AN ANALYSIS OF EQUITY IN INSURANCE. THE MATHEMATICAL APPROACH OF RISK OF RUIN FOR INSURERS

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1 Iulian Mircea AN ANALYSIS OF EQUITY IN INSURANCE. THE MATHEMATICAL APPROACH OF RISK OF RUIN FOR INSURERS A.S.E. Bucure ti, CSIE, Str.Mihail Moxa nr. 5-7, Tel Paul T n sescu A.S.E. Bucure ti, F.A.B.B.v., Str.Mihail Moxa nr. 5-7, floritanasescu@yahoo.co, tel Abstract: The goal of the present paper is a short analysis for the insurers foreign equity in Roania and the developent of a atheatical approach for the chronological evolution of the study regarding the insurers equity fro the point of view of assessing the insolvency probabilities and the ris provision so the estiating insolvency ris will not over overcoe an accepted value. Key words: Insurer, broer, adjusting coefficient, overcharging factor, probability of ruin, copound Poisson repartition. Introduction Since 989, Roania faced a developent of private property and a concentration of indigene capital, facts that deterined continuous growth of the insurance aret. In a first phase, the perissible legal fraewor (Law no. 47/99) and the lac of the Roanian capital but also the attractiveness of the Roanian aret were the ain factors for the aret penetration of the foreign capital on the Roanian insurance aret. The sae phenoenon too place in the insurance ediation sector. The nuber of insurers reached a noral evolution, with a spectacular growth at the beginning of the insurance aret, followed by a decrease according to the concentration and centralization of the capital, as shown below: (P is the percent of foreign capital to equity of the insurers). Year Nuber of insurers P % , , , , , , , ,0 Source: - OSAAR Reports, Bucharest, years ; - ISC Reports, Bucharest, years ; As it can be seen, a significant share of equity is represented by the foreign capital, ainly Austrian or Geran. Since year 007, it is evident a foreign capital doination due to the fact that the first 6 insurers have a aret share of ore than 70%. Since year 000, there are changes in the sales anageent for the insurance products, so the insurance agent is replaced by the insurance broer. The law 3/000 sets a clearer view on ediation sector, especially for broers. This is supposed to be a legal person, to prove 0

2 cash equity, to have valid third party liability insurance, and the only subject of activity the insurance ediation, to have qualified huan resources. The evolution for the nuber of broers is shown below: Year Nuber of broers Source: O.S.A.A.R. and ISC Reports, Bucharest, years It is iportant to assess the ris that an insurance copany faces banruptcy. In order to explain this, we shall provide a atheatical odel regarding the insurer s capital and we shall deterine the ruin probability, this eans the probability that the insurer has no ore resources to pay the indenities (obviously this isn t a de facto ruin, but an attention for the financial anageent).. The odeling of ris of ruin In order to study the variation of insurer s equity, we shall note with U(t) the insurer s equity at oent t, with u the initial capital, with S(t) the total clai deand (ore precisely the aount of total clai deand) occurred till the oent t, with the nuber of insurance policies, with N ( t ) the nuber of individual clai deands till the oent t for a given policy type, with Y i the individual deand nuber i for the insurance type and with S ( t ) the clai deand occurred till the oent t for a given policy type. We have: = and ( ) ( ) ( ) S t S t = N ( t) S t = Y. i= Additionally, we presue the following hypothesis to be fulfilled: i. there are no other expenses than the paid indenities and no other incoes than the cashed preius.. the unit incoe is c. 3. the individual clai deands for every policy type are independent identically distributed rando variables. The individual clai deands for every insurance type are independent and identically distributed variables 4. the stochastic processes ( ( )),, N t =, for the nuber of deands are Poisson t processes of paraeter λ. It results that process {S(t)} t is a copound Poisson process N t coposed of paraeter λ λ S t = Xi, where rando variables = i= X i are independent and identically distributed, having the oent of order noted with, ( ) =, so we have ( ) ( ) = M X. The variables X i equalize atheatically the individual real clai deands M ( S( t)) Y i, but there are not identically distributed. In these hypothesis, we have = λ t and D ( S( t)) λ t =. 5. The insurance preius are deterined with respect to the ediu value principle, with an overcharging relative confidence : c = ( + θ ) M ( N ()).

3 We have: U ( t ) = u + c t S ( t) (). We shall nae ruin the situation this capital is negative and we shall consider the oent of ruin at the oent this happen for the first tie. So we are dealing with the proble of ruin for a given, undeterined period of tie, called finite horizon. So: T = inf{ t > 0 U ( t ) < 0} is the oent of ruin. In this context, the ruin probability (noted with Ψ ) is the probability that the oent of ruin is finite. Fro the point of view of initial capital u, the overcharging factorθ, the clai files rate λ and the ediu individual clai deand, the function is: Ψ ( u, θ, λ, ) = P( T < ). r X The equation (in r) strictly positive solution: λ + c r = λ M ( e ) This coefficient (when does exist) shall be noted as R. Considering an existing adjustent coefficient R, we have: ψ = M e e is called adjustent coefficient. R u R U ( T ) ( T ) If the adjustent coefficient R does exist, then ( ) R u <. Ψ u (this is nown as the Craer inequality). This is deterined by the fact that for a finite T (T < ), we have U(T) < 0, so R U ( T ) M ( e T < ) >. Generally speaing, because the adjustent coefficient is difficult to copute, we see for a convenient R we θ obtain: R <. Considering the function (r) = ( X (r) ) c r which is strictly convex and due to the fact that net profit condition is fulfilled ( '(0) = c < 0), the results are (R) = (0) + R R c λ ω' ( s) ds > R c R + and R <. 0 λ R X interval, whose argins are used in the Craer inequality. Because M ( e ) > + R + If the individual deands have an exponential repartition of paraeter (so = ), because the ediu incoe per unit tie is c ( θ ) ψ e = + λ, we find for the ruin probability the expression: λ θ u + θ λ u c + θ c ( u, θ, λ, ) = e = e Considering the initial capital as a β nuber of ediu individual clai deands, we obtain: θ β θ ψ ( β, θ ) = e + (3). + θ We could iprove this odel by setting the proble to copute a ris provision (V), in order to define the probability that paid indenities should not overcoe the incoes plus this provision, under an accepted value p. If the ediu nuber of clai deands (in our case n = λ t ) is big enough, we can use the central liit theore (TLC) and approxiate S ( t ) with a noral variable of ean λ t and dispersion t λ. So we want to deterine V in order that ( ( ) ) (). P S t u + c t + V p. Using TLC

4 V z λ t u θ λ t, where p we obtain p z is the quartile of order p of the noral repartition. If we choose the iniu ris reserve in order not to bloc higher financial resources, we have: p λ θ λ (4) V = z t u t In the hypothesis that individual clai deand follows an exponential repartition of paraeter, so =, =, we have: λ t θ λ t V = z p u (5) Choosing an initial equity to represent a nuber of individual ediu clai deands ( β ), we obtain: z p λ t β θ λ t V = 3. Nuerical results Fro relation (3) we have deterined the values for ruin probabilities for a few values of preius overcharging factor θ and paraeter β, that provide the nuber of ediu individual deands to cover the initial equity. The results are listed in the table below: θ \ β (6) Obviously, the higher the initial equity and the bigger the overcharging preius are, the saller the ruin probability is. The diinution of ris of ruin realizes exponentially copared to the growth of those two factors. The increasing overcharging factor is liited by the necessity to aintain an adequate preiu level for the copetitiveness of the insurance copany on aret. We also ephasize this odel is correlating to strong the incoes and the indenities, the ediu incoe per unit tie being proportional to the ediu nuber of individual clai deands ultiplied by the value of ediu individual clai deand. This is due to the principle of ediu value for the preius coputation. We consider the use of ediu annual clai index instead of clai deand in order to obtain a ore realistic odel. Using relation (6) we obtained the following results: for p = 0.005, = 0.00, λ =, β =, values of t and β, the ris reserve V ( u..) is provided in the following tabel: t \ θ

5 for p = 0.00, = 0.0, λ = 5, θ = 0., values of t and β, the ris reserve V ( u..) is provided in the following tabel: t \ β We underline that the negative values for the relations (4)-(6) ean a ris reserve equal to zero. We also ascertain the ris reserve to be proportional to the clai ediu individual deand, decreasing when the initial equity covers any clai individual deands and decreasing when the requested argin for the insolvency ris decreases. References. Asussen, S. (000). - Ruin Probabilities, World Scientific, Singapore;. Dan-Constantin D nule iu (007)- Asigur ri coerciale, Ed. Risoprint, Cluj Napoca. 3. Mircea I. (006).- Mateatici Financiare i Actuariale, Ed. Corint, Bucure ti. 4. Mircea, I., erban, R. (006). - Evaluating the ruin probability of insurance, Econoic Coputation and Econoic Cybernetics Studies and Research, Vol. 40, no. 3-4, pp ; 5. T n sescu P., Dobrin M.(003) - Teoria i practica asigur rilor, Ed. Econoic, Bucure ti. 6. T n sescu P. i colectiv (007). Asigur ri coerciale oderne, Editura C.H.Bec, Bucure ti. 7. Zbaganu, Gh.(007). Eleente de teoria ruinei, Geoetry Balan Press, Bucharest. 4