Measuring the Services of Property-Casualty Insurance in the NIPAs

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1 1 Ocober 23 Measuring he Services of Propery-Casualy Insurance in he IPAs Changes in Conceps and Mehods By Baoline Chen and Dennis J. Fixler A S par of he comprehensive revision of he naional income and produc accouns (IPAs) ha is scheduled o be released on December 1, 23, a change in he definiion of propery and casualy insurance services will be inroduced. This definiional change will recognize he implici services ha are funded by invesmen income, will adop a reamen of insured losses ha is more consisen wih he economic behavior of he insurer, and will change he reamen of reinsurance. This change is briefly described in he June 23 issue of he SURVEY OF CURRET BUSIESS, and some of he associaed changes in he ables are described in he Augus 23 issue. 1 The Bureau of Economic Analysis (BEA) currenly measures services of he propery-casualy insurance indusry as is ne premiums earned minus ne losses incurred and dividend o policyholders, where ne premiums and losses refer o premiums and losses ne of reinsurance. However, he insurance oupu measured using he curren definiion does no include all he services provided by he propery-casualy insurance companies. Propery-casualy insurance companies provide hree ypes of services: Risk-pooling, financial services relaing o insured losses, and inermediaion. Insurance provides a mechanism for consumers, businesses, and governmen ha are exposed o propery-casualy losses o engage in risk reducion hrough pooling. The insurer provides a variey of real services for policyholders, such as loss selemens, risk surveys, and loss prevenion plans. The insurer collecs premiums in advance of he loss paymens and holds he funds in reserves unil he claims are paid. The insurer also provides inermediaion services hrough he inves- 1. See Moulon and Seskin (23, 19 23) and Mayerhauser, Smih, and Sullivan (23, 21). Karla Allen and Brad Gabel significanly conribued o he consrucion of he daa ses and he preparaion of he esimaes. Chrisian Ehemann provided valuable commens and suggesions for he projec. Arnold Kaz, Clinon McCully, and Bren R. Moulon paricipaed in many helpful discussions. men of he funds in reserves. e gains from he invesed funds in reserves are used o supplemen revenue from premiums o pay for claims or for reinsurance services; in oher words, policyholders pay a smaller premium in order o compensae for he opporuniy cos of heir funds ha are held by he insurer. According o various sudies ha focus on he performance of propery-casualy insurance services, he provision of hese services of financial proecion and financial inermediaion represens he oupu of he propery-casualy insurance indusry (Cummins and Weiss, 2). Replacing he acual losses incurred wih he normal losses in he calculaion of insurance services is a major innovaion in he definiional change. ormal losses represen he incurred losses ha he insurer expecs o pay (payable claims). This change in he reamen of losses recognizes ha because acual losses incurred are only known afer hey occur, insurance companies deermine he premiums for an upcoming period on he basis of heir percepion of he losses ha hey may incur. The new reamen eliminaes he large swings in measured insurance services ha are caused by caasrophes, such as he orhridge earhquake in 1994, Hurricane Andrew in 1992, and he erroris aacks on Sepember 11 h, 21. Anoher significan aspec of he definiional change is he use of expeced invesmen income as a measure of premium supplemens. Premium supplemens are he componen of implici services arising from he invesmen income earned from he invesmen in reserves. The inclusion of premium supplemens is found in he measure of insurance oupu in he Unied aions Sysem of aional Accouns (SA, 1993), bu is inclusion in he BEA measure of oupu is new. Economic models on he behavior of he insurer generally recognize ha insurance companies maximize heir profis by seing premiums ha are based on heir expecaions of fuure losses and invesmen reurns. The use of expeced, raher han he acual, invesmen income o measure premium supplemens is inended o beer capure he economic behavior of he insurer.

2 Ocober 23 SURVEY OF CURRET BUSIESS 11 A much-debaed issue abou he componens of invesmen income is wheher capial gains and he income on own funds should be included. In he SA, invesmen income is defined as he ineres and dividend income earned on echnical reserves, which are he unearned premiums plus unpaid losses. In he esimaion of expeced invesmen income, ne realized capial gains are included in invesmen income. Fixler and Moulon (21) argue ha capial gains should be included because he supply price of many services, such as financial services, is based on expeced capial gains. Hill (1998) also suggess ha capial gains should be reaed he same as invesmen income. Anoher issue in he compuaion of invesmen income is he reamen of mandaed reserves and ownfunds. In he Unied Saes, he saes have regulaory auhoriy over he operaions of insurance companies, and in many cases, hey mandae he holding of reserves and how he reserves mus be invesed. Such reserves do no appear as separae enries in he indusry consolidaed balance shees. In principle, he invesed mandaed reserves should be reaed he same as oher componens of echnical reserves. Invesmen income from he insurer s own-funds is no a componen in he premiums supplemen and is repored separaely from he invesmen income from he policyholders funds, or echnical reserves, on he insurer s annual saemen. However, because invesmen funds are fungible, he indusry-level rae of reurn o invesed funds is compued wih invesmen income from boh he insurer s own funds and he policyholders funds. Currenly, insurance services are calculaed from daa on premiums earned and losses incurred ne of reinsurance assumed and ceded. 2 This reamen of reinsurance is based on he assumpion ha reinsurance services are expors or impors or ha reinsurance assumed offses reinsurance ceded wihin a paricular line of insurance. However, his assumpion is incorrec because some domesic insurance companies specialize in reinsurance services and because he daa indicae ha reinsurance assumed seldom offses reinsurance ceded wihin a paricular line of insurance. Because insurance companies purchase reinsurance o reduce he risk ha hey mus bear in he even of greaer han expeced losses, such services will be reaed as an inermediae inpu o he insurance carriers indusry or as expors of services. Under he new definiion, services of he properycasualy insurance indusry will be measured as direc 2. Reinsurance is he purchase of insurance by an insurer. The buyer of he reinsurance is known as he ceding insurer and he seller of he insurance is he assuming insurer. premiums earned plus premiums supplemens minus normal losses incurred and dividends o policyholders. Direc premiums earned equal ne premiums earned plus premiums received from reinsurance assumed minus premiums paid for reinsurance ceded. Furher discussion on he definiional change and is impac on he naional income and produc accouns can be found in Moulon and Seskin (23). This aricle discusses he mehodology used o incorporae he expecaion behavior of he insurer ino he insurance oupu measure. Secion 2 focuses on he esimaion of normal losses and expeced invesmen income. I describes he expecaion behavior of insurers regarding heir fuure losses and heir fuure invesmen income, and i discusses he saisical mehodology for esimaing he normal losses and expeced invesmen income. Secion 3 discusses he effec of he definiional change on he measured propery-casualy insurance services. Secion 4 provides he concluding remarks. The aricle includes a echnical noe ha provides deails on he daa sources and daa preparaion for implemening he definiional changes. Esimaion of ormal osses and Expeced Invesmen Income To se premiums for a fuure period, profi-maximizing insurance companies mus esimae heir expeced invesmen income, heir normal losses, and heir operaing expenses. The imporance of expecaions is generally acceped, bu how expecaions on fuure losses and fuure invesmen income are formed is sill debaable. Two expecaions models ha may explain he insurer s behavior are he adapive expecaions model and he raional expecaions model. In a simple adapive expecaions framework, individuals adjus heir expecaions according o he deviaions of heir expecaions from heir acual experiences. In oher words, individuals adap heir expecaions according o he forecas errors. Specifically, he expecaion for he nex period is a weighed average of he acual experience in he curren period and he forecas error for he curren period. If expressed recursively, he expecaion for he nex period is a weighed average of he curren experience and of all pas experiences. The weighs on he lagged experiences decline exponenially, emphasizing he imporance of he more recen experiences in he formaion of expecaions. Adapive expecaions behavior seems consisen wih he observaion ha insurance companies esimaes of fuure losses are primarily based on heir pas

3 12 Measuring Insurance Services Ocober 23 losses. When evaluaing pas losses, he insurer accouns for facors, such as he characerisics of he insured, ha consisenly govern he general behavior of he insured over ime oward he insured risks. The insurer also accouns for recen regulaory and echnological changes ha may have affeced recen incurred losses. For example, if here were a recen change in he penaly for drunk driving, hen i would likely affec he recen number of accidens caused by drunk drivers. Recen advances in echnology in he insurance indusry have resuled in beer risk surveys and loss prevenion programs ha are likely o have helped reduce losses. Such facors sugges ha more recen loss experiences provide more informaion abou curren rends in losses, and hence, more recen loss experiences should carry more weigh in he formaion of expecaions on fuure losses. Similarly, curren and pas invesmen income provides a major source of informaion o insurance companies when hey esimae invesmen income for he fuure periods. However, because oher facors, such as he recen performance of he economy or recen changes in ax policy on invesmen income, may have more influence on he curren rend in invesmen income, recen invesmen experiences should be more imporan in he formaion of expecaions on fuure invesmen income. The adapive expecaions model is a sraighforward way o explain expecaions behavior, bu Muh (1961) poined ou ha his model lacks a heoreical basis, and he proposed a raional expecaions framework. Raional expecaions heory implies ha economic behavior underlies he formaion of expecaions, and expecaions are based on all he informaion ha is available when he expecaions are formed. To be consisen wih his heory, a srucural model ha seeks o explain he insurer s expecaions of fuure losses should include pas experiences and variables, such as he prices of maerials and services ha largely comprise loss paymens, number of policyholders, rends in rulings of cours oward legal liabiliies, and oher variables ha may affec fuure losses. Similarly, in addiion o curren and pas invesmen income, a srucural model ha seeks o explain an insurer s expecaion of fuure invesmen income should include variables such as ineres raes, he rae of change in echnical reserves, he rae of inflaion, sock indexes, he rae of growh in real GDP, and oher macroeconomic variables ha may affec fuure invesmen income. Raional expecaions models are echnically difficul o esimae. Firs, an economic opimizaion model mus be specified, and esimaion mus be preceded by an analyical soluion o he model. Even when he soluion is linear in he exogenous variables of he model, he coefficiens are ofen combinaions of he srucural parameers ha are generally no linear and are difficul o esimae. Second, because of he likely serial correlaion in he srucural disurbances, assumpions abou he auocorrelaion srucure are necessary. 3 Third, here is lile consensus on a srucure model ha correcly includes all relevan variables and ha properly explains heir ineracive roles in he formaion of he insurer s expecaions on fuure losses or on fuure invesmen income. Because of hese difficulies, he focus is on he roles of curren and pas losses and invesmen experiences, and he adapive expecaions model is used. Despie he heoreical weakness of his model, empirical evidence indicaes ha i works quie well in many economic applicaions. Esimaing normal losses or expeced invesmen income is essenially a forecasing problem. ormal losses are fuure losses ha are expeced o be paid by he insurer, and hence, saisically, hey are he forecass of fuure losses. Similarly, expeced invesmen income is he forecas of fuure invesmen income. A forecasing mehod ha is consisen wih he adapive expecaions framework is a weighed moving average model wih weighs on he lagged observaions declining exponenially. 4 An alernaive o his mehod is he n-poin simple moving average mehod, which has been used by he Ausralian Bureau of Saisics (1999). Time series mehods, such as Auoregressive Inegraed Moving Average mehods (ARIMA) are also alernaives for forecasing fuure losses and expeced invesmen income. 5 A common feaure of hese mehods is ha fuure values of a series depend only on is lagged values. Choices of saisical mehods Under he definiional change, he services of 22 lines of propery-casualy insurance is being remeasured. (A lis of he 22 lines is included in he echnical noe.) According o he published records, daa series for hese lines span from 18 o 72 years. Some lines of insurance services exhibi auocorrelaion and possible 3. Such assumpions are generally arbirary. Even when a simple auocorrelaion srucure of he disurbances is imposed, i may no be enough o simplify esimaion. Oher hypoheses abou he auocorrelaion funcion of he srucure disurbances may make i impossible o idenify he srucure parameers or complicae esimaion. 4. This weighed moving average mehod is also known as exponenial smoohing or exponenially weighed moving average mehod. In fac, Muh (196) shows ha if here is no rend and no seasonaliy, hen his model is an auoregressive inegraed moving average (ARIMA) model wih nonseasonal difference, an MA(1) erm, and no consan erm, oherwise known as ARIMA(, 1, 1). Thus, poenially more sophisicaed ARIMA modeling, or Box-Jenkins mehods, can be explored. 5. ARIMA mehods are developed for esimaing concise predicion models of ime series daa ha display complex paerns of auocorrelaions.

4 Ocober 23 SURVEY OF CURRET BUSIESS 13 heeroskedasiciy in he residuals in he daa series on losses and invesmen income. 6 Iniial experimenaion indicaed ha he search for an opimal ARIMA model o fi he daa for each of he 22 lines of propery-casualy insurance would be difficul and cosly. In addiion, o updae he esimaes annually for each line of insurance when new daa become available would add significanly o he coss of producing he naional income and produc accouns. The weighed moving average models focus on he rends and seasonal behavior of he daa. Because hese wo aspecs largely deermine he variance of he series, when chosen properly, he weighed moving average mehod performs well, relaive o more complicaed mehods, on a wide range of daa series. The weighed moving average model wih no rend and no seasonal facors requires he esimaion of a single parameer. Specifically, he mehod can be viewed as esimaing he value of α ha bes fis: Z = w 1 Z 1 + w 2 Z e, where w i = α(1 α) i 1, for i = 1,..., and e is a whie noise disurbance erm. This formula is idenical o ha derived from he adapive expecaions model developed by Cagan (1956). The n-poin simple moving average mehod is based on he assumpion ha he ime series is locally saionary wih a slow varying mean. Hence, he moving average of n mos recen observaions are used o esimae he curren value of he mean, and his mean is used as he forecas for he nex period. This mehod is a compromise beween he mean and random walk models. 7 The shor-erm averaging smoohs ou he bumps in he original series. By adjusing he degree of smoohing, n, one hopes o srike an opimal balance beween he mean and random walk models. The choice for he n-poin average is beween a lagged moving average or a cenered moving average. The Ausralian Bureau of Saisics (1999) chooses o use he cenered moving averages, implying ha he forecas of losses for period would be influenced by losses in he fuure periods. To avoid he influence of fuure evens on he formaion of expecaions, he lagged moving averages were used for forecasing fuure losses Auocorrelaions summarize emporal persisence of he ime series, such as rend, cycle, and seasonal variaions. 7. The mean model uses he mean of he enire sample as he esimaed value for each period in he sample. The random walk model predics ha one period s value will equal he previous period s value plus a consan represening he avearge change beween periods. 8. In a cenered moving average, he esimae for depends on values n/2 and values +n/2(wih n being an even number), and he +n/2 values would be inconsisen wih he esimaion of expecaions in. Compuaionally boh mehods are simple o implemen. An advanage of he weighed moving average mehod is ha he small se of model choices simplifies he process of choosing he bes model and makes i ideal for fairly small daa series. The disadvanage of he n-poin simple moving average mehod is ha he choice of n largely depends on subjecive judgmen because his mehod is no based on any saisical modeling. The common disadvanage of any moving average mehod is ha he forecass generaed from such a mehod will lag as he rend of he acual daa increases or decreases. Concepually, he weighed moving average mehod is superior o he n-poin simple moving average mehod because i places relaively more weigh on he mos recen observaions, whereas he n-poin simple moving average mehod places equal weigh on he n lagged observaions and excludes all observaions more han n periods back in ime. Moreover, he weighed moving average mehod relies on a smoohing parameer ha is esimaed from he enire ime series and ha is geared oward minimizing he mean square predicion errors. In order o evaluae he wo moving average mehods, normal losses and expeced invesmen income for five lines of insurance services were compued, and he summary saisics of he forecas or predicion errors were compared. The five lines of insurance services in he experimen are privae passenger auo liabiliy (PA), privae passenger auo physical damage (PAD), homeowners muliple peril (HMP), farmowners muliple peril (FMP), and workers compensaion (WCP). These lines were chosen because of heir significan shares in he propery-casualy insurance indusry. In 2, hese five lines accouned for 62 percen of he oal premiums earned by he indusry, and hey accouned for more han 85 percen of he premiums recorded in personal consumpion expendiures in he naional income and produc accouns. Compuing normal losses The daa series ha were available for he experimen were direc premiums earned and direc losses incurred from 1972 o 21. Time series daa on direc premiums and losses for almos all he lines of propery-casualy insurance services are highly nonsaionary. 9 In order o obain more saionary daa and o be able o incorporae informaion from direc premiums earned, he variable direc losses incurred, was redefined as he produc of direc premiums earned, P, and he direc loss raio, l = /P. Thus, he esimaes of normal losses were no compued only from direc 9. A nonsaionary ime series exhibis srong rend, and is mean and variance vary wih ime.

5 14 Measuring Insurance Services Ocober 23 losses incurred. Insead, expeced loss raios were firs esimaed from daa on direc premiums earned and direc losses incurred, and hen esimaes of normal losses were derived. e l +1 be he expeced, or he forecased, loss raio for period +1, given he informaion available in period, and le +1 be he normal losses for period +1. Formally, normal losses in period +1 can be expressed as: (1) + 1 = l + 1 P + 1, where l +1 is compued as: (2) l + 1 = El ( + 1 l, l 1,... ). The weighed moving average model discussed above akes he form (3) El ( + 1 l, l 1,...) = αl + ( 1 α)el ( l 1, l 2,...) = ασ i = ( 1 α) i l i, where α is he smoohing consan in he inerval (, 1). The expeced loss raio for period +1 can be calculaed as he weighed sum of he loss raio a period and he forecas of he loss raio for period, given informaion a 1. Expressed recursively, he loss raio a period can be calculaed as he exponenially weighed sum of loss raios of all previous periods. The smoohing parameer, α, can be esimaed fairly well if a daa series has a leas 3 observaions and is free of serial correlaion. The WinRATS 3.2 Version 5.1 program was used o esimae α, which chooses he esimae of α, αˆ, by minimizing in-sample, one-sep forecas errors. However, if he daa series is no long enough or if i exhibis serial correlaion, hen seing α o a reasonable value produces more reliable resuls han relying on imprecise esimaes. According o he saisical and engineering lieraure, he value of α is ofen chosen beween.1 and.3. Some sudies poin ou ha an esimaed value of αˆ greaer han.3 may sugges serial correlaion in he daa series. Esimaing normal losses wih he weighed moving average model involves wo seps. The firs sep is o esimae α and o generae forecass of loss raios. If he esimaed value of α, αˆ, does no sugges serial correlaions in he daa, hen αˆ is used o generae forecass of loss raios, l +1 ( αˆ ). If αˆ indicaes serial correlaions in he daa, hen α is chosen in he inerval (.1,.3) o generae loss raio forecass, and he chosen α value, α, is he one wih he minimum roo mean square predicion errors (RMSPE). 1 One may experimen wih many values of α in he specified range. The resuls wih α = (.1,.2,.3) indicae ha hese hree choices are sufficien. The second sep is o compue normal losses, +1 = l +1 P +1, and he summary saisics of he in-sample, one-sep forecas errors. In he experimen, esimaion resuls sugges ha αˆ = (.34,.19) for HMP and FMP, respecively. For PA, PAD, and WCP, αˆ indicaes serial correlaion in he daa, so he value of α was se. Based on he minimum RMSPE crierion, α =.3 was se for PA, PAD, and WCP. The n-poin simple moving average mehod is sraighforward o implemen. The expeced loss raio for period +1 is given by: (4) El ( + 1 l, l 1,..., l n+ 1 ) = 1 n 1 --Σ n i = l i. The main concern wih his mehod is he choice of n. An opimal n should smooh ou he bumps in he daa ha are generaed by shor-erm noise bu sill preserve he dynamic characerisics of he ime series. However, here is lile discussion in he lieraure on he crierion for choosing an opimal n, perhaps because he n-poin moving average mehod is no based on a formal saisical model. For he comparison of he wo ypes of moving averages, n = 5 was seleced for each line of insurance. This selecion is consisen wih he choice of α because four of he five lines of insurance in he experimen are eiher.3 or close o.3, implying ha he firs five lagged loss raios accoun for more han 83 percen of he forecased loss raios. An added consideraion is ha he Ausralian Bureau of Saisics ses n = 5 for is forecass of fuure losses. 11 Using eiher moving average mehod, he esimaion of expeced losses requires a plan for handling caasrophic losses. By definiion, hese caasrophes are unpredicable evens ha have significan effecs on losses. Some of he five lines of insurance ha were examined have experienced caasrophic losses. For example, homeowners muliple peril (HMP) experienced caasrophic losses in 1992 because of Hurricane Andrew, and he loss raio for 1992 reached Unless adjused for, caasrophic losses can have oo much influence on he compuaion of expeced losses and measured oupu. Accordingly, he following seps were aken o dampen he 1. Roo mean square predicion error is he square roo of he average of he squared differences beween he acual values and he prediced values for he sample period. 11. BEA s inernaional ransacions accouns recenly adoped a 6-year moving average because of he paricular feaures of heir daa series. (Bach 23).

6 Ocober 23 SURVEY OF CURRET BUSIESS 15 l -1 ( α =.3) yields he smalles RMSPE of all he choices of α values. The weighed moving average mehod ou performed he 5-poin moving average mehod in four of he five cases. To furher compare he wo moving average mehods, able 2 provides he summary saisics ha are ofen used o measure he performance of forecass: Mean error (ME), mean absolue error (MAE), mean absolue percenage error (MAPE), sandard deviaion of predicion error (SDPE), and roo mean square of predicion error (RMSPE). Since posiive deviaions end o offse negaive deviaions, MAE is ofen used o measure he accuracy of he forecased ime series values, in addiion o ME ha measures he average forecasing error. MAPE is a uni free measure of he accuracy of he forecass; i convers deviaions in any uni measuremen o average percenage deviaions. SDPE measures he dispersion of he forecas errors, and RMSPE ac- effec of caasrophic losses. Firs, he expeced loss raios using he sample daa were compued, and he daa for he year of he caasrophe were reaed as missing observaions. Second, he caasrophic loss was compued as he difference beween he acual loss raio and he esimaed loss raio. Third, he caasrophic loss was spread forward equally for 2 years, saring from he caasrophic year. For example, for HMP, using he weighed moving average mehod, he adjusmen for he caasrophic loss is compued as, l=(l 1992 l ( α )) / 2, and using he n- poin moving average mehod, he adjusmen is compued as l=(l 1992 l ,1987 (n = 5))/ 2. The adjusmen for caasrophic losses, l, is hen added o he forecass of loss raios for 1992 hrough 211. In able 1, he RMSPE for he lines of insurance in he experimen wih α = (.1,.2,.3) is compared wih he RMSPE for hose lines wih n = 5. oe ha if α can be esimaed as in he cases of HMP and FMP, l -1 ( αˆ ) yields he minimum RMSPE. If α canno be esimaed as in he cases of PA, PAD and WCP, Table 1. Roo Mean Square Predicion Errors (RMSPE) from oss Raio Forecass, Using Weighed and 5-poin Moving Averages α = αˆ α =.1 α =.2 α =.3 n = 5 Privae auo liabiliy Privae auo physical damage Homeowners muliple peril αˆ =.34 Farmowners muliple peril ( ) ( αˆ =.19) Workers compensaion *7.28 * * * * * Indicaes he lowes RMSPE in each column. Roo mean square predicion error (RMSPE) is he square roo of he average squared difference beween he acual value and he predicion value for he sample period. couns for boh he mean and he dispersion of he forecas errors. For each line, he summary saisics from he forecass were compared, using he weighed moving averages and choosing α based on he minimum RMSPE crierion. Summary saisics were also compued from he forecass using he 5-poin moving averages. Columns 2 and 3 conain he summary saisics from he forecass of loss raios and Columns 4 and 5 conain Table 2. Summary Saisics of Forecasing Errors from Weighed Moving Averages and 5-Poin Moving Averages [Forecas errors of normal losses are measured in millions of dollars] A. Privae passenger auo liabiliy insurance ( ) Forecas errors of Forecas errors of expeced loss raio expeced loss raio ( α =.3) ( n = 5) (percen) (percen) Forecas errors of Forecas errors of normal losses normal losses ( α =.3) ( n = 5) ME MAE , , MAPE SDPE , , RMSPE , ,1.67 B. Privae passenger auo physical damage ( ) Forecas errors of Forecas errors of expeced loss raio expeced loss raio ( α =.3) ( n = 5) (percen) (percen) Forecas errors of Forecas errors of normal losses normal losses ( α =.3) ( n = 5) ME MAE MAPE SDPE , ,25.65 RMSPE , ,25.66 C. Homeowners muliple peril ( ) Forecas errors of Forecas errors of expeced loss raio expeced loss raio ( αˆ =.34) ( n = 5) (percen) (percen) Forecas errors of normal losses αˆ =.34 Forecas errors of normal losses ( = ) ME MAE , ,738.7 MAPE SDPE , , RMSPE ,45.8 2, D. Farmowners muliple peril ( ) Forecas errors of Forecas errors of expeced loss raio expeced loss raio ( αˆ =.19) ( n = 5) (percen) (percen) Forecas errors of Forecas errors of normal losses normal losses ( αˆ =.19) ( n = 5) ME MAE MAPE SDPE RMSPE E. Workers compensaion ( ) Forecas errors of Forecas errors of expeced loss raio expeced loss raio ( α =.3) ( n = 5) (percen) (percen) ( ) n 5 Forecas errors of normal losses α =.3 Forecas errors of normal losses ( = ) ( ) n 5 ME MAE , , MAPE SDPE , ,5.74 RMSPE , ,5.87 Forecas errors of loss raio is l l 1. Forecas errors of normal losses is 1. ME is mean error of forecass MAE is mean absolue error of forecass MAPE is mean absolue percenage error of forecass SDPE is sandard deviaion of forecasing errors RMSPE is roo mean square predicion errors

7 16 Measuring Insurance Services Ocober 23 he summary saisics from he derived normal losses. The summary saisics indicae ha he weighed moving average mehod performed beer over all. If he smoohing parameer can be esimaed ha is, if he loss raio daa series does no exhibi serial correlaion he weighed moving average mehod clearly ou performs he 5-poin moving average mehod. The beer performance can be seen by comparing he summary saisics for HMP and FMP in pars C and D in able 2. For PA, PAD and WCP, where he smoohing parameer canno be esimaed, by seing α =.3, he weighed moving average mehod resuled in smaller MAE, MAPE, SDPE and RMSPE for PA and WCP from esimaed loss raios and derived normal losses. For PAD, he 5-poin moving average performed beer for esimaing loss raios, bu he weighed moving average produced a smaller RMSPE from compued normal losses, because he compued normal losses incorporae informaion from curren premiums. To illusrae he esimaion resuls obained from using he weighed moving averages and seing α according o he RMSPE crierion, in panels 1.1 o 1.5 in char 1, he acual loss raios are compared wih he forecass of loss raios for he five lines of insurance in he experimen. In panels 2.1 o 2.5 in char 2, he acual direc losses are compared wih he normal losses which are compued according o equaion (1). Compuing expeced invesmen income Daa on invesmen income are labeled as ne invesmen gain on funds aribuable o insurance ransacions, and hey are included in par II of he insurance expendiure exhibis (IEE) published in he Bes s Aggregae and Averages: Propery-Casualy by A.M. Bes Company. The ne invesmen gain on funds aribuable o insurance ransacions by line of insurance is defined as he produc of he indusry-level rae of reurn o invesed funds and he echnical reserves by line of insurance adjused for uncolleced premiums and for he expenses associaed wih unearned premiums. 12 The ne invesmen income for he curren year includes ne realized capial gains. The measuremen of invesmen income here is he same as ha used in he producer price index for propery-casualy insurance from he Bureau of abor Saisics (BS). Insurance companies ofen analyze heir invesmen experiences on he basis of he invesmen income o premium raios. e I denoe he invesmen income, and le i = I /P denoe he invesmen income o premiums raio in period. For each line of insurance, direc premiums earned plus premiums supplemens in 12. The compuaion of invesmen income used he formula developed by he aional Associaion of Insurance Commissioners. period, P + I, can be expressed as P (1 + i ), which corresponds exacly o he price measure used by BS in he producer price index for propery-casualy insurance. Using his characerizaion allows he BS index o deflae he measure of he curren-dollar insurance oupu. e i +1 be he expeced invesmen income o premiums raio for period +1, given he informaion available in period, and le I +1 be he expeced invesmen income for period +1 given by: (5) I + 1 = i + 1 P + 1. Char 1. Acual and Expeced oss Raios PRIVATE AUTO IABIITY 1.2 PRIVATE AUTO PHYSICA DAMAGE 1.3 HOMEOWERS MUTIPE PERI 1.4 FARMOWERS MUTIPE PERI WORKERS COMPESATIO OTE. /P is loss raio, E(/P) = l / 1 is expeced loss raio compued wih weighed moving averages. U.S. Bureau of Economic Analysis /P E(/P)

8 Ocober 23 SURVEY OF CURRET BUSIESS 17 In he weighed moving average model, he expeced invesmen income o premiums raio is compued as: (6) Char 2. Direc osses and ormal osses Million $ PRIVATE AUTO IABIITY HOMEOWERS MUTIPE PERI = where β is he smoohing parameer in (, 1). ike he experimen on normal losses, PA, PAD, HMP, FMP, and WCP are included in he experimen on expeced invesmen income. The esimaion experi + 1 Ei 1 = βσ i = 2.2 PRIVATE AUTO PHYSICA DAMAGE 2.4 FARMOWERS MUTIPE PERI WORKERS COMPESATIO OTE. is direc losses incurred, and = / 1 is normal losses. U.S. Bureau of Economic Analysis ( + i, i 1,... ) ( 1 β) i i i, imen used daa on invesmen income o premiums raios by line of insurance for Daa analysis revealed some degree of serial correlaion in he daa on i, for all five lines of insurance, which led o seing β = (.1,.2,.3). As shown in able 3, among he choices of β, β =.3 is associaed wih he minimum RMSPE. ike he compuaion of normal losses, he experimen included he n-poin moving average mehod wih he parameer n = 5. The esimaes ha used he weighed moving averages wih β =.3 yield smaller RMSPEs han hose used he 5-poin moving averages for four of he five lines. Table 3. Roo Mean Square Predicion Errors (RMSPE) from Expeced Invesmen Income o Premiums Raios, Using Weighed Moving Averages and 5-Poin Moving Averages Privae auo liabiliy Privae auo physical damage Homeowners muliple peril Farmowners muliple peril Workers compensaion β = β = β = *.361 *.589 *.547 *2.65 n = 5 * * Indicaes he minimum RMSPE in each column. To furher compare he esimaes from boh mehods, able 4 shows he summary saisics of he forecas errors from he forecass ha used boh moving averages. I is eviden ha using he weighed moving average mehod wih β =.3 resuls in smaller MAPE and RMSPE for PAD, HMP, FMP, and WCP. To illusrae he esimaion resuls obained from using boh mehods, panels 3.1 o 3.5 in char 3 provided a comparison of he esimaed invesmen income o premiums raio wih he acual invesmen income o premiums raios. Based on he resuls from he experimen, he weighed moving average mehod was chosen o compue he expeced loss raios and expeced invesmen income o premiums raios for all 22 lines of insurance. This mehod produced beer overall esimaion resuls, and i is consisen wih he adapive expecaions model, which concepually beer explains he behavior of he insurer han he n-poin moving average mehod. Because auocorrelaion is presen in he daa series on loss raios and invesmen income o premiums raios for mos of he 22 lines, α =.3 was used in he compuaion of expeced loss raios, and β =.3 was used in he compuaion of expeced invesmen income o premiums raios for all 22 lines. Effecs of Definiional Change on Insurance Oupu The definiional change in he oupu measures of he 22 lines of he propery-casualy insurance services has resuled in higher average levels of annual oupu. The increases derive from he inclusion of invesmen income as premium supplemens, bu hey are also

9 18 Measuring Insurance Services Ocober 23 Table 4. Summary Saisics of Predicion Errors from Expeced Invesmen Income o Premiums Raio, Using Weighed and 5-Poin Moving Averages [Percen] A. Privae passenger auo liabiliy (1978 2) Forecas errors of expeced invesmen income o premiums raio Forecas errors of expeced invesmen income o premiums raio B. Privae auo physical damage (1978 2) Forecas errors of expeced invesmen income o premiums raio Forecas errors of expeced invesmen income o premiums raio ( β =.3) ( n = 5) ( β =.3) ( n = 5) ME MAE MAPE SDPE RMSPE C. Homeowners muliple peril (1978 2) Forecas errors of expeced invesmen income o premiums raio Forecas errors of expeced invesmen income o premiums raio D. Farmowners muliple peril (1978 2) Forecas errors of expeced invesmen income o premiums raio Forecas errors of expeced invesmen income o premiums raio ( β =.3) ( n = 5) ( β =.3) ( n = 5) ME MAE MAPE SDPE RMSPE E. Workers compensaion (1978 2) Forecas errors of Forecas errors of expeced expeced invesmen income invesmen income o premiums raio o premiums raio ( β =.3) ( n = 5) ME MAE MAPE SDPE RMSPE Forecas errors of expeced invesmen income o premiums raio is i i 1 ME is mean error of forecass MAE is mean absolue error of forecass MAPE is mean absolue percenage error of forecass SDPE is sandard deviaion of forecasing or predicion errors RMSPE is roo mean square predicion errors aribuable, o a much lesser exen, o he use of daa on he direc basis, which includes daa on reinsurance services. The aggregaed average annual oupu of he 22 lines increased 35 percen; 32 percen of his increase is aribuable o he inclusion of daa on premium supplemens, and 3 percen is aribuable o he inclusion of daa on reinsurance services. As was expeced, he change o normal losses from acual losses and he use of expeced invesmen income raher han he acual invesmen income as premium supplemens did no significanly affec he aggregaed oupu. The increase in he aggregaed annual average oupu amouned o.8 percen. In heory, he aggregaed average annual oupu should no be affeced a all if he esimaion is conduced properly. The reason for he sligh effec is ha adjusmens for some caasrophic losses are allocaed o fuure years. In addiion, only he oupu of some lines are affeced by caasrophic losses, bu he aggregae measure is no affeced. The definiional change has also resuled in significanly less volailiy in he annual oupu of he insurance lines ha experienced caasrophic losses. The reducion in volailiy is largely aribuable o he use of normal losses raher han acual losses. To illusrae he effec of he definiional change and Char 3. Acual and Expeced Invesmen Income o Premiums Raios PRIVATE AUTO IABIITY 3.2 PRIVATE AUTO PHYSICA DAMAGE 3.3 HOMEOWERS MUTIPE PERI 3.4 FARMOWERS MUTIPE PERI WORKERS COMPESATIO OTE. I/P = i is invesmen income o premimiums raio, and E(I/P) = i / 1 is expeced invesmen income o premiums raio compued wih weighed moving averages. U.S. Bureau of Economic Analysis I/P E(I/P)

10 Ocober 23 SURVEY OF CURRET BUSIESS 19 using five insurance lines as an example, able 5 presens a comparison of he average annual oupu using he curren definiion wih ha using he new definiion, and i also shows a comparison of he volailiy in he acual daa series wih ha in he esimaed daa series. The sandard deviaion of a ime series measures he volailiy of ha series, and he raio of he sandard deviaions of wo series provides he relaive volailiy of he wo series. Column 2 shows he relaive volailiy in he expeced loss raios o he acual loss raios, and column 3 shows he relaive volailiy in he compued normal losses o he acual losses. Two observaions can be drawn from columns 2 and 3. Firs, he expeced loss raios and he normal losses show reduced volailiy. o surprisingly he reducion in volailiy is greaer for he lines ha experienced caasrophic losses. Allied lines had caasrophic losses in 1989, Table 5. Relaive Oupu and Relaive Volailiy in Acual and Esimaed Daa Insurance line Relaive volailiy of expeced loss raio versus acual loss raio Relaive volailiy of normal losses versus direc losses Relaive volailiy of expeced versus acual invesmen income o premiums raio Relaive volailiy of oupu using new definiion versus oupu using direc losses and acual invesmen income σ( l Relaive volailiy of expeced loss raio versus acual loss raio is 1) σ( l) σ ( Relaive volailiy of normal losses versus direc losses is 1) σ ( ) σ ( i Relaive volailiy of expeced versus acual invesmen income o premiums raio is 1) σ ( i ) Relaive volailiy of oupu using new definiion versus σ ( Y oupu using direc losses and acual invesmen income is ) D σ ( Y ) Y Relaive oupu using new definiion o oupu using curren definiion is C Y σ ( ) is he sandard deviaion of he ime series in he parenheses l -1 is expeced loss raio l is direc loss raio -1 is normal losses is direc losses incurred i -1 is expeced ne invesmen income o premiums raio i is ne invesmen income o premiums raio Y is oupu under new definiion, Y = P ( 1 d + i 1) 1 D D Y is oupu compued as Y = P (1 d + i ) C Y C is oupu under curren definiion, Y = P (1 d ) Relaive oupu using new definiion versus oupu using curren definiion Allied lines ( ) Homeowners muliple peril ( ) Privae auo liabiliy (193 21) Privae auo physical damage (193 21) Workers compensaion (193 21) , 1998, and 21, and homeowners muliple peril had caasrophic losses in Second, he reducion in volailiy in normal losses is less han ha in he esimaed loss raios. This is because normal losses are derived as he produc of esimaed loss raios and he direc premiums earned. Some volailiy in he direc premiums earned has been picked up in he compued normal losses. Similarly, column 4 shows ha he volailiy was reduced as a resul of using he expeced invesmen income o premiums raio raher han he acual invesmen income o premiums raio. The reducion in volailiy is greaer for allied lines; in recen years, he invesmen income for his line has swung down from an average of 3.78 percen in he 199s o 6.5 percen in 2 and o 2.3 percen in 21. Addiional volailiy from he daa on reinsurance may be added o he measured oupu by line of insurance. Therefore, comparing he volailiy in he oupu using he curren definiion wih he volailiy in he oupu using he new definiions does no provide accurae informaion on he effec of using normal losses and expeced invesmen income. In column 5, ha effec is measured by he raio of he sandard deviaion of oupu using he new definiion o ha of oupu measured wih direc losses and acual invesmen income as premium supplemens; clearly, he use of normal losses and expeced invesmen income reduces he volailiy in he oupu. In column 6 of able 5, he average annual oupu using he new definiion is compared wih average annual oupu using he curren definiion. The average oupu increased significanly, ranging from 8.6 percen for privae passenger auo physical damage o 73.4 percen for workers compensaion. Because he higher average annual oupu level is largely due o he inclusion of he expeced invesmen income as premium supplemens, he oupu measured using he curren definiion significanly underesimaes he conribuions of he financial inermediaion services provided by he propery-casualy insurance indusry. For he lines in able 5, he average expeced invesmen income is 3.1 percen of he direc premiums earned for allied lines, 3.9 percen for homeowners muliple peril, 4.6 percen for privae passenger auo liabiliy, 1.9 percen for privae passenger auo physical damage, and 7 percen for workers compensaion for heir respecive sample periods. In addiion o analyzing he effecs of he change in he definiion of insurance services on average annual oupu and volailiy in he esimaed daa series for he sample period, he effec of he change can also be

11 2 Measuring Insurance Services Ocober 23 illusraed from he esimaes for a paricular year as shown in able 6; 1992 and 21 were seleced o illusrae he effecs of he definiional change and o demonsrae how he adjusmens for caasrophic losses affec he levels and volailiy of he esimaed series. Par A of able 6 presens a comparison of he acual daa series wih he esimaed daa series and he oupu measured using he curren definiion and he new definiions for 5 lines of insurance for In 1992, Hurricane Andrew caused caasrophic losses in allied lines and homeowners muliple peril. In column 2, he acual direc loss raios are 1.2 for allied lines and 1.24 for homeowners muliple peril. In column 3, he corresponding esimaed loss raios, however, are.68 for allied lines and.73 for homeowners muliple peril. The significanly lower esimaed loss raios reflec he combined effecs of esimaing loss raios using he weighed moving averages and he adjusmens made for he caasrophic losses. Columns 4 and 5 in par A of able 6 show a comparison of he acual direc losses and he normal losses. o surprisingly, he relaive values of he acual losses o he esimaed loss raios are no equal o he corresponding relaive values of he acual losses o he normal losses. For example, he relaive values of he acual loss raios o he esimaed loss raios (dividing column 2 by column 3) are 1.76 for allied lines, 1.7 for homeowners muliple peril,.93 for privae auo liabiliy,.92 for privae auo physical damage, and.96 for workers compensaion. However, he relaive values of he direc losses o he normal losses (dividing column 4 by column 5) are 1.77 for allied lines, 1.7 for homeowners muliple peril,.92 for privae auo liabiliy,.92 for privae auo physical damage, and.97 for workers compensaion. The differenial relaive values of loss raios and losses are caused by he addiional informaion from direc losses ha is included in he compued normal losses. Columns 6 and 7 presen he acual and expeced invesmen income o premiums raios for he 5 lines. Columns 8 and 9 presen a comparison of he measured oupu using he curren definiion wih he oupu using he new definiion. Using he curren definiion, caasrophic losses resul in negaive oupu for allied lines and homeowners muliple peril. Qualiaively similar resuls are shown in par B of able 6 from esimaes for 5 lines of insurance for 21. Aircraf, fire, and allied lines suffered caasrophic losses as a resul of he erroris aacks on Sepember 11 h. In addiion o he caasrophic losses, allied lines also had an unusual negaive invesmen income in 21. This example again demonsraes ha using normal losses and expeced invesmen income grealy reduces he large swings in measured oupu. Using he curren definiion, he measured oupu for fire insurance is sill posiive despie he huge caasrophic losses, because he curren definiion uses premiums earned and losses incurred ne of reinsurance. The direc loss raio of 1.28 and he posiive oupu of fire insurance service measured using he curren definiion Table 6. A Comparison of Acual and Esimaed oss Raios, osses, and Invesmen Income o Premiums Raios, and Oupu Measured Using Curren Definiion and ew Definiion [osses and oupu measured in millions of dollars] Insurance ine oss raio (percen) A. A comparison of acual and esimaed daa for 1992 E (loss raio) (percen) Direc losses ormal losses Invesmen income o premiums raio (percen) E (Invesmen income o premiums raio) (percen) Oupu using curren definiion Oupu using new definiion Allied lines , , Homeowners muliple peril , , , ,545. Privae auo liabiliy , , , , Privae auo physical damage , , , , Workers compensaion , , , , B. A comparison of acual and esimaed daa for 21 oss raio (percen) E (loss raio) (percen) Direc losses ormal losses Invesmen income o premiums raio (percen) E (Invesmen income o premiums raio) (percen) Oupu using curren definiion Oupu using new definiion Aircraf , , Allied ines , , Fire , , , , Homeowners muliple peril , , , , Workers compensaion , , , , oss raio is l E (loss raio) is l -1, Direc losses is ormal losses is 1 Invesmen income o premiums raio is i E (invesmen income o premiums raio) is i 1 Oupu using curren definiion is C Y = P (1 d ) Oupu using new definiion is Y = P ( 1 d + i 1) 1

12 Ocober 23 SURVEY OF CURRET BUSIESS 21 suggess ha a significan porion of he unexpeced losses in 21 were recovered from he reinsurance services purchased. Fuure Research The objecive of he definiional change in he oupu measure of propery-casualy insurance services was o beer measure all he explici and implici services provided by he insurer. The esimaion resuls demonsrae ha he definiional change and he new saisical reamen of losses and premiums supplemens have a subsanial impac on he measured insurance services. However, furher research should coninue in order o improve he saisical mehodology. The adapive expecaions framework ofen works fairly well empirically, bu i lacks heoreical jusificaion. Fuure research should go oward he consrucion of a srucural model ha properly explains how he profimaximizing insurer uses all he informaion available o form expecaions of fuure losses and fuure invesmen income. Because a much longer ime series daa se for each line of insurance has now been consruced, more sophisicaed ime series modeling mehods ha can beer handle he auocorrelaions in he daa and ha could provide more robus esimaes should be explored. Technical oe: Preparing he Daa for he Definiional Change The new definiion of he propery-casualy insurance oupu can be expressed as: (T.1) Y = P ( 1 + i 1 d ) 1, where Y is oupu, P is direc premiums earned, -1 is normal losses, i -1 is expeced invesmen income o premiums raio, and d is dividend o premiums raio for period. Recall ha -1 = l -1 P, and l -1 is he expeced direc loss raio. Under he curren reamen, BEA uses ne premiums earned and ne losses incurred o measure insurance oupu. The change in he measure of insurance oupu requires he use of direc premiums earned and direc losses incurred. e premiums earned, P, equals direc premiums earned minus he ne purchases of reinsurance, P, and ne losses incurred, R, equals direc losses incurred minus losses recovered from ne purchases of reinsurance, R. The ne purchase of reinsurance is he difference beween he reinsurance ceded and he reinsurance assumed. Because published daa on he direc basis is unavailable before 1975, he preceding relaionships can be used o derive he needed daa by using ne reinsurance purchases and ne premiums earned and losses incurred. The definiional change in he measure of insurance oupu affecs he following 22 lines of propery-casualy insurance services: Aircraf, allied lines, boiler and machine, burglary and hef, commercial auo liabiliy, commercial auo physical damage, commercial muliple peril, earhquake, farmowners muliple peril, fideliy, fire, homeowners muliple peril, inland marine, medical malpracice, ocean marine, oher liabiliy, oher lines, privae passenger auo liabiliy, privae passenger physical damage, reinsurance, surey, and workers compensaion. The firs sep in he implemenaion of he definiional change is o consruc a daa se ha P conains he ime series daa on P,, P,, R, R,, and d for each line of insurance. i Daa sources and daa problems The main source of daa are he 194 o 22 ediions of Bes s Aggregae and Averages: Propery-Casualy by A.M. Bes Company. The ime series for direc premiums earned, direc losses incurred, ne invesmen income, and dividends o policyholders for are exraced from A.M. Bes s daabase. Daa series for years before 1975 are consruced from A.M. Bes s published daa. The firs, 194 ediion of A.M. Bes s daa on propery-casualy insurance services conained cumulaive daa for by line of insurance. Therefore, he longes span of he published imes series is 72 years, from 193 o 21. However, daa for all 22 lines of insurance for are no available; some are only available back o he 195s, and some dae back o he 197s or 198s. Table 7 displays he year when he daa on each of he 22 lines were eiher firs repored by Table 7. Saring Year of Daa Series on Insurance ines Insurance line Year daa sared Aircraf Allied lines Boiler and machine Burglary and hef Commercial auo liabiliy Commercial auo physical damage Commercial muliple peril Earhquake Farmowners muliple peril Fideliy Fire Homeowners muliple peril Inland marine Medical malpracice Ocean marine Oher lines Oher liabiliy Privae auo liabiliy Privae auo physical damage Reinsurance Surey Workers compensaion

13 22 Measuring Insurance Services Ocober 23 A.M. Bes or when he daa became consrucible from he available A.M. Bes daa. In addiion o he various saring years of he ime series for he lines of insurance, here are wo oher general problems wih he published daa. Firs, observaions in all of he series excep ne premiums earned are missing for he early years. As shown in able 8, some series have 2 missing observaions, and ohers have as many as 45 missing observaions. The daa are missing mainly because he daa were published in much less deail hen. Over ime, more deailed daa and beer qualiy daa have become available. Second, in he published daa, he classificaion of cerain lines of insurance has changed over ime. Some lines were iniially componens of oher lines for some years, bu laer, hese lines were repored as separae lines. Alernaively, some separae lines laer became componens of oher lines. The insurance lines ha were affeced by changes in classificaion consis of allied lines, boiler and machine, homeowners and farmowners muliple perils, oher liabiliy, oher lines, commercial and privae auo liabiliy and auo physical damage lines. Consrucing he daa se Given he problems wih he availabiliy and he qualiy of he daa, i is necessary o consruc a se of daa Variables P and P and PR and R d i Table 8. Availabiliy of Published Daa on Propery-Casualy Insurance Availabiliy of daa series : By-line and indusry oal daa available : By-line and indusry oal daa available, labeled as adjused direc premiums and adjused direc losses incurred : Daa unavailable a any level : By-line daa available on he basis of sock, muual, and reciprocal companies : Daa on losses unavailable a any level : Daa on indusry oal reinsurance daa available : Daa unavailable a any level : By-line daa available : By-line daa unavailable : Daa on indusry average dividend o premiums raio available : By-line daa on ne invesmen gain on funds aribuable o insurance ransacions available : By-line daa on ne invesmen gains or losses and oher income available : By-line daa unavailable : Daa on indusry oal ne invesmen gain or loss available P is direc premiums earned is direc losses incurred P is ne premiums earned is ne losses incurred R P is ne premiums earned from ne purchase of reinsurance R is ne losses recovered from ne purchase of reinsurance d is raio of dividend o policyholders o direc premiums earned i is raio of ne invesmen income o premiums earned for P,, i, and d for each line of insurance for he sample period. Direc premiums earned and direc losses incurred A.M. Bes began o repor business on he direc basis in 1992 in he insurance expense exhibi (IEE), par III allocaion o lines of direc business wrien, in Bes s Aggregaes and Averages: Propery-Casualy, so daa for P and have been available since hen. 13 For he years during which hese variables were no repored, hey mus be derived from oher daa: P can be derived from he relaion beween ne premiums earned and ne premiums for ne purchase of reinsurance, and can be derived from he relaion beween ne losses incurred and ne losses recovered from he ne purchase of reinsurance as follows: R R (T.2) P = P + P, = +. Thus, if daa on reinsurance, ne premiums earned, and ne losses incurred are available, P and can be derived for he years before Unforunaely, a complee daa series on ne losses incurred and on he by-line daa on reinsurance for he years before 1975 are also unavailable. Thus, exrapolaion echniques were used o esimae he missing observaions in hese series. There are wo problems in consrucing he complee series of ne premiums earned and ne losses incurred. Firs, ne loss raios were no explicily repored unil 195. Before 195, A.M. Bes repored loss and loss adjusmen expense raios joinly. Second, before 1971, ne premiums earned and ne losses incurred were repored on he basis of he sock, muual, and reciprocal companies. 14 To obain he by-line oal ne premiums earned and he oal ne losses incurred, he hree componens needed o be summed. However, daa on reciprocal companies were available only for 1971 and 1972 and only for allied lines, fire, homeowners muliple peril, oher liabiliy, and workers compensaion, and he daa were available only for 1972 for privae auo liabiliy and privae auo physical damage. o daa on reciprocal companies for he remaining lines were repored. Thus, he ne loss raios for and he ne premiums and ne losses for he reciprocal companies for need o be exrapolaed. For he sock and muual companies ne loss raios firs became available for 195; he shares of ne loss 13. For , P and were repored in IEE in par II allocaion o lines of business ne of reinsurance under adjused direc premiums earned and adjused direc losses incurred. Before 1975, hey were no repored a all. 14. A reciprocal company is an eniy formed by individuals, called subscribers, who underake all ypes of insurance aciviies.

14 Ocober 23 SURVEY OF CURRET BUSIESS 23 S raios, l 195 and l 195, relaive o he combined ne loss S M and loss adjusmen expense raio, l 195 and l 195, were calculaed for each line of insurance for 195, where S and M sand for he sock and muual companies, respecively. These shares were hen used as he exrapolaors o approximae he ne loss raios, l S and l M, for Specifically, for = 193,..., 1949, (T.3) lˆ S S 195 = l ,. S M M 195 lˆ = l S The ne losses incurred for he sock and muual companies are hen approximaed as ˆ S S S = lˆ P and ˆ M M = lˆm P. To obain he oal ne premiums earned and he oal ne losses incurred, an approximaion of he premiums and losses for he reciprocal companies was needed, bu daa on he reciprocal companies for some lines are available only for 1971 and For hese lines, he 2-year average raio of he oal ne premiums earned o he sum of ne premiums earned by he sock and muual companies, were compued. Similarly, he 2-year average raio of he by-line oal ne losses incurred o he sum of he ne losses incurred for sock and muual companies, were compued. These average raios were hen used o exrapolae he oal ne premiums earned and he oal ne losses incurred for = 193,...,197, (T.4) Pˆ = ˆ M P 1971 P S P M 1971 l S l 195 P P M P S S M M S + 2 S M ( P + P ) P P S P M For he lines ha repored ne premiums and ne losses from he reciprocal companies only for 1972, he exrapolaor is he 1-year raio of he oal premiums (losses) o he sum of he premiums (losses) from he sock and muual companies. For he oher lines, he oal premiums and oal losses are he sum of he premiums and losses from he socks and muual companies. As poined ou earlier, he by-line daa on reinsurance are no available for years before 1984, and he daa on indusry oal reinsurance have only been l S l 195 P 1972 P S P M , S M = ( + ) M M. S S available since To use he available indusry daa, by-line reinsurance daa for were approximaed by using he indusry oal reinsurance daa and he share of by-line reinsurance of he indusry oal. Because reinsurance daa are available for each line for , he shares of he ne premiums for he ne purchase of reinsurance and he ne losses recovered from he ne purchases of reinsurance for each line were compued for Then he median of each share series was consruced, and he median was used o exrapolae he by-line ne premiums for, and ne losses recovered from, ne purchases of reinsurances. Specifically, for = 1951, , Ri, RI, (T.5) Pˆ P m PRi, Ri, R, I =, ˆ, P R, I = m Ri, R, I where i and I in he superscrip index he insurance line and indusry oal, respecively, and where m( ) is he median of he shares for The median insead of he 1984 share was used in order o limi he impac of oulier years. R, i R, i Afer Pˆ and ˆ are compued, equaion (T.2) was used o approximae direc premiums earned and direc losses incurred for However, because no daa on reinsurance for are available, direc premiums earned and direc losses incurred for were exrapolaed. The exrapolaor is based on he assumpion ha direc premiums earned (direc losses incurred) grew a he same annual rae as ne premiums earned (ne losses incurred) from 193 o 195. This assumpion implies ha for = 193,...,195, P and can be exrapolaed according o (T.6) Pˆ P P = , ˆ = P The above discussion describes he consrucion of direc premiums earned and direc losses incurred for he insurance lines ha did no change classificaions over he years. However, he classificaions of some lines changed. Some classificaion changes did no require an adjusmen; for example, farmowners muliple peril was included in homeowners muliple peril unil 1973, when i became a separae line. On he oher hand, some adjusmens were necessary before compiling he daa. Classificaion changes and adjusmens The classificaion of he following lines changed: Allied lines, boiler and machine, oher liabiliy, oher lines, commercial and privae auo liabiliies and physical damage lines. As a resul of hese changes, some adjusmens were made. Allied lines. Allied fire and exended coverage were

15 24 Measuring Insurance Services Ocober 23 repored as wo lines for In 1971, hese wo lines were combined o form allied lines. To incorporae his change, allied lines for was compued as he sum of hese wo lines. Before 1992, muliple peril crop and federal flood insurances were included in allied lines, bu hey have become wo separae lines since hen. In 1997, glass was excluded from oher lines, and i has been included in allied lines since Boiler and machine. Seam boiler and engine machine were repored as wo separae lines of insurance from 193 o In 194, hey were combined as boiler and machine. In order o accoun for his change, boiler and machine for was compued as he sum of hese wo lines. Oher liabiliy. Oher liabiliy has been a separae line since From 193 o 1974, oher liabiliy was included in miscellaneous liabiliies, which became a separae line in From 193 o 197, miscellaneous bodily injury and miscellaneous propery damage were lised as separae lines, and hey joinly covered he liabiliies ha were laer included in miscellaneous liabiliies. To accoun for his change, miscellaneous liabiliies for was compued as he sum of miscellaneous bodily injury and miscellaneous propery damage. In 1975, oher liabiliy was formed from a major par of miscellaneous liabiliies. The remaining par of miscellaneous liabiliies coexised wih oher liabiliy for 3 years before i ceased o exis. To reflec his change, he average raios of oher liabiliy (OB) o miscellaneous liabiliies (MB) for 1975, 1976, and 1977 was compued, and hen he average raios were used as he exrapolaors o approximae ne premiums earned and ne losses incurred for oher liabiliy. Specifically, for = 193,...,1974, (T.7) ˆ OB, = P, MB , P ˆ OB, = MB, P OB, 1975 PMB, 1975 OB, MB, 1975 P OB, 1976 PMB, 1976 OB, MB, 1976 P OB, 1977 PMB, 1977, OB 1977 MB, Commercial and Privae Auo Insurances. Commercial auo liabiliy, commercial auo physical damage, privae auo liabiliy, and privae auo physical damage became individual lines in For , daa on privae and commercial auo insurances were combined in auo liabiliy and auo physical damage. From 193 o 197, he wo componens of auo liabiliy, auo bodily injury and auo propery damage, were wo separae lines, and he wo componens of auo physical damage, auo collision and miscellaneous auo lines, were also wo separae lines. Thus, for hose years, auo liabiliy and auo physical damage are represened by he sum of hese componens. In order o separae privae auo insurance from commercial auo insurance, he shares of hese insurances ha were accouned for by privae auo liabiliy and privae auo physical damage were compued. These privae auo shares have wo componens: The raio of privae auo insurance o oal auo insurance, and he raio of he share of household o oal moor vehicle sock in a given year, MVHS /MVS, o he share in 1972, MVHS 1972 /MVS For example, for = 193,...,1971, he privae share of auo liabiliy for he ne premiums earned, SP PA, is compued as: PA, PA P (T.8) SP 1972 MVHS MVHS = , A, P 1972 MVS MVS 1972 where P, PA is he ne premiums earned for privae auo liabiliy and P, A is oal premiums for auo liabiliy. The privae share of auo liabiliy for ne losses incurred is compued similarly. The privae auo shares are consruced o adjus he 1972 privae auo insurance o oal auo insurance raio by he changes in he relaive moor vehicle sock held by he households over ime. The ne premiums earned by privae auo liabiliy, P, PA, for , were approximaed as he produc of P, PA and SPPA. Specifically, for = 193,...,1972,, PA PA, A (T.9) Pˆ = SP P. e premiums earned for privae auo physical damage, ne losses incurred for privae auo liabiliy, and privae auo physical damage were approximaed in he same fashion as he ne premiums for privae auo liabiliy. The commercial auo share for auo liabiliy (auo physical damage) was compued as 1 minus privae auo share for auo liabiliy (auo physical damage). e premiums and losses of he commercial auo lines were approximaed accordingly. Oher lines. The oher lines caegory was creaed in 1973, and i includes a few small lines repored on he annual saemen of he propery-casualy insurance indusry. Since is creaion, he componens of oher lines have changed several imes. From 1973 o 1977, oher lines consised of facory muual, inernaional, reinsurance, and miscellaneous wrie-ins. Since 1978, i has included credi (iniially credi included morgage guaranee, which became a separae line in 1992). In 198, reinsurance became a separae line, and glass became a componen of oher lines unil 1997, when i

16 Ocober 23 SURVEY OF CURRET BUSIESS 25 became a componen of allied lines. Facory muual was eliminaed in he mid-198s. Currenly, oher lines consiss of credi, morgage guaranee, inernaional, and miscellaneous wrie-ins. As a resul of hese changes in oher lines, he only adjusmen made was o remove reinsurance from oher lines for , because reinsurance was he larges componen, and wihou an adjusmen, here would be a sharp decline in he daa series for oher lines. In addiion, separaing reinsurance from oher lines allowed a complee ime series for reinsurance for o be consruced. A.M. Bes repored oher lines wih and wihou reinsurance for Using hese repors, he shares of reinsurance in oher lines were calculaed, and he average of he shares was used o exrapolae reinsurance for Dividends o policyholders Since 1975, A.M. Bes has provided daa on dividends o policyholders by line of insurance. From 1975 o 1991, he daa were repored on he ne basis, and since 1992, he daa have been available on boh he ne basis and he direc basis. A.M. Bes also provided daa on he average dividends o policyholders as a raio of premiums earned a he propery-casualy insurance indusry level since From 193 o 195, daa on dividends were no available a any level, so he indusry average dividend raios for were used o approximae by-line dividend raios for For , he relaionship beween he by-line dividend raios and he indusry average dividend raios appeared o be relaively sable for mos of he lines. A simple regression was run for each line, using he log of dividend raios by line of insurance as he dependen variable and he log of indusry average dividend raios as he independen variable. The esimaed coefficien is saisically significan a he 5- percen level for 15 of he 2 lines (he 2 lines, earhquake and medical malpracice, ha sared afer 1975 were excluded). The regression resuls were hen used o projec he dividend raios for for hese 15 lines. The remaining 5 lines are aircraf, farmowners muliple peril, fideliy, surey, and burglary and hef. In erms of premiums earned, hese lines are among he smalles, and mos of hem have fairly low and fla dividend raios over ime. Thus, for hese lines, he average dividend raios for were used as he approximaed dividend raios for Unforunaely, no informaion on dividend raios for is available. Since dividend o premium raios accoun for less han 1 percen for mos lines for , he by-line average dividend raio for was used as he approximaed dividend raios for Premium supplemens A.M. Bes s daa on ne invesmen income by line of insurance have been available since For , he daa were labeled as ne invesmen gain or loss and oher income, and since 1992, he daa have been labeled as ne invesmen gain on funds aribuable o insurance ransacions. o daa on invesmen gain by line of insurance are available for years before However, daa on indusry oal ne invesmen gain or loss and oher income and daa on oal asses invesed for are available. To fill in he gaps in he series on ne invesmen income by line of insurance, he daa for were approximaed firs, using daa a he indusry level, and hen he daa for were approximaed. Using he indusry oal daa for , he ne invesmen gain by line of insurance was approximaed by muliplying he indusry-level rae of reurn by he echnical reserves for each line. The indusry-level rae of reurn was calculaed by dividing he oal ne invesmen gain or loss by he oal asses invesed, based on he assumpion ha each line of insurance had he same rae of reurn as he indusry oal for ha period. This assumpion is consisen wih he curren calculaion of he by-line invesmen income daa repored annually in he IEE able in Bes s Aggregaes and Averages: Propery-Casualy. Technical reserves, he sum of unearned premiums and unpaid losses, are no readily available by line of insurance. A.M. Bes provides daa on unearned ne premiums from 193, bu i does no provide daa on unpaid losses before Therefore, he median of he raios of unpaid losses o ne losses was compued and used o exrapolae he ne unpaid losses, ˆ U. Specifically, for = 193,...,1974, U (T.1) ˆ = m U , where m(.) is he median of he raios of unpaid losses o oal ne losses incurred from 1984 o To be consisen wih he curren definiion of invesmen funds used in A.M. Bes s repors, he echnical reserves for year were compued as he average of he sum of unearned premiums and unpaid 15. Because a consruced daa series on ne losses incurred is available for he enire sample period and because daa on unpaid loses for are available, he regression analysis could be considered o projec he byline unpaid losses for This approach was no pursued, because he sample size of 18 for unpaid losses is oo small o produce reliable resuls.

17 26 Measuring Insurance Services Ocober 23 losses in year and 1. Thus, ne invesmen income for = , 1974 can be approximaed as: (T.11) Î ri U P P U U U = [( + 1 ) + ( + 1 )] 2, where ri is he indusry-level rae of reurn o invesed funds and PU is he unearned ne premiums. o daa on ne invesmen income for are available. The by-line invesmen income daa for hese years was approximaed by muliplying he esimaed echnical reserves by he esimaed indusry-level rae of reurn. Because he indusry-level rae of reurn for was fla, mosly beween 2 and 2.5 percen, he average of he indusry-level rae of reurn for ha period was used as he esimaed indusry-level rae of reurn for References A.M. Bes Company Bes s Aggregaes and Averages: Propery-Casualy, Unied Saes. Oldwick, J. Ausralian Bureau of Saisics The Measuremen of onlife Insurance Oupu in he Ausralian aional Accouns. Paper presened a he OECD Meeing of he aional Accouns Expers, Paris, Sepember. Bach, Chrisopher. 23. Annual Revision of he U.S. Inernaional Accouns, SURVEY OF CURRET BUSIESS 83 (July): Cagan, Phillip D The Moneary Dynamics of Hyper-Inflaion. In Sudies in he Quaniy Theory of Money, edied by Milon J. Friedman. Chicago: Univer- siy of Chicago Press. Commission of he European Communiies, Inernaional Moneary Fund, Organisaion for Economic Co-operaion and Developmen, Unied aions, and he World Bank Sysem of aional Accouns Brussels/uxembourg, ew York, Paris, and Washingon, DC. Cummins, David J., and Mary A. Weiss. 2. Analyzing Firm Performance in The Insurance Indusry Using Fronier Efficiency and Produciviy Mehods. In he Handbook of Insurance, edied by G.D. Dionne, Boson: Kluwer Academic Publisher. Fixler, Dennis J., and Bren R. Moulon. 21. Commens on he Treamen of Holding Gains and osses in he aional Accouns. Paper presened a he OECD Meeing of aional Accouns Expers, Paris, Ocober. Hill, Peer The Treamen of Insurance in he SA. Paper presened a he Brookings Insiuion Workshop on Measuring he Price and Oupu of Insurance, Washingon, DC, April. Moulon, Bren R., and Eugene P. Seskin. 23. Preview of 23 Comprehensive Revision of he aional Income and Produc Accouns: Changes in Definiions and Classificaions. SURVEY OF CURRET BUSIESS 83 (June): Muh, John F Opimal Properies of Exponenially Weighed Forecass. Journal of he American Saisical Associaion 55 (June). Muh, John F Raional Expecaions and he Theory of Price Movemens. Economerica 29 (July):