Optimal Growth for P&C Insurance Companies

Transcription

1 Opimal Growh for P&C Insurance Companies by Luyang Fu AbSTRACT I is generally well esablished ha new business produces higher loss and expense raios and lower reenion raios han renewal business. Ironically, o add more new business, an insurer needs higher profiabiliy in order o generae he addiional capial needed o suppor is exposure growh. Irraional growh is one of he op reasons for he insolvencies of propery and casualy insurance companies. This sudy presens a mehod o balance he opposing forces of growh and profiabiliy. The proposed mehod is sraighforward and can be effecively employed by propery and casualy insurers in heir sraegic planning process. KEYwORdS Aging phenomenon, consrained maximum growh, opimal growh, combined raio, premium-o-surplus raio, enerprise risk managemen. 02 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

2 Opimal Growh for P&C Insurance Companies. Inroducion Long-erm profiabiliy and susainable growh are imporan goals of propery and casualy (P&C) insurers. Markeing plans such as segmenaion peneraion, new agency appoinmens, and new erriory enries are all subjec o he overall growh sraegy of an insurer. For an insurer o grow, i mus eiher explore new markes or arac new cusomers in exising markes. However, such new business generally produces boh higher loss and expense raios, and increases he overall operaional risk of he company. Numerous case sudies have shown ha rapid growh raes can cause serious financial problems for a P&C insurer, reduce long-erm value o is sakeholders, or even resul in bankrupcy. According o A. M. Bes (2004), 7.3% of P&C insolvencies from 969 o 2002 were caused primarily by rapid growh. D Arcy and Dohery (989; 990) and Cohen (2005) discuss he aging phenomenon of P&C insurance markes in which new business usually generaes a much higher loss raio han renewal business, ofen resuling in an underwriing loss in he firs year, bu improving loss raios in subsequen years for he reained porion of ha cohor of business. Wu and Lin (2009) examine eigh lines of business on 25 books wih $28.7 billion of premium from 995 o 2005 and demonsrae loss raio improvemens associaed wih his aging phenomenon. They find ha renewal business produces loss raios 7% o 8% lower han new business, wih an average loss raio difference of 3%. In addiion o larger expeced loss raios, he expenses associaed wih acquiring new businesses (such as adverising, markeing, and underwriing) are higher han for renewal business. Feldblum (996) suggess ha an insurance company should price risks o ake ino accoun he expeced profiabiliy over he lifeime of he policy, including he loss raio, expense raio, and reenion level a each renewal. An aggressive growh posure obviously means a higher proporion of a book of business is made up of new business, implying a higher combined raio and greaer underwriing risk. Therefore, an insurance company s planned pace of growh should de- pend on wheher he expeced profi from insurance operaions is high enough o suppor he growh. Wihou proper enerprise risk managemen, an aggressive growh sraegy canno be susained over a long period, and may resul in significan underwriing losses. P&C insurers may operae in he following cycle: reduce raes aggressively o be compeiive; grow he book rapidly; see loss raios deeriorae; increase he raes o alleviae underwriing losses; and wach sales go down because raes become less compeiive. Oher academic researchers have found ha increasing sales growh and improving per-uni profi margins can be conflicing goals. Aghion and Sein (2008) discuss ha, given he consrains on managemen ime and oher resources, doing more in one dimension ofen implies doing less in he oher. Harringon, Danzon, and Epsein (2008) invesigae medical malpracice markes and show ha insurance companies ofen sacrifice profi margins by cuing price excessively in he sof marke o mainain sales volume. Ma (2009) shows ha profiabiliy will be eroded significanly when a high growh arge is achieved by lowering underwriing sandards. Acuaries have long realized ha growh raes influence an insurer s loss, reserve, profi, and surplus, and have sudied hese effecs using radiional acuarial and accouning mehods. Mueeries (979) presens an accouning model o calculae he necessary profi margin o keep pace wih increasing premium growh. Based on raher conservaive assumpions, he concludes ha a leas a 5% before-ax underwriing margin is necessary o mainain he relaionship of premium o surplus. Niswander (984) measures he rade-off beween wo conflicing goals: surplus growh based on profiabiliy and exposure growh based on compeiive raes. Because he average age of loss wihin an exposure period may change over ime as a resul of growh, McClenahan (987) examines he impac of changes in exposure growh on loss developmen paerns, and proposes a mehod o adjus he developmen facors. To dae, only D Arcy and Gorve (2004) have sudied opimal growh from he perspecive of he managers of VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 03

3 Variance Advancing he Science of Risk insurance companies. They deermine he opimal growh rae by maximizing he marke value of a P&C insurance company. In heir work, a hree-facor economeric model is proposed in which an insurer s marke value is deermined by is surplus, ne wrien premium, and combined raio. The model parameers are derived hrough linear regressions using daa from fifeen publicly raded firms. They hen run a series of dynamic financial analysis (DFA) simulaions for a variey of growh raes using he Dynamo sofware. The D Arcy and Gorve (2004) sudy represens a significan milesone in pioneering he research on opimal growh for P&C insurers, bu several aspecs of heir work may limi he benefi of heir analysis for he enire P&C insurance indusry. Firs, heir mehod requires marke values, which are only available for sand-alone P&C companies ha are publicly raded. Muual, reciprocal, subsidiary, and privaely-held P&C companies do no have observable marke values. As D Arcy and Gorve poin ou, very few of he more han 3,000 P&C insurers are boh sand-alone and publicly-raded. Second, heir approach is sensiive o sock prices of insurance companies, which can be very volaile. For example, excluding daa from AIG, D Arcy and Gorve (2004) found ha he opimal growh rae does no exis (negaive growh will lead o higher marke values); while including AIG daa, he opimal soluion is abou 0%. Third, heir sudy is based on sophisicaed dynamic financial analysis (DFA), which requires significan resources and can be difficul o undersand. In his sudy we examine he numerical relaionship beween organic growh raes and he corresponding profiabiliy and capial needs using an approach ha requires less exensive daa. Analyical models are derived based on an economic equilibrium model. In he opimizaion process, we incorporae pracical consrains for he growh of P&C insurers. This The sofware is publicly available a Casualy Acuarial Sociey and Pinnacle Acuaries websies, and aries.com. paper exends and improves he previous sudy in hree major aspecs. Firs, i provides he maximum growh rae as well as he opimal growh rae under he predeermined consrains. Second, D Arcy and Gorve (2004) discuss ha opimal posiive growh may or may no exis, bu hey do no invesigae he condiions for he exisence of opimal growh. We invesigae his issue and find ha he exisence of an opimal growh rae depends on he relaive weighing assigned by a company s managemen o heir wo goals of increasing surplus and increasing sales. When he weigh on surplus is above a cerain hreshold, a posiive opimal growh rae does no exis: an insurer can increase he expeced value of he enerprise by increasing premium raes and improving profi margins while shrinking premium volume. Third, from he perspecive of implemenaion, he models are developed using simpler assumpions and formulas han hose deployed in Dynamo by D Arcy and Gorve (2004) and are herefore relaively easy o undersand and apply. Addiionally, all of he daa required o use his mehod should be readily available from a P&C company s acuarial daabase, and he calculaions involved are easy o program in a spreadshee, so ha he proposed mehods can be quickly implemened by P&C companies in heir sraegic financial planning. This paper is organized in a sraighforward manner. Secion 2 discusses he relaionship beween growh raes and combined raios, and inroduces he concep of an equilibrium new business percenage (ENBP). Secion 3 invesigaes he capial consrain on growh. Secion 4 develops a concepual framework for deermining he opimal growh rae ha maximizes he expeced enerprise value. Secion 5 provides a case sudy. The numerical relaionship beween growh rae and underwriing profi is presened. The consrained maximum growh raes and he opimal growh raes are calculaed under various scenarios of marke cycles, underwriing performances, and consrains. Secion 6 offers a summary of he main conclusions drawn from his analysis. The appendix exends he model by subdividing he renewal book ino muliple segmens. 04 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

4 Opimal Growh for P&C Insurance Companies 2. Growh raes and combined raios m Le P be he marke price a ime. If an insurance company does no deviae from he marke price, is renewal business (RB) loss raio would be L m, where subscrip r represens RB and superscrip r, m represens marke price. Le dp denoe he percenage difference of a company s price from P m. The m acual price of he insurer is P = P * ( + dp ). L r, is hen he company s acual RB loss raio a ime and is given by L r, = L m/( + dp ). The company s acual r, RB combined raio is hen C r, = L m/( + dp) + E, r, r, where E r, is he expense raio for renewal business. 2 Using he same noaion, he variables for loss, expense, and combined raios for new business (NB) are denoed by L n,, E n,, and C n, = L m /( + dp) + E, n, n, where L m is he NB loss raio if he company prices n, a he marke level. As noed in he inroducion, L n, > L r,, and E n, > E r,, meaning ha C n, > C r,. Le G be he exposure growh rae. By he law of demand in economics, insurance demand decreases as he price increases. Therefore he growh rae G for an insurance company is a decreasing funcion of is price differenial dp, G = α+ β dp, ( 2.) wih β < 0 o reflec decreasing demand wih higher prices. Depending on marke condiions, α can be posiive or negaive. In a hard marke, α is posiive, and an insurance company is able o rapidly grow is exposure a he marke price. Even if i prices moderaely above marke (0 < dp < α ), i can wrie more β accouns and grow is exposures. In a sof marke, α can be negaive. Under his scenario, if an insurance company prices a he marke rae level (dp = 0, G = α < 0), i will reduce is exposure. Le A and A denoe he proporions of he overall book of business ha NB and RB represen a ime, respecively. To grow in an absolue sense, an insurance company needs o arac more cusomers han hose ha fail o renew, so A is an increasing funcion of he growh rae G. Define R r, and R n, o be he reenion raios for renewal and new business. 3 According o Wu and Lin (2009), R r, > R n,. Again, by he law of demand in economics, reenion raios are decreasing funcions of a company s price differenial wih marke prices a ime, so R = µ + θ dp r, r r R = µ + θ dp ( 22. ) 4 n, n n Le he wrien exposure in period be Q. The wrien exposure in period wih annual growh G is Q = Q + G. ( 2.) 3 By expressing oal exposures as he sum of NB and RB exposures, he following formulaions will allow us o solve for he new business percenage, A, in erms of reenion and growh raes. To sar, he new business exposures wrien in period can be expressed as Q = Q A = Q + G A. ( 2. 4) n, The new business exposures wrien in underwriing period ha are reained in period are Q n, R n, = Q A R n, and he renewal business exposures wrien in underwriing period ha are reained in are Q r, R r, = Q ( A )R r,. The oal renewal business wrien in underwriing period is hen Q = Q A R + Q A R. ( 2.) 5 r, n, r, 2 The expense raio has wo componens: a variable expense raio which is generally independen of premium, and a fixed expense raio which is inversely proporional o he size of sales. To simplify he illusraion, we assume ha he enire expense raio is independen of premium. This assumpion ignores he concurren benefi from growh of reducing he fixed expense raio. The approach can be easily expanded so ha he fixed expense raio is a decreasing funcion of premium while he variable expense raio is a consan. 3 The renewal raes vary depending on he age of he policies. To simplify, we spli a book of business ino NB and RB. The model can be expanded o allow muliple segmens of he whole book by policy age, such as new business, firs renewal, second renewal, and so on. 4 Excep for price, many oher variables also affec reenion, including service, claims saisfacion, adverising, rewards (disappearing deducible), ec. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 05

5 Variance Advancing he Science of Risk Table. Required Nb percenage o achieve 5% overall growh when 0% of he curren book of business consiss of Nb Year () RB () NB (2) RB % (A ) (3) = ()/(5) NB % ( A ) (4) = (2)/(5) (5) = () + (2) % 0.0% % 22.6% % 23.7% % 23.8% % 23.8%.749 The oal exposure a ime is he sum of boh he NB and RB exposures: Q = Q + Q n, r,. ( 26. ) Subsiuing (2.3), (2.4), and (2.5) ino (2.6), we ge Q G Q G A Q A R ( + )= ( + ) + n, + Q A R. ( 27. ) r, In a cerain sae wih sable growh and reenion, NB and RB exposures could achieve an equilibrium where A will remain consan hrough ime: A = A. In his siuaion, we name A he Equilibrium New Business Percenage (ENBP). Subsiuing A for A in Equaion (2.7) and canceling Q from boh sides, we can derive he percenage of NB required o gain a specific growh rae G as + G Rr, A = + G + Rn, Rr, Rn, = + G + R R. (2.8) n, r, The following example illusraes he ENBP concep. Consider a company whose curren porfolio consiss of 0% new business and ha plans o grow is oal wrien exposure by 5% annually. The annual reenion raes for new and renewal business are 80% and 90%, respecively. The renewal business in he second year is 0. * 80% * 90% = If he growh follows he plan, he oal exposure in year wo is.5, and new business needs o be = The NB percenage is 0.26/.5 = 22.6%. The renewal business in he hird year is 0.89 * * 0.8 =.009. For 5% annual growh, he overall exposure is.5 2 =.323, and he volume of new business is = Following his procedure, Table shows he required new business percenages for he nex five years if he company wishes o grow is exposure by 5% annually. Similarly, Table 2 repors he required new business percenages if he curren NB percenage is 25% of he book. By comparing Tables and 2, i is clear ha in equilibrium, A is deermined by he planned growh rae and reenion raes of NB and RB, and ha i is independen of he curren NB and RB composiion of a book of business. However, his Table 2. Required Nb percenage o achieve 5% overall growh when 25% of he curren book of business consiss of Nb Year () RB NB RB % (A ) NB % ( A ) % 25.0% % 23.9% % 23.8% % 23.8% % 23.8% CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

6 Opimal Growh for P&C Insurance Companies Figure. Growh impac curve S Combined Raio Growh does no imply ha he curren mix of NB and RB is irrelevan o he growh plan. When he curren growh is slow (e.g., NB is 0% of oal business), he required NB growh in he second year is 0.26/0. = 60%. The 5% growh plan is aggressive. On he oher hand, when he curren growh is rapid (e.g., NB is 25% of oal business), he required NB growh in he second year is 0.275/0.25 = 0%. The 5% growh plan is relaively easy o achieve. Now le us evaluae he impac of he growh paern on he company s combined raio. Le b denoe he difference beween he NB and RB combined raios, m L L n b = C C = n, r, + dp m, r, + E E. ( 29. ) n, r, Thus, b is a measure of he performance of new business relaive o renewal business. The company s combined raio, C, can be wrien as = + C = AC + A C C Ab. ( 2. 0) n, r, r, Subsiuing (2.8) ino (2.0), he combined raio C a growh rae G is C = = C + + G R C R C + G + R R r, r, n, n, r, n, r, + G R b r, +. ( 2. ) + G + R R n, r, Equaions (2.0) and (2.) demonsrae he posiive relaionship beween he combined raio and he growh rae, which can be illusraed by he growh impac curve in Figure. 3. Capial consrain on growh The growh of an insurance company is generally consrained by inernal and exernal condiions such as surplus, sock marke valuaion, shareholder demands, and insurance marke condiions. This implies ha companies facing differen consrains will have differen growh sraegies. Hagsrom (98) discusses ha he availabiliy of surplus consrains he growh of an insurance company. Baker, Powell, and Vei (2003) show ha firms follow sock marke signals o adjus heir business sraegies. Wang e al. (20) demonsrae ha profiable growh is difficul o achieve in a sof marke, and cycle managemen is a crucial consideraion for an insurance company when deermining is growh sraegy. Pas sudies have shown ha surplus capaciy can consrain he growh of an insurance company and ha changes in insurance surplus affec he indusry s capaciy for bearing risk (Davis 979; Hagsrom 98; Gron 994; Winer 994; Cummins and Danzon 997). To suppor growh, regulaors and raing agencies require he insurance company o hold addiional capial o mainain is financial raing. In his secion, we illusrae our mehodology by assuming a simple capial consrain: o mainain a arge premium-o-surplus raio. 5 Consider an insurance 5 The mehod can be expanded o incorporae more complicaed capial consrains, such as mainaining a financial raing (a predeermined Bes s Capial Adequacy Raio, or BCAR) a a predeermined probabiliy. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 07

7 Variance Advancing he Science of Risk company in a growh equilibrium ha has a arge premium-o-surplus raio K. I wries one uni of premium in underwriing year and is growing is exposure a G annually. Assume he marke price a year is P m =. dp is he company s percenage price difference wih he marke. So is acual price a year is P = + dp. Wrien premium is he produc of wrien exposure Q and price P, ha is, = Q * ( + dp ). Moving Q o he lef side of he equaion, he wrien exposure a year is Q = + dp. ( 3.) The wrien exposure a year + is Q + = Q * ( + G + ). Subsiuing (3.) ino he equaion, Q + G+ = + + dp A ime +, he marke price is. ( 32. ) m m P = P + T, ( 3.) where T + is he marke rae change from year o +. To simplify he illusraion, we assume T + = 0, which implies P m + = P m. Le dp be consan wih ime: dp = dp + = dp. The insurer s acual price a + is hen P + = P m + * ( + dp + ) = + dp. The oal wrien premium in year + is G+ WP = Q P = + dp ( + + ) + dp = + G. ( 34. ) + The wrien exposure a year is Q Q = = + G + dp G +. ( 35.) The wrien exposures a and + are + dp and + G +, respecively. The wrien premium a year + dp is WP = Q P = ( + dp) ( + dp)( + G ) =. (3.6) + G The wrien premiums a and + are and + G +, respecively. Assuming annual policy erms and assuming uniform wriing of policies hroughou he year, he earned premium in calendar year is EP = 05. WP + WP = G, ( 37. ) and he earned premium in calendar year + is EP = 05. WP + WP + + ( + ) = G +. ( 38. ) The underwriing profi in year is EP * ( C ). We assume ha policyholder-provided funds (from unearned premium reserve and loss reserve) are proporional o wrien premium, and shareholder-provided funds are equal o surplus. Le S be he surplus a he beginning of year. The oal funds available for invesmen are I = λ WP + S, ( 39. ) where λ is he fund-generaing coefficien. Le Y be he yield on invesmen, u and I be he ax raes on underwriing and invesmen profi, respecively, and D be he dividend payou raio. The reained profi afer axes and dividends in year is ( ) EP C = + IY ( I ) π u ( D ). ( 30. ) The surplus a he beginning of year + is S + = S + π. To mainain he arge premium-o-surplus raio in policy year +, WP S + + ( + ) WP + G = S + π K. ( 3. ) 08 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

8 Opimal Growh for P&C Insurance Companies Figure 2. Maximum growh rae under he premium-o-surplus consrain Combined Raio M S D Growh Subsiue (3.0) ino (3.) and solve for C : C WP G K S ( + ) + K I Y D ( I) EP D K ( ) u. ( 32. ) Inequaliy (3.2) implies ha in order o grow he business while mainaining he arge premium-osurplus raio, an insurance company needs o achieve a profi rae beer han a cerain hreshold o suppor he growh. Inequaliy (3.) can also be rewrien as G K S + π. ( 3. 3) WP + Inequaliy (3.3) implies ha here is a cerain boundary ha consrains an insurance company s growh; beyond a cerain growh rae, an insurer will no be able o mainain is arge leverage raio. The growh limi curve D in Figure 2 shows his consrain. In order o suppor growh, an insurer has o keep is combined raio below curve D. The faser he growh, he higher he profiabiliy ha is required o suppor he growh. Recall from Figure ha he growh impac curve S indicaes ha faser growh will lead o a higher combined raio. The inersecion a poin M beween curves D and S is he maximum growh under he premium-surplus leverage consrain. All he poins below M on curve S are below he growh limi curve D. If an insurer grows more slowly han M, is premium-o-surplus leverage raio will remain under he arge. All he poins above M on curve S are also above he growh limi curve D. Therefore growh faser han M will make he expeced premium-osurplus raio higher han he arge level. 4. Opimal growh In general, an insurance company s value is an increasing funcion of is surplus, volume of sales, and profiabiliy. In he sock marke, raios such as price-obook, price-o-sales, and price-o-earnings are widely used parameers in a company s valuaion. Insurance companies combined raios are ofen volaile, so any valuaion based on earnings will also be unsable as a resul. D Arcy and Gorve (2004) run muliple regressions using observed marke values as he dependen variable and surplus, ne wrien premium, and combined raio as explanaory variables, and find ha he coefficien for he combined raio variable is significan for only one ou of 5 companies (Accepance). For he oher 4 companies examined in he sudy, all 5 companies combined, and all companies excluding AIG, he coefficien of he combined raio variable is no significan a he 5% level. However, as we all know, if he combined raio is low, earnings will be high and he surplus will increase accordingly. From his aspec, he impac of he combined raio on marke value is correlaed wih surplus. Afer conrolling for he effec of surplus, he combined raio is no longer a significan variable in esimaing he marke value of he insurer. As a resul, we assume ha an insurance company s expeced enerprise value is a funcion of surplus and ne wrien premium only, and VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 09

9 Variance Advancing he Science of Risk exclude he combined raio variable from D Arcy and Gorve s (2004) formula V = W φ S + W η WP, ( 4.) 6 + n + n + n where ϕ is he expeced price-o-book raio, η is he expeced price-o-sales raio, n is he planning or evaluaing horizon, and W is he weigh on surplusindicaed enerprise value. To simplify he formulaion, we assume he growh rae is consan wih ime, ha is, G = G. In Equaion (4.), surplus a year +n, S +n is a decreasing funcion of he growh rae, S +n = S(G,n); while wrien premium, WP +n = ( + G) n, is increasing wih G. The opimal growh rae is he one ha maximizes he expeced enerprise value a year-end +n: n MaxW φ SGn (, ) + ( W) η ( + G). (4.2) G Marke appeie o surplus and growh varies over ime. Aghion and Sein (2008) find ha he sock marke pricing rule impacs managemen s growh sraegy. When he sock marke is more ineresed in growh, managers may pay more aenion o sales volume. When invesors in he sock marke prefer profi margins, managers ake he cue and adap heir sraegies accordingly by cuing coss. The sudy demonsraed ha managemen s endency o give he sock marke wha i wans can lead o excess volailiy in business operaions. Ma (2009) examines Aghion and Sein s caering heory on public P&C companies and confirms ha insurers devoe more effor o growh when he sock marke places greaer value on growh, and ha he managerial shor-ermism from following sock marke preferences can exer a desabilizing influence on insurance pricing and he insurance marke cycle. Based on hose sudies, he auhors do no aemp o esimae empirical weighs implied by he sock marke. We sugges ha praciioners use a weigh ha rep- 6 The expeced enerprise value can also be viewed as he uiliy funcion of an insurance company s managemen. resens managemen s long-erm view of he balance beween exposure growh and surplus growh. 5. Numerical analysis To illusrae he framework, we perform a case sudy on a hypoheical insurance company. Key parameers are seleced based on he auhors experience. G = 2%. 5 dp. ( 5.) A he marke price level, an insurance company will grow is exposure 2% annually, which is close o he average annual GDP (gross domesic produc) growh. 7 The reenion raes of renewal and new business are R R r, n, = 84% 02. dp = 78% 03. dp. ( 52. ) 8 A he marke price level, he reenion raio of RB is 84%, 6% higher han ha of NB. This reenion difference beween RB and NB is consisen wih Wu and Lin (2009). A a 2% annual exposure growh rae, he equilibrium NB percenage by Equaion (2.8) is 078. A = = 8. 8%. The NB loss raio is assumed o be 75%, which is 3% higher han ha of RB. The difference in loss raios beween NB and RB is also consisen wih Wu and Lin (2009). The NB expense raio is assumed o be 37%, while he RB expense raio is 32%. A marke price levels, NB, RB, and whole book combined raios are 2%, 94%, and 97.4%, respecively. If he price is 4% below he marke, he company will improve reenion, arac more new business, and grow 8% a year. If i charges 4% above marke, i will reduce reenion, arac less new business, and shrink 4% annually. Table 3 exhibis 7 The average real GDP growh from 2000 o 2009 is.82% according o he Bureau of Economic Analysis, 8 New business conversion in general is sensiive o dp, a company s price difference relaive o indusry. Reenion is less sensiive o dp. This relaionship is refleced by he magniude of he coefficiens of dp in Equaions (5.) and (5.2). 0 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

10 Opimal Growh for P&C Insurance Companies Table 3. Equilibrium Nb percenages, loss and combined raios by growh Growh dp NB Reenion RB Reenion ENBP NB LR RB LR NB CR RB CR Whole Book CR 8.0% 4% 79.2% 84.8% 22.7% 78.% 64.6% 5.% 96.6% 00.8% 6.5% 3% 78.9% 84.6% 2.7% 77.3% 63.9% 4.3% 95.9% 99.9% 5.0% 2% 78.6% 84.4% 20.8% 76.5% 63.3% 3.5% 95.3% 99.% 3.5% % 78.3% 84.2% 9.8% 75.8% 62.6% 2.8% 94.6% 98.2% 2.0% 0% 78.0% 84.0% 8.8% 75.0% 62.0% 2.0% 94.0% 97.4% 0.5% % 77.7% 83.8% 7.7% 74.3% 6.4%.3% 93.4% 96.5%.0% 2% 77.4% 83.6% 6.6% 73.5% 60.8% 0.5% 92.8% 95.7% 2.5% 3% 77.% 83.4% 5.5% 72.8% 60.2% 09.8% 92.2% 94.9% 4.0% 4% 76.8% 83.2% 4.3% 72.% 59.6% 09.% 9.6% 94.% ENBP, NB, and RB reenion raios, and NB and RB loss and combined raios for dp values beween 4% and 4% (wih he corresponding growh assumpions). I is clear ha lower prices will drive faser growh, higher reenion, and higher combined raios. This is consisen wih he heoreical derivaion in previous secions and wih real-world observaions. The fund-generaing coefficien is assumed o be.2. So invesmen funds available from he unearned premium and loss reserves are 20% of wrien premium. The invesmen yield is assumed o be 4%. The ax rae on invesmen and underwriing income is 35%. 9 The dividend payou raio is assumed o be 30%, which is close o he acual S&P 500 payou raio in he 2000s. 0 Year 0 wrien premium is. The arge premium-o-surplus raio is assumed o be.5. The surplus a he beginning of year 0 is 0.667, so ha he premium-o-surplus raio a he saring poin is.5. The arge premium-o-surplus raio can vary by ime, and i may no be he iniial raio. For example, an insurer wih a low premium-o-surplus raio migh be willing o increase is leverage for a period of ime when i adops a high growh plan in a hard marke; 9 In pracice, insurance companies ofen hold a large amoun of municipal bonds exemp from ax. Praciioners can adjus he approach easily by applying separae ax raes on invesmen and underwriing incomes. 0 Dividend payou raio is no a consan in he growh decision making. For example, a company may be able o reduce he dividend level o suppor more rapid growh. i hen slows down he growh and arges a lower premium-o-surplus raio when he marke urns sof. The case sudy can be easily adjused o incorporae ime-varying arge leverage. Tables 4 hrough 6 show five-year underwriing and invesmen profis, surplus, premium, and premiumo-surplus raios if a company grows a 2%, 4%, and 8% (which correspond o scenarios ha he insurer consisenly prices a, 4% above, and 4% below he marke). Table 7 shows he same saisics a he consrained maximum growh rae of 5.52%. If he growh is below he hreshold, as shown in Tables 4 and 5, he company will generae more profi han is required o suppor he growh. Surplus will grow faser han premium; he expeced premium-o-surplus raio will go below.5 and decrease wih ime. Table 6 shows an example of wha happens if he growh rae is faser han 5.52%. In his case, he company will generae less profi han is required o suppor he growh. Surplus levels canno keep up wih he pace of premium growh, and he expeced premium-o-surplus raio will go above.5 and coninue o increase wih ime. Table 7 shows ha if he insurer grows a 5.52%, he expeced premium-o-surplus raio will remain a.5, which is wha we expec, as ha is he growh soluion o mainain a consan premium-o-surplus raio, given our inpu assumpions. Table 8 repors he combined raios on he growh impac and growh limi curves. Figure 3 shows he wo curves visually. To achieve faser growh, an VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY

11 Variance Advancing he Science of Risk Table 4. Five-year profis, surplus, and leverage raios a 2% growh Year Beginning Surplus WP EP Invesmen Yield Inv Profi UW Profi Toal Profi Tax Rae Afer-Tax Profi Payou % Reained Profi End Surplus Prem/ Surplus % % % % % % % % % % % % % % % % % % Table 5. Five-year profis, surplus, and leverage raios a 4% growh Year Beginning Surplus WP EP Invesmen Yield Inv Profi UW Profi Toal Profi Tax Rae Afer-Tax Profi Payou % Reained Profi End Surplus Prem/ Surplus % % % % % % % % % % % % % % % % % % Table 6. Five-year profis, surplus, and leverage raios a 8% exposure growh Year Beginning Surplus WP EP Invesmen Yield Inv Profi UW Profi Toal Profi Tax Rae Afer-Tax Profi Payou % Reained Profi End Surplus % % % % % % % % % % % % % % % % % % Prem/ Surplus Table 7. Five-year profis, surplus, and leverage raios a maximum allowable growh 5.52% Year Beginning Surplus WP EP Invesmen Yield Inv Profi UW Profi Toal Profi Tax Rae Afer-Tax Profi Payou % Reained Profi End Surplus % % % % % % % % % % % % % % % Prem/ Surplus % % % CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

12 Opimal Growh for P&C Insurance Companies Table 8. Combined raios on growh impac and limi curves Growh CR on Growh Impac Curve CR on Growh Limi Curve 9.00% % % % % % % % % % % % % % % % % % % % insurance company needs o reduce prices and arac more NB. Boh acions will resul in a higher combined raio. The combined raios on he growh impac curve, as shown in Table 8 and Figure 3, represen his relaionship. When he growh rae is zero, he combined raio is 96.3%. When he growh rae is 9%, he combined raio will increase 5.% o 0.4%. The company will swich from an underwriing profi o an underwriing loss if i grows faser han 6.5%. When growh is zero, an insurer can mainain he arge leverage raio a a 7.5% underwriing loss. Under his scenario, he invesmen income offses he underwriing loss, he ne income is zero, and he surplus says he same afer one underwriing period. The premium does no change eiher, so ha he insurer will mainain a sable leverage raio. If he insurer plans o grow a 9% annually, i needs o increase is surplus a he same pace. The corresponding combined raio on he growh limi curve is 94.0%, which implies ha he company has o achieve a 6% underwriing profi so ha i can reain a ne profi (afer ax and dividend) Figure 3. Empirical growh impac and limi curves Combined Raio M S D Growh Rae VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 3

13 Variance Advancing he Science of Risk ha is equal o 9% of is surplus. Unforunaely, a 9% growh in realiy will resul in an underwriing loss of.4%, according o he growh impac curve. A 5.52% growh, he combined raios on he wo curves are equal. If growh is faser han 5.52%, he growh impac curve is above he growh limi curve; ha is, he acual combined raios will be higher han hose required o mainain he arge leverage raio. Following such aggressive growh, he premiumo-surplus raio will penerae he arge leverage, as shown in Table 6. Therefore, 5.52% is he maximum allowable growh rae under he leverage consrain. Insurance capial requiremens vary by line of business. Volaile lines (such as coasal homeowners) in general require more economic capial o pay losses from wors-case scenarios, and he arge premium-osurplus raios can be less han one. Alernaively, sable lines (such as sandard personal auo) require less capial, and he arge leverage raios can be greaer han wo. Table 9 shows ha, if he insurer s arge leverage raio decreases from.5 o.0, is maximum growh under he consrain decreases from 5.52% o 4.54%. If he arge leverage raio increases o 3.0, i can grow 7.63% annually while sill keeping is expeced leverage raio a he arge level. I is well known ha companies wih superior risk selecion skills can ouperform he indusry in boh profiabiliy and growh. Assume an insurer can idenify he mos profiable segmens and can wrie ha business a a loss raio significanly lower han he indusry average. In his case, i can lower is price o be compeiive while sill being very profiable. As shown in Table 9, if he insurer can improve loss raios by 5% a marke price (NB LR = 70% and RB LR = 57%) hrough beer risk selecion, is maximum allowed growh rae increases from 5.52% o 8.00%. Alernaively, if he company is adversely seleced agains and loss raios deeriorae by 5% (NB LR = 80% and RB LR = 67%), he consrained maximum growh declines from 5.52% o 3.07%. If he insurer can improve loss raios by 0% (NB LR = 65% and RB LR = 52%), i can grow as fas as 0.5% wihou peneraing he arge.5 premium-o-surplus raio. Some personal auo insurers (such as Progressive) were able o achieve superior growh and profiabiliy using advanced analyics in risk selecion. Table 9 shows ha if an insurer can produce a loss raio 0% beer han he indusry average in he presence of a less resricive capial requiremen (wih a arge leverage raio of 3), i can obain a significan compeiive advanage, grow a a sellar 5.55% annual rae, and achieve an ousanding combined raio of 94.36% so ha surplus can grow a he same rae as sales. I is also well known ha insurance companies can produce lower loss raios and achieve faser growh in a hard marke. Wang e al. (20) shows ha loss raios Table 9. Consrained maximum growh under various loss raio assumpions and leverage consrains NB LR a Marke Price RB LR a Marke Price Targe Leverage Raio dp Combined Raio Consrained Maximum Growh 80.0% 67.0% % 02.8% 2.74% 80.0% 67.0% % 03.0% 3.07% 80.0% 67.0% 3.0.9% 03.43% 3.78% 75.0% 62.0%.0.69% 98.80% 4.54% 75.0% 62.0% % 99.36% 5.52% 75.0% 62.0% % 00.57% 7.63% 70.0% 57.0% % 94.68% 6.35% 70.0% 57.0% % 95.58% 8.00% 70.0% 57.0% % 97.55%.55% 65.0% 52.0%.0 4.2% 90.46% 8.8% 65.0% 52.0% % 9.67% 0.5% 65.0% 52.0% % 94.36% 5.55% 4 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

14 Opimal Growh for P&C Insurance Companies Table 0. Consrained maximum growh under exreme marke condiions Marke Cycle NB LR a Marke Price RB LR a Marke Price Targe Leverage Raio alpha dp Combined Raio Consrained Maximum Growh Sof 87.0% 74.0%.5 2.0% 0.5% 09.23%.24% Normal 75.0% 62.0%.5 2.0% 2.35% 99.36% 5.52% Hard 63.0% 50.0%.5 6.0% 4.6% 88.94% 2.23% in sof and hard markes can differ by as much as 25%. Consisen wih Wang e al., we assume ha he loss raios in exreme sof/hard markes are 2% worse/ beer han ha of he normal marke. If he marke is very hard, an insurer can grow is exposure by 6% while mainaining a marke price level (α = 0.06 in Equaion 2.). In a very sof marke, an insurer will shrink by 2% a he marke price (α = 0.02). Table 0 illusraes he impacs of marke cycles on loss performance and consrained maximum growh. In an exreme sof marke, he maximum allowable growh rae is negaive. The insurer has o reduce is size o mainain he arge leverage raio. In an exreme hard marke, he insurer can grow by over 2% annually while sill mainaining he leverage raio. Following D Arcy and Gorve (2004), we calculae he opimal growh rae ha maximizes he expeced enerprise value a he end of he fifh year. The expeced price-o-book raio and price-o-sales raio are assumed o be.2 and 0.8, respecively. Table shows he expeced enerprise value a he end of he fifh year by various growh raes and weighs on book value. If he weigh on book value is high (W 77%), a company can increase is value hrough rae increases while shrinking is business. This is because he profi and surplus growh will creae enough enerprise value o offse he reducion from he wrien premium loss. This case is equivalen o he regression scenario excluding AIG in D Arcy and Gorve (2004). Under his condiion, here is no posiive opimal growh and he insurance company would need o reduce is sales vol- As of July 2, 20, he sraigh averages of price-o-book raios and price-o-sales raios of four large P&C insurance companies (Travelers, Allsae, Progressive, and CNA) are.6 and 0.80, respecively. ume in order o increase is expeced enerprise value. When he weigh on book value is low (W 74%), he company can increase is value hrough aggressive growh and rae cus. In his case, he expeced enerprise value is increasing wih growh, leading o an opimal exposure growh greaer han 8%. An insurer can maximize is expeced value hrough aggressive growh because he gain in premium will more han offse he loss in surplus and profi. This is equivalen o he regression scenario including AIG in D Arcy and Gorve (2004), in which he opimal growh rae is abou 0%. When he weigh on book value is beween 74% and 76%, he opimal growh is beween 0% and 8%. As shown in Table, when W = 76%, a 2% exposure growh provides he highes expeced value of hose repored scenarios. When W = 75%, 5% growh offers he highes value. When W > 76.5%, a posiive opimal growh rae does no exis. Table 2 shows he opimal growh raes for weighs on book value beween 74.5% and 77.5%. Two consrains are added in he opimizaion: he expeced premium-o-surplus raio is no larger han.5 and he exposure growh is no less han 3%. When W = 74.5%, he leverage consrain is enforced so ha he opimal growh rae is he maximum allowable growh rae of 5.52%. When W = 77.5%, he minimum growh consrain is applied so ha he opimal growh is 3%. When 75% W 77%, he consrains do no impac he opimizaion. Figure 4 demonsraes he opimal growh rae visually for W = 76%. As he growh rae increases from 4% o.94%, he growh creaes more enerprise value han he loss from he reducion of book value. The expeced enerprise value increases seadily unil i reaches is peak value.092. When he growh is faser han.94%, he increase of enerprise value from growh canno offse he loss VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 5

15 Variance Advancing he Science of Risk Table. Expeced inernal enerprise value a he end of fifh year by growh rae Growh Expeced Enerprise Values by weigh on book value dp Surplus NWP 70% 7% 72% 73% 74% 75% 76% 77% 78% 79% 80% 8.0% 4% % 3% % 2% % % % 0% % % % 2% % 3% % 4% CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

16 Opimal Growh for P&C Insurance Companies Table 2. Opimal growh raes under consrains by weighs on book values Weigh on Book Value Growh dp 5h Year-end Surplus 5-year Cumulaive Growh Maximum Expeced Value 74.5% 5.52% 2.35% % % 5.06% 2.04% % % 3.53%.02% % %.94% 0.04% % % 0.3%.3% % %.44% 2.29% % % 3.00% 3.33% %.0982 from he reducion of book value, and so he expeced enerprise value decreases wih growh. 6. Conclusions Profi and growh are ofen wo conflicing goals of insurance companies. I is common knowledge ha profi and growh ofen move in opposie direcions. Rapid growh may diminish underwriing performance, reduce profi, and even cause bankrupcy. When facing underwriing losses, managemen may have o slow down he rae of growh. Slow or negaive growh may improve underwriing profi and increase surplus. A higher volume of surplus will reduce he premium-o-surplus raio and resul in a lower reurn on equiy, which may give managemen incenives o grow he business and increase he leverage raio. Insurance companies need o grow raionally by balancing hese wo conradicory forces. On one hand, growh will drive up he combined raio because new business generally produces higher loss and expense raios. Bu on he oher hand, rapid growh requires Figure 4. Opimal growh rae when w = 76% Expeced Enerprise Value Growh Rae VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 7

17 Variance Advancing he Science of Risk insurance companies o generae more underwriing profi and addiional capial. The faser he growh, he lower he combined raio needed o suppor such growh. In his sudy, he auhor proposes a sraighforward mehod o calculae a consrained maximum growh rae and opimal growh rae. Compared wih he previous sudy (D Arcy and Gorve 2004), he proposed approach has hree advanages: () i is easier o undersand and implemen; (2) all he required daa is readily available in he acuarial daabase of insurance companies; (3) i discusses furher he condiions for he exisence of a posiive opimal growh rae. The flip side of hose advanages is ha he approach is in a deerminisic framework 2 and herefore canno provide valuable sochasic insighs of DFA models, such as he risk fronier and disribuional saisics in D Arcy and Gorve (2004). Under growh equilibrium, he combined raio is an increasing funcion of he growh rae. This is he growh impac curve. To mainain a arge premiumo-surplus raio, higher capial and underwriing profi is needed so ha surplus can keep pace wih premium growh. The faser he growh, he lower he combined raio ha is required. This is he growh limi curve where he combined raio is a decreasing funcion of growh. The growh impac curve shows ha he profi margin will decrease wih growh from he perspecive of underwriing performance. The growh limi curve represens he need for he profi margin o increase wih growh from he perspecive of capial managemen. The growh limi curve enforces a consrain on growh: he combined raio needs o be below he curve so ha he expeced premium-o-surplus raio will no penerae a cerain arge level. The inersecion of he growh impac and limi curves is he maximum growh rae under he leverage consrain. To obain he opimal growh rae, an insurance company is assumed o maximize is expeced enerprise value, which is a weighed average of is 2 I is no difficul o add sochasic componens o he equilibrium model. However, by inroducing sochasic underwriing and invesmen performance, he approach will lose is appealing simpliciy. book value and sales volume. The mehodology and he resuls of our case sudy on opimal growh are consisen wih D Arcy and Gorve (2004). When he weigh on book value is beyond a cerain hreshold, he impac of surplus dominaes ha of wrien premium: an insurance company will increase is expeced enerprise value by raising premium raes and reducing is exposure. In his case, posiive opimal growh does no exis. This paper sudies he consrained maximum and opimal growh raes under cerain marke condiions for an insurer wih a sable pricing sraegy and a consan uiliy funcion. In pracice, marke condiions, pricing sraegies and uiliy funcions of insurance companies coninuously change wih ime. While growh is approaching is equilibrium under a specific environmen, he marke condiion changes. An insurance company may adjus is pricing sraegy accordingly. This implies ha growh may never reach equilibrium in he real world. Neverheless, knowing he heoreical boundary (he maximum growh rae under he leverage consrain) for growh can help insurance companies o reduce enerprise risks due o irraional growh. Knowing he opimal growh raes condiional on managemen s uiliy funcion can help managemen o make a growh plan ha is consisen wih heir evaluaions of he relaive imporance of surplus and sales volume. The calculaions of consrained maximum and opimal growh raes are sraighforward and can be effecively employed by propery and casualy insurers in heir sraegic planning process. Acknowledgmens The auhor hanks Professor Philip Garcia, Doug Pirle, FCAS, MAAA, and C. K. San Khury, FCAS, MAAA for heir valuable commens. References Aghion, P., and J. C. Sein, Growh versus Margins: Desabilizing Consequences of Giving he Sock Marke Wha I Wans, Journal of Finance 63, 2008, pp CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

18 Opimal Growh for P&C Insurance Companies A. M. Bes, Bes s Insolvency Sudy, Propery/Casualy US Insurers , A. M. Bes Special Repor, May Baker, H. K., G. E. Powell, and E. T. Vei, Revisiing Managerial Perspecives on Dividend Policy, Journal of Economics and Finance 26, 2003, pp Cohen, A., Asymmeric Informaion and Learning: Evidence from he Auomobile Insurance Marke, Review of Economic and Saisics 87: 2, 2005, pp Cummins, D. J., and P. Danzon, Price, Financial Qualiy, and Capial Flows in Insurance Markes, Journal of Financial Inermediaion 6, 997, pp D Arcy, S. P., and N. A. Dohery, The Aging Phenomenon and Insurance Price, Proceedings of he Casualy Acuarial Sociey 76, 989, pp D Arcy, S. P., and N. A. Dohery, Adverse Selecion, Privae Informaion and Lowballing in Insurance Markes, Journal of Business 63, 990, pp D Arcy, S. P., and R. W. Gorve, The Use of Dynamic Financial Analysis o Deermine Wheher an Opimal Growh Rae Exiss for a Propery-Liabiliy Insurer, Journal of Risk and Insurance 7:4, 2004, pp Davis, G. E., Underwriing Profis Necessary o Keep Pace wih he Increasing Premium Growh For Propery-Casualy Companies, [Review of Paper], Casualy Acuarial Sociey Discussion Paper Program, May 979, pp Feldblum, S., Personal Auomobile Premiums: An Asse Share Pricing Approach for Propery/Casualy Insurance, Proceedings of he Casualy Acuarial Sociey 83, 996, pp Gron, A., Capaciy Consrains and Cycles in Propery-Casualy Insurance Markes, RAND Journal of Economics 25:, 994, pp Hagsrom, D. S., Insurance Company Growh, Transacions of he Sociey of Acuaries 33, 98, pp Harringon, S. E., P. M. Danzon, and A. J. Epsein, Crises in Medical Malpracice Insurance: Evidence of Excessive Price- Cuing in he Preceding Sof Marke, Journal of Banking and Finance 32:, 2008, pp Ma, Y., Do Publicly-Traded Propery-Casualy Insurers Caer o he Marke? ARIA Conference Paper, McClenahan, C. L., Adjusing Loss Developmen Paerns for Growh, Proceedings of he Casualy Acuarial Sociey 74, 987, pp Mueeries, J. H., Underwriing Profis Necessary o Keep Pace wih he Increasing Premium Growh For Propery- Casualy Companies, Casualy Acuarial Sociey Discussion Paper Program, May 979, pp Niswander, R. E., The Relaionship Beween Underwriing Profi and he Surplus Raio: A Model, Casualy Acuarial Sociey Discussion Paper Program, May 984, pp Wang, S., J. A. Major, C. H. Pan, and J. W. K. Leong, U.S. Propery-Casualy: Underwriing Cycle Modeling and Risk Benchmarks, ERM Symposium Research Papers, 20. Winer, R. A., The Dynamics of Compeiive Insurance Markes, Journal of Financial Inermediaion 3, 994, pp Wu, C. S. P., and H. Lin, Large Scale Analysis of Persisency and Renewal Discouns for Propery and Casualy Insurance, CAS E-Forum, Winer 2009, pp VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 9

19 Variance Advancing he Science of Risk Appendix Model Exension o Muliple Levels of Renewals The proposed model in he main secion of his paper simplifies he aging phenomenon ino wo classes: new and renewal books of business. The loss raio of renewal business generally coninues improving wih policy age (D Arcy and Dohery 990). Reenion also increases as renewal policies become older. A naural exension of he model is o subdivide renewals ino muliple segmens. In he case sudy example, a he marke price level, he NB and RB reenions are 78% and 84%, respecively, and he NB and RB loss raios are 75% and 62%, respecively. Assume ha an acuarial praciioner would like o furher divide he RB book ino wo subgroups: firs wo renewals and he remainder. Le RB denoe he firs renewal, RB2 denoe he second renewal, and RB3+ denoe policies ha are of age hree years or older. Assume he reenion raios of RB, RB2, and RB3+ are 82%, 82%, and 85%, respecively; and he loss raios of RB, RB2, and RB3+ are 65%, 65%, and 60%, respecively. To exend he model, we need he equilibrium percenages for NB, RB, RB2, and RB3+. Using he same logic as in he main secion, suppose he exposures of NB, RB, RB2, RB3+ are 0.0, 0.09, 0.08, and 0.73, respecively. Again, assuming he book would grow a 2% per year a he marke price level, he convergences of equilibrium percenages are demonsraed in Table A. In year, he RB exposure (0.078) is equal o he prior year s NB exposure (0.) muliplied by he new business reenion raio (78%); he RB2 exposure (0.074) is he prior year s RB exposure (0.09) muliplied by he RB reenion (82%); he RB3+ exposure (0.686) is he sum of he renewals from he prior year s RB2 (0.08*82%) and RB3+(0.73*85%) books. The oal RB exposure (0.838) is he sum of RB, RB2, and RB3+. To grow he business by 2%, he NB exposure in year (0.82) needs o be he exposure from he arge growh (.02) less he oal RB exposure. Coninuing he process, he percenages of NB, RB, RB2, and RB3+ will converge a year 8. A equilibrium, NB is 8.7% of book, RB is 4.3%, RB2 is.5%, and RB3+ is 55.5% of book. If he saring exposures of NB, RB, RB2, and RB3+ are 0.20, 0.5, 0.0, and 0.55, respecively, he convergence will be faser, and will arrive a he same equilibriums a year 6, as shown in Table A2. Le A NB, A, A 2, and A 3 be he equilibrium percenages of NB, RB, RB2, and RB3+, respecively; and le R NB, R, R 2, and R 3 be he respecive renewal raes. RB is from he prior year s new business: A = A R + G. ( A) NB NB RB2 is from he prior year s RB: A = A R + G. ( A2) 2 RB3+ is from he prior year s RB2 and RB3+: ( + ) A = A R + A R G. ( A3) Finally, he sum of he percenages from various layers should be equal o one. A + A + A + A =. ( A4) NB CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

20 Opimal Growh for P&C Insurance Companies Table A. Equilibrium percenages convergence of case Year NB RB RB2 RB3+ RB Toal NB % RB % RB2 % % 9.00% 8.00% % 7.65% 7.24% % 3.65% 6.5% % 4.02% 0.98% % 4.5%.27% % 4.28%.38% % 4.30%.48% % 4.30%.49% % 4.3%.50% Table A2. Equilibrium percenages and he convergence of case 2 Year NB RB RB2 RB3+ RB Toal NB % RB % RB2 % % 5.00% 0.00% % 5.29% 2.06% % 4.36% 2.30% % 4.35%.54% % 4.33%.53% % 4.3%.52% % 4.3%.50% The four equilibrium percenages can be obained by solving he four equaions above: 3 ( + ) 3 ( 3 ) 2 + G G R3 A = NB 2 ( + G) ( + G R )+( + G) + G R + ( + G R3) R R + R R R NB NB 2 A R NB NB A = + G A R R NB NB A = 2 2 ( + G) A R RR NB NB 2 A = 3 2 ( + G) ( + G R3 ) R NB ( A5) Once he equilibrium exposure percenages for various layers of new and renewal business are derived, we can calculae he whole book s combined raio as a weighed average of he muliple layers. In his specific case, he equilibrium loss, expense, and combined raios a 2% growh rae are 64.%, 32.9%, and 97.0%, respecively. The new combined raio formula will replace Equaion (2.) in he main secion and he calculaions afer i will remain he same. 3 In pracice, i migh be easier o solve for he equilibrium percenages numerically in a fashion similar o ha used in Tables A and A2. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 2