LIFE INSURANCE PLANS AND STOCHASTIC ORDERS

Transcription

1 HE PUBLISHING HOUSE PROCEEDINGS O HE ROMANIAN ACADEMY, Series A, O HE ROMANIAN ACADEMY Volume 6, Numer /5, pp. - LIE INSURANCE PLANS AND SOCHASIC ORDERS Gheorghiţă ZBĂGANU Gheorghe Miho Cius Io Insiue of Mhemil Sisis nd Applied Mhemis of he Romnin Ademy, Cs Ademiei Române, Cle 3 Sepemrie no. 3, 57 Buhres, Romni. E-mil:zgng@sm.ro In risk heory severl sohsi order relions mong disriuions re ommonly used: inresing onvex, sohsi nd in morliy or hzrd re. We show heir mening in life insurne nd poin ou new ype of dominne sronger hn he sohsi dominne u weker hn he dominne in morliy.. LIE INSURANCE SCHEMES AIR IN EXPECAION. We del wih wo prners: he insured A nd he insurer B. here re mny kinds of life insurne shemes LIS). We del wih he simples ones whih run s follows: he wo prners gree ime = o sign onr: - momen or from o ) A pys o B sh moun of C monery unis MU) or sh flow ) in order h - ime B py o A S MU provided h A is sill live momen or B py o A sh flow s sring unil he deh of A). hey gree on given insnneous ineres re IIR) denoed y δ. So we del wih four ypes of LIS: i) ii) LIS of ype Π, C,S;, ; δ,): A pys o B C MU nd ime B pys o A S MU provided h A is sill live. he index, mens h i involves pymen from A o B nd pymen from B o A. We ll his n LIS of ype -. is he lifeime of A; he pymen is mde only if >. LIS of ype Π, C,s;, ; δ,): A pys o B C MU nd from unil he rndom ime when A dies, B pys o A sh flow of inensiy s MU. A sh flow is funion s:[,) [,) wih he mening h he sh moun pid on he ime inervl [,] is dx MU. he index, mens h his ype involves pymen from A o B nd sh flow pymen from B o A. We ll his n LIS of ype -. iii) LIS of ype Π,,S;,, ; δ,): in he inervl [, ] A pys o B sh flow of inensiy MU nd ime B pys o A S MU provided A is sill live. he index, mens h his ype involves one sh flow from A o B nd single py from B o A. We ll his n LIS of ype -. iv) LIS of ype Π,,s;,, ; δ,): in he inervl [, ] A pys o B sh flow of inensiy MU nd from unill, B pys o A sh flow of inensiy s MU. he index, mens h his ype involves one sh flow from A o B nd single py from B o A. We ll his n LIS of ype -. Someimes i is lled pension pln. Reomended y Mrius IOSIESCU, memer of he Romnin Ademy

2 Gheorghiţă Zăgnu Definiion.. An insnneous ineres re IIR) is ny funion δ:[,),) whih is righoninuous nd hs limi on he lef. Is mening is h MU orrowed ime = oss σ ) = exp δ x )dx MU ime =. he funion σ is he fruifiion for. hroughou his pper we e δ u) du shll ep h σ) =. he funion :[,) [,] defined y = = is lled he σ ulizion for. or ny funion whih is righ oninuous nd wih finie limis o he lef we shll use he reviion CADLAG. All he sh-flows will e supposed o e CADLAG, oo. We shll use he nlogy eween he IIR denoed δ nd he filure re of lifeime τ onsidered in [8]. he ide is sine is non inresing, righ-differenile, ) = nd )=, i n e onsidered o e he survivl funion of some soluely oninuous lifeime denoed y τ. If he densiy of τ is denoed y ϕ hen δ = ϕ / is he filure re of τ. Le us denoe he filure re of n soluely oninuous lifeime y r, is disriuion funion y nd is survivl funion > y. hus δ = r τ. Noie h he uries ll r he morliy fore of momen see [], p. 49) nd denoe i y µ. Insed of hey wrie p ; however his noion is umersome for mhemiins. In generl he onneion eween he survivl funion of some lifeime nd r is given y ) := = exp r dx dx..) Remrk. or he ske of shorening proofs, ll he lifeimes in he sequel will e supposed o e soluely oninuous nd wih finie momens of ny order. All he lifeimes uries use re like h. However, mny resuls re rue wihou suh drsil ssumpions. When is his LIS fir? he word fir hs mny inerpreions. We men fir wih respe o he greed IIR δ. In his inerpreion we know wh fir mens: suppose we del wih n LIS of ype -. As MU momen hs he vlue momen, hen if A is sill live ime fir would men h C ) = S ). If we dd he ineriude ou he lifeime of A, hen he prolem hnges i. Wh is h o lim he firness when deling wih rndom vriles? A lo hs een wrien ou h. One wy o nswer he quesion is o use he priniple of he expeed uiliy of Von Neumnn Morgensern see for insne [7] ). Bu if we ep h he insurer is risk neurl - mening h his esimion of risk X is is expeion h my hppen if he insurer hs mny insured people) hen we ould hink s follows. Any LIS involves wo rndom vriles: X nd Y. X is he ol sh moun pid y A o B ulized momen nd Y is he ol sh moun pid y A o B ulized he sme momen =. hen we hve Definiion.. An LIS is fir in expeion IE) iff EX = EY. Now, we shll see wh his definiion implies in he se of our four ypes of LIS : i) LIS of ype Π, C,S;, ; δ, ): X = C ) > ) nd Y = S ) > ) ; ii) LIS of ype Π, C,s;, ; δ, ): X = C ) > ) nd Y = d. ; min, ) min, ) iii) LIS of ype Π,,S;,, ; δ, ): X = d nd Y = S ) ) min, ) min, ) > ; iv) LIS of ype Π,,s;,, ; δ, ): X = d nd Y = d. hus we hve he following resul: min, ) min, )

3 3 Life insurne plns nd sohsi orders Proposiion.. Le us denoe y he survivl funion.hen i) n LIS Π, C,S;, ; δ, ) is IE iff C ) ) =S ) ) ; ii) n LIS Π, C,s;, ; δ, ) is IE iff C ) ) = d ; iii) n LIS Π,,S;,, ; δ, ) is IE iff d = S ) ); iv) n LIS Π,,S;,, ; δ, ) is IE iff d = d. Proof. he firs lim is ovious sine EX = C ) > ) = C ) ) nd EY = C ) ). or he oher ones, we use he following formul whih is esily proved using uini s heorem: if f is oninuous nd righ-differenile, hen Ef) f) = f dx, where f snds for he righ-derivive of f. If we ke f = x s min x, d hen f = for ny x, f = d for ny x >, hene ) f = for x >, f = for x < ; hus, EY = Ef) f) = d. o ompue EX in ses iii) nd iv), we use he funion f =, f = x d = ons for x >, f = nd f = on [, ),). hus EX = min, ) min, ) x min x, ) min x, ) f dx = d; his ime f = for x d for < x < hene f = for < x < d = d. Now we show he proilisi mening of hese relions. Rell h if is he lifeime of A nd >, hen he new rndom vrile *:{ > } [,) defined on he new proiliy spe { > } wih he new proiliy P {>} y he formul * := he usul noion of uries is >! ) is lled he residul lifeime ge or even ime-up-o deh ge see [3]). I is denoed y, u we prefer o denoe i y o void possile onfusions. Is survivl > + funion > = = is usully denoed y nd is hzrd re y r := > r+. Now remrk h if nd re survivl funions, hen is survivl funion oo. If = τ> nd = > hen = min,τ) >, provided h nd τ re independen. hus, if we suppose h our fiive τ desried y δ is independen of he rel lifeime we ould rese he ove proposiion s Corollry.. Le us denoe y he survivl funion.hen i) n LIS Π, C,S;, ; δ, ) is IE iff C min,τ) ) =S min,τ) ); ii) n LIS Π, C,s;, ; δ, ) is IE iff C min,τ) ) = ESmin,τ)) where S = dx; min, )

4 Gheorghiţă Zăgnu 4 iii) n LIS Π,,S;,, ; δ, ) is IE iff ECmin,τ)) = S min,τ) ) where C = min, ) min, ) iv) n LIS Π,,S;,, ; δ, ) is IE iff ECmin,τ)) = ESmin,τ)) wih C,S s ove. dx;. SOCHASIC ORDERS AND AIR LIS Definiion.. Le, e wo soluely oninuous lifeimes. Le, e heir survivl funions nd r, r heir hzrd res. We sy see, for insne [6] or [7]) - is sohsilly domined y nd wrie! s ) iff ; - is domined y in morliy nd wrie! m ) iff )! s ) ; - is inresing onvex domined y nd wrie! ix ) iff E + E +. I is well known see for insne [5], [6] or [7]) h! s Ew ) Ew ) for ny inresing nonnegive w; h! m r r nd h! ix Ew ) Ew ) for ny inresing onvex nonnegive w. However, we will no need h in he sequel. Our gol is o mke he onneion eween hese oneps nd he LIS whih re fir in expeion. o sr wih, suppose h Π, C,;, ; δ, ) is IE. his mens h C ) ) = ) ). As S is only proporionliy for we my s well ssume h S = MU; wrie hen he firness ondiion s C = C δ,, ;, ) ) ) ) )..) Suppose now h he insurer B dels wih wo insured persons A nd A. Suppose h he knows heir survivl funions nd h relly hppens if he knows heir ges, residene or se. When is i fir o hrge A less hn A for he sme reimursemen S? Proposiion.. Le, e wo soluely oninuous lifeimes. hen i) C, s,; δ, ) C, s,; δ, ) for every, > s iff ) s)! s ) s) ; ii) C,,; δ, ) C,,; δ, ) for every iff! s ; iii) C, s,; δ, ) C, s,; δ, ) for every s,, s < iff! m. Proof. or insne, C, s,; δ, ) C, s,; δ, ) s) s) s) s) s) nd he ls inequliy n e wrien s ) s) > -s) ) s) > -s) s, proving lim i). In priulr, if s= hen we hek he seond lim nd if he inequliy holds for every s < his is he very definiion of he dominion y morliy. Le us onsider now n LIS of ype Π, C,s;, ; δ, ). We know h his ype of LIS is IE iff C ) ) = d ; wrie h s C = C, ; s, ; δ, ) = s ) ) )d.) ) ) s) he nlog of Proposiion.. is Proposiion.. Le, e wo soluely oninuous lifeimes. hen

5 5 Life insurne plns nd sohsi orders i) C, s; x, ; δ, ) C, s; x, ; δ, ) for ny sh flow s nd ny, > x, iff )! s ) ; ii) C, s;,; δ, ) C, s;,; δ, ) for ny sh flow s nd ny, iff! s ; iii) C, s;, ; δ, ) C, s;, ; δ, ) for ny sh flow s nd ny,, >, iff! m ; iv) C, ; x, ; δ, ) C, ; x, ; δ, ) for every, >, iff min,τ))! ix min,τ) ; v) C, ;,; δ, ) C, ;,; δ, ) for every, iff min,τ))! ix min,τ) vi) C, ; x, ; δ, ) C, ; x, ; δ, ) for ny IIR δ nd ny, > x, iff )! s ) ; vii) C, ;,; δ, ) C, ;,; δ, ) for ny IIR δ nd ny, iff! s viii) C, ;, ; δ, ) C, ;, ; δ, ) for ny IIR δ nd ny,, >, iff! m. Proof. i). We hve C, s; x, ; δ, ) C, s;, x; δ, ) d d for ny ounded CADLAG s:[,) [,). If s = [,) for some, >, we ge d d. As he inegrnds re oninuous, his mens h ) x > nd, s n e ny momen greer hn x, we ge he inequliy > x whih is he sme s )! s ). Clims ii) nd iii) re esy onsequenes of i). We prove iii). Now, = is onsn, hene he inequliy eomes ) x d ) x d for ny > x. Le j * = min j,τ), j =,. Rell h τ is supposed o e independen of nd. hen j * > = j, hene he ove inequliy eomes * > ) x * > d * > ) x * > d for ny > x. Le h = x. Remrk h for ny lifeime we hve E h) + = E-x-h) + > = > d E x h) + ; > E x h; > x + h) E ; > ) = = =. hus he inequliy > > > >.4) n e wrien s E *) h) + E *) h) + for every h. his is he very definiion of he ix dominion. Now, v) is n esy onsequene when x =. o prove vi), wrie insed of nd le some >. Choose sequene δ n ) n of ineres res wih he propery h n [,] [,]. Here n = exp δn y) dy. or insne, ke δ n = /n if < nd δ n.3).4)

6 Gheorghiţă Zăgnu 6 = n for > ). Aording o.3) we hve n d n nd pplying Leesgue s dominion priniple we ge for ny x,, suh h x< <. As j re oninuous, i folllows h d d n d for every n. Leing.5) > x )! s ). he onverse impliion is onsequene of i). Now, vii) nd viii) re onsequenes of vi). As n ineresing yprodu, we noie Corollry.3. Le nd e wo life imes on proiliy spe in whih soluely oninuous disriued rndom vriles do exis. hen! s min,τ)! ix min,τ) for ny τ independen of nd ;! m min,τ))! ix min,τ)) for ny τ independen of nd nd for ny. Noie h he is indeed ovious vi he well known f h! s here exis versions of j, sy j suh h.s. see for insne [7]) Now we del wih n LIS of ype -. Aording o Proposiion.iii), n LIS Π,,S;,, ; δ,) is IE if i) S = S, ;,, ; δ, ) = d / ) )).6) Proposiion.4. Le, e wo soluely oninuous lifeimes. hen S, ;,, ; δ, ) S, ;,, ; δ, ), [,), < iff ) < ; ) ii) S, ;,, ; δ, ) S, ;,, ; δ, ), [,), < iff ) ) < ; iii) he following sserions re equivlen: ) S, ;,, ; δ, ) S, ;,, ; δ, ),, suh h < < nd for ny ; ) S, ;,, ; δ, ) S, ;,, ; δ, ),, suh h < < ; )! m. iv) if S, ;,, ; δ, ) S, ;,, ; δ, ), suh h < hen! s ; v) S, ;,, ; δ, ) S, ;,, ; δ, ), suh h <, for ny IIR δ iff! m. Proof.: i) We hve S, ;,, ; δ, ) S, ;,, ; δ, ) d / ) d / ) for every < <. Le <, =, = + h, h > smll enough. If we divide he

7 7 Life insurne plns nd sohsi orders ls inequliy y h, le h nd use he f h he inegrnds re righ-oninuous we ge ). he onverse is ovious. ) Now, ii) is n esy onsequene. In iii), he impliion ) ) is ovious; for ) ) use ii) pu ) ) in he form < ) )! s ) )! m. As for ) ), i is esy. ) ) or iv), jus le nd divide y he inequliy d / ) d / ): i follows h / ) / ). In vi). he novely is h now =. Le = nd =. So, he hypohesis is h d / ) d / ).7) for every < nd for every IIR δ. Le < e fixed nd le,). Le δ n ) n e sequene of IIR suh h n [,] for insne, δ n = /n if < nd δ n = n for > ). Repling in.7) y n nd leing n, one ges he inequliy d / ) d / ), rue for ny < < <. Repling y +h, dividing y h nd leing h i follows h he inequliy ) / ) ) / ) holds for ny < ; u his is preisely he definiion of! m. inlly, we del wih n LIS of ype - wih pension plns. By Proposiion.iv), n LIS Π,,s;,, ; δ,) is IE iff d = d. his ime we shll suppose h nd s re onsn sh flows: =, = s. hen he firness ondiion eomes d = r d.8) Suppose h r = nd denoe he orresponding from he ove equliy y,, ; δ,). hus,, ; δ,) = ) d d.9) Proposiion.5. Le, e wo soluely oninuous lifeimes nd le τ e lifeime independen of hem wih δ s hzrd re. hen i),, ; δ, ),, ; δ, ) for ny < < iff min,τ))! ix min,τ)) ; ii) he following sserions re equivlen: ).,, ; δ, ),, ; δ, ) for ny < < nd for ny IIR δ; ).,, ; δ, ),, ; δ, ) for ny < nd for ny IIR δ; ).! m iii) If,, ; δ, ),, ; δ, ) for ny < nd for ny IIR δ, hen! s. Proof. By.8) he inequliy,, ; δ, ),, ; δ, ) n lso e wrien s

8 Gheorghiţă Zăgnu 8 ) d d ) d d.) or s ) d d ) d d. Le = + h. Divide he inequliy y h, wrie insed of nd le h ; one ges he inequliy ) d ) d.) for ny. Wriing he inequliy s E[min,τ)) ) + ] E[min,τ)) ) + ] >, we see h we oin he very definiion of he f h min,τ))! ix min,τ)). o ge he onverse impliion, jus inegre.) from o. ii) he only non-rivil impliion is ) ). Le > nd hoose sequene δ n ) n of IIRs, suh h δ n onverge o [,] s n. Pu in.9) =, = =, δ n insed of δ nd le n. One ges d d d d.) for ny < <. Wrie now = Λ, pu = + h, divide.) y h nd le h. I follows h ) / d ) / d Λ Λ )) d.3) for ny <. Repe he rik wih = h; one ges d / ) d / ) Λ ) Λ ) d.4) for ny <. As inequliy.4) holds for ny < we my s well pu insed of nd insed of, jus o ge he sme inegrion limis in.3) nd.4). Addding he wo inequliies, we ge Λ ) Λ )) d. As < he mening is h Λ is non-deresing. Bu his is noher wy o desrie he f! m. iii) In.9) do he sme rik wih δ n [,] s efore. Ge le, o oin ) ); s is rirry,! s. d / d / d nd d Remrk. he only ses wihou equivlene re iv) from Proposiion.4 nd iii) from Proposiion.5). Aully he following equivlenes hold: ) S, ;,, ; δ, ) S, ;,, ; δ, ) for ny < ),, ; δ, ),, ; δ, ) for ny < ) )/ d )/ ) d for ny < <

9 9 Life insurne plns nd sohsi orders hey imply! s nd re implied y! m. hus his new sohsi order lies somewhere eween sohsi dominion nd dominion in morliy nd is differen from oh of hem. or exmple, if = e ) nd = Λ wih Λ = + [,3) + 7- [3,4) + 3 [4,), hen he reder my hek h ) is fulfiled hene ) nd ) re fulfilled oo!) u i is no rue h! m. o give n exmple when! s u ) is no rue is even esier. REERENCES. BARLOW, R. E., PROSCHAN,., Sisil heory of Reliiliy nd Life esing. o egin wih, Silver Spring, MD, BOWERS, N.L., GERBER, H., HICKMAN, J., JONES, D., NESBI, C., Auril Mhemis, Is, Ill, he Soiey of Auries, BURLACU, V., CENUŞĂ, G., Bzele memie le eoriei sigurărilor,buhres, ASE,. 4. GNEDENKO, B., BELEAEV Y.,SOLOVIEV A., Mehodes mhemiques dns l heorie de fiilie, Mosow, Mir, GOOVAERS, M.J., KAAS R., HEERWAARDEN, A.E. BAUWELINCKX,., Effeive Auril Mehods,Amserdm, Norh Hollnd, SHAKED, M., SHANIKUMAR, J., Sohsi Orders nd heir Appliions, Boson, Ademi Press, ZBĂGANU, G., Meode memie în eori risului şi uri. Buhres, Universiy Press, ZBĂGANU, G., Insnneous ineres res nd hzrd res, o pper. Reeived erury 8, 5