One-year risk of lifelong mortality
trend uncertainty
G.A.A. Witteman
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: G.A.A. Witteman Student nr: 8620156
Email: gaawitteman@gmail.com Date: August 15, 2018
Supervisor: Dr. T. Boonen Second reader: Dr. S. van Bilsen
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman iii
Abstract
Uncertainty in future mortality causes uncertainty in the financial po-sition of a pension fund today. One of the main risks for a pension fund is the mortality trend uncertainty, the risk that participants live longer than expected, also known as longevity risk. Pension funds can make use of stochastic modeling to calculate a required capital needed to ab-sorb possible losses due to this risk. In this thesis this mortality trend uncertainty is calculated using stochastic modeling. The risk is there-after related to the scenarios of technical provisions using the same modeling. This relation can be helpful for pension funds because the required capital can be calculate straightforward out of the technical provisions without the estimation of mortality parameters.
Preface v
1 Introduction 1
2 Modeling mortality 3
2.1 Future payments and technical provisions . . . 3
2.2 Data . . . 5
2.3 Forecasting mortality . . . 5
2.4 Generating projection tables. . . 7
3 One-year longevity risk 8 3.1 Value-at-Risk . . . 8
3.2 Technical provisions of lifelong payments . . . 9
3.2.1 Data . . . 9
3.2.2 Forecasting mortality . . . 9
3.3 The one-year longevity risk . . . 10
3.3.1 Mortality in the first following year. . . 11
3.3.2 Mortality including a one-year shock . . . 11
3.4 Results. . . 12
4 Conclusions and further research 15 4.1 Conclusions . . . 15
4.2 Further research . . . 15
Appendix A: Closing of the projection tables 17
Appendix B: Three example portfolios 18
Appendix C: Risk factors proposed by DNB 19
References 21
Preface
The research for this thesis took place from April 2018 to July 2018, while working with PGGM in Zeist, The Netherlands. I like to thank PGGM and my colleagues for their support. I like to thank my supervisor for the help to get this work done. And last but not least I say thanks to my family for their flexibility during the last four months.
Chapter 1
Introduction
Running a pension fund is a risky business. Participants of pension funds pay premiums now in order to receive lifelong payments after retirement. The intention of the funds and the expectation of the participants is that the welfare after retirement is kept at the same level as before retirement. With the aim to achieve this with a relatively low premium, the premiums are invested. If the returns of the investments are lower than expected, the capital available may be to low to pay out the intended pensions. Apart from the uncertain returns on the investments, there are a number of other risks a pension fund faces. For instance, costs are higher than expected. Costs rise with inflation, which is hard to predict. The interest rate term structure to value the future payments is also a risk. A low interest rate makes the future payments expensive and with that the financial position of the fund lower. Other types of risks a pension fund can face are operational risks like fraud, mistakes or loss of data by various reasons.
The Dutch regulator, The Netherlandse Bank (DNB), distinguishes a number of risks pension funds have to take care of. DNB requires extra capital on the balance sheet to absorb the possible losses due to these risks. This extra capital is the sum of capital calculated for the individual risks, taken in account diversification between the risks. The size of this required capital must be such, that ”only once in 40 years” the loss due to the expected risks are higher than this required capital.
One of the risks on the list of DNB is the longevity risk, a typical risk for pension funds and insurance companies. The longevity risk is the risk that life expectancies increases faster than expected. If less people die than expected, the invested premiums have to feed more pensioners. The life expectancy of a Dutch citizen has increased now for many years, see for instance Antonio et al. (2017). Pension funds do account for further improvements of life expectancies. They make use of best estimate assumptions, including the best estimate (negative) trend in future mortality. The uncertainty in this trend is called mortality trend uncertainty (MTU). This risk will be the subject of this thesis. For the MTU, the DNB does not describe how the required capital must be calculated. But DNB does suggest a method how it can be calculated. DNB had made available tables of sex-neutral, age-depending factors. The required capital can easily be calculated by applying these factors on the best estimate assumptions. Most pension funds use this method to calculate the required capital.
In September 2016, the Royal Dutch Actuarial Association (2016) published the mortality tables AG2016. These projection tables consists of age-dependent death rates for future calendar years, for men and women. Pension funds use these tables for the calculations of the technical provisions, the amount of capital the pension fund has to set aside in order to pay the expected future pension payments. The AG2016 tables are based on stochastic modeling. With the stochastic properties of the tables, different scenarios for future mortality can be calculated, resulting in more possible outcomes, instead of just one outcome. This makes it possible to examine the MTU. Pension funds are not obliged to make use of the stochastic properties of AG2016 to investigate their
longevity risk but the information is available for all the funds. The pension funds has to decide for themselves to use this information or not to use it.
Figure 1.1: Example of the outcome of stochastic simulation, the life expectancy of 65 year old men. The realized life expectancy is shown with the red line and the best estimate prediction of the life expectancy with the blue line. The blue plume gives the spread in the predicted life expectancies with a certain probability. In this figure, with a certain probability, the prediction of the life expectancy in 2040 lies between about 21 years and about 24 years. The best estimate prediction of the life expectancy in 2040 is about 22.5 years. Source: Royal Dutch Actuarial Association (2016).
Koter (2013) and Polman (2017) use stochastic modeling to calculate the mortality trend uncertainty. Koter uses the model of Lee and Carter (1992), Polman uses the AG2014 modeling. The AG2014 models were the first models of the Royal Dutch Actu-arial Association based on stochastic modeling. In both studies, the stochastic modeling leads to much lower required capital for longevity risk, compared to the required capital calculated with the method of DNB. Both studies make use of the DNB assumption that the longevity risk can be quantified by taking the 75th percentile out of the scenarios for the technical provisions. In other words, to take the worst scenario that remains after removing 25% of the worst scenarios first. In this thesis, this assumption of DNB is examined for a modeled pension fund using Lee-Carter mortality modeling. In Chap-ter 2, this modeling of the mortality is described. In ChapChap-ter 3 the required capital for MTU is calculated and related to the percentiles of the technical provisions of the lifelong pension payments calculated with the same modeling. In Chapter 4, the results are discussed and suggestions for further research are given.
Chapter 2
Modeling mortality
The financial position of a pension fund is determined by its coverage ratio. The coverage ratio is the amount of capital available divided by the best estimate technical provision. The best estimate technical provision is the actual value of the expected future payments for pensions and costs. With a coverage ratio of 100%, there is just enough capital on the balance sheet to pay out the expected pension payments. The technical provision stands for the already build up pension obligations, it does not contain (expected) future obligations1.
Thus, to be able to calculate the financial position, an indication of future payments must be made. Future payments depend on the assumptions made for the future. The most important assumption is the future death rate of the participants of the pension fund. Lower expected death rates lead to more expected (lifelong) pension payments and more expected payments lead to a higher technical provision and therefore to a lower financial position.
Mortality modeling is needed to generate future payments. The research for this thesis is done with the Lee-Carter model (Lee and Carter, 1992). An overview of actual issues concerning mortality modeling is given by Barrieu et al. (2012). The Lee-Carter model is one of the most famous models, used in many mortality studies and is an inspiration for the development of other mortality models. The Li-Lee model, a mul-tipopulation model described by Li and Lee (2005), is an extension of the Lee-Carter model. The Li-Lee mortality model, as proposed in Boer et al. (2016), is used by the Royal Dutch Actuarial Association (2016). See also Antonio et al. (2017) for more back-ground information about the modeling of the Royal Dutch Actuarial Association.
This chapter describes the modeling of the mortality applied in Chapter 3. First, in Section 2.1, a pension fund is modeled and the calculation of the technical provisions of the lifelong payments of this modeled pension fund is explained. In Section 2.2, the data is described, needed for this research, followed by modeling of the mortality in Section 2.3. In Section 2.4 projection tables are generated for the calculation of the technical provisions of Section 2.1.
2.1
Future payments and technical provisions
The modeled pension fund only contains old age pension, spouse pension is not modeled. The payments for old age start at age x = 65 and are payed as long as the pensioners are alive. Future costs are added as a percentage of the payments and are included in the size of the payments modeled.
1
To measure the effect of new pension obligations on the technical provision the (pure) cost effective contribution can be calculated likewise the technical provision. If the premium collected is lower than the cost effective contribution, the impact of the premium on the financial position is negative.
The modeling is done for age cohorts τ = {15, 16, ..., 120} and future years t = {0, 1, 2, ..., 120}. The payments are modeled as continuous payments, which means that half of the first payment is made at the beginning of the year, when x = τ + t = 65, and the other half of the first payment is made at the end of the year, when x = τ + t = 66. At x = 66, also the first half payment of the second year is done. At x = 67, second half payment of the second year and the first half payment of the third year is done, etc.. At t = 0, every age cohort τ has Nτ participants. The accrued claims are considered
the same for all participants within an age cohort, denote by δτ. The size of the claims
of cohort τ is therefore Nτδτ. The probability that participants of age cohort τ are still
alive after t years, starting from the beginning of calendar year T0, is denoted bytpτ(T0) .
The size of the payments for age cohort τ after t years is given by
A(τ + t) Nτδτ tpτ(T0), (2.1) with A(τ + t) = ( 0 if τ + t < 65, 0.5 if τ + t = 65, 1 if τ + t > 65. (2.2)
To obtain the discounted values of the payments per age cohort, Pτ(T0), the sum of the
discounted future payments is calculated as
Pτ(T0) = 120−τ
X
t=0
A(τ + t) Nτδτ tpτ(T0) (1 + r)−t, (2.3)
where (1 + r)−t is the discount function with constant interest rate r.
The total technical provision, P calculated in calendar year T0, is given by
P (T0) = 120 X τ =15 Pτ(T0) = 120 X τ =15 120−τ X t=0 A(τ + t) Nτδτ tpτ(T0) (1 + r)−t. (2.4)
For age cohort τ , the probability of being still alive after t years,tpτ(T0), is the product
of the probability of not dying in t years:
tpτ(T0) = t−1
Y
i=0
(1 − qτ +i(T0+ i)) for t ≥ 1, (2.5)
where qτ +i(T0+i) is the probability for the age cohort τ to die in calendar year T = T0+i
(at the age of τ + i). These probabilities are collected from the projection tables where these so called death rates are related to age x and calendar year T , qx(T ). This qx(T )
is defined as the probability that someone alive at January 1st of calendar year T , born on January 1 of year T − x, dies before January 1 of year T + 1. For instance, the AG2016 tables contains these death rates for age x = {0, 1, ..., 120} and calender years T = {2016, 2017, ..., 2066}.
Formula (2.5) shows that with i both the age and the calendar year increases. For instance, for the cohort τ = 30, with age x = 30 in T0 = 2018, the probability to
die in year T = 2021 is given by q33(2021). One year later, this probability for the
same cohort is given by q34(2022). So, both the age and the calendar year (belonging
to this age) is raised with 1. This probability q34(2022) differs from the probability in
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 5
mortality, the probability q34(2022) is expected to be less than the probability q34(2021).
Formula (2.4), in combination with (2.5), shows that, if the interest rate r is taken constant and the size of the payments are given, the uncertainty in the technical provi-sions of the modeled pension fund is caused completely by the uncertainty of the death rates qx(T ). At the end of this chapter, projection tables can be filled containing this
uncertainty, which makes it possible to calculate the technical provisions for different scenarios, needed in Chapter 3.
2.2
Data
The prediction of future death rates is based on historical central death rates, collected from the Human Mortality Database (HMD). For age x ∈ {xbegin, ..., xend} and calendar
year T ∈ {Tbegin, ..., Tend}, the central death rate mx(T ) is calculated as the number of
deaths Dx(T ) divided by the exposure Ex(T ):
mx(T ) =
Dx(T )
Ex(T )
. (2.6)
The exposure Ex(T ) is the number of person years of age x who lived in the calender
year T . It is the sum of the individual fractions of years lived by the aged x in calendar year T . If a person becomes 35 on 1st of April 2012, the number of person years of age 34 in 2012 is 3/12. See Wilmoth et al. (2017) for more information about the HMD. In Figure2.1a small part of a Lexis diagram is presented. The blue lines at 45orepresent individual lifetimes. The exposure of a certain age in a certain calendar year is measured by the length of the line in a particular square divided by √2. E.g, if born on January 1, the lifetime starts in the left lower corner of a square and ends in the right upper corner of a square. Dividing this lifetime by √2 gives exactly one year. Dx(T ) is the
number of deaths of age x in calendar year T , equal to the number of ended lifetimes in the square with age between [x;x+1) in calendar year between [T ;T +1).
Figure 2.1: An example of a small part of a Lexis dragram. Source: Wilmoth et al. (2017)
2.3
Forecasting mortality
The Lee-Carter model is given by
E[mx(T )] = exp{αx+ βxκ(T )}, (2.7)
where mx(T ) is the central death rate, αx and βx are age-dependent parameters and
κ(T ) is the time-dependent parameter.
The expected number of deaths ˆDx(T ), predicted by the model, is given by
ˆ
Dx(T ) = E[Dx(T )]
= Ex(T ) E[mx(T )]
= Ex(T ) exp{αx+ βxκ(T )} (2.8)
The number of deaths Dx(T ) is assumed to have a Poisson distribution with mean
Ex(T ) exp{αx+ βxκ(T )}. According Villegas et al. (2017), the Lee Carter parameters
are obtained by maximizing the log-likelihood for a Poisson distribution, which is given by xend X x=xbegin Tend X T =Tbegin {Dx(T )log ˆDx(T ) − ˆDx(T ) − logDx(T )!}. (2.9)
with the following constraints for κ(T ) and βx applied: Tend X T =Tbegin κ(T ) = 0 and xend X x=xbegin βx = 0. (2.10)
The future mortality is determined by a random variable with drift, also known as ARIMA (0,1,0). The time-dependent parameters κ(T + 1) follow from κ(T ) with
κ(T + 1) = κ(T ) + θ + (T + 1) starting with T = Tend, (2.11)
with drift parameter θ and random variable (T + 1) ∼ N(0, σ2k).
The drift parameter is calculated as the historical average decline in the mortality rate over time ˆ θ = 1 (Tend− Tbegin) Tend−1 X T =Tbegin (κ(T + 1) − κ(T )) = 1 (Tend− Tbegin) (κ(Tend) − κ(Tbegin)). (2.12)
The measure of the uncertainty of the predictions is defined as the variance of the historical data ˆ σk2= 1 (Tend− Tbegin) Tend−1 X T =Tbegin (κ(T + 1) − κ(T ) − ˆθ)2. (2.13)
Future central death rates ˆmx(T ), can now predicted with the estimated Lee-Carter
parameters:
ˆ
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 7
2.4
Generating projection tables
In this section a projection table is constructed. The projection table consists of death rates qx(T ) for ages x ∈ {0, 1, 2, ..., 120} and future calendar years T ∈ Tend+ {1, 2, ..., 120}.
The death rates qx(T ) are calculated from the central death rates ˆmx(T ). With the
fol-lowing assumption,
ˆ
mx+ξ(T + ξ) = ˆmx(T ), (2.15)
for all 0 ≤ ξ < 1, saying that within a year the central death rates remain constant, the one year death rates qx(T ) can be found using
qx(T ) = 1 − exp(− ˆmx(T )). (2.16)
Up to this moment, the mortality modeling is done for ages up to xend. For ages
above xend up to age 120, the projection table is closed accordingly the method of
Kannist¨o. See Appendix A for the mathematical background of this method.
With the closing of the projection tables, all the ingredients are there to generate the technical provisions for different scenarios P (T0) with (2.4).
Finally, the best estimate technical provision, denoted by BEτ(T0), can be calculated
by ”switching off” the uncertainty. This is done by ignoring the error term (T + 1) in (2.11). The result is that the future values of the time dependent Lee-Carter parameter κ(T + t) only depend on the value of κ(Tend) and the drift parameter θ. This is shown
by applying (2.11) without the error term to calculate κ(Tend+ t):
κ(Tend+ t) = κ(Tend+ t − 1) + θ
= κ(Tend+ t − 1 − 1) + θ + θ
= ...
= κ(Tend+ t − t) + tθ
One-year longevity risk
In the introduction of this thesis it was mentioned that pension funds need to have extra capital on the balance sheet to be able to pay out the future pension payments, regardless losses due to different kind of specified risks. But what capital is enough? The Dutch legislator relates the size of the required capital, also known as solvency capital, to the Value-at Risk (VaR) measure. The VaR is a widely used risk measure in financial institutions. The VaR will be explained in Section 3.1. In Section 3.2 the technical provisions of the lifelong pension payments of the modeled pension fund are calculated with the data from the Human Mortality Database. In Section 3.3 the one-year longevity risk is calculated based on the VaR measure. This chapter ends with the relation between the one-year longevity risk of Section 3.3 and the technical provisions of the lifelong pension payments from Section 3.2. Both the calibration of the Lee-Carter model and the modeling of the future mortality is done with the R package StMoMo described in Villegas et al. (2017).
3.1
Value-at-Risk
Rules for the solvency capital for pension funds are described within the Financial as-sessment framework (Financieel toetsingskader, FTK). The size of the solvency capital is calibrated such that the probability that the full solvency capital is needed within one year is less than 2,5 % (”once in 40 years”)1. The used risk measure is the Value-at-Risk (VaR). The VaR can be explained very well by a measure of the loss of a portfolio of risky assets. Given a certain investment horizon, what is the maximum loss which is not exceeded with a given high probability? This high probability is called the confidence level. For instance, the confidence level is 90% and the maximum loss given this confi-dence level is e1 million. The probability that the loss is higher than e1 million is no larger than 10%.
Definition of Value-at-Risk: Given some confidence level α ∈ (0, 1). The VaR(L) of a portfolio at the confidence level α is given by the smallest number ` such that the probability that the loss L exceeds ` is no larger than (1 − α). Formally,
VaRα(L) = inf{` ∈ R : P (L > `) ≤ 1 − α} = inf{` ∈ R : FL(`) ≥ α}. (3.1)
So, the VaR is a percentile of the loss distribution. See also Gibbons and Chakraborti (1992). More information about VaR can also be found in, for example, McNeil (2005). 1For insurance companies within the the European Community (EC) Solvency II is introduced in
2016. Within Solvency II the size of the solvency capital is calibrated such that the possibility that the full solvency capital is needed within one year is less than 0.5 percent (”once in 200 years”). The solvency margin for insurance companies is higher than the solvency capital of pension funds because insurance companies can not lower the payments or increase the premiums. Insurance companies go bankrupt if they can not fulfill their obligations.
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 9
If we apply the VaR to the required solvency capital of the mortality trend uncertainty, the confidence level is 97.5% and the time horizon is 1 year.
3.2
Technical provisions of lifelong payments
3.2.1 Data
The central death rates mx(T ) are subtracted from the Human Mortality Database
(HMD) in May 2018. The selection of the data covers the Dutch male population with ages from 0 up to 90 years old in the years 1970 up to 2014. In the next sections we refer to this data as Dataset 2014.
3.2.2 Forecasting mortality
The Lee-Carter model, described in Section 2.3, is used to estimate the central death rates mx(T ) with formula (2.7). Figure3.1 shows the three Lee-Carter parameters αx,
βx and κx(T ) for the Dataset 2014. The high values of αx for very low ages reflect the
relatively high death rates of very young children. For ages above 30, the αx increases
with age. The κx(T ) show that death rates are declining for years.
Figure 3.1: The estimated Lee-Carter parameters for Dataset 2014.
After estimating the mortality with the Lee-Carter model, the prediction is done accordingly formula (2.11). The drift parameter ˆθ and the variance ˆσ2
κ follow from
(2.12) and (2.13) with Tbegin= 1970 and Tend = 2014. With (2.14), the future mortality
ˆ
mτ(T ) is estimated and a projection table is made. With claim size 1 (Nτδτ = 1) and
r =3%, the technical provision ˆPτ at the start of calendar year T0 = 2015 is calculated
with (2.3): ˆ Pτ = ˆPτ(2015) = 120−τ X t=0 A(τ + t)tpτ(2015) (1.03)−t. (3.2)
This forecasting of the mortality is done 40,000 times2, resulting in 40,000 techni-cal provisions ˆPs∗
τ , with s∗ = {1, 2, ..., 40000}3. The best estimate technical provision
BEτ(2015) is also calculated. Figure 3.2 shows the best estimate technical provisions
for a yearly pension payment of 1.
Figure 3.2: Best estimate technical provisions for payments of 1. The technical provi-sion rises up to retirement mainly due to the interest rate of 3%. After retirement the technical provision goes down because of payments.
3.3
The one-year longevity risk
The aim of this section is to find the required solvency capital for the longevity risk, based on the VaR97.5% at a one-year horizon for the modeled pension fund. How can
this one-year longevity risk be calculated? The risk examined is the following: due to uncertainty in the mortality trend in the coming year, the best estimate technical provision, including this risk, becomes to high. So, the best estimate technical pro-visions, calculated with best estimate cash flows including the one-year shock, need to be calculated.
Boonen et al. (2017) describes the influence of future mortality on the valuation of the payments now. For this thesis only the mortality in the first following year is deter-mined using the two steps described. First, death rates are predicted for the following year (in this case for year 2015). For this thesis, this is done for 1,000 scenarios4. In this step, every scenario creates a new set of data, the historical data from 1970 up to 2014, plus one year of data from the scenario. In the second step each of the 1,000 new datasets lead to newly estimated Lee-Carter parameters and to 1,000 best estimate technical provisions for the lifelong pension payments. These best estimate technical provisions are sorted by size. Since the confidence level is 97.5%, the 97.5th percentile of the 1,000 best estimate technical provisions is the MTU. This method is executed in the following two subsections.
2
The number of 40,000 is chosen because this number of calculations could be done overnight.
3
Here ˆP is used instead of simply P to distinguish between the lifelong scenarios in Section 3.2 and the one-year scenarios in Section 3.3. For the same reason s∗is used here instead of simply s.
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 11
3.3.1 Mortality in the first following year
With the set of Lee-Carter parameters from Subsection 3.2.2, based on Dataset 2014, the death rates for calender year T = 2015 are determined. For scenario s = {1, 2, ..., 1000}, the time-dependent parameter κs(2015) with the drift parameter ˆθ from (2.12), and the s(2015) with variance from (2.13) is given by:
κs(2015) = κ(2014) + ˆθ + s(2015), (3.3) The central death rates for 2015 are calculated for each scenario by
ˆ
msx(2015) = exp(αx+ βxκs(2015)), (3.4)
where αx and βx are the same for all the scenarios.
The central death rates ˆmsx(2015) are added to Dataset 2014, making 1,000 datasets Dataset 2015(s). Each Dataset 2015(s) contains the same historical data from 1970 up to 2014 and a prediction for 2015. Only the prediction for 2015 makes each dataset different from the other 999 datasets.
3.3.2 Mortality including a one-year shock
All Dataset 2015(s) are fitted with the Lee-Carter model, resulting in newly estimated parameters αsx, βxs and κs(T ). The drift parameters ˆθs are calculated the same way as earlier but now with Tbegin= 1970 and Tend= 2015.
Since the best estimate technical provisions (including the one-year shock) are needed, the future values of the time-dependent parameter κs(T ), are deterministic predicted out of the former value, using (2.11) without the uncertainty. Using (2.17), these values are given by
κs(Tend+ t) = κs(2015 + t)
= κs(2015) + tθs(2015), (3.5) with j = {1, 2, ..., 120}.
For all s = {1, 2, ..., 1000} projection tables are made and all the projection tables are closed with the method of Kannist¨o. The result is a set of 1,000 best estimate projection tables, including the one-year shock. With each projection table a best estimate technical provision Pτs, including the one-year shock, is calculated using (2.3). Again, this is done for claim size 1 (Nτδτ = 1), r =3% and T0= 2015:
Pτs= Pτs(2015) =
120−τ
X
t=0
A(τ + t)tpsτ(2015) (1.03)−t. (3.6)
The stochastic tpsτ(2015) in (3.6) is calculated with (2.5), using (2.16) to calculate
the death rates qsτ +t(2015 + t) out of ˆmsτ +t(2015 + t). ˆmsτ +t(2015 + t) is calculated with (2.14) out of the parameters ατs, βτs and κs(2015 + t). Accordingly (3.5), κs(2015 + t) follows from the in (3.3) calculated κs(2015) and the drift parameters θs.
The 1,000 best estimate provisions Ps
τ are sorted by size. The 97.5th percentile of
the technical provisions is the capital that must be kept on the balance sheet for MTU. The probability that a shock in 2015 leads to a higher provision than this 97.5th, is no larger than 2.5%.
3.4
Results
The solvency capital for MTU, the extra capital to be held for this risk, is given by the 97.5th percentile of the stochastic Pτ from (3.6) minus the best estimate technical
provision BEτ(2015) from Section 3.2,
Required capitalτ,M T U = VaR97.5%(Pτ) − BEτ(2015). (3.7)
For all age cohorts τ , a risk factor fτ is calculate as the ratio of the required capital
and the best estimate provision without the one-year shock BEτ(2015).
fτ = Required capitalτ,M T U BEτ(2015) = VaR97.5%(Pτ) − BEτ(2015) BEτ(2015) . (3.8)
This risk factor fτ can be compared with the DNB risk factors of Table 4.4, applied
in formula (4.1). If the risk factors are known, the required capital can be obtained by simply multiplying the best estimate technical provision by this risk factor. The risk factors fτ are shown in Figure3.3.
Figure 3.3: The risk factors per age cohort, the required capital for the one-year longevity risk divided by the best estimate technical provision for the pension payments without the one-year shock. The V-shape is caused by the sharp peak in the best estimate technical provision as shown in Figure 3.2. In the calculation of the factors with (3.8), the best estimate technical provision is the denominator.
For a whole pension fund the factor f is defined as: f = Required capitalM T U
BE(2015) =
VaR97.5%(P ) − BE(2015)
BE(2015) . (3.9)
As shown in earlier studies, the results of this thesis also show that the risk factors are lower than the risk factors in the DNB instructions5. Using (3.9) and (4.2), the risk factors of a young, an average and an old pension portfolio are calculated. Table 3.1
gives the results for these three portfolios. See Appendix B for the description of the three portfolios used.
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 13
Table 3.1: Risk factors calculated from the one-year risk and calculated with the risk factors proposed by DNB. The results are shown for a young, an average and an old portfolio. The risk factors based on the one-year risk are much lower than the risk factors based on the method proposed by DNB.
Portfolio Risk factor f Risk factor DNB
Young 1.26% 4.10%
Average 1.23% 3.66%
Old 1.19% 3.14%
In Section 3.2 the technical provisions ˆPτs∗, based on Dataset 2014 were calculated 40,000 times. In this section the one-year longevity risk is compared to these technical provisions. The comparison is done in the following way. The percentile of the technical provisions from Dataset 2014 is found that equals the one-year MTU risk. So, the confidence level ατ is calculated such that
VaRατ( ˆPτ) = VaR97.5%(Pτ). (3.10) The results are plotted in Figure 3.4. The factors are age dependent and increase with age from just below 75% for low ages up to 95% for age 90 years. Since the con-fidence level ατ is increasing with age, a pension fund with an older average age is
expected to have a higher confidence level than a younger pension fund.
Figure 3.4: The VaR confidence levels ατ for the one-year longevity risk, applied to the
technical provisions ˆP (without the one-year shock).
The confidence level for a whole pension fund is given by
VaRα( ˆP ) = VaR97.5%(P ). (3.11)
Table 3.2gives the impact on age for the three portfolios using (3.9) for the factor and (3.11) for the matching confidence level.
Table 3.2: Factors f and confidence levels α for the young, the average and the old portfolio. As expected from the confidence levels per age cohort in Figure 3.4, the confidence levels increases with the average age of the portfolios.
Portfolio Risk factor f Confidence level α
Young 1.26% 78.7%
Average 1.23% 79.7%
Chapter 4
Conclusions and further research
In Chapter 3 the following two results are presented. First, the size of the required capital for longevity risk is given as a risk factor to be multiplied with the best estimate technical provision of the lifelong pension payments. Secondly, the confidence level is determined which relates the required capital to the technical provisions of the lifelong pension payments, using the uncertainty within the stochastic modeling. In this chapter these results are discussed and suggestions for further research are given.4.1
Conclusions
The goal of this thesis was to find a relationship between the one-year longevity risk and the technical provisions produced with a stochastic mortality model. The relationship was found with the VaR confidence level. The size of the confidence level appears to be different from the DNB instructions and shows a dependence on age. The percentiles of the three portfolios lie all above the 75th. Table 4.1 shows the risk factors where the percentiles of the one-year longevity risk are compared with the risk factors using the 75th percentile. The risk factor for the young fund is 1.26%. If the 75th percentile is taken, a risk factor of 1.07% is taken, about 15% to low. For the average fund the factor is about 19% to low and for the old fund about 22% to low. That means that the required capital, based on the 75th percentile, is 22% to low for the old fund. So, if a pension fund wants to use stochastic modeling to calculate the required capital, making use of the already estimated mortality parameters, age depending confidence levels are needed.
Table 4.1: Comparison of the factors f modeled in Chapter 3 and the factors following the instruction of DNB by taking the 75th percentile. Following the instruction leads to risk factors which are to low.
Portfolio Percentile Risk factor f Percentile Risk factor f
Young 78.7 1.26% 75.0 1.07%
Average 79.7 1.23% 75.0 1.00%
Old 80.7 1.19% 75.0 0.93%
4.2
Further research
The pension fund modeled for this thesis only contains old age pension for men. The addition of spouse pension to the modeled pension fund will be an improvement. If life expectations increase, the expected spouse pension decreases because of a later start.
But on the other hand, like old age pension, an entered spouse pension is expected to run longer if life expectancies increase. With the introduction of the spouse pension it is also interesting to study the longevity of both male and female.
Another improvement will be the use of the AG2018 mortality model instead of the Lee-Carter model1. The Lee-Carter model in combination with the data from the HMD is used for this thesis because of the simplicity of the modeling. Pension funds can use the more complicated AG2018 projection tables to calculate their technical provisions, applying the stochastic properties of the tables. This can be attractive because it can lead to much lower required capital for the longevity risk. This thesis indicates that simply applying the instruction of DNB can lead to an underestimation of the longevity risk. It is an interesting question if this is also the case with the AG2018 modeling. If pension funds want to calculate the required capital without estimating the model parameters themselves, the supply of age depending risk factors can be very helpful.
1
In September 2018 the AG2018 projection tables will be published by the Royal Dutch Actuarial Association
Appendix A: Closing of the
projection tables
The modeling in this thesis is done for ages up to 90 years. For the calculation of the technical provisions ages up to 120 are needed. The calculation of the death rates of the highest ages of a projection table is called closing. The projection tables in this thesis are closed with the same method as the AG2016 tables are closed. Source: Royal Dutch Actuarial Association (2016).
For ages above 90, (x, t) ∈ ˜X x T with ˜X = {91, 92, ..., 120}, the method of ta-ble closing of Kannist¨o is applied. The method is based on a logit regression for ages y ∈ XKan= {80, 81, ..., 90}. So, the number of ages yk where the regression is based on
is n = 11, the average of these ages is ¯y = n1 Pn
k=1yk= 85 and the sum of the squares
of the deviationPn k=1(yk− ¯y)2 = 110. For x ∈ ˜X ˆ mx(t) = L n X k=1 wk(x)L−1 ˆ myk(t) ! . where L(x) = 1 1 + e−x, L −1 (x) = −log 1 x − 1 , The weights are given by
wk(x) = 1 n+ (yk− ¯y)(x − ¯y) Pk j=1(yj− ¯y)2 = 1 11+ (yk− 85)(x − 85) 110 . 17
portfolios
The three portfolios described here are copied from the documentation of the AG2016 tables. Source: Royal Dutch Actuarial Association (2016).
Table 4.2: Weighted average age of the portfolios Participants Young Average Old Below 65 (not retired) 49.3 50.8 53.4 Above 65 (retired) 71.1 72.9 73.7
Table 4.3: Number of participants of the portfolios
Age Young Average Old
30 500 300 100 40 1200 850 500 50 2000 1400 800 60 1800 1800 1800 70 1500 1650 1800 80 300 550 800 90 0 50 100 18
Appendix C: Risk factors
proposed by DNB
DNB has suggested the following method to calculate the required capital for MTU. First, for the different pension types i, the best estimate technical provision is calcu-lated per age cohort τ , denoted by BEτ,i2. These provisions are multiplied by an age
depending factor fτ,i. The sum over all the ages and all the pension types gives the
required capital for the mortality trend uncertainty: Required capitalM T U =X
i
X
τ
(fτ,i∗ BEτ,i). (4.1)
Since only the pension type old age pension (i = 1) is considered in this thesis, only the factors for old age pension, are given in Table4.4. The factors are the same for men and women. For ages between the ages given in the table, DNB suggests to interpolate.
Table 4.4: DNB factors fτ,1 for old age pension.
Age cohort Risk factors fτ,1
≤30 10% 35 9% 40 8% 45 7% 50 5% 55 4% 60 3% 65 2% 70 2% 75 2% 80 2% 85 1% >90 1%
In Table4.5the risk factors for the portfolios, described in Appendix B, are presented for the old age pension. The best estimate technical provisions BEτ,1 are obtained by
discounting the best estimate payments with a constant interest rate of 3%3. The risk factors f for the portfolios are calculated with
f =X
τ
fτ,1∗ BEτ,1/BE1. (4.2)
2
There are 5 pension types mentioned by DNB. Of the 5 pension types, 4 pension types are a combination of the retirement pension with a spouse pension.
3The method of modeling and discounting payments is described in Chapter 2
Table 4.5: DNB risk factors for the three portfolios. As expected from the risk factors of Table 4.4, the youngest fund has the highest risk factor.
Portfolio Risk factor f
Young 4.10%
Average 3.66%
One-year risk of lifelong mortality trend uncertainty - G.A.A. Witteman 21
References
Antonio, K., Devriendt, S., de Boer, W., de Vries, R., De Waegenaere, A., Kan, H., Kromme, E., Ouburg, W., Schulteis T., Slagter, E., van der Winden M., van Iersel, C., Vellekoop, M. (2017), ”Producing the Dutch and Belgian mortality projections: a stochastic multi-population standard.” European Actuarial Journal 7, 2017, 297-336.
Barrieu, P., Bensusan, H., El Karoui, N., Hillairet C., Loisel S., Ravanelli, C., and Salhi, Y. (2012), ”Understanding, modelling and managing longevity risk: key issues and main challenges.” Scandinavian actuarial journal, 2012, 203-231.
Boer, B.L., de Boer, W., van Iersel, C.A.M., de Mik, J., Plat, H. J., Schulteis T.J.W., Vellekoop, M.H., Werker, B.J.M., van der Winden M.R. et al. (2016), Prognosetafel AG2016. Koninklijk Actuarieel Genootschap.
Boonen, T.J., De Waegenaere, A., Norde, H. (2017), ”Redistribution of longevity risk: The effect of heterogeneous mortality beliefs.” Insurance: Mathematics and Eco-nomics 72, 2017, 175-188.
DNB (2017), Handreiking verzekeringstechnisch risico (S6). http://www.toezicht. dnb.nl/2/50-202316.jsp, consulted on November 2, 2017.
Gibbons, J and Chakraborti, S (1992). Nonparametric Statistical Inference. Marcel Dekker, 4 edition.
Koter, L. de, (2013) ”Het effect van een intern model voor langlevenrisico op het Vereist Eigen Vermogen van Pensioenfondsen.” Master scriptie Universiteit van Amsterdam. Lee, R. and Carter, L. (1992), ”Modeling and forecasting U.S. mortality.” Journal of
the American Statistical Association. 87, 659-671.
Li, N. and Lee, R. (2005), ”Coherent mortality forecasts for a group of populations: An extension of the lee-carter method.” Demography, 42, 575-594.
McNeil, A., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management: Con-cepts, Techniques and Tools. Princeton University Press.
Polman, F. M., Krijgsman, C., Dajani K., Hemminga, M. A. (2017), ”Modelling a Dutch Pension Fund’s Capital Requirement for Longevity Risk.” De Actuaris, May 2017. Polman, F. M. (2017) ”Calculating a Dutch Pension Fund’s Capital Requirement for
Longevity Risk.” Master scriptie Universiteit van Utrecht. Value at Risk https://en.wikipedia.org/wiki/Value_at_risk
Villegas, A. M., Millossovich, P., Kaishev, V. (2017), ”StMoMo: An R Package for stochastic Mortality Modeling.” Demography, 42, 575-594.
Wilmoth, J.R., Andreev, K., Jdanov, D., Glei, D.A., Riffle, T, et al. (2017) Methods Protocol for the Human Mortality Database (Version 6). http://www.mortality. org/.
Royal Dutch Actuarial Association (2016). Projection Table AG2016. https://www. ag-ai.nl/
