Forecasting the Inevitable Consequence of Life:
On the Implications of the Choice of a Mortality Model
for the Pricing of Annuities
Master’s Thesis Econometrics, Operations Research and Actuarial
Studies
Forecasting the Inevitable Consequence of Life:
On the Implications of the Choice of a Mortality Model for the
Pricing of Annuities
Shady el-Gewily
Abstract
Underestimation of future longevity can have adverse consequences for the providers of life benefits, such as an annuity. Therefore, mortality models that produce high quality forecasts and that can appropriately quantify the uncertainty associated with forecasts are important for the financial health of such institutions. The pricing of an annuity requires accurate mortality forecasts and associated levels of uncertainty. There is a wide array of mortality models available, which can produce substantially different point forecasts and levels of uncertainty associated with forecasts. The main objective of this thesis is to evaluate to what extent model choice affects the price of annuities and how the similarities and differences can be attributed.
Age-specific all-cause mortality rates for the England and Wales male population by single year of age are obtained from the Human Mortality Database. The models employed in this thesis are the Lee-Carter, Cairns-Blake-Dowd M7 and Hyndman-Ullah models and a two-dimensional kernel regression model proposed by Li et al. (2016). These models are calibrated on the England and Wales male mortality expe-rience encompassing the calendar years 1961-2004 and the pensioner ages 60-89. The calibrated models are first compared on the basis of a set of selected qualitative and quantitative criteria that reveal relative merits and idiosyncratic characteristics of the models. The models are then used to estimate the density of the random present value of several annuities. The estimated density functions are compared and simi-larities and differences analyzed with the insights derived from the model comparison.
Contents
1 Introduction 5
2 Methods and Data 8
2.1 Terminology, Notation and Assumption . . . 8
2.2 Mortality Models . . . 9
2.2.1 Lee-Carter under a Poisson Setting . . . 9
2.2.2 Cairns-Blake-Dowd M7 Extension . . . 10
2.2.3 Hyndman and Ullah . . . 11
2.2.4 Li, O’hare and Vahid . . . 13
2.2.5 Characteristics of the Mortality Models . . . 14
2.2.6 Modeling Mortality at Very Old Ages . . . 15
2.3 Description of the Data . . . 16
1
Introduction
Studies have revealed a persistent downward trend in mortality rates and near linear im-provements in life expectancy (McDonald et al., 1998; Oeppen and Vaupel, 2002; de Beer et al., 2017). The improvements in longevity, while good for society as a whole, have lead to a range of social, political, economic and regulatory challenges (Barrieu et al., 2012). In particular, improved longevity brings with it increased costs for social security and health care benefits as well as higher pension liabilities. One of the consequences has been an increased retirement age in some countries, such as the United Kingdom and the Netherlands. The improvements in longevity were underestimated by forecasts produced by researchers in the previous century, as pointed out by e.g., Wong-Fupuy and Haberman (2004) and Pitacco et al. (2009). When improvements in longevity are unanticipated the balance sheets of public retirement systems, private annuity providers and insurers are threatened, because they will have to pay out more in social security benefits and pen-sions than expected (International Monetary Fund, 2012). The risk of underestimating longevity, longevity risk, is thus an important factor in decision-making regarding pricing and reserving for pensions and life annuities.
Life annuities are an important investment product that offers retirees the opportunity to insure against the risk of outliving their assets. Annuities involve the transfer of longevity risk from the policyholder to the annuity provider. A life annuity provides a guaranteed income for the remainder of the policyholders’s lifetime in exchange for a lump-sum pay-ment. In order to ensure its financial health, the firm that sells the annuity has to ensure it is able to pay out its future liabilities. The firm can protect itself by setting annuity prices sufficiently high, ensure it has adequate financial buffers and cross-hedge or transfer part of the longevity risk to reinsurance or the financial markets. The price of an annuity is directly related to the descriptive statistics (i.e., the expected value, 90th or 95th per-centile) of the random present value of its future cash flows. The random present value is a function of future survival probabilities, which have to be accurately forecast by an appropriate mortality model.
Mortality models vary according to a number of important elements: forecasting method-ology, number of sources of randomness driving mortality improvements at different ages, assumptions of smoothness in the age and time dimensions, inclusion or not of year of birth effects and the calibration method. As a result, mortality models may produce substantial differences in forecasts, which are propagated to the annuity price. A number of studies have been performed to compare mortality models, both on quantitative (e.g., goodness-of-fit, forecasting performance) and qualitative aspects (e.g., parsimony, transparency, ability to produce simulated sample paths and forecast percentiles, plausibility of forecasts and prediction intervals).
Dowd et al. (2010a,b) carried out a range of formal, out-of-sample backtesting and goodness-of-fit tests for various mortality models. Cairns et al. (2009) compared eight stochastic mortality models based on their general characteristics and ability to explain historical patterns of mortality. Cairns et al. (2011) introduced a number of additional qualitative criteria focused on the plausibility of forecasts and prediction intervals and compare six stochastic mortality models. It is argued that the proposed qualitative criteria are impor-tant because a model might perform well in terms of quantitative performance, yet produce forecasts exhibiting features which are clearly implausible. Other comparative studies are e.g., Booth et al. (2006); Hyndman and Ullah (2007); Plat (2009) and Li et al. (2016). There have also been several studies that discuss and evaluate the impact of model choice on the pricing of annuities. These studies primarily focus on quantitative aspects, see e.g., Brouhns et al. (2002b); Yang et al. (2010). The exceptions seem to be Cairns et al. (2009) and Cairns et al. (2011), which also attribute differences and similarities using insights obtained from their proposed qualitative criteria. The criteria put forth by Cairns et al. (2009, 2011) provide a framework to attribute similarities and differences in annuity prices. The numerous available mortality models and the differences in their characteristics in-spires the research question:
Research Question 1. To what extent does model choice affect the price of annuities and how can the similarities and differences be attributed?
Based on the forecasts produced by the selected mortality models, the density function of the annuity present value is estimated. Annuity prices are calculated using three common pricing strategies, namely using the mean and the 90th and 95th percentiles of the present value. The annuity prices will be compared quantitatively to shed light on the economic significance of model risk when pricing annuities. Moreover, the qualitative criteria put forth by Cairns et al. (2009) and Cairns et al. (2011) assist in attributing similarities and differences between annuity present values produced by different mortality models. This not only sheds light on the extent that annuity prices are similar or different, but also how such similarities and differences are generated.
This thesis adds to the previous literature in three ways. To my knowledge, the crite-ria put forth by Cairns et al. (2009) and Cairns et al. (2011) have not been applied to the Hyndman and Ullah (2007) and Li et al. (2016) models. Furthermore, in these studies the age effects are not extrapolated to include ages above 89. Given the recent improvements in longevity, it seems appropriate to include ages 90+ in order to avoid underestimation of annuity prices. In this thesis, the Kannist¨o (1994) law of mortality is used to extrapolate age effects to the age range 90-120. Finally, it appears that an implementation of the two-dimensional kernel model was not publicly available. An implementation in R is made available alongside this thesis, which facilitates replication and can assist in future research that makes use of this model.
2
Methods and Data
Section 2.1 introduces the terminology, notation and assumption for fractional ages and durations, which are used throughout this thesis. A more formal mathematical treatment can be found in Pitacco et al. (2009). Section 2.2 gives a description of the mortality models applied in this thesis. Section 2.3 gives a description of the data used in the empirical analysis. The research method used to answer the research question is presented in Section 2.4.
2.1
Terminology, Notation and Assumption
The one-year death probability qx,t denotes the probability that an individual of exact age
x at time t, dies before reaching age x + 1, where x, t ∈ R. The corresponding one-year survival probability is px,t = 1 − qx,t. The force of mortality µx,t represents the
instanta-neous rate of mortality at a given age x and time t. The behaviour of the force of mortality on the age interval (x, x + 1) and the time interval (t, t + 1) can be summarized by the (central) mortality rate mx,tat age x and time t. The central mortality rate is the weighted
arithmetic mean of the force of mortality over the area defined by the intervals (x, x + 1) and (t, t + 1), the weighting function being the probability of being alive at age x + u and time t + s, 0 < u, s ≤ 1.
Mortality statistics are generally published for integer ages and calendar years only. There-fore, an assumption for fractional values of x and t is needed. Throughout this thesis, a piece-wise constant force of mortality is assumed, which is frequently adopted in actuarial calculations. This assumption entails that the force of mortality remains constant over squares of age and time, but allowed to vary from one square to the next. Specifically, we assume that
Assumption 1. µx+ξ1,t+ξ2 = µx,t for 0 ≤ ξ1 < 1 and 0 ≤ ξ2 < 1.
Under assumption 1, we have for integer age x and calendar year t that qx,t = 1 − exp[−µx,t], and µx,t = mx,t,
see Pitacco et al. (2009) for a proof. From now on, age x and calendar year t are meant to be integer values. The fact that µx,t = mx,t under Assumption (1) is useful because
mortality rates are much easier to estimate from observed mortality statistics than forces of mortality.
The initial exposure to risk refers to the total number of individuals aged x alive at the start of the calendar year t. The central exposure-to-risk at age x last birthday during year t, denoted by CETRx,t, refers to the total time lived by people aged x last birthday in
calendar year t. Empirical mortality rates can be estimated from observed data as follows: mx,t =
dx,t
CETRx,t
,
2.2
Mortality Models
2.2.1 Lee-Carter under a Poisson Setting
The method proposed by Lee and Carter (1992), henceforth LC, assumes the following log-bilinear model for the age-specific force of mortality µx,t,
log µx,t = αx+ βxκt. (1)
The age effects are captured by the sequences αx and βx, whereas the calendar year effects
are captured by the single time index κt. The time index κt is modelled as a multiplicative
factor and estimated together with the βx. The age effect αx is assumed to be constant
over time. Since there are no known covariates on the right hand side of (1), the param-eters cannot be obtained by using ordinary regression techniques. Lee and Carter (1992) proposed to obtain a least-squares solution by using the first element of a singular value decomposition (SVD) in combination with a set of identifiability constraints. Once the parameters have been estimated, a time-series model for κt is specified and forecasts of
mortality rates can be obtained. The SVD calibration method requires the assumption that the random errors are homoskedastic. This is an unrealistic assumption, because the logarithm of the observed mortality rates is much more variable at older ages than at younger ages due to the much smaller number of exposure-to-risk, and thus death numbers, observed at high ages.
To allow for heteroskedastic errors, alternative calibration techniques have been inves-tigated in e.g., Brouhns et al. (2002a,b). Instead of an additive error structure, deaths are assumed to be independently distributed as Poisson random variables. The Poisson assumption on the random number of deaths at age x in calendar year t, Dx,t, implies
Dx,t ∼ Poisson CETRx,tµx,t.
The method assumes the same log-billinear model for the force of mortality µx,t specified
in equation (1). By virtue of Assumption (1) the model can be calibrated using mortality rates mx,t.
The parameters αx, βx and κt are calibrated by Maximum Likelihood Estimation (MLE).
The parameters are only identifiable up to a set of parameter constraints, which need to be specified. Following Lee and Carter (1992), the following constraints are imposed:
X x βx = 0 and X t κt = 0.
These constraints imply that the sequence of αx’s describes the time averages of the
age-pattern of log mx,t for ages x. The time index κt describes the change in the level of
mortality rates over time. The βx profile describes which mortality rates decline rapidly
To obtain forecast logarithms of age-specific mortality rates, an appropriate ARIMA model is specified for the calibrated calendar year effects ˆκt. Following Lee and Carter (1992),
the time index κt is modelled as a random walk with drift. Then for future calendar years,
point forecasts and prediction intervals of logarithms of mortality rates can be obtained by applying the forecasting procedure described in the Appendix. For ease of notation, we use the mathematical notation ˆmx,t for both the estimated mortality rates and the forecast
mortality rates.
2.2.2 Cairns-Blake-Dowd M7 Extension
The Cairns-Blake-Dowd model (henceforth, CBD) proposed by Cairns et al. (2006) models changes in logit-transformed age-specific one-year death probabilities. It is appropriate for pensioner ages only, say x > 60. The model is based on the empirical observation that the logit-transformed death probabilities, log qx,t
1−qx,t, are reasonably linear in x for this age
range. In contrast to the LC method, the CBD model treats age as a continuous covariate and incorporates two time indices for the calendar year effects. Empirical studies have shown that death probabilities have an imperfect correlation at different ages from one year to the next. A model coherent with this observation requires a minimum of two time indices.
Sometimes the logit-transformed death probabilities exhibit a slight curvature after the retirement age. This curvature can be captured by including a quadratic age term. For some countries, individuals with the same year of birth experience share similar mortality patterns (see e.g., Willets (2004), MacMinn et al. (2005) and Richards et al. (2006)). For these countries, age and calendar year effects are not sufficient and cohort effects must be incorporated. The CBD model can be generalized to include both a quadratic age term and a cohort effect, see Cairns et al. (2009). This specification (henceforth, CBD M7) assumes the following model for age-specific death probabilities:
logit qx,t = κ (1) t + κ (2) t (x − ¯x) + κ (3) t (x − ¯x)2− ˆσx2 + γc, (2)
where κ(1)t , κ(2)t and κ(3)t are the calendar year effects, γc is the cohort effect and ˆσx2 is the
mean value of (x − ¯x)2. The age effects assume a functional form and do not have to be estimated from historical data. This implies that the model produces estimated and forecast one-year death probabilities that progress smoothly with age.
The CBD M7 model can be calibrated using a variety of statistical methods, such as OLS or Maximum Likelihood, see e.g., Pitacco et al. (2009). In this thesis, the approach of Villegas et al. (2015) is followed and death counts Dx,t are assumed to be independent
and distributed as a Binomial random variable,
Dx,t ∼ Binomial(IETRx,tqx,t),
Following Villegas et al. (2015), the initial exposures-to-risk are obtained by using the ap-proximation IETRx,t ≈ CETRx,t+ 0.5dx,t, with dx,t the observed death counts. The model
is calibrated on empirical one-year death probabilities, calculated as qx,t = dx,t
IETRx,t.
The parameters are only identified up to a transformation. Cairns et al. (2009) suggested the following parameter constraints to ensure identifiability:
X c γc= 0 and X c cγc= 0 and X c c2γc= 0,
where the summation is over all observed cohorts c. These constraints ensure that the cohort effect γc fluctuates around zero and has no detectable linear or quadratic trend.
Forecasts of the future death probabilities ˆqx,t+h can be obtained by specifying time-series
dynamics for ˆκ(1)t , ˆκ(2)t , ˆκ(3)t and ˆγc. The calibrated calendar year effects ˆκ (1) t , ˆκ (2) t and ˆκ (3) t
are modelled as a multivariate random walk with drift, which is the standard approach in the literature, see e.g., Cairns et al. (2006, 2011) and Haberman and Renshaw (2011). Following Cairns et al. (2011) an AR(1) processs with a constant is used for the cohort index, which is independent from the dynamics of the time indices. The assumption of in-dependence of the dynamics of the period and cohort indices follows previous studies (e.g., Renshaw and Haberman (2006) and Continuous Mortality Investigation Bureau (2006)). Sample paths of future mortality rates can then be simulated and prediction intervals can be obtained. Details of the forecasting procedure are presented in the Appendix. For ease of notation, the mathematical notation ˆqx,t is used for both the estimated one-year death
probabilities and the forecast one-year death probabilities. 2.2.3 Hyndman and Ullah
Essentially, the HU method assumes that there is some underlying smooth function ft(x)
that is observed with error, relating the logarithms of mortality rates to age. To obtain ft(x), age-specific mortality rates mx,t are first smoothed using weighted penalized
regres-sion splines (Wood, 2003), for each calendar year t separately. This smoothing method is univariate and non-parametric in nature and it allows for monotonicity constraints. The resulting mortality rates progress smoothly over age and are monotonically increasing after some age threshold.
Once the smoothed mortality rates ft(x) have been obtained, they are decomposed
us-ing a basis function expansion usus-ing the followus-ing model:
log(fx,t) = θx+ K
X
k=1
βt,kφx,k + εx,t,
where θx is a measure of location of fx,t, {φx,k} is a set of orthonormal basis functions
with corresponding coefficients {βt,k} and εx,t ∼ N (0, v(x)). Robust principal components
(Robust PCA) is used to estimate the age effects θx and {φx,k}, to avoid difficulties with
outlying years. The calendar year effects βt,k are not estimated robust, so that any
out-lying years will be modelled by outliers in the time series. There are several methods to obtain robust estimates of {φx,k}, as described in Hyndman and Ullah (2007). The hybrid
algorithm proposed by Hyndman and Ullah (2007) is employed, which combines the best features of the available methods, making it an efficient and robust method for obtaining the basis functions {φx,k}. The model parameters are fully identified without imposing
parameter constraints.
Hyndman and Ullah (2007) proposed to select the number of basis functions K by mini-mizing the Integrated Squared Forecast Error (ISFE). Since then, Hyndman et al. (2017) have recommended to choose a K that is more than enough. The argument is that fore-casting performance is not affected by using too many basis functions, but using too few basis functions has adverse effects on forecasting performance.
Because the calibration method yields basis functions {φx,k} that are mutually
uncor-related, time-series for {βt,k} can be modelled independently. This greatly simplifies the
forecasting procedure. The calibrated calendar year effects { ˆβt,k} are modelled as
indepen-dent robust ARIMA models, which allows the fitted ARIMA models to contain outliers of various types so that unusual observations do not contaminate the forecasts (Chen and Liu, 1993). The ARIMA models are chosen optimally on the basis of the corrected ver-sion of the Akaike Information Criterion (AICc). Point forecasts, simulated sample paths and prediction intervals of future mortality rates ˆmx,t can be obtained using the approach
2.2.4 Li, O’hare and Vahid
Li et al. (2016) proposed a two-dimensional local constant kernel regression (henceforth, 2D KS) for the logarithms of age-specific mortality rates. This model allows for smoothness over age, time and cohort. The assumed model is as follows:
log(mx,t) = Γx,t+ εx,t,
where Γx,t is an unknown smooth function of age x and time t and εx,t random errors.
Kernel regression is a non-parametric regression method, which means that in contrast to parametric regression methods, no functional form is specified a priori. The idea behind local constant kernel regression is that Γx,t is assumed continuous and can be approximated
by a constant over a local neighbourhood around (x, t). This boils down to the estimate ˆ
Γx,t being a weighted average of the observed values of log(mx,t) in a local neighbourhood
around (x, t).
To obtain the smooth surface ˆΓx,t, the following weighted least squares minimization
prob-lem is solved for x ∈ [0, 1] and t ∈ [0, 1],
N X i=1 T X j=1 log(mxi,tj) − Γx,t 2 Kh1,h2,ρ(xi− x, tj − t), (3)
where the summation is over all ages and calendar years used to calibrate the model. The ages xi, i = 1, ..., N, and calendar years tj, j = 1, ..., T , are normalized so that they span
the interval [0,1]. The function Kh1,h2,ρ(·) is the bivariate normal kernel function and its
role is to serve as the weighting function. The solution to the minimization problem (3) at (x, t) is given by the Nadaraya-Watson estimator (Nadaraya, 1964; Watson, 1964):
ˆ Γx,t = PN i=1 PT j=1Kˆh1,ˆh2, ˆρ(xi− x, tj − t) log(mx,t) PN i=1 PT j=1Kˆh1,ˆh2, ˆρ(xi− x, tj − t) .
The bivariate normal kernel function is dependent on three bandwidth parameters, h1 > 0,
h2 > 0 and ρ ∈ [0, 1), which determine the degree of smoothing in the age, time and cohort
dimensions. One of the benefits of capturing cohort effects as a smoothness parameter is that it allows for the assessment of the strength of cohort effects between countries, see Li et al. (2016).
bandwidth parameters are then found by the solution to: (ˆh1, ˆh2, ˆρ) = argmin h1,h2,ρ 1 5N N X i=1 T X j=T −4 log mx,t− ˆΓx,t 2 .
Once the smoothed surface ˆΓx,t is estimated, one-step ahead forecasts can be obtained
sequentially. The proposed method of forecasting is deterministic in nature, which implies that simulated sample paths and prediction intervals that properly reflect the stochas-tic nature of mortality are hard to obtain. A more detailed discussion of the forecasting procedure can be found in the Appendix.
2.2.5 Characteristics of the Mortality Models
The mortality models described in Sections 2.2.1-2.2.4 differ in a number of ways. To provide an overview of these differences, the general characteristics of the various mortality models are summarized in Table 1.
Element LC CBD M7 HU 2D KS
Forecasting methodology Stochastic Stochastic Stochastic Deterministic
Effects Age, Calendar year Age, calendar year, cohort Age, calendar year Age, calendar year, cohort No. of sources driving
mortality improvements at different ages
1 4 4 1
Assumptions of smoothness None Age Age
Age, calendar year,
cohort Inclusion of cohort effects No Yes No Yes
2.2.6 Modeling Mortality at Very Old Ages
In order to prevent the underestimation of annuity prices, accurate forecasts of mortality for ages beyond 89 are required. Empirical mortality rates exhibit high variability in this age segment, due to the scarcity of lives at the advanced ages. Therefore, a method that extrapolates mortality rates to higher ages is required. The standard approach in the lit-erature is to use some parametric law of mortality.
In a comparative analysis by Antonio (2012), the mortality law by Kannist¨o (1994) is chosen as the best among a set of various extrapolation approaches, at least for the Bel-gian population. The mortality law of Kannist¨o (1994) models in a particular calendar year t, the force of mortality as:
µx,t =
ceωx
1 + ceωx, (4)
where the parameters c and ω have to be estimated from mortality data in the calendar year t. The Kannist¨o (1994) model is a parametric law of mortality of the logistic type, having the property that as age x tends to infinity, the force of mortality converges to 1. The age range used to calibrate the Kannist¨o model has a slight effect on the extrapolated rates and an appropriate choice must be made. Following Antonio (2012), the Kannist¨o model is calibrated to the age range 75-89 and extrapolated to ages 90-120. The same extrapolation method is used for the LC, CBD M7, HU and 2D KS models to remain as consistent as possible.
The parameters c and ω can be estimated by means of non-linear (weighted) least squares regression or by maximum likelihood (see e.g., Gavrilova and Gavrilov (2014) and Thatcher et al. (1998)). In the monograph by Doray (2008), it is shown that a reparametrized version of equation (4) makes it possible to estimate the parameters using ordinary or weighted least squares. One of the benefits of estimating the parameters by least squares is the ease of implementation. Following Doray (2008), the parameters c and ω are calibrated for each calendar year t separately by rewriting equation (4) as the linear model
logit(µx,t) = ˜c + ωx + εx,t,
where εx,t is a random error. Assuming homoskedastic error terms, the parameters c and
2.3
Description of the Data
The data consists of recorded death numbers and central exposures-to-risk for the Eng-land and Wales male population, which are provided by the Human Mortality Database (HMD)1 and are publicly available. The England and Wales male population is chosen
because a lot of literature on this population is available, which facilitates comparison of the results in this thesis with the established literature. The data encompasses the period 1861-2014 and age range 0-109. The recorded death numbers and central exposures-to-risk are used to calculate empirical unsmoothed all-cause mortality rates and one-year death probabilities by single year of age.
Figure 1 shows the log-transformed empirical mortality rates log mx,t in the calendar years
1961, 1990 and 2014. Each curve gives a snapshot of the mortality experience in a partic-ular year. In each year, the mortality experience has the characteristic shape describing a typical mortality curve over age. Mortality rates are comparatively high in the first year after birth and decrease rapidly to a minimum at around age 10. Thereafter, mortality rates increase in an approximately exponential fashion, before decelerating at the end of the life span. The excess mortality observable in the so-called accident hump (ages 18-25) is primarily caused by accidents, injuries and suicides (Pitacco et al., 2009). The mortality curves show an erratic variation at the highest ages, which is caused by relatively large sampling errors due to small exposure sizes at this age range. While there are marked improvements in the mortality rates over time for all ages, the strength of improvement differs with age. Trends in one-year death probabilities are very similar and are not shown to save space.
Figure 1: Logarithms of observed empirical mortality rates log mx,t for calendar years
1961, 1990 and 2014 and age range 0-89.
Only part of the data is used to calibrate the selected mortality models, namely the period between 1961-2004 and the age range 60-89 years old. The age range starts at age 60, because the mortality forecasts are used to price annuities, which are generally applicable to retirees only. A second reason is that the CBD M7 model is applicable only to ages above 60. Due to contamination of mortality data by noise at the advanced ages (say x > 89), the mortality models are calibrated to the maximum age of 89. The Kannist¨o (1994) mortality law is used to extrapolate to higher ages. The calibration period 1961-2004 is chosen to be in line with Cairns et al. (2009) and Cairns et al. (2011). This period does not encompass the Second World War and the influenza pandemic of 1957, rare events that do not reveal structural risks to the annuity provider but have a large impact on forecasts. The last ten years of data (2005-2014) are not used for model calibration, but are used to assess the out-of-sample forecasting performance of the various mortality models.
2.4
Research Method
The objective of this thesis is to answer the research question:
To what extent does model choice affect the price of annuities and how can the similarities and differences be attributed?
The price of an annuity is directly related to summary statistics (i.e., the expected value, 90th or 95th percentile) of the random present value of its future cash flows. The random present value of an annuity that pays one euro at the end of every calendar year conditional on the survival of an annuitant aged x in calendar year t is defined as:
ax,t = X k≥0 k Y j=0 px+j,t+j vk+1. (5)
The formula for the random present value depends on a set of future survival probabilities {px,t+h} and a discount factor v. The discount factor is equal to the inverse of the expected
interest rate, which is set to i = 0.02 = 1v for convenience. The future survival probabilities are unknown and need to be forecast. These forecasts are provided by the the LC, CBD M7, HU and 2D KS models.
Applying formula (5) using point forecasts of survival probabilities yields accurate esti-mates of ax,t only if future mortality evolves according to the point forecast with certainty.
This is clearly unrealistic, because there is uncertainty about the future evolution of mor-tality. To properly reflect the stochastic nature of mortality, formula (5) needs to be considered under many different sample paths that are likely to occur. When the mor-tality model allows it, simulated sample paths can be used to estimate the entire density function of ax,t rather than just a point estimate. To evaluate the implications of model
risk on annuity prices, three common pricing strategies are considered:
1. Pricing the annuity without any safety loading, i.e., the mean value of ax,t is used.
2. A safety loading is added by setting the price of the annuity to equal the 90th percentile of ax,t.
3. A safety loading is added by setting the price of the annuity to equal the 95th percentile of ax,t.
Section 2.2 has made clear that the mortality models differ in a number of ways. This implies that forecasts produced by the various models might differ substantially as a re-sult, which clearly impacts the estimated density function of ax,t and its price. In order to
explain the similarities and differences in annuity prices produced by the various mortality models, a more formal model comparison is needed.
To evaluate the quality of forecasts, many evaluation criteria on both qualitative and quan-titative aspects have been suggested in the literature, see e.g., Cairns et al. (2008, 2009, 2011), Plat (2009), Dowd et al. (2010a,b), Haberman and Renshaw (2011) and Cairns et al. (2011). A particular model may outperform alternative models both in terms of in-sample goodness-of-fit and out-of-sample forecasting performance, yet produce implau-sible forecasts or prediction intervals (Cairns et al., 2011). Therefore, it is important to study and compare the mortality models both qualitatively and quantitatively. The cri-teria suggested in the literature provide a framework to understand and explain potential similarities and differences that might arise in annuity prices. Not all of the criteria put forth in the literature contribute to explaining the potential similarities and differences between annuity prices to the same extent. Therefore, only the criteria that contribute the most in answering the research question are considered. The selected criteria are:
• The model should have high goodness-of-fit to the calibration sample. • Residuals by age, calendar year and year of birth should be pattern-free. • At least for some countries, the model should incorporate a cohort effect. • The model should be transparent.
• Point forecasts should be plausible and consistent with historical trends. • Point forecasts should have good out-of-sample forecasting performance.
• The structure of the model should make it possible to incorporate parameter uncer-tainty in simulations.
• Forecast levels of uncertainty should be plausible and consistent with observed vari-ability in mortality data.
A good mortality model is consistent with historical mortality data and captures all the relevant features of the mortality experience. For this reason, the models are evaluated in terms of goodness-of-fit and residual patterns and it will be investigated whether or not a model should allow for cohort effects. Unrealistic model output might be traced back to the calibrated model parameters, which is why it is important that a mortality model is sufficiently transparent to interpret the calibrated parameters. The produced point fore-casts should display no obviously implausible behaviour (such as a kink, an increasing rather than a decreasing trend, or predetermined convergence to a constant value) and has to perform well in terms of out-of-sample forecasting performance. To obtain the complete density function of a life annuity present value, it is necessary that sample paths and ac-curate predictions intervals can be produced. Such prediction intervals should take into account all sources of uncertainty, including parameter uncertainty. A mortality model that is not parsimonious might produce prediction intervals that are implausibly wide, due to the estimation of many parameters in combination with a short time series. It is impor-tant that prediction intervals do not consistently underestimate levels of uncertainty for certain age groups. Therefore, prediction intervals should be both plausible and consistent with observed variability.
The method used to evaluate goodness-of-fit and forecasting performance requires some choices. Performance can be evaluated on the basis of survival probabilities, mortality rates, life expectancy or annuity prices, depending on the purpose (Dowd et al., 2010a). In this thesis, logit-transformed mortality rates are chosen as the evaluation measure, be-cause three out of four models provide forecasts of mortality rates directly. The logit rather than the log transformation is chosen because the observed variability in mortality rates is larger on the logit scale which suggests that it provides a more vivid picture of relative goodness-of-fit and forecasting performance between mortality models. As an evaluation of goodness-of-fit and out-of-sample forecasting performance, the mean squared error (MSE), mean percentage error (MPE) and mean absolute percentage error (MAPE) of observed versus fitted logit-transformed age-specific mortality rates are calculated.
3
Empirical Results
This Chapter presents the results of the empirical study. The results of the model compar-ison are discussed in Section 3.1. Section 3.2 evaluates and compares the density functions of the annuity present values produced by the various mortality models and the resulting annuity prices.
3.1
Model Comparison
Figure 2 shows the observed versus fitted log-transformed mortality rates log ˆmx,t for the
LC, CBD M7, HU and 2D KS models. The left and center panel show the evolution of log-transformed mortality rates at age 65 and age 85 over the period 1961-2004. The four models produce similar log-transformed age-65 mortality rates, but the center panel shows that log mortality rates at age 85 are consistently underestimated by the CBD M7 model. The right panel shows observed and fitted log-transformed mortality for the ages 60-89 in the year 2004. It appears that the mortality rates at the advanced ages are comparatively low for the CBD M7 model. This can be attributed to the underestimated mortality rates around age 85 for this model, which are part of the age range used to calibrate the Kan-nist¨o (1994) law of mortality.
The model should have high goodness-of-fit to the calibration sample.
To evaluate the goodness-of-fit of each mortality model, the MSE, MPE and MAPE cor-responding to the logit-transformed mortality rates log mˆx,t
1− ˆmx,t are calculated on the age
range 60-89 and time period 1961-2004. The results are shown in Table 2. All four models obtain great in-sample fit, having at most mean absolute percentage error of 1.757%. The 2D KS model attains the best goodness-of-fit, whereas the LC method performs relatively weakest in-sample. It is no surprise that the HU and CBD M7 models outperform the LC model in-sample, because these models consist of more parameters and the MSE, MPE and MAPE do not penalize for the number of parameters.
Model MSE MPE MAPE
Period 1961-2004
LC 0.027 -1.073 1.757 CBD M7 0.0159 -0.597 0.919 HU 0.079 -0.786 1.158 2D KS -0.019 -0.369 0.593
Table 2: Goodness-of-fit of the LC, CBD M7, HU and 2D KS models based on the cali-bration period 1961-2004.
Figure 3: The scaled deviance residuals produced by the LC model. From left to right: by age, calendar year and year of birth.
Figure 5: The unstandardized residuals produced by the HU model. From left to right: by age, calendar year and year of birth.
At least for some countries, the model should incorporate a cohort effect. The observed patterns in the residuals by year of birth produced by the LC and HU models imply that cohort effects are present in the mortality data, which are not captured appropriately by these models. It appears that cohort effects are an important feature of the England and Wales male population in the period under study. This has been estab-lished in the literature, see e.g., Government Actuary’s Department (1995, 2001, 2002); Continuous Mortality Investigation Bureau (2002); Willets (2004); MacMinn et al. (2005); Richards et al. (2006). In particular, Willets (2004) and Richards et al. (2006) found that the largest improvements in mortality rates in England and Wales have been consistently experienced by individuals born between 1925 and 1945 (centered around 1930). Several explanations of this cohort effect have been suggested, namely smoking behaviour, diet in early life and prenatal conditions, see e.g., Willets (2004) and Gavrilov and Gavrilova (2004). Thus, for the England and Wales male population, mortality models should incor-porate a stochastic cohort effect to capture all relevant features of the mortality experience. The model should be transparent.
The calibrated parameters ˆαx, ˆβx and ˆκt for the LC method are shown in Figure 7. It can
be observed that the fitted sequence of ˆαx’s is nearly linear and monotonically increasing
with only slight curvature. Under the identification constraints that are imposed, the se-quence ˆαx reflects the time averages of log mx,t for ages x. The fitted ˆαxprofile implies that
the overall level of mortality is increasing with age, under the LC method. The estimated calendar year effect ˆκt decreases quite linearly over time, reflecting the declining trend of
observed mortality over time. The ˆβx profile indicates that improvements in mortality
rates are declining in age for ages 60-89.
The calibrated calendar year effects for the CBD M7 method are shown in Figure 8. The calendar year effects ˆκ(1)t , ˆκ(2)t and ˆκ(3)t affect logit-transformed death probabilities at different ages in different ways. The first time index ˆκ(1)t exhibit a downward trend, which expresses the improvement over death probabilities over time for all ages. The second and third time indices modulate the improvements in age-specific logit-transformed death probabilities over time, according to the quadratic function of age specified in equation (2). The estimated cohort effect ˆγt−x can be analyzed to determine which cohorts
experi-enced comparatively high and low death probabilities. For instance, the sequence of ˆγt−x’s
suggests that males born around the year 1910 have experienced consistently higher death probabilities than generations born both earlier and later than 1910. The peak observed in ˆγt−x for the oldest cohorts might be attributed to the fact that there are only few
Figure 9 shows the calibrated location function ˆθx and the K = 4 basis functions ˆφx,k along
with the respective coefficients ˆβt,k. The fitted location function ˆθx describes the mean age
pattern and suggests that the logarithms of mortality rates are monotonically increasing with age for the age range 60-89. The first fitted basis function ˆφx,1 models changes in
mortality over time for the entire age range, but the sensitivity to changes in the calendar year effect ˆβt,1 is decreasing in age. The second basis function ˆφx,2 primarily models the
differences in mortality for males between ages 60-74 and those aged 75-89, which can be seen from the fact that ˆφx,2< 0 for ages below 75 and ˆφx,2> 0 for older ages. Differences
between ages 65-80 and those younger or older than this age segment are captured by the third basis function ˆφx,3. The ˆφx,4is harder to interpret, but the peaks and troughs indicate
which ages are most sensitive to changes in ˆβt,4. The calibrated calendar year effects ˆβt,1
suggest that mortality rates have declined for all ages. The remaining ˆβt,k sequences show
an erratic evolution over time, making it hard to interpret the differences in mortality at various ages over time.
For the 2D KS model, the only parameters available are the optimal bandwidths, ˆh1 =
0.014, ˆh2 = 0.0132 and ˆρ = 0.554. These parameters do not lend themselves well to
inter-pretation of age or calendar year effects. Given the relative size of the calibrated cohort bandwidth ˆρ = 0.554 compared to the values reported in Table I of Li et al. (2016) for various countries, it can be concluded that there is a comparatively strong cohort effect present in the observed mortality data.
Point forecasts should be plausible and consistent with historical trends. A mortality model may produce point forecasts that are clearly unrealistic. The unre-alistic features of a forecast may only become visible for forecasting horizons exceeding the time period that is used in a quantitative back test. Therefore, it is important to evaluate the point forecasts visually, to determine whether the forecasts are plausible and are consistent with historical trends. Figure 10 shows the observed mortality rates mx,t
and the point forecasts at ages 60, 75, 85 and 95 produced by the various models. The first observation that can be made is that point forecasts produced by all models are consistent with historical trends. The point forecasts adequately follow the differing rate of decline at different ages. The models produce forecasts of mortality rates that decline faster at age 85 than at age 65, implying that these mortality rates tend to converge in future years. The point forecasts produced by the LC and HU models are reasonably smooth and show no sudden changes or kinks and therefore pass the plausibility criterion. For the CBD M7 model, some ‘wobbly’ behavior in the point forecasts at all ages can be observed. The point forecasts are linked to the estimated cohort effect and the wobbly behavior occurs in regions where the mortality rate is still influenced by the estimated cohort effect. From the period that the mortality rates are only dependent upon the (smooth) forecasts of the cohort effect, the progression no longer shows the wobbly behavior. Cairns et al. (2011) observed similar features of the point forecasts and concluded that these can still be consid-ered plausible. However, the age-95 mortality rates produced by the CBD M7 model show that a very pronounced kink can be observed around 2030. This feature of the forecast is clearly implausible. A robustness check showed that the kink disappears when calibrating the CBD M7 model on the time period 1981-2004, see the Appendix.
Figure 10: point forecasts of ˆmx,t produced by the various models at ages x = 60, x = 75
and x = 85 and x = 95. Note that the vertical axis is plotted on a log scale. Crude mortality rates mx,t for 1961-2004 are shown as black dots.
The model should have good out-of-sample forecasting performance.
The models are calibrated to the ages 60-89 and calibration period 1961-2004 and are then used to produce forecasts for the periods 2005-2009 and 2005-2014. Figure 11 shows the forecast logit-transformed mortality rates log mˆx,t
1− ˆmx,t for ages 65, 75 and 85 produced
by the LC, CBD M7, HU and 2D KS models. The logit-transformed observed mortality rates are also shown. It seems that there is a tendency for the models to overestimate mortality rates, which is least pronounced for the 2D KS model.
Figure 11: Point forecasts of log ˆmx,t and 95% prediction intervals produced by the LC
model at ages x = 65, x = 75 and x = 85. Empirical log mortality rates log mx,t for
1961-2004 are shown as black dots.
Model MSE MPE MAPE
Period 2005-2009 LC 1.877 -2.347 2.856 CBD M7 1.484 -1.606 2.311 HU 1.826 -2.085 2.577 2D KS -0.161 -1.039 1.593 Period 2005-2014 LC 3.039 -3.385 4.149 CBD M7 2.221 -2.297 3.417 HU 1.914 -2.442 2.969 2D KS -0.365 -1.257 1.734
It should be possible to use the model to generate sample paths and calculate prediction intervals.
The LC, CBD M7 and HU models use stochastic time series processes to model the evo-lution of mortality over time. These models allow for the generation of sample paths by simulating forecasts using the fitted time-series. Quantiles of these simulations can be used to obtain prediction intervals that properly reflect the stochastic nature of mortality over time. The 2D KS model uses a deterministic forecasting method which neglects the random nature of mortality. This implies that sample paths cannot be generated and that prediction intervals that are clearly too narrow.
The structure of the model should make it possible to incorporate parame-ter uncertainty in simulations.
For all four models it is possible to incorporate parameter uncertainty. For the LC, CBD M7 and 2D KS models, bootstrapping methods are employed. Parameter uncertainty for the HU model is incorporated by assuming normally distributed sources of error, which can be used to generate sample paths and calculate prediction intervals analytically. The details of the exact method used to incorporate parameter uncertainty are given in the Appendix.
The model should be relatively parsimonious.
The number of effective parameters for each model is shown in Table 4. The effective number of parameters takes account of the constraints on parameters for the LC and CBD M7 models. For the LC, CBD M7 and 2D KS models, the number of effective parameters is fixed for a particular calibration sample. The number of effective parameters used in the HU model depends on the choice of K, the number of basis functions. Following Hyndman and Booth (2008), K is chosen to be sufficiently large, namely K = 4.
LC CBD M7 HU 2D KS Effective of parameters 102 202 210 3
Forecast levels of uncertainty should be plausible and consistent with observed variability in mortality data.
Cairns et al. (2011) calculated empirical volatilities for historical mortality rates for the England and Wales male population. These authors defined the volatility νx at age x
by the empirical standard deviation of δ1962,x, ..., δ2004,x, with δt,x = log mx,t − log mx,t−1.
Historical volatilities thus obtained are shown in Figure 12. For the England and Wales male population, the historical volatilities vx are generally increasing with age.
Figure 13 shows point forecasts and fan plots of mortality rates ˆmx,t produced by the
various models. Each fan chart depicts the 50%, 80% and 95% prediction intervals for mortality rates produced by the various mortality models. They are plotted on a logarith-mic scale for ages x = 65, x = 75 and x = 85 so that the forecast levels of uncertainty can be compared with historical variability, as shown in Figure 12.
It can be observed that the LC model produces prediction intervals that are wider at age 65 than at age 85. This is inconsistent with the greater observed volatility in age-85 mortality rates between 1961 and 2004. This observation can be explained by the fact that the LC model has only one period index κt, which implies that the prediction intervals
are proportional to the estimated age effects ˆβx. Cairns et al. (2011) deemed that such
observations make the prediction intervals implausible. In contrast, the CBD M7 model does show prediction intervals that become progressively wider with age, reflecting the observed volatility.
Looking at the prediction intervals for the HU model several observations can be made. The prediction intervals are increasing with age, but expand very rapidly and without limit. Cairns et al. (2011) deemed similarly rapid expansions of prediction intervals produced by other models as implausible. The fact that there is so much uncertainty associated with forecast mortality rates produced by the HU model can be attributed to two factors. The first is that the model is relatively non-parsimonious; it requires four time series mod-els (see Table 1). The second factor is that the time series modmod-els are selected using an automatic selection procedure, which lead to high order ARIMA models that have to be estimated using a short time series.
Figure 13: Point forecasts of ˆmx,t and fan plots depicting the uncertainty associated with the forecast, produced by the various
models at ages x = 65 (green), x = 75 (red) and x = 85 (pink). Note that the vertical axis is plotted on a log scale. Observed mortality rates mx,t for 1961-2004 are shown as black dots.
Summary of Model Comparison
Table 5 summarizes the models on the basis of the selected criteria discussed in this section. Each of the models fails to meet at least one important criteria. This makes it difficult to appoint one mortality model as the most desirable. The table also suggests that it is fruitful to combine the insights from multiple mortality models, because the models compensate for each others’ weaknesses.
Criterion LC CBD M7 HU 2D KS
Ranking (Goodness-of-fit) 4 2 3 1 Inclusion of cohort effects - + - +
Transparency + + +/-
-Pattern-free residuals - + - + Sample paths and prediction intervals + + +
-Parameter uncertainty + + + +
Plausible point forecasts + - + +/-Ranking (Forecasting performance) 4 2 2 1 Plausible prediction intervals - + -
3.2
Annuity Pricing
The purpose of Section 3.1 was to compare the output produced by the mortality mod-els using a set of criteria that will help attribute similarities and differences in annuity prices. This Section presents the results related to the annuity present values and prices. Figure 14 shows estimated present values of a65,t and a85,t over the time period 1961-2030.
The estimated annuity present values are calculated using estimates (1961-2004) and point forecasts (2005-2100) of the age-specific survival probabilities px,t. The estimated annuity
present values appear to be diverging over time, but are quite consistent in the beginning of the time period shown. In the years where annuity present values are consistent between models, they are primarily based on fitted, rather than forecast, survival probabilities. The analysis in Section 3.1 indicated that all mortality models produced high goodness-of-fit to the data and that the fitted mortality rates (one-year death probabilities) were quite similar, which explains the consistency between the estimated annuity present values in those years. The estimated annuity present values start to diverge when the present val-ues become more dependent on forecast survival probabilities. The estimates of a65,t are
diverging sooner than those of a85,t. This is because annuity present values are based on
survival probabilities up to age 120 and thus start to depend on forecasts sooner at age 65 than at age 85.
It can be observed that the slope of the point forecasts of a65,t and a85,t varies over time,
reflecting the varying rates of mortality improvements produced by the different models at different ages and over time. The 2D KS model consistently produces the highest point forecasts of a65,t and the LC method the lowest. For a85,t the LC method also produces
Figure 14: Estimated densitiesad65,t and ad85,t over time, produced by the LC (black), CBD M7 (red), HU (blue) and 2D KS (green) models.
The point forecasts of a65,t and a85,t produced by the LC and HU models progress
rea-sonably smoothly over time. The point forecasts produced by the CBD M7 and 2D KS models show an evolution over time that is more erratic. The CBD M7 and 2D KS models incorporate cohort effects and the LC and HU models do not. This is an indication that the erratic evolution over time of a65,t and a85,t might be attributed to the inclusion of
cohort effects. The ‘wobbly’ behavior exhibited in the point forecasts of the CBD M7 model provides substantiating evidence that the erratic progression in the point forecasts of a65,t and a85,t is indeed caused by the inclusion of cohort effects.
The analysis based on point estimates of annuity present values ax,t thus far has indicated
that annuity present values between the various mortality models only show marked differ-ences when they are primarily dependent on forecast survival probabilities. The analysis in Section 3.1 has shown that the LC, CBD M7 and HU models are able to produce simulated sample paths of future mortality rates (one-year death probabilities). It was found that the 2D KS model is not suitable for producing simulated future mortality rates. When simulated sample paths are available, they can be can be used to obtain estimates of the entire density function of ax,t rather than just point estimates. This allows for the
assess-ment of the uncertainty associated with present values. Figure 15 shows the estimated densities of the annuity present values a65,2005, a85,2005, a65,2030and a85,2030produced by the
Figure 15: Estimated densities \a65,2005, \a65,2040, \a85,2005 and \a85,2030 produced by the LC
\ a65,2005 LC CBD M7 HU 2D Kernel Point forecast 14.05 14.461 14.565 14.979 Mean 14.046 14.489 14.857 -Median 14.046 14.458 14.854 -Variance 0.104 0.336 0.084 -Interquartile range 0.443 0.741 0.394 -90th percentile 14.465 15.242 15.230 -95th percentile 14.574 15.512 15.338 -\ a85,2005 LC CBD M7 HU 2D Kernel Point forecast 4.503 4.277 4.508 4.826 Mean 4.504 4.290 4.521 -Median 4.503 4.272 4.522 -Variance 0.005 0.0740 0.007 -Interquartile range 0.098 0.359 0.117 -90th percentile 4.593 4.644 4.631 -95th percentile 4.62 4.767 4.663
-Table 6: Summary statistics for \a65,2005 and \a85,2005 for the LC, CBD M7, HU and 2D KS
\ a65,2030 LC CBD M7 HU 2D Kernel Point forecast 15.847 16.371 17.575 17.213 Mean 15.829 16.414 18.32 -Median 15.85 16.355 18.314 -Variance 0.382 1.220 1.258 -Interquartile range 0.825 1.461 1.532 -90th percentile 16.605 17.836 19.76 -95th percentile 16.826 18.300 20.168 -\ a85,2030 LC CBD M7 HU 2D Kernel Point forecast 5.209 5.505 5.856 6.698 Mean 5.209 5.582 6.098 -Median 5.209 5.498 6.08 -Variance 0.059 0.999 0.330 -Interquartile range 0.320 1.305 0.770 -90th percentile 5.517 6.853 6.835 -95th percentile 5.607 7.291 7.076
-Table 7: Summary statistics for \a65,2030 and \a85,2030 for the LC, CBD M7, HU and 2D KS
Some general observations can be made that hold for all models. The mean value of \a65,2005
is consistently higher than that of \a85,2005. This reflects the fact that liabilities are expected
to be higher for an annuity sold to an annuitant aged 65 than for an annuitant aged 85, because the former is expected to live longer. A second general observation is that the mean of the estimated densities is higher in the year 2030 than in 2005, for both ages 65 and 85. This reflects the improvement in longevity over time that is forecast by all of the models. Furthermore, the estimated variance and interquartile range are increasing over time. This reflects the increasing uncertainty associated with the mortality forecasts produced by the various mortality models, which is propagated to the annuity present val-ues. The estimated densities are symmetric, which can be attributed to the distributional assumptions on the error structures for each of the models.
The comparison of the various mortality models in Section 3.1 sheds light on the dif-ferences in forecast mortality rates and associated prediction intervals produced by the models. These differences are reflected in the density functions of the annuity present values. The mean and median values of the annuity present values show considerable dif-ferences between models, which can be attributed to the difdif-ferences in point forecasts of mortality rates produced by the various models. The variance, interquartile range, 90th and 95th percentiles also show marked differences between models. This is the result of the considerable differences in the prediction intervals produced by the various mortality models. In Section 3.1 it became clear that the differences in point forecasts and predic-tion intervals between mortality models become increasingly prominent as the forecasting horizon grows. This is reflected in the estimated density functions of the annuity present values, since the differences tend to be larger for ax,2030 than for ax,2005.
Furthermore, it was found that the LC model underestimates volatility at high ages. This is reflected in the comparatively low variances, interquartile ranges and 90th and 95th percentiles associated with the annuity present values. The HU model produced rapidly expanding prediction intervals, which explains why annuity present values that depend pri-marily on long-term forecasts (e.g., \a65,2030) show particularly high values for the variances,
interquartile ranges and 90th and 95th percentiles. The prediction intervals produced by the HU model expand at such a rapid pace without limit. This implies that it can be expected that the measures of uncertainty produced by the HU model are expected to be even larger relative to the LC and CBD M7 models for annuities sold further in the future than the year 2030.
The observed differences in the estimated mean, 90th and 95th percentiles produced by the various mortality models yield considerable differences in annuity prices. The annuity price based on the estimated mean present value of \a65,2005 between the lowest (LC model)
and the highest price (2D KS model) is 6.6%. For \a85,2005 and \a65,2030 and \a85,2030 the
largest relative differences are 14.1%, 8.6% and 28.6%, respectively. These differences are economically significant: if the annuity a85,2030 is priced based on the mean present value,
the price based on the LC method is almost 30% lower than the price based on the 2D KS method. Economically significant differences are also found when setting prices equal to the estimated 90th and 95th percentiles. Based on the 90th percentile, the largest relative differences between prices for a65,2005, a85,2005, a65,2030 and a85,2030 are 5.4%, 3.2%, 19.0%
4
Conclusion
The objective of this thesis was to evaluate the extent to which model choice affects annuity prices and how the similarities and differences between annuity prices can be attributed. It turns out that the LC, CBD M7, HU and 2D KS models produce economically significant differences in annuity prices. Prices differ by as much as 28.6%, 24.2% and 30.0% based on the mean, 90th percentile and 95th percentile of the annuity present value, respectively. In some cases annuity prices are quite similar, but overall more differences than similarities can be found.
A comparative analysis of the LC, CBD M7, HU and 2D KS models indicated that none of the models satisfies all the desirable criteria for a mortality model (i.e., high goodness-of-fit, pattern-free residuals, incorporation of all relevant features of the mortality data, transparency, plausible point forecasts and prediction intervals, and good out-of-sample performance). Therefore, it cannot be concluded that one mortality model performs bet-ter than the others in an absolute sense. This implies that calculating annuity prices using multiple mortality models is fruitful, because the insights derived from the various models can be combined and the different models compensate for each others’ undesirable prop-erties.
The differences in annuity prices are attributable to some extent. Differences in annu-ity prices based on the mean present value can be explained by mortalannu-ity forecasts that are substantially different between mortality models. The 2D KS method produces consis-tently highest prices based on the mean present value, which is due to the comparatively strong mortality improvements that it forecasts. Differences in annuity prices based on the 90th and 95th percentiles can be attributed to differences in the prediction intervals pro-duced by the various mortality models. The LC model underestimates mortality at high ages, thus producing annuity prices that are comparatively low. The HU model produces prediction intervals that increase rapidly in width, explaining the high prices of annuity values that are dependent on long-term mortality forecasts. As annuity prices become more dependent on long-term forecasts (i.e., ax,2030 compared to ax,2005), differences in annuity
prices get larger. This can be attributed to the fact that point forecasts and prediction in-tervals produced by the various models have been found to be diverging as the forecasting horizon increases.
Limitations and Future Research
The mortality models have been calibrated to the same age range and time period. Mortal-ity forecasts and annuMortal-ity prices can be sensitive to the time period and age range that are used to calibrate the mortality models. The extent to which forecasts and annuity prices are robust to changes in the calibration sample can be investigated through a robustness analysis. In this thesis, no robustness analysis was applied because of time constraints. Ideally, annuity prices should be computed using various calibration samples. A model that produces annuity prices that are more robust to the calibration sample might be favored over mortality models that do not. One of the limitations of this thesis is that such a robustness analysis was not considered. It is also possible to determine the optimal calibration period for each model separately, see for instance Pitacco et al. (2009).
The data is not sufficiently rich to establish which models prices annuities most ‘cor-rectly’. Ideally, data from an annuity provider is used, because it allows back testing of annuity present values. On the basis of such an analysis, a particular mortality model might be more confidently chosen as the winner, in the sense that it gave the best ex-post assessment of the risks held by the annuity provider. Another limitation of the data that is used, is that the general population is considered. Studies have shown the evidence of selection effects in the annuities market, see e.g., Finkelstein and Poterba (2002, 2004). In particular, annuitants are longer-lived than non-annuitants. Furthermore, other fac-tors such as socio-economic status and lifestyle have been associated with differentials in mortality, see e.g., Balia and Jones (2008). The population that is used to calibrate the mortality models should reflect the portfolio of the annuity provider as much as possible, to prevent underestimation of the firm’s risk exposure. One extension of this thesis is to compare annuity prices based on a population of insureds, which more accurately matches a typical risk portfolio of an annuity provider. Such a data set could be provided by an annuity provider, but unfortunately was not available to me.
In this thesis, the male population of England and Wales is considered to calibrate the mortality models. The models have also been applied to other developed economies that have reliable mortality data available (i.e., the United States, Japan, Australia, the Nether-lands, France and Spain) and female mortality. Since these populations may exhibit dif-ferent mortality features than the England and Wales male population, the conclusions of this thesis cannot necessarily be generalized to other countries. A possible direction for future research is to assess the extent to which the findings in this thesis hold for other populations.
5
References
Antonio, Katrien (2012). Sluiten van een periodetafel GBM/V 2005-2010, extern rap-port als bijlage bij Prognosetafel AG 2012-2062. https://www.ag-ai.nl/download/ 26411-Sluiten_van_de_periodetafel_GBMV_2005-2010.pdf.
Antonio, Katrien, Anastasios Bardoutsos, and Wilbert Ouburg (2015). Bayesian Pois-son log-bilinear models for mortality projections with multiple populations. European Actuarial Journal 5 (2), 245–281.
Balia, Silvia and Andrew M. Jones (2008). Mortality, lifestyle and socio-economic status. Journal of Health Economics 27 (1), 1–26.
Barrieu, Pauline, Harry Bensusan, Nicole El Karoui, Caroline Hillairet, St´ephane Loisel, Claudia Ravanelli, and Yahia Salhi (2012). Understanding, modelling and managing longevity risk: key issues and main challenges. Scandinavian Actuarial Journal 2012 (3), 203–231.
Beekman, John A. and Clinton P. Fuelling (1990). Interest and mortality randomness in some annuities. Insurance: Mathematics and Economics 9 (2), 185–196.
Beekman, John A. and Clinton P. Fuelling (1992). Extra randomness in certain annuity models. Insurance: Mathematics and Economics 10 (4), 275–287.
Beekman, John A. and Clinton P. Fuelling (1993). One approach to dual randomness in life insurance. Scandinavian Actuarial Journal 1993 (2), 173–182.
de Beer, Joop, Anastasios Bardoutsos, and Fanny Janssen (2017). Maximum human lifes-pan may increase to 125 years. Nature 546 (7660), E16–E17.
de Beer, Joop and Fanny Janssen (2016). A new parametric model to assess delay and compression of mortality. Population Health Metrics 14 (1), 46.
Booth, Heather, Rob J. Hyndman, Leonie Tickle, and Piet De Jong (2006). Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions. Demo-graphic Research 15, 289–310.
Brouhns, Natacha, Michel Denuit, and Ingrid Van Keilegom (2005). Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Jour-nal 2005 (3), 212–224.
Brouhns, Natacha, Michel Denuit, and Jeroen K. Vermunt (2002a). Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries 2, 105–130.
Cairns, Andrew J.G., David Blake, and Kevin Dowd (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk and Insurance 73 (4), 687–718.
Cairns, Andrew J.G., David Blake, and Kevin Dowd (2008). Modelling and management of mortality risk: a review. Scandinavian Actuarial Journal 2008 (2-3), 79–113.
Cairns, Andrew J.G., David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, and Marwa Khalaf-Allah (2011). Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics 48 (3), 355–367.
Cairns, Andrew J.G., David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, Alen Ong, and Igor Balevich (2009). A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal 13 (1), 1–35.
Cairns, Andrew J.G., David Blake, Kevin Dowd, Guy D. Coughlan, and Marwa Khalaf-Allah (2011). Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin: The Journal of the IAA 41 (1), 29–59.
Chen, Chung and Lon-Mu Liu (1993). Joint estimation of model parameters and outlier effects in time series. Journal of the American Statistical Association 88 (421), 284–297. Continuous Mortality Investigation Bureau, (2002). An interim basis for adjusting the “92” series mortality projections for cohort effects. Technical report, The Faculty of Actuaries and Institute of Actuaries.
Continuous Mortality Investigation Bureau, (2005). Projecting future mortality: Towards a proposal for a stochastic methodology,. Technical report, Working paper 15.
Continuous Mortality Investigation Bureau, (2006). Stochastic projection methodolo-gies: Further progress and p-spline model features, example results and implications,. Technical report, Working paper 20.
Currie, Iain (2006). Smoothing and forecasting mortality rates with p-splines. Talk given at the Institute of Actuaries.
Currie, Iain (2016). On fitting generalized linear and non-linear models of mortality. Scandinavian Actuarial Journal 2016 (4), 356–383.
Currie, Iain, Maria Durban, and Paul H.C. Eilers (2004). Smoothing and forecasting mortality rates. Statistical Modelling 4 (4), 279–298.
Doray, Louis G. (2008). Inference for logistic-type models for the force of mortality. Living to 100 and beyond, SOA Monograph M-L108 1, 18.
