Professional Liability Insurance Contracts: Claims Made Versus Occurrence Policies

Transcription

1 ARICLES ACADÉMIQUES ACADEMIC ARICLES Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012, Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012, Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces by Mar Boyer ad Kare Gober résumé Nous préseos ue approche héorque qu s éresse aux déermas du chox de ype de polce d assurace ere les coras Clams-made e les coras Occurrece. La grade dfférece ere les deux ypes de coras es que le premer red resposable l assureur acuel pour des cdes passés qu so rapporés aujourd hu, alors que le secod red resposable l assureur acuel pour les cdes aujourd hu qu e sero rapporés que das le fuur. Le bu de l arcle es juseme de regarder das quelles crcosaces u assuré devra préférer u cora pluô que l aure. Mos clés: Assurace resposablé, assurace basée sur la dae des réclamaos, aséleco. Classfcao JEL : G absrac We prese hs paper a heorecal approach ha argues whe oe should op for clams-made or for occurrece-based polces lably surace. Occurrece-based coracs cover he polcyholder for losses curred a gve he auhors : Mar Boyer: CEFA Professor of Face ad Isurace, HEC Moréal (Uversé de Moréal) ad Crao, Moréal, Caada; [email protected] Kare Gober: HEC Moréal (Uversé de Moréal), Moréal, Caada. hs research s facally suppored by he Fods pour la Formao des Chercheurs e d Ade à la Recherche (FCAR - Québec) ad by he Socal Scece ad Humaes Research Coucl (SSHRC - Caada). he coug suppor of Crao s also graefully ackowledged. 251

2 year, o maer whe he clams s acually repored he fuure. I clams-made coracs, losses are covered he year whch hey are repored o maer whe hey occured he pas provded a clams-made surace polcy was vald he. he major dfferece bewee he wo ypes of corac s hus ha occurrece coracs are forward lookg whereas clams-made coracs are rerospecve. he goal of hs paper s o aalyze wha crcumsaces polcyholders would prefer oe corac of he oher. Keywords: Lably surace, clams made ad repored, adverse seleco. JEL classfcao: G 1. INRODUCION he roduco of clams made ad repored (CMR hereafer) surace polces has bee oe of he ma ovaos ha resuled from he uceray surroudg he lably crss of he lae 1970s. As chages he legal evrome was a udversfable rsk for surers, he law of large umbers o loger appled compleely o hese surace producs, so ha premums eeded o be creased o cover legal evrome rsk. Dohery (1991) cocludes ha he creased ecoomc mporace of muual surace compaes resuled drecly from hs lably crss. hese coracs complemeed radoal occurrece-based (OB hereafer) coracs whereby a polcyholder s sured for losses ha are curred durg he surace polcy year eve f he loss s o repored for may more years. I coras, CMR coracs sure polcyholders for losses ha are repored durg he polcy year eve f he loss was curred may years before (subjec o a rerospecve dae or me lm). he appare domace of clams-made polcy was such ha eve S. Paul Fre ad Mare, a major medcal malpracce surer he 1980s, swched s ere medcal malpracce book of busess o clams-made aroud ha me. Poser (1986) acpaed ha CMR polces would accou for sevey o eghy perce of he medcal malpracce surace premum eared durg oday, approxmaely 75% of medcal malpracce surace s beg sold uder a clams-made approcah. he paper focuses o he dfferece he polcyholder s expeced uly from each corac ype. A uderlyg assumpo we shall use s ha he surace marke s compeve so ha he premum pad s equal o each polcyholder s expeced loss. As a resul, we are able o cocerae o he mpac of he corac srucure s dffereces raher ha surer profably. 252 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

3 o beer udersad how he wo ypes of coracs work, we cocerae o hree feaures of he coracs: he way losses develop over he years, how surace premums are calculaed, ad he mpac of rsk averso o he decso o purchase oe corac or he oher. Our paper preses wo ma resuls. Frs, we show ha whe everyoe he ecoomy s rsk eural, he he oly reaso why sured ages would prefer he CMR corac s ha he dscou facor hey apply o fuure cash flows s lower ha he surer s, whch meas ha he prese value of fuure cash flows s worh less o he age ha o he surer. hs resul holds rue wheher we allow udversfable shocks o mpac losses or he clam over me. Our secod ma resul s ha whe sured ages are rsk averse, he hey are more lkely o prefer he CMR corac over he OB corac whe he al of he loss s loger. hs meas ha for shoral les, such as auomoble ad homeower surace, clamsmade ad repored surace coracs do o domae as much radoal occurrece based coracs ha log-al les such as medcal ad professoal malpracce lably surace. Our heorecal approach offers predcos ha oe could brg o he daa, such as he fac ha for rsk eural ages, CMR coracs should be preferred o OB coracs whe he sured age has mpora lqudy cosras so ha he dscous fuure cash flows much more ha f he were o lqudy cosraed. he paper s orgazed as follows. We frs prese he ecoomc mporace of each ype of surace corac, how sured losses develop ad how pure premums should be calculaed gve hese losses. We he prese he problem ha a rsk eural age faces a ecoomy where here are o sysemac shocks o he dsrbuo of losses. We roduce shocks o he loss dsrbuo Seco 4 ad move o a rsk averse age Seco 5. Fally, we dscuss he emprcal predcos of he model ha oe could brg o he daa ad coclude he las seco of he paper. 2. ECONOMIC IMPORANCE Clams-made coracs are mosly popular lably les where he damage has bee caused by a dvdual who exercses hs professo. I parcular CMR coracs are hghly popular, or eve he orm, s he case of medcal malpracce lably surace coracs, drecors ad offcers lably surace coracs ad oher suao of professoal lably surace coracs desged Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 253

4 o proec lawyers ad archecs case of a professoal error. 1 he rsg mporace of CMR polces has led he NAIC o clude a separae aggregae repor for les of busess where CMR polces are mpora sarg hese les of busess are medcal malpracce lably surace, produc lably surace ad oher lably surace, whch clude drecors ad offcers surace ad oher ypes of professoal lably surace. able 1 preses he oal of premums eared he Ued Saes for he case of he medcal malpracce surace le as a fuco of he ype of corac from 1997 hrough year ABLE 1 PREMIUM EARNED (IN HOUSANDS OF CURREN DOLLARS) IN MEDICAL MALPRACICE INSURANCE BY YPE OF CONRAC, oal premums eared OB corac premums eared CMR corac premums eared Perceage of oal CMR coracs ,032,842 1,441,057 3,591, % ,128,893 1,465,654 3,663, % ,267,617 1,591,712 3,675, % ,351,526 1,861,044 3,490, % ,780,544 1,718,908 4,061, % ,157,351 2,430,379 6,726, % ,302,736 2,496,678 5,806, % ,784,556 2,145,038 6,639, % ,629,529 2,065,908 6,563, % ,140,990 2,355,646 7,785, % Source: Bor ad Boyer (2011) usg he Naoal Assocao of Isurace Commssoers aual daa Propery ad Casualy Isurers, Uderwrg ad Ivesme Exhb. he able cludes premum from all surers reporg ozero premums eher ype of medcal malpracce surace polces. As we see, he marke mporace of clams-made coracs has rse over he years ad accous for close o 77% of he oal premums eared he medcal malpracce surace le of busess hs compares o 71% 1997 ad 70% he early ees accordg o Bor ad Boyer (2011). I s clear ha CMR polces have creased populary ad ha hey are gag groud amogs polcyholders he medcal malpracce surace dusry (see 254 Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

5 Harrgo, Dazo ad Epse, 2008, for a dscusso of he marke for CMR ad OB polces). I s o obvous ha CMR coracs have creased populary oher les of busess where CMR coracs are avalable. I Pael A of able 2 we prese he oal premums eared by corac ype for he medcal malpracce, produc lably ad oher lably les of busess for he years 2003 hrough We see ha he populary of clams-made coracs has creased relave o occurrece-based coracs oly he medcal malpracce area; he ABLE 2 PREMIUM EARNED (IN MILLIONS OF CURREN DOLLARS) AND NUMBER OF INSURERS BY YPE OF CONRAC IN PRODUC LIABILIY AND OHER LIABILIY LINES OF BUSINESS, Pael A: Eared premum Le of busess Corac Medcal malpracce Produc lably Oher lably OB $ 2,497 $ 2,145 $ 2,065 $ 2,356 CMR $ 5,806 $ 6,639 $ 6,564 $ 7,785 %CMR/oal 69.9% 75.6% 76.1% 76.8% OB $ 2,165 $ 2,754 $ 2,936 $ 3,042 CMR $ 323 $ 391 $ 512 $ 527 %CMR/oal 13.0% 12.4% 14.8% 14.8% OB $ 20,121 $ 23,698 $ 23,739 $ 26,407 CMR $ 12,323 $ 14,781 $ 15,402 $ 15,237 %CMR/oal 38.0% 38.4% 39.3% 36.6% Pael B : Number of surers Le of busess Corac Medcal malpracce Produc lably Oher lably Source: Bor ad Boyer (2011) OB CMR OB CMR OB CMR Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 255

6 wo oher les where clams-made coracs are avalable, here does o seem o be a large moveme he corac prefereces of he polcyholders. Eve whe we look a he umber of surace compaes ha provde each ype of corac hese hree les as Pael B of able 2, we do o see surers offerg CMR corac beg much more umerous relave o he umber of surers offerg OB coracs. Aoher eresg comparso we ca make bewee OB ad CMR coracs s wh respec o he loss rao (losses curred dvded by premums eared) for each ype of corac. able 3 preses hose loss raos by corac ype for he medcal malpracce surace dusry. ABLE 3 MEDIAN LOSS RAIO (LOSSES INCURRED DIVIDED BY PREMIUMS EARNED) IN HE MEDICAL MALPRACICE INSURANCE INDUSRY BY YPE OF CONRAC, year OB loss rao CMR loss rao CMR loss rao vs. OB loss rao Isgfca Sgfca* Sgfca* Sgfca* Sgfca* Sgfca* Sgfca* Sgfca* Sgfca* Sgfca* Source: Bor ad Boyer (2011). he able cludes oly surers wh posve premums. Losses curred clude defese ad cos coame expeses. I s eresg o oe ha CMR polces have a hgher loss rao ha OB polces over he pas e years. Ad excep for he year 1997, loss raos for he wo ypes of corac are sgfcaly dffere from each oher, wh he loss rao he CMR le beg hgher ha he OB le. hs suggess ha uderwrg profably s lower he CMR le ha he OB le. A aural queso we ca ask afer lookg a he ecoomc mporace of OB ad CMR coracs ad he ype of losses hey are desged o cover s why here 256 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

7 are wo ypes of coracs ha coexs hese surace les of busess. Wha makes hese markes aracve for boh CMR ad OB coracs o coexs whereas he radoal OB corac s he orm oher persoal les? 3. RISK NEURAL AGENS AND SHOCKLESS ECONOMY We ow suppose ha a rsk eural dvdual lves for K perods ad s exposed o some rsk durg he frs + 1 perods, ha s, he acve perods of hs lfe. I perod [0,] he age faces a poeal loss wh a vara dsrbuo f( ) ad mea E( ) = L. We suppose ha here s o chage loss developme paers ({a } 0 ) over me ad ha ay loss s fully developed before he dvdual des, + 1 K. Because he dsrbuo ad developme of he loss s he same every perod, he OB premum wll be he same every perod. he age ca, herefore, be sured wh a seres of OB coracs such ha he pays a premum P every perod from 0 o ad ohg afer ha, wh j P = P = δ Iα jl. j=0 I he case of CMR polces, he premum ha s pad oday mus cover losses ha are pad oday, o maer whe such a loss was curred. I he frs year of he CMR polcy (a = 0), oly frs year s repored losses eed o be sured. hs meas ha he frs year s premum s gve by Pˆ0 = a 0L. I he secod year, a me = 1, he premum s gve by Pˆ1 = a L + a 0 1L; he frs erm s he amou ha wll be pad o cover losses ha are curred perod = 1, whereas he secod erm s he amou ha wll be pad o cover losses ha were curred perod = 0, bu are o repored ul perod = 1. A me = 2, he premum s Pˆ2 = (a + a + a )L. Before perod, premum s growg every year o accou for he growg umber of poeal pas losses. I perod =, he perod 0 loss has fully developed ad he premum s j P = P = δ Iα jl. j=0 Afer he frs years of he age s professoal lfe ad ul he ed of hs career, he ca buy each year a CMR corac surg Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 257

8 for loss repors from he precedg years. Premum Pˆ = Pˆ s he pad so log as he dvdual s workg, for perods {,}. Afer rereme he dvdual sll eeds o purchase surace o cover losses ha were curred he pas, bu have o bee repored ye. Such a dvdual does o eed o purchase surace for losses ha are o be curred hs year sce he s o loger exposed o such losses. I perod + 1, hs premum he sars o decrease ad becomes P = α jl, = + 1,, + j= ul = +. Afer ha dae, he o loger pays ay premum (Pˆ = 0 for all + + 1). If dvduals acualze fuure premums a rae r A vara over me so ha her dscou facor s gve by δ = 1 A, he he 1+ ra prese value of all premums pad over he dvdual s lfeme wh a OB surace polcy s gve by A Π δ P δ δ = = α L A =0 =0 =0 I I he case of a CMR surace polcy, he prese value of he premums pad s + + Π δ = P = δ α L + δ α L + δ α L. A A A A =0 =0 =0 = + 1 =0 = + 1 = Comparg Π wh Πˆ, we ca sae he followg proposo. Proposo 1 : If he loss developme paer (a 0,,a ) ad he dsrbuo of he loss do o chage over me, he rsk eural polcyholders wll prefer a CMR corac o a OB corac f ad oly f δ I > δ A (or equvalely, r I < r A ). Proof. All proofs are relegaed o he Appedx. he dfferece bewee he wo premum profles sads her mg ad hs maers for a rsk eural age oly f dffere dscou facors ca be affeced o he paymes. Suppose we look a he loss ha s curred perod 0. he OB premum for hs loss dscous fuure poeal losses a he surer s rae: P 0 = = 0 d a I L. I he case of he CMR corac, he surer receves a premum each perod for he payme he s lkely o make he same perod. he surer s dscou rae does o fluece he CMR premum whereas 258 Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

9 he age acualzes fuure premums usg dscou facor d A. hs meas ha he CMR corac cos perceved by he age s Πˆ = 0 = 0 d a A L. herefore, a age wll prefer he CMR corac o he OB corac f ad oly f δ I > δ A (.e., f ad oly f r I < r A ). We ca fer from hs proposo ha f he surer ad he polcyholder use he same dscou rae, he he rsk eural polcyholder wll be dffere bewee he wo polces. If he eres rae used by he ages (r A ) o dscou fuure cash flows s greaer he he surer s eres rae (r I ), he he age wll surely prefer a CMR polcy o a OB polcy. As a resul, myopc or mpae ages prefer o purchase CMR polces; because hey value he prese proporoally more ha he fuure, hese polcyholders prefer o pay lower premums he shor ru eve f meas payg more he log ru. hs allows us o make our frs predco. Predco 1. For losses ha are o flueced by exeral shocks ad for rsk eural polcyholders, a CMR polcy wll be preferred o a OB polcy f ad oly f he sured s cos of capal s greaer ha he surer s cos of capal. 4. SHOCKS O HE LOSS DISRIBUION Proposo 1 assumed ha he dsrbuo of losses was vara me. Dohery (1991) clams, however, ha s because of he uceray fuure losses ha surers sared offerg CMR polces. We explore hs possbly hs seco. We roduce perodc shocks affecg he loss dsrbuo. We mus he dsgush bewee wo cases. If he dsrbuo of he shock s kow o all so ha he expeced value of he shock s he same perod 0 as he perod where he shock occurs, here s o uceray o he loss dsrbuo, oly a acpaed red s he expeced loss. O he oher ed, f he dsrbuo of he shock s ukow, he expeced value perod 0 of he shock, s o he same as he expeced value of he shock. I hs case, here s uceray o fuure loss dsrbuos sce shocks are uacpaed. We show ha wheher he shock s acpaed or o has o mpac o he rskeural comparave evaluao of he coracs. 4.1 Acpaed shocks o he loss dsrbuo Le us assume ha he clam pad by he surer ca evolve depedely of he eve because, for sace, he sze of he clam Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 259

10 s relaed o he regulaory evrome place a he mome he clam s fled raher ha he mome he loss s curred. he loss dsrbuo s he dsrbuo of he clam regardless of he perod whch he eve ook place. ha formao s kow o he polcyholder ad he surer. he shocks o he loss dsrbuo are acpaed he sese ha E( + ) s vara, ha s, here ca be shocks o he loss dsrbuo bu he expeced value of hese shocks are perfecly kow from perod 0 o. A OB corac sged perod akes accou of fuure expeced clams ha may arse followg a eve. he OB premum, he, s forward lookg: P = = 0 d I a E( + ). O he oher had, a CMR corac, he premum perod oly depeds o he loss clamed perod regardless of whe he pas he eve occurred. he premum wres: P l αe ( ) =, = k where k = 0,l < f <, k = 0,l = f ad k =,l = f >. Premums are o loger cosa eher form of corac. Makg a choce bewee CMR or OB corac perod, a age cosders he dfferece premums. he ex Proposo esablshes ha from perod o, oly dscoug ca make a dfferece bewee he wo ypes of coracs. Proposo 2 : If he dsrbuo of loss chages over me, rsk eural polcyholders wll prefer a CMR corac o a OB corac f ad oly f δ I > δ A (or equvalely, r I < r A ). Suppose a sgle eve occurrg perod. he OB premum s pad oly oce ad covers all fuure clams relaed o hs loss: P = = 0 d I a E( + ). he sequece of premums ha wll be pad usg a CMR corac o cover for he perod loss s gve by Pˆ + = a E( + ), = 0,,. I perod, he age correcly acpaes ha he CMR premum wll be +. here s, he, o uceray o fuure Pˆ + premums, eher OB or CMR arragemes. he dscoued sum of fuure CMR premums relaed o a eve s, he, = 0 d A Pˆ + = = 0 d A a E( + ) 260 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

11 If δ I = δ A, hs amous o he same. I s clear ha a age prefers a CMR arrageme oly f hs dscou facor s lower ha he surer s oe. Wh equal dscou facor, a rsk eural age s dffere bewee payg P ow or acpag o pay a sequece of premums Pˆ +, = 0,, ha he dscous. As log as he dsrbuo of he shocks s kow, here s o uceray ad he expeced value of a perod loss s correcly acpaed perod 0 by boh he surer ad he sured. Proof of Proposo 4.1 makes obvous, oce he erms are rearraged o hghlgh he expeced dscoued sums of premums, ha oly he dscou facors maer comparg he wo premum profles. Accoug for he assocaed fuure loss dsrbuo of losses does o chage a rsk eural age s evaluao of he eremporal value of each ype of corac. 4.2 Uacpaed shocks o he loss dsrbuo I hs subseco, we explore he case of a ucera dsrbuo of fuure losses o see f ha roduces a dfferece bewee he wo ypes of coracs. Uceray meas ha he expeced value of a perod loss s o he same f s evaluaed perod wh he formao avalable perod as f s evaluaed a earler perod. Pu dfferely, we wll assume ha eve f ages kow ha he expecao hey have of fuure losses s wrog, hey cao mprove. I ha case, OB premums, ha are forward lookg, rely o mperfec formao abou fuure losses. Le us deoe E ( + ), = 0,,, he expeced value of a clam + codoal o formao avalable. Uacpaed shocks o he loss dsrbuo mply ha he dsrbuo vares from perod o perod so ha E ( + ) E ( + ). he OB premum for a eve curred perod s pad ad equal o P = δ Iα E ( + ). =0 If he age decdes o eer a sequece of CMR coracs, he should expec o pay a sequece of premums Pˆ +, = 0,, such ha Pˆ + = a E + ( + ). he bes predcor a age ca use perod o evaluae E + ( + ) s E (E + ( + ) = E ( + ). Hece, perod, he polcyholder s expeced dscoued sum of fuure premums pad a CMR arrageme s Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 261

12 E δ A P+ = E δ α A E+ ( + ) = δ AαE ( + ). =0 =0 =0 Aga, he dscoued value of surace evaluaed s he same boh coracs f δ A = δ I. However, he surer ad he sured kow ha he OB premum s compued wh mperfec formao. he OB surer s he bearer of hs error s cosequeces sce he s he oe o pay for poeal fuure losses he rue expeced value of whch he does o kow. A CMR corac shfs he burde of uceray o he sured who cao compue expeced fuure premums he wll have o pay he fuure o sure a curre eve. However, a CMR ype of corac s a beer way of dealg wh uceray he sese ha allows o wa for accurae formao before premums are compued. I sadard propery/casualy surace coracs, a premum s pad each perod ad corporaes all ew formao arrved he perod. I s, he, surprsg ha OB coracs sll exs ha lock he surer o fuure oblgaos afer premums have bee pad oce ad for all. hs leads o our secod predco. Predco 2. For losses ha are flueced by uacpaed shocks a CMR polcy should be preferred o a OB polcy because allows o accou for releva formao a he me accrues. 5. RISK AVERSION Gve ha OB ad CMR coracs offer premums ha do o dffer prese value, follows ha f sured ages dscou he fuure a he same rae as he surers, a mpora move for choosg oe of he coracs mus be ha hey are somehow rsk averse. A rsk averse age edowed wh a cocave uly fuco has a preferece for he smoohg of hs cosumpo. Sce he dfferece bewee CMR ad OB coracs s all abou he mg of cash flows, a rsk averse age may prefer oe corac over he oher because of hs smoohg effec. Suppose a age has a cosa wealh Y per perod. Hs cosumpo perod s Y P where P s he premum pad o buy a surace corac. he age values cosumpo wh a cocave uly fuco u(y P ) wh u' > 0 ad u" < 0. We assume ha he dsrbuo of losses s vara such ha E( ) = L. he age ow compares eremporal ules uder each corac. Wh OB ad CMR premums respecvely equal o P ad as defed seco 3, he age s eremporal uly over hs ere lfe Pˆ s 262 Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

13 K U = u Y L + OB u Y τ δ δ α δ τ A I A ( ) τ=0 =0 τ = + 1 wh a sequece of OB coracs. Wh a sequece of CMR coracs, hs eremporal uly from perod 0 o s: τ τ δ α ( ) + δ τ U = u Y L u Y CMR A A L τ=0 =0 τ = K τ + δ α ( ) + δ τ Au Y L Au Y. τ = + 1 = τ τ = + he CMR sequece of premums s characerzed by paymes made over a loger perod compared o he OB sequece of coracs. I he frs perods of oe s professoal lfe, he CMR corac charges lower premums. he age s cosumpo a early perod < s hgher uder a CMR arrageme for all such ha τ α I =0 =0 δ α. Afer ha dae, ad for he remader of he age s lfe, he CMR premum s ever lower ha he OB premum. No oly s he CMR premum equal o Pˆ = L > a = 0 d I L = P from he mome losses become erely developed ul he rereme dae, bu, more mporaly, a CMR corac requres premums pad afer he age has ceased ay professoal acvy ha could geerae a loss. ha s, for perods afer perod + 1, cosumpo coues o be lower uder a CMR because he age sll pays a premum uder a CMR corac whereas he o loger pays ayhg uder a OB corac. he age s preferece for oe or he oher of he wo ypes of coracs shows he dfferece bewee he eremporal ules. Le us deoe DU = U CMR U OB he dfferece bewee he lfeme uly a age ges from a CMR corac ad he lfeme uly a age ges from a OB corac. A rsk averse age wll prefer a CMR corac f ad oly f DU s posve. We have: 1 τ δ α α δ τ DU = A u Y L u Y I L τ=0 τ + δ A ( ) α δ u Y L u Y I L τ= =0 + + δ α ( ) τ A u Y L u Y τ = + 1 = τ =0 =0 Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 263

14 Sce he evaluao of cosumpo wh a cocave uly fuco mples a preferece for smooher cosumpo pahs, ad sce he age s parly myopc ad dscous fuure uly wh facor d A, he aural smoohg advaage of a CMR corac over a OB s creased f d A s relavely low. Moreover, hs advaage s also greaer, he loger he delay () before a = 0 becomes larger ha = 0 d a I, ha s, he loger he delay before he CMR premum Pˆ = a L becomes larger ha he OB premum P = 0 a = 0 d L. I hs delay has wo major compoes: he developme paer (a 0,,a ) ad he dscou raes (d I, d A ). hese wo compoe are key o comparg he CMR sequece of coracs o he OB sequece of coracs. he drec effec of he sze of he al (he value of ) o hs delay s ambguous, however. O oe had, he loger he al, he loger he umber of early perods over whch he CMR premum s lkely o be lower ha he OB oe. O he oher had, he hcker he al (he hgher he value of a for dsa perods ), he sroger he dscoug effec ha reduces he OB premum. Hece, we ca summarze he CMR advaage as follows. he dfferece DU s more lkely o be posve 1. Whe d A s low; 2. Whe d I s low ad he loss has a shor ad h al; 3. Ad whe d I s hgh ad he loss has a log ad hck al. Aga, dscoug s a core deerma of each corac evaluao. he dffereal dscou facors s, however, o loger suffce o expla a age s preferece for oe ype of surace corac or he oher. Because of s mpac o he mg of cosumpo, he shape of he developme paer cao be gored. As a resul, we would lke o solae hs mg effec. o do so, we shall assume ha all players he ecoomy have he same dscou rae so ha d A = d I = d. hs allows us o remove he dscou facor dffereal as he source of he age s preferece for oe corac raher ha he oher ad o cocerae exclusvely o he rsk averso effec. I a frs approach, we smplfy he problem cosderg a sgle eve ( = 1). Our secod smplfyg approach wll be o reduce he framework o a shor al loss whereby he loss developme paer wll be lmed o oly wo perods ( = 1). I he las subseco, we llusrae our resuls wh some umercal compuaos uder hese wo combed approaches ( = 1 ad = 1). 5.1 A pure smoohg effec Wh a sgle eve framework ( = 1) ad equal dscou facors, he dfferece uly of purchasg he CMR sequece of coracs over he OB sequece of coracs wres 264 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

15 ( ) DU = u( Y α L) u( Y α δ L) + δ u( Y α L) u( Y ) 0 =0 =1 DU + DU As we saed earler, he advaage of he CMR sequece of coracs s ha offers a beer smoohg of he premums whereas he advaage of he OB sequece of corac s ha he surer s dscoug decreases he OB premum he frs perods. he observao of DU for ay paer (a 0,,a ) leads o he followg proposo Proposo 3 : For a sgle eve perod 0 ad o eve aferwards ad a developme paer of ay legh, a rsk averse age wh he same dscou facor as he surer prefers a CMR corac o a OB corac f here s o dscoug (d = 1). He s, however, dffere bewee he wo coracs f he fuure s oally dscoued (d = 0). Obvously, f eher he age or he surer akes accou of he fuure, he OB premum s reduced o P = a 0 L ad fuure paymes are o cosdered he eremporal uly. I ha case, he rsc dfferece bewee he wo ypes of coracs s gored ogeher wh he mg of paymes. O he oher had, whe he fuure s o dscoued (d = 1), hgher fuure CMR premums are compleely ake o accou. Proposo 5.1 esablshes ha prefereces for a smooh cosumpo s suffce o make a rsk averse age prefer a CMR corac over a OB corac. hs argume s mpora ad assers ha he key explaao for he choce of a CMR corac ca be a srog preferece for cosumpo smoohg over he age s lfe. Whe here s dscoug, however, he frs perod OB premum s lowered. hs lowerg of he OB premum could be such ha he beef of a lower al perod premum ca overcome he CMR s smoohg advaage so ha a rsk averse age could prefer he OB corac. he CMR s pure advaage s a frs perod advaage: DU + = u(y a 0 L) u(y ) = 0 d a L > 0. For a gve paer (a 0,,a ), DU + couously creases wh d. However, DU + s grealy depede o he sze of a 0 ad he dspachg of he a, = 0,, he smplex = 0 a = 1. Aoher advaage of he sequece of OB surace coracs s a absece of premum he las perods of he age s lfe so ha DU = = 1 d (u(y a L) u(y)) < 0. Obvously, absolue erms, hs parcular advaage of he OB corac creases wh d. hs Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 265

16 meas ha he effec of d o DU s ambguous sce DU + ad DU become larger, absolue erms, as d becomes larger. hs meas ha he mpac of a chage he dscou facor o a age s preferece for a CMR corac over a OB corac wll deped o he loss developme paer (a 0,a 1,,a ). Cosequely, he srucure of he al s a mpora deerma of a dvdual s surace purchasg decso bewee a OB ad a CMR corac. he ex proposo preses how a age s decso s affeced by he developme paer ad he dscou facor. Proposo 4 : here s a d * [0,1] such ha, for a gve developme paer (a 0,,a ), he CMR advaage DU s creasg d for d > d *. Dscoug remas a mpora facor for he choce of a ype of corac whe he sured s rsk-averse, eve whe surer ad sured share he same dscou facor. Ad eve hough a rsk averse age values he mg of he surace paymes he makes, he dscou facor has a ambguous mpac o he advaages of purchasg oe ype of corac over he oher. We kow from Proposo 5.1 ha a sequece of CMR coracs s preferred whe d = 1, ad we kow from Proposo 5.1 ha hs advaage s creasg whe measured close o d = 1. We ca, he, coclude ha here are values of d he eghborhood of 1 for whch a CMR corac s preferred by a rsk averse age. No oher geeral paer ca be exraced from our aalyss excep for he fac ha he advaage of he CMR corac s creasg he dscou facor for large eough values of hs dscou facor. he mpac for smaller values of d s ambguous, however, because depeds o he dsrbuo of he a. he a deerme he legh of me durg whch CMR premums mus be pad afer rereme. he loger, he hgher he CMR dsadvaage due o he exeded legh of paymes. I he ex subseco, we cocerae o he effec of he developme paer. We compare hck ad h als a case where he loss fully develops wo perods. 5.2 A wo-perod surace le We cocerae hs seco o a smplfed wo-perod developme paer o solae resuls relaed o he hckess of he al raher ha legh. We wll he have a suao whch = 1 so ha a 1 = 1 a 0. For a poeal eve each of he -perod age s professoal lfe, he CMR e advaage (.e., he uly of havg a CMR corac mus he uly of havg a OB corac) wres: 266 Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

17 DU = u( Y α L) u( Y ( α + δ(1 α )) L) ( u( Y L) u( Y ( 0 (1 0)) L) ) + δ α + δ α = ( u( Y (1 ) L) u( Y )) δ α 0 Oly he frs perod uly s greaer a CMR corac wh a wo perod le. Afer he al perod = 0, he OB premum s cossely lower ha he CMR premum. Hece, f DU were o be posve, would mea ha he dfferece u(y a 0 L) u(y (a 0 + d(1 a 0 )L) mus be hgh eough o compesae for all subseque dscoued egave dffereces. A frs predco s ha a low a 0 combed wh a hgh d would crease he OB premum ad, he, he advaage assocaed wh he CMR corac he frs perod. he problem s ha a low a 0 combed wh a hgh d also creases he las perod OB advaage: d + 1 (u(y (1 a 0 )L) u(y)) < 0. A low dscou facor d (parcularly a low d A ) o he oher had wll decrease he wegh of hese egave fuure dffereces. he frs perod advaage of he CMR creases whe a 0 decreases, hs s all he more rue f d s hgh (parcularly f d I s large, o crease he OB premum). We ca, he, expec ha he case of a shor al surace le of busess (as a wo-perod le), a decrease of he al, as measured by a crease a 0, wll crease he advaage of he OB corac over he CMR. hs s saed he ex proposo. Proposo 5 : If he al of he loss s shor ( = 1), he, a he marg, f he al s hck (a 0 (1 d)/(2 d)), he CMR sequece of corac becomes more advaageous for a rsk averse age whe he al becomes her (a 0 creases); f he al s h (a 0 close o 1), he OB sequece of coracs becomes more advaageous for a rsk averse age whe he al becomes eve her (a 0 creases). hs allows us o fer ha he CMR advaage s o moooc he hckess of he al, a leas based o he dffereces bewee he sured age s eremporal ules uder each form of corac. here s fac a value α δ ( 2 0 ),1] aroud whch he 1 δ CMR advaage DU s a s maxmum. Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 267

18 5.3 Numercal compuaos I has o bee possble o prove ha he CMR advaage DU s cossely posve excep whe d = 1. o exame he sesvy of eremporal ules uder each ype of corac o he varaos of d ad a 0, we ra some umercal compuaos o llusrae our 1 γ resuls. For a sadard CRRA uly fuco Y u( Y ) = wh γ = 1 2, 1 γ ad a sgle eve ( = 1) developg over wo perods ( = 1) so ha a 1 = 1 a 0, we compue DU = (u(y a 0 L) + du(y a 1 L)) (u(y a 0 L da 1 L) + du(y)) DU depeds o he values of d ad a 0. For a gve a 0, Pael A FIGURE 1 EXPECED UILIY OF HAVING A CLAIMS-MADE CONRAC COMPARED O AN OCCURRENCE CON- RAC AS HE DISCOUN RAE (d) VARIES, WIH AN AGEN WHO LIVES ONLY HROUGH ONE POSSIBLE EVEN ( = 1), WHEN LOSSES FULLY DEVELOP OVER WO PERIODS ( = 1), DEPENDING ON HE PROPOR- ION OF HE LOSS PAID IN HE INIIAL PERIOD, a 0. Pael A. he proporo of he loss ha s pad he al perod s a 0 = 1 9 ad he proporo ha s pad he fal perod s a 1 = 1 a 0 = Dfferece bewee he Uly of he CMR corac ad he uly of he OB corac Dscou facor δ 268 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

19 Pael B. he proporo of he loss ha s pad he al perod s a 0 = 1 3 ad he proporo ha s pad he fal perod s a 1 = 1 a 0 = Dfferece bewee he Uly of he CMR corac ad he uly of he OB corac Dscou facor δ of Fgure 1 shows ha DU s U-shaped d f a 0 s small (srcly smaller ha 1 3 our compuaos). I our compuaos, DU ca be egave ad decreasg for low values of d, creasg ad he posve for hgher values of d. hs llusraes Proposos 3 ad 4. Whe a 0 creases, he value of d for whch DU s mmum decreases. For a as Pael B of Fgure 1, DU s posve for ay value of d, whch meas ha a clams-made corac should be preferred by all ages who are exposed o a loss whereby more ha oe hrd of he losses are pad he al corac perod. I Fgure 2, we le he proporo of he loss pad he al perod vary whle keepg all oher parameers cosa. We see boh Pael A (where he dscou facor s hgh) ad Pael B (where he dscou facor s low) ha for a sgle eve, he relaoshp bewee DU ad a 0 s ha of a vered parabola. hs llusraes Proposo 5. For hgh values of d as Pael A, we see ha he age s always beer of wh a CMR corac ha a OB corac, o maer wha he loss developme paer s. Proposo 5 says ha DU s creasg a 0 for a 0 < (1 d)/(2 d) = 0,09, ad decreasg for a 0 close o oe. hs holds rue hs Pael A. Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 269

20 FIGURE 2 EXPECED UILIY OF HAVING A CLAIMS-MADE CONRAC COMPARED O AN OCCURRENCE CON- RAC AS HE PROPORION OF HE LOSS HA IS PAID IN HE INIIAL PERIOD (a 0 ) VARIES, WIH AN AGEN WHO LIVES FOR WO PERIODS ( = 2), WHEN LOSSES FULLY DEVELOP OVER WO PERIODS ( = 1) AND HE DISCOUN RAE IS RELAIVELY HIGH (d = 0.9). 1.4 Pael A. he dscou facor s relavely hgh (d = 0.9). Dfferece bewee he Uly of he CMR corac ad he uly of he OB corac Proporo of he loss pad he al perod α Pael B. he dscou facor s relavely low (d = 0.5). Dfferece bewee he Uly of he CMR corac ad he uly of he OB corac Proporo of he loss pad he al perod α Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

21 For low values of d as Pael B, however, DU s egave for low values of a 0, bu sll creasg a 0 for a 0 < (1 d)/ (2 d) = 0,33. Hece, ages are beer off purchasg a OB corac whe he proporo of he loss ha s pad he al perod s low ad he age dscous he fuure a lo. As a 0 creases, DU eveually becomes posve, whch meas ha he CMR corac becomes preferred o he OB corac, ad eveually reaches a maxmum posve value. he advaage he CMR corac over he OB corac he decreases, bu remas posve ul a 0 = 1. I our compuaos for hs smplfed framework where he developme paer s shor, he value of DU s posve for mos combaos of a 0 ad d. Fally, we fd ha he CMR advaage decreases whe he umber of possble eves,, creases. Wh more ha oe poeal eve, ha s whe > 0 ad he age sures for all hs professoal lfe, he creasg wegh of he dscoued OB premum plays for he OB corac ad DU s more ofe egave. 6. CONCLUSION he goal of hs paper was o compare he relave effcecy of wo ypes of surace coracs offered o ages who seek facal proeco agas lawsus brough upo hem as a resul of accdes whle exercsg her professoal acves. hese wo ypes of coracs are kow as he radoal ad well-kow occurrece based surace corac, whereby sured ages are covered for losses ha her cur he polcy year o maer whe he clam s fled he fuure, ad he clams-made ad repored surace corac, whereby sured ages are covered for losses ha are repored durg he polcy year o maer whe he losses was curred he pas. Clams-made coracs are popular lably les ad accou for 75% of he oal eared premums he medcal malpracce lably surace le of busess. A he same me, clams-made coracs are mosly exse propery surace les. Clamsmade corac are also he defaul ype of corac he case of drecors ad offcers lably surace ad may oher professoal lably surace coverages. Ad alhough clams-made coracs also exs o cover produc lably losses, hey are much less popular ha he radoal occurrece based corac sce oly 15% of produc lably surace coracs are clams-made. We were he faced wh he followg wo emprcal quesos for whch we waed o buld a model ha explaed he coexsece of he wo ypes of corac: Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 271

22 Why are clams-made corac more popular log-al les ha shor-al les? Ad Why are clams-made corac more popular persoal lably les ha commercal lably les? he heory proposed by Dohery (1991) suggess ha clamsmade corac are a aswer o a crease he uceray of he legal evrome whch surers operae. hs heory explas well why clams-made coracs should be more prevale logal les, bu does o expla why clams-made coracs should o be more prevale persoal lably les. I fac, rsk averse dvduals should be more reluca o assume he legal evrome uceray ha rsk eural frms. I coras, he heory we propose based o he loss developme paer of lably clams ad he dfferece bewee he dscou facors of sured ages ad surace compaes s able o expla he wo sylzed facs of shor-al versus log-al les ad of persoal versus commercal sured age. Aoher possble explaao for he prevalece of CMR over OB coracs cera ecoomc coex, bu oe we dd o exame he curre paper, s he uceray, a he me he clam s fled, as o whe he loss was curred exacly he pas. Whe s hard o ppo he exac me a loss was curred becomes dffcul o defy whch pas surer s resposble for he clam ha s fled oday. I may he be opmal erms of rasaco coss o have a CMR polcy raher ha a OB polcy. Wh a CMR polcy, here s o uceray as who s facally resposble for he clam sce s he surer uderwrg he corac a he me he clam s fled. If o he oher had s easy o ppo he surer who s facally resposble, he s less obvous ha a CMR corac would be preferred. Fally, f here are solvecy ssues wh pas surers, a sured age could prefer o purchase a seres of CMR coracs over hs lfeme raher ha face he possbly ha, he fuure, he surer who uderwroe he corac becomes bakrup ad uable o cover he loss. Usg a smple Markov-swchg approach wh bakrupcy beg a absorbg sae, s clear ha he loger a loss akes o fully develop, he more lkely he surer wll be bakrup ad uable o pay whe he clam s fally fled. Oe oher heory developed by Poser (1986) suggess ha surers swchg o clams-made coracs were wllg o coue uderwrg he rsk of pae jures bu dd o wa o assume he mg rsk (.e., whe he compesao s be pad) ad he correspodg flao ad vesme rsks. hs Poser approach s somewha lked o our approach he sese ha he swch o clams-made could be explaed by a reduco he surers dscou facor d I. 272 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

23 he heorecal approach we used hs paper o address why ad whe sured ages should prefer a clam-made ad repored surace corac o a occurrece-base surace corac allows us o draw hree ma coclusos. Frs, f ages seekg surace are rsk eural (ad herefore purchase surace oly because, for sace, hey are madaed o do so by he legslao), he, oly he dffereces he dscou rae of he surer ad of he age wll deerme he purchase of such or such corac. Secod, f here s some level of uceray as o wha fuure losses wll be, he a CMR corac ca deed help surers ge rd of some level of uceray as Dohery (1991). Because a CMR surace corac allows o prce surace usg all he formao ha s avalable a he curre me, eher he age or he surer are locked a corac ha was sged a pas perod uder codos ha may o loger apply. Fally, f ages are rsk averse, hey may prefer he clams-made surace corac o defer paymes he frs perods of her professoal lves o fuure perods so ha her come s a b smooher ha uder a occurrece surace corac. Refereces 1. Berger, Lawrece A., J. Davd Cumms, ad Sharo eyso (1992). Resurace ad he Lably Isurace Crss. Joural of Rsk ad Uceray 5(3), Bor, Parca ad M. Mar Boyer (2011). Clams-Made ad Repored Polces ad Isurer Profably Medcal Malpracce. Joural of Rsk ad Isurace 78(1): Cumms, J. Davd ad Sharo eyso (1992). Corollg Auomoble Isurace Coss. Joural of Ecoomc Perspecve 6(2), Dohery, Nel A. (1991). he Desg of Isurace Coracs Whe Lably Rules Are Usable. Joural of Rsk ad Isurace 58(2), Dohery, Nel A. ad Georges Doe (1993). Isurace wh Udversfable Rsk: Corac Srucure ad Orgazaoal Form of Isurace Frms, Joural of Rsk ad Uceray, 6: Harrgo, Sco E., Parca M. Dazo, ad Adrew J. Epse (2008). Crses Medcal Malpracce Isurace: Evdece of Excessve Prce- Cug he Precedg Sof Marke, Joural of Bakg ad Face, 32: Nye, Blae F. ad Alfred E. Hofflader (1987). Ecoomcs of Olgopoly : Medcal Malpracce Isurace as a Classc Illusrao. Joural of Rsk ad Isurace 54(3), Nye, Blae F. ad Alfred E. Hofflader (1988). Experece Rag Medcal Professoal Lably Isurace. Joural of Rsk ad Isurace 55(1), Poser, James R. (1986). reds Medcal Malpracce Isurace: , Law ad Coemporary Problems, 49 (2): Sloa, Frak A., Radall R. Bovbjerg, ad Pey B. Ghes (1991). Isurg Medcal Malpracce. New York: Oxford Uversy Press. Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 273

24 APPENDIX: PROOFS Proof of Proposo 1 Because ages are rsk eural, hey wll purchase he corac whose cos s smaller. I oher words, he CMR corac wll be preferred o he OB corac f ad oly f Πˆ < Π. We have Π = δ A P = δ Aδ IαL = δ AL δ α I, =0 =0 =0 =0 =0 ad Πˆ ca be smplfed o + = A A A Π δ α L + δ α L + δ α L =0 =0 = + 1 =0 = + 1 = + α δ α δ δ = A L = A A L = =0 = =0 =0 L δ A =0 =0 δ Aα (1) (2) (3) Hece, Πˆ < Π f ad oly f d A < d I. QED. Proof of Proposo 2 Wh a poeal eve each of he frs perods of he age s lfe, he sequece of OB premums s P = α δ I E( + ), =0 ha s, each perod OB premum depeds o he fuure expeced losses. he CMR premums ake oly curre losses o accou. However, hese losses evolve wh me. P = =0 =0 = ( ) α E f < ( ) α E f ( ) α E f (4) A rsk eural age wll choose he corac ha offers he lowes expeced dscoued sum of premums. We have 274 Isurace ad Rsk Maageme, vol. 79(3-4), Ocober Jauary 2012

25 P = δ A P = δ A α δ I E( + ) =0 =0 =0 α = δ δ E( ) =1 =0 A I + (5) ad + δ α + δ α + δ α P = A E( ) A E( ) A E( ) =0 =0 = + 1 =0 = + 1 = (6) + = δ E( ) α A =0 =0 + he, P > Pˆ f ad oly f d I > d A. QED Proof of Proposo 3 Suppose a poeal eve perod 0 wh a vara loss dsrbuo developg over + 1 perods ad such ha E( ) = L regardless of he perod he clam s made. Suppose here are o poeal eve afer perod 0. Wh he same dscou facor d A = d I = d [0,1], he age s eremporal uly over perods 0 o s U OB (d) a OB coracs ad U CMR (d) a CMR corac, such ha δ α α δ + δ DU( ) = u( Y L) u( Y L) ( u( Y α L) u( Y )) 0 =0 =1 I s easy o verfy ha DU(0) = 0 sce U CMR (0) = u(y a 0 L) = U OB (0). We he show ha DU(1) > 0. U (1) = u( Y L) + u( Y ) OB U (1) = u( Y α L) CMR =0 sce a = 0 = 1. By cocavy of u, we have ha α u( Y L) + α u( Y ) < u( α ( Y L) + α Y ) = u( Y α L), j so ha j =0, = / j =0, = / j u( Y L) u( Y ) < u( Y α jl), α j + α j=0 =0, = / j j=0 ad he, j Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 275

26 u( Y L) + u( Y ) < u( Y α L). j=0 Hece, DU(1) > 0. QED. Proof of Proposo 4 j Le us dffereae DU wh respec o d. We have DU δ j 1 ' j j 1 = jα δ Lu ( Y α δ L) + jδ ( u( Y α L) u( Y )) j=1 =0 j=1 j 1 ' = jδ [ α Lu ( Y α δ L) ( u( Y ) u( Y α L))] j=1 j =0 he cocavy of u mples ha a j Lu'(Y a j L) > [u(y) u(y a j L)] (lear approxmao of he dfferece s larger ha he dfferece). Hece, DU s posve a d = 1. For d [0,1], he sg s ambguous δ however. Le us deoe dˆj he value of d for whch = 0 a dˆj = a j. he, for d > dˆj we have = 0 a d > a j ad he, ' ' α Lu ( Y α δ L) > α Lu ( Y α L) > u( Y ) u( Y α L). j =0 j Le us deoe d * = max dˆ j = 0 j. We have, he, ha a j Lu'(Y = 0 a d L) > a j Lu'(Y a j L) > u(y) u(y a j L) for all d d *. Hece DU( δ) > 0 for d d*. δ QED. Proof of Proposo 5 Le us see wha happes o DU whe we slghly aler a 0. DU ' + 1 ' = L ( α ) δ ( ( α ) ) α u Y 0L u Y 1 0 L 0 + ( 1 δ) L δ u Y α + 1 α δ L =0 DU ( ) ( ( ) ) α ' α δ + 1 ' ( 1 / L) = u Y L u Y 1 α L j ' ( ( 0 ( 0 ) ) ) ( ( 0 ( 0 ) ) L) + 1 ' (1 δ ) u Y α + 1 α δ, j j j 276 Assuraces e geso des rsques, vol. 79(3-4), ocobre javer 2012

27 because = 0 d = (1 d + 1 )/(1 d). ha s, DU s posve f α0 [u(y a 0 L) u'(y (a 0 + (1 a 0 )d)l)] d + 1 [u'(y (1 a 0 )L) u'(y (a 0 + (1 a 0 )d)l)] 0 he erm he frs bracke s always egave because a 0 L (a 0 + (1 a 0 )d)l for a 0 < 1 ad u' s decreasg. he erm he secod bracke s posve f 1 a 0 a 0 + (1 a 0 )d. ha s, DU α0 1 s posve f α δ 1 δ 0. Noe ha he fraco s decreasg 2 δ 2 δ d, wh maxmum value 1/2 whe d = 0 ad mmum value 0 whe d = 1. 1 δ For ay a 0 greaer ha, he dervave DU ca be posve or egave. Le us fd he lm value of hs dervave as 2 δ α0 a 0 1. We have DU ' ' α = L δ α u Y L u Y + 1 δ L δ u Y L ' ( ) ( ) ( ) ( ) =0 + 1 ' ' = L δ u ( Y ) u ( Y L) δ + δδ 1 =0 =0 + 1 ' ' = δ u ( Y ) u ( Y L) L < 0 For hgh values of a 0, he CMR advaage becomes lower as he al becomes eve her. QED. Noes 1. he followg professoal proecos are offered o a clams-made bass: Drecors ad offcers lably, Prvae compay lably, Employme pracces lably, Fducary lably, Bakers professoal lably, Isurace compay lably, Secury & prvacy lably, ad Employed lawyers professoal lably. Professoal Lably Isurace Coracs: Clams Made Versus Occurrece Polces 277

28 Reproduced wh permsso of he copyrgh ower. Furher reproduco prohbed whou permsso.