FUTURE LIFE-TABLES BASED ON THE LEE-CARTER METHODOLOGY AND THEIR APPLICATION TO CALCULATING THE PENSION ANNUITIES 1

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1 ACA UNIVERSIAIS LODZIENSIS FOLIA OECONOMICA 250, 20 Agnieszka Rossa * FUURE LIFE-ABLES BASED ON HE LEE-CARER MEHODOLOGY AND HEIR APPLICAION O CALCULAING HE PENSION ANNUIIES Suary. In the paper a new recursie approach to the ortality forecasting is proposed based on the well-known Lee-Carter stochastic odel. he standard Lee- -Carter ethod and its odified ersion are presented and copared using ortality data for Poland in the tie period he results obtained indicate that the recursie approach gies ore accurate forecasts in ters of the ean squared error. Stochastic forecasts of age-specific death rates are also used to predict death probabilities and life epectancy being the ain paraeters of the life-tables. As an eaple, future life-tables for 2020 are calculated. Applications of Lee-Carter ethodology in pension annuity calculations are presented. Key words: ortality forecasting, Lee-Carter odel, future life-tables, pension annuity.. Introduction Long-standing obserations of different characteristics of the surial distribution in huan population show that the characteristics change in tie. For instance, obserations of annual probabilities of death giing surial to an age, conducted in the deeloped countries, show that such probabilities hae been decreasing in the recent decades, although the changes are irregular. hus, the considered probabilities can be therefore iewed as processes that are characterised by certain stochastic ariability, in addition to their general tendency. he other indicators hae been showing a siilar, stochastic character, for instance, life epectancy, ortality rates or rectangularization indices of the surial functions [Wiloth, Horiuchi, 999]. he trends and regularities obsered in the deeloped countries in the second half of the 20 th c. with respect to soe of the aforeentioned indicators can be sued up as follows: the ode of the death intensity cure oes towards ery old ages, concentration of deaths around the ode is increasing, * Associate Professor, Uniersity of Lodz, Unit of Deography and Social Gerontology. he paper was prepared as a part of the research project no. 4404/B/H03/2009/37 granted by Minister of Science and High Education. [3]

2 32 Agnieszka Rossa the surial cure is undergoing rectangularization which is a result fro the tendencies outlined aboe, accidental deaths at young ages especially ale twenty-year-olds hae been growing injuries, accidents, poisoning, both ale and feale life epectancy hae been increasing. he phenoena produce arious socio-econoic consequences. One of the is the growing nubers of people suriing to the retireent age, as well as an etending period of tie in which the pension annuity proiders need to pay out the benefits. herefore, iproing ortality rates hae a direct effect on the present alue of future liabilities and its related leel of reseres held by institutions paying the benefits. A reliable estiation of ortality rates in the future periods becoes therefore a key approach. his paper uses the Lee-Carter ethodology to derie ortality forecasts for Poland up to he recursie ersion of the Lee-Carter stochastic approach is proposed. he standard and recursie Lee-Carter ethod are presented and copared using ortality data for Poland. he results obtained indicate that the new approach gies ore accurate forecasts in ters of the ean squared error. Stochastic forecasts of age-specific death rates are also used to predict death probabilities and life epectancy being the ain paraeters of the life-tables. he forecasts of probabilities of death will be used to calculate the pension annuities to be paid out under the so-called second and third pillar of the Polish pension syste. Pension annuities are calculated fro aounts accuulated on indiidual funds kept by the Open Pension Funds OPF, the Eployee Pension Schees EPS or the Indiidual Pension Accounts IPA. 2. he Lee-Carter ethodology he fist attept at atheatical odelling of the intensity of deaths in dynaic ters was undertaken by Blaschke [923] who considered so-called dynaic Makeha s law. he odel assues that the ortality intensity µ t is function of not only age, but also of calendar tie t. he effect of the tie ariable t was epressed in the odel through soe deterinistic function. he stochastic character of ortality-related processes justifies the need of reaching for the stochastic ethods for odelling, forecasting and describing the phenoena. his is ery iportant, especially with respect to the accuracy of the forecasts of their future deelopent. Aong the stochastic odels for forecasting ortality that are popular today, there is the Lee-Carter odel. In the 990s, Lee and Carter [Lee, Carter 992] attepted to apply the theory of rando walk with a drift to odelling and subsequently forecasting the

3 Future Life-ables Based on the Lee-Carter Methodology 33 crude age-specific death rates t, cross-classified by age, and by calendar year t. he odel has the for or equialently ln t a b k t ε,t for 0,,...,ω, t,2,...,, t ep{a b k t ε,t } for 0,,..., ω, t,2,...,, 2 where {a } and {b } are sets of soe constants that are different for different age groups, and k t is an inde iewed as a discrete-tie stochastic process. Doubleinde ters ε,t represent the rando errors reflecting particular age-specific influences not captured by the odel. It is assued that ε,t are independent rando ariables, norally distributed with the ean equal 0 and ariance σ 2. he odel or 2 is undeterined without additional constraints. Let us assue, for instance, that we hae an epirical data atri M of logariths of specific ortality rates, i.e. a atri with eleents ln t in the body, where 0,,...,ω denotes the age group atri rows, whereas t,2,..., are calendar years atri coluns. Let for a set of paraeters {a }, {b }, 0,,... ω and {k t }, t,2,..., the odel be alid. It is easy to erify that for any constant c and the set of paraeters {a cb }, {b } {k t c} or {a }, {cb } {k t /c} the odel also holds. Hence, the paraeters k t are deterined to the transforation k t c or k t /c, paraeters b are deterined to the ultiplicatie constant, whereas the paraeters a to the additie constant. o ensure the unique paraeters of the odel it is necessary to define certain additional constraints. o this end, Lee and Carter assued that the su of the paraeters b for all age groups indeed by equals, whereas the su of the paraeters k t indeed by t equals 0. hus, the paraeters b and k t satisfy the following constrains ω b, kt 0, so that a ln t. 0 t t It follows, that under these constrains the paraeters a describe the age pattern of ortality aeraged oer tie, whereas b describe deiations fro the aeraged pattern when k t aries. he paraeters k t describe the effect of the calendar tie t on a change in the ortality. d t he odel is fitted to the crude death rates t 000, where K t d t denotes the nuber of deaths obsered at age and tie t, and K t

4 34 Agnieszka Rossa denotes the eposure to the risk of death at age. In our approach, we will replace crude death rates by their estiates ˆ t obtained fro the epressions 2q t ˆ t 000, where q t denotes the probability of death in year t, 2 q t giing surial to age. Probabilities q t will be taken fro the period lifetables published for years by the Central Statistical Office. 3. Forecasting fro the Lee-Carter odel Let us assue that the paraeters a and b in the odel are constant in tie t, which eans that estiates of the paraeters, once deried, can be used in the future, and the ortality forecasts can be obtained by odelling k t as a tie series. he forecasts concerning the predicted alues of k t together with the estiates of the paraeters a and b allow, based on the odel, forecasting long-ter ortality, and ore specifically, forecasting the logariths of deaths rates for future periods t>. As proposed by Lee and Carter, finding the alues of k t for t,2,..., proides a starting point for odelling a tie series {k t } as a rando walk with drift that can be described using the forula k t ck t e t, 3 where c stands for a constant a drift, while e t is an error ter with noral distribution with the ean 0 and a finite ariance. his approach allows reducing accuulation of error ters resulting fro the short-ter ariability in ortality rates, which largely influences the accuracy of the forecasts. he estiator of the drift c has the for cˆ k k, 4 whereas the ariance estiator of ĉ is gien by the forula 2 2 ˆ σ c kt ˆ. kt c 5 t 2 Estiation of the constant c allows aking forecasts concerning k t for t>. Inserted in odel, where the paraeters a and b are replaced by their estiates, allows aking the forecasts of future log-central ortality rates for t>.

5 Future Life-ables Based on the Lee-Carter Methodology 35 hen it is also possible to estiate other paraeters for future periods in order to obtain the so-called future life-tables. 4. Model fitting In the Lee-Carter ethodology the Singular Value Decoposition SVD is applied to derie the paraeters a, b, and k t. Let us consider the atri [ ln ˆ t ]. A 6 a ω SVD allows representing each eleent of the atri A as the following su r ln ˆ t a λ u t, 7 i where r rank {A}, λ i, i,2,, r are the ordered increasingly singular alues, and u i, i are the corresponding left and right singular ectors of A. Let us consider the following representation ln ˆ t a λ u ν t ε, i, t i i r where ε, λ u t. t i 2 i i i Such a representation leads directly to the Lee-Carter odel ln ˆ t a b kt ε,t, with b k λ u and b, 0. t t ω k t 0 t hus the Lee-Carter odel is siply the SVD approiation of the atri A [ ln ˆ t a ]. he percentage of the total ariance eplained by the Lee-Carter odel is r 2 2 λi i λ. Since the first singular alue λ is usually uch larger than all the others, therefore the percentage of the total ariance eplained by the odel is

6 36 Agnieszka Rossa usually ery high. For instance, uljapurkar et al., 2000 found that for soe countries with low ortality oer 94% of the ariance was eplained by the odel. he Lee-Carter ethodology and the subsequent odifications are broadly discussed in the literature [e.g. Carter, 996; Lee, 2000; Booth et al., 2002; Brouhns et al., 2002a,b; Renshaw, Haberan, 2003a,b,c; Li et al., 2004; Lundströ, Qist, 2004; Brouhns et al., 2005; Koissi, Shapiro, 2006; Koissi et al., 2006; Denuit, Dhaene, 2007]. 5. he recursie procedure of the Lee-Carter approach In this section a odification of the Lee-Carter approach based on a recursie algorith is proposed. k Let A be a atri with eleents obtained in the k-th step of the recursie k procedure. If k then A reduces to the atri A with eleents defined in 6. he procedure can be described as follows: [.] Apply the standard Lee-Carter approach using ln ˆ t, 0,,,ω, i.e. for one year ahead. k A and derie forecasts of [2.] Attach the alues of ln ˆ t obtained in the step to the atri k additional colun, giing the atri A. k A as an [3.] Repeat the steps -2 taking k:k, until k reaches the assued horizon of forecast. As it will be shown, the recursie Lee-Carter procedure gies quite siilar estiates of a, b, for 0,,, ω as the standard Lee-Carter approach SLC. Howeer, the estiates of kt for t,2,, differ substantially. What is ore, the longer is the tie horizon of forecasting the greater is the difference in the estiates of k t. Both approaches and SLC will be used to forecast the crude death rates for Poland ale and feale. he forecasts will be then used to derie the ale s and feale s future life-tables for the year 2020.

7 Future Life-ables Based on the Lee-Carter Methodology Estiation of the Lee-Carter paraeters for Poland o estiate the Lee-Carter s paraeters for Poland we deried data fro life-tables for tie period that are aailable at the website of Central Statistical Office he estiates of paraeters a and b for 0,,,00, obtained by eans of the SLC and approaches, are presented on Figures 2 separately for en and woen. Based on the estiates of k t the paraeter was projected for future periods. o this end, the forula 3 was applied, where the constant c was estiated using 4. he estiates of k t together with forecasts are presented in Figure age recursie LC standard LC Fig.. Estiates of a for [0,00] fro the recursie and standard Lee-Carter odel odel fitting based on the period ale s life-tables for Source: Deeloped by the author.

8 38 Agnieszka Rossa 0,035 0,025 0,05 0, recursie LC age standard LC Fig. 2. Estiates of b for [0,00] fro the recursie and standard Lee-Carter odel odel fitting based on the period ale s life-tables for Source: Deeloped by the author. 40,0 30,0 20,0 0,0 projection 0, ,0-20,0-30,0-40,0-50,0 recursie LC standard LC Fig. 3. Estiates of k t fro the recursie and standard Lee-Carter odel for the years and its forecast odel fitting based on the period ale s life-tables for Source: Deeloped by the author.

9 Future Life-ables Based on the Lee-Carter Methodology recursie LC age standard LC Fig. 4. Estiates of a for [0,00] fro the recursie and standard Lee-Carter odel odel fitting based on the period feale s life-tables for Source: Deeloped by the author. 0,030 0,020 0,00 0, age recursie LC standard LC Fig. 5. Estiates of b for [0,00] fro the recursie and standard Lee-Carter odel odel fitting based on the period feale s life-tables for Source: Deeloped by the author.

10 40 Agnieszka Rossa 50,0 40,0 30,0 20,0 0,0 projection 0, , ,0-30,0-40,0-50,0 recursie LC standard LC Fig. 6. Estiates of k t fro the recursie and standard Lee-Carter odel for the years and the forecast odel fitting based on the period feale s life-tables for Source: Deeloped by the author. he projections of the paraeters k t for t> were then used to forecast soe lifetable paraeters within the tie horizon ables and 2 present annual probabilities of death q t giing surial to and their estiates q t and q SLC t for the year t2009 separately for en and woen obtained by eans of both the recursie and standard Lee-Carter approach. Based on these estiates the ean squared errors for both types of estiators in relation to the real alues of q t, for 0,,..., ω were calculated. Means squared errors were defined as follows: MSE t q t q t ω 2, ω 0 ω SLC SLC 2 MSE t q t q t, ω 0 where q t, q SLC t represent estiates of q t for the age group 0,,...,00 receied by eans of the recursie and standard Lee-Carter ethod for the year t2009. ables 3 and 4 present the ale s and feale s life epectancies e 2009 and their forecasts e t for t2009 and t2020.

11 Future Life-ables Based on the Lee-Carter Methodology 4 ab.. Forecasts of ale s death probabilities fro the recursie and standard LC odel 2009 SLC SLC Age q 2009 q 2009 q 2009 Age q 2009 q 2009 q MSE 0, ,00748 Source: Deeloped by the author.

12 42 Agnieszka Rossa ab. 2. Forecasts of feale s death probabilities fro the recursie and standard LC odel 2009 SLC SLC Age q 2009 q 2009 q 2009 Age q 2009 q 2009 q MSE 0,0043 0,0059 Source: Deeloped by the author.

13 Future Life-ables Based on the Lee-Carter Methodology 43 As we can see fro ables and 2, the death probabilities predicted by eans of the and SLC ethods are siilar for sall, but the differences increase for old ages. Due to these differences the ean squared errors attached in the last rows of SLC ables and 2 satisfy the inequality MSE 2009 < MSE ab. 3. he ale s life epectancy and its forecasts fro the recursie LC odel for 2009 and 2020 Age e 2009 e 2009 e 2020 Age e 2009 e 2009 e 2020 a b c d e f g h

14 44 Agnieszka Rossa ab. 3. cont. a b c d e f g h Source: Deeloped by the author. ab. 4. he feale s life epectancy and its forecasts fro the recursie LC odel for 2009 and 2020 Age e 2009 e 2009 e 2020 Age e 2009 e 2009 e 2020 a b c d e f g h

15 Future Life-ables Based on the Lee-Carter Methodology 45 ab. 4. cont. a b c d e f g h Source: Deeloped by the author. 7. Application of the future life-tables to calculation of pension annuities Let us consider the aount of a retireent incoe to be paid out by a pension proider to a person aged years we assue that age is rounded to an integer in the for of a life annuity. o deterine the aount of onthly payents, we shall use in this inestigation an actuarial forula eployed to calculate the present alue of the life annuity payable at the beginning of each onth [see Skałba, 2002]. Let us assue then that life annuity is paid out ties within a year for the onthly payents we hae 2, at the beginning of each subperiods the length of which is gien by / of a year, with the instalent aount being / zlotys so that the total annual aount due will be zl; this is a so-called noralized case. Let us assue further that the last payent is effected at the beginning of the subperiod in which the recipient dies. Let be a non-negatie rando ariable. representing the reaining lifetie of the -year-old person, while * [ ]. Besides, let S denote a rounded-up portion of the last year of recipient s life with accuracy to a subperiod lasting /. Note that S is a discrete rando ariable taking its alues fro the set 2,,,,. 8 he present alue Y of the payents is a rando ariable, which can be written using the following forula

16 Agnieszka Rossa 46 S Y * 2. where / i is the so-called discount rate and i is an aerage rate of return calculated for seeral periods [see Skałba, 2002, p. 3]. Using the forula for the su of a geoetric series q q a aq aq aq aq 2 0, the ariable Y can be epressed as follows. * * * 2 0 S S S Y We shall calculate the epected alue of Y which is denoted by a. Assuing that * and S are independent we hae. E E E E * * S S Y a 9 he epected alue * E represents the so-called present alue of a benefit equal zloty, payable at the end of year of death. In other words, it is an actuarial alue of whole life insurance on being zloty, payable at the end of the year of death. In the actuarial notation, it is usually denoted by A. he alue is 0 *, E k k k k q p A. 0

17 where probabilities Future Life-ables Based on the Lee-Carter Methodology 47 > k, q P k p P k k denote, respectiely, the probability of suriing k consecutie years giing surial to, and the probability of dying within a year giing surial to k. he other epected alue S E can be deterined assuing that the ariable S takes alues fro 8 with identical probabilities equal /. hen we hae E k S k k k 0 or after straightforward transforation S E. 2 Forulae 9, together with 0, 2, and using probabilities allow 2 calculating the epected alue a also a. Let K denote the aount of funds accuulated at the OPF by a person retiring at the age years. Le B denote the onthly pension annuity benefit that a pensioner receies fro the annuity proider. We will assue that the pension aount B is deried fro the following equation, related to the present alue of the life annuity paid onthly in adance [see e.g. Szulicz. ed., 2007] 2 K 2 B γ a. 3 where γ is a share of charges for the annuity proider. In order to derie a benefit B fro 3 it is necessary to take certain assuptions about the alue of the accuulated capital K and the share of charges 2 γ, etc. Besides, it is necessary to find the actuarial alue of life annuity a. which is calculated using the forulae 9, 0, 2, and by using estiates of probabilities. he probabilities will be deried for future periods using the Lee-Carter ethodology. he results will be copared with analogous calculations that were ade using probabilities deried fro life-tables published for 2007 year by the Central Statistical Office [ ].

18 48 Agnieszka Rossa 8. Scenarios of pension annuity calculations o calculate illustratie aounts of future old-age pension annuity using 3 the following assuptions were ade: the iniu retireent age : 60, 65 or 70 years; the calendar year at retireent 2008; the pension capital K equal or zlotys; the share of charge γ7%; the rate of return i equal 3% or 5%. he probabilities were used in two ways: case I based on the future probabilities future life-tables deried using the Lee-Carter ethodology and case II applying a period life-table for the year Besides, to analyse the ipact of gender on the leel of the benefit, the calculations were ade separately for en and woen. Result obtained are presented in tables 5-8. ab. 5. he effect of gender and iniu retireent age on the benefit leel K00000 zlotys, γ7%, i3% he iniu retireent age Monthly pension annuity zlotys woen en 60 years Case I 65 years years years Case II 65 years years Source: Deeloped by the author. ab. 6. he effect of gender and iniu retireent age on the benefit leel K zlotys, γ7%, i3% he iniu retireent age Monthly pension annuity zlotys woen en 60 years Case I 65 years years years Case II 65 years years Source: Deeloped by the author.

19 Future Life-ables Based on the Lee-Carter Methodology 49 ab. 7. he effect of gender and iniu retireent age on the aount of benefit K00000 zlotys, γ7%, i5% he iniu retireent age Monthly pension annuity zlotys woen en 60 years Case I 65 years years years Case II 65 years years Source: Deeloped by the author. ab. 8. he effect of gender and iniu retireent age on the benefit leel K zlotys, γ7%, i5% he iniu retireent age Monthly pension annuity zlotys woen en 60 years Case I 65 years years years Case II 65 years years Source: Deeloped by the author. 9. Conclusions he results obtained reeal substantial ariations in the benefits calculated using the future and current life tables see cases I and II and indicate that the pension annuities grow when the iniu retireent age is oed upwards. Lower alues of onthly payents are proided fro using future life tables in the calculations case I instead of period life-tables for the year 2007 case II. he illustratie results presented in tables 5 8 show that calculating the annuities using the period life-tables ay epose the annuity proider to a risk of the considerable oerestiation of benefits and thus ay cause troubles with coering future liabilities. We can also see that the feale s pension annuities are lower both in the case I and II. his can be eplained through the fact that epected lifetie in the feale population is longer than in the ale population, which directly contributes to lower aounts of benefits. It is also worth noting that een though gender is a distinct deterinant of different benefits, in practice they are calculated using the coon life-tables. his leads to the oerestiation of benefits for feales and underestiation of benefits for ales.

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