INTERACTION BETWEEN ASSET LIABILITY MANAGEMENT AND RISK THEORY.

Transcription

1 INTERACTION BETWEEN ASSET LIABILITY MANAGEMENT AND RISK THEORY. GRISELDA DEELSTRA ENSAE and CREST, 3 av. Pierre-Larousse, Malakoff-CEDEX, France JACQUES JANSSEN CADEPS and SOLVAY, Universié Libre de Bruxelles, 5 av. Roosevel CP 194/7, 15 Brussels, Belgium SUMMARY We sar from he model of Janssen 1 (1992) and he papers of Ars & Janssen 2,3 (1994, 1995), in which hey developed some applicaions of he Janssen model of Asse Liabiliy Managemen (ALM) o real life siuaions. We sudy an exension of he Janssen model in which he asse fund A akes ino accoun fixed-income securiies. Therefore, we ake ino accoun he raes of reurn of he asse porfolio, which we model by a Vasicek 4 process. The liabiliy process B is defined by a geomeric Brownian moion wih drif which may be correlaed wih he asse process. In his generalized Janssen model, we sudy he relaions beween he asse process A and he liabiliy process B in order o poin ou some managemen principles. More exacly, we sudy he probabiliy ha he asses and liabiliies of a company have no good maching and we propose a degree of he mismaching. Therefore, we look a he process a = ( a, ) defined by a = ln A and a he firs mismaching ime B τ = inf{ : T, a() }. The deerminaion of he probabiliy of mismaching leads o he calculaion of crossing probabiliies P[ τ < T]. Only in special cases, explici resuls are obained and we urn o he approximaions proposed by Durbin5,6 (1985, 1992) and Sacerdoe & Tomasei7 (1992). The degree of mismaching follows from opion heory. These resuls are imporan as hey are useful o deermine ALMsraegies for insurance companies. Keywords: sochasic differenial equaion, Ornsein-Uhlenbeck process, probabiliy of mismaching, ALM, opion heory.

2 1. INTRODUCTION The las few years, he increasing imporance of various risks associaed wih heir financial aciviies has led many insurance companies o pay more and more imporance o modern echniques of asse liabiliy managemen largely inroduced in banks as i is well known ha he bad managemen of ineres raes risks can lead o heavy financial losses and possibly requires a significan increase in he free reserve of he enerprise. Our goal is o measure he riskiness of he insurance company by using a sochasic model of boh he asse and he liabiliy side of he balance and we consider he possibiliies of perfec maching and parially maching. We propose an indicaor of riskiness which we call he mismaching probabiliy or mismaching degree. This informaion is ineresing for he managemen of he company who can check wheher hey say wihin he risk limis and can approve heir sraegies wih respec o invesmen, reinsurance, pricing and accepaion of policies. This kind of informaion would also be useful for he deerminaion of a coningency reserve or he solvency of a porfolio of insurance policies. We do no propose hese measures as an alernaive of oher ALM approaches bu raher as a complemen. Saring from a good daabase, we advise o use differen ALM-ools like duraion analysis, gap managemen, simulaion and our mismaching probabiliy and/or mismaching degree in order o obain more useful informaion and a more complee idea of he siuaion of he company. The proposed measures of risk are also useful from he poin of view of regulaing auhoriies. In fac he goals of an insurance company and regulaory bodies are he same o a cerain degree. We sar from he model of Janssen1 (1992) which is symmeric in A and B and assumes geomeric Brownian moions boh for A( ) and B(). We sudy an exension of he Janssen model in which he asse fund A akes ino accoun fixed-income securiies and his inroduces asymmery for A and B. This is paricularly useful for insurance companies whose invesmens are more in bonds han in shares. We suppose ha he asse porfolio can be modeled by a fund conaining only purediscoun bonds which reflec he raes of reurn of he asse porfolio in he pas and wih mauriy he ime horizon of he period ha we are ineresed in. In his paper, we assume ha he raes of reurn follow an Ornsein-Uhlenbeck process. Then he sochasic differenial equaion of he asses follow from he paper of Vasicek4 (1979). We furher assume ha he liabiliy process B is defined by a geomeric Brownian moion wih drif, which is correlaed wih he asse process in a consan way. 2

3 In his generalized Janssen's model, we sudy he perfec maching and final maching of asses and liabiliies by deermining he probabiliy of mismaching and he degree of mismaching. To begin wih, we presen he generalized Janssen model in secion 2. Secion 3 is devoed o he sudy of probabiliies of mismaching. In secion 4, we concenrae on a degree of mismaching beween he asses and he liabiliies. Secion 5 concludes he paper. 2. THE GENERALIZED JANSSEN MODEL The mos realisic model is o look a a porfolio of asse pools A 1, A 2,..., A n wih some segmens conaining only ineres rae sensiive securiies and some only shares. This model will be called he mulidimensional model and will be reaed in anoher paper8. Firs, we concenrae on a less realisic bu more reaable model in order o obain an increased undersanding of differen influences. Insead of dividing he asses up in differen classes, we suppose ha we can model he asses as one group of ineres rae sensiive securiies, reflecing he raes of reurns of he asse porfolio in he pas. Since insurance companies inves paricularly in bonds, we model he asse porfolio by assuming ha i conains N zero-coupon bonds which are modeled by he raes of reurns which have been obained by he porfolio over he las years. The mauriy T of he bonds represening he asse porfolio cerainly should be larger han or equal o he ime horizon T if [,T] is he period ha we are ineresed in. In order o simplify he noaions, we choose T =T. The resuls abou he mismaching probabiliies can easily be generalized o longer mauriies. In case of he proposed risk measure of final mismaching for sochasic raes of reurn, however, i makes a difference wheher T = T or wheher T > T. The raes of reurn are assumed o follow an Ornsein-Uhlenbeck process of he form dr = (θ r )d + ηdz, where ( Z ) 1 is a Brownian moion and where,θ,η R +. This model has he realisic propery of being mean revering owards he long erm value θ where he speed of adjusmen is deermined by he parameer. The raes in he Vasicek model can be negaive bu in our opinion, negaive raes of reurn are possible since asses can be invesed in many differen financial insrumens. We assume ha financial markes are complee and fricionless and ha rading akes place coninuously. In his seing, Harrison and Kreps9 (1979) have shown ha here exiss a unique risk-neural probabiliy. 3

4 Under hese assumprions, he asses A, modeled by he invesmen in N purediscoun bonds wih mauriy T, are modeled by (see e.g. Vasicek's paper): da = A r + ηλ 1 ( e T ) ( ) d A η 1 e ( T ) ( )dz wih λ he parameer of marke risk and wih A T =N. We model he liabiliy process B by a lognormal process wih posiive consans µ B,σ B which is correlaed wih ( Z ) 1. Cummins and Ney1 (198) argue ha he lognormal disribuion is a reasonable model for insurer liabiliies if here is a good reinsurance program o hedge caasrophic jumps in he liabiliies. We now consider he process a = ( a, ), which has been defined in Janssen1 (1992), namely a = ln A and a B = ln A. This process has he same meaning as B he surplus process in risk heory. The sochasic differenial equaion of a = a, follows from Io's lemma: Theorem 1 The sochasic process a = a, da = µ ( r,,t)d + σ (,T)dW where µ ( r,,t)= r + ηλ 1 e ( T ) ( ) µ B η2 1 e ( T ) 2 2 σ 2 (,T)= η2 1 e ( T ) T ( 1 e ( ) )ϕσ B 2 and where W = W, ( ) ( ) is a soluion of he sochasic differenial equaion ( ) 2 + σ B 2 + 2η ( ) 2 + σ 2 B ( ) denoes a sandard Brownian moion. In he nex secion, we use his heorem o derive he probabiliies of mismaching PROBABILITIES OF MISMATCHING 3.1 Perfec maching Using he generalized Janssen model presened in he previous secion, we sudy he relaions beween he asses process A and he liabiliies process B in order o poin ou some managemen principles. We say ha he asses and liabiliies have no perfec mach if for some he asse value A( ) becomes lower han he liabiliy value B() or equivalenly if a() becomes negaive (see Janssen1 (1992) and Ars & Janssen2,3 (1994,1995)). Therefore, we define he firs mismaching ime in he period [,T] as τ = inf : T, a() { } 4

5 or in case of he Ornsein-Uhlenbeck process : T, a + r s ds + σ s dw s + ηλ η2 2 µ 2 B + σ 2 B 2 τ = inf ηλ +e T η η2 4 3 e 2T T + e ( ) η 2 ηλ 3 2 η2 T e 2 ( ) 3 4 where we resric ourselves o imes smaller han T. We now concenrae on he crossing probabiliies P[ τ < T], which canno be obained expliciely in he general model. To obain more insigh, we firs rea deerminisic raes of reurn Special case: Non-sochasic raes of reurn Firs, le us assume ha he volailiy coefficien η equals zero so ha he raes of reurn are deerminisic r = e r θ ( ) + θ. In his case, we can rewrie he firs mismaching ime τ as τ = inf : T, W 1 a + r θ µ B θ σ 2 B σ B 2 + e ( θ r ). a/ Consan raes of reurn In order o be able o use he nice and well-known resuls in case of a Brownian moion, we concenrae firs on he special case of consan raes of reurn r. Clearly, if r = θ, hen a = a, ( ) is a Brownian moion wih drif and denoing µ = θ µ B + σ 2 B 2 and σ = σ B, he resuls of Ars-Janssen2,3 (1994, 1995) hold. Indeed, da = µd + σdz and he firs mismaching ime equals τ = inf{ : T, a() } = inf : T, a σ W µ σ. The probabiliy of no perfec mach in he period [, T] urns ou o be (see for an overview e.g. Deelsra11 (1994)): P[ τ < T]= P sup W µ T σ a σ. =1 if a σ = 1 a σ T + µ T σ e u 2 /2 du + e 2 a µ / σ e u 2 2π du if a 2π σ >. a σ T + µ T σ 2 /2 5

6 Noice ha he formula in case of a > can be expressed in erms of he cumulaive σ Normal disribuion funcion Φ( ): P[ τ < T]= 1 Φ a σ T + µ T + e 2a µ / σ 2 Φ a σ σ T + µ T. σ An ineresing measure of he mismaching risk is o calculae he probabiliy of mismaching in he period [, [. For T ending o infiniy, we consequenly find ha: P[ τ < ] = e 2µa σ 2 µ, a > = 1 µ or a. Therefore, if µ is negaive or a is negaive, hen here will be no perfec mach wih probabiliy 1. Oherwise, he probabiliy of having a leas once a mismach equals e 2µa σ 2. This probabiliy decreases if a = ln A or µ = r µ B B + σ 2 B increases 2 and/or σ = σ B decreases. So, he iniial asses should be as large as possible in comparison wih he liabiliies. The insananeous inflaion and he volailiy of he liabiliies should be as low as possible. Indeed, if one sars wih very low asses and high liabiliies or wih liabiliies which are increasing very quickly, one can expec a mismach. As moivaed before, his informaion number, his indicaor of mismaching can be ineresing for he managers of he company, he regulaors as well as he cliens and everyone who has o deal wih he insurance company because i is a measure of he risk posiion of he company. Noice ha even if µ is negaive, hen he probabiliy of mismaching over he period [,T] does no equal 1. Bu of course, in order o lower he probabiliy of mismaching, he company should increase µ and ry o keep µ posiive. b/ Time-dependen raes of reurn If r θ, he deerminaion of he crossing probabiliy P[ τ < T] is no so easy since he drif erm of a = a, ( ) is ime-dependen and herefore, we canno rely on resuls abou Brownian moions crossing (piecewise) linear boundaries. The ime of firs mismaching is he crossing ime of a sandard Brownian moion o a boundary l() which is wholly convex for r < θ, and wholly concave for r > θ. Therefore, we can apply he resuls of Durbin5 (wih an appendix by Williams) (1992). I was shown in Durbin6 (1985) ha he firs-passage densiy p() of W(u) o a boundary l(u) a ime u = is p() = b(). f () < < T where f() is he densiy of W() on he boundary, i.e. 6

7 1 l()2 f () = exp 2π 2 and where 1 b() = lim s s EI [ ( s,w )( l(s) W(s) )W() = l() ] wih I ( s,w ) an indicaor funcion which is equal o 1 if he sample pah does no cross he boundary prior o s and equal o oherwise. As b() usually is no compuable in a direc way, Durbin (1992) expands he firs-passage densiy p() of W(u) o l(u) a u = as a series of muliple inegrals, namely p() = k j=1 ( 1) j 1 q j () +( 1) k r k () k =1, 2,... where q 1 () = l() l'() f (), l() q 2 () = l'() l() l( 1 ) l'() 1 f ( 1,)d 1, 1 j 2 l( j 1 ) q j () =... l'( j 1 ) j 1 j 1 l( i 1 ) l( i ) l'( i 1 i 1 ) i f ( j 1,..., 1, )d j 1...d 1 j>2 i =1 wih f ( j 1,..., 1,) is he join densiy of W( j 1 ),...,W( 1 ),W() on he boundary, i.e. a values l( j 1 ),...,l( 1 ), l() and where 1 k 1 r k () =... b( k ) k l( i 1 ) l( i ) l'( i 1 i 1 ) i f ( k,..., 1,)d k...d 1. i =1 By runcaing he series, one obains he successive approximaions: k p k () = ( 1) j 1 q j () k =1,2,... j =1 If l() is concave everywhere, hus r > θ, he error r k () in he k-he approximaion p k () is less han he las compued erm q k () and less han he nex erm q k+1 (). If he boundary is wholly convex, hen he error is bounded from above: r k ( ) u k () k =1, 2,... where 1 k 1 k l() l( u k () =... i 1 ) l( i ) l'( i 1 i 1 ) i f ( k,..., 1,)d k...d 1. i =1 k 7

8 The probabiliy ha a sample pah of W() crosses he boundary a leas once in he inerval [,T], namely P = T p()d, can be approximaed by T P k = p k ()d k =1, 2,.... As for he firs-passage densiy, in he wholly concave case he error R k is bounded by Q j = T q j ()d for boh j=k and j=k+1, while for he wholly convex case R k is bounded by T u k ( )d. These formulae easily can be programmed. Sacerdoe & Tomassei (1996) propose also approximaions for he firs passage probabiliies and indicae error bounds by using a series expansion for he soluion of he inegral equaion for he firs-passage ime probabiliy densiy funcion. However, in order o apply hese resuls, a hypohesis has o be fulfilled which clearly depends on he parameers General case: Ornsein-Uhlenbeck process Le us now concenrae on he firs mismaching ime τ in he case of sochasic raes of reurn, modeled by an Ornsein-Uhlenbeck process wih η. Some simple calculaions show ha τ can be rewrien as : T, y() ( Er [] s r s )ds σ s dw s a + r ϑ τ = inf ηλ + e T η η2 4 3 e 2T 2 µ + ϑ + σ 2 B 2 B 2 + ϑ r e T e 2 ( ) l() 3 + ηλ η2 +e ( T ) η 2 ηλ 3 2 η2 4. wih σ 2 (,T)= η2 1 2 e ( T ) ( ) 2 + σ 2 B + 2η 1 e ( T ) ( )ϕσ B. Expressed his way, we see ha we have o compue he firs-passage densiy from below of he coninuous Gaussian process y = y(); T s ( ) wih y( ) = e s ηe l dz l ds σ s dw s o he boundary l(). Neiher he process ( ) nor l() saisfies he assumpions of Durbin6 (1992) or Sacerdoe & y = y(); T 8

9 Tomassei7 (1996). Therefore, we urn o he approximaions of Durbin5 (1985), alhough in his paper no error-bounds are given. Under mild resricions on l() and on he covariance funcion cov(y(u),y(v)), Durbin (1985) derives approximaions for he crossing probabiliies and he firspassage densiy p() of a coninuous Gaussian process y() a a boundary l() a u =. Long calculaions show ha he covariance funcion cov(y(u),y(v)), which we denoe by ρ(u,v), equals: u s u v v ρ(u,v) = E e s ηe l dz l ds σ s dw s. e ηe j dz j d σ dw = η2 + 2ησ Bϕ e T η e 2T 2 + min(u,v)σ B + e min(u,v ) η e2 min(u,v ) η e 2T η2 e (u+ T) + e (v+t ) ( ) 2 3 η2 e max(u,v ) [ e min(u,v) + e min(u,v) 2] 2ησ Bϕ e ( min (u,v) T ) (e v + e u ) η2 e min(u,v ) 1 + e T σ Bηϕ e min ( (u,v) 1) 2. I is easy o verify ha he assumpions of Durbin's paper (1985) are fulfilled and herefore, we may apply he approximaions proposed in his paper. A firs approximaion Pg for he crossing probabiliy is: P g = ρ(u,) u u= ρ(,) l'() l(). 2ρ(, )/l2 () ρ(u,u) l 2 (u) d 2 du 2 u= 1/2.exp l2 (,) 2ρ(,) where is such ha ρ(u,u) is maximized. l 2 (u) Anoher approximaion P1 of he no-perfec-mach probabiliy P[τ < T] proposed by Durbin (1985) is: P 1 (T) = T p 1 ( )d where p 1 () = l() ρ(u,) l'() u. ρ(,) u u= A las approximaion uses his expression, namely where P 2 (T) = T p 2 ()d 9

10 wih p 2 () = p 1 () + [ l'() β 1 (r,)l(r) β 2 (r,)l(r)] f ( r)p 1 (r)dr β 1 (r,) β 2 (r,) = ρ(r,r) ρ(r, ) ρ(r,) ρ(,) ρ(r,u) u u = ρ(s,) s and where f( r) is he condiional densiy of y() a l() given ha y(r) = l(r). The firs approximaion Pg is he leas accurae and here may arise some problems wih finding he maximizing value. The las approximaion P2 appears o be he mos accurae bu involves more calculaions. Therefore, we sugges o use he approximaion P1. 1 s= 3.2 Final mismaching In pracice, perfec maching of insurance liabiliies migh be oo demanding since lowrisk invesmen sraegies associaed wih he highes degree of maching possible usually produce lower expeced reurns. Therefore, we also observe final maching which means ha we only check wheher he asses cover he liabiliies a he end of he period [,T]: A(T)>B(T). Therefore, he probabiliy of no final maching is he probabiliy PA [ T < B T ] = Pa [ T < ] where ( a ) is he process of mismaching defined above and his probabiliy follows from he disribuion of a T. From heorem 1, i is easy o see ha ( a ) is a Gaussian process since a = a + r s ds + ηλ 1 e ( T s) ( )ds µ B η e ( T s) ( ) 2 ds + σ 2 B 2 + σ ( s,)dw s wih he raes of reurn following an Ornsein-Uhlenbeck process, hus r s ds ~ N θ + r θ ( 1 e ), η2 + 2η2 e η2 e 2 3η Therefore, a T has a Normal disribuion wih mean 1

11 and wih variance m(t) = a + r ϑ ηλ + e T 2 η2 3 + η2 4 3 e 2T + ηλ η2 2 µ 2 B + ϑ + σ 2 B 2 T + ϑ r e T + η2 ηλ 3 2 η2 4 3 σ 2 (T) = η2 + 2ησ Bϕ e T η e 2T + σ 2 B T + e T η 2 η η e T [ e T + e T 2] 2ησ Bϕ 2 η 2 + 2e T e T 1 + e T σ Bηϕ ( e T 1) 2. Remark ha he mean is l(t) of he previous secion and ha he expression of he variance follows from a subsiuion of u=v=t in he covariance funcion ρ(u,v) also presened in he previous secion. We conclude ha he probabiliy of no final maching equals PA [ T < B T ] = Pa [ T < ] = 1 Φ m(t) σ(t) wih m(t) and σ(t) as above and where Φ(z) denoes he cumulaive sandard normal disribuion funcion in z. In case of deerminisic raes of reurn, he expressions for he mean and variance simplify and he probabiliy of no final maching equals T 1 e a + (r θ) + θ µ B + σ 2 B 2 T PA [ T < B T ] = Pa [ T < ] = 1 Φ. σ B T Remark ha perfec maching implies final maching since final maching pus only a resricion on he porfolio a ime T and herefore he probabiliy of no final maching is always lower han he probabiliy of no perfec mach. 4. MISMATCHING DEGREE In he case of no final maching, we propose a risk measure of final maching which gives an idea of he difference beween liabiliies and asses a he ime horizon T. We use he approach of Cummins12 (1988) in his calculaion of risk-based premiums and 11

12 of Kusakabe13 (1995) in his discree ALM model; and we propose as a measure of risk a ime : M (B T ) = E ( B T ) + e T i u du F wih () i modeling he shor-erm ineres raes, wih F he sigma-field of informaion unil ime and where he condiional expecaion is aken wih respec o he riskneural probabiliy. In he case ha he asses are higher han he liabiliies, he risk measure hus equals zero. A ime T iself, we know ha he measure M T equals (B T ) + = max(b T,). The value a ime can be obained by using echniques from opion heory and in paricular from he formulae of Black & Scholes14 (1973), Meron15 (1973) and/or Rabinovich16 (1989). Indeed, i is well-known ha he value of a call opion a ime which gives he righ (bu no he obligaion) "o buy" a ime T he liabiliies B T, modeled by he geomeric Brownian moion db = µ B B d + σ B B dw, a he exercise value K= A T of he asses a ime T, equals E ( B T ) + e T i u du F where he condiional expecaion is aken wih respec o he risk-neural probabiliy. If we assume ha he ineres raes are consan and T = T, we can use he wellknown Black & Scholes (1973) formula: M (B T ) = e i (T ) E[ ( B T ) + F ]= B Φ(z) e i(t ) K Φ(z σ B T ) wih log B K + i + σ 2 B 2 (T ) z = σ B T wih K= A T =N and where Φ(z) denoes he cumulaive sandard normal disribuion funcion in z. If we are ineresed a ime in he risk measure of no-final-mach, we jus have o plug in =. Remark ha he assumpion of consan ineres raes is no necessary. The ineres raes may be sochasic. Then he value of he risk measure follows from generalizaions of he Black & Scholes formula obained by e.g. Meron15 (1973) and Rabinovich16 (1989). 12

13 Meron15 (1973) exended he Black & Scholes formulae o he case of sochasic ineres raes which are such ha he zero-coupon bonds are deermined by a sochasic differenial equaion of he form dp = P ν( )d + P δ( )dz wih he bond and sock prices correlaed by EdZ [ dw ]= ρd. Using his noaion and wih T = T, he Meron (1973) formula implies ha he risk measure a ime equals M (B T ) = B Φ(z) P(T )Φ(z V (T )) wih log B K log ( P(T ) ) z = + 1 V(T ) V(T ) 2 and where T V(T ) = σ 2 B (T ) + δ 2 (s)ds 2σ B ρ δ(s)ds. In case he shor-erm ineres raes are modeled by a mean-revering Ornsein- Uhlenbeck process of he form di = qm ( i )d + ωdz wih EdZ [ dw ]= ρd describing he correlaion beween he shor-erm ineres raes and he reurn on he liabiliies, his formula leads o an explici expression (see Rabinovich (1989)). Indeed, a defaul-free discoun bond P ha maures a he ime horizon T is priced in his model by he formula (see e.g. Vasicek (1977)): P(T ) = G exp i H [ ] where H H(T ) = 1 exp [ q(t ) ] q and G G(T ) = exp m + ωλ q ω 2 2q 2 (H T + ) ω 2 H 2 4q wih consan marke price of risk λ. Using Iô's lemma, i is known ha he insananeous reurn variance of he bond δ is a funcion of ime, namely δ() = ωh(). Using his expression for δ, Rabinovich rewries Meron's (1973) formula for he call value wih given exercise price K= A T =N for ineres raes i modeled by a Vasicek process: M (B T ) = E ( B T ) + e T i u du F = B Φ(z) A T P(T )Φ(z V(T )) wih T 13

14 log B K log ( P(T ) ) z = + 1 V(T ) V(T ) 2 and where V(T ) 2 = σ 2 B (T ) + ω2 1 exp( 2q(T ) ) T 2H + q 2 2q 2ρσ Bω (T H). q Subsiuing =, delivers us he risk measure a ime = of he expeced defici a ime T, i.e. he expeced value of he difference beween he liabiliies and he asses when here is no final mach. If T > T, hen he resuls remain he same in he case wih deerminisic rae of reurn wih K= A T = A exp θt + r θ ( 1 e T ). In he general case, however, he risk measure of no final mach has o be deermined numerically since now no only he liabiliies B T bu also he asses A T a ime T are random. 4. CONCLUSIONS We have sucessfully exended he Janssen model in such a way ha he asse fund A akes ino accoun fixed-income securiies. This is imporan for insurance companies whose invesmens are more in bonds han in shares, especially for life-insurance companies. We have considered a reaable model in which we assume ha he asses can be represened by only zero-coupon bonds which reflec he hisorical raes of reurn. Those raes of reurn of he porfolio in he pas are supposed o be presened by an Ornsein-Uhlenbeck process. In his generalized Janssen model, we have sudied he probabiliy of mismaching of he asses and liabiliies of he company in a period [,T] by inroducing he firs mismaching ime τ = inf{ : T, a() } where a = ( a, ) is defined by a = ln A and where T can be assumed o be infiniy. B Furher, we have proposed a risk measure of no final maching which indicaes he difference beween he asses and he liabiliies a ime T. These resuls are imporan as hey are useful o deermine ALM-objecives o be achieved by he company. In a forhcoming paper, we will sudy a more realisic mulidimensional model and develop some ools needed o encouner hese objecives. 14

15 ACKNOWLEDGEMENTS The auhors would like o hank he Sociey of Acuaries of he USA for he CKER Research Gran for compleing he projec "Ineracion Beween Asse Liabiliy Managemen and Risk Theory". REFERENCES 1 Janssen J., 1992, "Modèles sochasiques de gesion d'acif-passif pour les banques e les assurances", Transacions of he 24h Inernaional Congress of Acuaries, ICA- ACI, Monréal, Ars P. and J. Janssen, 1994, "Operaionaliy of a Model for he Asse Liabiliy Managemen", Proceedings of he AFIR 4h Session, , Orlando. 3 Ars P. and J. Janssen, 1995, "Sochasic Model wih Possibiliy of Ruin and Dividend Repariion for Insurance and Bank", Proceedings of he AFIR 5h Session, , Brussels. 4 Vasicek O., 1977, "An Equilibrium Characerizaion of he Term Srucure", Journal of Financial Economics, 5, Durbin J., 1985, "The Firs-Passage Densiy of a Coninuous Gaussian Process o a General Boundary.", J. Appl. Prob., 22, Durbin J., 1992, "The Firs-Passage Densiy of he Brownian Moion o a Curved Boundary.", J. Appl. Prob., 29, Sacerdoe L. and F. Tomassei, 1996, "On Evaluaion and Asympoic Approximaions of Firs-Passage-Time Probabiliies", Adv. Appl. Prob., 28, Deelsra G. and J. Janssen, 1998, "Some new resuls on he ineracion beween ALM and risk heory", working paper. 9 Harrison J.M. and D.M. Kreps, 1979, "Maringales in Muliperiod Securiies Markes", Journal of Economic Theory, 2, Cummins J.D. and D.J. Ney (198), "The Sochasic Characerisics of Propery- Liabiliy Insurance Profis", Journal of Risk and Insurance, Deelsra G., 1994, "Remarks on 'Boundary Crossing Resul for Brownian Moion'", Blaeer, 21 (4), Cummins J.D., 1988, "Risk-Based Premiums for insurance Guarany Funds", Journal of Finance, 43, Kusakabe T., 1995, "Asse Allocaion Model for Japanese Corporae Pension Fund from Liabiliy Aspecs", 25h Inernaional Congress of Acuaries, , Brussels. 15

16 14 Black F. and M. Scholes, 1973, "The Pricing of Opions and Corporae Liabiliies", Journal of Poliical Economy, 81, Meron R.C., 1973, "Theory of Raional Opion Pricing", Bell Journal of Economics and Managemen Science, 4, Rabinovich R., 1989, "Pricing Sock and Bond Opions when he Defaul-Free Rae is Sochasic", Journal of Financial and Quaniaive Analysis, 24 (4). 16