal to cx,t,s/nx,t,s, but assumptions
need to be made about the variation and skewness of insured amounts within an [x, t, s] group in order to estimate the relief of amounts due to deaths of the policyholders.
For a group of policies we assume that some variation and skewness in the insured amounts exists among participants, because pension wealth is linked to salary height. Also, when the number of deaths are known, the relief of insured amounts should be in the range [0, cx,t,s]. We can use a beta distribution to model this because its support is
bounded, but it is not straightforward how the hyperparameters should be chosen and moreover, it cannot be fitted using a glm function in R.
Thus we choose to model the relief as simple as possible, because there is little information. We stay close to the approach of aggregate claim amount modeling, hence we assume that the relief of insured amounts is a sum of dx,t,s independent gamma
distributed random variables. When the number of deaths dx,t,s is much smaller than
the number of policyholders nx,t,s, this assumption is appropriate, for the mass of the
gamma distribution that crosses cx,t,swould be rather small. Therefore, we assume that
when a person dies, the insured amount is given by a gamma distribution with index parameter αx,t,s and scale parameter βx,t,s.
Since the expectation of a gamma distributed random variable is equal to α/β and the variance is α/β2, in GLM literature the following re-parametrization is often made,
described for example byOhlsson and Johansson (2010). αx,t,s= 1 φ , βx,t,s= 1 φ ηx,t,s . (3.22)
Here is ηx,t,s the expected relief per person in group [x, t, s] and φ is the dispersion
parameter. A sum of independent gamma random variables is again gamma distributed, where the index parameter is the sum of the individual index parameters and the scale parameter stays the same. Then for a sum of dx,t,sindependent gamma random variables,
the following holds.
Rx,t,s | Dx,t,s= dx,t,s∼ Gamma dx,t,s 1 φ, 1 φ ηx,t,s , (3.23) where ηx,t,s= θx,t,s cx,t,s nx,t,s (3.24) The sum of the individual reliefs in the [x, t, s] group is Rx,t,s, which has the following
variance.
Var(Rx,t,s | Dx,t,s= dx,t,s) = dx,t,sφ η2x,t,s, (3.25)
= φ
dx,t,s
E(Rx,t,s)2. (3.26)
The number of deaths dx,t,s can be seen as an exposure, so we add this as an extra
argument weights to the glm function in R. Using these weights and a log link in the glm function, we fit the model where θx,t,sis a function of age and status. As a result we
obtain maximum likelihood estimates for ηx,t,s and φ. For ηx,t,s we use the same form
as in3.15.
log Rx,t,s= log cx,t,s
dx,t,s
nx,t,s
+ log θx,t,s. (3.27)
Now we have obtained a model for a stochastic experience factor 3.12, which is the product of the factors ξx,t,sand θx,t,s. The expected value of the relief in a one-year time
horizon is then given by:
E(Rx,t,s) = E(E(Rx,t,s | Dx,t,s= dx,t,s)) (3.28) = E Dx,t,sθx,t,s cx,t,s nx,t,s (3.29) = (nx,t,s qcx,t,sξx,t,s) θx,t,s cx,t,s nx,t,s (3.30) = cx,t,s (qcx,t,s ax,t,s). (3.31)
This shows that the expectation of the relief of insured amounts is indeed the initial amount cx,t,s times the mortality rate qx,t,sb , which is the product of the country
mor-tality rate and the experience factor. As the underlying of a longevity linked security probably only involves qx,t,sc , the estimation and forecasting of ax,t,sbecomes crucial in
constructing an effective hedge.
3.3
The complete model
We combine the model for the stochastic experience factor to the AG 2014 model simi-larly to the approach ofPlat(2008). The time parameters in model3.18must therefore
be extrapolated simultaneously with the time parameters of the AG 2014 model in a way that takes the correlation structure into account.
Plat(2008) assumes that for the time parameters wta Vector Autoregressive (VAR)
model is appropriate, because he states that it is reasonable that the process is sta-ble and does not increase or decrease unlimitedly. When combining wt with the time
parameters Kt and κt in the country population model, the complete model is fitted
using a Seemingly Unrelated Regression (SUR, see Zellner (1962)). This is a method to estimate coefficients in a system of equations, where the error terms are correlated. Here coefficients are estimated simultaneously by applying generalized least squares to the whole system, instead of equation for equation.
Recall the mortality rate per insured amount qx,t,sb :
qx,t,sb = qcx,t,s ax,t,s. (3.32)
Here is the country mortality rate qx,t,sc estimated by the AG 2014 model with parameters Ktand κt. Assume that ax,t,shas m time parameters, so we can write ax,t,sin the form
3.33.
qx,t,sc = 1 − exp(− exp[Ax+ BxKt+ αx+ βxκt]),
ax,t,s= exp ˜w + wt0rx . (3.33)
Hence, for every year t, we have the following normally distributed time parameters:
wt= Θwt−1+ ψ + υt. (3.34)
Kt= Kt−1+ θ + εt. (3.35)
κt= φκt−1+ δt. (3.36)
Now we aim to estimate the parameters ψ, θ, φ and the covariance matrix for υt, εt, δt.
For the portfolio is a small subset of the country population, which is a small subset of the European population, we assume that wt and Kt are only correlated via κtand
not directly, so the drift parameter and variance of Kt will not be changed in the SUR.
This leads to the new system of equations, where only the correlation structure of wt
and κt are taken into account.
wt= Θwt−1+ ψ + υt. (3.37)
κt= φκt−1+ δt. (3.38)
We cannot fit the parameters using the built-in functions for vector autoregression mindlessly, because wt and κt have only a few years in common, while we have
esti-mates for κt from 1970 on. Using SUR, we all historic data of these common years,
corresponding to equations 3.37and 3.38, into one matrix. w(1)t .. . w(m)t κt = X1w 0 . . . 0 0 . .. ... .. . . . . Xw m 0 0 . . . 0 Xκ ζ1w .. . ζmw ζκ + 1 .. . .. . m+1 . (3.39) Y = Xζ + . (3.40)
Suppose the available experience data contains the years t = 1, ..., n. Then all X’s on the diagonal are matrices of n rows. According toPlat(2008) the parameters are estimated by using the following steps.
1. Fit equation 3.35 and 3.36 simultaneously according to 2.43, 2.44 and 2.45 and with all the appropriate data.
2. Fit3.37 equation by equation using OLS.
3. Save the residuals of equations 3.37 and 3.38 corresponding to the n years they have in common.
4. Now the variance of the error terms in the m + 1 equations is given by covariance matrix of the residuals:
b σi,j = 1 n − m − 1 n X t=1 ei,tej,t. (3.41)
5. The variance of is estimated by the matrix bΩ = ΣN In, where Inis the identity
matrix. The error terms i,tand j,tare therefore correlated withbσi,j and the error terms i,t and i,s are uncorrelated if t 6= s, because they occur at different times.
6. The estimator for ζ is given by:
b ζ = X0Ωb−1X −1 X0Ωb−1Y . (3.42)
Now κt has a slightly different parameter φ and a different variance, while we keep
the drift and variance of Kt and the correlation cor(Kt, κt) unchanged. Given the new
estimates, we can now simulate the European and national mortality trend, as well as the portfolio mortality trend, with all the appropriate correlations.
3.4
Application in risk mitigation
With this information we are able do get an estimate of the exposure to longevity risk by doing a Monte Carlo simulation in which the death rates qx,tc and the experience factors ax,t,sare simulated for future years. Then the value of the liabilitiestV is a sum
of (deferred) annuities, denoted by 3.1 and 3.4, where the k-year survival rate for age x, year t and status s is given by:
kpx,t,s= k−1 Y j=0 px+j,t+j,s= k−1 Y j=0 (1 − qx+j,t+j,sb ). (3.43) After a number of simulations, we have obtained estimates for tV and for the country
mortality rate qx,t,sc , so we can assess how the these mortality rates of the national pop-ulation affect the pension liabilities. This may help the hedger to choose the underlying rates for the longevity hedge. Also, given a certain hedging strategy we can estimate how effective this hedge is using the Monte Carlo simulation.
The hedge is only effective when the basis risk is not too large, so the mortality rate per insured amount qb
x,t,s should show a high positive correlation with qcx,t,s. Since qx,t,sb
is the product of qx,t,sc and the experience factor, we suppose the basis risk is small when ax,t,s has a low volatility and when the correlation with the country mortality rate is
high.
When the expected value of a cash flow to the pensioners is known, there is still some risk left since the realized payment may differ from the expected value. This is the idiosyncratic longevity risk and can be mitigated by simply having more policyholders. To obtain a measure for the total risk it is appropriate to estimate this type of risk too, but we expect that the idiosyncratic risk does not affect the decision making for the hedge or its effectiveness, so we choose to ignore it when constructing the hedge.
Now we have obtained a full profile of the systematic longevity risk in the given pension portfolio. The longevity hedge should be adjusted optimally to this profile to maximize its risk mitigation property.
Longevity linked securities
When subjected to systematic longevity risk, the insurer has several possibilities of how to respond. The insurer can for instance bear the risk itself by holding a sufficient capital buffer or seek an internal hedge by diversifying the longevity risk across different insurance products or different underlying cohorts (Blake et al., 2006). Also, instead of traditional annuities where the payments are based on predicted future mortality rates, annuity providers can offer annuity products where survivor credits are paid to annuitants based on experienced mortality rates within the pool of annuitants. That way, a part of the exposure to longevity risk is transferred to the surviving policyholders. And of course, longevity risk can be transferred to other parties like reinsurers or investors in exchange for a risk premium.
This longevity transfer exists in the form of a security where the underlying is a mortality metric like a survivor rate or an annuity. If the cash flows from such a contract are customized to the actual cash flows from the portfolio, it is an ‘indemnity based’ hedge and if the cash flows depend on a mortality index, it is an ‘index based’ hedge. The index based hedge has the disadvantage that the index does probably not correlate perfectly with the mortality experience of the portfolio. Despite this basis risk, the index type of hedge may still be preferable to an indemnity hedge. This is because an index hedge is based on objective mortality data and has lower costs, especially when deferred annuities are involved (Coughlan et al., 2011). This thesis focuses entirely on the comparison of the strategies of bearing the risk and participating in an index hedge. The objective of our hedge is to reduce the uncertainty of the expected liability value due to systematic longevity risk. In the following subsections we summarize the different possibilities when it comes to longevity linked securities, we calibrate the hedge using Monte Carlo simulation and we define hedge effectiveness.
4.1
Types of contracts and counterparties
Blake et al.(2006) enumerates the possible counterparties who are willing to participate in the index hedge. A natural counterparty is of course the life insurer, since this agent benefits from longevity improvement as the deaths of the policyholders occur later than expected. But also general investors may be interested in investing in a longevity linked security, because it has a low correlation with other risk factors in the financial market. This leads to a well diversified investment portfolio and a relatively high return when the expected market return is low.
The government and the regulators are also important stakeholders in the immature longevity market, since they probably want to promote a liquid market for longevity risk in order to enhance financial stability. Furthermore, since the government provides state pensions via the pay-as-you-go system, it may be interested in managing its own longevity risk exposure.
According to Blake et al. (2006) the first mortality linked securities were bonds.
Swiss Re issued a three-year life catastrophe bond in 2003 where the principle payment reduced when the underlying mortality index would exceed 1.3 times the 2002 base level of mortality rate in any of the years until maturity date. It was well received by investors, unlike the bond that BNP Paribas announced in 2004 and withdrew later due to insufficient demand. This bond was an annuity bond, intended for pension insurers, where the coupon payments depended on a realized survivor index of English and Welsh males aged 65 in 2002. The new security stranded, because the bond was thought to be expensive since large capital was needed and the basis risk encountered due to differences between the underlying population and insured population was substantial.
Meanwhile other types of mortality linked securities have been designed.Blake et al.
(2006) describe roughly four types of longevity linked securities: the longevity bond, the mortality swap, the mortality futures, and the mortality options. We now discuss the other three types shortly.
• Mortality swap
A mortality swap is an agreement between two parties to swap the floating leg for the fixed leg in future years, where the floating leg depends on a mortality index. Comparing to longevity bonds, the swap has some advantages. For instance, they have lower transaction costs and are easily cancelled.
Also, mortality swaps do not require a liquid market, but can be traded over-the-counter (OTC). This enables the security to be tailor made to the needs of the user, since the fixed leg, notional value and underlying can be chosen optimally and is therefore more flexible than the longevity bond. The drawback of OTC trading is however that both parties are subject to credit risk whereas when trading in organized exchanges credit risk is handled by the exchange itself. The simplest form of a longevity swap is a q-forward, that includes only one such payment at a future time.
• Mortality futures
A futures contracts involves an underlying price process and a delivery date of the contract. Then the futures price for the underlying is such that the price of the contract is 0 by definition. In order to keep credit risk low, both parties place collateral into a margin account, such that price movements are absorbed. For mortality futures, a liquid market is necessary since futures are normally traded in organized exchanges. Futures contracts can only be successful if cer-tain requirements are met, for instance the existence of a large, active and liquid underlying spot market.
• Mortality options
Hedgers may be interested in options, because an option can be used as insurance against high losses owing to the underlying variable while they can still profit from advantageous outcomes. Also speculators on the volatility of the underlying may take an interest in options.
This can be an extension to the liquid mortality futures market, but also OTC contracts are possible, so that a hedger might buy a cap to a survivor rate tailored to his own mortality experience.
We restrict our research by only considering the q-forward contract, because it is a simple security and it is the building block of more complex longevity derivatives, like mortality swaps. For the hedging application of other derivatives, we refer to the analysis ofDawson et al. (2010), who derive a framework for pricing of swaps, futures, forwards, swaptions, caps and floors and test the application to hedging pension fund liabilities.
4.2
Pricing of longevity linked securities
If the hedger has found a counterparty who is willing to participate in a q-forward contract, they have to agree on the fixed rate. The hedger pays the floating rate and the counterparty pays the fixed rate, such that the value of the contract is 0 and the only cash flow is the difference between fixed and floating at the maturity date. For now, we assume the counterparty is a risk averse investor who is not subject to any longevity risk. For the hedger, this construction leads to a compensation when the underlying mortality rate turns out to be lower than the fixed rate and a positive expected return for the investor if the mortality rate becomes higher than the fixed rate.
The q-forward contract is defined by the underlying mortality rate Q, the fixed rate q, the notional value z and the time to maturity tm. At maturity, the net payoff amount
(NPA) for the hedger is given by4.1.
NPA(tm) = z(q − Q). (4.1)
For the investor it is the other way around. He pays a fixed rate and receives the floating rate, so we define the return on the investment by 4.2. The value of contract at time 0 should be equal to 0, but the question rises how to value the random variable Q. Therefore we discuss four pricing principles for Q in order to define an appropriate fixed rate.
RQ=
Q
q − 1 (4.2)
1. Fair premium
We find the fair premium for Q by simply taking the expectation under the real world probability measure, according to:
q = E(Q). (4.3)
No risk averse investor will agree on participating on the contract if the fixed leg to be paid is equal to the expectation. To make the contract attractive, the fixed rate must be lower than this, such that the expected return is positive. Then the difference between E(Q) and q is the risk premium.
2. Capital asset pricing model
As an investment product, the q-forward has an expected return E(RQ) and a
volatility σQ. We can therefore use the capital asset pricing model (CAPM,
de-scribed byWang(2002) among others) to define the fixed rate. The risk premium is defined by comparing the q-forward return with the risk-free rate r and the return RM from investing in the general market, where βQ is the sensitivity of the
investment return to the market return. This model makes the assumption that all asset returns are normally distributed.
E(RQ) = r + βQ(E(RM) − r). (4.4)
and βQ=
Cov(RQ, RM)
σ2M
CAPM can be restated using the ‘market price of risk’, given by equation 4.5. If the market price of risk is known, we find the fixed rate q based on the expected excess return and volatility of the investment according to4.6.
E(RQ) = r + λQ σQ. (4.5) where λQ= Cor(RQ, RM) E(RM) − r σM . q = E(Q) 1 + r + λQ σQ . (4.6)
Friedberg and Webb (2007) suggest that it is more appropriate to use the con-sumption capital asset pricing model (CCAPM) instead, where the risk premium is determined by the marginal utility of consumption. They state that income from investments is more valuable in times when consumption suffers than at other times. According to Bauer et al.(2010) this approach leads to very low risk premiums and it is likely that a ‘mortality premium puzzle’ exists, which implies that risk premiums for mortality risk are much higher than the models would suggest.
3. Risk neutral pricing
In 2002, Wang proposed a framework for quantifying insurance risks which can be used for both risks and assets. In the special case of an asset with a normal rate of return, the pricing method coincides with the CAPM and option pricing is equivalent to the Black-Scholes option pricing formula.
Wang(2002) finds risk-adjusted prices by distorting the CDF of the risk or asset with the market price of risk, such that the risk-adjusted price is simply the expectation under the risk-adjusted CDF. Wang (2001) states that this measure is a coherent risk measure and that it is preferable to the coherent tail value-at-risk (TVaR), since the Wang transform adds extra weight to high severity losses with low probability into account instead of only taking an average.
Then for the CDF F (x), the risk-adjusted CDF F∗(x) is given by equation 4.7, where λ is the market price of risk and Φ is the standard normal cumulative distribution. Dowd et al.(2006) show that the price does not heavily depends on λ, but rather on the distribution of the risk. Also Lin and Cox (2005) use this method and they find λ = 0.1476 for males and λ = 0.2024 for females, based on annuity prices.
F∗(x) = Φ[Φ−1(F (x)) + λ]. (4.7) Equation 4.7 is the Wang transform, where normal and lognormal distributions are preserved such that:
Normal(µ, σ2) 7→ Normal(µ − λσ, σ2). (4.8) Log-normal(µ, σ2) 7→ Log-normal(µ − λσ, σ2). (4.9) Here we can see that the normal transform coincides with the CAPM when λ is defined as in 4.5, because then the risk-adjusted price equals the risk-free rate. The Wang transform is therefore a generalization of CAPM, because it relaxes the normality assumption. In our case we find the fixed rate of the q-forward contract by choosing q such that the risk-adjusted expected return on the investment equals the risk-free rate corresponding with the time to maturity.
