D I S C U S S I O N P A P E R

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1 I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worst-case actuarial calculatios cosistet with sigle- ad multiple-decremet life tables CHRISTIANSEN, M.C. ad M. DENUIT

2 WORST-CASE ACTUARIAL CALCULATIONS CONSISTENT WITH SINGLE- AND MULTIPLE-DECREMENT LIFE TABLES MARCUS C. CHRISTIANSEN Istitute of Isurace Sciece Uiversity of Ulm D Ulm, Germay MICHEL M. DENUIT Istitut de Statistique, Biostatistique et Scieces Actuarielles Uiversité Catholique de Louvai B-1348 Louvai-la-Neuve, Belgium Jue 15, 2012

3 Abstract The preset work complemets the recet paper by Barz ad Müller (2012. Specifically, upper ad lower bouds are derived for the force of mortality whe oe-year death probabilities are give, assumig a mootoic or covex shape. Based o these bouds, worst-case scearios are derived depedig o the mix of beefits i case of survival ad of death comprised i a specific isurace policy. Key Words: life isurace, multistate models, first-order life tables.

4 1 Itroductio I actuarial studies, life tables are based upo a aalytical framework i which death is viewed as a evet whose occurrece is probabilistic i ature. I the sigle-decremet case, the differet causes of death are ot distiguished. Life tables create a hypothetical cohort of, say, 100,000 persos at age 0 ad subject it to all-cause age-specific oe-year death probabilities q x (the umber of deaths per 1,000 or 10,000 or 100,000 persos of a give age observed i a give populatio. I doig this, researchers ca trace how the 100,000 hypothetical persos (called a sythetic cohort would shrik i umbers due to deaths as the group ages. Deotig as l x the expected umber of survivors at age x, we have the recurrece relatio l x+1 = l x (1 q x, startig from a fixed umber l 0 of ewbors. Here, p x = 1 q x is the oe-year survival probability at age x. Multiple decremet life tables distiguish betwee differet causes of death (traffic accidet, suicide, cacer, etc.. They are useful whe the beefit i case of death may deped o the cause of death (typically, higher beefit i case of accidetal death, or o beefit i case death is due to some particular cause. Such life tables display the expected umber l x of idividuals alive at age x together with the expected umber of death due to cause 1 to m, say. The cause of death is defied as the disease or ijury that iitiated the sequece of evets leadig directly to death. It is selected from the Iteratioal Classificatio of Diseases (ICD, for istace. All-cause death rates are the replaced with cause-specific death rates. The 4 leadig causes of death are geerally heart disease, stroke, accidets, ad cacer. See, e.g., Jemal et al. (2005 for more details. As explaied above, a life table ca be see as a sequece of oe-year survival probability p x or death probability q x idexed by iteger age x ragig from birth (or some other iitial age α to a ultimate age ω. The kowledge of the life table itself does ot characterize the probability distributio of a radom lifetime T coformig to it. For iteger x, we have Pr[T > x] = lx l 0 but these probabilities are ot kow for fractioal ages. This is why actuaries ad demographers developed so-called fractioal age assumptios allowig to iterpolate survival fuctios betwee cosecutive values lx l 0 ad l x+1 l 0 for iteger x = α, α + 1,..., ω 1. Some recet developmets i this regard are Joes ad Mereu (2000, 2002, Frostig (2002, 2003, ad Hossai (2011. Barz ad Müller (2012 ordered the differet fractioal age assumptios ad built radom variables that are stochastically larger tha other lifetimes subject to icreasig force of mortality ad give oe-year survival probabilities. Assumig that the preset value of beefits is a mootoic fuctio of policyholder s remaiig lifetime (as i the whole life isurace cover or i a auity cotract, for istace, Barz ad Müller (2012 derived bouds o actuarial quatities uder the assumptio that the force of mortality is icreasig ad coheret with a give sequece of oe-year survival probabilities. The icreasigess of the forces of mortality seems to be i lie with observatios (at least at adult ages. Typically, oe-year death probabilities are relatively high i the first year after birth, declie rapidly to a low poit aroud age 10, ad thereafter rise, i a roughly expoetial fashio, before deceleratig (or slowig their rate of icrease at the ed of the life spa. Except aroud the accidet hump due to violet deaths (suicide ad traffic accidets, maily at youg adult ages ad some cotroversy about mortality of supercetearias, it is reasoable to assume that all-cause forces of mortality icrease mootoically with age. 1

5 The preset paper aims to relax the two mai assumptios made i Barz ad Müller (2012: icreasig death rates ad isurer s liabilities mootoic with respect to the policyholder s remaiig lifetime. Ideed, the majority of isurace products mix beefits i case of death ad i case of survival so that the preset value of beefits may ot be mootoic i the policyholder s remaiig lifetime. Moreover, whe multiple causes of death are cosidered, the mootoicity coditio imposed to death rates does ot apply to certai causes (like suicide or traffic accidets, for istace while it remais plausible for others (like cacer, for example. The preset paper relaxes these rather strog assumptios ad derives worst-case scearios i lie with a specific life table ad give the mix comprised i a specific isurace product. Precisely, we show how the worst-case techiques of Christiase ad Deuit (2010 ad Christiase (2010 ca be used to obtai bouds where the methods of Barz ad Müller (2012 are ot applicable. The ext table summarizes the extesios to Barz ad Müller (2012 derived i the preset paper, stressig the techical differeces arisig betwee the two studies: Barz ad Müller, 2012 preset paper patter of states sigle-decremet sigle- or multiplealive-dead model decremet model cotract desig liability mootoe with all mix of survival respect to lifetime ad death beefits bouds (partially sharp ot sharp but accurate eough for practical purposes The paper is orgaized as follows. Sectio 2 presets the multistate model used i this paper ad makes the lik with the approach of Barz ad Müller (2012. Sectio 3 cosiders the 2-state alive-dead model, or sigle-decremet case whereas Sectio 4 exteds the results to the multiple-decremet case. Based o the mootoicity or covexity of forces of mortality, bouds are derived o every age iterval. Cosiderig the sig of the sum-at-risk, worst-case sceario ad best-case sceario for the reserves correspod to these lower or upper bouds. Some umerical illustratios demostrate the accuracy of these values. Note that lapses ca also be accouted for by addig a state m + 1 correspodig to the exit from the portfolio, with appropriate lapse rates. This is aother major improvemet to Barz ad Müller ( Survival model 2.1 Multiple-decremet model Assume that the history of each policyholder is described by a right-cotiuous Markovia process {X t, t 0} with state space {0, 1,..., m}, where 0 is the iitial state alive ad 1 to m correspod to m differet causes of death. As metioed i the itroductio, oe of the states 1 to m may correspod to lapse or surreder. Here, time t measures the seiority of the policy (i.e., the time elapsed sice policy issue. Policyholder s age at policy issue is deoted as x, so that the age at time t is x + t. Heceforth, all quatities are idexed by t (the correspodig age beig x + t. 2

6 The probability distributio of the Markov process {X t, t 0} is uiquely described by the forces of mortality µ 0j, j = 1,..., m, defied as µ 0j (t = lim t 0 Pr[X t+ t = j X t = 0]. t We cosider a geeral life isurace cotract icludig beefits i case of survival (sojour i state 0 ad i case of death (trasitio from state 0 to some state 1 to m. We write B 0 (t for the aggregated survival beefits mius premiums o [0, t] ad b 0j (t for the beefit paymets that are due i case of a death due to cause j occurrig at time t (b 0j may also represet the surreder value. For mathematical techical reasos we assume that the fuctios t B 0 (t ad t b 0j (t, j = 1,..., m, have fiite variatio o compacts ad that B 0 (t is right-cotiuous. The cotract termiates at time ω x = ω x <, at the latest. The policyholder s remaiig lifetime T ca be defied by meas of the process {X t, t 0} by T = sup{t 0 X t = 0}. I words, T is the istat whe the process {X t, t 0} jumps from state 0 to oe of the absorbig states 1 to m. Let ϕ(t be the iterest itesity (or spot rate. The preset value of future beefits mius premiums for a policyholder alive at time t is the give by A(t = (t,t ( exp s 0 ϕ(udu db 0 (s + The so-called prospective reserve at time t i state 0 is defied as 2.2 Sigle-decremet model m j=1 V (t = E[A(t X t = 0] = E[A(t T > t]. ( T exp ϕ(udu b 0j (T. 0 I the two-state model with state space {0, 1}, 0= alive ad 1= dead, Barz ad Müller (2012 assumed that the preset value of future beefits mius premiums at time 0 is of the form f(t, where f is a mootoe fuctio. By choosig B 0 (t = 0 ad ( t b 01 (t = exp ϕ(sds f(t 0 we obtai A(0 = f(t, which meas that we ca itegrate the modelig framework of Barz ad Müller (2012 ito our framework. Note that f has fiite variatio o compacts sice it is mootoe. 3

7 3 Worst-case sceario i the sigle-decremet model 3.1 Icreasig force of mortality I this sectio, we deal with a two-state space {0, 1} correspodig to the sigle-decremet model. We assume that ( +1 the values of p = exp µ 01 (t dt are kow for every iteger, (3.1 the trasitio itesity µ 01 (t is mootoic icreasig. (3.2 Note that the quatities are idexed by cotract seiority, ot by age (this meas that p is the oe-year survival probability at age x +. Let M 01 be the set of all trasitio itesities µ 01 that satisfy (3.1 ad (3.2. The iformatio cotaied i (3.1 ad (3.2 does ot suffice to uiquely defie the trasitio itesity µ 01, ad so M 01 has ifiitely may elemets. We wat to fid upper ad lower bouds for the prospective reserve V (t, give that µ 01 M 01. From the two coditios (3.1-(3.2 above we ca coclude that we must have for iteger times, which implies that µ 01 ( l p µ 01 ( + 1 l p 1 µ 01 ( l p for all positive itegers. Sice µ 01 (t is mootoic icreasig, we get l p 1 µ 01 (t l p +1, t + 1. (3.3 Remark 3.1. The same approach applies to decreasig forces of mortality, eve if this situatio should ot happe except i some very particular cases (life settlemets for idividuals who just udergo serious surgery, whose chaces of survival icrease as time passes, for istace. If µ 01 is decreasig, we get µ 01 ( l p µ 01 ( + 1. Thus, we ca show that, provided the force of mortality is mootoic (either decreasig or icreasig, the iequalities hold true for t + 1. mi{ l p 1, l p +1 } µ 01 (t max{ l p 1, l p +1 } 3.2 Covex or cocave forces of mortality Eve if icreasigess is geerally a reasoable assumptio for all-cause death rates, let us ow derive upper ad lower bouds for other shapes of mortality itesity. This assumptio may apply to some cause-specific death rates, like suicide or traffic accidets, for istace. The ext results allow us to derive bouds o covex forces of mortality. Propositio 3.2. Assume that µ 01 is covex. If µ 01 (t l p + c for t [, + 1], the µ 01 (t l p c for all t [, + 1]. 4

8 Proof. I order to get if t [,+1] µ 01 (t uder the coditio +1 µ 01 (u du = l p, the optimal shape for the covex fuctio µ 01 is a triagle with the apex poitig dowwards. As all triagles have a upper boud of l p + c ad as the areas where µ 01 is above ad below l p must be equal, the apex of the triagles caot be below l p c. Propositio 3.3. If µ 01 is covex the the iequality µ 01 (t max{ l p 1, l p +1 } holds for all t + 1. Proof. If the iterval where the covex fuctio µ 01 (t is miimal is to the left of, the µ 01 (t is icreasig for all t ad µ 01 (t l p +1 for all t [, + 1]. Aalogously, if the iterval where µ 01 (t is miimal is to the right of + 1, the µ 01 (t l p 1 for all t [, + 1]. If ξ [, + 1] is a miimum of µ 01, the µ 01 is decreasig till ξ ad icreasig from ξ o, which implies that o [, ξ] ad [ξ, + 1] the fuctio µ 01 is bouded by l p 1 ad l p +1, respectively. Combiig the two propositios above, we get the followig iequalities o a covex force of mortality: 2( l p max{ l p 1, l p +1 } µ 01 (t max{ l p 1, l p +1 } (3.4 for all t [, + 1]. Assumig that µ 01 is cocave, the same reasoig yields mi{ l p 1, l p +1 } µ 01 (t 2( l p + mi{ l p 1, l p +1 } (3.5 for all t [, + 1]. 3.3 Worst-case ad best-case scearios for the reserve Let us start with the case where the all-cause force of mortality µ 01 is icreasig. Let us deote as N 01 the set of all icreasig µ 01 that satisfy (3.3. Let l 01 (t ad u 01 (t be the lower ad upper boud (accordig to (3.3 o µ 01 (t at time t. Clearly, M 01 N 01, ad, hece, upper ad lower bouds with respect to N 01 are also upper ad lower bouds with respect to M 01. With the help of the worst-case techique of Christiase ad Deuit (2010 for mortality itesities, we obtai the followig result. Propositio 3.4. The best-case reserve V (t = if µ01 N 01 V (t; µ 01 ad the worst-case reserve V (t = sup µ01 N 01 V (t; µ 01 uiquely solve the itegral equatios V (t =B 0 (ω x B 0 (t V (s ϕ(s ds (t,ω x] + 1 ( (b 01 (s V (s (u 01 (s + l 01 (s b 01 (s V (s (u 01 (s l 01 (s dt, 2 (t,ω x] V (t =B 0 (ω x B 0 (t V (s ϕ(s ds (t,ω x] (t,ω x] ( (b 01 (s V (s (u 01 (s + l 01 (s + b 01 (s V (s (u 01 (s l 01 (s dt. 5

9 Furthermore, applyig the results i Christiase (2010 allows us to associate V (t ad V (t to specific mortality itesities, as show ext. Propositio 3.5. Defie µ 01 (t = 1 b01 (t V (tl 01 (t + 1 b01 (t<v (tu 01 (t, µ 01 (t = 1 b01 (t V (t u 01(t + 1 b01 (t<v (t l 01(t. The, V (t = V (t; µ 01 ad V (t = V (t; µ 01 for all t. Depedig o the sig of the sum-at-risk at time t (differece of the reserve ad b 01 (t, or et cost of a trasitio at time t for the isurace compay i case of a death occurrig at time t, the worst-case ad best-case reserves the correspod to the lower or to the upper boud o the all-cause force of mortality µ 01. While the best-case ad worst-case itesities of Barz ad Müller (2012 are elemets of M 01, the itesities µ 01 (t ad µ 01 (t are either i M 01 or i N 01 \ M 01. Thus, our bouds for the prospective reserve are less tight tha the bouds give by Barz ad Müller (2012. However, Barz ad Müller (2012 assume that A(0 is mootoe with respect to the remaiig lifetime T, whereas our modelig framework does ot eed that mootoicity, allowig to study also mixed cotracts with ay combiatio of sojour beefits, premium paymets ad trasitio beefits. The same approach applies to decreasig, covex or cocave µ 01. It suffices to replace the bouds i (3.3 with those of Remark 3.1, (3.4 or (3.5. The followig umerical example compares the bouds derived from Propositios to those i Barz ad Müller (2012. Example 3.6. As i Barz ad Müller (2012, we assume that µ 01 (t = t, but that oly the p at iteger times (see (3.1 are actually kow. Cosider a whole-life isurace cover with a uit death beefit ad 6% aual iterest rate. Table 1 shows upper ad lower bouds for the prospective reserves A 30 ad A 50 for mootoic or covex force of mortality. Compared to the values reported i Barz ad Müller (2012, our bouds are less accurate but remai evertheless sharp eough for practical purposes. I the last two colums the table gives bouds for the prospective reserves of edowmet isuraces that pay a survival beefit of 2 at age 65 or a death beefit of 1, whichever occurs first. For the edowmet isuraces the preset values of future paymets are ot mootoe with respect to the policyholder s remaiig lifetime, ad so the method of Barz ad Müller (2012 is ot applicable here while the method developed i the preset paper still is. The coclusios stated before still apply i this case: the bouds are accurate eough to be iformative. 6

10 mootoicity assumptio A 30 A A E A E 50 upper boud true value lower boud covexity assumptio A 30 A A E A E 50 upper boud true value lower boud Table 1: Worst-case ad best-case scearios for whole-life isurace cover ad edowmet isurace described i Example Worst-case sceario for multiple-decremet models Let us ow exted the ideas of the previous sectio to multiple-decremet models. Defie the oe-year death probability due to cause j ( +1 ξ m q (j = Pr[X +1 = j X = 0] = exp µ 0k (ηdη µ 0j (ξdξ ad the oe-year survival probability m p = 1 j=1 q (j = exp ( +1 Clearly, q = m j=1 q(j. Istead of (3.1 ad (3.2, let us ow assume that k=1 m µ 0j (ηdη. j=1 the values of q (j are kow at iteger times. (4.1 As explaied above, whe multiple causes of death are cosidered, icreasigess may become urealistic for some specific decremets. This is why we cosider death rates with various shapes i the ext result, all complyig with assumptio (4.1. Propositio 4.1. (i If µ 0j is either icreasig or decreasig the { } mi{q 1, (j q +1} (j q (j 1 q (j +1 µ 0j (t max,, t q 1 1 q +1 (ii If µ 0j is covex the ( { } q (j q (j 1 q (j +1 max, q(j µ 0j (t (4.2 1 q 1 1 q +1 1 q { } q (j 1 q (j +1 max,, t [, + 1]. 1 q 1 1 q +1 7

11 (iii If µ 0j is cocave the mi{q (j 1, q (j +1} µ 0j (t ( q(j + q (j mi{q (j 1 q 1, q +1} (j, t [, + 1]. Proof. Let us establish (i. Sice the itesities µ 0j (u are o-egative, we have (4.3 ad hece +1 µ 0j (u (1 q du q (j +1 µ 0j (u du, q (j +1 µ 0j (u du q(j, (4.4 1 q whece the aouced iequality follows. Let us ow tur to (ii. As i the alive-dead model, covexity implies that we first have decreasigess ad the icreasigess. Hece, { } q (j 1 q (j +1 µ 0j (t max,, t q 1 1 q +1 Also similar to the alive-dead model, we have that if µ 0j (t q(j 1 q + c o t [, + 1], the µ 0j (t q (j c for all t [, + 1], which leads to the lower boud i (4.2. The proof for (iii is similar. Let N be the set of all trasitio itesities µ 0j, j = 1,..., m, that satisfy (4.1. Furthermore, we write l 0j (t ad u 0j (t for the lower ad upper boud o µ 0j (t at time t (accordig to Propositio 4.1(i, (ii or (iii. By applyig the worst-case method of Christiase (2010, we obtai the followig geeralizatio of Propositio 3.4. Propositio 4.2. The best-case reserve V (t = if µ N V (t; µ ad worst-case reserves V (t = sup µ N V (t; µ uiquely solve the itegral equatios V (t =B 0 (ω x B 0 (t (t,ω x] V (s ϕ(s ds + 1 ( (b 0j (s V (s (u 0j (s + l 0j (s b 0j (s V (s (u 0j (s l 0j (s dt, 2 j=1 (t,ω x] V (t =B 0 (ω x B 0 (t V (s ϕ(s ds j=1 (t,ω x] (t,ω x] ( (b 0j (s V (s (u 0j (s + l 0j (s + b 0j (s V (s (u 0j (s l 0j (s dt. 8

12 mootoy assumptio reserve at age 30 reserve at age 50 upper boud true value lower boud covexity assumptio reserve at age 30 reserve at age 50 upper boud true value lower boud Table 2: Worst-case ad best-case scearios for the combiatio of critical illess cover with edowmet isurace described i Example 4.4. For the worst-case reserve the proof is give i Christiase (2010. The proof for the best-case reserve is aalogous. From the best-case ad worst-case reserves we ca derive the correspodig trasitio itesities, as stated ext. Propositio 4.3. Defie µ 0j (t = 1 b0j (t V (tl 0j (t + 1 b0j (t<v (tu 0j (t, µ 0j (t = 1 b0j (t V (t u 0j(t + 1 b0j (t<v (t l 0j(t. The, V (t = V (t; µ ad V (t = V (t; µ for all t. The followig umerical example illustrates the accuracy of the bouds derived i Propositio 4.2. Example 4.4. Cosider a mixture of a edowmet isurace ad a critical illess isurace. If the policyholder gets a critical illess before age 65, a paymet of 2 is made ad the cotract termiates. If the policyholder does ot icur a critical illess but dies before age 65, a death beefit of 1 is paid. If oe of these two evets occurs, a survival beefit of 2 is made at age 65. Oly the trasitios active to dead ad active to ill are relevat here. We assume that the mortality itesity has the same form as i Example 3.6 ad that the morbidity itesity has the form µ 02 (t = exp(0.065t, but that the isurer has oly iformatio o q (1 ad q (2 at iteger times (see (4.1. Table 2 shows lower ad upper bouds for the prospective reserve i state active. We see that the bouds derived from Propositio 4.2 are accurate eough for practical purposes. Ackowledgemets Michel Deuit ackowledges the fiacial support of AG Isurace uder the K.U.Leuve Health Isurace Chair. 9

13 Refereces Barz, C., Müller, A., Compariso ad bouds for fuctioals of future lifetimes cosistet with life tables. Isurace: Mathematics ad Ecoomics 50, Christiase, M.C., Biometric worst-case scearios for multi-state life isurace policies. Isurace: Mathematics ad Ecoomics 47, Christiase, M.C., Deuit, M.M., First-order mortality rates ad safe-side actuarial calculatios i life isurace. ASTIN Bulleti 40, Frostig, E., Compariso betwee future lifetime distributio ad its approximatio. North America Actuarial Joural 6(2, Frostig, E., Properties of the power family of fractioal age approximatios. Isurace:Mathematics ad Ecoomics 33, Hossai, S.A., Quadratic fractioal age assumptio revisited. Lifetime Data Aal 17, Jemal, A., Ward, E., Hao, Y., Thu, M., Treds i the leadig causes of death i the Uited States, Joural of the America Medical Associatio 294, Joes B.L., Mereu J.A., A family of fractioal age assumptios. Isurace:Mathematics ad Ecoomics 27, Joes B.L., Mereu J.A., A critique of fractioal age assumptios. Isurace:Mathematics ad Ecoomics 30,

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