New Approximations of Ruin Probability in a Risk Process

Transcription

1 Quality echnology & Quantitative Management Vol 7, No 4, pp , 00 QQM IQM 00 New pproximations of Ruin Probability in a Ris Process Seung Kyoung hoi, Moon ee hoi, ye Sun Lee and Eui Yong Lee Department of Statistics, Soomyung Women's University, Seoul, Korea (Received January 009, accepted May 009) bstract: continuous-time ris process is considered, where the premium rate is constant and claim process forms a compound Poisson process We introduce new approximations of the ruin probability of the ris process, which extend ramér's and ijms' approximations We also introduce an extended formula of the well-nown exponential approximation hese new approximations give closer values to the true ruin probability than the existing ones Keywords: ontinuous-time ris process, ramér's approximation, exponential approximation, ruin probability, ims' approximation Introduction classical continuous-time ris process in insurance business is defined as follows: N() t U() t = u+ ct X, where U() t is the surplus at time t > 0 with initial value u> 0, c is the premium rate, and N() t is the number of claims by time t > 0 with X i being the amount of the i -th claim It is assumed that { N( t), t > 0} is a Poisson process of rate λ > 0 and X i 's are independent and identically distributed with general distribution function G and mean μ Moreover, c is given by c = ( + θ ) λμ, where θ > 0 is called the relative security loading, which insures that the premium rate is larger than that of payment he ruin probability of the ris process is defined by i = ( u) = Pr{ U( t) < 0, for some t > 0 U(0) = u}, as a function of initial surplus u We say that a drop occurs if the surplus goes below u Let Y be the amount below u after the drop, then, it can be shown that the probability that the drop occurs is given by (0) =, +θ i

2 378 hoi, hoi, Lee and Lee and the distribution function of Y is given by y Ge ( y) = 0[ G( x)]d x, μ the equilibrium distribution function of G See Klugman et al ([9], pp 39-4) We say that another drop occurs if the surplus goes again below the level of the previous drop, and so on If we denote Y, Y 3, to be the subsequent amounts below the levels of the previous drops, then we can show that the probability that there is another drop is still (0) and the amounts below the levels of the previous drops are independent and identically distributed with G e Define L = Y + Y + + YK as the maximum aggregated loss with K being the number of drops, then the ruin probability is now given by θ ( ) ( u) = Pr{ L > u} = Ge ( u), = + θ + θ ( ) where G e is the -fold convolution of G e and ( ) ( ) Ge = Ge he ruin probability is, however, very complicate to calculate even though the distribution G of the claim size is explicitly given, because of the recursive convolutions and infinite sum ence, many authors have suggested approximation formulas of the ruin probability, for examples, ramér [], adwiger [8], Lundberg [0], Beeman [], De Vylder [3], Grandell [5, 6], ijms [] and so on See Grandell [7] for the comparisons of these approximations ramér's approximation and ijms' approximation which extends ramér's are nown to wor better than the other approximations, when the tail of G decreases exponentially fast, for examples, when G is an Erlang distribution or the mixture of exponential distributions Specially, when the initial surplus u is large, ramér's and ijms' approximations are very close to the true ruin probability See Klugman et al ([9], pp 44-50) When u is relatively small, ijms' approximation improves ramér's, however, both approximations are still not satisfactory, even though G has exponentially decaying tail ence, we, in this paper, extend ijms' approximation to give closer values to the true ruin probabilities when u is both small and large In the following section, after summarizing ramér's and ijms' approximations, we introduce two new approximation formulas of the ruin probability, compare our approximations with ramér's and ijms' and show that our approximations wor better than ramér's and ijms' numerically when G is an Erlang distribution Finally, after introducing the well-nown exponential approximation, we mae use of the idea of ijms' to produce the extended version of the exponential approximation pproximations of the Ruin Probability ramér [] introduced an approximation formula for the ruin probability, thereafter, many authors generalize the formula in various ways See Grandell [7] ramér's approximation formula is given by κu ( u) = e,

3 New pproximations of Ruin Probability in a Ris Process 379 where = μθ /[ MX ( κ) μ( + θ)], MX( t) is the moment generating function of the claim size, and κ > 0 satisfies + ( + θ ) μκ = ( κ ) ramér's approximation approaches the true ruin probability as u he convergence rate is very fast when G has exponentially decreasing tail owever, when u is relatively small ramér's approximation is not so good ence, ijms [] suggested an extension of ramér's approximation by adding an exponential term to the approximation formula of ramér's ijms' approximation formula is given by u/ α M X κu ( u ) = e + e, u 0, where = /( + θ ) which satisfies (0) = /( + θ) = (0) Moreover, α is obtained by solving equation which is given by E( X) 0 ( u)d u= E( L) = E( K) E( Y) =, μθ E( X)/( μθ) / κ α = /( + θ ) ijms' approximation gives closer value to the true ruin probability than ramér's owever, there is still some difference between ijms' approximation and the true ruin probability, when u is relatively small See able ence, we introduce new approximations which extend ijms' to give closer values to the true ruin probabilities when u is both small and large We consider the following approximation formula: u/ α u/ α κu ( u) = e + ue + e, u 0, u/ α where we add term ue to the approximation formula of ijms', which goes to 0 as u goes to either 0 or is determined to satisfy (0) = θ / μ( + θ) = (0), which enables ( u ) to approximate the true ruin probability better than ( u ) when u is small is given by = θ κ μ( + θ) + α + α, now, satisfies equation solution of 0 ( udu ) = E ( X )/ μθ It can be shown that α is a positive θ E( X ) + κ α α + + = μ( + θ) + θ κ μθ

4 380 hoi, hoi, Lee and Lee We prefer the smaller α when there are two positive solutions, since with this α the u/ α added term ue goes faster to 0 as u goes to It is nown that ijms' approximation gives the exact ruin probability when G is either a mixture of the exponential distribution and Erlang distribution with shape parameter or a mixture of two exponential distributions So is our ( u ) heorem ( u ) gives the true ruin probability whenever ( u ) does Proof Suppose that ijms' approximation gives the true ruin probability hen, we must have (0) = (0), which is equivalent to θ + κ = α + θ μ( + θ) With this condition, in our approximation becomes 0 fter some algebras, we can show that the equation for α has single root which is same as that of ijms' ence, ( u) = ( u) = ( u) nother way of determining α and is to mae these satisfy E( X) 0 ( u)d u= E( L) = and μθ 3 E( X ) [ E( X )] 0 u u u= E L = + ( )d ( ), 3μθ ( μθ ) that is, to mae the first two moments of the approximation be equal to those of ( u) We can show that α and satisfying the above equations are given by and 4α 3μθ ( μθ) κ 3 E( X) [ E( X)] = 3 α + 3 E( X ) E( X ) [ E( X )] α α = 0 κ μθ 3μθ ( μθ ) κ gain, we prefer smaller α when there are two positive solutions Let ( u ) denote the approximation formula with these α and, then it also satisfies the following property: heorem ( u ) gives the true ruin probability whenever ( u ) does Proof: If ijms' approximation gives the true ruin probability, then ( u ) should satisfies that 3 E( X ) [ E( X )] 0 u u u= E L + α = + ( )d ( ) κ 3μθ ( μθ)

5 New pproximations of Ruin Probability in a Ris Process 38 In this case, we can show that in our approximation becomes 0 and the equation for α has one positive solution which is same as that of ijms' ence, ( u ) = ( u ) = ( u ) From the definitions of ( u ) and ( u ), we expect that ( u ) performs better than ( u ) when u is small, otherwise, ( u ) approximates the true ruin probability closer than ( u ) It is nown that ramér's and ijms' approximations wor well when the tail of G decreases exponentially fast, for examples, when G is an Erlang distribution or the mixture of exponential distributions Klugman et al ([9], pp49-50) give a numerical example when G is an Erlang distribution with shape parameter 3 and mean to show how well do ramér's and ijms' approximations, specially ijms' approximation, wor With the same example, we show numerically that our two approximations give closer values to the true ruin probabilities than ramér's and ijms' approximations in most cases Example Suppose that the probability density function of the claim size is given by 3x gx ( ) = 7 xe /, x> 0 We define the relative error of an approximation ( u ) to the true ruin probability ( u) as ( u) ( u) ε ( u) = 00 ( u) In able, the values of ( u), ε( u), ε( u), ε ( u) and ε ( u ) are listed for various values of θ and u able omparisons of the approximations θ u ( u) ε ( u ) ε ( u) ε ( u) ( u) 05 4 ε s we expected, our ( u ) and ( u) approximate the true probability ( u) better than ramér's ( u ) and ijms' ( u ) in most cases of able Specially, ( u ) wors very well when u is small, meanwhile, ( u ) wors very well when u is relatively large

6 38 hoi, hoi, Lee and Lee For the values of u larger than 5, all of ( u), ( u), ( u) and ( u ) give very close values to the true ruin probability, since they all converge to ( u) as u increases Exponential pproximation ijms' idea which was used to extend the approximation formula of ramér's can be applied to extend the well-nown approximation formula of De Vylder [4], called exponential approximation De Vylder [4] derived the following approximation formula for the ruin probability ( u) as θu τ ζθu ζ E ( u) = exp = exp, τ + τθ ζ + (4 /3) θζζ3 where ζ = E( X ) and τ = EY ( ) = ( ζ + )/( + ) ζ Notice that ( u ) is of the form E u ( u E ) = e κ, where = exp{ + τ/ τ + τθ} and κ = exp{ θ / τ + πθ} ence, applying ijms' idea to E ( u ) gives the extended version of the exponential approximation, which is given by u/ α κ u E ( u ) = e + e, where = /( + θ ) which satisfies E (0) = (0) Moreover, α is obtained by solving equation 0 E ( udu ) = E ( X )/ μθ, that is, τ α = ( ) θ κ cnowledgments We than anonymous referees for their helpful comments his wor supported by Korea Science Engineering Foundation (KOSEF) grant funded by the Korea government (MOS)(R ) and by the Soomyung Women's University grants 009 References Beeman, J (969) ruin function approximation ransactions of the Society of ctuaries,, 4-48, ramér, (930) On the Mathematical heory of Ris Sandia Jubilee Volume, Stocholm In: Martin-Löf, (Ed), arald ramér ollected Wors, Vol I Springer, Berlin, 994, De Vylder, F E (978) practical solution to the problem of ultimate ruin probability Scandinavian ctuarial Journal, De Vylder, F E (996) dvanced Ris heory Self-ontained Introduction Editions de l'université de Bryxees and Swiss ssociation of ctuaries

7 New pproximations of Ruin Probability in a Ris Process Grandell, J (977) class of approximations of ruin probabilities Scandinavian ctuarial Journal (Suppl), Grandell, J (99) spects of Ris heory Springer, New Yor 7 Grandell, J (000) Simple approximations of ruin probabilities Insurance: Mathematics and Economics, 6, adwiger, (940) Über die Wahrscheinlicheit des ruins bei einer grossen Zahl von Geschäften riv für mathematische Wirtschaft-und Sozialforschung, 6, Klugman, S, Panjer, and Willmot, G E (004) Loss Models: From Data to Decisions, Second Edition John Wiley & Sons, oboen 0 Lundberg, O (964) On Random Processes and their pplication to Sicness and ccident Statistics, st Edition lmqvist & Wisell, Uppsala ijms, (994) Stochastic Models-n lgorithmic pproach, Wiley, hichester uthors Biographies: Seung Kyoung hoi is a researcher of the department of Statistics at Soomyung Women s University in Seoul, Korea She received her MS in 00 and PhD in Statistics in 006 from the same university Moon ee hoi received her BS in 004 and MS in Statistics in 009 from Soomyung Women s University in Seoul, Korea ye Sun Lee received her BS in 007 and MS in Statistics in 009 from Soomyung Women s University in Seoul, Korea Eui Yong Lee is a Professor of the department of Statistics at Soomyung Women s University in Seoul, Korea e received his PhD in Statistics from the SUNY at Stony Broo in 988 e was an ssistant Professor at Wright State University from 988 to 990 fter coming bac to Korea he taught at POSE for nine years until he joined Soomyung Women s University in 999