ANNUITIES AND LIFE INSURANCE UNDER RANDOM INTEREST RATES

Transcription

1 ANNUITIES AND LIFE INSURANCE UNDER RANDOM INTEREST RATES Bey Levikso Deparme of Saisical Uiversiy of Haifa Haifa, 395 ISRAEL Rami Yosef Deparme of Busiess Admiisraio Be Gurio Uiversiy, Beer Sheva ISRAEL Absrac I his paper we sudy he effec of radom ieres raes o life isurace programs ad o auiies boh cerai ad o-cerai. This is doe uder several assumpios o he sochasic srucure of he ieres raes. We fid he momes ad he disribuios of several radom acuarial fucios of ieres, such as a : A :. Comparisos wih he case of fied ieres raes are give. This is doe usig umerical resuls ad graphical represeaios of our model.. Iroducio ad lieraure review A impora problem facig he isurace idusry is esimaig fuure ieres raes. I paricular he fuure ieres raes are impora i log erm isurace coracs such a life isurace. The liquidiy i fiacial markes make his problem eve more sigifica. Risks i life isurace are due o wo facors (i) Radomess i he remaiig lifeime of he isured (ii) Uceraiy i ieres raes. The law of large umbers guaraees ha he

2 2 risk due o deahs ca be reduced by sellig may coracs. Risks due o flucuaig ieres raes are difficul o reduce. This observaio moivaed may researches o sudy he effec of volailiy o pricig ad reservig life isurace i radom evirome. I acuarial lieraure here is a disicio bewee aalyzig he effec of radomess i he wo cases above. As early as 969 A.H. Pollard. ad J.H. Pollard published a paper i which hey reaed acuarial fucios as radom variables. The radomess beig caused oly by variaios i he age a deah. Specifically hey aalyze A, a ad v as radom variables givig heir firs wo momes ad he correlaio bewee pairs of hese radom variables. laer o De Peril (989) gave a survey of he disribuio fucios (d.f.) ad he probabiliy desiy fucios (p.d.f.) of he beefi fucio of mos commo life isurace s ad auiies. Boyle (976) sudied he effec of he sochasic aure of ieres raes o acuarial fucios, assumig ha he force of ieres is geeraed by a whie oise, ha is forces of ieres i successive years are assumed o be ucorrelaed ad ormally disribued radom variables. Pajer ad Bellhouse (98,98) developed a geeral heory for auiies ad isurace fucios assumig ha he force of ieres follows auoregressive process. The heory is furher worked ou for ucodiioal ad codiioal auoregressive processes of orders oe ad wo. Beekma ad Fuellig, (99,99) preseed a model evaluaig auiies whe ieres raes ad fuure life imes are radom. Epressios for mea values ad sadard deviaios of prese value of fuure paymes are obaied. This is doe assumig he force of ieres rae behaves eiher like Oresei-Uhlebeck process or a Wieer process.

3 3 2. The model We sudy impora radom acuarial fucios i discree ime, whe he ieres raes form a sequece of radom variables. The acuarial fucios iclude: & s - Accumulaed auiy - cerai due uder sochasic ieres raes. A : - Temporary life assurace (wih erm - ) uder sochasic ieres raes. A - -year edowme assurace uder sochasic ieres raes. : IA - Icreasig whole- life assurace uder sochasic ieres raes. : & a - Temporary life auiy uder sochasic ieres raes. : I a - Icreasig emporary life auiy uder sochasic ieres raes. : Hereafer above a acuarial fucio deoes he value of a acuarial fucios uder radom ieres raes. I his work he ieres raes are assumed o be eiher i.i.d r.v s or markovia sream. We ge he cumulaive disribuio fucios of hese acuarial fucios as well as heir momes. Graphs ad umerical soluios are give for he disribuios ad momes of he above acuarial fucios. 2. Disribuios ad momes of auiy cerai - & s Le R be he aual ieres rae durig [-, -+) for,2,...,. Ad le X + R (i.e X is he value of $ a -+ if deposied a -.

4 4 $ $ $ $ $ $ X X - X -2 X 2 X Hece he radom value of a auiy cerai for years is: && s i X i X ( + X + 2 X ( + && s * ) X X KK + X X 2 3 L X ) () Where & s * + k 2 j k X j 2 Noe ha i) & s * is idepede of X. ii) & s * is disribued as & s&. Le F ( y) p( & s y), Codiioig o && s X we ge: y F (y) F ( )dg () && s && s (2) Where G is he disribuio of X. Observed ha (2) is a recursive equaio for F (y) && s wih iiial codiio F && s (y) p(x y) G (y). E[( & s ) k ] k y df (y) (3) & s Le µ E( && s ), The µ * E( & s ) E E( && s X ) E E{(( + & s )X ) X } (4)

5 5 This yields he recursive relaio: µ ( + µ ) µ, µ (5) So 2 µ µ + µ µ + µ + L + µ µ (6) Similarly if we deoe by σ 2 he variace of & s, Var (& s ). we ge σ 2 ( + µ ) 2 σ 2 + σ 2 ( µ ) 2 2 (7) Noe ha he above recursive relaios ( (2), (6), (7) ) applies for ay disribuios of i.i.d ieres raes. Numerics Firs case X ( + R ) U(,.), F() Cerai auiy uder U (,.) for

6 6.2 Cerai auiy uder U (,.) for.8 F() Noe he larger he he seeper he graph of he disribuio. For he momes we ge firs wo momes of,... & s assumig U(,. ) R Firs momes Secod momes 2 4 µ.5 σ µ σ µ σ µ 4.97 σ

7 7 Secod ad hird case 2 d case X + ep( λ 69),... So wih probabiliy.999, X falls i (,.)..2 Cerai auiy uder Ep (69 ) for 4.8 F() Cerai auiy uder Ep (69 ) for.8 F()

8 8 3 rd case X + ep( λ 2),... So ha mea ieres rae is 5%. Cerai auiy uder Ep (2 ) for 4.8 F() Cerai auiy uder Ep (2) for.2.8 F() We ca see he slopes of he X has uiform disribuio. & s whe X is epoeial, seeper i he case whe For eample if we compare he uiform case wih ep(λ2) for ui ime:

9 9 F() uiform F() ep F() We see ha he epoeial disribuio is more risky he he uiform disribuio (i.e. oe cross over), see Kaas(994) ch. III. Waers (978) gave recursive formulas for he momes of he auiy cerai uder i.i.d radom ieres raes, calculaed a he begiig of he aum ad, solved i umerically, whe he X s are i.i.d r.v s havig logormal disribuio. We compare he epeced value of & s whe (i) Ieres raes have logormal disribuio ad (ii) Whe ieres raes have a uiform disribuio, assumig he wo disribuios have he same firs wo momes. We fid for & s i he logormal case ha µ , µ while i he uiform case, we ge µ , µ Clearly i boh cases he firs momes of & s is.5 for boh disribuios.

10 2.2 Temporary assurace uder radom ieres raes - A : I his case he radomess is due o : (i) The lifeimes of he isureds. (ii) The ieres raes. More specifically he remaiig life, T,is clearly radom. Moreover T, has kow disribuio as give by a appropriae life-able (such as A(967-7)). The sequece 2.. {R } assumed o be radom, havig properies as lised i subsecio Specifically le R be i.i.d radom variables which represe he ieres raes a he year [ -, ) for,2,...,. R R 2 R 3... R Recall ha A q v k + : k (8) k uder radom ieres raes oe has o replace k v i (8) by k E( + R ) + hus we ge : k E(A ) q E( R ) : k + (9) + k The ree diagram for E (A ) is: :

11 q E( + R ) E (A ) : p. ( $ ) q E( + R ) 2 E (A 2) : q E( + R ) + + p p +. ( $ ) The backward ad forward recursive equaio for E (A ) are give respecively by : () ad () below 2 E(A E(A + ) ) + q ( p ) ( E( R ) ) : : k + () + + k 2 E(A ) : p E(A )E( + + : R ) + q E( + R ) 2 () Wih boudary codiios. E (A ) :, (2) wih boudary codiios give by (2). Codiioig o he saus of he idividual a he ed of he firs year (i.e - acive or dead) we ge:

12 2 P(A : Where z) P(A : P(( + R ) z I )P(I ) + P(A : z)q + P(( + R ) z I )P(I ) A + : z)p (3) - if he isured, aged, dies I durig he firs year i.e durig (,+). - oherwise Le α : be he disribuio fucio of A :, so by codiioig o R we ge from (3) he followig equaio: (z( + r)) (r) α q P (R ) + p f dr : z α R r + : 2 (4) wih boudary codiios: z < α : z,,2, (5) Numerics Le R U (,. ) i. i. d r.v ' s, 2,... Usig life able A(967-7) ad equaio (4) we ge he followig graphs for α ad 55 : α 45 : 2

13 3 Temporary life assurace disribuio uder radom ieres - alpha(55:) alpha(55:) z.2 Temporary life assurace disribuio uer radom ieres - alpha(45:2) alpha(45:2) z We ca see ha alpha is moooe icreasig fucio of. Noe ha α () : is he probabiliy ha T.

14 4 Remark We would like o compare emporary isurace wih discou facor v where v E( + R ), wih emporary isurace uder radom ieres raes. The by Jese iequaliy E(A ) A. I paricular for 55 ad, we ge: : : E (A ).328 A : 55 : 2.3 whole life assurace uder radom ieres raes - A For whole life assurace wih radom ieres raes, oe ca derive formulas for he momes of A ad for is disribuio by leig i he formulas obaied for A. I paricular we have: : k E(A ) q E( R ) k + (7) + k Le ζ be he cumulaive disribuio fucio of A. By codiioig o R we ge from he followig recursive equaio: (z( + r)) (r) ζ q P (R ) + p f dr z ζ (8) + R r 2.4 Term edowme assurace uder radom ieres raes - A : For his radom edowme assurace, we have: E(A ) : p E(A )E( + R + : ) + q E( + R ) 2 (9) wih iiial codiio E(A ) : q E( + R ) + p E( + R ) E( + R ) (2)

15 5 i.e A β : be he cumulaive disribuio fucio of β Le : P( A z) he we ge: : : β (r) ( z ( + r )) q P ( R ) + p β dr f z : + : R r 2 (2) uder he boudary b'd()tj47(e) (tj4t9 T TD8)Tj-5w( )Tj/F2 Tf2BT6 TD5

16 6.2 Term edowe assurace uder radom ieres raes - beha(45:2) beha(45:2) z Here we ca see ha as he isured is youger, z - he l.h.s of he suppor of beha is closer o, ad he graph, he eighborhood of is seeper year life auiy uder radom ieres raes - a : Cosider he case where he isured aged pays $ per aum i he ed of each year uil he miimum bewee his remais life ad he e years. We would like o sudy such a program uder sochasic ieres raes. We have he followig basic relaio a : ( a + )( + R ) + :,, I I (23) where I is give i (3). Codiioig o I we have he followig recursive relaio for ψ :

17 7 (24) ψ : P( a : z) P( a : z I )P(I ) + P( a : z I )P(I ) q + P [ ( + R ) a + z ] + : p q + p (z( + r) ) (r) ψ f dr R r + : 2 uder he boudary codiio: ψ (z ) : q + p P (R z ) z (25) Numerics Le R U (,. ) i. i. d r.v ' s, 2,... Usig life able A(967-7) ad equaio (24) we ge he followig graphs for ψ ad 55 : ψ 6 : 5 :

18 8 The disribuio of -year life auiy - psi(55:;z).2 psi(55:;z) z The disribuio of -year life auiy - psi(6:5;z).2 psi(6:5;z) z We ca see here ha as he isured is youger, z - he l.h.s of he suppor of psi is closer o.

19 9 We have derived also formulas for he disribuios ad he momes of : A :

20 2 Pollard, A.H. & Pollard, J.H.A (969). A sochasic approach o acuarial fucios. Joural of he Isiue of Acuaries.95, Waers, J.A. (978). The momes ad disribuios of acuarial fucios. Joural of he Isiue of Acuaries 5, 6-75.