N = N t ; t 0. N is the number of claims paid by the

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1 Iulan MICEA, Ph Mhaela COVIG, Ph Canddate epatment of Mathematcs The Buchaest Academy of Economc Studes an CECHIN-CISTA Uncedt Tac Bank, Lugoj SOME APPOXIMATIONS USE IN THE ISK POCESS OF INSUANCE COMPANY Abstact: In an nsuance company, the sk pocess estmaton and the estmaton of the un pobablty ae mpotant concens fo an actuay: fo eseaches, at the theoetcal level, and fo the management of the company, as these nfluence the nsue stategy. We consde the evoluton on an extended peod of tme of an nsue suplus pocess. In ths pape, we pesent some pocesses of estmatng of the un pobablty. We dscuss the appoxmatons of un pobablty wth espect to the paametes of the ndvdual clam dstbuton, wth the load facto of pemums and wth the ntensty paamete of the numbe of clams pocess. We analyze the model whee the pemums ae computed on the bass of the mean value pncple. We gve numecal llustaton. Keywods: un pobablty, sk pocess, adjustment coeffcent. JEL Classfcaton: C, G, G3. INTOUCTION The mathematcal model of an nsuance sk busness s composed of the followng objects: X = K of ndependent and dentcally dstbuted andom a) A sequence { },,3, vaables (..d..v.), havng the common dstbuton functon F. X s the cost of the -th ndvdual clam. N = N t ; t. N s the numbe of clams pad by the b) A stochastc pocess { } company n the tme nteval [,t ]. The countng pocess N and the sequence { X } ae ndependent objects.

2 Iulan Mcea, Mhaela Covg, an Cechn-Csta The total amount of clams pad by the company n the tme nteval [,t ] s ( t) N ( t ) = X ( ( t ) = { ; } t beng the clam pocess). The sk pocess = { ; } s defned by Y( t) c t ( t) Y Y t t =, whee c s a postve eal constant numbe coespondng to the aveage pemum ncome pe unt of tme. We shall consde that the pocess N s a homogeneous Posson pocess wth ntensty λ and that we wll use the mean value pncple n ode to compute the net pemums, thus c = ( + ) λ m, whee s the elatve safety loadng and m s the expected value of the ndvdual clam o the expected cost of a clam. We denote k mk = E X, k =,,3, K. We shall denote by ntal captal, C( t ) - the captal of the company at moment t, hence = C( ) and C( t) = + Y( t). We defne the un as beng the stuaton when the captal of the company takes a negatve value. The τ = nf t C t <. { } un moment τ s defned as and g( ) ( ) λ m x Let h( ) = ( e ) df( x) = +. We denote by (, ) the un pobablty up to moment n and by (, ) n pobablty on an nfnte tme hozon, so n(, ) = P τ < n C ( ) =, g ( ) = c, (, ) = P τ < C( ) =, g( ) = c and (, ) lm (, ) =. n n the un The adjustment coeffcent (o Lundbeg exponent) s the postve soluton of the equaton λ h( ) c =. A well-known esult s that: f the adjustment coeffcent exsts, the un pobablty S( τ) s (, ) = e ( E e ), ( ) whee S( τ ) = ( C( τ ) τ < )) epesents the sevety of the loss at the moment of un. In case the ndvdual clam follows an exponental dstbuton, X Exp( α), α > then λ α λ g( ) (, ) = e α g

3 Some Appoxmatons Used n the sk Pocess of Insuance Company FI z F x dx m. It s known the The ntegated tal dstbuton s = ( ) z Beekman s convoluton (o Pollaczek-Khnchne) fomula: n n (, ) = F % I ( ) ( 3) + n= + F% = F. If F s the cumulatve dstbuton functon fo a whee I exponental dstbuted loss, then F = F. I. SOME APPOXIMATIONS OF UIN POBABILITY I e Vylde (978) poposed the followng appoxmaton, whch s based on the dea to eplace the sk pocess Y by a sk pocess Y % wth exponental dstbuted clams such that k k E Y ( t) E Y ( t) % =, k =,,3. The sk pocess Y % s % λ, %, % α. Snce ln z ln = +, whee z= e, detemned by the paametes we have 3 Y t 3 ν λ λ λ 3 ( ν ) ln E ν ν ν e = t m m + m + and we get 3! 3 m3 m% = α =, % m m3 = and % 9 m λ= λ. Thus, we obtan the % 3 m 3 m m appoxmaton (, ) (, ) e + % = ( 4) 3 m = 3 m + m m n If N n ': n X, X ": a % 3 + % 6 m m + % % α 3 m m m, 3 e ( 5) 3 X :, p >, p =, we ntoduce p I I I E( X ") E( X ) I % 9 E X λ = λ and ' and we obtan m% b E( X ) a m m b = n p, = n p, ( ') E X = ', % =, 3 3 E X "

4 Iulan Mcea, Mhaela Covg, an Cechn-Csta 3 ( ") ( ') (, ) E X + E X = e 6 3 E X " + ' 3 E( X ") E( X ) 6 The most famous appoxmaton s the Came-Lundbeg appoxmaton: λ m (, ) CL(, ) = e ( 7) λ h ' c Let SE be the class of subexponental dstbuton,.e. F SE f s shown by Embechts and Veovebeke (98), that, f = % + We showed that f SE then ( x) F% lm =. It x F% x FI SE, then (, ) F ( ) ( 8) FI (, ) = I Anothe well known esult s lm (, ) C= x x e F% x dx m I λ m lm 9 F g λ m e C =, whee s Came constant and s Lundbeg exponent. m m 3m Fo small we have < <, so that we may have m m m. t t = nf nf >. Let H( ) P Y( t) Y( t) + We have (, ) = ( H( ) ) ( + ) m m ( ), µ H = ( + ) m m 3 m σ H = +, whee µ H and σ H ae the mean and the m 3m m vaance coespondng to H. The dea of Beekman-Bowes appoxmaton s to eplace H( ) wth a Γ -dstbuton functon G( ), such that the two fst moments of H and G should match. It ths way, t s obtaned the appoxmaton fomula and

5 Some Appoxmatons Used n the sk Pocess of Insuance Company whee a x (, ) (, ) = x e dx + Γ ( ) 3 + m + 3 a= 3m 4m m 3m ( ) ( a) and b 6m m b= 3m 4m m 3m + 3 In the case of exponentally dstbuted clams we have a=, α. α b= and + (, ) = e + = (, ). + The smplest appoxmaton, whch only depends on some moments of F, seems to be the dffuson appoxmaton: m m (, ) (, ) e ( ) = As the Lundbeg exponent s small fo small values of we have m 3 h( ) = m + + o( ) whch leads to m o = m + and λ m CL(, ) = e = e λ h ' c + o ( ) m + o m Thus the dffuson appoxmaton may be egaded as a smplfed Came-Lundbeg appoxmaton. Snce the dffuson appoxmaton s not vey accuate, Gandell () m 4m m3 3 poposed to use fo h( ) thee moments. Thus = + o 3 ( ), m 3m 3m 3 λ m = + o λ h ' c 3m + m m ( ) 3m = and ( ) m 4m m3. 3 m 3m, e ( ) G 3m + mm 3 4m m3 6mm m Snce = + o( ) the e Vylde appoxmaton m m m m m may be egaded as a smplfed Gandell appoxmaton. Anothe appoxmaton s obtaned usng eny s theoem about the p-thnnng of the pont pocess. Thus

6 Iulan Mcea, Mhaela Covg, an Cechn-Csta m m ( + ) (, ) = e 3 + 4mm 3 sup,, 3m + Kalashnkov (997) showed that fo all >. 3. NUMEICAL ESULTS AN CONCLUSIONS ( ) In ths secton, ou pupose s to compae the dffeent appoxmatons lsted above, though a numecal example. We have to deal wth the absolute eo (δ ) and the elatve eo ( ε ). Thus, fo appoxmaton of we have A (, ) = (, ) (, ) and ε ( ) δ A A A (, ) (, ) δa, =. In ode to compae an appoxmaton A wth anothe appoxmaton B, we wll use δab(, ) δab(, ) = A(, ) B(, ) and ε AB(, ) =., Let : 5 ( ) A X be a dscete andom vaable descbng a clam (a loss) whch takes the value of monetay unt a hgh pobablty, and a elatvely lage value of 5 monetay unts (n the case of a natual dsaste, fo example) wth a elatvely low pobablty. In the followng, we lst the appoxmatons of un pobablty, AP (see 6 table -4). Fo gaphc llustaton (see fgue -4) we compute MAP= AP, whch s shown n fgues.

7 Some Appoxmatons Used n the sk Pocess of Insuance Company = AP fo. Table MAP fo = M M M M Fgue

8 Iulan Mcea, Mhaela Covg, an Cechn-Csta Table AP fo = MAP fo = M M M M Fgue

9 Some Appoxmatons Used n the sk Pocess of Insuance Company AP fo =.5 Table MAP fo = M M M M Fgue 3

10 Iulan Mcea, Mhaela Covg, an Cechn-Csta Table 4 AP fo = MAP fo = M M M M Fgue 4 5 As we llustate elatve eo (see tables 5-6) we compute ME= ε,, 5 ME ε, 5 = and ME3= ε, whch ae shown n fgues 5-6.

11 Some Appoxmatons Used n the sk Pocess of Insuance Company Table 5 ε, fo.3 ε, ε, ε, ME fo = ME ME ME3 Fgue 5

12 Iulan Mcea, Mhaela Covg, an Cechn-Csta Table 6 ε, fo =.5 ε, ε, ε, ME fo = ME ME ME Fgue 6 We consde that the appoxmatons, and =, up to.3 fo.3 ae bette. They ae maxm =, up to.9 fo absolute eos up to.5 fo. =.5 and maxm elatve eos up to.7% fo =., up to 5.5% fo =.3 etc. We obseve that the elatve eos ncease when ntal captal nceases whle the

13 Some Appoxmatons Used n the sk Pocess of Insuance Company un pobablty deceases. The dffuson appoxmaton s not vey accuate because ts elatve eos ae vey hgh, up to 99% f =.5. Fo the epatton of clam, whch we used thee ae no exact un pobabltes avalable. EFEENCES [] Asmussen, S. (), un Pobabltes, Wold Scentfc, Sngapoe; [] Badescu, A., et al. (5), Phase-Type Appoxmatons to Fnte un Pobabltes n the Spae-Andesen and Statonay enewal sk Model, ASTIN Bulletn, vol. 35, no., pp. 3-44; [3] Badescu, Ande, Badescu Adan (7), Saddlepont Appoxmatons fo un Pobabltes, Jounal of Economc Computaton and Economc Cybenetcs Studes and eseach, ssue 3-4, ASE Publshng House, Buchaest; [4] e Vylde, F. (978), A Pactcal Soluton to the Poblem of Ultmate un Pobablty; Scandnavan Actuaal Jounal, pp. 4-9; [5] ckson,., Wllmot, G. (5), The ensty of the Tme to un n the Classcal Posson sk Model, ASTIN Bulletn, vol. 35, no., pp. 45-6; [6] ekc, S., et al. (4), On the stbuton of the efct at un when Clams ae Phase-type, Scandnavan Actuaal Jounal, 4, pp. 5-; [7] Embechts, P., Veavebeke, N. (98), Estmates fo the Pobablty of un wth Specal Emphass on the Possblty of Lage Clams,, Insuance: Mathematcs and Economcs,, pp. 55-7; [8] Gandell, J. (), Smple Appoxmatons of un Pobablty of Insuance; Mathematcs and Economcs, 6, pp ; [9] Mcea, I. (6), Matematc fnancae ş actuale (Fnancal and Actuaal Mathematcs), COINT Publshng House, Buchaest; [] Mcea, I., Şeban,. (6), Evaluatng the un Pobablty of Insuance, Economc Computaton and Economc Cybenetcs Studes and eseach, Vol. 4, no. 3-4, pp ; [] Stanfod,., et al. (), un Pobabltes Based on Clam Instants fo some Non-Posson Clam Pocesses Insuance. Mathematcs and Economcs, 6, pp. 5-67; [] Zbaganu, Gh. (7), Elemente de teoa une (Elements of un theoy) Geomety Balkan Pess, Buchaest.