LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

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1 LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac. Pricing of insurance produc is usually evaluaed on a basis where ineres rae is assumed o be fied over ime. To obain a more realisic assessmen of he pricing of is produc i would be benefi if he ineres raes are flucuaing. This paper compares acuarial quaniies which calculaed based on fied ineres rae o sochasic ineres rae using he Vasicek and he Co, Ingersoll and Ross financial valuaion models. In his case, ime o mauriy in financial valuaion models is adjused wih T(), a coninuous random variable represening fuure lifeime of a life-aged-. T() is obained hrough simulaion based on Gomperz Moraliy Law. By means of Mone Carlo simulaion we calculae acuarial quaniy under whole life insurance and give empirical resuls. Furhermore, we quanify Wang s Transform risk measure wih respec of loss disribuion. Key-words: life insurance pricing, ineres rae derivaives, Gomperz moraliy law, Mone Carlo simulaion. 1 Inroducion During he las year, insurance pricing models involving sochasic ineres rae have become more and more ineresing. I is worhwhile o undersand ha one has o be careful when working wih sochasic ineres rae. ew environmen, no known in a world of deerminisic ineres rae especially when combining wih finance which have used more sophisicaed mahemaical conceps, such as maringales or sochasic inegraion in order o describe he economic behavior or o derive compuing mehods such as he absence of arbirage opporuniy and equilibrium heory.. Enormous lieraure in finance such as Vasicek (1977) and Co, Ingersoll and Ross (CIR) (1985) have documened ha ineres rae should be followed by a sochasic process. The remainder of he paper is organized as follows. Secion is sared wih moraliy as he basic building block in acuarial science. In secion 3, we deermine he acuarial quaniies based on ineres rae derivaives models. In secion, 4 we simulae and give empirical resuls. Secion 5 is closed wih a conclusion. Moraliy Symbol () will be used o denoe a life-aged- (Bowers e.al. 1997). Moraliy can be saed by K() discree random variable represening he number of compleed fuure years lived by () or T() coninuous random variable represening fuure

2 L. OVIYATI, M. SYAMSUDDI lifeime of (). The moraliy able describes no only compleely K() disribuion bu also can describe T() disribuion hrough he approimaion. The hree characerisics of Gomperz Moraliy Law are as follows The force of moraliy; + μ + BC B >, C >, > The densiy funcion; BC f () d ( BC ) ep + ( C 1) ln C The disribuion funcion; F () PT ( ( ) ) BC ( C 1) log C 1 e (1) M () 1 e Parameer B and C are esimaed by nonlinear leas square based on he U.S. Moraliy Table (oviyani, L. and Syamsuddin, M., 3). I is known ha M() random variable in equaion (1) has eponenial disribuion (λ1). To simulae M(), le U be a uniform (,1) random variable. According o Ross (1977), M - ln(u) consiues random variable of eponenial disribuion (λ 1). T() can be simulaed using he inversion mehod (Pai, 1997) based on M() random variable. I will be obained T random variable of Gomperz Moraliy Law M ln( C) ( ) T ln + 1 / ln( C) BC 3 Acuarial Quaniy The presen value of a benefi a ime is defined by z b v where b is he amoun of benefi a ime and v is he discoun facor. Calculaing a presen value usually assumes ha is value depends on he lengh of he ime period and he force of ineres / he ineres rae is consan. We denoe Z as he random variable wih oucome z. The epecaion of he presen value random variable, E(Z), is called he acuarial presen value for he whole life insurance wih a uni payable a he momen of deah of (), denoed by A.

3 LIFE ISURACE WITH STOCHASTIC ITEREST RATE ( ) A E v v f ( ) d () The acuarial presen value for a coninuous whole life annuiy is a v p d Quaniy he discoun facor v ep (-δ) represens he presen value a ime zero of one uni of accoun a ime discouned by consan ineres rae. The discoun facor discouned by sochasic ineres rae is v () ep rsds () We recall he ineres rae derivaive models. Securiies wih payoffs ha depend on ineres raes are called ineres rae derivaives. Such securiies are imporan because almos every financial ransacion enails eposure o ineres rae risk and ineres rae derivaives provide he means for conrolling ha risk. Ineres rae derivaive securiies are relevan o many forms of invesmen and he Vasicek model and he CIR model are based on zero coupon bond. In his approach, i is specified ha he insananeous shor rae r() saisfies an equaion of he Io equaion, showing he relaionship beween derivaive price changes and he ineres rae and ime changes. dr() μ (r,)d + σ ( r, ) dw() Given an iniial condiion r(), he equaion defines a sochasic process r() (Lamberon, D., and Lapeyre, B., ). Evoluion of ineres raes is driven by he shor rae r() and shor raes are revering wih a consan reversion rae. I is assumed he ineres raes follow he sochasic process suggesed by Vasicek and Co, Ingersoll and Ross (Brigo, D. and Mercurio, F., 1) and can be epressed as follows The Vasicek Model r() c θ() σ W() dr() c( θ -r())d + σ dw() : he shor erm ineres rae : he speed of adjusmen in mean revering process : he long run average value of r() : he sandard deviaion of he ineres rae process : a sandardized Weiner process

4 L. OVIYATI, M. SYAMSUDDI The Co, Ingersoll and Ross Model dr() c( θ-r())d + σ r() dw() The process for r() involves only sources of uncerainy driving all raes. This usually means ha in any shor period of ime all raes move in he same direcion. The drif [c(θ-r())] and σ are assumed o be funcions of r, bu are independen of ime. According o he CIR model, he sandard deviaion of he change in he shor rae in a shor period of ime is proporional o r() 1/. This means ha, as he shorerm ineres rae increases, is sandard deviaion increases. Differs from he Vasicek model, he CIR model eliminaes he possibiliy of negaive ineres raes. Le P() denoes he curren of a one-dollar zero-coupon bond a period. From he wo models, where P () E ep rsds ( ) B( ) r A () e σ ( B ( ) ) θ c σ B() 1 e AVasicek () ep ; B () Vasicek c 4c c A B CIR CIR ( c+ c + σ ) c + σ e (, ) c + + c e 1 + c + σ (, ) c + σ ( σ )( ) c + σ ( e 1) c + σ ( σ )( ) c + + c e 1 + c + σ cμ / σ Furhermore, he acuarial presen values for he whole life insurance wih a uni payable a he momen of deah of () and a coninuous whole life annuiy based on he wo models of ineres rae derivaives are denoed by A p and a. p c

5 LIFE ISURACE WITH STOCHASTIC ITEREST RATE A p E( E( v( )) E( v( )) f ( ) d (3) T P(, ) f ( ) d T σ r () μ σ c c σ c ApVasicek ep μ ( 1 e ) 3( 1 e ) f ( ) d c c 4c d ( e ) d ( + c)/ cμ/ σ de 1 ApCIR ep ( ) ( ) ; d r f d d c + σ d d c+ e ( d+ c) d c+ e ( d+ c) a P() p d p P() μ + f () d (4) G() f () d The premium is calculaed based on equivalence principle. P A / a p p p Afer having he acuarial quaniies, we calculae a loss funcion; L v( T) P. at The risk of company loss will be occurred when he value of loss funcion (L) is posiive; his means ha he value of benefi which will be paid is greaer han premium obligaion received. The las sep is o quanify Wang s Transform risk measure wih respec of loss disribuion; For a loss variable L wih disribuion F, Wang (1) defined a new risk measure for capial requiremen as follows: For a pre-seleced securiy level α, le λ Φ -1 (α)

6 L. OVIYATI, M. SYAMSUDDI Apply he Wang Transform : F*(l) Φ[Φ -1 (F(l)) - λ] Se he capial requiremen o be he epeced value under F* : WT(α) E*(L) 4 Empirical Resuls In his secion, we presen empirical resuls and sensiiviy analysis wih respec o parameers of insurance conrac. To calculae A (equaion ()), A p (equaion (3)) and a p (equaion (4)) each depends on E(v ), E(v()) and E(G()). By using Mone Carlo Inegraion Mehod in Malab 6. which uilizes he resuls he Law of Large umber, he acuarial quaniies are δ T 1 δ Ti E ( e ) e i E ( PT ( )) ep rs ( ) ds P ( ) i 1 i 1 1 P ( ) EGT ( ( )) μ i 1 + where Ti consiues i h T simulaion, i 1,,,. Table 1 and Figures 1 o 4 display paern he acuarial quaniies under whole life insurance and Wang s Transform risk measure wih changes in parameers of he speed of adjusmen in mean revering prices (c), iniial volailiy of ineres raes (σ) and he long run average value (θ) ( 5, 35 and 45 ; r.5). Figure 1. Acuarial Presen Values (35) Figure. Annuiies (35)

7 LIFE ISURACE WITH STOCHASTIC ITEREST RATE Figure 3. The Premiums (35) Fig. 4. Wang s Transform Risk Measures (35) Figures 1 and show ha he values of APV and Annuiy based on sochasic ineres raes give less values han fied ineres rae alhough heir premium values (Figure 3) are relaively close o each oher ranges from.999 o.118 (Table 1, 35). Furhermore he obainable informaion (Figures 4) indicaes ha he Wang s Transform Risk Measures (35 and also for 5 and 45) of he Vasicek Model and he CIR model give also less values. I will make a grea influence relaed o he amoun of loss ha will be borne by he company. Table 1. Acuarial quaniies and Wang s Transform (WT) risk measure. 5 ; T51.16 APV Annuiy Premium WT k μ The Vasicek Model The CIR Model

8 L. OVIYATI, M. SYAMSUDDI 35 ; T41.76 APV Annuiy Premium WT k μ The Vasicek Model The CIR Model ; T3.86 APV Annuiy Premium WT k μ The Vasicek Model The CIR Model The simulaion resul (Table 1) shows, as epeced, he premium values of he Vasicek Model and he CIR model are increasing in age. Table gives adjused parameer values wih respec o he close premiums beween fied and sochasic ineres raes.

9 LIFE ISURACE WITH STOCHASTIC ITEREST RATE Table. The parameers of The Vasicek Model and he CIR Models (σ.5;.15;.5) Ages Parameers The Vasicek Model The CIR Model c ;.55 θ.51 ;.5 ;.53.4 ;.43 c ;.55 ;.6 θ.6 ;.61 ;.6.5 ;.51 c ;.8 θ.8 ;.85 ; ;.67 According o Table, i can be seen ha parameer of he speed of adjusmen in mean revering process (c) of he Vasicek Model is 1.1 for all he ages, whereas parameer c of he CIR Model are varying from.5 o.8. I shows ha he CIR Model is sensiive o he change in he speed of adjusmen in mean revering process (c). 5 Conclusion In his paper we have shown he combinaion of he use of financial and acuarial approaches o price life insurance conrac. The Vasicek and The Co, Ingersoll and Ross financial valuaion models are used in order o calculae acuarial quaniies and Wang s Transform risk measure. Simulaion resul shows a significanly differen risk value in relaion beween fied and sochasic ineres rae. References [1] Brigo, D. and Mercurio, F. (1), Ineres Rae Models, Theory and Pracice, Springer-Verlag, Germany. [] Bowers,.L., Gerber, H., Hickman, J., Jones, D. and esbi, C. (1997), Acuarial Mahemaics, nd ed., Iasca, Illinois, he Sociey of Acuaries. [3] Co, J.C., Ingersoll, J.E. and Ross, S.A. (1985), A Theory of he Term Srucure of Ineres Raes, Economerica 53, [4] Devroye, L. (1986), on-uniform Random Variae Generaion, Springer- Verlag. [5] Lamberon, D., and Lapeyre, B. (), Inroducion o Sochasic Calculus, Applied o Finance, Chapman Hall, UK. [6] Mao, H., e.al. (4), Pricing Life Insurance: Combining Economic, Financial and Acuarial Approaches, Journal of Insurance Issues, 7,, pp.xxx-xxx. [7] oviyani, L. and Syamsuddin, M. (3), Some Risk Measures for Capial Requiremens in Acuarial Science, Proceedings of Inernaional Conference on Mahemaics and Is Applicaions, Souh Eas Asian Mahemaics Sociey - Gadjah Mada Universiy, Yogyakara,

10 L. OVIYATI, M. SYAMSUDDI [8] Pai, J. S. (1997), Generaing Random Variaes wih a Given Force of Moraliy and Finding a Suiable Force of Moraliy by Theoreical Quanile- Quanile Plos, Acuarial Research Clearing House, 1, [9] Ross, S. (1997), Simulaion, Harcour/Academic Press, USA. [1] Vasicek, O.A. (1977), An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics, 5: [11] Wang, S. (1), A Risk Measure ha Goes beyond Coherence, www. sas.uwaerloo.ca/sas\_dep/iipr/1-repors/iipr-1-18.pdf L. oviyani: Deparmen of Saisics, Universias Padjadjaran. Jl. Raya Bandung-Sumedang km 1 Jainangor, Indonesia. lienda@dns.mah.ib.ac.id, & liendany@yahoo.com M. Syamsuddin: Deparmen of Mahemaics, Insiu Teknologi Bandung. Jl. Ganesha o. 1 Bandung, Indonesia. muhia@dns.mah.ib.ac.id