Mortality Improvement Rates: Modeling And Parameter

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Mortality Improvement Rates: Modeling and ParameterUncertaintyAndrew HuntPacific Life Re, London, United KingdomAndrés M. VillegasSchool of Risk and Actuarial Studies andARC Centre of Excellence in Population Ageing Research (CEPAR)UNSW Sydney, AustraliaPresented at the Living to 100 SymposiumOrlando, Fla.January 4–6, 2017Copyright 2017 by the Society of Actuaries.All rights reserved by the Society of Actuaries. Permission is granted to make brief excerpts for apublished review. Permission is also granted to make limited numbers of copies of items in thismonograph for personal, internal, classroom or other instructional use, on condition that the foregoingcopyright notice is used so as to give reasonable notice of the Society’s copyright. This consent forfree limited copying without prior consent of the Society does not extend to making copies for generaldistribution, for advertising or promotional purposes, for inclusion in new collective works or forresale.-1-

Mortality Improvement Rates: Modeling andParameter UncertaintyAndrew Hunta , Andrés M. VillegasbaPacific Life Re, London, United KingdomSchool of Risk and Actuarial Studies andARC Centre of Excellence in Population Ageing Research (CEPAR)UNSW Sydney, AustraliabAbstractRather than looking at mortality rates directly, a number of recent academic studies havelooked at modeling rates of improvement in mortality when making mortality projections.Although relatively new in the academic literature, the use of mortality improvement rateshas a long-standing tradition in actuarial practice when allowing for improvements in mortality from standard mortality tables. However, mortality improvement rates are difficult toestimate robustly, and models of them are subject to high levels of parameter uncertainty,since they are derived by dividing one uncertain quantity by another. Despite this, thestudies of mortality improvement rates to date have not investigated parameter uncertaintydue to the ad hoc methods used to fit the models to historical data. In this study, we adaptthe Poisson model for the numbers of deaths at each age and year, proposed in Brouhnset al. (2002), to model mortality improvement rates. This enables models of improvementrates to be fitted using standard maximum likelihood techniques and allows parameter uncertainty to be investigated using a standard bootstrapping approach. We illustrate theproposed modeling approach using data for the U.S. population and the England and Walespopulation.Keywords: Mortality improvements; Mortality forecasting; Parameter uncertainty1. IntroductionSome of the most far-reaching social and economic challenges of the current age are causedby the rapid increases in longevity and aging of populations across the world. One strandof efforts to meet these challenges has been the development of a wide range of models inorder to forecast the future evolution of mortality rates, based on a combination of statisticalextrapolation of historical data and expert judgment.Email addresses: a.o.d.hunt.00@cantab.net (Andrew Hunt), a.villegas@unsw.edu.au (Andrés M.Villegas)This project has received funding from the ARC Center of Excellence in Population Ageing Research(grant CE110001029). The work in this study was started before Dr. Hunt commenced work at Pacific LifeRe, and any opinions expressed in this paper are held in a personal capacity and should not be construed asthe views of Pacific Life Re or related companies.1

However, one of the subtle differences between academic models for forecasting and thoseused by actuaries in the life insurance industry is over what variable to model. Academicmortality models usually focus on modeling mortality rates at age x and time t, denotedvariously as µx,t (the instantaneous force of mortality), mx,t (the central rate of mortality)or qx,t (the one-year probability of dying). Many of these models have been inspired by theseminal paper of Lee and Carter (1992) and operate in the generalized age/period/cohortframework described in Hunt and Blake (2015c) and implemented in Villegas et al. (2016).More specifically, as discussed in Hunt and Blake (2015c), much of the recent actuarialliterature looking at the modeling and forecasting of human mortality builds on the Poissonlog-bilinear modeling approach introduced in Brouhns et al. (2002), in which the numbersof deaths at age x and year t are modeled as independent Poisson variables and wherethe central rate of mortality mx,t is taken as the response variable linked to a parametricpredictor structure ηxt by means of a log-link function, i.e.,ln mx,t ηx,t .(1)In contrast, practitioners are often interested primarily in the mortality improvementmx,tqx,tµx,t. This is because the changesrates, usually defined by ln µx,t 1 , ln mx,t 1 or 1 qx,t 1in mortality rates are what is of interest when assessing longevity risk for an insurer or pension scheme. However, improvement rates are usually estimated using the largest data setavailable over a long time period—often the national population—in order to give reliable estimates. Such a data set will usually have mortality rates very different from the populationof interest. Nonetheless, by considering mortality improvement rates, inferences made usingthese large data sets can still be used for smaller sub-populations, albeit potentially subjectto longevity “basis risk” (see Haberman et al. (2014)). Furthermore, the discussion of mortality improvement rates also allows practitioners to compare the evolution of mortality inpopulations with very different levels of mortality—for instance, men and women or populations in different countries. In the United Kingdom, the concept of mortality improvementrates became widely adopted among actuaries as a result of the Continuous Mortality Investigation (2002) and has continued with the development of the CMI Mortality ProjectionModel (Continuous Mortality Investigation (2009) and subsequent developments). Similarly,the Scale AA improvement rates were introduced by the Society of Actuaries in the UnitedStates in 1995, and the Scale BB improvement rates in 2012, for use when projecting mortality rates (Society of Actuaries Group Annuity Valuation Table Task Force, 1995; Societyof Actuaries, 2012).However, the modeling of improvement rates is more challenging than the modeling ofmortality rates themselves. Since improvement rates are effectively the first derivatives ofthe mortality rates, any uncertainty in the measurement of mortality rates is magnifiedsignificantly in the measurement of improvement rates. On the one hand, as illustratedby Figures 1a and 1b, the general trend in generally improving mortality rates in the raw(or “crude”) data is far clearer when looking at mortality rates themselves than at theimprovement rates, where the noise around the signal is far more prominent. On the otherhand, as Figures 1c and 1d illustrate, the age shape of mortality rates is very clear and wellunderstood, while the age shape of mortality improvement rates is very noisy and displaysconsiderable heteroskedasticity across ages.2

0.050.00 0.10 0.05Improvement rate at age 700.060.050.040.030.02Mortality rate at age 2010year(b) Crude Improvement Rates at Age 700.20.10.0 0.2 0.1Improvement rate in 20111e 011e 021e 031e 04Mortality rate in 2011 (log scale)0.3(a) Crude Mortality Rates at Age 700204060801000age20406080100age(c) Crude Mortality Rates in Year 2011(d) Crude Improvement Rates in Year 2011Fig. 1. England and Wales Male Mortality and Improvement RatesIn recent years, a number of academic studies have modified the structure in Equation(1) to look at the modeling and forecasting of mortality improvement rates. This has meantusing response variables and link functions such as mx,t 1ηx,t lnmx,tin Mitchell et al. (2013) andηx,t 2mx,t 1 mx,tmx,t 1 mx,tin Haberman and Renshaw (2012). This is usually thought of as using a new responsevariable with the log or identity link, respectively, rather than keeping mx,t as the responsevariable with a nonstandard link function.Such an approach does not present any theoretical problems; however, some practicalissues need to be considered. First, the distribution of the response variables is highly3

nonstandard, so the use of the Poisson distribution is no longer appropriate. In practice,a Gaussian error structure is often assumed with suitable modifications to allow for thecomplex relationship between the variance of an observation and the underlying exposures.Second, as illustrated before, the variance of the response variable is likely to be farhigher as a proportion of the mean than when modeling mortality rates and with a highdegree of heterogeneity across ages and years. The parameter error in the measurementsof the free parameters in the predictor structure will therefore be far higher than for thecorresponding model of mortality rates. This means we must adopt far simpler predictorstructures than would be the case for models of the mortality rate. For these reasons, moreresearch is needed before such mortality improvement models become widely adopted.The academic studies of improvement rates to date, while trailblazing in their approachto the topic, have been forced to make ad hoc modeling assumptions in order to deal with thechallenges associated with the direct modeling of mortality improvement rates. In contrast, awell-developed theoretical framework for the class of generalized age/period/cohort models ofmortality rates has been developed. Therefore, this paper tries to apply some of the structuredeveloped for the study of mortality rates to the modeling of mortality improvements, toreduce the need for some of the ad hoc modeling assumptions and allow a more rigorousexamination of mortality improvement rates. More specifically, we adapt the Poisson modelfor the numbers of deaths at each age and year, proposed in Brouhns et al. (2002), to modelmortality improvement rates. This approach enables models of improvement rates to befitted using standard maximum likelihood techniques, which has several advantages:i. The Poisson structure for death counts accounts automatically for heterogeneity acrossages due to exposures (see , Haberman and Renshaw (2012)).ii. It allows parameter uncertainty to be investigated using the standard bootstrappingtechniques considered in Brouhns et al. (2005) and Koissi et al. (2006).The reminder of this paper is organized as follows. In Section 2, we introduce some of thenotation used throughout the paper. In Section 3, we investigate the connections betweenmodels of mortality and improvement rates, as well as the potential to allow for constantimprovement rates in mortality models. We then develop techniques for fitting improvementrate models to data and apply them to the mortality experiences of England and Wales andof the United States in Sections 4 and 5. In doing so, we note some of the differences inthe definition of improvement rates in previous studies and the impact these have on therobust estimation of the parameters within improvement rate models. We also investigate theimpact of parameter uncertainty on the age and period terms in improvement rate modelsand briefly look at projections from improvement rate models. Finally, in Section 6, wesummarize our findings and provide some conclusions.2. Data and NotationThroughout this paper, we assume that the available data comprise a cross-classifiedmortality experience containing observed numbers of deaths at age x in year t, dx,t , withmatching central exposures ex,t . We assume that age x is in the range [1, X], calendar yearor period t is in the range [0, T ] and, therefore, that year of birth y t x is in the range[ X, T 1].4

We denote the force of mortality and the central rate of mortality by µx,t and mx,t , respectively, with the crude (empirical) estimate of the latter being m̂x,t dx,t /ex,t . Furthermore,we assume that the force of mortality is constant over each year of age x and calendar year t,implying that the force of mortality and central death rate coincide, i.e., µx,t mx,t . Finally,consistent with Brouhns et al. (2002), we assume that the random number of deaths Dx,t atage x in year t is a Poisson-distributed random variable with distributionDx,t Poisson(ex,t mx,t )(2)and, hence, that mx,t E(Dx,t )/ex,t . Observed death counts dx,t are the realization of therandom variable defined in Equation (2).3. Poisson Improvement Rate ModelsIn this section, we exploit the connections between improvement rate models and mortality rate models to produce a Poisson formulation of mortality improvement rate models. Wethen discuss how this formulation can be used to assess parameter uncertainty in mortalityimprovement rate models and to obtain forecasts of mortality rates.3.1. PreliminariesSimilar to Mitchell et al. (2013), we start from a model of the annual improvement rate,given by mx,t mx,t ηx,t ,(3) lnmx,t 1where the negative sign is for presentational purposes to ensure that improvements (i.e.,declines) in mortality rates are positive and that ηx,t can be interpreted as a the continuousrate of improvement at age x in year t.To add structure to this, we then define the predictor structure ηx,t , using the generalage/period/cohort structure described in Hunt and Blake (2015c), i.e.,ηx,t αx NX(i)βx(i) κt γt x ,(4)i 1where αx is a static function of age, which gives the average (constant) rate of improvementin mortality at each age x;(i) κt are period functions governing the change in improvement rate in year t;(i) βx are age functions that modulate the corresponding period functions;2 and2These age functions can be nonparametric (having form determined entirely by the data) or parametric(having a predefined functional form), as discussed in Hunt and Blake (2015c).5

γy is a cohort function describing systematic differences in the rate of improvementwhich depend upon a cohort’s year of birth, y t x.Unlike Mitchell et al. (2013) and Haberman and Renshaw (2012), we do not model mx,t directly, since the mortality improvement rates in this specification do not followa standard probability distribution. They are also highly heteroskedastic, meaning thatstandard estimation techniques are problematic. Instead, we iterate Equation (3) to giveln (mx,t ) ln (mx,0 ) tXηx,τ .τ 1By defining Ax ln (mx,0 ) as the initial mortality curve, this can be rewritten asln (mx,t ) η̃x,t Ax tXηx,τ .(5)τ 1In this form, it is natural to use a Poisson model for the death counts, such that thenumber of deaths observed at age x and for year t follows a Poisson distribution with meanex,t mx,t . Under this assumption and with the log-link function,Dx,t Poisson(ex,t exp(η̃x,t )),(6)as per Brouhns et al. (2002) and Hunt and Blake (2015c), but with the modified predictorstructure η̃x,t , which gives us a model of mortality improvement rates directly rather than amodel for mortality rates.3We also see that, since we can use the Poisson model for the death counts in this formulation of an improvement rate model, we are able to estimate the parameters using maximum likelihood techniques and estimate their parameter uncertainty using the techniquesof Brouhns et al. (2005) and Koissi et al. (2006). This, therefore, overcomes some of the keylimitations of the methods in Mitchell et al. (2013) and Haberman and Renshaw (2012, 2013),which used more ad hoc fitting techniques and did not investigate parameter uncertainty.43One drawback of using a Poisson model for the death counts, common to models of mortality ratesand improvement rates, is that it assumes that the variance of an observation is equal to its expectation.Such overdispersion can be dealt with by using an overdispersed Poisson model in a generalized nonlinearmodeling framework or by allowing for heterogeneity in the population via the use of the negative binomialdistribution, such as in Delwarde et al. (2007); Li et al. (2009). However, we do not investigate this furtherin this study.4In the case of Mitchell et al. (2013), least squares estimation was used to fit the improvement rates,while in Haberman and Renshaw (2012), an iterated generalized linear model procedure was used to allowfor overdispersion in the observed improvement rates. However, in neither case were these distributionsselected on the basis of providing an appropriate distribution for the observed death counts. Consequently,this means that many common methods for assessing parameter uncertainty are not appropriate, as discussedin Section 3.5.6

3.2. Estimation and Equivalent Mortality Rate StructureWe now exploit the connection between improvement rate models and traditional mortality rate models to devise an estimation approach for the Poisson improvement rate modeldefined by Equations (5) and (6).From Equations (4) and (5), the predictor structure in this latter equation can be rewritten asln(mx,t ) η̃x,tln(mx,t ) Ax ln(mx,t ) Ax tXηx,ττ 1tXαx τ 1ln(mx,t ) Ax αx t ln(mx,t ) Ax αx t NX!(i)βx(i) κt γτ xi 1NXi 1NXβx(i)tX(i)κtτ 1 tXγτ xτ 1(i)βx(i) Kt Γt x(7)i 1with(i)K0 0 and Γ X 0,and(i)Kt tXτ 1(i)κtand Γt x tXγτ x ,(8)for 1 t T .(9)τ 1In Equation (7), it is clear that αx is determining the constant trend rate of mortalityimprovement in the historic data at each age. We also see that, if the αx term is notincluded, Equation (7) is equivalent to a standard age/period/cohort model (see Hunt andBlake (2015c)). Therefore, we see that conventional mortality rates models are identicalto improvement rate models without constant improvement terms, and merely differ in thepresentation of the parameters (i.e., thein Equation (8), as opposed to theP constraints Pconventional identifiability constraints t Kt 0 and c Γc 0).In contrast, we see that including an αx constant improvement term in Equation (7)extends the family of generalized age/period/cohort models discussed in Hunt and Blake(2015c) with a term that is nonparametric in age and linear in time. Therefore, everymortality rate model discussed in Hunt and Blake (2015c) has an extended version thatincludes a constant improvement rate term, which is equivalent to using the same predictorstructure for mortality improvement rates rather than mortality rates.To estimate the improvement rate model in Equations (5) and (6), we can then estimatethe equivalent mortality rate model defined by (7) with the constraints in (8) and recoverthe parameters of the improvement rate model using the relationships in (9). Hence, we canuse standard techniques to fit mortality rate models to data and convert these to models ofthe improvement rate. In this paper, we follow such an approach and estimate the modelsusing the R package StMoMo (Villegas et al., 2016), which enables the fitting of generalage/period/cohort mortality rate models.7

3.3. Models With Constant Improvement RatesA key question when deciding on the form of the predictor structure in Equation (4) iswhether to include an αx term. This term represents the average rate of improvement ateach age over the period of the historic data and, when the model is projected, will give aconstant component to the rate of improvement in mortality in the future.Some authors take issue with this and believe that it conflicts with the requirements ofbiological reasonableness.5 In particular, Haberman and Renshaw (2012) show that the αxstatic age function in a standard mortality rate model disappears when the first derivative istaken to obtain an improvement rate model. Furthermore, there are legitimate questions as towhat form any constant rate of improvement should take. Including a nonparametric αx termin the predictor structure of Equation (4) will assume that the average rates of improvementobserved over the period of the historic data at each age will continue indefinitely into thefuture. Inst

bSchool of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR) UNSW Sydney, Australia Abstract Rather than looking at mortality rates directly, a number of recent academic studies have looked at modeling rates of improvement in mortality when making mortality projections.

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