STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW. Ocke Kurniandi. PT. MAA Life Assurance, Indonesia

Transcription

1 STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW Ocke Kurniandi PT. MAA Life Assurance, Indonesia Absrac. This paper inroduces he alernaive mehod o calculae premium ha is compliance under syariah law. There is disincion beween convenional mehod and syariah principles abou ineres rae. The mehod is based on sochasic model and uses sochasic differenial equaion. I s easier use numerical approach, han analyic approach. As case sudy we examine his mehod o fiveyear life insurance coverage by Mone-Carlo simulaion. Key-words: syariah principles, sochasic model, sochasic differenial equaion numerical approach, Mone-Carlo simulaion. 1 Inroducion Many life insurances in Indonesia have been selling he producs under syariah law. Principles of syariah law for insurance are he insurance produc ha mus no conained ghoror (unexplainable rule), maisir (gambling), and riba (ineres rae or guaraneed invesmen reurns) (Dewan Syariah Nasional Majelis Ulama Indonesia, 2003). In pracice he acuaries always assume he ineres rae when calculaing he premium. If we assume ineres rae or give he guaraneed invesmen reurn hen he premium calculaion can be acceped under syariah law. Now, we have disincion beween convenional mehod and syariah principles. Many books in acuarial science pu ineres rae as a deerminisic variable bu now some books inroduce ineres rae as a random variable. If he ineres rae as a random variable han his assumpion is mached wih syariah principles. However, ha we sar o hink he invesmen reurn or ineres rae as a random variables. Laer we can find many differen formulaions and percepion beween convenional and syariah produc. We sar o find he premium calculaion formula as firs sep o formulaing he syariah acuarial mehod. Basic economic heory would sugges ha ineres rae, like prices, are esablished by supply and demand. If he demand for funds o borrow is srong in relaion o he availabiliy of funds, ineres raes will rise. Conversely, if he demand for funds o borrow is weak in relaion o be availabiliy of funds, ineres raes will fall. In syariah principles, he heory s a lile bi differen because hey use profi sharing concep hen borrow and lender have same posiion on invesmen reurn. The major facors which have an influence on level of invesmen reurn 1. Produciviy growh in economy 2. Inflaion 3. Risk and uncerainy 4. Governmenal policy 5. Random flucuaion 532

2 STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW In sochasic models he major facors will be expressive as drif and volailiy facor. The drif and volailiy facors are measuremen of economy and governmenal condiion. 2 Sochasic Process Sochasic process used in sock marke and he model doesn give a guaraneed ineres rae. In many cases sochasic process have been used o predic he invesmen reurn or ineres rae and his mehod assume he ineres rae or invesmen reurn is a random variable (Kellison, 1991; Baxer, 1996) Definiion 2.1 Sochasic Process { r, 0 } as Browns moion wih drif µ and variance σ if 1. r0 = 0 2. { r, 0 } saioner and independen incremen 3. r normal disribuion wih µ and variance σ Generally, one assumes ha he value of random variable of invesmen reurn will grow wih ime. Le us consider he difference equaion r = µ + σε (1) Here µ (he drif) is a measure of growh rae and σ (he volailiy) is a measure of he variabiliy of he process as ime increases. ε is normal disribuion wih mean zero and variance 1. In he limi, as goes o zero, such a process is uniquely defined and commonly refereed o as a Brownian process. Le µ and σ are funcion wih wo variable, and r. dr = µ d + σ dw (2) ( r, ) d σ ( r )dw dr µ +, = (3) The equaion (3) is called a sochasic differenial equaion (SDE) for r (Baxer, 1996; Rolski, 1999; Fabozzi 2002) As in ordinary differenial equaion (ODE), an SDE need no have soluion, and if i does i migh no be unique. We model he invesmen reurn as ( θ α r ) d + dw dr σ = (4) for some consan θ, α andσ. The SDE is composed of Brownian par and resoring drif which pushes i upwards when he process is below θ / α and downward when i is above. The magniude of he drif is also proporional o he disance away from his mean. We can use Io s formula o check ha he soluion o his, saring r a r0, is 533

3 O KURNIANDI θ α θ α αs r = + e r0 + σe e dws α α 0 As i happens r can be rewrien in erms of a differen Brownian moion W as (5) θ = + e α r θ + σe α 2α e 1 W 2α α α r 0 (6) θ so ha r has a normal marginal disribuion wih mean e θ ( r ) 2 2α variance σ ( 1 e )/ 2α + α 0 α and. As ges large, his converges o an equilibrium normal 2 disribuion of mean θ / α and variance σ / 2α. This does no mean ha he process r converges bu he disribuion of r converge o he normal disribuion. 3 Premium Calculaion We shall define he presen value funcion, Z, by z = b v (7) Here b is benefi funcion and v is a ineres discoun facor from he ime of paymen claim o he ime of policy issue. The v deermine from r, invesmen reurn as 1 v =, s, r > 1 (8) s= 1 ( 1+ rs ) is lengh of he inerval from issue o deah. Therefore, z is he presen value, a policy issue, of he benefi paymen. The elapsed ime from policy issue o deah of he insured is he insured s fuure-lifeime random variable, denoe as T and V is random variable of ineres discoun facor. Thus, he presen value, a policy issue, of benefi paymen is he random variable zt. We shall denoe his random variable by Z and base he model for he insurance on equaion. Z = b V T (9) An n-year erm life insurance provide for a paymen only if he insured dies wihin he n-year erm of an insurance commencing a issue. If a uni is payable a he momen of deah of (x), hen 1 b = 0 n > n Then expecaions of he presen-value random variable Z is called he ne single premium 1 A x: n = E[ Z] = b vs dfr ( ) dfx ( ) (11) 0 0 (10) 534

4 STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW if we make a simple approximaion o equaion (4) by assuming ha every claim will pay a he end of year, hen we will have he ne single premium or presen value of probabiliy claim amoun. [ ] 1 A x: n = E Z = b vs qx (12) = 0 s= 0 We use ne premium principles and sandard deviaion principles (Kaas; 2001) o calculae he ne premium 1. Ne Premium Principles : NP = E[Z] 2. Sandard Deviaion Premium Principles : NP = E[Z] + [Z] where is posiive number. 4 Mone-Carlo Simulaion As we know he disribuion of r converge o he normal disribuion bu equaion (8) only accep r more han negaive one (in Indonesian siuaion more han zero). I s no easy o handle his siuaion. The sandard approach o handle his siuaion in pracice is o use simulaion. Le we consider he equaion (4) wih discree version r = θ α r + σ ε (13) ( ) where ε is normal disribuion wih mean zero and variance 1. In he limi, as goes o zero. We noe ha ( r) = ε σ + erms of higher oher in Now we have ( θ α r ) + σ r + = r + ε 1 (14) if we make a simple approximaion o equaion (4) by assuming ha every claim will pay a he end of year and = 1, hen we will have r + 1 = r + ( θ α r ) + σ ε (15) We consider discoun facors as equaion (8). The Mone-Carlo simulaion procedure for n years insurance coverage and m replicaion is as follows : 1. Generae random numbers from normal sandard disribuion. 2. Nex find sequence r1, r2,, rn using equaion (15) and discoun facors v1, v2,, vn using equaion (8). 3. Calculae he ne single premium by equaion (12). 4. By m replicaion, we can have he m possible value of ne single premium Suppose he m replicaion is sochasic independen hen we can have an average value as expecaions of he presen value random variable Z (Ross; 1996). 535

5 O KURNIANDI Le us denoe he m possible value of ne single premium are X1, X2,., Xm, because for each ne premium is one of he possible value of he ne premium hen X1, X2,., Xm are sochasic independen random variable wih mean and variance 2. Alhough he sample mean X and variance S 2 m 2 X = X i / m S = ( X i X )/ ( m 1) i = 1 are effecive esimaor of mean and variance 2., hus from he cenral limi heorem, i follows ha for large m X Φ n ( ) S ~ N ( 0,1) By using he boosrapping echnique for esimaing mean square error from he sample value of ne single premium X1, X2,., Xm. We can esimae he underlying probabiliy disribuion funcion F of ne single premium by he socalled empirical probabiliy disribuion funcion Fe, he probabiliy ha presen value of claim or ne single premium value is less han or equal o x ha is Numb of i, X i x Fe ( x) = (17) m If m is large hen Fe is close o F. In probabiliy heory, we know he srong law of large numbers implies ha Fe converges o F as m wih probabiliy 1. 5 Case Sudy : Five-Year Life Insurance In his paper we will examine ne single premium of five-year life insurance wih benefi 1000 a he momen of deah of (x) and probabiliy risk for each years coverage. This includes he profi sharing of invesmen 70% o policy owner a he end of policy years. We assume he parameer models are θ = 0. 06, α = 0. 5, σ = 0.15 and iniial/curren invesmen reurn is 9% per annual. The mos imporan echnique in Mone-Carlo simulaion is o ge large number sample. Now, we use 500 replicaion o ge large number, is enough number o ge he approximaion of expecaion of random variable Z. Firs sep of Mone-Carlo Simulaion is generae n m random numbers (n years and m replicaion), in his case we generae 2500 random number for 5 years wih 500 replicaion. By equaion (15), we generae 2500 number of ε from normal disribuion wih mean 0 and variance 1. m i = 1 Table 1. Random Numbers ~ N(0,1) Replicaion Year (16) 536

6 STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW Using equaion (15), We can find sequence { r } of invesmen reurn for 5 years and 500 replicaions. Table 2 is invesmen reurn from able 1. Table 2. Invesmen Reurns Replicaion Year % 9.0% 9.0% 9.0% 9.0% % 9.1% 10.9% 8.3% 9.2% % 11.7% 14.1% 11.8% 10.3% % 10.2% 12.3% 11.2% 10.3% % 10.1% 12.4% 12.1% 11.9% % 7.9% 11.9% 14.7% 9.5% Using equaion (8) and (12), We can find presen value of insurance benefi for each year. This value afer we ake he profi sharing facor for he insurance company, in his case he invesmen reurn will be given o he company 30% and he 70% o he policy owner. Table 3. Presen Value of Insurance Benefi Replicaion Year Sum of presen value of insurance benefi for each year is he ne premium of five years insurance coverage. Le s see able 4, 5 possible ne premium of five-year life insurance. Table 4. Five Ne Premium of 5 Years Life Insurance NP Le us consider able 5, expecaion of invesmen reurn r is close o he mean of our model θ / α =12%. Table 5, Saisic of Invesmen Reurn Mean of sample = 11.34% Variance sample = Sandard Deviaion =

7 O KURNIANDI Consider able 6, expecaion of ne premium wih 500 replicaion Table 6. Saisic of Ne Premium from Random Variable Z Mean of sample = Variance sample = Sandard Deviaion = Le us consider figure 1, a hisogram of 500 possible ne premium of five-year life insurance wih deah benefi Using boosrapping echnique we can approximae he probabiliy ha ne premium will be cover he possible of he invesmen reurn happen in five years. If we ake x (posiive value) as a premium hen we have he number of ne premium less or equal han x, denoe y, I mean x possible o cover y/500 of he possible invesmen reurn happen in five years Figure 1. Hisogram of 500 possible ne premium. Consider able 7, Then ne premium of five-year life insurance use sochasic models and he ne premium principles and sandard deviaion principles (Kaas, 2001), where is facor ha we choose as safey margin of he ne premium. The ne premium use various safey margin and relaes o he probabiliy ha ne premium can cover he possible invesmen reurn happen in five years. 538

8 STOCHASTIC MODELS FOR PREMIUM CALCULATION UNDER SYARIAH LAW Table 7. Ne Premium Five Years Life Insurance use sochasic models Facor Premium x - value Prob % % % % % % % Now we compare his echnique o he classic echnique which convenional produc has been developed. Le us consider able 9, ne premium rae which various fixed ineres rae from 6% o 14%. Table 8. Ne Premium Five Years Life Insurance use classic mehod In. Rae Premium Rae Prob. 6% % 7% % 8% % 9% % 10% % 11% % 12% % 13% % 14% % When we assume, he iniial or curren invesmen reurn is 9% per annual. Ineres rae from 6% o 9% have ne premium more han 4.02 (ne premium using sochasic models). Inerval from 12% o 14% have ne premium rae less han 4.02 bu saring he firs year company has negaive spread (from -3% o -5% pa) and in his inerval he premium can cover maximum 21% of he possible invesmen reurn happen in five years. I is clear ha using sochasic models, he five years life insurance syariah will be cheaper and more compeiive han convenional produc. Moreover insurance company who sells syariah produc does no have he negaive spread and more solvency han sell convenional produc. Finally sochasic models can be applied ino syariah principles and using his mehod he premium can be more compeiive han convenional mehod. We hope he Indonesian regulaor will accep his mehod as one of sandard premium calculaions mehod of he syariah produc. I is more ineresing concep when we use sochasic model o define he reserve valuaion mehod for syariah produc and mos imporan sudy is how o measure he solvency of syariah insurance company. 539

9 O KURNIANDI References [1] Dewan Syariah Nasional Majelis Ulama Indonesia, (2003), Himpunan Fawa Dewan Syariah Nasional, Edior: I. Sam, Hasanuddin, PT. Inermasa, Jakara. [2] Fabozzi, F. J. (2002), Ineres Rae, Term Srucure, and Valuaion Modeling, John Wiley & Sons, Inc, New Jersey. [3] Thompson, J.R. (2000), Simulaion, A Modeler s Approach, John Wiley & Sons Inc, New York. [4] Baxer, M. & Rennie, A. (1996), Financial Calculus, An inroducion o derivaive pricing, Cambridge Universiy Press, Cambridge. [5] Kellison, S.G. (1991), The Theory of Ineres, Richard D. Irwin, Inc. Boson. [6] Ross, S.M. (1997), Simulaion, Academic Press, California. [7] Ross, S.M. (1999), An Inroducion o Mahemaical Finance, Cambridge Univerciy Press, Cambridge. [8] Bowers, N.L. JR. (1986), Acuarial Mahemaics, The Sociey of Acuaries, Illinois. [9] T. Rolski, T. H. Schmidli, V. Schmid, and J. Teugels (1999), Sochasic Process for Insurance and Finance, John Wiley & Sons, Inc, England. [10] Kaas, R. Goovaers, M. Dhaene, J. and Denui M. (2001), Modern Acuarial Risk Theory, Kluwer Academic Publishers, The Neherlands. OCKE KURNIANDI: Associae Acuary, PT. MAA Life Assurance Kawasan Bisnis Granadha, Jl. Jenderal Sudirman Kav 50 L 9, Jakara 12930, Indonesia. Phone: / Fax: ocke@maa.co.id 540