Critical Approach of the Valuation Methods of a Life Insurance Company under the Traditional European Statutory View

Transcription

1 Crcal Aroach of he Valuao Mehods of a Lfe Isurace Comay uder he radoal Euroea Sauory Vew Dr. Paul-Aoe Darbellay ParerRe Belleresrasse 36 C-8034 Zürch Swzerlad Phoe: Fa: E-mal: aulaoe.darbellay@arerre.com Absrac A surace comay has a alue jus as s coracs do. he effcecy of a comay or of a Prof Cere s measurable ad he rofably of a surace roduc ca be quafed. By usg recse crera s ossble o assess boh he comay ad s roducs ad o ae arorae decsos. Mehods es ha sasfy hs classc eed o ealuae. I he frs ar of hs aer we wll eame he mehods le Prof esg Embedded Value Value Added ad oal Rae of Reur he coe of a edowme surace roduc. he coug wh he same eamle we wll rese a crcal aroach of hese mehods. We wll see ha hese mehods are o always easy o udersad ad ha somemes he Value Added ad he oal Rae of Reur are o able o show drecly he rofably. Fally he las ar of hs arcle we wll do a sesy aalyss order o deerme he mos flueal arameers. hs sudy s doe wh he eamles of a edowme a erm surace ad a auy. Key words Profably Prof esg Embedded Value Value Added Varably

2 sory radoally lfe assurace comaes hae reored facal resuls o shareholders o he bass of he sauory requremes of he surace comaes' legslao. So he mos commo measure of a lfe surace comay's facal year was he sauory eargs from oerao. hs has bee a coee measure sce also rereses he amou of moey whch ca be ad o olcyholder or ad he form of ddeds. he major dsadaage of relyg uo sauory eargs as a measure of how well a comay s dog s ha sauory accoug eds o be desged o roec agas solecy ad herefore by s ery aure suffers from oer coserasm. Sauory eargs do o measure how well a comay s dog o a gog cocer bass. For eamle caal esed acqurg busess acquso eeses ad aluao sra s mmedaely wre off. Successful acquso of rofable ew busess resuls a mmedae loss followed by a subseque ehaced seres of rofs. Alhough suable for solecy esg he sauory aroach by chargg he caal cos of ew busess o reeue ad gorg he fuure surlus sream arbuable o ew busess fals o dslay ay accoug erod a meagful accou of he radg acy of ha erod. For mos roducs a slowdow sales wll resul a mmedae crease sauory eargs ad geerally mos would o regard a slowdow sales as beg a sg of a healhy comay! So s clear ha sauory eargs are he wrog mehod o measure he healh of he comay see for eamle Burrows ad Fces [3] Colls [7] ad he ublcao of Waso & Sos [24] Largely as a resul of he adequaces of sauory accoug US surers were requred by he Secures Echage Commsso he early 970 s o beg o reor eargs o shareholders o a Geerally Acceed Accoug Prcles GAAP bass see Garder [] ad Faga [0]. he major adaage of GAAP accoug s ha does aem o roduce eargs ha reflec how well or how badly he surace comay had erformed a form whch s useful o maageme. Wh GAAP geerally a crease sales wll o deress GAAP eargs o he same degree as would sauory eargs. Uforuaely because 00% of acquso coss are o deferred creased sales wll sll deress GAAP eargs o some ee. Addoally margs for coserasm are ormally roduced o he assumos ad GAAP mgh suffer from he loc- rcle. Oce assumos are se for a arcular geerao or brach of busess he assumos cao be chaged uless fuure losses are lely. Aoher major dsadaage of GAAP s ha GAAP eargs may ary sgfcaly bewee wo decal comaes deedg o he objeceess of maageme esablshg assumos. herefore oerall GAAP s o a good rogoscaor for how well a comay s dog. Durg he erods of flucuao eres raes whch occurred he US durg he md- 970 s ad early 980 s some US comaes bega o loo a cash flows as a measure of how well her comaes were dog. he real adaage of usg cash flows as a measureme ool s ha s he oly bass ha loos a real moey. Bu s really oly a mmedae solecy es. Cash flow s o a measure of how well a comay s dog. I he 80 s here were ew roducs for eamle u led roducs he crease comeo mergers ad resrucurg decrease rof margs due o creased comeo ec. he resul has bee a greaer eed o be able o maage ad corol 2

3 surace oeraos he face of creasg flucuaos ad uceray. he aers of Faga [0] ad Smh [22] ela hs eoluo. I s eresg o oe see Faga [0] ha solecy ad rofably cao be showed by he same mehod. Solecy s a cosra whch deermes he secury margs ad mehods o use. Profably res o show how well a comay s dog elmag he cos effec of he frs year ad ag he fuure rofs o accou. I shor we ca say ha four os hae corbued o he adoo of aluao mehods ard comeo bewee surers Iesors' ressure o hae comrehese resuls Producs' eoluo owards greaer flebly Deregulao facal corol of solecy ad o more arff aroal such as sadard moraly ables 2 Produc assessme : Prof esg We ow ha he acquso of rofable ew busess resuls a mmedae loss due o he acquso eeses ad he frs ayme o echcal resere. Aferwards we hoe ha a seres of rofs wll follow. he Prof esg aes fuure rofs o accou ad he decreases he effec of he esme of he frs year. o calculae or o esmae hese fuure rofs we use eeced alues. I fac hese fuure rofs come from he rof ad loss accou see auser [5] ad Lahme [7]. ere s a eamle of a surace accou. We hae o he lef he charges ad o he rgh he come see Chuard [6]. Prof ad loss accou Deah clams ad Release of reseres Amous ad o maury Premums receed Surreder alues ad Ieres ad gas receed Eeses curred ad commssos ad Bous ad Icrease reseres aes Prof durg he year ycally we hae he followg cure for he Prof esg whch rereses he eeced rofs of he rof ad loss accou each year. 3

4 Prof esg I s ery commo o hae a loss followed by a seres of rofs. Bu where does come from? Is he Prof esg relaed wh he acuaral rules? Yes s ad we wll see hs he e chaer. 2. he Prof esg 3 ses We wll see he relaosh bewee he Equalece Prcle ad he Prof esg 3 ses. Sarg from he Equalece Prcle we oba he radoal Marg ad he afer some modfcaos we oba he Prof esg. 2.. Equalece Prcle Equalece Prcle has he followg defo: Acuaral Prese Value APV of Premums equals o APV of Beefs lus APV of Charges. Usg a radoal edowme for a male aged of durao years we hae he followg alcao P' ' ä A α γ ä Where ä he e sgle remum for a -year emorary lfe auy-due whch rodes for aual aymes of u as log as he beefcary les. A he e sgle remum for a edowme whch rodes for a ayme of a he ed of he year of deah f occurs wh he frs years oherwse a he ed of he h year. α acquso eeses γ admsrao eeses 4

5 5 he we hae our eamle ha he oal remum equals o he edowme beef lus he acquso eeses oly he frs year lus he oal admsrao eeses. We suose ha he colleco eeses are egraed he admsrao eeses radoal Marg Usg he bes esmae sead of rude assumos we ca rewre he formula as follows we symbolse he bes esmae bass wh he sar. 0 0 '' B u L ä A M ä P γ α Where Bous Lase beef robably o lase Marg radoal B L u M he wo las eressos are he surreder beef ad he bous. hs s radoally how he rof s calculaed. he Prof s o recogsed yearly bu a he begg of he corac for he whole durao Prof esg Iserg reseres rewrg he formula wh sums o he me ad regroug we oba 0 '' 0 α γ G I V V B u L q P Where eres eared Reseres Prof I V G I he sum we ca see he rof from he accoug G. he surace rofs are dscoued wh he facor. he eleme braces a he ed of he sum s he dfferece bewee he resere a he ed of he year ad he oher a he begg of he year. I fac each erm of he sum s a le of he Prof ad Loss accou. So he Prof esg uses he radoal elemes of he acuaral scece. Of course ay ew eleme ca be sered le for eamle a resurace remum see Smar [2]. We ca see ha he Prof esg ca be used as rofably ool or as a rcg ool.

6 he aer of are ad McCucheo [4] shows dfferely he relao bewee he radoal formulas of he remum ad he Prof esg. 2.2 Commo crera o ealuae he rofably hese crera are elaed for eamle by Leasseur ad Quar [8] ad by Delau ad Magée [9] Ne Prese Value he Ne Prese Value wh Rs Dscou Rae s defed as follows see Sharo [20]: 0 G NPV If he Ne Prese Value s ose he roduc s rofable. Profably occurs whe he roduc geeraes a leas o he frs year sra. Rs Dscou Rae he Rs Dscou Rae s defed as he rae of reur led o busess rs of he surace comay. I realy hs rae s ery subjece. I deeds o he maageme he shareholders ad he esors see for eamle Chuard [5] Geddes [2] ad Merda [9] Ieral Rae of Reur he Ieral Rae of Reur s defed as he rae whch ges a Ne Prese Value of zero. 0 G IRR 0 If he Ieral Rae of Reur s hgher ha he Rs Dscou Rae he he roduc s rofable Relao bewee IRR ad NPV IRR NPV 0 IRR 6

7 3 Comay assessme : Mehods We wll sudy fe mehods whch ealuae a surace comay. I hs chaer all he umercal eamles are based o he daa ge he aee "Numercal alcao ". 3. Embedded Value Embedded Value rereses a he aluao dae a esmae of he asse alue ad he soc of he surace comay. More recsely he Embedded Value of a lfe offce a a arcular aluao dae s ae o be he sum of he shareholders e asses ad he alue of he busess -force a he aluao dae. he alue of -force busess a he aluao dae s he rese alue of fuure rofs eeced o emerge from olces already wre. Colls ad Keeler [8] Geddes [2] Kameec [6] Merda [9] ad Smar [2] roose he followg formula: Where Embedded Value Shareholders Ne Asses he Soc: he alue of he busess -force a he aluao dae Ne Asses share caal solecy caal rof hdde reseres surlus Le us see ow hs formula more deal. If we deoe z- aluao year EV EV a he ed of he year NA Ne Asses a he ed of he year S Soc ealuaes a me year of rof for > sauory rof eeced for he year ealuaed for he whole force busess a he me for sauory rof realsed for he year dscou facor wh he Rs Dscou Rae We hae he followg formula for Embedded Value EV NA z z z soc 7

8 We ow ha z rereses he fuure rof eeced o be geeraed resec of resely force busess ad o be rasferable afer allowg for all relea aes o he rof ad loss accou. I he sum we use o ae oly he fuure rofs o accou. Bu we ca wre more elcly. s ew busess year aluao year sauory rof year for s < sauory rof eeced for he year for he busess wre he year s ealuaed G s for s < e sauory rof realsed he year for he buses wre he year s forr s sauory rof realsed he year for he ew busess Summg he G s o he whole years s we oba he sauory rof of he rof ad loss accou: G s s for If we roduce a a rae ha s ad whe he sauory rofs are ose ad a rae f > 0 0 oherwse We ge he followg formula for Embedded Value [ ] EV > where s called dsrbuable rof of he year. hs s he sauory rof afer aes. 8

9 3.. Numercal eamle Le us ae he followg eamle order o llusrae he Embedded Value. Edowme surace C 00'000 α 4% o Caal β.5% o Premum γ 0.3% o Caal SM 978/83 4% P 3970 α 3.8% β % 6% 8% Ieres o e asses.5% γ 0.2% ae rae 0% he eres o e asses s he afer a esme reur eared by he asses ha suor shareholders e asses. I s o uusual for hs rae of reur o be subsaally below he reur eared o asses suorg surace lables because of asses allocao echques used by comaes. I s commo o assg o-eres-bearg asses or lower-yeldg asses o suor shareholders e asses ad o allow he hgher-yeldg asses o bac olcyholder lables. Before showg he chars we hae o oce ha we wll rese a rojeco of he aluao mehods. I realy hese mehods are calculaed each year wh he daa of ha year ad used o comare wh fgures of reous years. I our eamle we are eresed he deelome of he resuls hrough me ad he we suose ha we do o chage he assumos ha he assumos mach wh he realy ad ha here s o ew busess. Wh hese assumos he real rof equals he eeced rof. Soc rofle % We oce ha he rofle s decreasg. hs s because he aual rofs are early cosa ad he each year he soc has oe rof less because here s o ew busess. Embedded Value rofle 9

10 % Each year he Embedded Value creases because of he eres eared o e asses ad he aual rof Embedded Value adaages Embedded Value aes he fuure rofs of he Prof esg o accou ad he he alue s o oly flueced by he loss of he frs year. Embedded Value allows comarg facal resuls of dffere ds of aces. Vsual rereseao whch seems o be easy o udersad Embedded Value dsadaages Whou ublshg he assumos he resul of he mehod has o real sese. Rs Dscou Rae s o a geeral coce ad furhermore Embedded Value s esecally sese o Rs Dscou Rae. I radoal boo eeg we eer use hs rae. hs rae s oly used Prof esg ad Embedded Value. Embedded Value s sesy o Embedded Value 8% Embedded Value 0% Year 0 Embedded Value wh of 8% equals Embedded Value wh of 0% equals

11 3.2 Arasal Value Arasal Value s he eeso of he Embedded Value o he mare alue see Bager []. So o oba hs alue we hae o cosder he goodwll. Bu he goodwll s a subjece alue whch rereses he alue of he cles he fuure ew busess ec. So hs alue s ery dffcul o ealuae. Burrows ad Whehead [4] roose he followg formula: Arasal Value Embedded Value Goodwll 3.3 Value Added Value Added s a dyamc alue. I s he crease he Embedded Value bewee wo me erods. VA EV EV We saw ha he Embedded Value s he alue of he comay he Arasal Value s he mare alue of he comay ad ow we see he Value Added whch s a dyamc measure cosdered o show how well he comay s dog durg a secfed erod see for eamle Burrows ad Fces [3] ad Merda [9]. We wll use ow oly Embedded Value because Arasal Value s more comlcaed o alue because of he goodwll. Bu of course hs mehod ad he followg ca also be calculaed wh Arasal Value. o llusrae he Value Added we ca ae he followg eamle. Suose ha a comay has 00 of equy ad es eeryhg ew busess we could hae for eamle: Begg of year Ed of year Ne asses 00 0 Busess force 0 5 Value Added Prof ad loss accou could hae show a loss of 00. hrough hs eamle we oce ha he Value Added seems o be more able o show he rofably ha he sauory rofs.

12 3.3. Numercal eamle Value Added rofle 60 8% Frs year he Value Added s esecally hgh because he comay sells a rofable roduc a hs me. Aferwards here s o ew busess ad he Value Added decreases. Comarso bewee Value Added ad sauory eargs: % Sauory eargs Value Added hs grah s ery eresg because we ca see he dfferece bewee Value Added ad sauory eargs. I hs eamle he wo mehods behae comleely dfferely. he sauory eargs show frs a loss followed by a seres of gas ad he Value Added shows frs a ga followed by a seres of smaller gas. he Value Added seems much more able o show rofably ha he sauory eargs. 2

13 Value Added comoes Sarg wh he defo of he Value Added we wll sl he formula fe comoes EV EV VA S S VA we oe eres o e asses Rs Dscou Rae a rae D dded rae ad we rewre he soc a me - o aoher way order o reare he calculao S D S S S VA we use he formula of he soc ad we oba > > D S VA ad regroug D S VA We use he formula of sauory rof ad we solae he aes s s D s G s G S VA

14 4 s D s G s G G S VA we arrage he elemes [ ] s D s G s G G G S VA ad we ge fally hs formula [ ] s D s G s G G G S VA where he fe comoes of he Value Added ca be see [ ] eerece acual assumos ad emergeceof chage of adjusemes caal corbuo from ew busess ae relef year ew busess of soc of essseme year for he ew busess of year sauory rof of corbuo from forcebusess eres o e asses s D s G s G G G S VA We hae foud fe comoes of he Value Added whch are he followg: Ieres o e asses Corbuo from force busess Corbuo from ew busess Caal adjusmes Chage of assumos ad emergece of acual eerece

15 3.4 oal Rae of Reur oal Rae of Reur eresses he crease of Embedded Value as a erceage see Merda [9]. RR VA D C EV where D shareholders ddeds a he ed of me C shareholders corbuos a he ed of me I s ecessary o do correcos for ddeds ad shareholder corbuo because he Embedded Value s chaged by hese alues. Whe dded s ad he e asses decrease ad he Embedded Value oo. he shareholder s ew s ha he recees dded ad he Value Added. he we hae o ae accou of he dded he oal Rae of Reur. Whe shareholders mae corbuos he Embedded Value creases bu he shareholder s ew s ha he recees he Value Added less wha he corbues Numercal eamle oal Rae of Reur rofle 8% 6% 8% 4% 2% 0% 8% 6% 4% 2% 0% hs rofle seems o be easy o udersad ad allow fg a smle objece. For eamle oe could say e year we hae o reach a oal Rae of Reur equal o he Rs Dscou Rae. 5

16 4 Crcal Aroach Ul ow we hae see he aluao mehods. Eeryhg seems o be smle ad we are able o show he rofably of a comay. Bu some asecs of hese mehods could be crcsed. I hs chaer all he umercal eamles are based o he daa ge he aee "Numercal alcao 2". 4. he roblem he resuls of he aluao mehods deed o he model ad he arameers le he reur o esme he bes esmae moraly ec. hese mehods use always he mea of fuure rss. So he adaage s o oba a uque alue bu he dsadaage s o hde he flucuaos. I realy he flucuaos are due o he model he radomess ad he uceray. o choose a model wh a ew o coceualsg real heomea s a smle aroach whch cao reflec eacly he realy. he radomess s he uforeseeably of ees whch follow a robably fuco. Ee whe we erfecly ow he robably law he effece realsao of he ees cao be deermed. I our model arcular he radomess corresods o he flucuaos of he reur o esme ad he flucuaos of he real umber of deahs. he esmaos we calculae oday are oly mea alue of fuure realsao. he uceray comes from he merfec owledge of he arameers ad her ossble eoluo hrough he me. For eamle we do o ow eacly he real amou of eeses ad hs amou ca follow a fuure ueeced eoluo. Besdes hese flucuaos roblems here s a erreao roblem. he Value Added ad he oal Rae of Reur are ofe cosdered as rofably ools ee hough hese mehods ca ge a false sgal o he rofably of he comay. o dscuss hs erreao roblem we ae he followg umercal eamle Edowme surace Caal 00'000 α 5% o Caal γ 0.4% o Caal β cluded γ SM 988/93 4% P 332 α 5800 Muao eeses 333 8% Ieres o e asses.5% γ 80 6% ae rae 0% We wll use wo secal cases as follows: A rofable case: Prooro of rof reaed 3% A o rofable case: Prooro of rof reaed 5% he rooro of rof reaed s he erceage of rof ha he surace comay was o ee. hs meas he ar whch s o dsrbued o he sured. Le us see ow he chars hese wo cases. 6

17 Value Added Profable case No-rofable case he frs year loos sesble here s o erreao roblem. he surace comay sells a rofable corac whch creases he alue of he comay he frs case. I he secod case he corac s o-rofable ad he alue of he comay decreases. Secod year ad laer. For he frs case we do o oce ay roblem. he Value Added decreases because he comay does o sell ay oher corac. I he secod case he comay does o sell ay oher corac ad he Value Added s ose alhough had sold reously a o-rofable corac. he se esecally bewee he frs ad he secod year s o easy o udersad for he urose of rofably. Profable case 9% 8% 7% 6% 5% 4% 3% 2% % 0% -% % -3% -4% -5% -6% oal Rae of Reur 9% 8% 7% 6% 5% 4% 3% 2% % 0% -% -2% -3% -4% -5% -6% No-rofable case We hae eacly he same erreao roblem wh he oal Rae of Reur. I order o udersad wha haes le us aalyse he fgures of he Value Added. 7

18 Profable case Ieres o e asses Corbuo from force busess Corbuo from ew busess Caal adjusme dded Value Added No-rofable case Ieres o e asses Corbuo from force busess Corbuo from ew busess Caal adjusme dded Value Added For he frs year here s o erreao roblem. he dfferece comes from he ew busess ose for he rofable case ad egae for he o-rofable case. For he secod year we hae a erreao roblem. Le us aalyse each comoe of he Value Added: -Ieres o e asses s he same because bous begs oly he hrd year our eamle. -Corbuo from ew busess s he same because we do o sell ay ew coracs. -Caal adjusme s he same because we do o ge ay bous before he hrd year. -he dfferece comes from he corbuo from force busess. Wha s surrsg s ha he o-rofable case hs corbuo s ose ad hs s le a reur effecely he calculao. ha s why he Value Added s ose he o-rofable case. Fally he roblem of erreao s due o he corbuo from force busess ad more esecally o he Rs Dscou Rae. Le us see ow he fluece of Rs Dscou Rae o Value Added Profable case wh 8% Profable case wh 9% he Rs Dscou Rae s a requreme of reur he frs year bu aferwards hs rae becomes a reur o he soc. Afer he frs year he Value Added calculaed wh a of 9% s hgher ha wh a of 8%. hs s really he source of he erreao roblem. 4.2 hree ds of aluao We hae o adm ha he Value Added ad he oal Rae of Reur are o able o show drecly he rofably of he comay. Bu he erreao roblem dsaears f we use he aluao mehods he correc coe. 8

19 Value of a surace comay Icrease of he alue of a comay Profably of a comay 4.2. Value of a comay Of course he alue of a comay s drecly ge by Embedded Value or Arasal Value Icrease he alue of a comay For measurg he crease he alue of a comay we hae o aalyse he Value Added ad s comoes eres o e asses corbuo from force busess corbuo from ew busess caal adjusmes chage of assumos ad emergece of acual eerece ad he oal Rae of Reur. We ca aalyse each o o see f s ose or egae. Ad he we ow whch comoe crease decrease or sagae he alue of he comay. he Value Added ges he real crease. he oal Rae of Reur ges he relae crease ad he comoes of he Value Added allow searao of he comoso of he crease. I our reous eamle dese he fac ha he coracs sold do o mee he crera of Ieral Rae of Reur he reur s ose bu smaller ha he Rs Dscou Rae. ha s why he comay creases s alue wh hs roduc Profably of a comay he roblem of erreao comes also because we do o hae a clear defo of rofably. We wll use he followg defo of rofably. I our coe he rofably s he measure of he acy of a comay coeco wh he ew busess ealuaed hrough he fuure rofs eeced o emerge from he ew busess durg a secfed erod of me. Whe we are eresed he rofably of a surace comay seems o be logcal o be esecally eresed he ew busess creaed durg he secfed erod of me. I our coe we ca aalyse he Ieral Rae of Reur of he ew busess ad s NPV. We ca he hae he followg coclusos: IRR NPV of ew busess Cocluso > >0 Am of reur reached 0<IRR< <0 Am o reached bu ose reur <0 <0 Loss 4.3 Fal remars abou crcal aroach We hae see ha he Value Added ad he oal Rae of Reur cao be drecly cosdered as ools for measurg he rofably. We hae see ha he aluao mehods mus be used he rgh coe. We hae o use he comoes of he Value Added. hs s 9

20 why we hae defed wha we call alue crease of he alue ad rofably of a surace comay or a rof cere. I our coe we ge a defo of rofably. We hae o foud ay ew mehod we jus ela ha he aluao mehods do o always drecly ge he rgh aswer ad ha we hae o use he rgh coe. 5 Sesy Aalyss As we hae already see he resuls of he aluao mehods deed o he model ad he arameers le he reur o esme he bes esmae moraly ec. hese mehods use always he mea of fuure rss. So he adaage s o oba a uque alue bu he dsadaage s o hde he flucuaos. I realy he flucuaos are due o he model he radomess ad he uceray. he uceray comes from he merfec owledge of he arameers ad her ossble eoluo hrough he me. For eamle we do o ow eacly he real amou of eeses ad hs amou ca follow a ueeced fuure eoluo. 5. Aroach he sesy aalyss s used o aalyse he fluece of he uceray he arameers' alues. For hs urose we creae some scearos wh omsc ad essmsc alues of arameers. We suose ha cera rages of arameer alue are ossble ad we es he reacos of he resul of he mehod wh hose modfcaos of arameer alue see Leasseur ad Quar [8] ad Sharo [20]. wo ways are ossble he successe sesy aalyss ad he smulaeous sesy aalyss. I he successe sesy aalyss he alue of a uque arameer s chaged. he alue of he oher arameers does o chage. he am of hs aalyss s o erac he more flue arameers o he resul of he aluao mehod. I he smulaeous sesy aalyss he alue of he whole arameers s chaged. he am of hs aalyss s o aalyse he wors ad he bes ossble suao ag resecely he essmsc alue of seeral arameers ad ag he omsc alue of seeral arameers. Bu we hae o oce ha some erdeedeces are ossble bewee he arameers. So we hae o model whe s ossble ad relea. I s rue ha he leel of arao of he arameers alue wll fluece he arao he resul of he aluao mehod. Bu o hae a homogeeous comarso we ca ae a cosa erceage of arao for each arameer alue. Of course hs erceage s eerheless arbrary. 5.2 Sesy esmao We wll use curre ecoomc oo: he elascy see Baumol Blder e Scarh [2] 20

21 heorec elascy E R R R Emrcal aromao ˆ E R R R hs s he elascy of he resul R of he aluao mehod eeg wh he arameer. Aferwards we wll oly use he emrcal aromao of he elascy bu o smlfy he erm we wll oly sea abou elascy ad o more abou emrcal aromao of he elascy. We say ha he arameer s elasc f he resul of hs formula s hgher ha ad resecely elasc f he resul of hs formula s lower ha. hs elascy formula s used he successe sesy aalyss bu s ossble o show ha we ca geeralse hs formula for successe sesy aalyss. 5.3 Ierdeedece bewee arameers I our model we hae ae he followg erdeedeces bewee arameers. - We suose ha he eeses deed o he flao ad are erdeede. - he eres rae deeds o flao. - he Rs Dscou Rae ad he eres o e asses deed o he eres rae. - he bous deeds o he bes esmae bass. he followg able shows he arameers whch are chaged he scearos. Parameer Symbol Ierdeedace formula Rs Dscou Rae λ Ieres o e asses Prooro of rof reaed Acquso eeses a me Admsrao eeses a me Muao eeses a me Surreder alue a he ed of year ρ R R J CP CP a&& FA c FA δ MFA CA I FG FG δ MFG I FM S ae rae Dded rae D Bes esmae moraly q Aual flao rae I Surreder rae u FM δ MFM I 5.4 Alcao A sesy aalyss s aled o he Prof esg Embedded Value ad s crease wh four ds of roducs a edowme wh aual remums a edowme wh oe sgle 2

22 remum a erm surace wh aual remums ad a emorary mmedae lfe auy. I hs chaer all he umercal eamles are based o he daa ge he aee "Numercal alcao 3". A successe ad a smulaeous sesy aalyss are doe wh araos of arameer alue of ± 0% Successe sesy aalyss of he Prof esg Edowme wh aual remums 400 Prof esg for all scearos Each bar rereses a scearo. Frsly we cao see ay sgfca arao hs char. hs s he radoal rofle of a edowme wh aual remums. he oly scearo whch s clearly dffere from he ohers s he eese scearo. he ohers seem o be close o he referece scearo. Bu f we aalyse he NPV we hae aoher suao. 22

23 NPV NPV Elascy IRR IRR Elascy Referece scearo 462.2% Omsc eeses % Pessmsc eeses % Omsc Rs Dscou Rae % 0.00 Pessmsc Rs Dscou Rae % 0.00 Omsc rooro of rof reaed % 0.48 Pessmsc rooro of rof reaed % 0.5 Omsc moraly rae % Pessmsc moraly rae % Omsc aes % Pessmsc aes % Omsc flao % 0.37 Pessmsc flao % 0.35 Omsc surreder alue % Pessms surreder aluec % Omsc surreder rae % Pessmsc surreder rae % hs elascy of eeses meas ha for eamle a arao of 0% he eeses causes a arao of -60.6% he NPV! I hs able he eeses are clearly he arameer o whch he NPV s mos sese he case of a edowme wh aual remums. As he acquso eeses are mora he frs year ad because oly a ar of s zllmersed he dfferece s arcularly relea for he frs year. Aferwards he dfferece s smaller bu s becomes greaer hrough me. A he ed of he corac he dfferece s aga hgher because flao creases he eeses ad a he same me he aual remums remas always he same. he Rs Dscou Rae he rooro of rof reaed ad he moraly are also cosdered elasc arameers. 23

24 Edowme wh sgle remum 700 Prof esg for all scéaros he rofle of hs sgle remum edowme roduc s uusual. We do o hae ay al esme ad we hae some losses a he ed of he corac. As before we cao see ay hgh dfferece hs char. he oly scearo ha s clearly dffere from he ohers s he eese scearo. he ohers seem o be closed o he referece scearo. NPV NPV Elascy Referece scearo 2452 Omsc eeses Pessmsc eeses Omsc rooro of rof reaed Pessmsc rooro of rof reaed Omsc flao Pessmsc flao Omsc surreder alue Pessms surreder aluec Omsc Rs Dscou Rae Pessmsc Rs Dscou Rae Omsc aes Pessmsc aes Omsc moraly rae Pessmsc moraly rae Omsc surreder rae Pessmsc surreder rae I hs case we see ha o arameer ca be cosdered elasc. hs s because he NPV s ery hgh he referece scearo. Furhermore he rofle of he sauory eargs are for eamle o sese o he because we hae a hgh ga a he begg ad less hgh amous aferwards. 24

25 erm surace wh aual remums 000 Prof esg for all scéaros I hs case we hae aga a radoal rofle of sauory eargs. he eese scearo s aga he mos flueal. NPV NPV Elascy IRR IRR Elascy Referece scearo 49 2.% Omsc eeses % Pessmsc eeses % Omsc Rs Dscou Rae % 0.00 Pessmsc Rs Dscou Rae % 0.00 Omsc rooro of rof reaed % 0.5 Pessmsc rooro of rof reaed % 0.54 Omsc moraly rae % Pessmsc moraly rae % Omsc flao % Pessmsc flao % Omsc aes % Pessmsc aes % Omsc surreder rae % Pessmsc surreder rae % Omsc surreder alue % Pessms surreder aluec % We hae eacly he same elasc arameers ha he edowme roduc wh aual remums. 25

26 emorary mmedae lfe auy Prof esg for all scéaros he rofle of sauory eargs s oce aga dffere. A hgh ga he frs year follows by a seres of losses ad he a seres of gas. hs s rcally due o he bous whch decreases. he eese scearo s always he remarable oe. NPV NPV Elascy Referece scearo 643 Omsc rooro of rof reaed Pessmsc rooro of rof reaed Omsc Rs Dscou Rae Pessmsc Rs Dscou Rae Omsc flao Pessmsc flao Omsc eeses Pessmsc eeses Omsc aes Pessmsc aes Omsc moraly rae Pessmsc moraly rae I hs case he rooro of rof reaed ad he Rs Dscou Rae are elasc arameer. We oce ha here he eeses scearo s o he mos flueal. hs s because he admsrao eeses are lower ad here s o muao eese because surreder s mossble. 26

27 5.4.2 Smulaeous sesy aalyss of he Prof esg By dog he smulaeous sesy aalyss we oba araos aromaely 2 or 3 mes hgher ha he successe aalyss. he Ne Prese Value could ee become egae Successe ad smulaeous sesy aalyss of Embedded Value ad s crease We cosder ow a rof cere whch sells he frs year a orfolo of decal coracs. Aferwards he rof cere sells each year a ew orfolo of he same roduc wh a olume creasg by 5% a. We cosder ha here are o sochasc flucuaos radomess ad ha we do o chage he assumos. So he frs year s he year where he rof cere lauches he roduc ad he ffh year corresods o he suao whch he rof cere s afer fe years of sellg he roduc wh he assumos of he ge scearo. We oce ha all he dffere scearos derge from he referece scearo. Each year he ew coracs sold acceuae hs dfferece. he followg geeral cocluso ca be made o he successe ad smulaeous aalyss: he flueal arameers Prof esg are also flueal o he Embedded Value ad s crease. No ew arameer s cosdered as elasc. I he essmsc scearo of he smulaeous aalyss he Embedded Value decreases. 5.5 Cocluso o he sesy aalyss hrough hs aalyss we oce hree flueal arameers: he eeses he Rs Dscou Rae ad he rooro of rof reaed. here are flue o as well as he Ne Prese Value ad as he Embedded Value ad s crease. I arcular he roduc wh aual remums ad roduc wh oe sgle remum do o hae he same sesy o he arameers. he sesy of he roduc wh aual remums s always hgher ha he roduc wh oe sgle remum. hs s due o he rofle of sauory eargs. Prof esg of a aual remums roduc Prof esg of a oe sgle remum roduc

28 Wh aual remums we hae a loss he frs year follows by a seres of gas. So hs rofle s more rsy ha he oe of he sgle remum where we hae a ga he frs year followed by a seres of rofs. For eamle he Rs Dscou Rae s of course more flue o such a rofle because he gas come he fuure o recoer he frs year loss. he same cosderao s aalable for he rooro of rof reaed. hs s more surrsg for he moraly. We may hae hough ha he roduc a erm surace whch sured oly he rs of deah or a edowme hae more fluece o he resul of he aluao mehod ha he way of ayg he remums. Bu fac hs s logcal. hs rs ae by he comay s hgher wh aual remums roduc ha wh sgle remum roduc ad hs s clear he char of he sauory eargs of course wh a whole lfe auy he sesy could be hgh oo. Fally he fluece of he arameers deeds esecally abou he rofle of he sauory eargs. Oe could hae hough ha he eres rae s a flue arameer. We are a deermsc model where o sochasc flucuaos es ad our model here s erdeedece bewee he eres rae he flao ad he Rs Dscou Rae whch reduces he effec of chagg he eres rae. hs aalyss shows ha he aluao mehods are sese o he chages he arameer alue. hs s arcularly clear he smulaeous aalyss. A modfcao of 0% of he alue of arameers he smulaeous aalyss could comleely chage he ew ha we hae of a roduc of a rof cere or ee of a comay. he sesy s so hgh ha we ca doub abou he relably o use he Prof esg as a rcg ool! Furhermore hs aalyss we hae oly aalysed he flucuaos due o he uceray of he arameer alue. Ceraly ag ogeher he uceray he radomess ad he error of he model o accou would acceuae he roblem of relably of he mehod. A sudy shows ha he flucuaos due o he radomess are hghly relea. For eamle o smulae he eres rae sead of usg a mea eres rae ad o smulae also he umber of deah ad surreders ges he followg char see Veraguh [23]. Prof esg 0.0% 8.0% 6.0% 4.0% 2.0% 0.0% -2.0% % -6.0% -8.0% -0.0% 28

29 hs s a eamle of a radoal edowme. We suose ha he eres rae follows a logormal law ad ha he umber of deah ad surreders follow boh a Beroull law. I bold he mddle we see he cure of he Prof esg calculaed wh he mea alue of he arameers. Wh hs char we really see ha ee f we ow ery well he law of he ees he realy s comleely dffere from he mea. hs s he effec of he radomess. he oucomes are comleely dffere from he eeced alue. 6 Cocluso We hae see wha are he aluao mehods. he ma adaages of he aluao mehods are o be mahemacally smle a m of acuaral ad accoug oos he sual rereseao seems easy o be udersood he aluao mehods ae accou of he fuure rofs ad hey allow comarg facal resuls of dffere ds of aces. We hae see ha he aluao mehods deed o a lo of assumos. I meas ha whou ublshg he assumos made order o do he calculao he resul of a aluao mehod has o real sese. We hae see ha hese mehods are o always easy o udersad. I order o aod a erreao roblem we hae o use always he correc coe. We hae also see ha araos due o he uceray are relea. I meas ha usg for eamle he Prof esg as a rcg ool o oba he mos comee remum by reducg he secury margs s ery dagerous. Of course he radomess ad he model error we soe abou before acceuae he o-relably of he aluao mehods used as rcg ools order o ge he mos comee remum. he aluao mehods show more he rofably ad he sauory eargs show more he solecy. Bu we hae o be aware ha correcly usg hese aluao mehods s o smle care abou he assumos ad awareess of he ossble flucuaos are ecessary. 29

30 7 Aee 7. Basc acuaral formulae dscou facor robably ha a lfe aged wll sure bewee age ad rcg bass q robably ha a lfe aged wll de bewee age ad rcg bass q robably ha a lfe aged wll de bewee age ad bes esmae bass u robably o surreder bewee age ad bes esmae bass u q robably for a led aged o say he orfolo a he age bes esmae bass 2... age ad bes esmae bass robably ha a lfe aged wll sure bewee A Ne rese alue of he ayme of f he sured s ale a he ed of years ad also of he ayme of u a he ed of he year of deah f deah occurs wh years edowme surace A q 0 see Gerber [3] ä Ne rese alue of aual aymes of u durg years as log as he beefcary les A -years emorary lfe auy-due. ä 0 see Gerber [3] 30

31 7.2 Numercal alcao hs aee shows he daa used chaers 2 ad 3. Beefs ad daa o he assured A male aged 30 buys a edowme for durao of 20 years. A creasg bous s ad each year as a deduco from he gross remum. I case of surreder he assured recees 90% of he Zllmersed resere. Prcg bass Acquso eeses a he begg of he frs year: α 4 % o he sum assured Colleco eeses begg of year: β 5 % o he gross remum Admsrao eeses begg of year: γ 03 % o he sum assured echcal rae: 4 % Moraly able: SM 978/83 Acuaral formula of a edowme: P ä A α β P ä γ ä P Bes esmae bass Bes esmae bass are symbolsed by Colleco coss begg of he year: β % o he gross remum Acquso coss begg of he frs year: α 38% o he sum assured Admsrao coss begg of he year: γ 02% o he sum assured Eeced reur rae o he asses: 6% Bes esmae moraly rae: q 85% q of he rcg moraly calculaed wh he moraly of he able: SM 978/83 Probably of surreder u Surreder beef V s he Zllmersed resere he frs hree years he followg years S ad a he ed of year s as follows S 90 V where % 3

32 0 f V 0 V α A P ä ä oherwse ä where P e remum he bous s creasg as follows PE I s ad aually deduco of he gross remum a he begg of he year from he secod remum. Oher assumos A orfolo of 0000 les s cosdered he reseres used are he reseres wh he admsrao eeses he al e asses equal o 350 Rs Dscou Rae 8 % a rae 0% reur rae o he e asses 5% 7.3 Numercal eamle 2 he chars ad umercal alcaos showed he chaer 4 come from a uque eamle whch s a edowme wh he followg secfcaos. Beefs ad dae o he assured A male aged 40 buys a edowme for durao of 25 years. A creasg bous s ad each as a deduco from he gross remum. I case of surreder he assured recees 90% of he Zllmersed resere. Prcg bass Sum assured: C 00'000 Acquso eeses a he begg of he frs year: α 5 % o he sum assured Admsrao eeses begg of year: γ 04 % o he sum assured Colleco eeses : β cluded γ echcal rae: 4 % 32

33 Moraly able: SM 978/83 Acuaral formula of a edowme: CP' ' ä C A α C γ C ä where A &&a 557 he he gross remum s : P CP where P gross remum for a sum assured of Bes esmae bass Bes esmae bass are symbolsed by Acquso coss begg of he frs year fed coss commsso o he ages 5974 Admsrao coss begg of he year 80 Surreder coss begg of he year Eeced reur rae o he asses: 6 % Bes esmae moraly rae: q 85% q Surreder robably u 02 MAX ; Surreder beef S ad a he ed of year s as follows S 90 % V f where V V >0 oherwse S 0 Zllmersed resere V 05 α ä C A P ä ä where P e remum he bous PE s ad aually deduco of he gross remum. he bous creased yearly ad s ad from he hrd remum. We use he followg formula where J rereses he yearly bous erceage of he gross remums R J CP CP CP a&& where b a&& b Ia&& j j 2 j j P gross remum calculaed wh he bes esmae bass 33

34 symbolsed he bes esmae bass j s he umber of remums from whch he bous s ad We choose b 4 ad b 2 3 ad we hae hrd remum deduced by: b b J 2 fourh remum deduced by: b 2b J ec. 2 Oher assumos We cosderer a orfolo of decal olces beefs ad eeses We use he Zllmersed reseres Zllmersed reseres : 50% of he acquso eeses are Zllmersed Ial e asses equal o 4000 Rs Dscou Rae 8 % a rae 0% reur rae o he e asses 5% R rooro of rof reaed rofable case 3% R rooro of rof reaed o rofable case 5% D dded rae erceage of he dsrbuable rofs 0% 7.4 Numercal alcao 3 hs aee shows he daa used chaer 5. Four ds of roduc are used a edowme wh aual remums a edowme wh sgle remum a erm surace wh aual remums ad a mmedae emorary auy-due. he assured are male aged 40 ad he durao of he corac s 25 years Le us sar wh he daa of he edowme wh aual remums ad he we wll oly dcae for he oher ds of roduc he dffereces comared wh he edowme wh aual remums. Edowme wh aual remums 34

35 Beefs Sum assured equals o A bous s ad aually deduco of he gross remum a he begg of he year from he secod remum. Surreder beef S ad a he ed of year s as follows S V 90 % f V >0 oherwse S 0. Prcg bass Acquso eeses a he begg of he frs year: α 5 % o he sum assured Admsrao eeses begg of year: γ 04 % o he sum assured Colleco eeses : β cluded γ echcal rae: 4 % Moraly able: SM 988/83 Acuaral formula of a edowme: CP ä C A α C γ ä where A ä 5633 he he gross remum s : : P for a sum assured of ad CP for he sum assured of Bes esmae bass Bes esmae bass are symbolsed by Acquso coss begg of he frs year: fed coss MFA 000 ad commsso o he ages CA 38% of he sum assured Admsrao coss begg of he year MFG 250 Surreder cos begg of he year MFM 50 Eeced reur o he e asses : 6 % Bes esmae moraly rae: q 85% q Surreder robably u 02 MAX ; Surreder beef S S ad a he ed of year s as follows 90 % V f V >0 oherwse S 0 35

36 " where : V Zllmersed resere V 05 α ä C A P ä ä where P e remums he bous PE s ad aually deduco of he gross remum. he bous creased yearly ad s ad from he hrd remum. We use he followg formula where J rereses he yearly bous erceage of he gross remums R J CP CP CP a&& where b a&& b Ia&& j j 2 j j P gross remum calculaed wh he bes esmae bass symbolsed he bes esmae bass j s he umber of remums from whch he bous s ad We choose b ad b 2 5 ad we hae hrd remum deduced by: b b J Oher assumos 2 fourh remum deduced by: b 2b J ec. 2 We cosderer a orfolo wh decal olces beefs ad eeses We use he Zllmersed reseres Zllmersed reseres : 50% of he acquso eeses are Zllmersed Ial e asses equal o 4000 he erdeedece bewee he arameers are ge he umercal alcao Rs Dscou Rae 8 % a rae 0% reur rae o he e asses 5% R rooro of rof reaed rofable case 7% I aual flao rae 2% D dded rae erceage of he dsrbuable rofs 0% 36

37 Le us see ow he oher ds of rodus. We oly dcae he arameers whch do o hae he same alue ha he edowme wh aual remums. Edowme wh sgle remum Acquso coss begg of he frs year: fed coss MFA 54 ad commsso o he ages CA 48% of he sum assured he bous s calculaed erceage of he sum assured ad creased wh he followg facors b 5 ad b2 erm surace wh aual remums R rooro of rof reaed rofable case 80% Acquso coss begg of he frs year: fed coss MFA 3054 ad commsso o he ages CA 8% of he sum assured he bous creased wh he followg facors b 0 ad b2 emorary mmedae lfe auy-due Acquso eeses a he begg of he frs year: α 5 % o he gross remum Admsrao eeses begg of year: γ 2 % o he aual auy ad R rooro of rof reaed rofable case 3% Bes esmae moraly rae: q 20% q Acquso coss begg of he frs year: fed coss MFA 04 ad commsso o he ages CA 48% of he gross sgle remum Admsrao coss begg of he year MFG 80 No surreder cos because o surreder s mossble hs roduc he bous s calculaed erceage of he aual auy ad ad decreased wh he followg facors b 25 ad b2 -. I begs a he ed of he secod surace year. 7.5 Bous formula I he umercal alcao 2 ad 3 we use a bous formula whch s elaed more deal ow. he e rese alue of he fuure bous ad s as follows: 37

38 R P P C a& R rooro of rof reaed age of he assured a he begg of he corac durao of he corac symbolsed he bes esmae bass I hs formula P P C rerese he dfferece bewee he gross remum ad he remum calculaed wh he bes esmae bass. By mullyg by - R we ge he ar ge o he assured ad usg he e rese alue a&& o dscou we ge he e rese alue a he begg of he corac of he fuure bous. We wa ow o dsrbue hese bous durg he whole durao of he corac ad eress he bous erceage of a referece called characersc as for eamle he remum ec. J R c CP CP a& where J bous rae fuco of age a he ceo ad durao e rese alue rereseg he way of dsrbug he bous scheme for u c characersc. Value o whch we wa o aly he bous rae for eamle remum sum assured ec Now we ca solae he bous rae. J c CP CP R a&& Le us see more deal hs formula he case of he edowme wh aual remums. A a&& R J α γ C CP c a&& a&& where α acquso eeses γ admsrao eeses I hs formula we hae a& A 0 q 0 38

39 39 ] [ Z A a P && where C q A 0 { { o he ages Commsso cos Ial CA IC he admsrao coss Ne rese alue of 0 FG a & & S u Z where S surreder beef a he ed of year I our model we cosder ha he bous PE s ad aually deduco of he gross remum. he bous creased yearly ad s ad from he hrd remum. So we hae hrd remum deduced by: 2 b b J fourh remum deduced by: 2 2 b b J ec. I he formula we hae : 2 j j j j a I b a b & & & & 2 j j j j j j a I E b a E b & & & & j j j j b 0 j j j j j j a b 0 2 & & where j rereses he umber of remums from whch he bous s ad

40 8 Refereces [] Bager R. M. "Valuao of a Lfe Assurace Comay for Purchase" Joural of he Isue of Acuares ol. 99 ar [2] Baumol W. J. Blder A. S. Scarh W. M. "L écoomque Prces e olques mcro-écoome" ages 5-23 Edo Eudes Vaes Moréal 990 [3] Burrows R. P. ad Fces S. W. "Measurg he Facal Performace of Isurace Comaes" Ieraoal Cogress of Acuares ol 988 [4] Burrows R. P. ad Whehead G.. "he Deermao of Lfe Offce Arasal Values" Joural of he Isue of Acuares ol 4 ar [5] Chuard M. "Wer- ud Erfolgsmessug der Lebeserscherug" DGVM-Semar Leggres o ublé [6] Chuard P. "Aalyse du résula e réaro de l'ecéde das l'assurace sur la e" Isu de sceces acuarelles de l'uersé de Lausae 988 [7] Colls S. A. "Coug he Wrog Beas" he Acuaral Dges Volume 6 No. Feb. /Mar. 997 [8] Colls S. A. ad Keeler D. J. "Aalyss of Lfe Comay Facal Performace" Preseed o: Sale I Acuaral Socey o 2d Noember 993 o ublé [9] Delau. ad Magée M. "Les oueau rodus d'assurace-e" Edos de l Uersé de Bruelles Colleco Acuara 99 [0] Faga C. "Prof Recogo Reorg ad Aalyss Lfe Assurace" Preseed o Sale I Acuaral Socey; Lodo Jauary o ublé [] Garder A. M. "G.A.A.P. Accoug for Lfe Offces" rasacos of he 2s Ieraoal Cogress of Acuares ol [2] Geddes. "Recogo of Lfe Assurace Profs - he Embedded Value Aroach" Reor by Isue of Acuares Worg Pary Oe day semar - he Albay oel Brmgham 0 July 990 o ublé [3] Gerber. U. "Lfe Isurace Mahemacs" Srger-Verlag 990 [4] are D. J. P. ad McCucheo J. J. "A roduco o Prof-esg" ublcao of he Uersy ero-wa Edburgh U. K. 99 [5] auser. "Prof esg o Lebeserscherugsrodue" Verscherugswrschaf ef 8/993 [6] Kameec J. "Ealuer ue comage d'assuraces" L'argus ju

41 [7] Lahme D. "Erragsweraalyse-Wadel der arfewclug der deusche Lebeserscherug" Verscherugswrschaf ef 2/994 [8] Leasseur M. ad Quar A. "Face" Edo écoomca Pars 990 [9] Merda G. A. "Value-Based Facal Measureme" e Verzeergs-Archef LXX [20] Sharo A. C. "Moder Cororae Face" ages Busess & Ecoomcs New Yor 99 [2] Smar I. C. "Prcg ad Profably a Lfe Offce" Joural of he Isue of Acuares ol 04 ar [22] Smh P. R. "More Iformae Dsclosure of Lfe Offce Affars" Joural of he Isue of Acuares ol 99 ar [23] Veraguh C. "Eude de la arablé des méhodes d éaluao de l assurace e das u coee sochasque" hèse réseée à l Ecole des aues Eudes Commercales de l Uersé de Lausae 999 [24] Waso R & Sos "Lfe Prof Recogo-he Ne Sage" Waso Comme 99 o ublé 4

42 Oher refereces o meoed he e Bardola J. "Embedded Value" réseao Zurch Assuraces 996 o ublé Berle J.-P. ad Colls S. A. "Isurace Sraegy Grou : Maagg Ecoomc Value Added For Lfe ad Proery/Casualy Isurers" February o ublé Bowers N. L. Gerber. U. cma J. C. Joes D. A. Nesb C. J. "Acuaral Mahemacs" Socey of Acuares 986 Colls S. A. "Usg core slls alue added aalyss" he Newsleer of he Socey of Acuares Volume 29 No. 2 February 995 Darbellay P.-A. "Ealuao d'ue comage d'assuraces e e de ses rodus: méhodes aroche crque e aalyse de sesblé" hèse réseée à l Ecole des aues Eudes Commercales de l Uersé de Lausae 999 Edwards C. "Prof esg racce" Swss Acuares Worg Sesso Jue o ublé erz D. B. "Rs Aalyss Caal Iesme" arard Busess Reew Jauary- February 964 erz D. B. "Iesme olces ha ay off" arard Busess Reew Jauary- February 968 Rohde M. "De Erfolg m oraus besmme-kezahlegesüzes Prof esg" Producorollg ud arferug m deregulere Verscherugsmar. ud 2. Dezember 995 o ublé Köhler M. "Embedded Value" sémare Werhur Assuraces 995 o ublé Meha J. B. "Allowg for Asse Lably ad Busess Rs he Valuao of a Lfe Offce" Joural of he Isue of Acuares ol 9 ar Prauzsch A. "Shareholder alue : Réoluo ou éard moullé?" Joural Bacassurace Fé./Mars 998 Ross S. M. "Iao au robablés" Presses olyechques romades 987 Reed O. A. "Facal Reorg I he U.K." A memo Ocober o ublé Wyss R. " Wer ud Erfolgsmessug der Lebeserscherug" DGVM-Semar 2. Seember 996 o ublé 42