A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates

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1 A comparison of he Lee-Carer model and AR-ARCH model for forecasing moraliy raes Rosella Giacomei a, Marida Berocchi b, Svelozar T. Rachev c, Frank J. Fabozzi d,e a Rosella Giacomei Deparmen of Mahemaics, Saisics,Compuer Science and Applicaions, Bergamo, Universiy, Via dei Caniana, 2, Bergamo 24127, Ialy (rosella.giacomei@unibg.i) b Marida Berocchi, Deparmen of Mahemaics, Saisics,Compuer Science and Applicaions, Bergamo, Universiy, Via dei Caniana, 2, Bergamo 24127, Ialy (marida.berocchi@unibg.i) c Svelozar T. Rachev, School of Economics and Business Engineering, Universiy of Karlsruhe, Posfach 6980, Karlsruhe, Germany, Deparmen of Saisics and Applied Probabiliy, Universiy of California, Sana Barbara, CA , USA, and Chief-Scienis, FinAnalyica Inc. (rachev@ki.edu) d Frank J. Fabozzi, Yale School of Managemen, 135 Prospec Sree, Box , New Haven, Connecicu , USA (frank.fabozzi@yale.edu and fabozzi321@aol.com) e CONTACT AUTHOR: Frank J. Fabozzi, Yale School of Managemen, 135 Prospec Sree, Box , New Haven, Connecicu , USA (frank.fabozzi@yale.edu and fabozzi321@aol.com) 1

2 A comparison of he Lee-Carer model and AR-ARCH model for forecasing moraliy raes Absrac: Wih he decline in he moraliy level of populaions, naional social securiy sysems and insurance companies of mos developed counries are reconsidering heir moraliy ables aking ino accoun longeviy risk. The Lee and Carer model is he firs model o consider he increased life expecancy rends in moraliy raes and is sill broadly used oday. In his paper, we propose an alernaive o he Lee-Carer model: an AR(1)-ARCH(1) model. More specifically, we compare performance of hese wo models wih respec o forecasing age-specific moraliy in Ialy. We fi he wo models, wih Gaussian and -suden innovaions, for he marix of Ialian deah raes from 1960 o We compare he forecas abiliy of he wo approaches in ou-ofsample analysis for he period and find ha he AR(1)-ARCH(1) model wih -suden innovaions provides he bes fi among he models sudied in his paper. Key words: moraliy raes, Lee-Carer model, auoregression-auoregressive condiional heeroskedasiciy model, AR(1)-ARCH(1) model JEL Classificaion: C51, C52, C53, C59, G22 2

3 A comparison of he Lee-Carer model and AR-ARCH model for forecasing moraliy raes 1. Inroducion Moraliy risk is he risk of having a higher percenage of deahs han expeced, which implies a higher probabiliy of deah. Facors ha impac moraliy are wars, epidemics or pandemics, leading o a higher moraliy rae (i.e., people living less han expeced). Longeviy risk is he risk of people surviving longer han expeced or observed deah raes being lower han expeced. Advances in medical science, echnological improvemens and lifesyle changes end o reduce he number of deahs. From an economic perspecive, he decline of moraliy has a significan adverse impac on pension plans and annuiy insurers (i.e., eniies who provide old age benefis). A significan overesimae of moraliy he rae implies a high risk profile for pension funds and annuiy insurers. Longeviy risk can be hedged wih reinsurance conracs and wih longeviy derivaives. A ypical conrac is a longeviy bond which pays a coupon ha is proporional o he number of survivors in a seleced birh cohor. The pricing of such producs requires handling he uncerainy of life expecancy going forward. The prevailing lieraure deals wih differen scenarios of moraliy risk and/or sochasic deah disribuions. 1 The model presened by Lee and Carer (1992) appears o be he firs model ha considers he increased life expecancy rends in moraliy raes. Alhough originally applied o U.S. moraliy daa, i is now applied o all-cause and cause-specific moraliy daa from many counries. 2 Moreover, many of he recen approaches ha have been proposed in he lieraure are consisen wih Lee-Carer model (see Lifemerics (2007) and he reference herein) and he consensus in he lieraure in he las decade appears o consider he Lee-Carer model he leading saisical model for forecasing moraliy. 3 The Lee-Carer mehod combines a demographic model wih a saisical model of ime series o forecas moraliy raes. As poined ou by Girosi and King (2007), he Lee-Carer model can be viewed as a special ype of a mulivariae process in which he covariance marix depends on he drif vecor and he innovaions are ineremporally correlaed. In his paper, we presen an alernaive economeric model o he Lee-Carer model for 1 See Milevsky and Promislow (2001), Balloa and Haberman (2006), and Giacomei e al. (2009, 2010). 2 See, for example, Tuljapurkar e al. (2000). 3 See, among ohers, Lee and Miller (2000), Lee (2000), Deaon and Paxson (2004), and Denui (2009). 3

4 forecasing moraliy and hen empirically compare he forecasing properies of he wo models. The model we propose is he auoregression (AR)-auoregressive condiional heeroskedasiciy (ARCH) model. More specifically, we propose he AR(1)-ARCH(1) model. The paper is organized as follows. In he nex secion, we review he main characerisics of he Lee-Carer model. Our economeric approach is described in Secion 3, followed by a descripion of our daabase and he resuls of esimaes of he models in he Secion 4. In Secion 5, we compare he resuls of all models. 2. The Lee-Carer model Le m x, be he deah rae for age x in year. Lee and Carer (1992) suggesed a log-bilinear form for he force of moraliy x,, ha is ln m k x, x, x x x, x=1,, A; =1,,T, (1) where x, x are age-specific parameers, k is a ime-varying parameer represening a common facor risk, and x, is a zero mean Gaussian error N(0, 2 ). The random erm x, reflecs a paricular age-specific hisorical influence. The coefficiens x are age-specific consans ha describe he general shape of he age-moraliy profile. The index k serves o capure he main emporal level of moraliy. Since he parameerizaion in (1) is invarian wih respec o he ransformaions: (, k ) ( c, k c) for (, k ) ( c, k c) for some c\0, (2) x x x x x hen he parameers x, k should saisfy he consrains: A x1 1; x T k 0 (3), 1 in order o ensure he idenifiabiliy of he model. The consrain T k 0 1 implies ha he esimaes of parameers x, are given by he averages of he force of moraliy over he ime period, T 1 ha is, ˆ x x,. T 1 2 Considering ha ˆ x, x xk x, Nxk, are Gaussian disribued wih mean x k and variance σ 2, hen he parameers x and k can be esimaed via maximum likelihood. In paricular, as remarked by Lee and Carer (1992), he opimal soluion can be found using he Singular Value Decomposiion (SVD) of he marix of he cenered age profiles z ˆ,,. x x x Given he marix Z [ zx, ] x1,..., A, 1,..., T, we can compue he normalized eigenvecor 4

5 u 1 =[u 1,1,.,u 1,T ] (respecively v 1 =[v 1,1,.,v 1,A ]) of he marix ZZ ' (respecively ZZ ' ) corresponding o he larges eigenvalue λ 1. Then he opimal esimaes saisfying he consrains (3) imposed on he parameers are given by he vecors: β ˆ ˆ ˆ v1 [ 1,..., A]' A v and ˆ [ k ˆ 1,..., k A ˆ T]' v 1 j1 1, j j1 1, j k u 1. Typically for low-moraliy populaions, he approximaion Zλ 1 v 1 u 1 accouns for more han 90% of he variance of ln(m x, ). A furher re-esimaion sep for he parameers k is required because wih he above procedure he number of fied deahs does no equal he number of observed deahs. The parameers k ˆ are adjused (aking esimaes ˆ x, ˆx as given) such ha he new esimaes k solve he equaions where D and x, wih age x in year. A ˆ ˆ x, exp x x x1 D N k =1,,T, N are, respecively, he oal number of deahs in year and he oal populaion In order o forecas fuure moraliy raes, Lee and Carer assume ha x and x remain consan over ime and he ime facor k is inrinsically viewed as a sochasic process. They sugges using he following random walk wih drif model for k : kˆ ˆ k 1, (4) 2 where N 0, rw are independen and idenically disribued (i.i.d.) Gaussian disribued 2 wih null mean and variance. The maximum likelihood esimae of he drif parameer is rw given by ˆ ˆ ˆ 1 /( 1) 2 1 T 1 kt k T and he variance esimae is ˆ ˆ ˆ ˆ 2 rw k 1 1 k esimae k ˆT a ime T we ge kˆ kˆ ˆ T T expeced log-moraliy can be approximaed as follows: T 1 2 where N 0, rw kˆ ˆ T k 1 ˆ ˆ ˆ, ˆ ˆ ˆ ˆ ˆ xt xx kt x x kt ( T 1). (5). To and he 3. The AR-ARCH model 5

6 In his secion, we propose as an alernaive o he Lee-Carer model esimaing an AR(1)-ARCH(1) model for forecasing he force of moraliy x,. We analyze separaely columns and rows of he moraliy able. As a firs sep, we analyze he daa by rows; ha is, we fix a specific age x and consider a ime series process wih = 1,,T. Assuming he ime series presens a x polynomial rend of degree n, we esimae he following univariae AR(1)-ARCH(1) on he residuals, for each age x, wih x=1,..,a: x, px() 1 x, 1 x, 2 2 x, 0 1 x, 1 where x, is he innovaion of he ime series process x, wih x, z x,, z an i.i.d process wih zero mean and consan variance, and p x () is polynomial of degree n. As a second sep, we analyze he daa by columns: ha is, we fix a specific year and consider he age process wih x=1,,a. We assume a polynomial rend and we esimae for each year, wih =1, T x x, p( x) 1 x1, x, 2 2 x, 0 1 x1, where η x, is he innovaion of he age process for a fixed, wih x, zx x,, z x is an i.i.d. process wih zero mean and consan variance, and p (x) is a polynomial of degree n. x (7) (6) 4. An empirical analysis based on he Ialian moraliy rae In his secion, we repor he resuls of fiing he Lee-Carer model and he AR(1)-ARCH(1) model o Ialian moraliy daa aken from he Universiy of California, Berkeley Human Moraliy Daabase available from 1922 o We choose an opporune range of daa (from 1960 o 2006) in order o have a reliable and complee daa se. In Figure 1 we can observe he surface of he moraliy daa for years from 1960 o 2006 and ages from 0 o 94. However, we resric he age from 40 o 94 in order o avoid he hump around age 0 and 39. In Figure 2 we repor he surface of he daase used in our analysis. We divided he daase ino he following hree subsamples: Subsample 1: From 1960 o 2003 and from age 40 o 91. Subsample 2:- From 2004 o 2006 and from age 40 o Available a 6

7 Subsample 3: Year 2004 o 2006 and from ages 92 o 94. I is from subsample 1 ha we esimae he parameers of he models. Using subsample 2 we compare he Lee-Carer and AR-ARCH models. Recall ha he classical Lee-Carer model can forecas only in one direcion (i.e., he ime). The AR(1)-ARCH(1) models wih differen innovaions are compared using subsample 3. Tha is, we forecas in wo direcions, ime and age in order o assess which of he wo models performs he bes. Our empirical analysis involves he following hree seps: Sep 1: Esimae Lee-Carer model and discuss how o model he ime facor k. For hree consecuive years, k and he force of moraliy for he period are forecased. Sep 2: Fi he AR(1)-ARCH(1) model wih Gaussian and -innovaions and forecas in he wo dimensions he force of moraliy for he period and for ages Sep 3: Compare he forecas from he Lee-Carer model, he AR(1)-ARCH(1) model wih Gaussian and -innovaions, and he acual daa on he subsample 2 and invesigae he forecasing capaciy of AR(1)-ARCH(1) models wih differen innovaions for subsample 3 for he period and for he ages We discuss he resuls of hese hree seps below. Sep 1 We fi he Lee-Carer model using he mehodology presened in Secion 3. In Figure 3 we repor he esimae of emporal level of moraliy k ˆ. We esed for he presence of he uni roo by applying he Adjused Dickey-Fuller (ADF) es and found ha we could no rejec a null hypohesis for he presence of he uni roo for differen lags and differen significance levels (from 0.01 o 0.05). We model k ˆ wih a random walk wih drif. Using equaion (4), we esimae he drif = We observe ha he residuals of he model are subsanially i.i.d. Gaussian (no auocorrelaion is revealed, he residual mean is 0, he sandard deviaion is , and he kurosis is ). We esed for he presence of he uni roo in he residuals and we could rejec he null hypohesis. Saring from he observaion for he las year, we forecased hree years k i wih i=1,2,3 and we reconsruced he forecased marix. Sep 2 Our analysis of he daa indicaes he presence of a nonlinear deerminisic rend in boh direcions hrough ime and age. To remove he rend, we deermined he coefficiens of a 7

8 polynomial of degree 2 ha fis he daa in a leas squares sense. Then we analyzed he derended daa. We esed for he presence of he uni roo and we could always rejec he null hypohesis for boh direcions. However, we observed he presence of a single jump on he diagonal of he force of moraliy marix. Considering he jump as a possible oulier, we smoohed he de-rended daa in order o remove he jump. Smoohing he daa implies removing par of he kurosis bu he ime series characerisics remain unchanged. In order o have robus resuls, we coninued our analysis using boh he de-rended original daa and he de-rended smoohed daa. In Table 1 we repor he main saisics (volailiy, skewness, and kurosis) and he p-value of he Kolmogorov-Smirnov and he Jarque-Bera ess for he de-rended original daa We From he kurosis figures we see ha almos 30% of he ime series are lepokuric and for 20% we rejec he null hypohesis of a sample drawn from a normal disribuion. Inspecing he daa s auo-correlogram and he squared daa, we deec occasionally he presence of auocorrelaions and heeroskedasiciy, especially when we consider he age process wih x=1,,a. x We fi an AR(1)-ARCH(1) model assuming alernaively a Gaussian innovaion and a - suden innovaion. In Table 2 we repor he esimae of he AR(1)-ARCH(1) for he age process and in Table 3 he ime process x wih -suden innovaion process. x In each row of Table 2 we repor for a differen year (wih = 1960,..,2003) he coefficiens, he asympoic -saisics, he order of he inegraed process, and he esimae of degrees of freedom of he innovaion process. We observe ha he AR coefficiens are significanly differen from zero for 31 of he 44 years. However, we canno rejec he null hypohesis for he ARCH coefficien for mos of he age process. The esimaed degrees of freedom reveal a non-gaussian innovaion process for 70% of he ime series. In each row of Table 3 we repor he resul for differen ages, wih x = 40,.., 91 obaining similar resuls as repored in Table 2. The AR coefficiens are significanly differen from zero for 16 of he 52 ages and we canno rejec he null hypohesis for he ARCH coefficien for mos of he age processes. The esimaed degrees of freedom confirm a non-gaussian innovaion process for 46% of he ages. For he period , we forecas he moraliy rae using equaion (7) under he wo differen disribuional assumpions. Sep 3 8

9 Finally, we esed ex-pos he abiliy of each model o forecas fuure Ialian moraliy raes. In Figure 4 we repor he acual daa, he Lee-Carer forecas, and he AR(1)-ARCH(1) forecas wih innovaions (he 95% confidence levels is shown in red). We analyzed he forecass for hree consecuive years for ages ranging from In order o compare he models, we consruced a modified version of Theil s U index. Theil s U index is a measure for forecasing qualiy in an ou-of-sample analysis. 5. I can be inerpreed as he mean-squared error of he proposed forecasing model divided by he meansquared error of a naïve predicion model used as a benchmark. For saionary ime series, he naïve model is generally given by eiher he previous observaion or a no-change model. In our analysis, we modified he index using as a naïve model he linear deerminisic rend esimaed for each age x. Index s values less han uniy show an improvemen over he simple naïve forecas. In order o have a value of he index for each year wih = 2004,..,2006, we compued 2 x, x, TheilU x wih x, px() x In Table 4 we repor Theils U index for each of he hree years for boh he de-rended daa and he smoohed daa, as well as he index for he overall period. We observe ha in he forecasing period, he inroducion of he economeric model lead o a more accurae forecas wih compared o he Lee-Carer model. Theil s U index for he Lee- Carer model ranges beween 0.60 and In conras, he AR(1)-ARCH(1) ranges beween and when we consider a Gaussian innovaion, and beween and when we consider a -suden innovaion. This means ha he sum of he squared errors of he AR(1)-ARCH(1) model is less han he sum of he squared errors of he Lee-Carer model of he naïve model. These resuls 5 y F TheilU T 2 y y1 1.. T where F and y sand for a pair of prediced and observed values, wih =1,,T and y 0 is he las observaion used for he esimaion. 9

10 srongly sugges ha he AR-ARCH model wih -suden innovaion is superior o he Lee-Carer model in forecasing. If we consider he index for each single year, we observe ha he forecasing abiliy decreases wih he increase of he ime period forecas. Once again, AR(1)-ARCH(1) model assuming a -suden innovaion provides he bes fi o boh he original daa and he smoohed daa se. Finally, we forecas for each year he force of moraliy for ages 92 o 94, and on he enlarged daase, we fi again model (7) obaining he forecas for years and ages In Table 5 repor he resuls of Theil s U index. We do no provide he Lee-Carer model values since his model does no provide forecass for ou-of-sample ages. The AR(1)-ARCH(1) wih -suden innovaions confirm is superioriy using boh he original and he smoohed daa. In Figure 5, we show he acual daa, he AR(1)-ARCH(1) forecas wih -suden innovaion (in red he 95% confidence levels) for ages 92,93, and Conclusions In his paper, we propose wo AR(1)-ARCH(1) models for forecasing he moraliy rae and compare heir forecass o he classical Lee-Carer model. We find ha an AR(1)-ARCH(1) model wih -suden innovaions provides he bes fi among he models invesigaed because i is able o capure he non-gaussian behavior of he dynamics associaed wih he ime and age processes. Moreover, his model is capable of enlarging he moraliy marix in wo dimensions: ime and age. 10

11 Acknowledgemens The auhors grealy acknowledge he naional Ialian gran from he Minisry of Educaion and Research: Financial innovaions and demographic changes: new producs and pricing insrumens wih respec o he sochasic facor aging (local coordinaor M. Berocchi) and he local gran from he Universiy of Bergamo 2009 (coordinaor M. Berocchi). Svelozar Rachev graefully acknowledges suppor by grans from Division of Mahemaical, Life and Physical Sciences, College of Leer and Science, Universiy of California, Sana Barbara, he Deuschen Forschungsgemeinschaf and he Deuscher Akademischer Ausausch Diens. References Balloa, L., Habermann, S The fair valuaion problem of guaraneed annuiy opions: he sochasic moraliy environmen case. Insurance: Mahemaics and Economics. 38, Deaon, A., Paxson, C Moraliy, income, and income inequaliy over ime in Briain and he Unied Saes. Technical Repor, N. 8534, Naional Bureau of Economic Research Cambridge, MA. Denui, M. M An index for longeviy risk ransfer. Journal of Compuaional and Applied Mahemaics. 230, Giacomei R., Orobelli, S., Berocchi, M Impac of differen disribuional assumpions in forecasing Ialian moraliy. Invesmen managemen and financial innovaions. 3, Giacomei R., Orobelli, S., Berocchi, M A sochasic model for moraliy raes on Ialian daa. Journal of Opimizaion Theory and Applicaions. To appear. Girosi F., King, G Undersanding he Lee-Carer moraliy forecasing mehod. Technical Repor, Rand Corporaion. Lee, R.D., Carer, L.R Modelling and forecasing U.S. moraliy. Journal of he American Saisical Associaion. 87, Lee, R. D The Lee-Carer mehod for forecasing moraliy, wih various exensions and applicaions. Norh American Acuarial Journal. 4, Lee, R. D., Miller, T Assessing he performance of he Lee-Carer approach o modeling and forecasing moraliy. Available a LifeMerics A oolki for measuring and managing longeviy and moraliy risks. Technical Repor. Milevesky M. A., Promislow, S.D Moraliy derivaives and he opion o annuiise. Insurance: Mahemaics and Economics. 29, Tuljapurkar S., Li, N., Boe, C., An universal paern of moraliy decline in he G7 counries. Naure. 405,

12 Figure 1 Surface of he moraliy daa for he years from 1960 o 2006 and he ages from 0 o 94: Ialian k ˆ esimaion ( ) 12

13 Figure 2 Surface of he moraliy daa from year 1960 o year 2006 and from age 40 o age 94 Figure 1 This figure gives surface of he moraliy daa for he years from 1960 o 2006 and ages 0 o 94. Ialian k ˆ esimaion ( ) 13

14 Figure 3 Esimaion of he empirical level k ˆ of moraliy for he Ialian populaion ( ) for he Lee-Carer model: Ialian k ˆ esimaion ( )

15 Figure 4 Forecass for each age of he force of moraliy for he years from 2004 o 2006 for he Ialian populaion 15

16 Figure 5 Forecass for each year of he force of moraliy of he ages from 92 o 93 for he Ialian populaion ( ). 16

17 Table 1: Summary saisics for he de-rended original daa: Volailiy, skewness, kurosis,and p- value of he Kolmogorov-Smirnov (KS) and Jarque-Bera (JB) ess Age Volailiy Skewness Kurosis KS JB Year Volailiy Skewness Kurosis KS JB

18

19 Table 2 Esimae of he coefficiens of he AR(1)-ARCH(1) and -saisic for he age process Coefficiens -saisic Order of AR K ARCH AR K ARCH Inegraion DoF

20 Table 3: Esimae of he coefficiens of he AR(1)-ARCH(1) and -saisic for he ime process Coefficiens -saisic Order of AR K ARCH AR K ARCH inegraion DoF

21

22 Table 4 Theil s U index for he hree differen years for he de-rended daa and he smoohed daa and he index for he overall period Original Smoohed Original daa Smoohed daa daa daa Model Lee Carer AR-ARCH Gaussian Innovaion AR-ARCH -suden Innovaion Table 5 Theil s U index for he hree differen years for he de-rended daa and he smoohed daa, and he index for he overall period using he enlarged sample se Original Smoohed Original daa Smoohed daa daa daa Model AR-ARCH Gaussian Innovaion AR-ARCH -suden Innovaion