Analysis of Premium Liabilities for Australian Lines of Business

Transcription

1 Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao

2 Acknowledgements I am grateful to the Australan Prudental Regulaton Authorty (APRA) and the Reserve Bank of Australa for ther fnancal support through the Bran Gray Scholarshp. I would also lke to thank my supervsor Dr Allen Truslove, Mr Jacke L and Ms Anna Jones for ther helpful advce. Dsclamer The materal n ths report s copyrght of Emly Tao. The vews and opnons expressed n ths report are solely that of the author s and do not reflect the vews and opnons of the Australan Prudental Regulaton Authorty. Any errors n ths report are the responsblty of the author. The materal n ths report s copyrght. Other than for any use permtted under the Copyrght Act 1968, all other rghts are reserved and permsson should be sought through the author pror to any reproducton. Emly Tao

3 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper Background ollowng the ntroducton of APRA s new prudental regulatory framework n July 00, nsurers are now requred to value ther unexpred rsk prospectvely. A reserve referred to as Premum Lablty values the estmated losses that are expected to arse n the future from busness whch have already been underwrtten. Under General Prudental Standard (Australan Prudental Regulaton Authorty 00 (a)) general nsurers are requred to report the central estmates and rsk margns assocated wth ther Premum Lablty, and Outstandng Clams Lablty for all lnes of busness underwrtten. 1 As Premum Lablty s a lablty whch has not yet occurred, the estmaton of ts value wll requre some form of modellng. Ths future lablty s dependent on two man varables: the frequences of future clams and future clam costs for the polces currently n force. Both varables are random n nature, so there s a hgh level of uncertanty assocated wth ther predcton. Currently, there s lmted lterature avalable explorng the statstcal nature of Premum Lablty. Am and Sgnfcance of Research Ths research paper supplements the lmted pool of current lterature by presentng a theoretcal model that can be adopted to examne the statstcal behavour of future clams labltes. A smple model s constructed and used to estmate the frst two central moments of both Premum Lablty and Outstandng Clams Lablty. The results from the model wll be used to compare the varablty of Outstandng Clams Lablty to the varablty of Premum Lablty for long-taled and short taled lnes of Australan busness. Ths paper wll argue that the co-effcents of varaton of Outstandng Clams Lablty and Premum Lablty dffer and the magntude of the dfference depends on whether the lne of busness s long or short-taled. The sgnfcance of ths result s that the rato between the volatlty assocated wth the estmaton of Premum Lablty to the volatlty assocated wth the estmaton of Outstandng Clams Lablty should not be constant for all lnes of busness. The research results conclude that ths rato s larger for long-taled lnes of busness. Ths s a consequence of the averagng or poolng effects across the ndependent accdent years. The poolng effect plays a more promnent role n reducng the varablty of the Outstandng Clams Lablty of longer-taled lnes of busness. 1 Premum Lablty refers to an nsurer s clam labltes arsng from future clamable events for polces whch are currently n force. Outstandng Clams Lablty refers to an nsurer s clam labltes for clamable events whch have already occurred, but have not been notfed to the nsurer. Emly Tao

4 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper Research Approach Model for calculatng future clam payments: The model adopted requres the nput of run-off clams data and s desgned to calculate the mean and varance for total clams payments n any sngle accdent year. Usng ths approach, the model can be appled to a range of dfferent stuatons and allows further assumptons to be ncorporated on a case-by-case bass. It has been assumed that the volatlty of future clam payments n each accdent year s affected by only two ndependent sources of uncertanty: 1. The number of clams that wll occur, whch wll be denoted by the random varable N and;. The average sze of future clam payments, whch wll be denoted by the random varable X. The model s constructed as follows where the random varable S s used to denote the total dollar amount of payments made for clams ncurred durng accdent year,. Total Clam Payments (S ) = Total number of clams (N ) x Average cost per clam (X ) N(µ 1, σ 1 ) N(µ, σ ) Expected value = µ 1 µ Varance derved by smulaton The model s based on the assumpton that the total clam payments made n any accdent year s equal to the product between the total number of clams whch occurred and the average clam payments made n respect of those clams. or smplcty s sake, the model assumes that the behavour of the two ndvdual random varables follow a normal dstrbuton. Usng run-off data collected from reported clams, the values of N and X can be calculated usng the Chan Ladder Method for the prevous number of accdent years for whch data has been collected. A trend lne can be ftted across these data ponts and then projected forward to obtan the future expected values of N and X. The volatlty can be measured by calculatng the devaton of Emly Tao

5 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper the data ponts from the trend lne. N and X can now be modelled as two ndependent normal random varables wth a mean and varance estmated by the past data. An example of ths analyss s seen below n gure 1. Motor Vehcle 1,500,000 Total Number of Clams Reported 1,400,000 1,300,000 1,00,000 1,0,000 1,000, , , , ,000 y = 4468x - 9E , Accdent Years gure 1 Ths method wll be adopted to fnd the mean and varance of Premum Labltes and wll requre the estmaton of ( N ), V ( N ), E( X ), V ( X ) for the subsequent accdent year. E and As N And, and X are assumed to be ndependent, t follows that: ( S ) E( N ) E( X ) E = ( S ) V ( N X ) V = Values for the mean and varance of N and X have already been estmated by the model. The assumpton of normalty allows values of N and X to be generated by smulaton and multpled wth each other to produce smulated values of S. The varance of a large sample of smulated values s used as an approxmaton to the varance of S. Assumng that ndvdual accdent years are ndependent and dentcally dstrbuted, the model can also be used to estmate the mean and varance of the correspondng Outstandng Clams Lablty. Ths allows for comparson between the co- Emly Tao

6 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper effcents of varaton of Premum Lablty and Outstandng Clams Lablty for dfferent lnes of busness. Data The Insurance and Superannuaton Commsson, many of the functons of whch now form part of APRA, released detaled clams analyss data for general nsurance lnes of busness over the perod coverng three lnes of busness: motor vehcle, publc lablty, and compulsory thrd party (Insurance and Superannuaton Commsson ). Premum Lablty Total Number of Clams - N The trendlne s a functon of the accdent years and s representatve of the expected total number of clams reported n each accdent year. It s expressed as Y = ax + b, where Y s the total number of clams reported and X s the accdent year. Once the values of a and b are calculated, the trendlne can be used to forecast the expected value of N for the subsequent accdent year. In the model used for ths research, X s replaced by the year 1997 (nsurance data s only provded up to 1996). Ths value s the expected total number of clams n the followng accdent year E N. ( ) As the data collected represents the aggregate experence of the prvate nsurance sector, t s reasonable to assume that all un-systematc rsks have been dversfed away, so that the devatons from the trendlne (the mean) correspond only to the systematc rsk or the nherent uncertanty assocated wth estmaton of the lablty. On the assumpton that the ndvdual accdent years are ndependent, the varance of the total number of clams for each ndvdual accdent year can be defned as: ( N ) V = ( devaton of 1 data pont 14 from the trendlne for = 1983, 1984,,1996 Two degrees of freedom are lost n the denomnator due to the estmaton N V N. of the two parameters for E ( ) and ( ) Average Payment per Clam - X Emly Tao

7 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper The average payment per clam n accdent year s defned as: X = total clam payments for accdent year total number of clams settled n accdent year P δ = The clam payments are recorded as nomnal amounts. In order to consstently compare the payments made at dfferent tmes, the clam payments are adjusted to a stable currency. Average Weekly Earnngs (AWE) s used as the nflaton ndex. Payments are assumed to be made n the mddle of each development year. The AWE ndex s preferred over the Consumer Prce Index, the typcal measure of nflaton, due to the large for average payment per clam are calculated n the same manner usng process whch calculated E ( N ) and V ( N ) for total number of clams n the prevous secton. proporton of the clam costs beng wages related. ( ) Outstandng Clams Lablty E X and V( X ) Whle Premum Lablty estmates the clam payments wth respect to a sngle accdent year n the future, Outstandng Clams Lablty estmates future clam payments for clams whch occurred over a number of earler accdent years. or each of these accdent years, only a porton of the total clam payments remans outstandng; the other porton has already been reported at the tme of valuaton. The number of accdent years that the Outstandng Clams Lablty spans s dependent on the nature of the labltes. or long-taled lnes of busness such as publc ndemnty, ths number s qute hgh (typcally n excess of 7 years) and for short-taled clams such as motor vehcle, ths number s much smaller (1 to years). Assumng that the ndvdual accdent years are ndependent, the varablty of the Outstandng Clams Lablty wll be reduced comparatvely to the correspondng Premum Lablty due to the effects of poolng across the ndependent accdent years. It can be shown that ths averagng affect becomes more domnant when the Outstandng Clams Lablty contrbutes to a bgger proporton of the total clam payments or when the Outstandng Clams Lablty spans over a hgher number of accdent years. In other words, there s a larger comparatve reducton n volatlty between Outstandng Clams Lablty and Premum Lablty for longer-taled lnes of busness compared to short-taled busness. Avalable data covers 14 accdent years and development years. Based on ths data, the ratos between the co-effcents of varaton between Premum Lablty and Outstandng Clams Lablty are compared for short taled and long taled lnes of busness usng the followng formulas. Emly Tao

8 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper Let S be defned as the total clam payments of accdent year, whch s a random varable. Let f,j be defned as the proporton of S that s pad n development year j. nally s defned as the sum of f,1, f,11,, and (.e. = f, j = outstandng proporton). f, j= 1 It can be seen that the total Outstandng Clams Lablty, OS at a pont n tme after the 14 accdent years s equal to 14 S and ts coeffcent of = varaton s: CV ( OS ) ( OS ) ( OS ) Var = = Ε = = Var Ε ( S ) ( S ) Let P be the Premum Lablty, the coeffcent of varaton at the same tme s: CV ( P) = ( P) ( P) Var Ε Assumng that for each accdent year Ε ( S ) = μ and Var( S ) σ can be derved as follows: =, the results It then follows that: CV( P) σ μ = and CV ( OS ) = σ μ = = CV ( P ) > CV ( OS ) Results The three lnes of busness under consderaton are compulsory thrd party, publc lablty and motor vehcle. The frst two lnes are consdered to produce long-taled clams whle the motor vehcle busness s short-taled n nature. The Chan Ladder method (Buchanan, Hart et al. 1996) s adopted to estmate the Outstandng Clams Lablty and the followng table presents the results. Emly Tao

9 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper = = CV ( P ) (%) CV ( OS ) (%) ( ) CV P ( ) CV OS Publc lablty CTP Motor vehcle The results clearly show that the averagng effects are much greater for the long-taled busnesses. In other words, the rato between the co-effcents of varaton of Premum Lablty to that of Outstandng Clams Lablty for long-taled labltes s greater than the rato for short-taled rsks. Implcatons for Prudental Supervson The Rsk Captal actors are prescrbed by APRA for the calculaton of the Insurance Rsk Captal Charge (Australan Prudental Regulaton Authorty 00 (b)). The Rsk Captal Charge needs to reflect the nherent uncertanty assocated wth the estmaton of the underlyng lablty and should therefore be proportonal to the co-effcent of varaton of the future lablty. The results suggest that the Premums Lablty Rsk Captal actors prescrbed by APRA to calculate the Insurance Rsk Captal Charge are nadequate for the more volatle lnes of busness. The Premums Lablty Rsk Captal actors set by APRA are 150% of the correspondng Outstandng Clams Lablty Rsk Captal actors for all lnes of busness. Ths constant 50% loadng would only be reasonable f the varablty of the Premums Lablty exceeds the varablty of the Outstandng Clams Lablty by the same amount for all lnes of busness. Ths s not the case as the results show that the co-effcent of varaton of the Premum Labltes exceeds the co-effcent of varaton of Outstandng Clams Lablty by a much greater amount for the longer-taled rsks. The Insurance Rsk Captal Charge s a margn held to buffer the rsk that the actual value of the lablty s greater than the expected value calculated by the nsurer. Emly Tao

10 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper Concluson The rapd advancement of technology has allowed for the constructon of more ntrcate and realstc models. Although these models can produce results wth hgher degrees of accuracy, they are often much more complex and data ntensve. If the rght assumptons are not used or the data s unrelable, these models may smply produce more statstcal nose. Ths paper has provded a startng pont to Premum Lablty modellng by ntroducng a basc model wth relatvely few assumptons. Ths paper has also hghlghted a relatonshp between the co-effcents of varaton for Outstandng Clams Lablty and Premum Lablty. The results show that the co-effcent of varaton of the Premums Labltes for some lnes of busness s naccurately reflected by Premums Lablty Rsk Captal actors whch all have a constant 50% loadng. Ths s a result of the averagng or poolng effects across the ndependent accdent years, and plays a more promnent role n long-taled lnes of busness. The predcton of future clams lablty s hghly speculatve by nature and ths paper has only provded an ntroducton to ts estmaton. It s hoped that further research wll construct and test more sophstcated models whch ncorporate addtonal assumptons and parameters. Such analyss wll mprove our understandng and further our ablty to predct and manage the random varaton of these labltes. Emly Tao

11 Summary of Analyss of Premum Labltes for Australan Lnes of Busness / Honours Research Paper References Australan Prudental Regulaton Authorty (00(a)). General Insurance Prudental Standards GPS - Lablty Valuaton for General Insurers. Australan Prudental Regulaton Authorty (00(b)). Gudance Note GGN Insurance Rsk Captal Charge. Buchanan, R. A., D. G. Hart, et al. (1996). The Actuaral Practce of General Insurance. Sydney, Insttute of Actuares Australa. (pp. 9 3) Insurance and Superannuaton Commsson ( ). Selected Statstcs on the General Insurance Industry for the year ended Emly Tao