Lu Xia Effectivity in Hedging Longevity Risk A Life Insurance Scheme of a Child Plan

Transcription

1 Effeciviy in Hedging Longeviy Ris A Life Insurance Scheme of a Child Plan MSc Thesis

2 Tilburg Universiy e EFFECTIVITY IN HEDGING LONGEVITY RISK: A LIFE INSURANCE SCHEME OF A CHILD PLAN MASTER THESIS FOR ECONOMICS AND FINANCE OF AGING February 27, 2013 Absrac: In his aricle, we discuss a life insurance scheme-child Plan. We use he Lee-Carer and he CBD Models o forecas moraliy daa of US and UK in order o quanify he Plans. We compare he Child Plan wih a Single Life Insurance Plan, and find ha he Child Plan has a lower price (ne premium) and is less sensiive o moraliy ris. So, we draw a conclusion ha he Child Plan has an inner hedging sysem owing o a join deah probabiliy design. Auhor: Supervisor: Berrand Melenberg We also find ha addiional benefi condiions-income benefi and boh parens insured will increase he price of he Child Plan and cause a higher ris o he insurer. And he older he insurers are, he higher he price and he degree of sensiiviy o moraliy riss are. K e y w o r d s : agi n g, m o r a l i y r a e, l i f e i n s u r a n c e, C h i l d P l a n, h e d g i n g

3 Maser Thesis of Economics and Finance of Aging 2011 Conen 1 Inroducion Bacground and Benchmar Plan Noaion Aging in UK and US Benchmar Plan Summary Modeling Moraliy in US Lee-Carer Model Specificaion Fiing he Lee-Carer Model Forecass for US daa Daa Period Adjusmen Cohor Life Table in US Modeling Moraliy in UK Cairns-Blae-Dowd Model Specificaion Fiing he CBD Model Finess Comparison for he CBD Model Forecass for UK Daa Cohor Life Table in UK Child Plan Exension Se up New Plans Older Paren(s) Insured... 39

4 Maser Thesis of Economics and Finance of Aging Sensiiviy o Moraliy Model Ris Sensiiviy Analysis under Lee-Carer Model Sensiiviy Analysis under CBD Model Summary and Discussion on Hedging Summary and Conclusion Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Reference... 67

5 1 Inroducion 1 1 Inroducion Child Plan is an Insurance produc o ensure children s fuure by financing he ey milesones in heir lives (college educaion, marriage, ec.). Usually, i includes wo pars. The firs par is he normal mauriy benefi: he insured receives he accumulaed amoun of guaraneed fixed benefi a he mauriy dae. The second par is he deah benefi: if he insured paren dies, he remaining premium paymens will sop, and he beneficiary will sill receive he guaraneed fixed benefi immediaely or on he mauriy dae. Furhermore, some companies have addiional policy ha an amoun of money which equals o he annualized premium will be paid o he beneficiary saring from he dae of deah ill mauriy dae. This benefi is nown as income benefi which is a par of deah benefi. As many aricles saed (for example: Bija, Kupiszewsa, e al., 2007; Bovenberg, 2008; Chrisensen, Doblhammer, e al., 2009), European counries sep ino an era of aging. Longer life expecancy and lower feriliy rae accelerae he aging problem. And an increase of mean age of European mohers a he firs childbirh is a reason of lower feriliy. For Child Plan, hese wo facors affec is pricing in opposie direcions. Declining deah probabiliy of parens reduces he probabiliy of he premium loss for he insurance company, and he delay of he firs childbirh means a moraliy able wih higher age groups should be used in he calculaion of premium. And higher age groups in mos cases indicae a higher moraliy ris. In his paper, we analyze how he Child Plan reacs o he moraliy rae change, and figure ou how differen benefi condiions wor in order o hedge he longeviy ris. The layou of his paper is presened as follows. In chaper 2, we sar wih a bacground of he recen aging rend and hen ouline a basic Child Plan as he benchmar. US daa and UK daa will be used for comparison. Tha is because he hisorical daa shows ha US and UK boh encouner a similar large decline of moraliy rae in recen years, whils he mean age of mohers a he firs childbirh in UK increased o 30 while he daa in US ep around 25. This five-year gap will lead o a differen life able for he insured. In chaper 3 and chaper 4 of his paper, I fi Lee-Carer model (Lee and Carer, 1992) o US daa and fi CBD model (Cairns, Blae, and Dowd, 2006) o UK daa respecively, and hen discuss he plausibiliy of he fi. This sep help us o ge he furher forecasing of moraliy rae, which will be a quanified ool for us o assume moraliy rae change. In chaper 5, we go bac o he benchmar Child Plan and invesigae some exensions of his Child Plan. We add wo benefi condiions ino he Plan: income benefi and boh parens insured. We also increase he age of he insured paren(s) o discuss he impac of insured age. We analyze sensiiviy o moraliy rae of each benefi condiions in Chaper 6. We simulae hree

6 1 Inroducion 2 moraliy ris scenarios and draw sensiiviy curves o show he degree of sensiiviy. In conclusion, we find ha he Child Plan has an inner hedging sysem by a join deah probabiliy design. When a moraliy-lined issue comes and he moraliy raes of he aduls and he children change a he same ime, he Child Plan will suffer less loss or gain less compared wih a Single Insurance Plan. The condiions of income benefi and boh parens insured will increase he ne premium of he Child Plan respecively, and have larger exposure o moraliy riss. Also, he older he insured is, he higher he ne premium and degree of sensiiviy o moraliy riss are, which can be found in US and UK daa boh. This aricle is a discussion abou moraliy ris in one ind of life insurance plan-child Plan. The approach o quanify or simulae he moraliy ris in his aricle can also be applied o oher moraliy lined producs lie annuiy, pension, and oher life insurance producs. And he sensiiviy curve is an effecive ool o measure he ris exposure o moraliy riss.

7 2 Bacground and Benchmar Plan 3 2 Bacground and Benchmar Plan 2.1 Noaion We analyze Child Plan wih moraliy and survival relaed daa, and we use wo moraliy models o obain such daa. Thus, i would be beer o idenify a consisen and clear noaion hroughou. Since our original daa of moraliy relaed are all from Human Moraliy Daabase, we reference he noaion and daa explanaion of HMD o define he noaion in his paper. Calendar year is defined as running from ime o ime + 1. Deah raes consis of deah couns divided by he exposure-o-ris. We use m x, as cenral deah rae for age x in calendar year (shown as mx in HMD). More specifically, m x, # deahs during calendar year aged x in las birhday average populaion during calendar year aged x in las birhday (2.1) This cenral deah rae will be used in chaper 3 for Lee-Carer model. Bu we will use anoher measure of moraliy more frequenly, which is called deah probabiliy q x, (or q x for shor, and shown as qx in HMD). This is he probabiliy ha an individual aged exacly x a exac ime will die beween and + 1. In some aricles his measured daa is also called moraliy rae. To disinguish wih he formal daa, we call i deah probabiliy or one-year deah probabiliy. Cenral deah rae and deah probabiliy are he only wo original daa in his paper: we use hisorical dae from HMD and fuure daa from he forecas resuls of Lee-Carer and CBD model. In chaper 2 and 5, here is also a hird measure of moraliy called survival probabiliy T p x,. This is inerpreed as he probabiliy ha he individual aged x a ime survives a leas anoher T years. Bu we always ge his moraliy daa by calculaion wih he following equaion: p ( 1 q ) ( 1 q ) ( 1 q ) ( 1 q ) (2.2) T1 T x, x, x1, 1 xt 1, T 1 x+i,+i i Relaionship beween m x, and q x, The deah rae, m x,, and he deah probabiliy, q x,, are wo differen ways o measure moraliy. The approximae relaion beween hese wo coefficiens is as following: q m x, x, (2.3) 1 ( 1 ax, ) mx, Here a x, refers o he average lengh of survival beween ages x and x+n for people dying in he inerval during year. For he 1*1 year period life able we used from HMD, n is 1 and ax equals o 0.5 for age from 1 o 109. This can be inuiively inerpreed lie his: for an individual die beween age x and x+1 (0<x<109), he deah could happen in any ime of a calendar year wih equal probabiliy, hus for a large populaion of people dying in his year, he middle of year is liely o be he ime poin ha half of he people in his populaion have died.

8 Bacground and Benchmar Plan 4 However, when i comes o new born babies younger han one-year old and raher old elders, he resuls are no he same. Considering he relaive high deah probabiliy of infans, half year is no he average lengh of survival for babies dying before one-year-old birhday. In HMD, a x, usually aes a value below Since infan moraliy is remarably improving, a x, for age 0 are also differen among years and shows an increasing rend in recen 50 years. And for age above 110, since here is no yearly daa bu a summary deah moraliy rae as age of 110+, he a x, for his age is hus bigger han 1 year and varies for differen years. In chaper 3, when we ge a fied model of he Lee-Carer approach for cenral deah rae, we use he above approximaion mehod o ransfer he resuls ino deah probabiliy. Since in his aricle, we only need he deah probabiliy forecas for individuals above one-year old and less han 43 (we assume Plans wih laes enry year of 2006 and he laes hisorical daa we have for US is daa in year 2007), so we always use a x, equal o Aging in UK and US An unprecedened improvemen in populaion longeviy has been observed worldwide (see for example, Benjamin and Soliman 1993, McDonald 1997, and McDonald e al. 1998). In 1960, he Average life expecancy a birh varied in OECD counries beween 48.3 in Turey and above 73 in Norway and Neherlands. During he pas 50 years, average life expecancy a birh for OECD as a whole has increased by abou 2.5 years per decade (Bovenberg, 2008) (Figure 1). This longeviy rend, combined wih lower feriliy rae are he main causes of aging (Bovenberg, 2008) and will change he age srucure of he populaion in OECD counries in 2050 (Figure 2). Figure 1 34 OECD Counries Life Expecancy a Birh: Toal Populaion Ausralia Ausria Belgium Canada Chile Czech Republic Denmar Esonia Finland France Germany Greece Hungary Iceland Ireland Israel Ialy Japan Korea Luxembourg Mexico Neherlands New Zealand Norway Poland Porugal Slova Republic Slovenia Spain Sweden Swizerland Turey Unied Kingdom Unied Saes Source: OECD Healh Daa 2011, November 2011

9 2 Bacground and Benchmar Plan 5 Figure 2 OECD Toal Populaion by Age Group, Gender, in 2000 and 2050, in Percenage of Toal Populaion in Each Group in 2000 in 2050,10,8,6 MEN,4,2,0 OECD-oal WOMEN,0,2,4,6,8,10 in 2000: 1,129.6 Toal populaion (in millions) in 2050: 1,334.0 Source: OECD Demographic and Labour Force daabase, used in OECD (2007), Sociey a a Glance: OECD Social Indicaors 2006 To measure he degree of aging and he change of he age srucure, an indicaor named Age-dependency Raios is applied: he Youh-dependency Raio (number of individuals aged less han 20 over o he populaion aged 20 o 64) and he Old-age-dependency Raio (number of individuals aged 65 over o he populaion aged 20 o 64). From 1980 o 2050, he Old-age-dependency Raio is projeced o more han double (from 20% o 47%) in OECD area. In UK and US, his raio is projeced o rise sharply from 27% o 47% and 20% o 39%, respecively. Conversely, he Youh-dependency Raio will decline in he fuure. By 2050, his raio is projeced o reach a level of 40%, wih a decline of 24 percens (around 1/4) in OECD average. The raio in UK and US will drop less sharply, bu will also fall below 50%, wih a level of 38% and 49% (Figure 3). Figure 3 Age-dependency Raio, in 1980, 2000 and 2050, in OECD Toal, UK and US Old-age-dependency raio 47% 47% 39% 27% 27% 20% 20% 22% 21% 60% 40% 20% Youh-dependency raio 56% 53% 47% 48% 49% 43% 40% 38% OECD Toal UK US 0% OECD Toal UK US Source: same o Figure 2

10 2 Bacground and Benchmar Plan 6 The fall in he youh-dependency raio may lower public expendiures in educaion, bu hese declines are no large enough o offse higher spending owards he elderly 1. The rend of aging drives a subsanial growh of annuiy mare in life insurance companies. However, moraliy improvemen also poses a challenge o life insurance companies (Olivieri 2001 and Coppola, Di Lorenzo, and Sibilo 2002), because i delays he payou period and increases he liabiliy for providing he annuiy. Therefore, i raises he imporance of hedging longeviy ris. Many sudies have invesigaed he issue of hedging longeviy ris in radiional life annuiy, life insurance produc or oher moraliy derivaives (see for example, Milevsy and Promislow 2001, 2002, Blae and Burrows 2001, Charupa and Milevsy 2001 ) (Wang, Huang, e al., 2010), in his paper, a sor of insurance scheme named Child Plan will be invesigaed on hedging longeviy ris issue. The life expecancy increase didn rise smoohly across ages. Compared o he one-year deah probabiliy of period life able 2 a 1980 and 2006, i can be found in boh UK and US daa ha he survival probabiliy changes remain less han 0.5% beween hese wo generaion from age 1 o age 50, excep a significan improvemen in survival rae of new born infans (below one-year old). Meanwhile, from age 50 o around 90, he improvemen percenage is projeced o increase faser and faser o a level of 6% in UK and 4% in US (Figure 4). To sum up, he one-year deah probabiliy declines are projeced o be significan among he infans and he elderly, and he one-year deah probabiliy eep relaively sable for oher groups (children, he young and he middle-aged). 1 SOCIETY AT A GLANCE: OECD SOCIAL INDICATORS 2006 EDITION 2 A period life able is supposed o represen he moraliy condiions a a specific momen in ime, whereas a cohor life able depics he life hisory of a specific group of individuals.

11 One-year deah probabiliy Survival probabiliy Improvemen % One-year deah probabiliy Survival probabiliy Improvemen % 2 Bacground and Benchmar Plan 7 Figure 4 One-year Deah Probabiliy, in UK and US, birh in 1980 and 2006, and Survival Probabiliy Improvemen Percenage UK Age 7% 6% 5% 4% 3% 2% 1% 0% Improvemen % US Age 4% 3% 2% 1% 0% -1% -2% Improvemen % Source: Human Moraliy Daabase 3 To avoid a negaive improvemen rae, one-year survival probabiliy, insead of deah probabiliy is used o calculae he improvemen. The formula is used lie his: deah probabiliy a age x wihbirh year1980 deah probabiliy a age x wihbirh year % 1 deah probabiliy a age x wihbirh year 1980

12 2 Bacground and Benchmar Plan Benchmar Plan Plans Ouline In his paper we will discuss los of plans, and we will mae a number of hem as following: Table 1 Child Plans and Conras plans Ouline No. P0 P1 P2 P3 P4 Plan Name Saving Accoun: deposi in insallmens and wihdraw in lump sum Single Life Insurance Plan Child Plan: single female insured wihou income benefi Child Plan: single female insured wih income benefi Child Plan: boh parens insured wihou income benefi The firs hree plans will be explained in his secion laer and we will discuss he las wo plans in Chaper Child Plan A Child Plan usually aes one paren or he parens joinly as Life Assured. The produc policies vary in he real mare. In his paper, we sar he plans wih basic policies. Since we only focus on he moraliy change among ime periods, he gender difference is no a big issue. To simplify he firs Child Plan, he siuaion of insuring one female paren will be discussed. The goal of his plan is o calculae he ne premium under differen moraliy rae ables and o analyze he differences. Before he calculaion, here are some assumpions ha need o be se up. This plan is a benchmar for furher discuss in Chaper 5 and will be labeled as P2. Child Plan is a join produc ha conains life insurance and an individual accoun. In he life insurance par, he child will be he beneficiary and he life of he paren is insured. In he individual accoun par, he mauriy period is 18 years. The amoun of he payou is guaraneed and is assumed a 10,000 USD. The paymen period sars a he birh day of he baby and erminaes when he child reaches 18-year old or when he siuaion of deah benefi happens. Deah benefi refers o he issue ha he paren dies before mauriy (he child reaches 18-year old) and hereby he premium paymen erminaes. When he child survives a he policy mauriy, he guaraneed benefi will be paid. If deah benefi siuaion doesn happen, he guaraneed benefi will be paid on he mauriy when he child survives. If he child dies before mauriy, he premium paymen erminaes and benefi will no be paid. The annual ineres rae is assumed o be fixed a a level of 3%. The premium will be paid on a yearly basis a he beginning of each year. To simplify he plan, he paren and he child are assumed boh born in 1 s he January. (Table 2)

13 2 Bacground and Benchmar Plan 9 Table 2 Child Plan (P2) Assumpions P2: Child Plan Iem Condiion Annual Premium o be calculaed acuarially Premium Paymen Frequency Yearly Annual Ineres Rae 3% I is a fixed benefi guaraneed of 10,000 USD which will be paid a Mauriy when he child survives. Mauriy Benefi If he child dies before mauriy dae, he premium paymen erminaes. No benefi will be paid a end of policy erm. If he paren dies before mauriy dae while he child survives hen, he premium paymen erminaes afer he Deah Benefi paren die. And he guaraneed fixed benefi can sill be received a he end of policy erm Income Benefi No Child (Nominee): 0 Enry Age Moher (Life Assured): equal o he average female age a he firs childbirh Mauriy 18 years Ne Premium Measuremen Based on he assumpion above, he ne premium can be calculaed when he deah probabiliy is nown. In his chaper, 1*1 year period life able is used. I is no accurae o use if longeviy ris exiss, bu i is a very easy way for saring. Laer in his paper, we will use cohor life able and show he adjusmen. According o he assumpions above, he presen value of every dollar guaraneed benefi payou (Y) equals: Y 1 ( ) T 1 r (2.4) Here r is he annual ineres rae 3%, T refers o mauriy period of 18 years. This benefi can only be received when he child survives a mauriy. The probabiliy of receiving benefi (Y) can be wrien as an expression of deah probabiliy of he child: 1 T a a a a P[ Y ( ) ] T px ( 1 1qx ) ( 1 1qx1) ( 1 1qxT 1) 1 r (2.5) Where a T p refers o he probabiliy ha he child (a) a age x survives a leas anoher T years x and a 1q x refers o he one-year deah probabiliy for he child (a) a age x. The expecaion of presen value of uni benefi payou (Y) equals: 1 EY ( ) T a T px 1 r (2.6)

14 2 Bacground and Benchmar Plan 10 Similarly, if we assume yearly premium as 1 US dollar, he presen value of oal premium ( ) is a cumulaed value of yearly premium: S 1 ( ) ( ), S<T 1 r 1 r 1 r T 1 1 ( ) ( ), S T 1 r 1 r 1 r (2.7) Here S refers o he minimum curae fuure lifeime of he child (a) and he paren (b) wih beginning age of y. If he paren (b) dies or child (a) dies before mauriy (i.e. S<T), he remaining premium before he mauriy doesn need o be paid. If hey boh survive a leas T year (i.e. S T), all he premiums of T years have o be paid. To simplify he equaion, we use a o n express he presen value: a n ( ) ( ) n 1 r 1 r 1 r (2.8) Therefore, equaion of oal premium ( ) can be shor for: a,? S T S = a,? S T T1 (2.9) The expeced value of oal premium ( ) can be expressed by deah probabiliy of he paren (b) and child (a): T 1 1 b a y x 01 r (2.10) E p p Liewise, b p refers o he probabiliy ha he paren (b) a age y survives a leas anoher y years. In order o calculae he ne premium, we yield he Loss of he insurer (L) given by: L C Y (2.11) Here C is he amoun of he guaraneed benefi payou 10,000 USD and π is he annual premium amoun charged by insurance company. When he expeced loss of insurer equals o 0 (i.e. EL=0), his annual premium is called ne premium. In fac, he real premium charged by insurer o he cusomer will be added a ris premium called safey loading. This is because he ne premium doesn ae ino accoun ris. However, for our cases, we jus need an acuarially fair price o judge he characers of Child Plan. Tha is also he reason why we are seing a plan and assumpions o calculae he ne premium oher han using acual mare produc s price. The laer conains safey loading and insurance company s profi which are difficul o be separaed. Finally, we ge he ne premium expression as following:

15 2 Bacground and Benchmar Plan 11 a b f ( r, T, p, p ) C 1 T a ( ) T px 1 r 1 p 1 r 2 x y T1 a b 0 x p y (2.12) Here he subscrip 2 of f 2 refers o he number of Plan. And he funcions o calculae ne premium for differen plans below is analogous Change of Ne Premium under Aging Trend The plan uses he mean age of mohers a he firs childbirh as he Enry Age of he paren. In 1980, a woman gave he firs childbirh a average age of 25 in UK and 24 in US. However, by 2006, he daa increased sharply in UK o a level of 30. In he meanime, he US daa only increased slighly o 25. To calculae he ne premium, female life able is used o calculae he paren s survival probabiliy and he gender oal life able is used for he child. Based on he deah probabiliy decline and childbirh delay, here are 3 groups o be compared. Group A calculaes he case ha he child born in 1980 and Group B and C refer o child born in Group A and C uses he mean age of mohers a he firs childbirh a ha ime as he enry age of paren, while he Group B assumes he mean age unchanged and uses he previous age daa o calculae (Table 3 lef). On he one hand, because of he deah probabiliy declined from 1980 o 2006, he ne premium drops in boh UK and US (Table 3 righ). UK daa in Group B dropped slighly by 0.55 USD (abou 0.13%) compared wih Group A. The US daa fell even more slighly abou only 0.01%. On he oher hand, he mean age of moher a he firs childbirh delay increased he deah probabiliy of paren, herefore he ne premium in Group C rose compared wih Group B, wih an increase level of 0.33% in UK and 0.06% in US. Table 3 Ne Premium Value, in 1980 and 2006, in UK and US UK US Group Descripion Resul: Premium Value Group No. A B C A B C Child Birh Year Paren Age Child Birh Year Paren Age The changes of he premiums are quie small. I is because he Child Plan embedded an inner hedging sysem of moraliy ris. The benefi payou relies on he probabiliy ha he child survives. If he child dies before mauriy, he insurance company will gain from he previous premium paymens wihou providing mauriy benefi payou. Namely, in his case, L. When he deah probabiliy of he child declines, his gain discussed above is less liely o ge. To he conrary, deah benefi causes a loss o he insurance. When he deah probabiliy of he paren drops, he loss of he deah benefi will reduce. However, he survival probabiliy of he paren and he child change in he same direcion (Table 4). The benefi and loss will be parly offse.

16 2 Bacground and Benchmar Plan 12 Table 4 Survival Probabiliy Change beween 1980 and 2006 Moher Child Group No. A B A B UK USA Conras Plans To prove he exisence of he inner hedge sysem, we inroduce anoher wo plans as conras. The firs one is a saving accoun ha is deposied in insallmens and wihdrawn in lump sum. I is oally ris-free wih same ineres rae of 3%. To be sric, i is no a life insurance plan so we label i as P0. The assumpions are defined in Table 5. Table 5 Saving Accoun Plan (P0) Assumpions P0: Saving Accoun Plan Iem Condiion Annual Deposi Fixed: Deposi Frequency Yearly Annual Ineres Rae 3% Mauriy Benefi I is a fixed benefi guaraneed of 10,000 USD which will be wihdrawn a Mauriy. Mauriy Same o P2: 18 years Unlie oher life insurance plan, he annual paymen (deposi) is fixed wih an amoun of , and calculaed as following equaion: f ( r, T) C 1 T ( ) 1 r T r 0 (2.13) I is no lin o any individual s moraliy and is forced o deposi yearly before wihdraw. And he deposi amoun is he same wih UK and US counries. The second conras plan is a single Life Insurance Plan and labeled as P1. This Life Insurance Plan insures he life of paren only. Similar o he Child Plan (P2), his produc conains yearly premium paymen and guaraneed benefi payou in a lump sum a he end of policy period. Bu he condiion of he premium and benefi payou doesn associae o he deah probabiliy of he child or he beneficiary. If he insured dies, he premium will sop and he beneficiary will receive benefi payou a mauriy. If he insured survives, guaraneed benefi can also be paid. The assumpions of Life Insurance Plan are summarized as following:

17 2 Bacground and Benchmar Plan 13 Table 6 Life Insurance Plan (P1) Assumpions P1: Life Insurance Plan Iem Condiion Annual Premium o be calculaed acuarially Premium Paymen Frequency Same o P2: Yearly Annual Ineres Rae Same o P2: 3% Mauriy Benefi I is a fixed benefi guaraneed of 10,000 USD which will be paid a Mauriy. If he insured dies before mauriy dae, he premium Deah Benefi paymen erminaes. And he guaraneed fixed benefi can sill be received a he end of policy erm. Income Benefi Same o P2: No Enry Age Same o P2 Mauriy Same o P2: 18 years According o he assumpions above, he ne premium can be easy o calculae: b f ( r, T, p ) C 1 T ( ) 1 r 1 1 r 1 y T1 0 b p y (2.14) For a Life Insurance Plan, he ne premium declines due o deah probabiliy changes from 1980 o 2006 are much more significan han changes in Child Plan. (Table 7) In UK daa, he evoluion percenage of Life Insurance Plan is almos riple o ha of Child Plan (-0.35% vs. 0.13%) and he US daa is even more obvious (-0.28% vs %). Table 7 Premium Comparison of Saving Accoun, Single Life Insurance Plan and Child Plan Saving Accoun Life Insurance Plan (P1) Child Plan (P2) (P0) A B Evl. % A B Evl. % UK % % USA % % Wih Saving Accoun Plan we can explain why he ne premium changes seem so small. Alhough our life insurance plans are all risy producs, hey are belong o a life insurance caegory called Endowmen Insurance. An endowmen policy is a ype of life insurance which is payable o he insured if he/she is sill living on he policy's mauriy dae, or o he beneficiary oherwise. And i is much less risy han normal insurance conrac in which he insured will for sure loss premium if he/she survives. When we compare he ne premium wih P0 and P1, we find ha he ne premium of P1 isn much higher han P0. I is because he survival probabiliy of he insured is very high, prey close o 1 (see Table 4). The price for moraliy ris ha he insured need o pay will no be oo high. If we increase he assumed age of insured, he premium difference will become larger. Therefore, when he dead probabiliy decreases, he ne premium of a life

18 2 Bacground and Benchmar Plan 14 insurance produc will no drop below he corresponding price of ris-free saving accoun. Namely, he decline rae from group A o group B of UK in life insurance Plan P1 will no be higher han -0.79% (decline rae from o ). Furhermore, he disance of ne premium beween P0 and P1 have se he boundary ha he Child Plan will be ranged. For example, under age and year choice of group B, he ne premium of Child Plan will be neiher higher han ne premium of P1 (416.76) nor lower han deposi amoun of P0 (414.65) Reference Discoun Rae Afer he comparison of P0, P1 and P2, we draw a conclusion ha amoun of yearly deposi of saving accoun is he boom line of he ne premium. However, he number isn simple, convenien and inuiive enough for he judgmen and analysis. So we inroduce a beer indicaor and named i as reference discoun rae. Here is he inerpreaion: if he value of ne premium is used as he yearly deposi in anoher saving accoun wih same mauriy period and final wihdraw as P0, he discoun rae of such saving accoun mus be differen, and his new ineres rae is a ris-free ineres rae. So if an individual wans o buy a cerain ind of life insurance produc, his person can easily now how many poins of ineres rae he or she will lose if buying a sandard saving accoun produc insead. And he losing poins of ineres rae are he price for he riss covered by insurance conras. I should be noiced ha he reference discoun rae is no a real discoun rae, in his aricle, he real discoun rae always equal o 3% as we assume for our plans. And he soluion of reference discoun rae (r * ) is he inverse funcion of following funcion beween π and r * : 1 ( ) T * * 0 (, ) 1 r T1 f r T C 0 1 * 1 r (2.15) If we calculae ne premium for Single Life Insurance Plan and oher Child Plans, we will use funcions f 1, f 2, f 3 and f 4 (f 3 and f 4 will be presened in Chaper 5). Bu he calculaion of reference discoun rae will always use he same funcion f 0-1, which is he inverse funcion of he one calculaing ne premium in a Saving Accoun Plan and hasn any variable of survival probabiliy. * 1 Wih a program in an excel sofware, we calculae he inverse funcion r f, T resuls are showed in he following able: and he Table 8 Reference Discoun Rae of Saving Accoun, Single Life Insurance Plan and Child Plan Saving Accoun Life Insurance Plan (P1) Child Plan (P2) (P0) A B C A B C UK 3.00% 2.92% 2.96% 2.92% 2.96% 2.98% 2.94% USA 3.00% 2.92% 2.94% 2.94% 2.97% 2.98% 2.97% 0

19 2 Bacground and Benchmar Plan 15 We can also define reference discoun rae ino wo caegories. One is from he premium esimaed in he conrac before he insurance policy begins. So we call i Sold Reference Discoun Rae. The oher one is calculaed a he end of policy based on acual deah rae and herefore named as Realized Reference Discoun Rae. In Table 8, we can easily now how many ineres rae poins he insurance company will earn if he insured all survive. Tha is, he realized reference discoun rae equals 3% and he difference from he sold reference discoun rae is he benefi for he insurance company. Bu i is no he profi ha an insurance company aims o. As we discussed before, his ne premium is no he final offer price, he insurance company will add a safey-loading which is where heir majoriy of income comes from. However, when he moraliy rae changes beyond heir esimaion, he realized reference discoun rae will probably be lower han sold reference discoun rae and hey will suffer a loss. 2.4 Summary Boh ne premium analysis and ris-free analysis show a small change of Child Plan compared wih single Life Insurance Plan. I is because he Child Plan is a scheme wih an easy inner hedging sysem by a more complicaed conrac design-join deah probabiliy design. However, in his chaper, we jus show a general frame of our research, so we use period life able insead of cohor life able o easily calculae he ne premium, which implies ha he longeviy ris has been ignored in some exen. To aain he cohor life able, deah probabiliy forecas model is essenial. In he nex wo chapers, wo models will be used o UK and US daa.

20 3 Modeling Moraliy in US 16 3 Modeling Moraliy in US 3.1 Lee-Carer Model Specificaion A variey of approaches have been proposed of modeling moraliy rae over ime and he leading wor is he Lee-Carer mehod. In heir 1992 JASA paper, Lee and Carer posulaed a simple relaionship for he cenral deah rae m x, as following: log ( m )? a b e (3.1) x, x x x, Here he m x, refers o he cenral deah rae for age x a ime. The a x coefficiens describe he average shape of he age profile, and he b x coefficiens describe he paern of deviaions from his age profile when he parameer varies. To idenify he soluion of he model, he following normalizaion consrains are imposed: s.. log ( m )? a b e x, x x x, n 1 xm 0 x xx1 b 1 (3.2) Tha is because for any soluion a, b,, here exiss anoher soluion a-bc, b, +c and a, bc, /c for any scalar c. Therefore, is deermined only up o a linear ransformaion, variable b is deermined only up o a muliplicaive consan, and variable a is deermined only up o an addiive consan (Lee and Carer, 1992). In his paper, we follow he Lee-Carer forecas model o fi US daa in order o exend period life able of US daa afer year Thus, we can ransfer i ino a cohor life able o improve he Child Plan. 3.2 Fiing he Lee-Carer Model We sar wih he US female daa of 1*1 year period forma from 1933 o I is also from Human Moraliy Daabase. Bu according o Lee-Carer model we use cenral deah rae his ime insead of deah probabiliy. Firsly, we buil a marix A wih: x xx1,, xm,? 1,, n x, xx1,, xm,? 1,, n A ( A ) log m (3.3) I is easy o esimae a x under firs consrain of variable. Tha is: n 1 aˆ A log ( m ) (3.4) x x x, n 1 To esimae b x and, we need Singular Value Decomposiion (SVD).

21 3 Modeling Moraliy in US 17 For A mn, here exiss a facorizaion as following: T A UDV s.. U mm orhogonal marix D mn diagonal marix V nn orhogonal marix (3.5) If we denoe columns of U by be rewrien as: ui m R and columns of V by i v n, he equaion above can also A T diiuivi (3.6) i1 refers o he ran of marix A. The SVD mehod can be used o find a leas squares soluion when applied o he marix of he logarihm of he rae afer he averages over ime of he (log) age-specific raes have been subraced (Lee and Carer, 1992). Therefore, we apply SVD on ( A a ˆ x x ) x, and ge he esimaors of b x and as following: u (3.7) u ˆ 1 bx m i1 i m ˆ ui v1 d11 (3.8) Table 9 and Table 10 show he resul of our fied value of coefficiens of a x, b x and. Table 9 Fied Value of a x and b x for of Lee-Carer Model i1 Age(x) a x b x Age(x) a x b x

22 3 Modeling Moraliy in US 18 Age(x) a x b x Age(x) a x b x

23 3 Modeling Moraliy in US 19 Table 10 Fied Value of for of Lee-Carer Model Year Year Year Using hese esimaors, we choose beginning year 1933 and ending year 2007 as examples o compare he acual daa wih model esimaion (Figure 5). The fi loos good, and he shapes of he fied value almos mach he age profile of acual daa.

24 Esimaed log(mx) 3 Modeling Moraliy in US 20 Figure 5 Comparison of Acual and Esimaion for 1933 and 2007 of Lee-Carer Model Age logm1933 logm2007 fi1933 fi Forecass for US daa In heir paper, Lee and Carer found empirical evidence ha he variable are decreasing approximaely linearly. As a resul, Lee-Carer model furher assumes he variable follow a sochasic process as a random wal wih drif. Specifically, C e (3.9) 1 The e is independen and idenically disribued (i.i.d.) wih mean zero and variance σ 2. The drif is esimaed by: n n ˆ ˆ ˆ 1 ( ˆ ˆ 1 ) ˆ C i 1 i i n 1 n 1 n 1 n 1 (3.10) i 1 i 1 And hen we calculae he sandard deviaion of he error erm σ 2 by: n 2 1 ˆ ( ˆ ˆ 2 C) (3.11) i n 1 i11 The approximae linear decline paern of also can be observed in our esimaion (Figure 6). However, from he figure, we also found ha he slope is becoming smaller since around 1953 and remains relaively sable unil around Afer 1973, esimaed variable goes bac decreasing bu he decline is no as seep as he firs wo decades. This slope change suggess a period adjusmen for our model. I seems ha less hisorical daa will mae he forecas beer.

25 3 Modeling Moraliy in US 21 We will go bac o his discussion laer in his aricle. Figure 6 Fied Value of for of Lee-Carer Model Year Following he Lee-Carer model, we ge a yearly decline esimaor Ĉ equal o , and ˆσ equal o This sandard deviaion is bigger han hose calculaed in Lee and Carer s paper wrien in They go Ĉ as and sandard error of equaion as only for fiing US daa from 1930 o 1989 (Lee and Carer, 1992). The reason for he difference is parly because we use one-year deah rae daa and have 111 observaions for b x while hey only use five-year deah rae daa and have only 23 observaions for b x. Considering he second consrain ha sum of b x always equals o 1, our b x is liely o be one fifh (23/111=0.207) of heirs. As a resul, our soluion of may probably be five imes o heirs. I is because our soluion of a x is close o heirs due o he same approach of calculae and i resrics he soluion of b x imes should also be close o heirs. So he difference of variables ( Ĉ ) and he variance ˆσ imes of heir resuls. is also more or less five Since we have go an expression of variable wih ime, we can use he model o exend life able for he fuure years. Figure 7 show an example of forecas for To creae a cohor life able, a forecas is needed for every year from 2008 o 2050.

26 Esimaed log(mx) 3 Modeling Moraliy in US 22 Figure 7 Moraliy Forecas for 2050, Wih 95% Confidence Inerval of Lee-Carer Model Age 3.4 Daa Period Adjusmen Considering he slope change of variable in around 1953 and 1973, we repea he process of Lee-Carer model wih daa saring from 1953 and 1973 insead of Table 11 shows how he slope and sandard deviaion of variable change. Since he number of b variable for hese hree choices is he same (111), he slope and sandard deviaion of variable are comparable. As we discussed before, recen years slope (from year 1973 o 2007) is less seep han he hisorical slope (from year 1933 o 1953), so Daa Choice 2 and 3 is beer for forecasing fuure variable if we assume his slower decline rend coninues. Table 11 K Variable Changes for Processing Differen Daa Choice of Lee-Carer Model K variable Daa Choice 1 Daa Choice 2 Daa Choice 3 Ĉ ˆσ Choice 1 uses cenral deah rae daa from year 1933 o 2007; Choice 2 uses daa from year 1953 o 2007; Choice 3 uses daa from year 1973 o 2007 In order o choose he fies daa period, we compare hese hree resuls by using Sum of Square Residuals (SSR) 4 wih following equaion: x, x x x SSR log m aˆ bˆ ˆ (3.12) The common period for hese hree fiing rials is from 1973 o We separae his 35-year 4 This SSR is no exacly he same wih he common used SSR in saisics erm. I is because we don summarize all he residuals of he sample model. Insead, we sum up a cerain par of he whole populaion range in order o compare he siuaion of his range.

27 3 Modeling Moraliy in US 23 period ino four pars: 1973 o 1979, 1980 o 1989, 1990 o 1999 and 2000 o We firsly calculae every year esimaed log(m x, ) using differen esimaed variables a ˆx, b ˆx and ˆ ha we go from hese hree rials. Afer ha we calculae he SSR for he yearly fiing resuls compare wih acual daa. Then we summarize he yearly SSR ino four periods. From he comparison of hree daa period choice, we find ha he hird daa choice fis he recen year daa beer han he oher wo in all four periods (Table 12). Especially for he las period, which is closes o he fuure, he firs and he second rial don perform well enough and increase he SSR very quicly. Table 12 Model Finess Resuls for Differen Daa Choice of Lee-Carer Model SSR Daa Choice 1 Daa Choice 2 Daa Choice s s s s Toal From he wo mehods comparison above, we decide o adjus our model by using more recen daa period ( ) for fuure forecas. The new paern of is smooher decreasing han he previous rial (Figure 8). Figure 8 Adjused Fied Value of for Year Since he sandard deviaion becomes less, he revised moraliy forecas for 2050 has a narrower 95% Confidence Inerval (Figure 9).

28 Esimaed log(mx) 3 Modeling Moraliy in US 24 Figure 9 Adjused Moraliy Forecas for 2050, Wih 95% Confidence Inerval Age 3.5 Cohor Life Table in US Afer he daa period change, we have new esimaion of a, b and variables of Lee-Carer Model (see deail daa in Appendix A). Now we have period life able for US female daa from 1933 o 2050, i is long enough o ransfer ino cohor life able for our Child Plan o calculae ne premium. And we also need US Gender Toal daa o calculae survival probabiliy of children. So we repea he same process o he daa of US Gender Toal (see deail daa in Appendix B). Afer changing he life ables of US, we go new resuls of ne premium for he hird group we menioned in chaper 2 (i.e. Group C: Child is born a 2006 when he moher is 25). And we also calculae ne premium for a Life Insurance Plan (Table 13). The ne premium of Life Insurance Plan decreases because he deah probabiliy we use from cohor life able is smaller han ha from period life able. Smaller deah probabiliy will lead o a bigger survival probabiliy of insured and a higher expeced value of premium. To eep he Expeced Loss of he insurer equal o zero, he insurer will decrease he annual premium value. Table 13 Comparison of Ne Premium Value and Reference Discoun Rae beween Period Life Table and Cohor Life Table in US Daa Life Insurance (P1) Ne premium Reference discoun rae Child Plan (P2) Ne premium Reference discoun rae Period Life Table % % Cohor Life Table % % According o he plan assumpions in chaper 2, he difference beween Child Plan and Life Insurance Plan is wheher anoher individual s survival (he child) is involved or no in he premium and benefi payou condiions. So we can figure ou he effec of his join deah probabiliy design by comparing hese wo plans.

29 4 Modeling Moraliy in UK 25 4 Modeling Moraliy in UK 4.1 Cairns-Blae-Dowd Model Specificaion Afer Lee and Carer (1992) developed a sochasic moraliy approach fiing o US daa, many researchers have exended Lee-Carer model (see, for example Brouhns, Denui, and Vermun, 2002; Renshaw and Haberman, 2003; Currie, Durban, and Eilers, 2004). If raning hose models based on he saisical qualiy of fi, hen Cairns, Blae, and Dowd (2006) proposed an exension model (CBD) ha fis he UK moraliy daa bes (Cairns, Blae, e al., 2007). CBD model for moraliy rae is defined as: logi( q ) ( x x) e (4.1) ( ) ( ) x, x, In his equaion, x is he mean age in he sample range, ( 1) and ( 2) are sochasic processes ha are assumed o be measurable a ime, e is he error erm. And logi( q x, ) refers o logisic regression o deah probabiliy q x, : logi q q x, ( x, ) log( ) 1 qx, (4.2) So i can also be rewrien as followed for he moraliy curve: q e 1 e ( 1) ( 2) ( xx ) ex, x, ( 1) ( 2) ( xx ) ex, (4.3) 4.2 Fiing he CBD Model Before fiing UK deah probabiliy o he CBD model, we need o choose he range of he sample daa. Firsly, we choose he age range from 0 o 60. In his paper, we only need moraliy daa for he children and he middle-aged, so an age range below 60 is sufficien. Secondly, we use he longes sample ime range of year 1922 o 2009 we can ge from HMD. Afer ha, we do logisic regression o dead probabiliy q x wih age x for each year and sore he resuls of esimaors of ( 1) and ( 2) year by year. Table 14 show he fied value of variable changed by year. Table 14 Fied Value of ( 1) and for of CBD Model ( 2 ) Year () ( 1) ( 2) Year () ( 1) ( 2)

30 4 Modeling Moraliy in UK 26 Year () ( 1) ( 2) Year () ( 1) ( 2) Alhough we have model daa from year 1922, we show he fi resul from year 1933 in order o cross-compare he fi resul figure of Lee-Carer model (Figure 5). And he laes year daa o be displayed is he year 2009 resul (Figure 10) which is wo year laer han he fi resul figure of Lee-Carer model. The fi of CBD model is clearly no good. Especially for age 0, he CBD model fails o reveal a relaive high deah probabiliy of infans and a dramaically sharp decrease o he nex age. In adverse, he fied value of CBD model sars a a very low poin and eeps increasing smoohly and sricly. Besides, he fi resul of higher age above 50 is also fairly unaccepable. Alhough, he fied value has prediced an acceleraed increase as age grows, he prediced

31 Esimaed deah probabiliy qx 4 Modeling Moraliy in UK 27 increase is obviously no quic enough. Figure 10 Comparison of Acual and Esimaion for 1933 and 2007 of CBD Model 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Age q1933 q2009 fi1933 fi2009 The reason for such bad fi is because of he model frame of CBD approach iself. The CBD model jus assumes a linear relaion beween logi(q x ) and age x in each year. However, oher models lie Lee-Carer model have variables wih age x and are hus able o describe a non-linear shape of age profile. The linear age profile of CBD model is really a srong assumpion and hus raher unrealisic. In fac, in heir JRI paper of 2006, Cairns, Blae, and Dowd (2006) only consider pos-age-60 moraliy curve and he model fis good for age above 60. In heir laer four aricles (Cairns, Blae, and Dowd, 2006; Cairns, Blae, e al., 2007; Dowd, Cairns, e al., 2008; Cairns, Blae, e al., 2009), his age range choice hasn been exended. However, he laes year daa (2009) do show a beer resul of model fiing han far previous year daa (1933). I is he resul relaed o our major wor indeed. Therefore, we need o find more deails on he finess progress of CBD mode across he ime, so ha we can judge wheher he CBD model is appropriae for forecasing. 4.3 Finess Comparison for he CBD Model The reason we choose age range from 0 o 60 insead of whole range 0 o 110 is because we have

32 4 Modeling Moraliy in UK 28 learned from he fiing resul of las chaper ha wider range of daa doesn always resul in a beer fi. If we choose he whole daa, he average age of sample will increase, and hen he model qualiy of fi will be weaened. To chec ou his hypohesis, we need o fi he model for a second rial and compare he resuls. Therefore, we have wo rials of CBD model fiing by using Daa Choice 1* (age from 0 o 60) and Daa Choice 2* (age from 0 o 110+). Furhermore, we found in las secion ha he CBD model has an innae defec in modeling infan deah probabiliy pea, so we aemp o cu down he daa of age 0 and chec ou if here will be a beer fi for he res ages. This is idenified as Daa Choice 3* (age from 1 o 60). Similar o las chaper, we use Sum of Square Residuals (SSR) o compare differen daa range choice. In his case, he equaion of SSR is rewrien as following: SSR ˆ( 1) ˆ( 2) 2 n xx e qx, ˆ( 1) ˆ( 2) (4.4) xx x1 1 e In order o quanify he finess across he ime, we follow he same period division as las chaper and summarize he SSR by following 4 periods (Table 15). However, i should be noed ha 1970s refers o year for consisency, and he 2000s includes year 2000 o 2009 in CBD Model and Lee Carer Model. In addiion, we only choose age range from 1 o 60 ha is he cross among he hree rials. The uni of Table 15 is 1% for sae of clariy. The hird rial wih he leas sample age range excels he oher wo ousandingly. The oal SSR of four periods is merely abou one half of he firs and he second rial (52% and 58%). Among all of four periods, he hird rial eeps an absoluely leading posiion. The only problem is ha we lose he moraliy model for age 0. Bu, in our Child Plan, we in fac don need forecas daa for age 0 since he younges subjec is born in year Therefore, we will choose he hird rial for UK moraliy model (see esimaors in Appendix D). Table 15 Model Finess Resuls for Differen Daa Choice of CBD Model, Age Range from 1 o 60 Uni: 0.01 CBD Model Lee-Carer Model Daa Choice 1* Daa Choice 2* Daa Choice 3* Final Choice 1970s s SSR 1990s s Toal However, his amoun of SSR value only has relaive meaning for comparing he hree rials. To display he finess resul of he whole CBD model, we also mae a reference wih Lee-Carer model. We use Lee-Carer model o calculae UK female daa in order o mae sure ha he SSR 5 Here 1970s refers o 1973 o 1979 for boh CBD and Lee-Carer Model

33 4 Modeling Moraliy in UK 29 of boh models are comparable. I can be clearly observed ha Lee-Carer model has a much lower SSR resul han CBD model, a only around 1.35% o 2.58% of he SSR of CBD model (Table 15). However, we sill use CBD model for UK daa because i is he mos popular model used in many oher papers, alhough CBD is used for populaion aged above 60 in hese papers. Period SSR Summary Table above is a beer approach o provide a sandard for us o mae daa range choice, bu i can show he finess across age and ime. Hence we draw a new marix graph o show he goodness of fi boh a an age and a year level. Firsly, we consider he direcion of model bias. We define he following funcion of S x, which describe he sign of model residual: -1, if q qˆ S 0, if q qˆ 1, if q qˆ x, x, x, x, x, x, x, (4.5) Figure 11 shows he sign of he residual of he CBD model rial we choose (Daa Choice 3*). Dar grey area (S x, =1) mars he area ha he model is underesimaed compared wih acual daa. A he same ime, he ligh grey area (S x, =-1) shows he opposie siuaion ha he esimaed value is bigger han acual daa. Figure 11 Sign of Residual of UK Female Daa Fiing by CBD Model We noice ha he funcion S x, shows a similar paern across he years, i is shown as he

34 4 Modeling Moraliy in UK 30 horizonal exension of he wo color blocs. Furhermore, he blocs of color also remain relaively complee in verical level of age. I almos underesimaes all he poins above age 50 and age below 5, and in he res age range overesimaes. We suppose ha he big color bloc suggess he goodness of fi is no perfec; oherwise he dar grey and ligh grey colors will probably mix ino smaller blocs. Secondly, we draw anoher marix graph o show he goodness of fi. Similar o SSR Table we eep on using square of residual (e x, 2 ) as he indicaor of he goodness of fi. Bu in order o presen a graph for a beer undersanding, we mae a ransform of (e x, 2 ) as following: 1 2 lg r2x, log 10 ( e x, ) (4.6) 2 Hence, we calculae a new variable of lgr2 ranged from -7.1 o I can be inerpreed inuiively as he power of 10 ha measured he accuracy of he modeling. In Figure 13, we can see is value in a marix range. If we define a perfec mach as a condiion ha he residual is less han , i means lgr2 should be less han -4. In oher word, he green and blue blocs in Figure 13 are he perfec mach areas. We also draw a marix graph for he acual deah probabiliy, and we use he following lgq x, for comparison wih lgr2 : lg q log ( q ) (4.7) x, 10 x, However, he deah probabiliy iself varies a lo in he marix (Figure 12). The value of lgq x, change from -4.3 o So i is no wise o judge he residuals wih a boundary of absolue value. If we define a 1% level as a perfec mach (i.e., he model residual is less han 1 percen of he acual daa), he value of lgr2 x, should be a leas wo unis smaller han logarihm of deah probabiliy (lgq x, ).

35 4 Modeling Moraliy in UK 31 Figure 12 Log Deah Probabiliy log 10 (q x, ) for UK Female Daa from 1922 o 2009, Age from 1 o 60 Figure 13 Goodness of Fi of UK Female Daa Fiing by CBD Model

36 4 Modeling Moraliy in UK 32 In Figure 12, he boundary of differen moraliy daa (lgq x, ) level is very clear. Bu we draw approximae curves for simpliciy insead of he real boundary o divide he moraliy daa ino five pars and copy hese curves o Figure 13. A bigger area number refers o a relaively higher deah probabiliy. We discuss he finess by hese five areas each and visually esimae he average value of each area in Figure 12 and Figure 13 ino he following able: Table 16 Model Finess Resuls of CBD Model by Deah Probabiliy Volume Area Area No lgq lgr2 Difference Average From he able, we noice ha he difference is far away from perfec 2. Our esimaion of he average residual weigh will be more han 10% (Difference equals 1 means he residual is abou 10% of acual daa). This migh no be accepable in a relaive level. However, in an absolue level, he area 4 and 5 show a small logarihm value (less han -4). I means he difference beween acual daa and fied value is only abou , which can be regarded as an accepable fi. I is also remarable ha he mos fied area is he recen year daa. Therefore, we decide o coninue he forecas wih CBD model. 4.4 Forecass for UK Daa According o he CBD model, wo variables ( 1) and ( 2) follow a random wal wih drif. We have already made he regression year by year and ge he yearly esimaors of hese wo variables. I shows a relaive seady drif respecively (Figure 14 and Figure 15). Figure 14 Fied Value of (1) Variables for 1922 o (1) Year

37 4 Modeling Moraliy in UK 33 Figure 15 Fied Value of (2) Variables for 1922 o (2) Year Cairns, Blae, and Dowd modeled (Cairns, Blae, and Dowd, 2006): K ( 1) ( 2) (, )' as a wo-dimensional random wal wih drif K K CZ (4.8) 1 1 in which μ is a consan 2*1 drif vecor, C is a consan 2 * 2 upper riangular volailiy marix (o be precise, he Cholesi square roo marix of he variance-covariance marix ), and Z is a wo dimensional sandard normal variable, each componen of which is independen of he oher (Dowd, Cairns, e al., 2008). I is very similar o Lee-Carer model wih linear moves assumpion of variable. So we decide o follow he same formula as Lee-Carer Model o ge he esimaors of he drifs and he variances of he error erms. From Figure 14 and Figure 15 we found boh variables have slowed down he upward or downward rend since abou 1970s. To follow a similar ime period of US daa we fi wih Lee-Carer model, we sar from year 1973 and ge: ˆ (4.9) ˆ ˆ ˆ T CC (4.10) We also mae an example of forecas for 2050 by using esimaors above (Figure 16). And we use daa log(qx) in order o ge a similar shape wih Lee-Carer Model (Figure 7). To calculae his forecas, we add esimaed 41-year drifs of wo variables o he acual daa of year So he forecas line in Figure 16 is no very smooh. If we choose anoher approach ha uses formula direcly o calculae prediced log(qx) for 2050 by esimaors of variables, he forecas line will be oo smooh o be real. The reason replies on he assumpion in CBD model ha he relaion beween logiq x and age x in cerain year is linear.

38 Esimaed log(qx) 4 Modeling Moraliy in UK 34 Figure 16 Moraliy Forecas for 2050, Wih 95% Confidence Inerval of CBD Model Age 4.5 Cohor Life Table in UK We follow he same process for UK daa of gender oal o see life able for child (see resuls of esimaion in Appendix E). And ransfer he period life able o cohor life able and mae comparison of he resuls: Table 17 Comparison of Ne Premium Value and Reference Discoun Rae beween Period Life Table and Cohor Life Table in UK Daa Life Insurance (P1) Ne premium Reference discoun rae Child Plan (P2) Ne premium Reference discoun rae Period Life Table % % Cohor Life Table % % Similar o US daa, he resuls of ne premium decrease and he reference discoun raes increase. Bu he changes in UK daa are bigger han US daa, wih abou 4% changes in reference discoun rae, while he US daa only changes abou 1%. I is because he paren age ha we use in UK is 5 year higher han ha in US. And he cohor effec is hus larger in UK han US.

39 5 Child Plan Exension 35 5 Child Plan Exension 5.1 Se up New Plans Since we now have he cohor life able for boh counries, i is ime o do furher analysis for Child Plan. Firs, we need o se up anoher wo plans: Child Plan wih income benefi (P3) and Child Plan wih boh parens insured (P4) Child Plan wih Income Benefi Income benefi is a special condiion as a supplemen of he dead benefi. Tha is: if he insured dies before he end of mauriy period, aside from erminaing annual premium paymen in he res years of he policy period, he beneficiary will also receive addiional payou from he insurer as an income benefi. Someimes, he income benefi payou has he same amoun as he annual premium paymen, bu i is no a regulaion. In our hird plan, we assume ha he income benefi equals o he ne premium (See Table 18 for all he assumpions). This benefi condiion will beer proec he beneficiary agains a fuure income flow ris.

40 5 Child Plan Exension 36 Table 18 Child Plan wih Income Benefi (P3) Assumpions P3: Child Plan wih Income Benefi Iem Condiion Annual Premium o be calculaed acuarially Premium Paymen Yearly Frequency Annual Ineres Rae 3% I is a fixed benefi guaraneed of 10,000 USD which will be paid a Mauriy when he child survives. Mauriy Benefi If he child dies before mauriy dae, he premium paymen erminaes. No benefi will be paid a end of policy erm. If he paren dies before mauriy dae while he child survives hen, he Deah Benefi premium paymen erminaes afer he paren die. And he guaraneed fixed benefi can sill be received a he end of policy erm If he paren dies before mauriy dae, an annual income benefi Income Benefi paymen will be received a he same amoun of annual premium every year ill he end of policy erm as long as he child survives. Child (Nominee): 0 Enry Age Moher (Life Assured): equal o he average female age a he firs childbirh Mauriy 18 years To calculae he ne premium of P3, we use Tha is: I as oal value of uni payou of income benefi. I a = 0, T 1 a S,? S T S T (5.1) The expeced value of oal income benefi can be expressed by deah probabiliy of he paren (b) and child (a): T1 I 1 b a E ( 1 py ) px 1 r (5.2) 0 If we assume he same amoun for premium and income benefi paymen as π, he oal loss of he insurer can be wrien as:

41 5 Child Plan Exension 37 And he ne premium can be calculaed lie his: I L C Y (5.3) a b f ( r, T, p, p ) C 3 x y T1 0 1 T a ( ) T px 1 r 1 b a ( 2 py 1) p 1 r x (5.4) From he equaion above, since he survival probabiliy can be above 1, i can be concluded ha he ne premium of income benefi produc is always higher han normal Child Plan produc. I can also be explained by inuiive judgmens: he income benefi produc should charge a higher price han normal Child Plan because of addiional ris proecion service. This resuls can also be seen in Table 19. The ne premium value in boh US and UK increase significanly from P2 o P3 and are even higher han P1. Alhough a higher price implies a bigger ris for he insurer, i s oo early o ell wheher selling a policy of Child Plan wih income benefi is a more risy liabiliy han Single Insurance Plan. We will analyze he riss in nex chaper. Table 19 Comparison of Ne Premium Value and Reference discoun rae among Single Life Insurance Plan (P1), Child Plan (P2) and Child Plan wih income benefi (P3), in US and UK Daa Counry Resuls P1 P2 P3 US Ne premium Reference discoun rae 2.95% 2.98% 2.93% UK Ne premium Reference discoun rae 2.96% 2.98% 2.94% Child Plan wih Boh Parens Insured In he real mare, he number of he insured can be one or more han one. For Child Plan produc, i someimes lins boh parens ogeher in one policy. In our forh plan, we assume wo-year age gap beween he faher and he moher (he faher is older), and he deah benefi will be acivaed by he deah of he paren(s) before mauriy. All he condiions are assumed below:

42 5 Child Plan Exension 38 Table 20 Child Plan wih Boh Barens Insured (P4) Assumpions P4: Child Plan wih Boh Barens Insured Iem Condiion Annual Premium o be calculaed acuarially Premium Paymen Yearly Frequency Annual Ineres Rae 3% I is a fixed benefi guaraneed of 10,000 USD which will be paid a Mauriy when he child survives. Mauriy Benefi If he child dies before mauriy dae, he premium paymen erminaes. No benefi will be paid a end of policy erm. If a leas one of he parens dies before mauriy dae while he child survives hen, he premium paymen erminaes afer he deah of Deah Benefi paren(s). And he guaraneed fixed benefi can sill be received a he end of policy erm Income Benefi No Child (Nominee): 0 Moher (Life Assured): equal o he average female age a he firs Enry Age childbirh Faher (Life Assured): wo year older han he moher Mauriy 18 years In his plan he expeced value of oal premium can be wrien as following equaion wih deah probabiliy of child (a), he moher (b) and he faher (c): T1 1 b c a y z x (5.5) 01 r E p p p Similarly, c p refers o he probabiliy ha he faher (c) a age z survives a leas anoher years. z We use Lee-Carer model and CBD model once again o obain he life able for he male in US and UK daa (see Appendix C for he fiing resuls of Lee-Carer model and Appendix F). And he ne premium can be calculaed lie his: a b c f ( r, T, p, p, p ) C 1 T a ( ) T px 1 r 1 c p pz p 1 r 4 x y z T1 b a 0 y x (5.6)

43 5 Child Plan Exension 39 Obviously, he price of produc of boh parens insured will be more expensive han one paren insured produc. The resuls in our plans have shown his (Table 21), he ne premiums in boh counries increase from around 415 USD o 420 USD. The reference discoun raes fall below 2.9%. This increase of ne premium is even sronger han he increase impaced by income benefi of P3. Table 21 Comparison of Ne Premium Value and Reference discoun rae among Child Plan (P2), Child Plan wih Income Benefi (P3) and Child Plan wih Boh Parens Insured (P4), in US and UK Daa Counry Resuls P2 P3 P4 US Ne premium Reference discoun rae 2.98% 2.93% 2.86% UK Ne premium Reference discoun rae 2.98% 2.94% 2.89% 5.2 Older Paren(s) Insured The enry age of he moher we used for P1 o P4 are assumed as he mean age of mohers a he firs childbirh boh in UK and US. However, in real world, he proporion of babies wih older moher in heir lae 30s or older has gone up. A figure in 2007 revealed ha abou one-fifh of he new mohers were over 35 in UK (Figure 17). This older group can be ignored in Child Plan Insurance producs. The higher-age parens have more incenive o purchase an insurance produc. On one hand, higher aged parens have lower survival probabiliy and insurance for heir children are more essenial. On he oher hand, higher aged parens are more liely o be higher income group and have financial advanage for he purchase. Therefore in his secion, we will use age 35 as he enry age of moher and 37 as he enry age of faher, while he child is sill assumed born in Follow he age group number in Chaper 2, his age group will be named as Group D.

44 5 Child Plan Exension 40 Figure 17 UK Age-specific Profiles of Feriliy in % 7% 30% 45% Younger han o o 34 Older han 35 From he able below, we can see he difference beween average age group and older group. Firsly, since he age gaps beween hese wo groups are 5 year in UK and 10 year in US, he US daa decreases more in reference discoun rae. Secondly, he declines in P3 and P4 are dramaic in boh counries and he reference discoun rae of P4 in US daa even falls down o 2.71% below 2.80% level. Table 22 Reference Discoun Rae Comparison of Average Age Group (C) and Older Age Group (D) among Single Life Insurance Plan (P1), Child Plan (P2), Child Plan wih Income Benefi (P3) and Child Plan wih Boh Parens Insured (P4), in US and UK Daa Counry Age Group P1 P2 P3 P4 C 2.95% 2.98% 2.93% 2.86% US D 2.90% 2.92% 2.82% 2.71% Diff 0.05% 0.05% 0.11% 0.14% C 2.96% 2.98% 2.94% 2.89% UK D 2.94% 2.96% 2.90% 2.85% Diff 0.02% 0.02% 0.03% 0.04%