STOCHASTIC LIFE ANNUITIES Daniel Dufresne

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1 SOCHASIC LIFE ANNUIIES Daniel Dufresne Absrac his paper gives analyic approximaions for he disribuion of a sochasic life annuiy. I is assumed ha reurns follow a geomeric Brownian moion. he disribuion of he sochasic annuiy may be used o answer quesions such as Wha is he probabiliy ha an amoun F is sufficien o fund a pension wih annual amoun y o a pensioner aged x? he main idea is o approximae he fuure lifeime disribuion wih a combinaion of exponenials, and hen apply a known formula (due o Marc Yor) relaed o he inegral of geomeric Brownian moion. he approximaions are very accurae in he cases sudied. JEL Classificaion: C Keywords: Sochasic life annuiies; Inegral of geomeric Brownian moion; Sums of lognormals; Combinaions of exponenials 1. Inroducion Defined conribuion pension arrangemens, wheher in he form of employer-sponsored pension plans or individual savings, have enjoyed grea populariy laely. he main drawback of hese arrangemens is he uncerainy of fuure lifeime and raes of reurn. As a consequence, reirees face wo difficul decisions: how much o draw form heir savings every monh, and how o inves wha is lef. A mahemaical difficuly is ha he disribuion of a life annuiy including random duraion of life and random raes of reurn is no known in closed form. he main goal of his paper is o presen an apparenly new way o approximae he disribuion of such sochasic life annuiies. his problem has already appeared in he acuarial lieraure. Pollard (1971) was apparenly he firs o consider he problem of finding he disribuion of a life annuiy, when boh moraliy and he discoun raes are random. Several auhors have sudied he same problem since, including Boyle (1976), Waers (1978), Wilkie (1978), Panjer & Bellhouse (198), Bellhouse & Panjer (1981), Wesco (1981), Dufresne (1989, 199, 1992), Beekman and Fuelling (199), Frees (1991), De Schepper e al. (1992abc, 1994), Vannese e al. (1994, 1997), Parker (1994a, 1994b), and Hoedemakers e al. (25). he majoriy of he conribuions focused on calculaing he momens of he annuiy, he goal being o reconsruc or approximae he disribuion from is momens; a few acuarial papers deal wih he disribuion of annuiies wih random raes of ineres, or wih heir comonoonic approximaion. his paper proceeds differenly. he disribuion of a sochasic life annuiy wih an exponenial lifeime disribuion is already known explicily, in he case where raes of discoun are independen over ime (more deails below). I will hen be seen ha he same holds if he lifeime has a densiy given by a combinaions of exponenials. Nex, because combinaions of exponenials are dense in he se of disribuions on R +, any lifeime disribuion may be approximaed o any degree of precision by a combinaion of exponenials. Consequenly, he disribuion of a sochasic life annuiy, wih a given (arbirary) lifeime disribuion, may be approximaed by he disribuions Daniel Dufresne, FSA, is Professor a he Cenre for Acuarial Sudies, Universiy of Melbourne, Melbourne VIC 31, Ausralia, dufresne@unimelb.edu.au.

2 of sochasic life annuiies wih lifeimes disribued as combinaions of exponenials; moreover, he error of his approximaion ends o zero as he approximaing lifeime disribuions converge o he rue lifeime disribuion. Each approximaing sochasic life annuiy disribuion is known exacly, and will be idenified. Numerical illusraions are given. Some of he papers cied above assumed ha raes of discoun form an auoregressive process, probably o reflec he well-known serial dependence of ineres raes. In his paper, he discouned values of annuiies are no based on an ineres rae model, bu are insead he amoun of money required o fund he annuiy, given a paricular invesmen policy. Le U be he ime- value of one dollar invesed a ime. his dollar may have been invesed in socks, bonds, and so on. he amoun required a ime o fund a $ C cash-flow a ime is hus $ C/U. he amoun iniially required o fund several cash flows {C j } a imes { j } is hen S = j C j U j. his amoun is of course random, and is no a price in any sense of he word. (No-arbirage pricing (as described, for insance, in Harrison & Pliska (1981)) does no lead o a unique price in cases where cash flows depend on moraliy, as he marke is hen incomplee.) Nowihsanding his, here is an ineres in knowing he probabiliy disribuion of S, for insance when comparing invesmen sraegies, or, possibly, when informing insurance conracholders of he likely amouns of heir benefi paymens, if here is a variable componen in heir conracs. I will be assumed ha he raes of reurn on he invesmens chosen are serially independen. his is he usual assumpion for socks, adoped by Black & Scholes (1973), hough heir lognormal model goes back o Osborne (1959); he firs appearance of Brownian moion was also in he conex of a financial model (Bachelier, 19), and also feaured reurns which are independen over ime, hough i is he asse values hemselves (and no heir logarihm) which were normally disribued. Bond yields, however, are highly correlaed over ime. Neverheless, i can be argued ha a managed porfolio of bonds would ofen have reurns which are less correlaed over ime. Le {W ; } be a sandard Brownian moion, and define U = e m+σw. he process {U ; } is called geomeric Brownian moion. As indicaed above, U will represen he ime- value of one dollar invesed a ime. he parameers (m, σ) depend on he available securiies and on he invesmen sraegy followed. he variable S defined above is hen a sum of lognormals, a opic sudied from he poin of view of Asian opions by several auhors (see Dufresne (24) for references). here is anoher source of randomness in a sochasic life annuiy. he variable will represen he fuure lifeime of an insured or pensioner. We will always assume ha invesmen resuls and survival do no affec one anoher, or, expressed mahemaically, ha and he Brownian moion W are sochasically independen. he amoun required a ime o fund an annuiy of 1 per annum payable coninuously while he pensioner is alive is D = U 1 s ds = e ms σw s ds. (1.1)

3 his formula corresponds o a level annuiy, bu noe ha obvious modificaions allow for (1) paymens ha increase exponenially (say, wih inflaion), by including an exponenial facor e y in he inegral; and (2) guaranee periods, by replacing wih max(,n). I does no seem possible o apply our resuls o discreely paid annuiies, hough he difference beween he discouned values of annuiies paid monhly and annuiies paid coninuously is no very grea. Observe ha he disribuion of does no have o be coninuous (see Secions 3 and 4). Secion 2 recalls some known resuls abou he inegral of Brownian moion which are used in he sequel. Secion 3 deals wih he approximaion of disribuions on R + by combinaions of exponenials, and Secion 4 gives a proof ha he disribuion of a sochasic life annuiy can be approximaed o any degree of precision by a sochasic annuiy for a duraion ha has a combinaion of exponenials as densiy. Secion 5 looks a ransforms and momens of sochasic life annuiies, and shows ha he disribuion of a sochasic life annuiy is in general NO deermined by is momens (however, i may be deermined by is reciprocal momens). Secion 6 liss he assumpions made in he numerical examples in he subsequen secions. Secion 7 idenifies he Jacobi approximaions used for he disribuion of he fuure lifeime. Numerical disribuions of sochasic life annuiies are given in Secion 8. Secion 9 shows how, in cerain cases, i is numerically feasible o use a more direc roue o find he disribuion of a sochasic life annuiy, and one such case is used as a check on he mehod described in Secion 8. Secion 1 is abou deferred annuiies, and Secion 11 shows how he resuls presened may be applied o sochasic annuiies-cerain. Noaion and vocabulary. he cdf of a probabiliy disribuion is is cumulaive disribuion funcion, he ccdf is he complemen of he cdf (one minus he cdf), and he pdf is is probabiliy densiy funcion. An aom is a poin x R where a measure has posiive mass, or, equivalenly, where is disribuion funcion has a disconinuiy. A sequence of random variables {X n } is said o converge in disribuion o a random variable X if P(X n x) converges o P(X x) for all x where he disribuion funcion of X is coninuous; convergence in disribuion is also called weak convergence (see Billingsley, 1986). Finally, use will be made of he Pochhammer symbol (a) = 1, (a) n = a(a +1) (a + n 1), n = 1, 2, Fundamenal resuls on he inegral of Brownian moion his secion collecs some known resuls abou he inegral of Brownian moion. More deails can be found in he references; in paricular, Yor (21) conains several papers on hese opics, while Dufresne (25) gives a summary of several of he resuls, wih deailed references. he expression in equaion (1.1) depends on, m and σ. he scaling propery of Brownian moion makes i possible o ransform equaion (1.1) in such a way ha any one of hese parameers has a prese value. he connecion wih Bessel processes (Yor, 1992) makes he choice σ =2more convenien in many developmens. We hus adop Yor s noaion A (µ) = e 2(µs+W s) ds,, (2.1)

4 where W is a sandard Brownian moion. From he scaling propery of Brownian moion, he conversion rule beween formulas (1.1) and (2.1) is: has he same disribuion as D = e ms σw s ds 4 σ 2 A(µ), where = σ2,µ= 2m 4 σ 2. (2.2) he process A (µ) has been sudied in several conexs, noably in relaion o Asian opions (for insance, Geman & Yor (1993)). is, Momens of A (µ). he disribuion of A (µ) has all momens (posiive and negaive) finie, ha E(A (µ) ) r < r R. Ramakrishnan (1954) showed ha, for n =1, 2..., E(A (µ) ) n = n / n b n,k e ak, a k = 2kµ +2k 2, b n,k = n! (a j a k ). (2.3) k= Proofs of his resul may also be found in Dufresne (1989) and in Yor (1992). he conversion rule (2.2) may be used in an obvious way o find corresponding formulas for ( n E e ms σw s ds), n =1, 2... j= j k (see Secion 5). Formulas for E[(A (µ) ) r ], r>, can be found in Dufresne (2). he disribuion of A (µ) sampled a an independen exponenial ime. he resul below firs appeared in Yor (1992). Le S λ Exponenial(λ), ha is, le S λ be a random variable wih densiy f Sλ (x) = λe λx 1 {x>}. Suppose, moreover, ha S λ is independen of he Brownian moion W in definiion (2.1). A summary of he proof of he following heorem is given in he Appendix. heorem 2.1 (Yor). he disribuion of A (µ) S λ has a densiy given by equaion (2.4), and is he same as ha of B 1,α /2G β, where B 1,α and G β are independen, B 1,α Bea(1,α), G β Gamma(β,1), wih α = γ + µ, β = γ µ, γ = 2λ + µ his is equivalen o f λ (u) = λ(2u) (µ γ)/2 1 Γ( ) µ+γ ( 2 Γ(γ +1) 1 F γ µ ,γ+1; 2u) 1 1{u>}, (2.4)

5 where 1 F 1 is he confluen hypergeomeric funcion: 1F 1 (a, b; z) = n= (a) n z n (b) n n!. he disribuion of A (µ) S is also known for hree oher disribuions for S, see Yor (21, pp.14-15). Laplace ransform of he disribuion of A (µ). I has been proved ha he disribuion of A (µ) is no deermined by is momens (Hörfel, 25); his makes sense inuiively, as he same siuaion prevails for he lognormal disribuion. Moreover, E exp(sa (µ) )= for all s>, as for he )) < if s<1, is deermined by is momens (Dufresne, 21). he. A formula is given for he Laplace lognormal disribuion. By conras, and a lile surprisingly, E exp(s/(2a (µ) which implies ha he disribuion of 1/A (µ) same reference shows formulas for he momens of 1/A (µ) ransform of A (µ) in Yor (21, p.16). he disribuion of A (µ). Several expressions have been found for he densiy of he inegral of geomeric Brownian moion a a fixed ime, bu none of hose expressions is simple. We give only wo examples, he reader is referred o Dufresne (25) for more deails. Oher expressions for he densiy are given in De Schepper e al. (1992ab) (see he commens in Deelsra & Delbaen (1992)). he following expression for he densiy of A (µ) is due o Yor (1992): /2 g(, x) = e µ2 x e µu 1 2x (1+e2u) θ eu /x() du, where θ r () = re π 2 2 2π3 exp( y 2 /2) exp( r cosh y) sinh(y) sin ( ) πy dy. Anoher expression for he disribuion of A (µ) is given in Dufresne (21). If q(y, ) = e π 2 8 y2 2 π 2 cosh y, hen he densiy of 1/(2A (µ) ) is 2 µ x µ+1 2 e µ 2 /2 e x cosh2 y q(y, ) cos ( π 2 ( y µ)) H µ ( x sinh y)dy. (2.5) Here H µ is he Hermie funcion (Lebedev, 1972, Chaper 1). he above expression reduces o a single inegral when µ =, 1, 2,..., bu is oherwise a double inegral

6 he disribuion of A (µ). For any m, σ > (Dufresne, 199), If m, hen ( 2 1 σ 2 e m σw d) Gamma ( 2m σ, 1 ) m, σ >. (2.6) 2 lim e ms σw s ds = a.s. 3. Approximaing disribuions on R + by combinaions of exponenials By combinaion of exponenials we mean a funcion of he form f() = n a j λ j e λj 1 {>}, (3.1) j=1 where {a j }, {λ j } are consans. his funcion is a probabiliy densiy funcion if (1) n a j = 1; (2) λ j > j; (3) f(x) x. j=1 Condiions (1) and (2) imply ha he funcion f( ) inegraes o one over R +, bu hey do no imply (3); for example, consider f(x) = e x 16e 2x +24e 3x = e x [24(e x 1 3 )2 5 3 ]. If a j > for all j, hen (3.1) is called a mixure of exponenials. A proof of he following resul is given in he Appendix. heorem 3.1. (a) Suppose is a non-negaive random variable. hen here exiss a sequence of random variables { n } each wih a pdf given by a combinaion of exponenials and such ha n converges in disribuion o. (b) If he disribuion of has no aom, hen lim sup n < F () F n () =. A mehod will now be presened for approximaing probabiliy disribuions by combinaions of exponenials. More deails can be found in Dufresne (26) (wo oher mehods are presened in ha paper). For α, β> 1, he shifed Jacobi polynomials are defined as R n (α,β) (x) = (α +1) n 2F 1 ( n, n + α + β +1,α+1;1 x) = n! n ρ nj x j, j=

7 where 2 F 1 is he Gauss hypergeomeric funcion and ρ nj = ( 1)n (β +1) n ( n) j (n + λ) j (β +1) j n!j!. he shifed Jacobi polynomials are orhogonal on [, 1], for he weigh funcion w (α,β) (x) = (1 x) α x β. Based on he properies of he Jacobi polynomials, i is possible o prove ha for a wide class of funcions φ( ) defined on (, 1) (including all coninuous and bounded funcions), φ(x) = h n = n= 1 c n R n (α,β) (x), c n = 1 1 h n (1 x) α x β [R (α,β) n (x)] 2 dx = φ(x)(1 x) α x β R n (α,β) (x) dx Γ(n + α + 1)Γ(n + β +1). (2n + λ)n!γ(n + λ) his can be applied o probabiliy disribuions on R + in he following way. Le and define, for some r>, F () = 1 F () = P( >) ( g(x) = F 1 ) r log(x), <x 1, g() =. (If represens he ime-unil-deah of a life currenly aged x, hen F () = p x.) his maps he inerval [, ) ono (, 1], =corresponding o x =1, and corresponding o x +; since F ( ) =,weseg() =. Ifα, β, p and {b k } are found such ha hen g(x) = x p k= F () = e pr b k R (α,β) k (x), <x 1, k= b k ρ kj e jr. If p>, a combinaion of exponenials is obained when his series is runcaed. he consans {b k } can be found by b k = 1 h k 1 = r h k x p g(x)r (α,β) k (x)(1 x) α x β dx j e (β p+1)r (1 e r ) α R (α,β) k (e r )F () d

8 his is a combinaion of erms of he form Sochasic life annuiies e (β p+j+1)r (1 e r ) α F () d, j =, 1,...,k. (3.2) If α =, 1, 2..., hen his inegral is a combinaion of values of he Laplace ransform of F ( ); he laer may be expressed in erms of he Laplace ransform of he disribuion of : e s F () d = 1 s F () de s = 1 s [1 Ee s ], s >. (3.3) he following resuls are consequences of he classical heory of orhogonal polynomials (see Dufresne (26) for proofs). heorem 3.2. Suppose α, β> 1, ha F ( ) is coninuous on [, ), and ha he funcion e pr F () has a finie limi as ends o infiniy for some p R (his is always rue when p ). hen F () = e pr k= b k R (α,β) k (e r ) for every in (, ), and he convergence is uniform over every inerval [a, b], for <a<b<. No all disribuions saisfy he condiion in heorem 3.2 for some p>. he nex resul does no need his assumpion. heorem 3.3. Suppose α, β> 1 and ha for some p R and r> (his is always rue if p<(β +1)/2). hen [ lim F () e pr N e (β+1 2p)r (1 e r ) α F () 2 d < N k= b k R (α,β) k (e r )] 2 e (β+1 2p)r (1 e r ) α d =. heorem 3.2 gives convergence a every poin in (, ), while heorem 3.3 gives mean square convergence. he runcaed series obained using his mehod are no rue disribuion funcions. he funcion may be smaller han or greaer han 1 in places, or i migh decrease over some inervals. his is

9 good enough for he purposes of his paper, bu he mehod can be modified in such a way ha a rue disribuion funcion resuls. See Secions 3 and 4 of Dufresne (26). 4. he disribuion of a sochasic life annuiy Given ha is independen of he Brownian moion W, he disribuion of A (µ) by condiioning on : P(A (µ) x) = can be found P(A (µ) x) df (). (4.1) However, in ligh of he known expressions for he densiy of A (µ) (see Secion 2), i appears ha formula (4.1) may no be easy o use in numerical applicaions. In fac, he compuaion of he densiy of A (µ) for fixed is iself a problem, see Dufresne (24), Ishiyama (25). he only excepions are when µ =, 1, 2,..., in which case he pdf of A (µ) may be expressed as a single inegral (see equaion (2.5)); his is illusraed in Secion 9 below. he approach suggesed in his paper is o approximae he disribuion of by a combinaion of exponenials, and hen apply heorem 2.1. he following resul says ha he error made in approximaing he disribuion of A (µ) 1 by A (µ) 2 is never larger han he error made in approximaing 1 by 2, which is a very good hing numerically. heorem 4.1. Le 1, 2 be random variables wih disribuion funcions F 1 ( ),F 2 ( ), respecively, and ha are independen of W. hen If µ<, hen P(A (µ) 1 x) P(A (µ) 2 x) sup F 1 () F 2 () x. (4.2) < P(A (µ) 1 x) P(A (µ) 2 x) C(x) sup F 1 () F 2 (), < where C(x) = 1 1 2x y µ 1 e y dy < 1. Γ( µ) Proof. Inegraion by pars in formula (4.1), we ge, for x>, P(A (µ) x) = = 1 P(A (µ) x) d[1 F ()] [1 F ()] d P(A (µ) >x)

10 he funcion P(A (µ) Sochasic life annuiies >x) is non-decreasing. Hence P(A (µ) 1 x) P(A (µ) 2 x) ( F 1 () F 2 () d P(A (µ) ) sup < F 1 () F 2 () >x) P( lim A (µ) >x). his proves inequaliy (4.2). If µ<, we may hen apply resul (2.6), which says ha, if Γ µ Gamma( µ, 1), P(A (µ) >x) = P(Γ µ < 1 2x ) = C(x). Remark. his heorem does no res on he paricular definiion of he process {A (µ) }, ashe same proof holds more generally: if 1, 2 are independen of a non-decreasing process {X } wih X =for <, hen, for all x, P(X 1 x) P(X 2 x) P( lim X >x) sup < F 1 () F 2 (). A quesion arises, however, in cases where he cdf of A (µ) is approximaed by a funcion which is no a rue disribuion funcion. Specifically, he approximaion for F ( ), call i G( ), may be smaller han or greaer han 1 in places, or may decrease in places. he approximaion for P(A (µ) 1 x) is hen H(x) = x) dg(). P(A (µ) Do he upper bounds in heorem 4.1 sill hold? A review of he proof of heorem 4.1 shows ha he resul becomes: heorem 4.2. Suppose G( ) has bounded variaion, wih G() =for <, and suppose also ha lim G() = G( ) exiss. hen P(A (µ) x) H(x) 1 G( ) P( lim A x) + sup F () G() P( lim A >x) < sup < F () G() x. An immediae consequence of heorems 3.1 and 4.1 is ha if,{ n } are independen of W, if F ( ) is coninuous, and if n converges in disribuion o, hen A (µ) n converges in disribuion o A (µ). A more general resul is easily obained. heorem 4.3. If,{ n } are independen of W, and if n converges in disribuion o, hen (e W (µ) n,a (µ) n ) converges in disribuion o (e W (µ),a (µ) )

11 Proof. Wrie he join characerisic funcion of he pair (e W (µ) n,a (µ) n ) as E e is 1e W (µ) +is 2 A (µ) df n (), s 1,s 2 R. he funcion f() =E exp(is 1 e W (µ) + is 2 A (µ) ) is coninuous and bounded, and hus he weak convergence of { n } o implies ha E f( n ) converges o E f( ) (his is a classical resul, see for insance Billingsley (1986), p.344, heorem 25.8). Corollary 4.4. Suppose { n } converges in disribuion o. hen Proof. Since he disribuion of A (µ) lim n P(A(µ) n x) = P(A (µ) x) x. is coninuous, his is a direc consequence of heorem 4.3. Le g(, x) be he he densiy funcion of A (µ).ifs λ Exponenial(λ) is independen of W, hen he densiy of A (µ) S λ is (see formula (2.4)) f λ (x) = λe λ g(, x) d = λ(2x) (µ γ)/2 1 Γ( ) µ+γ ( 2 Γ(γ +1) 1 F γ µ ,γ+1; 2x) 1 1{x>}. Recall ha γ = 2λ + µ 2 in his expression. Now, suppose ha has an arbirary disribuion on R +, ha i is independen of W, and ha F ( ) is approximaed by G( ), a combinaion of exponenials, wih dg() n = a j λ j e λj, >. (4.3) d he densiy of A (µ) j=1 is hen approximaed by n a j f λj (x). (4.4) j=1 I is possible o find he ccdf of he corresponding approximaion for A (µ) in closed form. heorem 2.1 implies ha, if x>, ( ) P(A (µ) B1,α S λ >x) = P > 2x G β 1 ( ) u = P > 2x d u (1 u) α G β = = 1 u/2x v β 1 1 (2x) β Γ(β) 1 Γ(β) e v dv d u (1 u) α u β 1 (1 u) α e u 2x du β Γ(α +1) ( = (2x) Γ(γ +1) 1 F 1 β,γ+1; 1 2x)

12 We have hus proved he following resul. Sochasic life annuiies heorem 4.5. If G is as in equaion (4.3), hen P(A (µ) >x) dg() = n j=1 a j (2x) µ γ j 2 Γ( µ+γ j 2 +1) ( γj µ 1F 1 Γ(γ j +1) 2,γ j +1; 2x) 1, (4.5) where γ j = 2λ j + µ Momens and Laplace ransform of sochasic life annuiies Using resuls (2.2)-(2.3) and he independence of and W, we obain formulas for he momens of sochasic life annuiies. Recall ha µ = 2m σ 2. heorem 5.1. If n N + and E exp[( km + k 2 σ 2 /2) ] < for k =1,...,n, hen E D n = ( ) n 4 n σ 2 b n,k E e ( km+k2 σ 2 /2). k= Noe ha E e δ is he acuarial value of an insurance of one uni payable a he momen of deah (usually wrien A x ), wih an insananeous rae of ineres equal o δ. In pracical applicaions i would be finie for any δ (observe ha km + k 2 σ 2 /2 may be posiive or negaive depending on k, m and σ). he Laplace ransform of A (µ) his is finie for all s>. may also be obained from ha of A(µ) E e sa(µ) = heorem 5.2. (a) If P( >) >, hen E e sa(µ) Ee sa(µ) = s>. df (); by condiioning on, (b) If he disribuion of has one or more aoms in (, ), hen he disribuion of A (µ) deermined by is momens. (c) If ɛ for some ɛ>, hen is no Ee s/(2a(µ) ) < s<1,

13 and he disribuion of 1/A (µ) Sochasic life annuiies is deermined by is momens. Proof. (a) Condiion on, and use he resul for fixed (Secion 2). (b) Hörfel (25) has shown ha he disribuion of A (µ) is no deermined by is momens, for any fixed >. Suppose is an aom of he disribuion of. hen here are wo disinc disribuions wih he same momens as A (µ) he same momens as A (µ). We can hen produce wo disinc disribuions wih, by changing he disribuion of A(µ) on he se { = }. (c) he resul is obvious if s, so le <s<1. I is known ha Ee s/(2a(µ) ) < for every > (Dufresne, 21). Since A (µ) is non-decreasing in, ɛ> implies Ee s/(2a(µ) ) Ee s/(2a(µ) ɛ ) <. I is well known ha a disribuion which has a finie Laplace ransform in a neighborhood of he origin is deermined by is momens. Par (b) above shows ha, a leas when he disribuion of has one or more aoms, here is no jusificaion in approximaing he disribuion of he sochasic life annuiy based on is momens only, as was suggesed by several auhors. I is plausible ha he same holds for arbirary disribuions for. 6. Model assumpions he nex secions will use he following moraliy assumpions. he lifeimes of he annuians obey Makeham s law, wih µ x = A + Bc x. he parameers used will be hose chosen by Bowers e al. (1997, p.78): A =.7, B =5 1 5, c =1.4. For he join and las survivor example boh annuians have he same fuure lifeime disribuion. On he financial side, one dollar invesed a ime grows o e m+σw a ime, where W is sandard Brownian moion. his is he same assumpion as for he risky asse in he usual Black-Scholes. he benchmark scenario will be: m =.6, σ = Jacobi approximaions of lifeime disribuions he echnique for finding an approximaion for he fuure lifeime disribuion is described in Secion 3. he parameers of he 2-erm Jacobi expansion of F () = P( 65 >)

14 [able 1abou here] are given in able 1. he oher parameers required (see Secion 3) are α =, β =, p =.2, r =.8. he same procedure was applied o he duraion of he las-o-die saus 65:65, and he parameers are also shown in able 1. he oher parameers are α =, β =, p =.1, r =.55. he precision of an approximaion ˆF of F was chosen as he sup norm of he difference: ˆF F = sup ˆF () F (). In he case of one life, he esimaed precisions are In he case of wo lives, F F 3 =.82, F F 5 =.43, F F 1 =.65 F F 2 =.24, F F 4 = F F 3 =.15, F F 5 =.82, F F 1 =.12 F F 2 =.3. [Figure 1abou here] Figure 1 shows some of he low-order approximaions of he ccdf of 65:65 ; he higher order approximaions canno be disinguished visually from he exac ccdf. able 2 compares he exac values of a 65 and a 65:65 wih he values obained using he 2-erm approximaions of he disribuions of 65 and 65:65. [able 2 abou here] Some pracical consideraions he numerical examples in his paper uses a Makeham survival funcion, bu in pracice he life able will mos likely be given by a able of l x or q x values a ineger ages x. If his is he case, hen he firs sep is o urn his able ino a survival funcion F () =P( >) for a coninuous parameer, in order o be able o compue he inegrals in expression (3.2). heorem 4.1 and Corollary 4.4 hold for any variable, bu, since a combinaion of exponenials is coninuous, he maximum error sup ˆF () F () < does no end o if has one or more aoms. his would make he error bound in heorem 4.2 useless for esimaing he error on he disribuion of he sochasic annuiy. We will hus consider coninuous funcions F ()

15 Any inerpolaion echnique is accepable here, provided he resuling funcion F () is a rue ccdf, bu linear inerpolaion is paricularly convenien, as will now be shown, a leas when he parameer α is an ineger. Given a reiremen age w (an ineger), le F () = (1 u) k p w + u k+1 p w, u = k (, 1), k =, 1... his is equivalen o he uniform disribuion of deahs assumpion. When α is an ineger, hen expression (3.2) may be expanded as a combinaion of inegrals of he form e γ F () d. Under he assumpion ha F () is linear beween inegers, his is k= 1 e γ(k+u) [(1 u)f (k)+uf (k + 1)] du = 1 e γk kp w [e γu + e γu ](1 u) du k= he second inegral in he las expression equals while he firs inegral is Hence, 1 γ 2 (eγ 1 γ), 1 2 cosh(γu)(1 u) du = 2 1 γ 1 sinh(γu) du = 2 [cosh(γ) 1]. γ2 e γ F () d = 2 γ 2 [cosh(γ) 1]ä w 1 γ 2 (eγ 1 γ), (1 u)e γu du. where he annuiy is compued a rae i = e γ 1. his agrees wih he limi as m ends o infiniy of formula (5.4.14) in Bowers e al. (1997, p.152). 8. Disribuions of sochasic life annuiies: numerical illusraions In his secion, we presen wo numerical examples which involve he disribuion of sochasic life annuiies. In all cases (excep when σ =) he disribuion of D is approximaed by formula (4.5), using he 2-erm approximaions for he disribuion of shown in Secion 7. Example 8.1. Disibuion funcion of D for σ =and σ =.2. Figure 2 compares he ccdf s of D (for a single-life annuiy) in he cases where σ =and σ =.2. A simple check on he accuracy of he approximaion is o calculae he firs and second momens of D. When σ =, he exac firs momen is 9.279, while wih he approximae disribuion funcion is also

16 9.279 o five decimal places; he exac and approximae sandard deviaions are boh o five decimal places. When σ =.2, he exac and approximae firs momens are boh 1.823; he exac and approximae sandard deviaions are and , respecively. he very small errors in momens are no surprising, since he momens of D are combinaions of annuiy values, and he Jacobi approximaion is based on he Laplace ransform of F, which is iself an annuiy value when α =(see equaions (3.2)-(3.3)). he sandard deviaion is significanly higher when σ =.2, which could have been guessed from he hicker ail of he disribuion of D. For insance, he probabiliy ha $ 15 is sufficien o fund a $ 1 per annum annuiy is.9993 when σ =, while i is only.7981 when σ =.2; he same compuaion for an inial amoun of $ 12 yields probabiliies of.7339 and.6739, respecively; recall ha he expeced value of D here is when σ =, and when σ =.2. [Figures 2 and 3 abou here] Example 8.2. In his example we look a he shorfall probabiliy P(D > (1 + q)k) for fixed q and K bu when σ varies. Figure 3 shows he shorfall probabiliy in he case of single life, if K = ā.6 65 = and q =,.5. I can be seen ha he probabiliy of no having enough funds is more or less consan when q =, bu ha i increases significanly as a funcion of σ when q =.5; his means ha a larger loading is required for a given shorfall probabiliy when reurns have greaer volailiy. Figure 4 applies he same reasoning o he join-and-las-survivor annuiy. Here K = ā.6 65:65 = 1.823, and he same conclusions are reached as in he case of a single-life annuiy. [Figure 4 abou here] 9. Check agains exac disribuion when µ = Expression (2.5) for he densiy of 1/(2A () ) simplifies o a single inegral if µ =, 1, 2,... (his is because he Hermie funcion H µ reduces o he usual Hermie polynomial in hose cases). he resuling pdf is no really simple, bu i can be used more easily in numerical calculaions han when he pdf is a double inegral. We will use he case µ =o perform a check on he resuls of he preceding secion. he pdf of A () is g(, u) = e π 2 8 πu 3 2 e 1 2u cosh2 y y2 2 (πy) cosh(y) cos dy. (9.1) 2 o obain he pdf of A (), we will now inegrae he pdf of A() imes he pdf of : f () A (x) = g(, x)f () d. (9.2)

17 A numerical problem arises when is small, however, because of he oscillaory naure of he funcion inside he inegral (9.1). In order o circumven his difficuly, he lognormal approximaion will be used for small. I is known (Dufresne, 24) ha he disribuion of he appropriaely scaled variable A (µ) ends o a lognormal as +. Figure 5 compares he pdf s of A ().5, as compued from equaion (9.1), wih is lognormal approximaion (he laer is obained by maching he firs and second momens). he difference is no very grea, and becomes smaller when <.5. I was decided ha, in compuing inegral (9.2), he inegral densiy (9.1) would be replaced by is lognormal approximaion for <.5; his should no cause a lo of difference in he numerical values of inegral (9.2). In order o find ou wheher he las claim is correc, he disribuion of A () was compued using his approximaion when is exponenial, and hen compared wih he known exac disribuion (heorem 2.1). Figure 6 shows boh curves (hey canno be disinguished visually). he larges difference beween he wo pdfs is approximaely.46, and is aained around x =.4. he absolue difference beween he wo pdf s quickly decreases as x increases, and is less han 1 9 for.2 x<25. he error inroduced in replacing he exac pdf wih he lognormal approximaion for x<.5 hus appears very small. Finally, a check was performed assuming has he Makeham disribuion of Secion 7 (for a single life). he pdf of A () was calculaed using formula (9.2), and hen inegraed numerically (using he rapeze rule) o give he ccdf of A (). his was compared wih he Jacobi approximaion described in Secion 8. he maximum absolue difference was esimaed (see hird column below). he second column gives he precision of he Jacobi approximaion of he disribuion of, from Secion 7. heorem 4.2 says ha, in heory, he maximum error in approximaing he disribuion funcion of A () is never bigger han he error made in approximaing he disribuion of. he able below is in agreemen wih he heorem, and appears o confirm ha he inegral formula (9.2) gave more precise resuls han he 2-erm Jacobi approximaion (of course a he cos of a greaer programming and compuing effor). No. of erms Precision Makeham ccdf Max. Difference (Inegral-Jacobi) Deferred annuiies Suppose he annuiy is due o sar being paid in n years, when he annuian, if alive, is age w.if is he fuure lifeime a he curren age, w n, hen he amoun of money required o fund his deferred annuiy is D = 1 {>n} U n max( n,) U n U n+s ds, U = e m+σw

18 Le be he random variable represening he fuure lifeime of he annuian a age w, given survival up o ha age, and le Y = e mn σw n, D = e ms σw s ds, where W is anoher Brownian moion, independen of W. he independen incremens propery of Brownian moion means ha, for x, P(D >x) = P( >n)p(y D >x), Hence, condiionally on survival o age w, he disribuion of D is as he produc convoluion of a lognormal and he disribuion of a sochasic life annuiy, he laer calculaed as in previous secions. 11. Oher echniques for approximaing he densiy of A (µ) ( fixed) he problem of approximaing he disribuion of A (µ) for fixed may also be solved by finding a combinaion of exponenials which approximaes he degenerae disribuion a. In paricular, i is known (Dufresne, 26) ha if X a LogBea(a, b, c) (see Appendix for definiion) wih c> fixed, b = κa for some consan κ>, hen d X a x = 1 ( c log 1+ c ) as a. κ Raher han o use a combinaion of exponenials, i is possible o use a gamma approximaion for a degenerae disribuion. Anoher way o look a his is o go back o he derivaion of he disribuion of A (µ) S λ in Secion 2. he proof of he following heorem is in he Appendix. heorem he densiy of A (µ) g(, x) ( 1) n 1 = lim n (n 1)! [ ( ( n ) n n 1 λ n 1 is (2u) (µ γ)/2 1 Γ( µ+γ 2 ) ( Γ(γ +1) 1 F γ µ ) )] ,γ+1; 1 2x λ= n. Conclusion his paper has proposed an analyic echnique for compuing he disribuion of sochasic life annuiies, when he reurns are lognormal. he numerical examples presened indicae ha he mehod can be very accurae. Some possible applicaions of he resuls are:

19 (1) Given paricular values for m and σ (which represen he ype of invesmen sraegy chosen), find he probabiliy ha a pension will be paid in full by he iniial amoun invesed. (2) Find he effec on he disribuion of he discouned annuiy of inroducing an n-year or a join-life guaranee. (3) Among he se of pairs (m, σ) available in he marke, find he one which maximise he probabiliy of a cerain amoun being sufficien o fund a pension. APPENDIX Proofs heorems 2.1 and 11.1 he idea of he proof of heorem 2.1 is o find a funcion h λ (, ) such ha for any non-negaive funcions f( ) and g( ), E[f(e W (µ) S λ )g(a (µ) S λ )] = f(r)g(u)h λ (r, u) dr du. (Here W (µ) = µ + W.) hen h λ (, ) is he join densiy funcion of (e W (µ) S λ,a (µ) S λ ). By he Cameron-Marin heorem, (A.1) E[f(e W (µ) )g(a (µ) )] = e µ2/2 E[e µw f(e W )g(a )], and so E[f(e W (µ) S λ )g(a (µ) S λ )] = E λe λ µ2/2 e µw f(e W )g(a ) d. Yor (1992) used a sochasic ime change and he properies of Bessel processes o show ha, if γ = 2λ + µ 2, h λ (r, u) = λ u rµ 1 e (1+r2 )/2u I γ ( r u )1 {r,u>}, where I ( ) is he modified Bessel funcion of he firs kind: (z/2) p+2n I p (z) = n!γ(n + p +1). n= (A.2) he densiy of A (µ) S λ may be found by inegraing ou r in formula (A.2); his yields formula (2.4). Now, urn o heorem Suppose S n,λ Gamma(n, λ); hen S n,λ converges in disribuion o he degenerae disribuion a if n and n/λ. Replacing S λ wih S n,λ in equaion (A.1), we ge E[f(e W (µ) S n,λ )g(a (µ) S n,λ )] = E = ( 1)n 1 λ n (n 1)! λ n n 1 2 Γ(n) e λ µ /2 e µw f(e W )g(a ) d n 1 λ n 1 E e λ µ2 /2 e µw f(e W )g(a ) d

20 hen he join densiy funcion of (e W (µ) S n,λ,a (µ) S n,λ ) is and he densiy of A (µ) S n,λ is h n,λ (r, u) = 1 u rµ 1 e (1+r2 )/2u ( 1)n 1 λ n f n,λ (u) = ( 1)n 1 λ n (n 1)! [ n 1 λ n 1 (n 1)! (2u) (µ γ)/2 1 Γ( µ+γ 2 n 1 λ n 1 I γ( r u )1 {r,u>} ) ( 2u)] Γ(γ +1) 1 F γ µ ,γ+1; 1. Leing n we ge heorem his resul may be seen as an applicaion of a classical inversion heorem for Laplace ransforms (see Feller, 1971, p.233). Proof of heorem 3.1 In he proof we use he logbea disribuion, which we now define. Definiion. For parameers a, b, c >, he logbea disribuion has densiy f a,b,c (x) = c Γ( b c + a) Γ( b c )Γ(a) e bx (1 e cx ) a 1 1 {x>}. his law is denoed LogBea(a, b, c). he name chosen comes from he fac ha X LogBea(a, b, c) e cx Bea(b/c, a). I can be seen ha when a =1, 2,..., he pdf of he logbea disribuion is a combinaion of exponenials. (a) A disribuion funcion F ( ) can be approximaed o any degree of precision (a all poins ) by some discree disribuion. For insance, one may define a new disribuion by firs giving i he same aoms as F ( ), and hen, beween hose aoms, making i a sep funcion close o F ( ). he problem hen reduces o approximaing discree disribuions ha have a finie number of possible values; he laer are convex combinaions of degenerae disribuions (ha is, ha ake only one value). Par (a) will hen be proved if we find how o approximae a degenerae disribuion a a poin by combinaions of exponenials. I is known (Dufresne, 26) ha if X LogBea(a, b, c), c> is fixed, b = κa for a fixed consan κ>, hen X d x = 1 ( c log 1+ c ) κ as a. Rescaling as necessary, and leing a N end o infiniy, we see ha here is indeed a sequence of combinaions of exponenials which converges in disribuion o any consan in R

21 (b) Suppose F ( ) is coninuous, and le { n } be a sequence of random variables converging o in disribuion. Le ɛ>, and find poins 1 < < m such ha F ( 1 ) < ɛ, F ( j+1 ) F ( j ) < ɛ j =1,...,m 1, F ( m ) > 1 ɛ. hen here exiss n such ha F n ( j ) F ( j ) < ɛ, j =1,...,m, n n. his implies F n () F () 2ɛ R. References Bachelier, Louis. 19. héorie de la Spéculaion. Annales scienifiques de l École Normale Supérieure III-17: Docoral hesis, Sorbonne. Reprined by Édiions Jacques Gabay, Paris (1995). English ranslaion in Cooner, P. (ed.) (1964). he Random Characer of Sockmarke Prices. MI Press, Cambridge, pp Beekman, John A. and Clinon P. Fuelling Ineres and Moraliy Randomness in Some Annuiies. Insurance: Mahemaics and Economics 9: Bellhouse, David R., and Harry H. Panjer Sochasic Modelling of Ineres Raes wih Applicaions o Life Coningencies Par II. Journal of Risk and Insurance 48: Billingsley, Parick Probabiliy and Measure (Second Ediion.) Wiley, New York. Black, Fisher, and Myron Scholes he Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81: Bowers, Newon L., Jr., Hans U. Gerber, James C. Hickman, Donald A. Jones, and Cecil J. Nesbi Acuarial Mahemaics. 2nd ed. Schaumburg, IL: Sociey of Acuaries. Boyle, Phelim Raes of Reurn as Random Variables. Journal of Risk and Insurance. 43: De Schepper, Ann, F. Éienne De Vylder, Marc Goovaers, and Rob Kaas. 1992a. Ineres Randomness in Annuiies Cerain. Insurance: Mahemaics and Economics 11: De Schepper, Ann, and Marc Goovaers. 1992b. Some Furher Resuls on Annuiies Cerain wih Random Ineres. Insurance: Mahemaics and Economics 11:

22 De Schepper, Ann, Marc Goovaers, and Freddy Delbaen. 1992c. he Laplace ransform of Annuiies Cerain wih Exponenial ime Disribuion. Insurance: Mahemaics and Economics 11: De Schepper, Ann, M. eunen, and Marc Goovaers An Analyical Inversion of a Laplace ransform Relaed o Annuiies Cerain. Insurance: Mahemaics and Economics 14: Deelsra, Griselda, and Freddy Delbaen Remarks on he Mehodology Inroduced by Goovaers e al. Insurance: Mahemaics and Economics 11: Dufresne, Daniel Weak Convergence of Random Growh Processes wih Applicaions o Insurance. Insurance: Mahemaics and Economics 8: Dufresne, Daniel he Disribuion of a Perpeuiy, wih Applicaions o Risk heory and Pension Funding. Scandinavian Acuarial Journal: Dufresne, Daniel On Discouning When Raes of Reurn Are Random. ransacions of he weny-fourh Inernaional Acuarial Congress, Monreal,1992, 1: Dufresne, Daniel. 2. Laguerre Series for Asian and Oher Opions. Mahemaical Finance 1: Dufresne, Daniel. 21. he Inegral of Geomeric Brownian Moion. Advances in Applied Probabiliy 33: Dufresne, Daniel. 24. he Log-Normal Approximaion in Financial and Oher Compuaions. Advances in Applied Probabiliy 36: Dufresne, Daniel. 25. Bessel Processes and Asian Opions. In: Ben Ameur, Haem, and Breon, Michèle (Eds.), Numerical Mehods in Finance. Dordrech: Kluwer Academic Publishers. Dufresne, Daniel. 26. Fiing Combinaions of Exponenials o Probabiliy Disribuions. o appear in Applied Sochasic Models in Business and Indusry. Feller, William An Inroducion o Probabiliy heory and is Applicaions II (Second Ediion). New York: Wiley. Frees, Edward W Sochasic Life Coningencies wih Solvency Consideraions. ransacions of he Sociey of Acuaries 42: Geman, Hélyee, and Marc Yor Bessel Processes, Asian Opions and Perpeuiies. Mahemaical Finance 3: Harrison, J. Michael, and Sanley R. Pliska Maringales and Sochasic Inegrals in he heory of Coninuous rading. Sochasic Processes and heir Applicaions 11: Hoedemakers, om, Grzegorz Darkiewicz, and Marc Goovaers. 25. Approximaions for Life Annuiy Conracs in a Sochasic Financial Environmen. Insurance: Mahemaics and Economics 37: Hörfel, Per. 25. he Momen Problem for Some Wiener Funcionals: Correcions o Previous Proofs (wih an Appendix by H.L. Pedersen). Journal of Applied Probabiliy 42:

23 Ishiyama, Kazuyuki. 25. Mehods for Evaluaing Densiy Funcions of Exponenial Funcionals Represened as Inegrals of Geomeric Brownian Moion. Mehodology and Compuing in Applied Probabiliy 7: Lebedev, Nikolai Nikolaevich Special Funcions and heir Applicaions. New York: Dover. Osborne, M.F.M Brownian Moion in he Sock Marke. Operaions Research 9: Panjer, Harry H., and David R. Bellhouse Sochasic Modelling of Ineres Raes wih Applicaions o Life Coningencies. Journal of Risk and Insurance 47: Parker, Garry. 1994a. wo Sochasic Approaches for Discouning Acuarial Funcions. ASIN Bullein 24: Parker, Garry. 1994b. Momens of he Presen Value of a Porfolio of Policies. Scandinavian Acuarial Journal: Pollard, John On Flucuaing Ineres Raes. Bullein de l Associaion royale des acuaires belges 66: Ramakrishnan, Alladi A Sochasic Model of a Flucuaing Densiy field. Asrophysical Journal 119: Vannese, M., Marc Goovaers, Ann De Schepper, and Jan Dhaene A Sraighforward Analyical Calculaion of he Disribuion of an Annuiy Cerain wih Sochasic Ineres Rae. Insurance: Mahemaics and Economics 2: Vannese, M., Goovaers, Marc, and E. Labie he Disribuion of Annuiies. Insurance: Mahemaics and Economics 15: Waers, Howard H he Momens and Disribuions of Acuarial Funcions. Journal of he Insiue of Acuaries 15: Wesco, David A Momens of Compound Ineres Funcions Under Flucuaing Ineres Raes. Scandinavian Acuarial Journal: Wilkie, A. David he Rae of Ineres as a Sochasic Process. Proceedings of he 2h Acuarial Congress of Acuaries 1: Yor, Marc On Some Exponenial Funcionals of Brownian Moion. Advances in Applied Probabiliy 24: (Reproduced in Yor (21).) Yor, Marc. 21. Exponenial Funcionals of Brownian Moion and Relaed Processes. Springer-Verlag, New York. (Conains en papers by Marc Yor.)

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26 Figure 5. Lognormal vs exac PDFs PDF 3 25 Lognormal approx. 2 Exac formula