ABSTRACT 1. INTRODUCTION

Transcription

1 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES FOR A LARGE PORTFOLIO WITH VARYING PREMIUMS BY LOURDES B. AFONSO, ALFREDO D. EGIDIO DOS REIS AND HOWARD R. WATERS ABSTRACT In ths paper we present a method for the numercal evaluaton of the run probablty n contnuous and fnte tme for a classcal rsk process where the premum can change from year to year. A major consderaton n the development of ths methodology s that t should be easly applcable to large portfolos. Our method s based on the smulaton of the annual aggregate clams and then on the calculaton of the run probablty for a gven surplus at the start and at the end of each year. We calculate the wthn-year run probablty assumng a translated gamma dstrbuton approxmaton for aggregate clam amounts. We llustrate our method by studyng the case where the premum at the start of each year s a functon of the surplus level at that tme or at an earler tme. 1. INTRODUCTION Ths paper presents a method for calculatng the probablty of run n contnuous and fnte tme for a compound Posson rsk process where the premum rate s constant throughout each year but can change at the start of each year. We are nterested n the case where the premum depends on the past aggregate annual clams experence and, n partcular, where t depends on the surplus level at the end of the prevous year or at some earler tme. Our method nvolves smulatng the aggregate clams for each year, calculatng the premum to be charged each year gven the past aggregate clam amounts, and then calculatng the wthn year probablty of run assumng a translated gamma dstrbuton approxmaton to the surplus process. The translated gamma approxmaton uses deas whch go back at least to Seal (1978a) and has been used more recently by Dckson and Waters (1993, 2006). It has ts roots n Bohman and Esscher (1963). In Secton 2 we set out our model and general procedure for calculatng run probabltes n contnuous and fnte tme. In Secton 3 we gve detals of the translated gamma approxmatons we use to calculate the probablty of run wthn each year, gven the surplus at the start and at the end of the year. Detals of our procedure to smulate run probabltes are gven n Secton 4. Astn Bulletn 39(1), do: /AST by Astn Bulletn. All rghts reserved.

2 118 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS Wkstad (1971) and Seal (1978b) gve values for the probablty of run n fnte and contnuous tme for some examples of a classcal rsk process (wth constant premums). In Secton 5 we check the accuracy of our methodology by applyng t to these examples and comparng our values wth thers. In Secton 7 we apply our method to a rsk process where the premum at the start of each year depends on the current or a past level of the surplus. We consder several cases. We start by assumng that the premum for the comng year depends on the surplus at the end of the precedng year,.e. on the current surplus. Ths s ntutvely appealng but may not be practcable, snce t requres the nsurer to determne and charge the new premum nstantaneously. In practce there may be some delay n settng a new premum rate so we also consder the case where the premum n the comng year depends on the surplus one year ago. In both cases, the hgher the surplus, the lower wll be the premum. The premum rate s set each year so that the probablty of ultmate run from that tme s always (approxmately) equal to a pre-determned value. We do ths usng De Vylder s (1978) approxmaton; detals are gven n Secton 6. Our model s set n such a general way to make t easly adaptable to dfferent stuatons. For nstance, changng the premum strategy or weakenng the homogenety of rsk condtons. The problem of calculatng the probablty of run when the premum s a functon of the surplus level at the end of the year has been studed by many authors, n most cases n nfnte tme. For example, Davdson (1969) let the safety loadng decrease wth an ncreasng rsk reserve. Taylor (1980) and Jasulewcz (2001) consdered the case where the premum rate vares contnuously as a functon of the surplus. Petersen (1989) llustrates wth a smple numercal method how the probablty of run can be calculated when the general premum rate depends on the reserve. Dckson (1991) consdered the case where the premum rate changes when the surplus crosses an upper barrer. More recently, Cardoso and Waters (2005) presented a numercal method for calculatng fnte tme run probabltes for the same problem. In a recent paper Malnovsk (2008) provdes an analytcal soluton to a problem smlar to ours, but only n the case where clam amounts are exponentally dstrbuted. The use of smulaton to estmate run probabltes s not new. Dufresne and Gerber (1989) observed that the probablty of run s related to the statonary dstrbuton of a certan assocated process and estmated t usng smulaton. Mchaud (1996) smulated the jumps and nterjump tmes for two models n order to approxmate the probablty of ultmate run. In hs frst model the surplus earns nterest. In hs second model the premum rate changes n contnuous tme, decreasng as the surplus ncreases, but wthout any ratonale gven for the dfferent levels of the premum rate. 2. THE MODEL We consder a rsk process over an n-year perod. We denote by S(t) the aggregate clams up to tme t, so that S(0) = 0, and by Y the aggregate clams n year,

3 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 119 =1,,n, so that Y = S() S( 1). We assume that {Y } =1 n s a sequence of..d. random varables, each wth a compound Posson dstrbuton whose frst three moments exst. We denote by l the Posson parameter for the expected number of clams each year and by f (, s) the probablty densty functon (pdf ) of S(s) for 0 < s # 1. Let P denote the premum charged n year and let U(t) denote the nsurer s surplus at tme t,0# t # n. We assume premums are receved contnuously at a constant rate throughout each year. The ntal surplus, u(= U(0)), and the ntal premum, P 1, are known. For =2,,n, we assume that P s a functon of {U( j)} 1 j=1, the surplus at the end of each of the precedng years. For any tme t, 0# t # n, U(t) s calculated as follows: U(t) =u P j j = 1! +(t +1) P S(t) (2.1) where s the nteger such that t! [ 1,), and where! P = 0. For $ 2, the premum P and surplus level U() are random varables snce they both depend on the clams experence n prevous years. Where we wsh to refer to a partcular realsaton of these varables, we wll use the lower case letters p and u(), respectvely. The probablty of run n contnuous tme wthn n years s denoted by c(u, n) and defned as follows: 0 j = 1 j c(u,n) = def Pr(U(t) < 0 for some t! (0,n]) Let c(u( 1),1, u()) be the probablty of run wthn year, gven the surplus u( 1) at the start of the year, the surplus u() at the end of the year and the rate of premum ncome p durng the year. We wll develop a formula for c(u( 1),1, u()) followng methods n Dckson and Waters (2006, Secton 3.2). Let D(u( 1),1,z) denote the probablty that, startng from a surplus of u( 1), run does not occur n the year and the surplus at the end of the year s greater than z. Then: D(u( 1),1,y) = 3 # (1 c(u( 1),1,z)) f (u( 1)+p z,1)dz y and so: c(u( 1),1,y) =1+ 1 d f^u] - 1g + p - y, 1h dy D(u( 1),1,y) Usng formulae (3.11) and (3.13) from Dckson and Waters (2006) and wrtng y = u(), we have:

4 120 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS c^u ] - 1g, 1, u ] gh = # 1 - u()/ p u() ] s = 0 ( 1 - s) f^ u - 1g + p s, sh f^p ] 1- sg - u] g, 1- shds f^u] - 1g + p - u] g, 1h f^ u] - 1g + p - u] g, 1- u] g/ ph exp^-lu] g/ ph + f^u] - 1g + p - u] g, 1h (2.2) In prncple, formula (2.2) can be used to calculate values of c(u( 1),1, u()). We requre values of the pdf f(, s) for values of s from 0 to 1. Although these values can be calculated usng well known recursve formulae, the number of values requred can be prohbtvely large, partcularly f l s large, and so some approxmate method of calculaton s requred. In the next secton we gve detals of an approxmate method for calculatng ths probablty. 3. APPROXIMATE CALCULATION OF THE WITHIN-YEAR PROBABILITY OF RUIN In ths secton we gve detals of our translated gamma method for approxmatng the wthn-year probablty of run, c(u( 1), 1, u()). Afonso (2008) gves more detals of ths method and also detals of a dfferent method nvolvng approxmaton usng a Brownan moton process. Snce the latter produces generally nferor numercal results, no further detals are gven n ths paper. The translated gamma approxmaton to the probablty c(u( 1), 1, u()) s based on replacng each pdf n formula (2.2) by an approxmatng gamma densty chosen to match the frst three moments. Ths s based on the methodology n Dckson and Waters (2006, formulae (4.7) & (4.9)). Let a, b and k be such that: a b 2 E9^Y - E6Y@ h C = a 32 / a Var b 2 = 6 + k = E6 3 and let H(s) be a random varable wth a gamma dstrbuton wth parameters as and b. Then for 0 < s # 1 the random varable H(s)+ks has a translated gamma dstrbuton whose frst three moments match those of S(s). We can approxmate c(u( 1),1, u()) by replacng each compound Posson pdf n formula (2.2) by the pdf of H(s)+ks, wth the approprate value of s.

5 Let F G (,s) and f G (, s) denote the cumulatve dstrbuton functon and pdf of H(s), respectvely. In partcular, we need to replace: f (x,s) byf G (x ks, s) exp( lt) byf G ( kt,t) For ths last relatonshp, note that for the compound Posson process exp( lt) s the probablty of no clams n a tme nterval of length t. We approxmate ths by the probablty that H(t)+kt s negatve, whch s F G ( kt,t). Our translated gamma approxmaton to c(u( 1),1, u()), whch we denote by c TG (u( 1),1, u()), s gven by: c TG = ^u ] - 1g, 1, u ] gh # CALCULATING CONTINUOUS TIME RUIN PROBABILITIES u()/ p u() s = 0 G ] ^ k ( 1 - s) f _ u - 1g + p - h s, s f _ ^ p -kh] 1- sg - u] g, 1- sds f ^u] - 1g + p - k - u] g, 1h G G + f b u] - 1g + ^p -kha1- k, 1- l F ^-ku] g/ p, u] g/ ph G f ^u] - 1g + p - k - u] g, 1h G u() u() p p G (3.1) The advantage of usng c TG (u( 1),1, u()) as an approxmaton to c(u( 1), 1, u()) s that there are well establshed and fast algorthms for calculatng gamma denstes so that the former can be calculated far more quckly and easly than the latter. 4. SIMULATION OF RUIN PROBABILITIES Our goal s to estmate c(u,n). To acheve ths we smulate N paths of the surplus process (2.1) at nteger ntervals,.e. for t=1, 2,, n. Each path starts at u (=U(0)). The premum n the frst year, P 1 (=p 1 ), s gven. Let c j (u,n), j =1,2,,N, denote the estmate of c(u,n) from the j-th smulaton. Our procedure for calculatng c j (u,n) s as follows: () Smulate the values of {Y } n =1 by assumng each Y s approxmately dstrbuted as H(1) + k, where H(1) and k are as defned n Secton 3, so that H(1) has a translated gamma dstrbuton wth parameters a, b and k. () From the smulated values of {Y } n =1,say {y } n =1, calculate successvely u(1) (=u+ p 1 y 1 ), then p 2 (as a functon of u and u(1)), u(2) (=u(1) + p 2 y 2 ) and so on untl the surplus at the end of each year, {u()} n =0, has been calculated.

6 122 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS () If u() < 0 for any, =1,2,,n, then we set c j (u,n) = 1 and we start smulaton j +1. (v) If u() $ 0 for all, =1,2,,n, we approxmate c j (u( 1), 1, u()) by c TG (u(u 1),1, u()), as descrbed n Secton 3, and then calculate c j (u,n) as follows: n c j (u,n) =1 % (1 c TG (u( 1),1, u())) = 1 The mean of our N estmates, {c j (u,n)} N j =1, s then our estmate of c(u,n) and we can use the sample standard devaton of the N estmates to calculate approxmate confdence ntervals for the estmate. We wll denote our estmate c TG (u,n). A pont to note about ths procedure s that the sze of the portfolo, as measured by the Posson parameter, l, does not affect the scale of the calculatons. Hence, ths methodology can be appled as easly to large as to small portfolos. In fact, the larger the value of l, the more accurate the translated gamma approxmatons to the wthn-year run probabltes and the annual changes n surplus are lkely to be General comments 5. ACCURACY OF THE PROCEDURE Our methodology for estmatng the probablty of run n fnte and contnuous tme s based on two approxmatons: () We smulate the annual aggregate clams usng a translated gamma approxmaton. () We estmate the wthn-year probablty of run, gven the startng and fnal surplus and the rate of premum ncome, usng a translated gamma approxmaton. We would expect our method to gve reasonable approxmatons f the expected number of clams each year, l, s large and the ndvdual clam sze dstrbuton does not have too fat a tal. Wkstad (1971) and Seal (1978b) provde values of run probabltes n fnte and contnuous tme for some compound Posson rsk processes, n all cases wth a fxed premum rate. We can test the accuracy of our method by applyng t to ther examples. The run probabltes n ther examples range from practcally zero to almost 1. We restrct our comparsons to cases where ther value for the probablty of run les between and 0.05, whch we consder covers all values of practcal nterest. In all the examples n ths secton we use smulatons.

7 5.2. Examples CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 123 Example 1: Wkstad (1971) n hs case IA consdered exponentally dstrbuted ndvdual clams wth mean 1 and wth one clam expected each year (l = 1). Table 5.1 shows hs values for selected cases and our estmates, c TG (u,n), of these values together wth the standard errors of these estmates. In Table 5.1, h denotes the premum loadng factor, so that the premum rate each year s (1 + h), and c(u,n) denotes Wkstad s value. TABLE 5.1 VALUES AND ESTIMATES OF c(u,n). WIKSTAD (1971, CASE IA). u n h c(u, n) c5tg(u, n) SD[c5TG(u, n)] Example 2: Wkstad (1971) n hs case IIA consdered a compound Posson surplus model where ndvdual clam amounts have the followng dstrbuton: P(x) = exp( x) exp( x) exp( x) Ths s descrbed by Wkstad as an attempt to model Swedsh non-ndustral fre nsurance data from Table 5.2 shows values of the probablty of run, wth l = 1, n the same format as Table 5.1. TABLE 5.2 VALUES AND ESTIMATES OF c(u,n). WIKSTAD (1971, CASE IIA). u n h c(u, n) c5tg(u, n) SD[c5TG(u, n)]

8 124 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS Example 3: Seal (1978b) also consdered exponentally dstrbuted ndvdual clams wth mean 1 and wth one clam expected each year (l = 1). Table 5.3 shows hs values for selected cases and our estmates usng the same format as n Tables 5.1 and 5.2. TABLE 5.3 VALUES AND ESTIMATES OF c(u,n). SEAL (1978B). u n h c(u, n) c5tg(u, n) SD[c5TG(u, n)] Example 4: Closed formulas for the probablty of ultmate run, denoted c(u), exst when ndvdual clams have ether an exponental or a mxed exponental dstrbuton. See, for example, Gerber (1979), pages We can test our methodology by calculatng c TG (u,n) for very large values of n and l for both clam sze dstrbutons and comparng the results wth the exact values of c(u). Results are shown n Tables 5.4 and 5.5 for n = l = 1000 for the followng two clam sze densty functons: f(x) =exp( x) x > 0 (exponental, mean = 1) f(x) = 2 3 exp( 3x) +2 7 exp( 7x) x >0 (mxed exponental, mean = , varance = ) TABLE 5.4 VALUES AND ESTIMATES OF c(u): EXPONENTIALLY DISTRIBUTED CLAIM AMOUNTS. u c(u) c TG (u,1000) SD[c5TG(u,1000)] h = h = h =

9 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 125 TABLE 5.5 VALUES AND ESTIMATES OF c(u): MIXED EXPONENTIAL CLAIMS. u c(u) c TG (u,1000) SD[c5TG(u,1000)] Conclusons The comparsons between c(u,n) and c TG (u,n) n Tables 5.1 to 5.5 are somewhat unfar snce n each case the exact value, c(u,n) s based on a process where one clam s expected each year whereas our approxmaton has been developed to work well wth portfolos where large numbers of clams are expected each year. Nevertheless, the approxmate values are very close to the exact values n almost all cases. The exceptons are values for u = 10, n =1 n Table 5.2. These are extreme cases snce not only s just one clam expected n the one year tme nterval, but also the clam sze dstrbuton, P(x), was descrbed by Wkstad (1971) hmself as extremely skew. 6. THE PREMIUM AS A FUNCTION OF THE SURPLUS In some of the applcatons of our methodology n Secton 7 we wll determne the premum rate at the start of each year as a functon of the surplus level. The hgher the surplus, the lower wll be the premum. More precsely, for $ 1 we wrte P, the premum rate to be charged n the -th year, as h(u(t )), where h s some functon whch we wll specfy below and t takes one of two values: t = 1; or t = max( 2,0) In the frst case, P depends on the surplus at the end of the precedng year,.e. at the current tme. Ths s the most ntutvely appealng case. However, t may not be reasonable to expect the nsurer to adjust the premum rate nstantaneously, as ths case requres. The other case allows for ths by determnng the premum as a functon of the level of surplus one year earler. We wll also show results for t = 0. In ths case the premum s fxed throughout the term. Our methodology wll work for any chosen functon h(u(t )). The functon used n our examples, and the ratonale underlyng t, are descrbed n ths secton. Gven the surplus u(t ), we wll determne P so that the probablty of ultmate run, assumng the premum rate does not change, s approxmately some pre-determned level, for example We want to keep the ultmate run probablty unchanged, say as a rsk mesure. We wll use De Vylder s (1978) approxmaton to acheve ths. For k = 1, 2, 3, let m k denote E[Y k ] and let:

10 126 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS a = 3 m 9lm 2 2 m, l =, m 3 3 and P = P lm 1 + l a Then De Vylder s approxmaton to the probablty of ultmate run gven ntal surplus u(t ), denoted c DV (u(t )), s gven by: c DV (u(t )) = l exp* - ea - l o u ^ a P P t h4 (6.1) Gven a pre-determned value for c DV (u(t )), we can calculate numercally for any ntal surplus the correspondng value of P and hence P and hence the premum loadng factor, h (=P / (lm 1 ) 1). Formula (6.1) does not gve a closed form soluton for P. Snce we are gong to have to calculate the premum for each year of each (of many) smulatons, t s convenent to have a smple formula for h n terms of u(t ). We acheve ths usng Excel by calculatng the values of u t for a range of values of the safety loadng usng formula (6.1) and then fttng a power functon (usng Add trendlne), so that; for some parameters A and B. h. Au B For small values of u(t ) De Vylder s approxmaton can gve uncomfortably large values for the premum loadng factor. De Vylder (1978, page 118) says that for very small values of u the accuracy (of hs approxmaton) s not so good. For ths reason we put an upper bound of 100% on the premum loadng factor, so that: h(u t ) = (1 + mn(au B t,1)) lm 1 (6.2) In our applcatons n Secton 7 we wll use two target probabltes of ultmate run, and 0.01, whch we wll denote T1 and T2, respectvely, and three dfferent ndvdual clam sze dstrbutons: exponental, gamma and lognormal wth moments as shown n Table 6.1. TABLE 6.1 MEAN, VARIANCE AND SKEWNESS: EXPONENTIAL, GAMMA AND LOGNORMAL. Exponental Gamma Lognormal Mean Varance Skewness

11 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 127 Table 6.2 shows for each of the three dstrbutons and for the two target probabltes of ultmate run the values of the parameters A and B to be used n formula (6.2). TABLE 6.2 PARAMETERS FOR THE POWER FUNCTION FOR FORMULA (6.2). c(u) = 0.01 c(u) = Dstrbuton A B Dstrbuton A B Exponental Exponental Gamma Gamma Lognormal Lognormal Formula (6.2) s the way we choose as an attempt to control the surplus process by adjustng the premum each year so that the probablty of ultmate run has a gven target value. Ths s necessarly a somewhat crude attempt snce a two parameter functon does not fully reflect the behavor of the premum loadng, h, as the surplus vares. We can check the accuracy of ths formula by calculatng the (approxmate) probablty of ultmate run usng formula (6.1) for selected values of the ntal surplus, u, and the ftted premum loadng, h(u) (=Au B ). Some results are shown n Tables 6.3 and 6.4 for some values of u of nterest n our examples. In prncple, the results n these tables should be close to the target value for c(u), ether or In practce, ths s often not the case. The dfferences between the target and calculated probabltes of ultmate run arse from the naccuracy of the ft of the two parameter power functon (over a large range of values of u) and the senstvty of c(u) to the premum loadng, h. A further pont to bear n mnd s that even f the values n Tables 6.3 and 6.4 are close to the target probablty of run, the actual probablty of ultmate run for the premum loadng factor h and the gven TABLE 6.3 EXPONENTIAL: VALUES FOR c(u) CALCULATED USING FORMULA (6.1) AND THE SAFETY LOADINGS OF TABLE 6.2. u T1 T

12 128 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS TABLE 6.4 GAMMA & LOGNORMAL: VALUES FOR c(u) CALCULATED USING FORMULA (6.1) AND THE SAFETY LOADINGS OF TABLE 6.2. T1 T2 u Gamma Lognormal u Gamma Lognormal clam sze dstrbuton may be very dfferent. Ths s because the values n Tables 6.3 and 6.4 are based on De Vylder s approxmaton, whch n turn s based on a process wth exponentally dstrbuted clam amounts. If the actual process has, for example, lognormally dstrbuted clam amounts, then the actual probablty of ultmate run may dffer from the value shown n Table 6.4. Despte these shortcomngs, we regard our approach to adjustng premums, as descrbed above, as havng a sounder bass than an ad hoc method Scenaros 7. APPLICATIONS In our numercal examples, we wll consder: () a fnte term of 10 years, whch we consder to be a reasonable plannng horzon, () two target probabltes of ultmate run: (T1) and 0.01 (T2), () three methods for calculatng the premum, all of them wth the same target for the probablty of ultmate run: P1: t = 0, so that the premum s fxed throughout the 10 years at the level whch would acheve (approxmately) the target probablty of ultmate run, P2: t = 1, so that the premum s adjusted at the start of each year accordng to the current level of the surplus, and, P3: t = max( 2, 0), so that the premum s adjusted at the start of each year accordng to the level of the surplus one year ago. (v) two algorthms for calculatng the Posson parameter for the expected number of clams. These are: N1: l = 1000 so that the Posson parameter s constant from year to year.

13 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 129 N2: l vares from year to year between 800 and More precsely, let l j denote the Posson parameter n the th year for the j th smulaton. Then {l j } s a set of ndependent and dentcally dstrbuted random varables, each wth a unform dstrbuton on the nterval [800, 1200]. In ths case, the premum s calculated usng the mean value of l j, so that: P = h(u t )=(1 + mn(au B t,1)) E[l j ] m 1 In our examples the Posson parameter s smulated for each of the 10 years for each of the smulatons and then these smulatons of the aggregate annual clams are used for a gven combnaton of T and P. Scenaro N1 s the classcal model for clam numbers. However, Daykn et al (1996) suggest that ths scenaro may not capture the full varablty of the clam number process n practce. They say on page 329, each (clam number) process s a supermposton of trends, cycles and short-term fluctuatons, and also that, It seems clear that busness cycles are so common n general nsurance, and ther mpact so profound, that any rsk theory model whch clams to descrbe real-lfe stuatons must permt the user to evaluate the mpact of any cycles whch may be present. Fgure of Daykn et al (1996) shows some examples of the varablty of the clam rato (clams/premums) for general nsurance. Our scenaro N2 s an attempt to produce ths varablty through a varable Posson parameter Exponental clam amounts Tables 7.1 and 7.2. show results for exponentally dstrbuted clam amounts separately for the two target probabltes of run, T1 and T2, and wthn each table for each combnaton of scenaros N and P. TABLE 7.1 EXPONENTIAL: VALUES OF c(u,10), SCENARIO T1. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10)

14 130 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS TABLE 7.2 EXPONENTIAL: VALUES OF c(u,10), SCENARIO T2. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) We make the followng comments about Tables 7.1. and 7.2: () Standard devatons of our estmates of c TG (u,10) (not shown) are all very small, rangng from 3.42E-09 (T1/N1) to 4.56E-06 (T2/N2). () For the combnaton N1/P1, the values of c TG (u,10) are very close to the values n Table 6.3, as we would expect them to be. () For lower values of u we have the followng orderng for run probabltes as functons of the premum scenaro: P2 > P3 > P1. As u ncreases, the orderng swtches to: P3 > P2 > P1, then to: P3 > P1 > P2, and eventually to: P1 > P3 > P2. We can see ths n detal n Afonso (2008). (v) Varyng the Posson parameter, scenaro N2, ncreases the probablty of run consderably, as we would expect. We can gan some nsght nto the causes of run, or at least end of year, as opposed to wthn-year, run, by recordng some nformaton from those smulatons whch result n end of year run,.e. for whch u() < 0 for some. Table 7.3 shows the followng nformaton for scenaros T1/N1 and T2/N2 and for a low and a hgh value of the ntal surplus: P The premum scenaro. Prop The proporton of the probablty of run n Table 7.1 or 7.2 attrbutable to end of year run. Avg The mean of the year n whch end of year run occurs. Avg u( 1) The mean of the surplus at the start of the year of run. Avg l The mean Posson parameter n the year of run. Ths s always for N1. Avg p The mean premum n the year of run. Avg y The mean aggregate clams n the year of run.

15 r y 1,y CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 131 The correlaton between the aggregate clams n the year before run and the year of run. All these tems, except r y 1,y, are recorded from the smulatons used as the bass for Tables 7.1 and 7.2. The correlaton coeffcent, r y 1,y, has been calculated from a separate set of smulatons usng data from smulatons where end of year run occurs after the frst year. Also, we used smulatons from the referred only for the N2 case because of the sze of the fle. TABLE 7.3 EXPONENTIAL: STATISTICAL INFORMATION FOR RUIN CASES, SCENARIO T1. u P Prop Avg Avg u( 1) Avg l Avg p Avg y r y 1,y N N N N These statstcs show the followng for scenaros N1/T1 and N2/T1 (results for T2 are smlar): (a) The proporton of the probablty of run due to end of year run does not depend very much on the premum calculaton scenaro, but does depend on the ntal surplus. Ths proporton wll depend on the Posson parameter: a smaller value for ths parameter, and hence fewer expected clams each year, wll ncrease the proporton of the probablty of run due to end of year run. (b) The average aggregate clams n the year of run s sgnfcantly hgher than the expected aggregate clams. (c) The correlaton between aggregate clams n the year before run, y 1, and the year of run, y, s negatve for all premum scenaros. (d) The average surplus at the start of the year of run does not depend to any great extent on the ntal surplus. It s comparable for P1 and P3, but notceably hgher for P2.

16 132 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS (e) The average premum n the year of run s comparable for all three premum scenaros. (f) The average number of years untl run s hgher for P3 than for P2 and almost always hgher for P2 than for P1. An addtonal pont for scenaro N2 s that: (g) The average value of the Posson parameter n the year of run, around 1 160, s near the upper end of ts range. It s not surprsng that, however the premum s calculated, heaver than expected clams are assocated wth run (pont (b)), or that, for N2, a large value for the Posson parameter can lead to a large value for the aggregate clams (pont (g)) and then to run. Where the premum depends on the surplus at the start of the current year, scenaro P2, the mplcaton of ponts (b), (c) and (d) s that a major factor causng (end of year) run s a year of relatvely lght clams, and hence a lower premum n the followng year, followed by a year of heaver than expected clams. The run profle statstcs for P1 and P3 are very smlar. Ths s not surprsng snce, typcally, the average number of years untl end of year run for P3 s four or less and for the frst two years the premums wll be the same for these two scenaros. More detals can be found n Afonso (2008) Gamma and lognormal clam amounts Tables 7.4, 7.5, 7.6 and 7.7 show results for gamma and lognormally dstrbuted clam amounts n the same format as Tables 7.1 and 7.2. TABLE 7.4 GAMMA: VALUES OF c(u,10), SCENARIO T1. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10)

17 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 133 TABLE 7.5 GAMMA: VALUES OF c(u,10), SCENARIO T2. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) TABLE 7.6 LOGNORMAL: VALUES OF c(u,10), SCENARIO T1. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) TABLE 7.7 LOGNORMAL: VALUES OF c(u,10), SCENARIO T2. N1 N2 P1 P2 P3 P1 P2 P3 u c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10) c TG (u,10)

18 134 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS We make the followng comments about Tables 7.4 to 7.7: () Standard devatons of our estmates of c TG (u,10) (not shown) are all comparable to those for exponental clams. () For the combnaton N1/P1, the values of c TG (u,10) are close to the values n Table 6.3, though not as close as n the case of exponental clam amounts. Ths s to be expected snce the values n Table 6.3 are based on De Vylder s approxmaton, whch s exact for exponental clam amounts. () As for exponental clams, for lower values of u we have the followng orderng for run probabltes as functons of the premum scenaro: P2 > P3 > P1. As u ncreases, the orderng swtches to: P3 > P2 > P1, then to P3 > P1 > P2, and eventually to: P1 > P3 > P2. We can see ths n detal n Afonso (2008). (v) Varyng the Posson parameter, scenaro N2, ncreases the probablty of run consderably, as t dd for exponental clams. Tables 7.8 and 7.9 show statstcs for end of year run for gamma and lognormal clams, respectvely, n the same format as Table 7.3. These statstcs lead us to the same conclusons about the major causes of run as those stated at the end of Secton 7.2. Compared to the case wth exponental clams, the ncrease n the average surplus at the start of the year of run s lnked drectly to the ncrease n the average aggregate clams, whch, n turn, s lnked to the greater varablty of aggregate clams when ndvdual clams have a gamma or lognormal dstrbuton. See Table 6.1. More detals can be found n Afonso (2008). TABLE 7.8 GAMMA: STATISTICAL INFORMATION FOR RUIN CASES, SCENARIO T1. u P Prop Avg Avg u( 1) Avg l Avg p Avg y r y 1,y N N N N

19 CALCULATING CONTINUOUS TIME RUIN PROBABILITIES 135 TABLE 7.9 LOGNORMAL: STATISTICAL INFORMATION FOR RUIN CASES, SCENARIO T1. u P Prop Avg Avg u( 1) Avg l Avg p Avg y r y 1,y N N N N ACKNOWLEDGEMENTS The authors gratefully acknowledge fnancal support from Fundaçao para a Cênca e a Tecnologa (programme FEDER/POCI 2010). Lourdes B. Afonso s grateful to Kenneth Wlder of the Unversty of Chcago for hs help to change hs C++ random number generator class n order to run n 64 bt machnes. Also, ths author gratefully acknowledges the fnancal support from Fundaçao Calouste Gulbenkan for the presentaton of part of ths work at the I:ME 2006 Conference n Leuven. REFERENCES AFONSO, L.B. (2008) Evaluaton of run probabltes for surplus processes wth credblty and surplus dependent premums. PhD thess, ISEG, Lsbon. BOHMAN, H. and ESSCHER, F. (1963) Studes n rsk theory wth numercal llustratons concernng dstrbuton functons and stop loss premums. Part I. Skandnavsk Aktuaretdskrft, 1963: CARDOSO, R.M.R. and WATERS, H.R. (2005) Calculaton of fnte tme run probabltes for some rsk models. Insurance: Mathematcs and Economcs, 37(2): DAVIDSON, A. (1969) On the run problem n the collectve theory of rsk under the assumpton of varable safety loadng. Skandnavsk Aktuaretdskrft, 1969(3-4 Suppl): DAYKIN, C.D., PENTIKÄINEN, T. and PESONEN, M. (1996) Practcal Rsk Theory for Actuares. Chapman and Hall, London. DE VYLDER, F. (1978) A practcal soluton to the problem of ultmate run probablty. Scandnavan Actuaral Journal, 1978(2): DICKSON, D.C.M. and WATERS, H.R. (1993) Gamma processes and fnte tme survval probabltes. ASTIN Bulletn, 23(2):

20 136 L.B. AFONSO, A.D. EGIDIO DOS REIS AND H.R. WATERS DICKSON, D.C.M. and WATERS, H.R. (2006) Optmal dynamc rensurance. ASTIN Bulletn, 36(2): DICKSON, D.C.M. (1991) The probablty of ultmate run wth a varable premum loadng a specal case. Scandnavan Actuaral Journal, 1991(1): DUFRESNE, F. and GERBER, H.U. (1989) Three methods to calculate the probablty of run. ASTIN Bulletn, 19(1): GERBER, H.U. (1979) An Introducton to Mathematcal Rsk Theory. Huebner Foundaton for Insurance Educaton Unversty of Pennsylvana, Phladelpha, Pa USA. JASIULEWICZ, H. (2001) Probablty of run wth varable premum rate n a Markovan envronment. Insurance: Mathematcs and Economcs, 29(2): KLUGMAN, S.A., PANJER, H.H. and WILLMOT, G.E. (2004) Loss models: From data to decsons. John Wley and Sons, Inc., 2nd edton. MALINOVSKII, V.K. (2008) Rsk theory nsght nto a zone-adaptve control strategy. Insurance: Mathematcs and Economcs, 42(2): MICHAUD, F. (1996) Estmatng the probabltes of run for varable premums by smulaton. ASTIN Bulletn, 26(1): PETERSEN, S.S. (1989) Calculaton of run probabltes when the premum depends on the current reserve. Scandnavan Actuaral Journal, 1989(3): SEAL, H.L. (1978a) From aggregate clams dstrbuton to probablty of run. ASTIN Bulletn, 10(1): SEAL, H.L. (1978b) Survval probabltes, the goal of rsk theory. Wley, New York. TAYLOR, G.C. (1980) Probablty of run wth varable premum rate. Scandnavan Actuaral Journal, 1980(1): WIKSTAD, N. (1971) Exemplfcaton of run probabltes. ASTIN Bulletn, 6(2): LOURDES B. AFONSO Depart. de Matemátca and CMA Faculdade Cêncas e Tecnologa Unversdade Nova de Lsboa Caparca Portugal E-mal: lbafonso@fct.unl.pt ALFREDO D. EGÍDIO DOS REIS Depart. of Mathematcs, CEMAPRE and ISEG Techncal Unversty of Lsbon Rua do Quelhas Lsboa Portugal E-mal: alfredo@seg.utl.pt HOWARD R. WATERS Depart. of Actuaral Mathematcs and Statstcs and The Maxwell Insttute for Mathematcal Scences Herot-Watt Unversty Rccarton Ednburgh EH14 4AS Scotland E-mal: H.R.Waters@ma.hw.ac.uk