The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time Surplus Processes Sootor Boota,, Siriya Progjit, Pesiri Sompog ad Wachara Teparos Departmet of Geeral Sciece, Faculty of Sciece ad Egieerig Kasetsart Uiversity, Chalermphrakiat Sako Nakho Provice Campus, Sako Nakho 47000, Thailad Abstract I this paper, we compare the differet of miimum iitial capital obtaied from two discrete time surplus processes: ivestmet ad o-ivestmet surplus processes. I case that a isurace compay is allowed to ivest i a risk-free asset with a costat iterest rate uder a rui probability which is ot exceed 0.. The rui probability of discrete time surplus process is obtaied by simulatio approach ad miimum iitial capital is approximated by regressio aalysis. From the study results, it is show that uder the rui probability which does ot exceed 0., miimum iitial capital ca be reduced by ivestmet surplus process compared with o-ivestmet surplus process. Keywords: discrete time surplus process, rui probability, miimum iitial captial. 200 MSC: 9B30. tatios. { Y, } is a sequece of idepedet ad idetically distributed radom variables (i.i.d.), Y is claim size occurs at time T for. 2. c is premium rate. 3. U is discrete time surplus process,. 4. Φ ( u ) is the rui probability, ad u is the iitial capital. Itroductio A surplus of o-life isurace defied i a probability space (, FP, ) Surplus = Iitial capital + Icome Outflow. Ω cab be described by Correspodig author. Speaker. E-mail address: sootor.bo@ku.th (S. Boota), pesiri.so@ku.th (P. Sompog) Proceedigs of AMM 207 MIS-04-
I 2006, Cha ad Zhag [] cosidered the discrete time surplus process at time T = k, 0 ;, () k = U = u + c Y U = u where u 0 is the iitial capital, c is icome at time T = ad { Y, } is a sequece of outflow. They cosidered two cases of Y. First, Y is assumed to be a sequece of expoetial radom variables with expoetial desity fuctio give by f( x) = βe β x. They proposed the recursive ad explicit formulas of rui probabilities such that U falls below zero for the first time as follows. For each, ( βc ( u) ) c u ( βc ( u) ) k βc( u) k βck ( u) ( ) c ( u) Φ ( u) = Φ ( u) + e = e (2) ( )! c ( u) ( k )! c ( u) where c ( u) = u + c. Secod, Y is assumed to be a sequece of geometric radom variables ad Φ ( u ) = P ({ Ui () < 0 for some i {,2,..., }} U (0) = u ). The geometric mass fuctio is defied by j p j = P( Y = j) = pq, j 0, where p+ q =, 0 < p <. For simplicity, they set premium rate c = i surplus process () ad proposed recursive formula of rui probabilities by the followig for 2, ad k ( ) + u is defied by k = Φ ( u) = q, Φ ( u) =Φ ( u) + k ( u), (3) u+ 2 + + k f( u)[ f( u) + ( + )][ f( u) + ( + 2)] [ f( u) + (2 )] k u pq f u u! f ( u) + + ( ) =, ( ) = + 2. (4) I 203, Sattayatham et al. [2] geeralized the results of Chag ad Zhag [] by cosiderig the discrete time surplus process () such that Y is ay sequece. The recursive formula of rui probability is defied by u+ c Φ ( u) =Φ ( u) + Φ ( u + c y) df ( y). (5) Y It ca be see that Φ ( u ) show i (2), (3) ad (5) are i recursive form ad difficult to compute. They approximated the miimum iitial capital uder the coditio that rui probability was ot greater tha a give quatity. This miimum iitial capital is approximated by applyig the bisectio techique which is a umerical method. If we set U : = UT ( ) for all = 0,, 2,3, where U0 = u i () ad defie the iter arrival time Zk = Tk Tk for k, the () ca be writte i the form of U = U + cz Y U = u (6),. 0 I particular, if Z = for all (claims happe everyday), the (6) is of the form U = U,. + c Y U = u (7) 0 I this paper, if isurace compay is allowed to ivest i a risk-free asset with a costat iterest rate r (buy bod or ope fixed deposit accout), a costat iterest rate r from ivestmet will be icluded i the surplus process (7) ad the surplus process ca be writte by U = U ( + r) + c Y, U = u, (8) 0 Proceedigs of AMM 207 MIS-04-2
where r is a daily iterest rate give by ( ) 365 r = + r0, r 0 is compoud iterest rate with r 0 = 2% per aum. The values of c i (7) ad (8) are computed by expected value premium priciple, i.e., c = ( + θ ) EY, θ is called safety loadig. The objective of this paper is to study the relatioship betwee iitial capital, rui probability ad miimum iitial capital (MIC) that a isurace compay has to hold to esure that the rui probability is ot greater tha 0. i surplus process (7) ad (8). 2 Mai Results 2. Data Aalysis Data set of motor isurace claims i 2009 of isurace compay i Thailad is used i this paper. Figure shows claims size that the claims happe everyday. The ull hypothesis H 0: data is assumed logormal distributio ( µ =.0630, σ = 0.9939 ). We foud that the chi-squared value (5.8476) is less tha the chi-squared critical value (4.07) for degree of freedom 7 at a sigificace level of 0.05. Thus claims size is accepted logormal distributio at 95% cofidece. Figure. Claim size Y Next, parameters of logormal distributio, µ ad σ, are improved by miimizatio of the chisquared value with radomized eighborhood search approach proposed by Boota et. al.[3]. By this approach, we obtai µ =.084 ad σ = 0.9388 at 95% cofidece. 2.2 Simulatio Results I this part, we cosider (7) ad (8) uder the coditio that days. A simulatio method is used to compute the rui probability, T T T whe T = 365 0 P { U < 0 }, Zi 365 U0 = u. i= The flowcharts for computig the rui probabilities of (7) ad (8) are listed i Figure 2. Iitial capital u = 0, 20,000, 40,000,...,,080,000 Baht are set ad 0,000 simulatios with = 365 are computed. Proceedigs of AMM 207 MIS-04-3
Start NN= the umber of simulatios uu=iitial capital, θθ = safety loadig A j=j+ j=n i= B yy = 0, uu = 0 i=i+ i 365 A yy ii = log ormal(,,.084, 0.9388) uu ii = uu ii + ( + θθ)eeyy yy ii (eq. (7)) uu ii = uu ii ( + rr) + ( + θθ)eeyy yy ii (eq. (8)) uu ii < 0 cout = cout+ j=n A B Rui Probability = cout/nn Ed Figure 2. Flowchart for computig the rui probability of equatio (7) ad (8). Proceedigs of AMM 207 MIS-04-4
2.3 Regressio Aalysis ad Miimum Iitial Capital Regressio Aalysis Liear regressio is method for calculatig the best fit straight lie. Let data poits ( xi, yi) for i =,2,,. The equatio of the straight lie that best fits these poits is y = mx + t where m is slope ad t is y -itercept. We obtai ad xy x y m = i i i i i= i= i= xi ( xi) i= i= t = 2 xi yi xi xy i i i= i= i= i= xi ( xi) i= i= Miimum Iitial Capital From the simulatio results (Usig method i Figure 2), relatio betwee iitial capital ad rui probability of o-ivestmet ad ivestmet cases are show i Figure 3 ad Figure 4, respectively. The top curve is plotted for θ = 0., the ext curve is plotted for θ = 0.2 ad so o. Figure 3. -ivestmet case Figure 4. Ivestmet case The relatios betwee rui probability ( u ) ivestigated by expoetial fuctio Φ,365 : = s that obtai from 2.2 ad iitial capital u is By takig the atural logarithmic fuctio to (9), we get s = a exp( bu). (9) w b is slope ad l a is y -itercept. l s = bu + l a. Proceedigs of AMM 207 MIS-04-5
Applyig the least squares liear regressio method (settig l s = y, u = x), the approximated parameters a ad b are obtaied as follow. a = exp 2 ui l si ui ui l si i= i= i= i= ui ( ui) i= i= (0) u l s u l s b = i i i i i= i= i= ui ( ui) i= i= () where u i is iitial capital ad s i is rui probability at iitial capital u i. If the rui probability does ot greater tha α ( (,365) Φ u = s α ), i.e., a exp( bu) α (from (9) we have s = a exp( bu) ), the α u l b a. Therefore, the miimum of u is α l. Recall the miimum iitial capital i this b a paper deoted by MIC, so α MIC = l. b a (2) MIC of o-ivestmet ad ivestmet cases with α = 0. are show i table. Table MIC (Baht) i case ivestmet ad o-ivestmet with α = 0.. safety loadig θ MIC (-ivestmet) MIC (Ivestmet r 0 =2%) Reducig of MIC (%) 0. 2,278,705,697,269 25.52 0.2,323,995 955,322 0.3 964,296 67,83 0.4 778,375 54,094 0.5 655,49 40,350 0.6 562,895 336,05 0.7 493,59 273,867 0.8 438,277 223,894 0.9 385,465 73,99 34,55 27,975 27.85 30.33 33.95 37.40 40.30 44.47 48.9 55.07 62.49 Proceedigs of AMM 207 MIS-04-6
3 Coclusio The results show i table idicate that a isurace compay uses strategy of ivestmet with iterest rate r 0 = 2% per aum. Uder the rui probability which does ot exceed α = 0., the MIC of ivestmet surplus process is less tha MIC of o-ivestmet surplus process for all safety loadig θ. It is cocluded that ivestmet surplus process ca be reduce the MIC compared with o-ivestmet surplus process. Refereces [] W. Cha ad L. Zhag, Direct derivatio of fiite-time rui probabilities i the discrete risk model with expoetial or geometric claims, rth America Actuarial Joural, 0 (2006), 268-279. [2] P. Sattayatham, K. Sagaroo ad W. Klogdee, Rui probability-based iitial capital of the discete-time surplus process, Variace: Advacig the Sciece of Risk, 7 (203), 74-8. [3] S. Boota, A. Sattayatham ad P. Sattayatham, Estimatio of Weibull parameters usig a radomized eighborhood search for severity of fire accidet. J. Mathematics ad Statistics, 9 (203), 2-7. [4] W. Klogdee, P. Sattayatham ad S. Boota, O approximatig the rui probability ad the miimum iitial capital of the fiite-time risk process by separated claim techique of motor isurace. Far East Joural of Mathematical Scieces, Special Vol (203), 579-604. Proceedigs of AMM 207 MIS-04-7