A Re-examination of the Joint Mortality Functions

Transcription

1 Norh merican cuarial Journal Volume 6, Number 1, p (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali analsis involving muliple lives is easil one of he more complicaed aspecs in he heor of life coningencies. In his paper, we re-invesigae join morali funcions and in paricular, we eamine an asserion ha relaes he join-life and lassurvivor random variables. This common asserion saes ha he sum of he lifeimes of he join- life and he las-survivor sauses is equal o he sum of he lifeimes of he single sauses. However, we show ha his asserion is no precisel correc. We herefore offer a modificaion o he definiions of he sauses so ha his common asserion holds. 1. Inroducion. The equaion relaing he join-life and las-survivor fuure lifeime random variables is given as (1) T ( ) T( ) = T() + T() +, wihou an independence assumpion. This asserion relies on cerain criical, e unsaed, assumpions. In Secion 4, we will presen a modified version of (1) wih proper assumpions. Since man formulae rel on his asserion, a number of basic relaionships in muliple life morali funcions urn ou o be wrong. For insance, 1

2 (2) p + p = p + p and (3) + = + are no valid wihou cerain independence assumpions. (ll necessar noaions and definiions are given in he following secion). Since hese relaions are used o price join life insurance producs, i is imporan o invesigae assumpions ha make hese relaions hold, and when hese assumpions do no hold, i is imporan o undersand he correc relaions. In Secion 3 eamples are given o illusrae ha hese equaliies do no hold. In Secion 4 he noaions are eamined in greaer deail, and valid equaliies are given. 2. Noaions and definiions. We follow closel he noaions in he ebook, Bowers, e al. (1997). We pu he cone in erms of spousal morali. Le X be he random variable represening female s age a deah, Y be he random variable represening male s age a deah, () is he saus denoing a person-aged-, ( ) is he join-life saus ha survives as long as boh () and () survive, ( ) is he join las-survivor saus ha eiss as long as a leas one of () and () is alive, and T(u) is he fuure lifeime for saus (u). 2

3 We define, for a saus (u), he condiional survival funcion p and he ne single u premium u for life insurance polic ha pas a he end of he ear he saus (u) fails. The annual ineres rae is denoed b i and 1 v =. 1 + i ( () > ) = P( X > + X ) p = P T >, ( () > ) = P( Y > + Y ) p = P T >, ( ( ) > ) = P( X > + Y > + X > Y ) p = P T and, >, ( ( ) > ) = P( X > + o r Y > + X > Y ) = P T, > p k+ 1, and u = v ( k pu k+ 1pu ), where u represens a saus,,, or. k 3. Eamples. The following are couner-eamples ha show equaliies (2) and (3) do no hold. Eample 1. ( p + p p + p ) Consider a pair of animals from a purel ficional species. Shorl afer birh he mae for life. probabili breakdown of heir ages a deah is given in he able below ( K m is he curae fuure lifeime of a newborn male animal, ec.): P( K f = 0) P( K f = 1) P( K f = 2) P( K m = 0) P( K m = 1) P( K m = 2)

4 Thus, for a newl maed pair, he probabili ha he male dies in is hird ear (K m = 2) while he female dies in is hird ear (K f = 2) is.20. Consider a pair, which have boh survived heir firs ear. Then we have he following: p 1:1 = P(boh survive a leas one more ear (given he survived hus far)) =.20/.50 = 2/5 p 1:1 = P(a leas one survives one or more ears (given he survived hus far)) =.40/.50 = 4/5 p 1 m = P(he male survives a leas one more ear (given i survived hus far)) =.40/.70 = 4/7 p 1 f = P(he female survive a leas one more ear (given i survived hus far)) =.40/.70 = 4/7 Clearl, , so p + p p + p. Eample 2. ( + ) For calculaing las-survivor acuarial presen values, = + is commonl used. However, i urns ou ha his formula does no hold in general. Consider a copula model (e.g., Hougaard s copula wih Weibull marginals): Le he bivariae survival funcion (, ) P{ X > and Y > } = C( S ( ) S ( ) S () S 1, 2 =, where m j = > = σ j P{ X } ep for j = 1, 2, and j m j 4

5 1 α [ ] α ( ) α C u, v = ep ( lnu) + ( lnv). Noe ha C ( u, 1) = u and C (, v) = v bivariae survival funcion ( ) 1, hus he marginal survival funcions derived from he S, coincide wih S 1 () and (), respecivel. Le X be he "female age a deah" random variable, and le Y be he "male age a deah" random variable as defined earlier. We consider he model wih m 1 = , σ 1 = 8. 99, m 2 = 85.98, σ 2 =11. 24, α = (hese parameers come from using he maimum likelihood mehod on eperience daa for join annui conracs from an insurance f compan see Youn and Shemakin, 1999) and compue he ne single premiums 65, S 2 m 70, 65:70, and 65:70 ( 65:70 and 65:70 were compued using he bivariae survival funcion wih all he compuaions done on Mahemaica). Wih i =. 05, we have he following resuls: f 65 = m, 70 = , 65 :70 = = :70, and f m :70 = The epression ields an error of more han 10 %! The following able demonsraes he values of he raio f 65 + m 70 65:70 65:70 for various values of he associaion parameer α and he ineres rae i. Table 1. The raio f 65 + m 70 65:70 65:70 for various values of α and i. 5

6 Ineres rae 3% 6% 9% 12% ssociaion α ssociaion α =1 corresponds o independence beween male and female lives. Higher values of associaion and higher ineres raes bring abou a subsanial discrepanc beween he eac value and is approimaion : :70 f m Proposiions and nalsis. We presen a modified version of equali (1) T ( ) T( ) = T() + T() + ha correcl relaes he lifeimes of join-life and las-survivor sauses o he lifeimes of single sauses. Firs, le us closel eamine wh equali (2) p + p = p + p is no rue in general. We sar wih absrac probabili argumens, raher han morali argumens. The reason is ha man of us are so familiar wih he equali (2), i is worhwhile o sep back a lile. Le, B, C, and D be random evens. Then he condiional probabiliies of B and B given C D are relaed b he equaion 6

7 (a) P( B C D) + P( B C D) = P( C D ) + P(B C D ). Noe ha he same idenical condiion (ha C D be rue) is presen in each of he four probabiliies; o change some of hese condiions would creae a saemen which is no alwas rue. For eample, (b) P( B C D) + P( B C D) P( C ) + P(B D ) in he general case (alhough i ma be fairl close in man cases). We will now ranslae his resul ino acuarial erms. Le = {X > + }, B = {Y > + }, C = {X > }, and D = {Y > }. Then P( B C D) = p, P( B C D) = p, P( C ) = p, and P(B D ) = p, so inequali (b) shows us ha p + p p + p. We see ha he equali (2) does no hold because p makes no assumpions on he curren saus of Y and p likewise makes no assumpions on wheher X survives o age or no, while p and p require boh X> and Y> as condiions. The same is rue for he relaion beween T(), T(), T() and T( ). To define T(), one makes no assumpions on he curren saus of Y and, o define T(), one makes no assumpions on wheher X survives o age or no, bu o define T() and T( ), boh X> and Y> are required. In order o relae join life funcions wih single life funcions, one needs o consider single sauses in a join cone wih spousal informaion. 7

8 Noaion. We inroduce ( ) o denoe a saus of a person-aged- whose spouse is a person-aged-. Thus he fuure lifeime random variables for he sauses ( ) and ( ) are given b T( )=X-, defined when boh X> and Y>, and T( )=Y-, defined when boh X> and Y>. Now we can sae ha T()=min(X-,Y-), defined when boh X> and Y>, and equals min(t( ),T( )), and T( )=ma(x-,y-), defined when boh X> and Y>, and equals ma(t( ),T( ). The condiional survival funcions for he sauses ( ) and ( ) would become ( ( ) > ) = P( X > + X > Y ) p = P T, > and ( ( ) > k) = P( Y > + X > Y ) p = P T, >. We now can properl sae (4) p + p = p + p. We sae he following proposiion wihou an furher proof. Proposiion 1. (5) T()=min(T( ),T( )), (6) T ( ) (7) T()+ T ( ) =ma(t( ),T( )), and = T( )+T( ). 8

9 We wan o noe ha, while T( ), T( ), T(),and T ( ) are all defined on he common domain X>, Y>, T() and T() are no: in a join cone, T() is defined when X>, Y>0 and T() is defined when X>0, Y>. In such a cone, T() and T() do no have a common domain and canno be added as random variables. Thus, (7) ma be considered as a correcion of (1). Proposiion 2. There eis random variables X and Y, ha for some,, and (8) p + p p + p (9) + +. Proof. Saemen (8) is demonsraed b Eample 1. Saemen (9) is demonsraed b Eample 2. We noe ha epressing p and p in a join cone would be ( ) ( ) (10) = P T() > = P X > + X >, Y > 0 p ( ) ( ) (11) = P T() > = P Y > + Y >, X > 0 p. In he same vein, idenifing T() wih T( 0) would be mahemaicall correc, alhough is inerpreaion ma seem unnaural. In he ppendi, we eamine he difference beween T () and T( ) as he relae o life insurance premiums. furher reamen of insurance premiums wih spousal saus is 9

10 given in "Pricing Pracices for Join Las Survivor Insurance" b Youn and Shemakin, RCH If lives X and Y are indeed independen, he saemens (2) and (3) become rue. Unforunael, recen sudies show ha survivals of pairs of husbands and wives are no independen. (See nnui Valuaion wih Dependen Morali b Frees, e. al.) The Proposiion 2 is no as absrac or rivial as i migh seem. n assumpion of equali in (2) and (3) is he basis of man imporan relaionships in muliple life morali funcion heor. The Third Eaminaion of he Socie of cuaries ess knowledge of muliple life morali funcions (among oher opics) and hisoricall has made use of hese formulae. 5. Conclusion. Wha kind of independence assumpion is required o creae equali in (8)? The answer is, as he condiions in he epressions (10) and (11) indicae, he morali rae of he female or he male should no depend on wheher he have a surviving spouse or no, nor on he surviving spouse s age. This is generall assumed in pracice. Insurance companies do no classif insurers according o wheher one has a surviving spouse or no, nor o spouse s age. I is worh noing ha, according o a copula model as illusraed b Tables 2-4 in he ppendi, he life insurance premiums wih spousal saus classificaion are lower han hose wihou he classificaion. The percenage differences are higher for older spouses and for higher ineres raes. 10

11 References BOWERS, N.; GERBER, H.; HICKMN, J.; JONES, D.; and NESBITT, C. (1997) cuarial Mahemaics, Schaumburg, Ill.; Socie of cuaries. FREES, E.; CRRIERE, J.; and VLDEZ, E. (1996) nnui Valuaion wih Dependen Morali, Journal of Risk and Insurance, Vol. 63, YOUN, H.; and SHEMYKIN,. (1999) Saisical specs of Join Life Insurance Pricing, 1999 Proceedings of he Business and Economic Saisics Secion of he merican Saisical ssociaion, YOUN, H.; and SHEMYKIN,. (2000) Pricing Pracices for Join Las Survivor Insurance", cuarial Research Clearing House,

12 ppendi To illusrae an effec he difference beween T () and ( ) insurance, we compare insurance premiums based on T () and ( ) T has in pricing life T. k+ 1 We noe ha = P( T() > k) = P( X > + k X ), = v p p ) and k p > ( ( ) > k) = P( X > + k X > Y ) ( k k+ 1 k p = P T, >, and compue he insurance k+ 1 ( k k+ 1 premium wih spousal saus = v p p ). The following ables demonsrae he raio / beween he premium values for females age wih and wihou spousal classificaion, spousal ages allowed o var. Thus, denoes female s age and denoes her spouse s age. Compuaions were preformed according o Hougaard s copula model wih Weibull marginals m 1 = , σ 1 = 8.99, m 2 = , σ 2 =11. 24, and α = Ineres rae varies from 3% o 7%. Table 2. The raio / wih ineres rae 3%. Male ge Female ge Table 3. The raio / wih ineres rae 5%. Male ge Female ge

13 Table 4. The raio / wih ineres rae 7%. Male ge Female ge