Comparison of the claims reserves methods by analyzing the run-off error

Transcription

1 Comparison of he claims reserves mehods by analyzing he run-off error AUTHORS ARTICLE INFO DOI JOURNAL FOUNDER Nicolino Eore D Orona Giuseppe Melisi Nicolino Eore D Orona and Giuseppe Melisi (06). Comparison of he claims reserves mehods by analyzing he run-off error. Insurance Mares and Companies, 7(). doi:0.5/imc.7().06.0 hp://dx.doi.org/0.5/imc.7().06.0 "Insurance Mares and Companies" LLC Consuling Publishing Company Business Perspecives NUMBER OF REFERENCES 0 NUMBER OF FIGURES 0 NUMBER OF TABLES 0 The auhor(s) 08. This publicaion is an open access aricle. businessperspecives.org

2 Nicolino Eore D Orona (Ialy), Giuseppe Melisi (Ialy) Comparison of he claims reserves mehods by analyzing he run-off error Absrac The variabiliy of claim coss represens an imporan ris componen, which should be aen ino accoun while implemening he inernal models for solvency evaluaion of an insurance underaing. This componen can generae differences beween fuure paymens for claims and he provisions se aside for he same claims (run-off error). If he liabiliy concerning he claims reserve is evaluaed using synheic mehods, hen he run-off error depends on he saisical mehod adoped; when i is no possible o sudy analyically he properies of he esimaors, mehods based on sochasic simulaion are paricularly effecive. This wor focuses on measuring he run-off error wih reference o claims reserves evaluaion mehods applied o simulaed run-off marices for he claims selemen developmen. The resuls from he numerical implemenaions provide he auhors wih useful insighs for a raional selecion of he saisical-acuarial mehod for he claims reserve evaluaion on an inegraed ris managemen framewor. The seing of he analysis is similar o ha adoped in oher sudies (Sanard, 986; Peniainen and Ranala, 99; Buhlmann e al., 980), however, i differs for esimaion and simulaion mehods considered and for he saisics elaboraed in he comparison. Keywords: run-off error, ousanding claims reserves, sochasic simulaion. Inroducion The random claim selemen regarding he acciden year i (i = 0,,, ) is given by he sum of a random number of claims, each one subec o a single claim selemen, and i can be represened as follows: N i = 0 X i = Y i, i = 0,,...,, () whereas N ~ i represens he oal number of claims ~ incurred in he year generaion i; Y i is he random selemen for he claim incurred during he acciden year i; symbolizes boh he ime of observaion of he porfolio and he oal number of generaions sill open. Since he selemen claimed for every acciden usually requires wo or more paymens, which can ae place during he acciden year or he subsequen years, he aggregaed claims cos for every acciden year can be represened as follows: X i = X i,, i = 0,,...,, () = 0 ~ whereas X i, represens he amoun paid for selemens regarding claims incurred during he acciden year i and seled afer years; represens he maximum number of defermen years considered for he oal selemen of a single claim. A he ime of observaion he recorded informaion from he company regards he amouns: X i, : i = 0,,...,; = 0,,..., - i, (3) while a forecas of fuure amouns should be done: ~ X i, : i,,..., ; i,...,. (4) The random amoun required for fuure selemens regarding claims no ye seled or repored (and IBNR), for each acciden year, is given by: R i = X i,, i =,,...,. (5) =-i+ The aggregae amoun required is hen given by he sum: i= R = R i. (6). The run-off errors The saisical mehods for he ousanding claims reserve evaluaion consis in he formulaion of a forecased value of he necessary reserve, based on a proeced analysis of he daa obained by he examinaion of relevan ime series. In oher words, an evaluaion mehod provides an esimaor R ˆ = f K 0,K,...,K of he expeced value for he ousanding claims reserve, which depends on he informaion a disposal K = f K 0,K,...,K for each acciden year, and of which a he ime here Nicolino Eore D Orona, Giuseppe Melisi, 06. Nicolino Eore D Orona, Associae Professor, DEMM Deparmen, Univerisiy of Sannio, Beneveno, Ialy. Giuseppe Melisi, Adunc Professor, Universiy of Sannio, Beneveno, Ialy. In general, for he disribuion of he ousanding claims reserve, oher han he expeced value we can esimae momens of order higher han or even paricular quaniles.

3 are some (parial) deerminaions (e.g., paid, claim number, closed wihou paymen, re-opened). The difference beween fuure paymens for claims selemens and he amoun of he relaive ousanding claims reserve, evaluaed using a specific esimaor, gives us he run-off error. The run-off error for each acciden year can be represened as follows: e i =Rˆ i R i = Xˆ i,x i,, i =,,...,, =-i+ 3 (7) while he run-off error relaive o he enire porfolio can be expressed by: ˆ e = Rˆ R = X i, X i,. (8) i= =-i+ The eniy of he run-off error depends on he differences beween he se of hypoheses on which he esimaion model is based and he acual characerisics of he porfolio 4 ; such differences condiion, evidenly, he properies of he esimaor of he claims reserve. The formula measures he run-off error, a he observaion period, for all he acciden years aen ino consideraion, compensaing he possible differences of opposie sign beween he run-off errors of he various acciden years. The esimaor ˆR is called unbiased if he expeced value of he esimaor equals he expeced value of he ousanding claims reserve for which he esimaor is used: ERˆ R =ERˆ E R =0. (9) For an unbiased esimaor, he expeced run-off error equals 0. You could say ha an unbiased esimaor provides esimaes of he provision for claims ha do no conain loadings (posiive or negaive, implici or explici). The ampliude of he disorion ha characerizes he esimaors is, however, only he firs crierion of comparison. In fac, a mehod ha can provide esimaes wih low For he purpose of he evaluaion reliabiliy, he informaion on which he proeced analysis is based should include sufficien, independen and homogeneous daa. 3 The formula measures he run-off error, a he observaion period, for he enire acciden year, compensaing he possible differences of opposie signs during he forhcoming developmen years. Knowing he gap beween expeced and acual iming of selemens is of crucial imporance for he reinsurance reaies ha compensaes he reserved claims of he ceden company. 4 If he amouns of fuure selemens are discouned a he ime of observaion, o he error in he forecas of he cash flow of selemens you mus add he error relaive o he forecased fuure raes of reurn. disorion, bu for which he individual forecass differ considerably from he acual values, may no be an appropriae mehod for he esimaion of reserves. I is useful hen o consider oher precise indicaors such as he mean percenage error, R-R ˆ E, and he mean square error, R ER-R ˆ 5. Moreover, since i follows: ˆ ˆ ˆ, (0) R R = R+ R R R a good esimaion mehod mus provide an esimaor wih high correlaion wih he reserve o esimae.. The claim reserve evaluaion mehods Beween he muliple procedures for he evaluaion of ousanding claims reserve proposed in lieraure, four of hem were chosen for his wor, considering heir widespread uilizaion in he professional environmen: he Chain-Ladder mehod, he Separaion Mehod (arihmeic and geomeric), he mehod and he Bornhueer- mehod. We give below a concise represenaion of he conen and how hey were applied in he analysis... The Chain-Ladder mehod. The Chain-Ladder mehod considers he run-off riangle of cumulaive paymens of selemens: C i, : i = 0,,...,; = 0,,..., - i, () = 0 C i, = X i,. whereas The underlying hypohesis is ha he disribuion of he selemens is consan for each acciden year, he developmen facors are esimaed as: C i,h+ i= 0 m ˆ h =, h = 0,,...,. -h- C i,h i= 0 -h- () Then, assuming ha developmen facors remain unalered also for he fuure, he cumulaive fuure paymens are calculaed: ˆ - C i, = C i, i m ˆ h, =,,...,. (3) h=-i 5 Le s recall ha, beween wo biased esimaors, Rˆ A and Rˆ B, we will say ha Rˆ A is more efficien han Rˆ B if and only if ~ ~ E R Rˆ ˆ A E R R B.

4 The difference beween he ulimae cos and he cumulaive cos unil he year of observaion will provide an esimae of he reserve for a single generaion: ˆ. Rˆ i = C i, C i, i, i =,,..., (4) The sum of hese differences, for all generaions, measures he esimaed amoun of oal claims reserve. Among he varians of he mehod, he one based on he riangles of he relaionship beween he cumulaive average coss heory was considered, which esimaes developmen facors as weighed averages of he raios observed wih he weighs obained by calculaing w i, = i+ +, ha depends on he acciden years of claims and he ime span of selemen... The Taylor mehod. The mehod of (arihmeic) elaboraes he riangle of he average coss of claims for he acciden year, assuming ha each of hese coss, ne of he random noise erm, is, on average, expressed as a produc of wo facors: E X i, =r +. (5) Facor r, as a funcion of only years of developmen and varies beween 0 and, is he way in which paymens per claim are disribued in ime, regardless of generaion; while he second facor, ha depends on boh he year of developmen and he acciden year, represens an index of exogeneiy, wih paricular reference o he inflaion, exrapolaed by log-linear regression. The availabiliy of an adequae informaion base allows he esimaion of he facors r ˆh and ˆh (h = 0,,, ), expressing hen he average cos per claim of generaion, according o he produc of he wo facors menioned above; while he facors ˆh (h = +, +,, ) are exrapolaed from he facors ˆh by log-linear regression. I is esimaed ha in his way he fuure average coss per claim for each generaion, muliplied by he corresponding number of claims, can predic he cumulaive amouns of fuure claims and, subsequenly, he oal claims reserve. Among he varians of he described mehod he socalled of geomeric ype was considered, wih exrapolaion of he index of inflaion, using a log-linear regression..3. The mehod. The mehod is based on he average coss of claims paid in previous generaions and heir relaive selemen speed. The wo ey assumpions are: ) he claim selemen speed is consan over ime; ) he average cos of claims paid is a funcion of he period beween he acciden dae and he ime of acual paymen. Hence, saring from he riangle of run-off in he number of claims seled: n i, : i = 0,,...,; = 0,,..., i. (6) The rae of selemen for developmen year is calculaed as: ni, v =, =,,...,, (7) a + n i= 0 i, wih which he number of seled claims is esimaed: a n ˆi, = nˆ i, v, =,,..., ; i = +,...,, (8) and he number of claims sill ousanding: a a n ˆ = nˆ n ˆ, =,,..., ; i = +,...,. (9) i, i, - i, Then, we consider he run-off riangle of he X i, : i = 0,,...,; = 0,,..., i average cos paid ha produces esimaes of fuure average coss, X ˆ i, : i =,,...,; = i+,...,, by log-linear regression of he average coss for each developmen year. Nex, muliplying he proeced average coss corresponding o he number of claims ha are expeced o be seled, you ge he esimaes of he oal amouns of claims sill ousanding. The sum of all hese fuure amouns represens an esimae of oal claims reserve: R= ˆ Rˆ i = nˆ X ˆ i,. (0) i, i= i= =-i+.4. Bornhueer- mehod. As par of Bornhueer- Mehod, he run-off riangle of accumulaed paymens C i, : i = 0,,...,; = 0,,..., i is considered, from which we esimae developmen facors: C i,h+ i= 0 m ˆ h =, h = 0,,...,. -h- C i,h i= 0 -h- () Benchmar values are deermined according o he cos of generaion by muliplying he premiums of each generaion for a suiable loss-raio (Bornhueer and, 97). In his case, he 3

5 benchmar value was calculaed using he following formula 6 : C ˆ = 4 ˆ - C i, i m +i h i= 0 h=-i i= 0 +i. () The esimaes of he ulimae cos for each generaion are obained by applying he facors of Bornhueer- o he benchmar values: ˆ ˆ C i, =C i,i +C, M i whereas M = i - h=-i reserve for each generaion: (3) m ˆ. Then, he esimae of he ˆ. h Rˆ i = C i, C i, i, i =,..., (4) 3. The simulaion mehods of he run-off marix In pracice, he run-off error can be measured only afer he compleion of he claims selemen process. In his wor, we will quanify he run-off error, simulaing he claims selemen process unil we obain all he members of he run-off error ~ e i, i,,...,. formula For his purpose, we represen he random selemens, in each cell of he run-off marix, wih he following (collecive) model: N i, X i, = Y i, ; i, = 0,,...,, (5) =0 whereas N i, represens he oal number of claims for he acciden year i, seled during he Y i, represens he random developmen year ; selemen for he claim incurred during he acciden year i and seled afer years. For simpliciy we will exclude he possibiliy of selemen in insallmens over several years of developmen. For he simulaion of he amouns X i, we have considered four mehods, which are disinguished for he developmen rule concerning he claims selemen inside he run-off riangle and are based on probabilisic assumpions regarding boh he disribuion of he number of claims 6 The formula modifies ha proposed in Bornhueer and (97), where an arihmeic average of he ulimae cos of he generaion is used. N i = random selemen of each claim N i, and he disribuion of he Y i,. 3.. Mehod of random developmen facors. The emporal disribuion of he selemens inside he run-off marix is governed by he developmen facors, as described in he Chain- Ladder mehod framewor wih he excepion ha he main hypoheses on which he Chain-Ladder mehod is based upon are no respeced in his case 7. This mehod simulaes he run-off marix hrough he following seps (Narayan and Warhen, 997):. A value n (i) of he random variable N i, number of claims, is generaed from a Poisson disribuion wih a prese parameer.. n (i) values, y i, of he random variable Y i are generaed from a lognormal disribuion wih a prese parameers and. 3. The sum of he claims coss is calculaed, obaining he ulimae cos for each generaion: ni ( ) = 0 C i, = y i. (6) 4. Pseudo-random numbers, H ( = 0,,, ), are generaed, calculaing: T=a+bH+cln +, (7) U = T +T +...+T. (8) 5. The cumulaive paymen for each generaion i, unil he developmen year, is calculaed according o -U Ci,=Ci, e. (9) 6. The value of he claims reserve for he generaion i resuls: R i = C i, C i, i. (30) 7 The assumpions implici in he Chain-Ladder model are: ) he developmen of selemen is made according o unnown developmen E C i C i,0,..., C i, C i, m facors m 0, m,... m -, wih, and i, 0 -; ) he variables C(i,0),,C(i,) and C i',0,..., Ci', relaed o differen acciden years i i`are independen; 3) here are consan unnowns as a 0, a -, so Var C i, C i,0,..., C i, C i, wih i, The probabiliy disribuions associaed wih he random quaniies are chosen in an arbirary manner, bu hey are consisen wih he acuarial lieraure. For he hypohesis of claims frequency wih he Poisson disribuion, see Buhlmann e al. (980); for he assumpion of lognormal disribuion of he amoun of selemens Hewi and Lefowiz (979), Hewi (970).

6 7. For each acciden year seps o 6 are repeaed, muliplying he ulimae cos of claims by he facor: infl i I i = +i, i =,...,, (3) whereas i inf l is an annual inflaion rae. 3.. Mehod of bacward calculaed random developmen facors. This mehod is similar o he previous one, wih he only difference ha he developmen facors are calculaed using bacward seps. The mehod simulaes he run-off marices hrough he following seps (Narayan and Warhen, 997): N i,. A value n (i) of he random variable number of claims, is generaed from a Poisson disribuion wih a prese parameer.. n (i) values, y i, of he random variable Y i are generaed from a lognormal disribuion wih a prese parameers and. 3. The sum of he claims coss is calculaed, obaining he ulimae cos for he acciden year i: ni ( ) = 0 C i, = y i. (3) 4. random variables are simulaed H ( = 0,,, -), a normal disribuion wih parameers = + ; = d +. (33) d 5. Developmen facors are calculaed: H m e ; = M = m, = 0,,...,. (34) The parameers d and d are assigned values such as o ensure ha he facors m are greaer han wih high probabiliy. 6. The cumulaive paymen a he end of he year ( = 0,,, ), for he generaion i, is calculaed as: C i, C i, =. (35) M 7. The claim reserve for he generaion i resuls: R(i) = C (i,) C (i, i) (36) 8. For each acciden year seps o 7 are repeaed, muliplying he cos of selemens for each acciden year by he facor: I(i) = ( + i infl ) i, i =,, (37) whereas i infl is an annual inflaion rae Mehod of single selemens. This simulaive mehod is derived from he esimaion models proposed by Sanard (986) and by Buhlmann, Schnierper and Sraub (980), which consider he selemen of a single claim as a sochasic process depending on hree parameers: he incurring year, he reporing year and he selemen year. In his framewor, he simulaing model assumes an exponenial disribuion for he defermen periods regarding he reporing year and he selemen year (McCleanahan, 975; Weissner, 978). Furhermore, he selemen amoun varies wih he variaion of he defermen period beween he selemen year and he incurring year. The mehod simulaes he run-off marices hrough he following seps:. A value n (i) of he random variable N i, number of claims, is generaed from a Poisson disribuion wih a prese parameer.. For each claim n (i), i is necessary o simulae:.. he defermen of he ime of acciden,, respec o he beginning of he year of generaion; wih an uniform random variable (0, );.. he ampliude of he deferral period from he ime repored,, measured from he ime of acciden; wih exponenial random variable wih prese mean ;.3. he ampliude of he deferral period from he ime of closing, 3, measured from he ime repored; wih 3 exponenial random variable wih prese mean 3 ;.4. Le s assume = min ( ; ) and 3 = min ( 3 ; ). 3. For he selemen relaed o he single claim a Pareo disribuion is assumed wih densiy: f ~ Y y y 0, 0, y ;, (38) whereas: a () = a a b a is he shape parameer and = +b +i infl scale parameer, dependen on he annual inflaion rae i infl. The model used o represen he dynamics of he parameers of he Pareo disribuion generaes values wih increasing selemen parallel o defermen period, his ensures ha he cumulaive amouns of selemen of a generaion, along he rows of he marix of developmen, have a posiive rend. In pracice, he assessmen of he amoun of a claim can in ime have eiher an 5

7 increase or a decrease, resuling in a non-monoonic cumulaive selemen. So, for each of he n (i) claims an amoun of selemen is associaed wih y, =,..., n(i), calculaed as: y y 0, se, (39) p; i, se p y p;i, =, p se > (40), (4) whereas is he smalles ineger greaer han or equal o he sum + + 3, while p is generaed by a random variable wih an uniform disribuion on (0, ). 4. Cumulaing he selemens observed in each cell, we obain he aggregae amoun X (i, ) while he cumulaive paymen for he generaion i, unil he developmen year, is given by C (i, ). 5. Cumulaing unil he las year of developmen he amoun of he final cos for he generaion considered C (i, ) is obained; while he loss reserve is R (i) = C (i, ) C (i, i). 6. For each acciden year seps o 5 are repeaed, inflaing he cos of selemens for each acciden year a he annual inflaion rae i infl Peniainen-Ranala mehod. This mehod simulaes he developmen of he aggregaed selemens for claims incurred during a given year, assuming ha he srucure funcion and he inflaion rae follow an auoregressive process. This mehod operaes hrough he following seps:. For acciden claims during he generaion of he mos remoe (base year, i = 0), we choose arbirarily he average number claims, n, and he firs hree momens from he origin of he single cos disribuion, respecively, m = a, a and a 3.. I simulaes he number of claims incurred in he base year, n (0), (using he inverse of he Anscombe ranformaion) and is aggregae cos of claims X (0, 0) (using he formula of Wilson- Hilfery, applicable o a compound Poisson random variable). 3. The number of claims for subsequen generaions is calculaed using he following: n( i) n(0) I ( i), (4) n whereas I n (i) = ( + i n ) i, i = 0,,...,, (43) while i n measures he annual rae of growh of he porfolio. 4. Represened, hen, he srucure funcion (funcion ha modifies he average frequency of claims paid annually) wih he auoregressive process of he firs order: q i, =a q+bq q i, + q. (44) I simulaes ~ q N0; q for each cell (i, ) and hen calculaes: assuming qi, 0 =,q i, : i =,,...,; = i+,...,, (45). 5. The emporal disribuion of he number of claims for each acciden year is deermined by he following: n(i, ) = n(i)q(i,)g n (), (46) whereas g n () is he funcion of he emporal disribuion of he number of claims (measures he probabiliy an incurred claim in he year i is liquidaed afer years). The values of he probabiliy g n () ( = 0,,.., ) have been hypohesized independen o he generaion year and assumed equal o he componens of he vecor: 0.; 0.8; 0.5; 0.; 0.0; 0.08; 0.06; 0.04; g= n. 0.07; 0.06; Assumed ha he cos of a single claim could grow due o inflaion, we can simulae he pahs of he inflaion rae, modeled wih he auo regressive process: i + =max i infl +binfl i i infl + infl ;i min, (47) whereas i min = minimum inflaion rae; i infl = average inflaion rae; while i is assumed ha i 0 =i N 0;. and infl infl infl So we derive he pahs of he inflaion facor: T = 0 I infl T = +i, T = i+ ; i, = 0,...,. (48) 7. In his way, for each marix of he run-off, he fuure selemen flows are obained: X i, : i,,..., ; i,...,, (49) wih 00 n infl n Xi,=X, I ii i+g qi,. (50) 8. Cumulaing unil he las year of developmen, he amoun of he ulimae cos for he individual generaions is obained. For each generaion, he ulimae cos, deducing he 6

8 cumulaive cos of he evaluaion year, resuls in he individual reserve. Is oal forms he claims reserve for he enire porfolio. 4. Numerical applicaion A comparaive analysis was se up for he examinaion of he run-off error ampliude regarding each esimaing mehod, considering differen ses of parameers, which were recursively modified predicing: a differen level of inflaion, a higher volailiy of he selemen amoun, a higher volailiy of he disurbing facors characerizing he selemen process, various emporal profiles for he claims developmen. For each se of parameers 4.00 selemen marices were generaed wih each one of he described simulaion echniques. The inferior riangle of he fuure selemens was obained from he superior riangle of every simulaed marix. Therefore, gap indicaors beween esimaed reserves and effecive (simulaed) reserves were calculaed. 4.. Mehod of random developmen facors. The numerical values aribued o he parameers were he following: number of claims: = 000; selemen: = 7.36; =.5; a = 0.; b = 0.; c = 0.5; inflaion rae: i infl = 4%. Table shows he saisics elaboraed o analyze he mehod. These saisics allow o now he sign of he error, and hen he endency of he evaluaion mehods o overesimae or underesimae he value of he reserve. Table. Mehod of random developmen facors Chain Ladder v. Chain Ladder v. Bias Mean square error Mean percenage error 0.3%.% 0.6% 0.64% 0.8% -7.3% Corr. coeff Toal claims reserve: mean = ; sandard deviaion = Considering all he acciden years, all he mehods for ousanding claims reserve predicion provide more or less biased esimaors, while showing a resrained mean percenage error (wih he excepion of he Bornhueer- mehod). According o he mean square error crierion, he mehod, whose esimaor is characerized by an adequae correlaion level wih he esimaed reserves, presens a higher level of preferabiliy. Analyzing he single acciden years, we deduce ha he Chain-Ladder mehod provides a less biased esimaor wih a lower mean square error, wih he relevan excepion of he las acciden year, which compromises, more han for any oher mehod, he overall efficiency of he esimaor. The following able shows he analysis of he saisics calculaed for each generaion: Table. Bias mehod of random developmen facors Toal Table 3. Mean square error mehod of random developmen facors

9 Table 3 (con.). Mean square error mehod of random developmen facors Toal Table 4. Mean percenage error mehod of random developmen facors.0%.0%.05%.47% 0.8% 50.0% 0.57% 0.57%.57%.66%.65% 4.08% % 0.74% 0.6% 0.7% 0.0% 3.60% % 0.47% 0.63% 0.66%.0% 5.37% % 0.56% 0.% 0.5% 0.64% 7.79% % 0.45% 0.% 0.3% 0.85%.0% 7 0.4% 0.4%.40%.66%.% 5.99% 8 0.4% 0.5% 0.3% 0.6% 0.75% -.05% 9 0.% 0.4% 0.0% 0.65% 0.84% -6.69% 0.03% 4.48%.08%.45%.4% -.7% Toal 0.3%.% 0.6% 0.64% 0.8% -7.3% 4.. Mehod of bacward calculaed random developmen facors. The numerical values aribued o he parameers were he following: number of claims: = 000; selemen: = 7.36; =.5; d = 00; d = 500; inflaion rae: i infl = 4%. Table 5 shows he saisics elaboraed for he second mehod. Table 5. Mehod of bacward calculaed random developmen facors Chain-Ladder v. Chain-Ladder v. Bias Mean square error Mean percenage error 0.6% 0.90% -0.37% -.0%.5% -3.74% Corr. coeff Toal claims reserve: mean = ; Sandard deviaion = The mehod esimaor shows he lower mean square error for boh he single acciden year esimaion and he whole porfolio esimaion. The Chain-Ladder esimaor resuls o be he less biased esimaor and shows he lower mean percenage error. The mehods are characerized by a sysemaic underesimaion of he reserve, which resuls o be a discriminaing characerisic for a mehod uilized in conrolling he reserves se aside by an insurance company. The Bornhueer- mehod, whose ousanding claims reserve predicion is based upon a benchmar value, which depends on he ulimae cos for each acciden year, presens a sysemaic overesimaion (underesimaion) of he reserve concerning he firs (las 3) acciden years, as i is eviden from he following analysis carried ou for single generaion: Table 6. Bias mehod of bacward calculaed random developmen facors

10 Table 6 (con.). Bias mehod of bacward calculaed random developmen facors Toal Table 7. Mean square error mehod of bacward calculaed random developmen facors Toal Table 8. Mean percenage error mehod of bacward calculaed random developmen facors 7.85% 7.85% 7.74% 8.0% 0.3% 60.6% 3.3% 3.5% 3.79% 3.37%.38% 45.7% 3.98% 3.00%.0% 0.73% -0.43% 35.4% 4.93% 3.00%.50% 0.94%.09% 8.% 5 3.6% 3.5%.07% 0.37% 0.56% 0.46% 6.5%.67%.60% -0.5% 0.87%.99% 7.%.33%.3% 0.54%.35% 7.4% 8.65%.0% 0.7% -.00%.0% -0.09% 9 0.9%.50% 0.30% -.39%.3% -6.7% %.% 0.7% -0.87%.98% -.3% Toal 0.6% 0.90% -0.37% -.0%.5% -3.74% 4.3. Mehod of single selemens. The numerical values aribued o he parameers were he following: number of claims: = 000; selemen: a = 000; b = 50; a =.5; b = 0.5; inflaion rae: i infl = 4%; defermen: =; 3 =. Table 9 shows he saisics elaboraed for he hird mehod. Chain-Ladder v. Chain-Ladder v. Table 9. Mehod of single selemens Bias Mean square error Mean percenage error 0.3%.% 0.6% 0.64% 0.8% -7.39% Corr. coeff Toal claims reserve: mean = ; Sandard deviaion =

11 The simulaing echnique appears o be raher coheren in srucure, wih he claims developmen model upon which he mehod is based, hus, resuling in an esimaor wih he lowes esimaion gap for boh he single acciden year es- imaion and he whole porfolio esimaion. All mehods provide esimaors wih high levels of correlaion wih he esimaed reserve. In addiion for more informaion, he following are he analysis for single generaion: Table 0. Bias mehod of single selemens Toal Table. Mean square error mehod of single selemens Toal Table. Mean percenage error mehod of single selemens Generaion Chain Ladder v. Chain Ladder v. Separaion Separaion r Bornhuee 8.% 8.% 7.8% 7.48% -0.55% 59.67% 5.5% 5.65% 4.65% 4.45% -0.65% 46.89% 3.5%.% 0.75% 0.4% 0.3% 33.54% % 0.56% -0.98% -.30% 0.54% 3.88% % 0.9% -0.% -0.5% 0.33% 7.7% % 0.50% -0.8% -0.35% 0.57%.39% 7 0.7% 0.4% -0.55% -0.58% 0.04% 4.79% 8.8%.0% -0.8% -0.4% -0.9% -0.7% 9 0.6% 0.3% -0.70% -0.5% 0.0% -6.8% % 5.5% -0.59% -0.34% 0.5% -.96% Toal 0.0%.34% -0.74% -0.69% -0.6% 0.49% 4.4. Peniainen-Ranala mehod. The numerical values aribued o he parameers were he following: number of claims: = 000; a q = 0.4; b q = 0.6; q = 0.05; i a =%; selemen: = 0.006; = 0.00; 3 = 0.000; inflaion rae: i infl =4%; i min =%; b infl =0.7; infl = Table 3 shows he saisics elaboraed for he las mehod. 0

12 Chain-Ladder v. Chain-Ladder v. Table 3. Peniainen-Ranala mehod Bias Mean square error Mean percenage error.%.76% 0.07% -0.07%.6% -.67% Corr. coeff Toal claims reserve: mean = ; sandard deviaion = In his case, according o boh he mean percenage error crierion and he dispersion crierion, he geomeric mehod presens he higher level of preferabiliy. Moreover, he analysis of individual generaions sill shows he characerisic underesimaion of mehods based on he. The following show he analysis for single generaion: Table 4. Bias-Peniainen-Ranala mehod Toal Table 5. Mean square error Peniainen-Ranala mehod Toal Table 6. Mean percenage error Peniainen-Ranala mehod.09%.09%.9%.58%.03% 50.89%.9%.3%.03% 0.65%.57% 8.0% 3.7%.% 0.8% 0.47%.57% 9.46% 4 0.8% 0.87% 0.36% 0.03%.6% 67.43% % 0.86% 0.0% -0.09%.6% 47.7% % 0.96% 0.05% -0.0%.66% 9.5% % 0.85% -0.6% -0.36%.70% 3.9% %.0% -0.9% -0.45%.73% -0.46% %.3% -0.39% -0.50%.74% -.55% 0.8% 3.55% 0.79% 0.73% 3.% -.8% Toal.%.76% 0.07% -0.07%.6% -.67%

13 The mehod proposed by Peniainen and Ranala has been used o es he sensiiviy of he esimaors when we change he emporal disribuion of he selemens. Figure and Figure 3 show ha, under he scenarios described in Figure, he precision obviously is higher when he selemens are concenraed in he early developmen years. The precision becomes very low in he mehods when he selemens occur during longer ime spans. The level of precision of he esimaors based on geomeric is superior in all he scenarios oulined. The mean percenage error and he mean square error of he esimaors provided by he mehods and mehod Bornhueer- achieve elevaed values when he emporal disribuion of selemens does no assume he canonical forms (scenarios A and B). Fig.. Scenarios of he emporal disribuion of selemens

14 Fig.. Mean square error of he esimaors by varying he emporal disribuion of he number of selemens Fig. 3. Mean percenage error esimaors produced by varying he emporal disribuion of he number of selemens Conclusion The numerical implemenaion resuls poin ou he following: he esimaing mehods produce a lower run-off error if applied o a developmen marix which, despie no respecing some of he probabilisic hypohesis of he mehod, provide a selemen disribuion according o he mechanism considered by he esimaing model; he sign of he run-off error may differ from generaion o generaion and, as a resul of compensaion, beween he individual generaions and he enire porfolio; some of he esimaing mehods, despie showing a minor disorion of he reserve esimaion for he enire porfolio, resul imprecise in he predicion of he run-off for single acciden years; he preferabiliy of he esimaing mehods did no show paricular sensibiliy o he choice of numerical values aribued o he parameers. The analysis has suggesed ha here isn a beer applicable mehod for each daa se and for each line of business. Therefore, before selecing he more coheren mehod, i is necessary o examine he daase, he run-off riangle, he underlying dynamics of he daa and he differen evoluion of he selemen mechanism of differen lines of business. References. Buhlmann, H., Sraub, E., Schnieper R. (980). Claims reserves in casualy insurance based on a probabilisic model. Mieilungen Vereinidung Schweizerischer Versicherungsmahemaier, 80 (), pp Bornhueer R.L.,, R.E. (97). The acuary and IBNR. PCAS LIX, pp Dayin C.D., Peniainen, T., Pesonen, M. (993). Praical ris heory for acuaries. Chapman & Hall. 4. England, P.D. (00). Analyic and boosrap esimaes of predicion errors in claims reserving. Insurance: Mahemaics and Economics, 3, pp Holmberg, R.D. (994). Correlaion and measuremen of loss reserve variabiliy. Casualy Acuarial Sociey Forum Spring, pp Insiue of Acuaries. (997). Claims reserving manual. Insiue of Acuaries, rd edn, London. 7. Mac, T. (994). Measuring he variabiliy of Chain Ladder reserve esimaes. Casualy Acuarial Sociey Forum Spring, pp Mac, T. (008). The predicion error for Bornhueer-. Asin Bullein, 38, pp

15 9. Narayan, P., Warhen, T. (997). A comparaive sudy of he performance of loss reserving mehods hrough simulaion. Casualy Acuarial Sociey Forum Summer, I, pp Peniainen, T., Ranala, J. (986). Run-off ris as a par of claims flucuaion. Asin Bullein, 6, pp Peniainen, T., Ranala, J. (99). A simulaion procedure for comparing differen claims reserving mehods. Asin Bullein,, pp Rubinsein, R. (98). Simulaion and Mone Carlo mehod. Wiley, New Yor. 3. Sanard, J.N. (986). A simulaion es of predicion errors of loss reserve esimaion echniques. PCAS LXXII, pp Taylor, G.C. (986). Claims reserving in non-life insurance. Norh Holland, Amserdam. 5. Van Eeghen, J. (98). Loss reserving mehods. Surveys of Acuarial Sudies,, Naional Nederland NV, Roerdam. 6. Verral, R.J. (994). Saisical mehods for Chain-Ladder echniques. Casualy Acuarial Sociey Forum Spring, pp Verral R.J. (004). A Bayesian generalised linear model for he Bornhueer- mehod of claims reserving. Norh American Acuarial J., 8, pp Wrigh, T.S. (990). A sochasic mehod for claims reserving in general insurance. Journal of he Insisue of Acuaries, 7, pp Zehnwirh, B. (994). Probabilisic loss developmen facor models wih applicaions o loss reserve variabiliy, predicion inervals and Ris Based Capial. Casualy Acuarial Sociey Forum Spring, II, pp