Value of formao sharg mare muual surace Kev L, Joh Lu, Ja Ya 3 ad Je M Deparme of Logscs & Marme Sudes, The Hog Kog Polechc Uvers, Hog Kog. Emal address:.x.l@polu.edu.h. Deparme of Logscs & Marme Sudes, The Hog Kog Polechc Uvers, Hog Kog. 3 School of Ecoomc Sceces, Washgo Sae Uvers, US. Absrac Wh emprcal evdece from mare muual surace (MMI), a mpulse feedbac model s cosruced o address how formao sharg ca help crease boh he socal welfare ad he effcec of operao of he MMI ssem. Focusg o formao sharg, hs paper cosders he premum polc opmzao of a pure (.e., o-soc cosderao) muual surace ssem wh a homogeeous mare of decal members. Our fdgs cofrm ha he prcple of formao sharg ca be aaed uder equal-rs poolg, bu o ecessarl uder uequal-rs poolg, ad reveal ha quafable dffereces exs he valuao of formao sharg uder he wo schemes of rs poolg. I dcaes ha he e o successful MMI s equal-rs poolg. Algorhms are developed o compue he value of formao sharg b solvg he HJB equaos ad quas-varaoal equales. I deermes ha formao sharg ca acheve bes socal welfare as well as effce operao of a P&I Club. The cocluso provdes a scefc bass for boh maageral sraeg ad compeo regulao. The fdgs are also applcable o a wde rage of reserve ad veor maageme problems. Ke words: Iformao sharg; Muual Isurace; P&I Club. Iroduco I he mare surace mare, muual Clubs are o he pure commercal surace eerprses as coveoall defed. Isurace s based o membershp of he Club. A dvdual shp-ower, who was o pool hs rs a muual Club, mus frs oba membershp hrough pame of a membershp fee, whch s deemed o be he premum as defed commercal surace (Cass, Chchls ad Wu, 996; Lamm-Tea ad Sars, 993). Numerous sudes o he formao of muual surers cocer adverse seleco (Smh ad Suzer, 99; Lgo ad Thsle, 5), moral hazards (Smh ad Suzer, 995) ad formao asmmer (Cabrales e al, 3). A he ouse of he surace dusr, here were ol muual surers, ad because of he exsece of adverse seleco ad formao asmmer, he orgal rs pool degeeraed o wo pes of sub-rs-pools formed b he homogeeous assureds. There are a oal 8 Mare Muual Clubs, 3 of whch are members of he Ieraoal Group of P&I Clubs (see able ). The 3 P&I Clubs ha comprse he Ieraoal Group of P&I Clubs (he IG) are muual o-for-prof surace orgazaos ha bewee hem cover hrd par lables, whch clude polluo, loss of lfe ad persoal jur, cargo loss ad damage ad collso rss. The Clubs are muual orgazaos, ha s, he shpower members are boh sured ad surers, as he members boh ow ad corol her dvdual clubs (Gold, ; Hazelwood, ). The da-o-da acves ad operaos of he Clubs are delegaed o maagers. The 3 P&I Clubs cover abou 95% of he world s ocea-gog vessels erms of oage (JLT, 3; Golsh, 5). Ths moopol poso, ad he wa ha P&I Clubs operae, have rggered wo Europea Commsso (EC) vesgaos ad rulgs, 985 ad 999 respecvel, he feld of compeo law (Gsele, 999; EC, 985, 999). 4
Name of Club Year Esablshed Headquarers IG member Aual Growh Sze (g) (mllo) UK 869 Lodo Yes.6% Gard 97 Aredal Yes 4.76% 98 3 Braa 854 Lodo Yes 5.68% 8 4 Seamshp 974 Lodo Yes.38% 65 5 Sadard 885 Lodo Yes 3.3% 58 6 Japa 95 Too Yes.4% 54 7 Suld 897 Oslo Yes -5.3% 5 8 W. Eglad 856 Lodo Yes 3.73% 46 9 N. Eglad 86 Newcasle Yes 9.7% 43 Lodo 866 Lodo Yes.46% 8 Amerca 97 New Yor Yes 9.33% 8 Swedsh 87 Goeborg Yes.36% 5 3 Cha 984 Bejg No 3.47% 9 4 Shpowers 855 Lodo Yes 9.86% 9 Average 9.% 46 Table : Major Mare Muual Isurace Noes:. Ths has bee combed from varous sources b he auhors.. IG Ieraoal Group of P&I Clubs; g.. gross oage of eered shps. A e message from he EC s ha he P&I Club ssem should be more raspare boh o s members ad o he ousde world, order o esure full mplemeao of he prcple of formao sharg (Garer, 999; Mace, 4). The Ieraoal Marme Orgazao (IMO) has also ased P&I Clubs o ope her boos, for he geeral eres of safer shps ad cleaer oceas. Such suggeso has alwas bee rejeced b P&I Clubs uder he defese of proeco of prvac, so he lac of daa from P&I Clubs corbues o he eed for academc research o be coduced hs area (Johsad, ; Bee, ). Much formao abou MMI geeral has bee veled aqu ad los obscur (Dover, 975). All of hese dscussos have poed o oe crcal legal argume wheher MMI should ope up or coue o eep closed formao o clams records ad oher formao. However, hs paper deermes ha formao sharg ca acheve bes socal welfare as well as effce operao of a P&I Club. Assumg he sadard Browa moo characerzao of a clam process assocaed wh each dvdual vessel of a club (Tapero ad Jacque, 987; Asmusse ad Tasar, 997; Segl ad Tch, 999), we develop for he MMI cos mmzao problem a mpulse feedbac model of a pure MMI (Besoussa ad Los, 984; ad Aub, ). We focus he aalss o a comparso of opmal surace posos (.e., fuds o had) uder wo formao srucures uequal-rs versus equal-rs. I pracce, P&I Clubs collec premums a he begg of each polc ear o 8 Februar. Therefore, we model a P&I Club as a mpulse corol ssem he sese ha he oal reserve of he Club s rese b a mpulse premum corol, so ha surace clams curred durg he polc ear ca be suffcel covered a a desrable level of rs poolg. Through a mpulse feedbac aalss, we frs calculae he opmal premum polces ( erms of oal-cos mmzao) uder dffere rs poolg srucures (uequal- versus equal-rs poolg). We he deermed a quafable exra value a uequal-rs poolg MMI ssem, as compared wh a equal-rs poolg oe. I hs paper, hs exra value erm s referred o as he value of 5
formao sharg. We develop algorhms o compue he value of formao sharg, b solvg he HJB equaos ad quas-varaoal equales. Heerogeeous Assureds A B C A B C Toal Isurace Cos of Assureds wh Asmmerc Iformao Value of Iformao Sharg Homogeeous Assureds X D E F Mmze he Expeced Isurace Cos Toal Isurace Cos of Assureds wh Smmerc Iformao Dsace of he Expeced Toal Isurace Coss wh/whou Iformao Sharg Fgure : Value of Iformao Sharg. Impulse Feedbac Model for Uequal-Rs Poolg Cosder a geeral muual surace Club of members over a me horzo [, T ] (whe T, [, T ] [, ) { } ), durg whch each member esablshes a accou of surace poso (.e., dvdual accou balace) a me, ad a record of he dvdual clams curred b member (deoed b ). The records of dvdual accous ad clams are ep cofdeal a he Club. Le x each Gaussa clam process x N(, ) be characerzed as a drfed Browa moo, as follows: dx d dw (,, ), () where d w represes a Weer dffereal. The Club wll revew he curre surace poso ad clams ouloo a he begg of each reewal perod (e.g., each ear,,, ). Followg he revew, he Club he deermes for each member a premum call (a mpulse corol), deoed as q, so as o rese he member s surace poso;.e., d q, where d s he dffereal of he surace poso durg a revew perod [, ). We characerze hs dffereal usg he followg dffereal characerscs: d h ( ) d dx () ( ) d dw (,, ), where h ( ) represes he rae of dvdual poso cosumpo (e.g., operaoal ad maageme coss) excludg he clam coverage coss, ( ) h ( ), ad dw dw. For he res of he paper, we adop he followg oao ssem: Deoe b small leer he vecor, e.g., 6
he poso vecor as (,, ), ad b a capal leer he sum Y. The damcs of a MMI ssem ca be cocsel preseed vecor form as d ( ) d dw, where ) s a vecor drf, ( j s a marx dsurbace wh ad for j, ad dw s a -dmeso Weer dffereal (Goller ad Wbau, 99; Tasar, ). As a fuco of he Club s sae, le L ) be a oegave o-decreasg maageme cos (Lagraga) assocaed wh member of he Club, ad j be he oal MI operag cos of he Club. Smlarl, here s a mpulse cos (cludg he premum pame) assocaed wh each premum call q for reewal ear, deoed b K q ), ad a aggregae mpulse cos for he ere Club ( L( ) K( q ) K ( q ) for all Tq. Thus, he Club s facg he L ( ) problem of mmzg he oal surace coss uder uequal-rs formao (Kavadas ad Loch, 3; Kular, Magaze ad Raur, 4), whch we formulae, vecor form, as a mpulse corol ssem (Besoussa ad Los, 984; Aub, ), as follows: T r r rt ( ) m E K( q ) e L( ) e d ( T ) e, for a q Tq (3) Subjec o: d ( ) d dw d q where ( T ) s a ermal fuco... Opmal Premum Polc uder Uequal-Rs: Impulse Feedbac Corol Accordg o mpulse corol heor, he value fuco ( ) s a soluo o he followg quas-varaoal equales (QVI) ssem (see Pola ad Zasev, 995; Aub, ; Lu, 4): ( ) r () A () H(,D ) ) () ( K)() ) ( K) (r A ) H (4) where: ) D ( ) (,, ) represes a grade of, where D cos shared b member. ) A s a secod-order dffereal ( vscos sese) operaor. ( ) 3) H(, D ) D, L s he Hamloa of he MI ssem (3). s he margal surace 4) ( K)( ) : f ( q) K( q) s ermed he f-covoluo of fucos ad K. q 7
Accordg o mpulse corol aalss, here exss for ssem (3) a opmal poso (,, ) ermed a mpulse feedbac corol, o whch he acual poso a he begg of each reewal perod mus be rese b collecg premums accordgl. For he sae of referece, we prese below he opmal mpulse feedbac polc for he dvdual posos, bu wh he proofs omed (Aub, ; Lu, 4). PROPOSITION. For he MMI ssem (3), le before he reewal premum s colleced;.e., poso q be he poso (vecor) realzed a revew me. The here exss a opmal, such ha he opmal reewal premum s deermed b he followg base-soc polc: f oherwse We shall oe ha he opmal mpulse feedbac polc of he base-soc pe descrbed Proposo s obaed whou regard o he pe of formao srucure (.e., uequal-rs versus equal-rs). Therefore, we shall cofe our aalss of MMI ssems o feedbac polc, uder eher he uequal-rs or equal-rs formao srucure. For coveece, we use he erm uequal-rs (or equal-rs) mpulse feedbac o dffereae base-soc feedbac pes uder a uequal-rs (or equal-rs) formao srucure... Heerogeeous Membershp: Uequal-Rs Poolg Whle complee formao abou posos ad clam records s ep a he Club, each member s ol formed of he dvdual opmal mpulse feedbac poso, ad her ow clams process x. The Club ow eeds o deerme a rs level for he MMI ssem (3). For hs purpose, we defe a opmal uequal-rs rs level as: Pr Gve x x ( ) for each N (, ), we ca he mmedael wre. (5) Uder he uequal-rs mpulse feedbac, each member s formed of ad, whch are ep cofdeal as s uequal-rs records. Noe ha a uequal-rs MMI ssem s geeral uequal-rs poolg. A he Club level, a average rs level ca be calculaed usg as a assessme of aggregae rs poolg. Le, feedbac ssem, as descrbed b Proposo. 3. Equal-Rs Poolg for Homogeeous Membershp d deoe he uequal-rs MI mpulse Suppose ha he maagers of he same Club are compelled o operae wh a equal-rs poolg scheme, uder whch a opmal base-soc polc, deoed b, s ow ope o ever member of he Club. We shall oe ha he smples mplemeao alerave of equal-rs poolg s o adop homogeeous membershp. Before we dscuss he deals of (,, ), le us roduce he 8
cocep of equal-rs poolg regardg surace clams. Le be he oal surace poso of he Club uder a equal-rs formao srucure. I hs case, he Club srves o maa he aggregae poso Y b collecg premums from he members, whch moe s he used o collecvel cover aggregae clams, surace poso Y Y (, ), where ad X x N. Because dvdual accou records have bee rs-equalzed, he oal mmzed Y allocao of oal fuds s expeced o be reduced comparso wh he oal uequal-rs poso, where s he uequal-rs opmal poso as gve Proposo. Thus, he Y amog heerogeeous members should o loger be deermed based o a acual dvdual clams record x. Isead, he dvdual premum raes are deermed accordg o he prcple of formao sharg;.e., he prcple of equal-rs sharg, as opposed o uequal-rs poolg as gve (5). The dea of he collecve coverage of clams b he equal-rs allocao of premum corbuos s ermed equal-rs poolg. 3.. Impulse Feedbac Model uder Equal-Rs Poolg Thus, he resul of equalzg rs poolg, (or equvalel homogezg membershp) s a poeall reduced effecve share of clams coverage. The dvdual s effecve share of clams coverage uder equal-rs poolg, deoed as x N(, ), has he same mea as he acual dvdual clam x, bu has a smaller varace (for ) as a resul of equal-rs poolg (or membershp homogezg);.e., E x ( ), ad var( x ). (6) Wh he above characerscs, a member s effecve share of he clams uder equal-rs poolg s x N(, ), as compared o s acual share of he clams x N(, ). The effecve ad acual shares have he same mea, ad ol dffer sadard devao, specfcall for. Replacg acual wh effecve, we ca derve from equao () he damcs for he effecve poso uder equal-rs poolg, ad express hem vecor form as follows: d ( ) d dw, wh ad for j. The, wh he same cos srucure, we where j j ca derve from ssem (3) a equal-rs mpulse feedbac MMI ssem as beg T r r rt ( E K( q e L( e d ( ) m ) ) T ) e, for a q (7) Tq subjec o: d ( ) d dw. d q The opmal equal-rs mpulse feedbac poso (,, ) ca he be obaed from he 9
same QVI ssem (4), excep for he operaor A (. ) A, whch s modfed wh as: 3.. Equal-Rs Poolg uder Impulse Feedbac I hs subseco, we show ha he opmal poso (,, eals a equal-rs poolg scheme amog all he members. For a equal-rs MMI ssem, a opmal aggregae rs level ca be defed as Pr Y X. Wh he opmal aggregae rs level, a opmal equal-rs MMI mpulse ssem ca be deoed as,. A opmal equal-rs mpulse feedbac MMI ssem, ca be deermed b solvg he correspodg QVI ssem. A opmal uequal-rs mpulse feedbac MMI ssem was prevousl obaed as,. Ule he uequal-rs poolg MMI mpulse ssem, we show below ha dvdual mpulse feedbac uder equal-rs poolg, whch s uquel deermed from he correspodg QVI ssem, ca be mplemeed hrough a equal-rs level for all members. PROPOSITION. Deoe b, a opmal equal-rs mpulse feedbac MMI ssem. The, uder a equal-rs poolg scheme a level, holds ha: ad ( ), (8) where s he effecve sadard devao uder equal-rs poolg as gve (6). Proof. B he cosra ha Pr( Y X ), we ca wre ha: Y ( ), or, equvalel: ( ). Leg each em of he summao above be zero, we oba: ( ). Leg, we ca wre he above equvalel as follows: ) Pr( ( x ), where x N(, ). Ths cocludes he proof of Proposo. 4. The Value of Iformao Sharg 4.. Defe he Value of Iformao sharg We have hus far obaed wo value fucos assocaed respecvel wh wo rs poolg srucures )
of a MMI ssem, amel, he value fuco of a uequal-rs ssem, versus he value fuco of a equal-rs ssem,. B defo, a value fuco represes he oal value (a cos hs case) of a Club opmall operaed uder a specfc rs poolg srucure, eher uequal-rs poolg or equal-rs poolg. I hs sese, he dfferece bewee he wo value fucos ca be used as a measure of he value (or he prze) for formao sharg, especall he dfferece he wo value fucos whe evaluaed a her respecve opmal base-soc posos. For hs purpose, we defe he value of formao sharg as: ( ) ( P ), (8*) muual where ad are he opmal base-soc posos respecvel uder uequal- ad equal-rs. Frs, we eed o be assured ha a oegave dfferece bewee he wo value fucos exss (.e.,, ad herefore P ). Iuvel, he dfferece s oegave (.e.,), muual sce equal-rs poolg should reduce he oal surace cos. Now le us ascera hs uo usg rgorous aalss. Sce for, we ca wre for some. The, he HJB equao of a uequal-rs, ca be derved from ssem (4) as follows: r ( ) H (, D). (9) Or, equvalel, we ca wre he above as: F(,, D, A ) ( ), where ( ) F r H (, D ) s a augmeed Hamloa for he equal-rs MMI ssem,, of whch he HJB equao ca be wre as: F (,, D, A ). Wh ad for some, he HJB equao (9) of a uequal-rs MMI ssem ca mmedael verf ha F(,, D, A ) F (,, D, A ). The fac ha he Hamloa H s decal for boh uequal-rs ad equal-rs ssems mples ha ad ca ol dffer b a addve fucoal erm;.e., for some fucoal erm. Wh hs, he value of formao sharg ca be aalcall measured b. I prcple, he exac expresso of hs prce erm ca be obaed b solvg for ad from he respecve HJB equaos of he uequal-rs MMI ssem, ad he equal-rs MMI ssem,, respecvel. However, closed-form soluos are ofe uaaable, ad eve umercal soluos are sll oo complex o be racable usg umercal mehods. I he Proposo below, we derve a more racable lower-boud fuco, whch gves he leas cos dfferece caused b formao prvac. PROPOSITION 3. Le ad be he respecve value fucos uder he uequal-rs ad equal-rs MMI ssems wh a srcl covex Lagraga L. The, for each o-mpulse erval
[, + ) for all Tq, he followg holds rue: ) The value fucos ϕ ad ϕ ϕ are srcl covex for [, + ) ;.e., > ϕ > for all. ) There exss a fucoal δ >, such ha ϕ = ϕ + δ. Therefore, P prvac = δ >, where ( ) ( δ = ϕ ϕ ). 3) Gve ϕ ad ϕ, he δ = ϕ ϕ has a oegave Hamloa, specfcall, F(, δ, Dδ, A δ ) = ( ) Δσ > ϕ, = where Δ σ σ σ. Proof. Iem above s a prove resul corol heor (Flemg ad Soer, 6), ad s proof s ϕ hus omed. Gve > for all, em of Proposo 3 ca be verfed from he HJB equao (9). To prove em 3 of he Proposo, we oba from (9) he followg varaoal equal: F (, ϕ, Dϕ, Aϕ) > F(, ϕ, Dϕ, A ϕ), Where F ( ) dffers from F ( ) ol he secod-order operaor A. The, usgϕ = ϕ + δ, we ca verf from HJB (9) ha F (, ϕ, D ϕ, A ϕ ) + F(, δ, Dδ, Aδ ) =. Nog ha (,,, F ϕ Dϕ A ϕ ) = ad A = A + ( Δσ ), we ca wre he above HJB = equao as follows: F(, δ, Dδ, A δ ) ( ) Δσ = ϕ. = ϕ Nog ha > ad Δσ > for >, he proof of em 3 of Proposo 3 s mmedae from he above equal. Wh hs, we coclude he proof. 4.. Compug he Value of Iformao Sharg I summar, he exac value of formao sharg, deoed b ( ) ( Vmuual ϕ ϕ ), ca be compued as follows: ) Solve he uequal-rs ad equal-rs HJB equaos respecvel for ϕ ad ϕ. ) Oba he respecve opmal base-soc posos, (uequal-rs) ad (equal-rs). 3) The compue he value of formao sharg,.e., ( ) ( V = ϕ ϕ ). The above soluo for he exac prce of prvac requres he solvg of wo HJB equaos, oe for he uequal-rs ssem ad he oher for he equal-rs ssem. These equaos mosl requre complex umercal mehods o solve hem. However, usg Proposo 3, we ca cosruc a approxmae soluo, whch requres much less compuao. The scheme for he approxmae mehod of soluo s descrbed below: muual ad
) Solve for he equal-rs for s HJB equao, ad oba. ) Deerme: V ( ) ( ). 3) Deerme a lear fucoal dfferece ˆ C wh D ˆ b solvg he followg frs-order PDE ssem: ˆ ˆ r H (, D ˆ) V, subjec o: for all. 4) The compue he approxmae value of formao sharg V ˆ( ). ˆmuual The raoale of he above approxmao s o see he approxmae dfferece he form of a lear fuco (.e., D ˆ ), so as o avod solvg he HJB equaos wce. 4.3. Value of Iformao Sharg ad Volal of Rs Compared wh he compuao of he value of formao sharg, a more mpora ad eresg opc cocers he characerscs of he value of formao sharg,.e., ( ) ( V ). B Iem 3 of Proposo 3, he characerscs of he formao sharg value are characerzed he correspodg Hamloa F (,, D, A ), whch has bee obaed Proposo 3 as: F(,, D, A ) ( ), where s he varabl dffereal bewee o-rs poolg ad equal-rs poolg. For he deal case of homogeeous membershp wh..d. dvdual clam processes wh decal for all,,, he varabl dffereal ca be deermed as ( ). I erms of he Hamloa characerzao of ( ) ( Vmuual ), ca mmedael be see ha he value of formao sharg creases alog wh he average varabl ad he sze of he Club. Boh parameers ad are measures or dcaors of he volal of uderlg rss erms of aggregae clams. Wh he fdg hs paper ha homogee faclaes opmal realzao of he value of formao sharg, s whou loss of geeral o coclude ha he more volale he surace rs s, he more compeve he muual surace becomes. 5. Fdgs ad Implemeao Frs, le us summarze he useful observaos ad maageral mplcaos ha ca be draw from he resuls we have obaed so far hs paper, especall from Proposo 5. 5.. Fdgs ad Implcaos ) The prce of formao prvac s mal depede o, where muual. If he clams are deermsc ( ), he he prce of prvac would vash, ha s, here would be o dfferece bewee a uequal-rs ad a equal-rs formao srucure whou regard o he degree of heerogee of he members. ) A ufed oage-based premum polc ca be jusfed ol f he oage of a vessel s learl ad 3
assocaed wh he varabl of he clams curred b he vessel. Ths fdg suggess ha a rgorous ad esve sascal sud of he correlao bewee oage ad s clams eeds o be coduced, so as o deerme wheher or o a oage-based premum polc s jusfable. 3) Uder a uequal-rs formao srucure, he oal cos mmzg premum polc eals a uequal-rs poolg scheme amog he heerogeeous members of a P&I Club. Uder he opmal uequal-rs premum polc, he dvdual s share of rs s solel deermed b s acual clams record. 4) Gve he fac ha he oage-based premum polc has bee pracced uder uequal-rs formao P&I Clubs sce her esablshme 5 ears ago, le us suppose ha he oage-base premum polc s jusfed (.e., ha oage s learl assocaed wh he varace of clams). The above fdgs ), ), ad 3) he expla he pheomeo mare surace ha vessels of a smlar oage ed o jo he same P&I Club. 5.. Implemeao of Equal-Rs Poolg The prcple of formao sharg mples he equal-sharg of rss. Nog ha he equal-poolg of rs ca be aaed wh a opmal premum polc uder a equal-rs formao srucure, we ol eed o exame how o mpleme a uequal-rs equal-rs poolg scheme;.e., equal-rs poolg a uequal-rs MMI ssem,. Two scearos for mplemeg equal-rs poolg a uequal-rs MMI ssem ca be mmedael cosdered: Oe wh a equal-average level of rs, where s as ha deermed (5), ad he oher wh a equal-level of rs. I wha follows, we exame he deals of he mplemeao of uequal-rs equal-rs poolg. The surace hreshold poso for each member uder he equal-average rs deoed b ( ), ca be deermed o be: ( ) ( ). scheme, I vecor form, we wre he equal-average rs poolg poso as ( ) ( ( ),, ( )). From equao (5), we ca deerme ha geeral he equal-average rs poolg poso ( ) ad he opmal uequal-rs poolg poso would have a hgher hreshold poso (.e., hreshold poso (.e., ) ( ) f ) dffer (.e., ( ) f. ) ( ). Idvduall, some, ad some ma have a lower Sce he oal surace cos s mmzed uder a opmal hreshold, we ca coclude ha he oal cos uder a equal-average rs scheme, deoed b (), ca be o less ha he uequal-rs value fuco ( ) (.e., ( ( )) ( ) ). Ths suggess ha, erms of oal Club cos, uder a uequal-rs MMI ssem s worse o mpleme a equal-average rs poolg scheme ha s o mpleme a opmal uequal-rs poolg scheme. Now le us exame wha happes whe a equal-equal-rs rs level s mplemeed for he uequal-rs MMI ssem. I hs case, he dvdual hreshold poso, deoed b ( ), ca be deermed as: ( ) ( ). Usg smlar argumes, we ca show ha ( ( )) ( ). Thus, we ca coclude ha erms of oal surace cos, a uequal-rs poolg ssem s beer for he allocao of premums a 4
of oal surace cos, a uequal-rs poolg ssem s beer for he allocao of premums a uequal-rs MMI ssem. 6. Cocluso I s cocluded ha a ope polc or equal-rs formao ca lead o a more effce MMI ssem overall for soce, ad o a greaer degree of faress ad formao sharg for he sured. The paper deermes ha formao sharg ca acheve bes socal welfare as well as effce operao of a P&I Club. The sud provdes a scefc bass for fuure legslao o MMI compeo law. The cocluso provdes a scefc bass for boh maageral sraeg ad compeo regulao. Acowledgeme I s o acowledge ha he research s suppored b The Hog Kog Polechc Uvers uder projec No. B-Q99. Refereces Asmusse, S. ad Tasar, M. (997), Corolled dffuso models for opmal dvded pa-ou. Isurace: Mahemacs ad Ecoomcs 997; ; 5. Aub, J. (), Opmal mpulse corol problems ad quas-varaoal equales hr ears laer: a vabl approach. Opmal Corol ad Paral Dffereal Equaos, 3-34. IOS Press. Bee, P. (), Muual a a Dsace? Rs ad Regulao Mare Isurace Clubs. Evrome ad Plag A ; 3; 47-63. Besoussa, A., Los, J. (984), Impulse corol ad quas-varaoal equales. Ke, Eglad: Tras-Ier-Scea, 984. Cabrales, A., Calvó-Armegol, A. ad Jacso, M.O. (3), La Crema: A Case Sud of Muual Fre Isurace. Joural of Polcal Ecoom 3; ; 45-458. Cass, D., Chchls, G. ad Wu, H. (996), Idvdual Rs ad Muual Isurace. Ecoomerca; 996; 64; 333-34. Dover, V. (975), A hadboo o mare surace. 8 h edo. UK: Wherb & Co Ld; 975. Europea Commsso (985), O.J.L376/ 985. Europea Commsso (999), O.J.L5/ 999. Flemg, W. H., Soer, H. M. (6) Corolled Marov processes ad vscos soluos. d Edo, New Yor, N.Y.: Sprger-Verlag, 6. Garer, B.A. (999), Blac s Law Dcoar. 7 h Edo. M.: Wes Group; 999. Gold, E. (), Gard Hadboo o P&I Isurace. Aredal: Gard;. Golsh, H. (5), Mare Isurace 5/6. Drewr Shppg Cosulas Ld: Lodo; 5. Goller, C. ad Wbau, S. (99), Porfolo Seleco b Muual Isurace Compaes ad Opmal Parcpag Isurace Polces. Isurace: Mahemacs ad Ecoomcs 99; ; 37-45. 5
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