SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE

Transcription

1 SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE BY KEVIN DALBY THESIS for he degree of MASTER OF SCIENCE (Modeling and Daa Anali) Facul of Mahemaic and Naural Science UNIVERSITY OF OSLO Ma 2011

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3 Abrac Thi hei decribe, analze and applie he Solvenc II on life and penion inurance b uing he andard formula in he Quaniaive Impac Sud 5 (QIS5) o calculae he Solvenc Capial Requiremen (SCR). We pecificall examine he conequence for he Norwegian occupaional defined benefi cheme, primaril for he privae ecor. The andard formula in QIS5 o ome exen pecif re cenario wihou giving explici formula a he hould be exac for he applicaion. We herefore ouline exac formula for he Norwegian occupaional defined benefi cheme. We do hi boh for he ne expeced cah flow and for he reed cah flow. The laer we do b giving a mehod for calculaing he reed urvival and hazard rae funcion. We alo price he embedded inere rae guaranee uing marke conien price from he Norwegian wapion marke. We dicu bond pecificall and rediribuion of cah flow generall o improve he preciion. Uing he conrac boundar principle in Solvenc II we bae our calculaion on ha all policie are convered o paid up policie. Thi ma primaril be relevan for penion cheme in he privae ecor. However, formula for acive policie are alo given. Addiionall one would onl need he fuure rik premium for marke rik. A he end we perform a full QIS5 conequence ud for a Norwegian penion fund, uing he oulined formula and dicuing all relevan ep. To uppor hi par we have developed algorihm in Mahemaica o perform he necear calculaion. III

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5 Acknowledgemen Thank o m upervior, Pål Lillevold, for preparing a ubjec of curren opic, for providing real life daa for a penion fund, and for he helpful guidance hroughou he projec. The pracical aignmen of he hei made i poible o gain realiic inigh ino acuarial ubjec. Thank o boh m paren and paren in law for pending a lo of ime wih m wo-ear old on during he final monh of he projec. Thank o m moher for helping wih proof-reading of he ex. La bu no lea, hank o m wife for making i poible o complee hi projec and for nuring our new-born on. V

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7 Conen 1 Inroducion Solvenc II and QIS General principle (QIS5) Ouline of he hei Life Inurance Survival ime The baic model The hazard rae The Gomperz-Makeham hazard rae model Benefi and ne ingle premium Ne expeced cah flow Biomeric rik Inere Rae Yield curve Zero-coupon bond and forward price Inere rae eniivi Rediribuion of cah flow Swapion Counerpar rik Spread rik Concenraion rik Counerpar defaul rik The Norwegian legilaion Ae and liabiliie Profi haring Rik pricing Solvenc II: The rucure of QIS Inroducion The Be Eimae Inurance liabiliie The inere rae guaranee VII

8 6.2.3 Fuure dicreionar benefi Expene The Solvenc Capial Requiremen The equivalen cenario approach The modular approach The Minimum Capial Requiremen The Rik Margin Own fund Financial Rik Inere rae rik Equi rik Proper rik Spread rik Currenc rik Concenraion rik Illiquidi rik Counerpar rik (module) Life Underwriing Rik Morali rik Longevi rik Diabili and Morbidi rik Lape rik Expene rik Reviion rik Caarophe rik (CAT) SLT Healh Rik: Diabili and morbidi Cae ud: QIS5 for a penion fund Overview Technical proviion Solvenc capial requiremen Life underwriing rik Financial rik The Baic Solvenc Capial Requiremen (BSCR) VIII

9 9.3.4 nbscr: The modular approach nbscr: The equivalen cenario The olvenc capial requiremen The penion fund olvenc II balance hee Concluion Appendix A Sreing urvival funcion Appendix B QIS5 correlaion marice Appendix C Mahemaica code Reference IX

10 Li of figure and able Figure 6.1: Solvenc and minimum capial requiremen Figure 6.2: Rik module in he andard formula Figure 6.3: Top level correlaion marix Table 9.1: Ae and liabiliie Table 9.2: Technical proviion ex. rik margin Figure 9.3: Zero-coupon ield curve (NOK) Figure 9.4: Forward one ear ield curve (NOK) Figure 9.5: Volaili urface for Norwegian wapion wih one ear enor Table 9.6: Inere pamen accoun Table 9.7: Equi expoure and rik Table 9.8: FX expoure from he equiie and FX rik Table 9.9: Helper ab preadhee for pread rik Table 9.10: BSCR calculaion able Table 9.11: nbscr calculaion modular approach Table 9.12: Calculaing he equivalen re cenario Table 9.13: The gro change in ne ae value under he equivalen cenario Table 9.14: nbscr under he equivalen cenario Table 9.15: Solvenc capial charge for operaional rik Table 9.16: Solvenc capial requiremen Table 9.17: Minimum capial requiremen Table 9.18: Calculaion of rik margin Table 9.19: Solvenc II balance hee Figure B.1: Top level correlaion marix Figure B.2: MarkeDown correlaion marix (lower inere rae cenario) Figure B.3: MarkeUp correlaion marix (higher inere rae cenario) Figure B.4: Equi ub module correlaion marix Figure B.5: Life module correlaion marix X

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13 1 Inroducion 1.1 Solvenc II and QIS5 The Solvenc Capial Requiremen (SCR) i he regulaor amoun of capial which inurance companie are required o hold in order o wihand unforeeen even. Under he curren regulaion, Solvenc I, hi i known a he olvenc margin. Solvenc I wa adoped b he European Parliamen and he Council in 2002 and wa a limied reform of he earlier EU Solvenc direcive. During he Solvenc I proce i wa acknowledged ha a more profound reform wa required o incorporae all apec of an underaking. A projec now known a Solvenc II wa ared o advance he horcoming. In December 2005 he fir conequence ud of he propoed olvenc regulaion wa compleed, known a he Quaniaive Impac Sud 1 (QIS1). Thi wa followed up b QIS2 in June 2006, QIS3 in Ocober 2007, QIS4 in Jul 2008, and finall QIS5 in November 2010, along wih he developing progre of he Solvenc II propoal. Implemenaion of Solvenc II follow he Lamfalu proce where a level 1 framework direcive e ou he general principle. The Solvenc II framework direcive, 2009/138/EU, wa approved b he EU- Parliamen and Council on April 22 nd 2009 (The European Parliamen and of he Council, 2009). Deailed implemening meaure are inroduced a level 2 in he Lamfalu proce, wih EIOPA 1 giving advice o he commiion on implemening meaure (e.g. EIOPA ha developed a draf of he QIS5 echnical documenaion). QIS5 hould according o he chedule be he la quaniaive impac ud before Solvenc II i implemened on Januar However, he echnical pecificaion in QIS5 are no necearil he final propoal for he level 2 implemening meaure in Solvenc II. Amendmen will be likel adoped baed on he reul from QIS5 reporing. Solvenc II ha hree pillar imilar o he Bael II banking regulaion. Pillar 1 cover he quaniaive requiremen for calculaing he echnical proviion and olvenc capial 1 European Inurance and Occupaional Penion Auhori (ranformed from CEIOPS Januar ) 1

14 requiremen. Pillar 2 cover he inernal rik managemen ubjec o upervior review. Pillar 3 enail marke dicipline eing dicloure requiremen. 1.2 General principle (QIS5) The echnical proviion and olvenc capial requiremen are baed on he ne expeced cah flow of underaking covering all ae, liabiliie and financial inrumen. Thi i a broad erm covering boh cah-in and cah-ou. The echnical proviion dicoun all cah flow on a marke conien bai. Addiionall a rik margin i included o cover he co of holding olvenc capial. Thi follow he principle of ha he echnical proviion hould cover he price an underaking would have o pa anoher underaking for auming he liabiliie. The Solvenc Capial Requiremen (SCR) i calculaed a he value-a-rik uing a one ear ime horizon and a 99.5 percen confidence level. An underaking will hen need o hold ae covering he um of he echnical proviion and he SCR. There i alo he Minimum Capial Requiremen (MCR) and he Abolue Minimum Capial Requiremen (AMCR) eing a floor on he MCR. The SCR ma calculaed in everal wa, uing (European Commiion, 2010a); a) full inernal model, b) andard formula and parial inernal model, c) andard formula wih underaking-pecific parameer, d) andard formula, and e) implificaion. The QIS5 echnical documenaion deail he andard formula and implificaion where allowed for. However, he andard formula i he principle rule. The andard formula ue a modular approach, pecifing he deail of a re cenario for each rik. A andard formula ma or ma no include an explici formula. Simplificaion ma omeime be ued if he implified formula i proporionae o he underling rik, and i i an undue burden for he underaking o perform he complee calculaion. The ep for aeing he proporionali aumpion are oulined. Inernal model ma be ued if he are approved b he naional regulaor auhoriie. 1.3 Ouline of he hei The projec aignmen i o decribe how he re cenario are deigned in QIS5 for life and penion inurance, and analze how hee are ued for calculaion of he olvenc capial 2

15 requiremen. Thi include; a) a verbal decripion, b) a mahemaical decripion, and c) a conequence ud of a penion fund. The plehora of life inurance produc i immene and we will herefore confine he dicuion o he Norwegian occupaional defined benefi cheme. Thi i relevan for he conequence ud of he penion fund. We will onl focu on he privae ecor cheme. Wih ome aleraion he dicuion and formula will alo appl o he public ecor cheme. Furhermore, ome of he dicuion ma be relevan for oher life inurance produc, and ma be compued if he ne expeced cah flow can be accouned for. We have however pecificall lef ou uni linked and defined conribuion cheme from he formula. Thee produc diverge ince he invemen rik i born b he policholder. Ae under managemen in Norwa for hee produc are alo low compared o he defined benefi cheme. Lal we will onl conider he andard formula, excep from one implificaion. In chaper 2 we ar ou wih giving he formula for calculaing he ne expeced cah flow for he Norwegian defined benefi cheme baed on he urvival model. We will alo need he reed cah flow, and hee ma be compued b uing he reed urvival funcion oulined in appendix A. In chaper 3 we inroduce he necear concep for calculaing cah flow for bond, rediribuing of cah flow, dicouning of cah flow, and handling he inere rae guaranee on a marke conien bai. Chaper 4 decribe he QIS5 formula for credi rik, inerpreed broadl. In chaper 5 we addre he Norwegian legilaion covering relevan apec of he Norwegian defined benefi cheme. Thee chaper ouline he necear acuarial and financial heor for calculaing he SCR. We proceed b decribing he rucure of QIS5 in chaper 6. We will need o calculae he echnical proviion, ince he ae in he inurance fund ma no cover he echnical proviion, and reduce he available amoun of own fund for covering he SCR. However, hi difference i echnicall no par of he SCR. We coninue b decribing boh he modular and equivalen cenario approach. In chaper 7 we decribe he financial re cenario and ue he financial heor from previou chaper o calculae he capial charge. Analogoul in chaper 8, we decribe he life underwriing hock and ue he acuarial formula (and dicouning formula) for calculaing he capial charge. Having explained and defined he necear ool, we proceed wih he pracical aignmen in chaper 9. We explain he algorihm, and dicu relevan iue and reul in relaion o he conequence ud. A la in chaper 10, we conclude b giving ome reflecion of he projec. 3

16 2 Life Inurance Life inurance ha exied for cenurie and can be raced back o Roman ime in he form of annuiie when marine inurance wa available. A conrac called an annua conied of a ream of pamen eiher for a fixed erm or for life and wa offered b hoe who old marine inurance. The earlie known guide o pricing of uch conrac i daed back o 220 AD (Cannon & Tonk, 2008). More recenl in relaive erm, he eveneenh and eigheenh cenurie proved o be an influenial period for he advancemen of life inurance cheme. The Scoih Minier Widow Fund 2, which effecivel ared March 25h , i ofen cied a he fir ucceful life inurance fund. The Church of Scoland had earlier alo made aemp o organize financial proviion for he widow and orphan of i minier in variou cheme, bu had failed due o inadequae uppor or poor organizaion (Dow, ). The Scoih Minier Widow Fund cheme le he minier chooe premium from four differen level. The premium were iniiall inveed in loan o member minier a a fixed 4 percen inere rae level 4. Thi implified reurn calculaion and enabled he fund o pa annuiie o new widow in he amoun and he orphan were able o receive ock capial. The acuarial calculaion were baed on he world fir life able conruced b Sir Edmund Halle in 1693 (Gerber, 1997), uing he Brelau aiic. Inereingl hee conrac are picall ill in place wih ome modificaion in he penion benefi cheme dicued in hi hei. The fir ecion in hi chaper inroduce he general baic urvival ime model followed b a decripion of he Gomperz-Makeham model which preumabl i he mo widel ued model among acuarie. The econd ecion pu he model in he conex of life inurance conrac b defining he ne ingle premium (and equivalenl he reerve) for each pe of inurance conrac conidered in hi hei. A decribed in he one of he hei he primar focu will be on he Norwegian privae occupaional defined benefi cheme. The hird ecion inroduce able of expeced cah flow for he remaining lifeime of a given life. 2 Church of Scoland Minier and Scoih Univeri Profeor Widow Fund 3 Approved b he Aembl in Ma Compulor loan o member wa ended in 1778, having proved o be hurful o member and he Fund 4

17 Thi i an alernaive approximaion o he coninuoul dicouned cah flow. However, we defer elaboraing on he mapping and dicouning mechanim for hi approach unil chaper 3. Finall in ecion four we will commen on underwriing rik and briefl dicu recen reearch ha model biomeric rik explicil o allow for an acual Value-a-Rik model in conra o he cenario baed approach in andard formula in QIS Survival ime In life inurance he underwrier agree o make a ingle pamen, or a ream of pamen, coningen on predeermined life even unfolding (or no) in relaion o he inured peron. Survival ime model are ued o model probabiliie of hee even occurring over ime and herefore have a naural repreenaion a a ochaic proce, ee e.g. (Aalen, Borgan, & Gjeing, 2010). We will mol refrain from hi approach alhough ecion four make ome reference o hi approach. In hi ecion urvival i ued a a generic erm which can have wo meaning, a) ha a life ha no deceaed, or b) ha a life i no diabled The baic model We look a a life of age x (ear) a a aring poin from a ubpopulaion (i.e. he populaion i ubdivided ino male and female). Le T repreen he fuure lifeime of he individual, o ha x + T repreen he ime of deah (or diabili). T i unknown in advance and we aume ha i i a ochaic variable wih a cumulaive deni funcion (2.1) which give he probabili of no urviving unil ime 0. The acuarial noaion for hi i (2.2), while he probabili of urviving pa ime 0 i denoed b (2.3). The laer i known a he urvival funcion. PT, 0 G (2.1) q x G (2.2) p x 1 G (2.3) In he cae of morali rae, he urvival funcion will end o go o zero a approache infini (or approximael zero for value greaer han below). In conra, a urvival 5

18 6 funcion for anoher pe of even ma converge o a value wihin he uni inerval a he full populaion ma no experience he even under conideraion (e.g. becoming diabled). Implicil G() i condiional on he individual having urvived pa age x ince G(0) 0 b conrucion. The one ear deah and urvival probabili, 1 q x and 1 p x, are ofen denoed b q x and p x repecivel. A life able, which i alo called a morali able, i a equence of oneear deah probabiliie q 0, q 1,, q, where i an old age ha i almo unaainable, e.g. 120 ear no being fuuriic abou advance in medicine. The complee diribuion of p x and q x for {1,2,3 } can be calculaed recurivel b he formula below uing an arbirar life able a a bai:, 1,2, p p p p p x x x x 2,3, if 1 if x x q p q q q x x x Ideniie (2.4) (2.6) will alo be ueful, ee (Gerber, 1997) for deail: x x x p p G G T P T p 1 1 (2.4) x x x x p q q G G G T P T q 1 (2.5) x x x q p G G G G T P q 1 1 (2.6) The hazard rae The urvival funcion give he uncondiional probabiliie for he even occurring afer ime 0. A relaed funcion i he hazard rae 5 implicil auming ha T ha a probabili deni funcion (i.e. G () 0 exi for all x 0 and 0). In conra o he urvival funcion, he hazard rae i condiional on he even no happening before ime 0. The produc of hazard 5 (Aalen, Borgan, & Gjeing, 2010) ue he erm he hazard rae for (2.4) and cumulaive hazard rae for he inegral expreion in (2.6).

19 rae and an infinieimal ime inerval, ield he inananeou probabili of an even happening over he nex infinieimal period. The hazard rae i formall defined in (2.4). 1 G 1 G 1 1 G' x lim P T T lim (2.4) G 1 G The relaionhip beween he wo funcion become more apparen afer rewriing he righ hand expreion of (2.4) a he derivaive of he inegraed expreion, which i hown below. G' d d x ln1 G lnpx (2.5) 1 G d d Finall olving for he urvival funcion in (2.5) ield (2.6) which eablih he well-known relaionhip beween he urvival and he hazard rae. p x exp x d (2.6) 0 A fir glance i ma eem onl complicaing inroducing he hazard rae, bu in urvival anali i i more common o eimae he hazard rae (or more ofen he cumulaive hazard rae) raher he urvival funcion direcl from empirical daa. A comprehenive decripion of model and aiical mehod can be found in (Aalen, Borgan, & Gjeing, 2010). We will briefl conider he Gomperz-Makeham hazard rae funcion which will be ued laer in he quaniaive par The Gomperz-Makeham hazard rae model The model wa in par developed b Gomperz who poulaed ha he hazard rae would grow exponeniall a a funcion of age, illuraed b he econd par of he righ hand ide in (2.7). In a morali model hi ha he inuiive inerpreaion of an average ageing facor. Makeham laer generalized he model b adding a conan o he hazard rae which i he fir par of he righ hand in (2.7). The conan rie o capure facor independen of age, e.g. like acciden. Pu ogeher hi ield he Gomperz-Makeham hazard rae model. x x c (2.7) The cumulaive hazard rae of Gomperz-Makeham can eail be calculaed a below. 7

20 0 x d c 0 x d c x c 1/ ln c Finall, he urvival funcion can be found b inering he cumulaive hazard rae ino (2.6). p x c / lnc x exp c 1 (2.8) The Gomperz-Makeham model uuall give a reaonable approximaion o morali rae, which preumabl i wh i i quie popular conidering i implici. However, i normall give an inadequae decripion of morali rae for older age, ielding oo high morali rae (Bølviken & Moe, 2008). In pie of hi, he ne calculaion bai for he Norwegian collecive defined benefi cheme (K2005), i baed on hi model among oher. 2.2 Benefi and ne ingle premium A ne premium for an inurance polic i calculaed in uch a wa ha he difference beween expeced preen value of he benefi and he expeced preen value of he ne premium are zero. Thi i known a he equivalence principle. Generall, he premium paid b he polic holder have loading compenaing for underwriing rik, operaing expene and profi, which o ome degree are dicreionar facor o individual inurance companie. The ne premium i a pure noion and exclude all loading and hould herefore be relevan for all underaking on a ne bai. A policholder picall agree o pa premium on a periodic bai e forward in he inurance polic erm, commonl on an annuall, quarerl or monhl bai. If he erm on he oher hand e forward a ingle premium, he premium i full paid up fron. Thu, when enering he inurance polic, he ne ingle premium and he preen value of he ne ingle premium i equal and furhermore alo equal o he expeced value of he benefi. Thi relaion i ueful, and we will ue he ne ingle premium approach o value he conracual obligaion of an underaking. In a full funded penion cheme he echnical reerve need a a minimum o cover he ne ingle premium for he earned benefi a all ime. In hi ecion we aim o decribe baic elemen of he worker benefi and la ou he formula for calculaing he ne ingle premium in he curren Norwegian em for occupaional collecive defined benefi cheme primaril for he privae ecor. However, we 8

21 noe ha he em will undergo ignifican change over he nex couple of ear in repone o he Norwegian ae penion reform which wa implemened on he 1 of Januar The ae penion reform i in principle an adapion o a defined conribuion cheme where 18.1 percen of a axpaer alar up o a limi i accumulaed ino a axpaer premium reerve each ear. The reiremen age i flexible wihin a pree range and i i alo poible o reire/work par ime. The bigge change i, however, he adjumen facor for life expecanc which ranfer a large par of he longevi rik from he axpaer ono he beneficiarie. The Governmen had earlier noiced ha a ignifican par of he worker in he privae ecor wa onl covered b he ae penion cheme, and herefore enaced a law which came ino force in 2006 requiring all emploer o run a penion cheme (OTP 6 ) a par of he emploee compenaion. The minimum required level i quie mode requiring onl a 2% conribuion of he alar ino a defined conribuion cheme. Alhough here i a endenc of emploer hifing oward defined conribuion cheme, he collecive defined benefi cheme are ill he mo widepread penion cheme in Norwa. The majori of he emploer wih penion cheme eablihed before OTP, have a defined benefi cheme and all worker in he public ecor are in principle covered b a defined benefi cheme. There are baicall four differen componen in he Norwegian collecive defined benefi model: reiremen penion afer urning 67 ear unil deah, diabili penion unil reiremen, widow penion and orphan penion unil he age of 18 and/or 21 ear which are all life annuiie paing a cerain fixed amoun on a regular bai (labeled ne benefi in he nex paragraph). A an addiional componen, Widow penion ma alo exend o regiered parner. I i alo quie common o include a whole life inurance polic. We diregard hi alo ince onl life inurance companie can be licened o offer hee conrac. Gro benefi in a penion cheme are defined a a percenage of he alar and are equal for all member wihin a penion cheme. The gro benefi include expeced ocial ecuri benefi under he prevailing ocial ecuri em. Thi i, however, uncerain due o he rik of poible legilaive reform or poible variaion of an aumed income pah. The ne benefi are impl he difference beween hee erm and coniue he acual obligaion of 6 Obligaorik jeneepenjon 9

22 an underaking for he privae ecor ince i i illegal o guaranee gro benefi 7. We will no dicu he formula defining he ocial ecuri benefi. Thee are omewha ediou and don involve acuarial calculaion. Secondl he gro and ne benefi have no been full compaible afer he ae penion reform came ino effec, and in pracice awaiing legilaive change for he privae collecive penion cheme. Thirdl a alread dicued inurance policie for he privae ecor onl cover ne benefi. In he coninuaion we aume ha he ne benefi readil available and ha hee are paid ou a a fixed coninuou pamen ream. Conequenl, we impl need o conider he cae of paing one uni coninuoul per ear and cale hee appropriael ince he ne ingle premium will be proporional o he ne benefi. We alo informall inroduce he dicouning funcion d n which we define a he forward price of a zero coupon bond from ime n and mauring a ime (n + ). We will dicu hi concep more in chaper 3. In he meanime we aume a fla inere rae erm rucure repreening he he echnical reerve rae, i. and he uual dicouning funcion d n = (1 + i) -. Below we follow he noaion ued in (Lillevold & Parner AS, 2010) wih ome ligh modificaion in addiion o inroducing he dicouning funcion above. E i he ne ingle premium for he relevan benefi wih a coninuou pamen ream of one uni. We define n a he number of ear unil reiremen and le i equal zero for he reired member. i inroduced in ecion (2.1.1), p i he morali urvival funcion and repreen he age of he inured individual. Now, uing he equivalence principle and inering he dicouning funcion which lead o (2.9), he ne ingle premium for a reiremen benefi can be calculaed raighforward b and mulipling he ne reiremen benefi b (2.9) (he laer i no hown). The morali funcion i implicil alo dependen on he gender of he inured. n n 0 n n E d0 p d p d d p d (2.9) n 0 0 n We now hif aenion o he widow penion righ which i paid ou in he even of he inured paing awa for he remaindering of he widow life ime. The ne ingle premium for hi benefi i hown in (2.10) and ma no be full inuiive. The fir expreion i fairl raighforward noing ha p + i he marginal probabili for he inured paing awa 7 In conra o hi he public ecor penion cheme i baed on gro benefi. 10

23 in ear 8, and imulaneoul defining K a he forward preen value of widow benefi in ear condiional on he inured paing awa exacl in ear. E K 0 d 0 p g( ) (( ) f ( )) 0 K d d p ( ) f ( ),2 d (2.10) The econd expreion in (2.10) however require ome more explanaion. The Norwegian archepe for he collecive defined benefi cheme for widow penion i baed on populaion demographic. Orphan penion i reaed imilarl below. No informaion abou marial au i acuall required. The funcion g( + ) i impl he probabili ha an individual of age + i married, and i condiional on he gender. Furhermore, f( + ) repreen he average age difference for a married couple where he inured i + ear old, alo condiional on he gender. Conequenl, he age of he inured poue i on average + f( + ) ear old. Finall, we need he poue urvival funcion. Thi i impl he urvival funcion for he inured oppoie ex condiional on he poue being alive when he inured pae awa. The la par follow ince g( + ) alread accoun for he poibili ha he poue ma have paed awa earlier. If he benefi exend o a parner hi i reaed imilarl replacing g( +) in (2.10) wih an appropriae analogou funcion. We omi furher deail o avoid repeiion. Calculaing he ne ingle premium for orphan penion righ hould be raighforward having worked hrough (2.10). The fir expreion in (2.11) below i perfecl idenical alhough he funcion K i acuall redefined. Now, hifing o he econd expreion, k( + ), which i he average number of children, while z( + ) i he children average age when he inured i ( + ) ear. The pamen ream end when he orphan urn S BP ear, uuall 18 or 21 ear or ome combinaion. Thi i earl in life and he urvival funcion i herefore approximaed b he conan 1. 8 Which i eail obained b mulipling (2.3) b (2.4) 11

24 E K 0 d 0 p k( ) BP z( ) 0 K d d d (2.11) In (2.10) and (2.11) we have defined he ne ingle premium for he widow and orphan penion righ. Thi i dependen on he inured paing awa. We alo need o conider he iuaion where he inured in fac ha paed awa and he widow and orphan are receiving benefi. Thi i repreened b expreion (2.12) and (2.13) repecivel where he pamen ream no longer i dependen on originall inured individual having paed awa. We now regard he inured peron having hifed focu from he originall inured o he widow or he orphan repreened b (2.12) and (2.13). E 0 d0 p d (2.12) E BP 0 d0 d (2.13) Finall we need o conider diabili penion benefi. So far we have onl conidered he inured having been able o ener wo ae, eiher being alive or having deceaed. In order o calculae he diabili benefi we need o inroduce a hird ae, which i being alive bu in a diabled condiion. A a implificaion we don allow an individual o recover from hi, i.e. being diabled unil deah. We aume ha morali urvival funcion i independen of he diabili ae, which leave expreion (2.9) (2.13) unchanged. Likewie, uing he aumpion of independence beween morali and diabili, he probabili of receiving diabili penion a a cerain ime i impl he produc of he morali urvival funcion and di he probabili of being diabled, he laer being (1 p ). A diabled individual will receive benefi unil reiremen in n ear. Thi lead o expreion (2.14). E n 0 di d0 p 1 p d (2.14) 12

25 When an individual i in a diabled ae, he probabili of being diabled i alwa one uing he aumpion ha he recover rae from he diabili ae i zero. Now, replacing (1 ) (2.14) b 1 immediael give (2.15). di p E n 0 d0 p d (2.15) Expreion (2.9) (2.15) are in principle he necear formula o calculae he echnical reerve and olvenc margin capial in ubequen chaper. I ma however be more convenien o conider he ne expeced cah flow which we inroduce below. 2.3 Ne expeced cah flow A cah flow i a nominal received or paid quani a a cerain ime or period of ime. The QIS5 dicouning helper ab ue he calendar ear a a bai. We will ake hi a a aring poin and define E a he expeced cah flow in ear = 1,, for each benefi under conideraion, when he inured i of age old. We coninue uppreing he implici condiionali on ex. We will alo ue an indicaor funcion I(logical expreion) which i equal o one if a logical expreion i rue and zero oherwie. The expeced cah flow in a given ear,, can hen eail be found b inering an indicaor funcion wih an appropriae logical condiion ino (2.9) (2.15) and removing he dicouning funcion everwhere. Thi i hown in (2.16) (2.22), repecivel. Noe ha expreion (2.16), (2.17) and (2.18) ma involve forward aring benefi, conequenl i hifed appropriael. Reiremen benefi: E p I n n n 0 1 d (2.16) Widow benefi: E K 0 p g( ) K (( ) f ( )) 0 d p ( ) f ( ),2 I 1 d (2.17) 13

26 14 Orphan benefi: ) ( ) ( z BP d I k K d K p E (2.18) Widow receiving benefi: d I p E 0 1 (2.19) Orphan receiving benefi: BP d I E 0 1 (2.20) Diabili benefi: n di x x d I p p E (2.21) Diabled receiving benefi: n d I p E 0 1 (2.22) The indicaor funcion are compuaionall inefficien. I i, however, raighforward o facor hem ou b implemening hem a in he lied Mahemaic code in appendix C. 2.4 Biomeric rik Survival ime anali aim o idenif imporan covariae ha ma affec urvival ime. An example i demographical facor for morali model like age, ex, marial au, moking habi, criical dieae ha run in familie, occupaion, exerciing habi. Thi i, however, beond he cope of he hei. We aw in he laer ecion ha he marial au, number of children, and age were reaed a biomeric average condiionall on he inured age and ex. I quie likel hi approach wouldn work in a volunar inurance cheme a he individual level ince pronounced elecion effec could evolve over ime. In he long run hi could ruin an inurance cheme.

27 3 Inere Rae Alber Einein i omeime quoed having aid ha he mo powerful force in he univere i compound inere 9. If o, hi can work wo wa depending on being a credior or a borrower. Life inurance companie and penion fund aume boh role, o i i obvioul imporan o have a good underanding of he inere rae rik in order o make necear adjumen o changing inere rae regime. Aenion o hi ha increaed dramaicall he la decade along wih inere rae falling o ver low level compared o he recen hiorical andard. We will onl cover he baic par and briefl dicu ield curve, bond and forward price, modified duraion and mapping of cah flow ino inere rae verice. 3.1 Yield curve A ield curve, or inere rae erm rucure, i eeniall a e of quoaion for bond or inere rae wap of imilar credi quali in he ame currenc, bu ored b increaing mauriie. The price ma be quoed a ield. Oherwie a price ma be convered o a ield b calculaing he inernal rae of reurn over he erm o mauri. Zero-coupon bond price are of paricular inere ince he dicouned value of a cah flow can be calculaed b impl aking he produc of he cah flow and he zero coupon bond price having he ame erm o mauri. The zero-coupon bond hould be of imilar credi quali a he cah flow when he dicouned value i ued for valuaion purpoe. On he oher hand, if he purpoe i o quanif he general inere rae rik i ma be meaningful o ue a benchmark zero-coupon ield curve, hock he ield curve appropriael and calculae he price change of he zerocoupon bond wih relevan ime o mauri. The change in value reuling from general inere rae rik ma hen be calculaed a he produc of he laer and he cah flow. However, hi aume ha he relevan credi pread are unchanged. Yield curve are a heoreical concep ince hee canno acuall be oberved direcl in he marke wihou making ome aumpion. Thi i even he cae when drawing a curve rough 9 See e.g. hp://eekingalpha.com/aricle/ compound-inere-he-mo-powerful-force-in-heunivere?ource=feed 15

28 a e of poin repreening ield for bond acro differen mauriie. Forunael, erm rucure model have been developed and reearched exenivel and differen model ma have diincive virue ha are uied for pecific purpoe. An imporan objecive in he Solvenc II direcive i o calculae he marke value of ae and liabiliie in underaking imulaneoul. Life inurance involve cah flow wih ver long erm o mauri which ma well exend beond he longe mauriie quoed in he marke. Thi inroduce ome iue. Firl, more aumpion are needed o conruc and exrapolae he ield curve. Secondl, here won be a marke where he inere rae rik can be immunized perfecl. In Solvenc II exrapolaion of ield curve i done b auming ha here exi a long erm real rae and a long erm inflaion rae for each currenc. Thi ield he long erm nominal rae. The porion of he ield curve ha exend beond he mauriie in he marke are hen aumed o (mean) rever o he long erm nominal rae. Thee are pre-calculaed ield curve and will be aken a given going forward. 3.2 Zero-coupon bond and forward price We informall inroduced he dicouning funcion d n in ecion (2.2) where we defined i a he forward price of a zero coupon bond from ime n wih mauri a ime (n + ). We hould alo add ha he principal of he zero-coupon bond i implicil aumed o be 1 currenc uni which i received b he bond bearer a mauri. Thu, leing n equal zero we can find he po value or preen value of a cah flow of one currenc uni received a mauri. When n i poiive, d n i a forward price. Leing r be a zero-coupon ield curve decribed above, we can calculae d n a (3.1). d n 1 1 r n rn n n (3.1) Thi follow from he definiion of he zero-coupon ield curve, i.e. he inernal rae of reurn of a zero-coupon bond a calculaed in (3.2), and he uual arbirage argumen. If he forward price differ from (3.1) here exi an arbirage opporuni in he marke which i exploied b buing he inexpenive bond and elling he expenive bond n 1 n r 1 and n rn 1 (3.2) n d0 n d0 1

29 When he ield curve i fla, i.e. a conan, he po price equal he forward price and i independen of n a alread aumed in ecion (2.2). Thi can be derived b replacing r n and r n+ b he conan, i, in (3.1). Now, having decribed he dicouning funcion properl we can calculae he ne ingle premium o marke value in ecion (2.2) b inering (3.1) ino (2.9) (2.15). We alo inroduced he expeced cah flow, E, in ecion (2.3) and ma calculae he ne ingle premium o marke value b uing (3.3). V d E (3.3) 1 Thi i however an approximaion ha will lighl undervalue he benefi perienl (auming ha all forward inere rae are poiive). The reaon for hi i ha he ream of expeced pamen during a period i allocaed o end of a period. I i, however, raighforward o improve hi approximaion which we will dicu in (3.4) Inere rae eniivi In he financial lieraure, modified duraion i he andard ool for managing inere rae rik. We hall in addiion ue hi quani o approximae he cah flow of a bond if we onl have informaion abou he marke value and he duraion. Thi will be covered below. We will again reric he dicuion o zero-coupon bond ince a coupon or inere paing bond compuaionall can be broken down ino a erie of zero-coupon bond (which acuall doe rade in ome marke, and are in he US known a STRIPS 10 ). The modified duraion of a bond i derived b differeniaing he (po) bond price wih repec o he ield and ubequenl dividing b he (po) bond price a hown in (3.4). In he derivaion we have aumed a earl compounding a e ou in he QIS5 dicouning cah flow helper ab. Thu, he bond price eniivi o he inere rae can be calculaed a a produc of i modified duraion and marke value. 1 d 0 d d dr 0 1 d 0 d dr 1 r 1 1 d 1 d 1 r 1 r 1 0 r d 0 0 (3.4) 10 Separae Trading of Regiered Inere and Principal Securiie 17

30 3.4 Rediribuion of cah flow JPMorgan inroduced he RikMeric 11 mehodolog in 1994 which wa a Value-a-Rik mehodolog for meauring marke rik (J.P. Morgan, 1996) and included a daae covering fixed income inrumen, equiie, foreign exchange and commodiie. The inereing par in ecion i he implificaion echnique ued for handling fixed income inrumen. The RikMeric mehodolog mapped cah flow from fixed income inrumen ono foureen differen verice each repreening a cerain mauri on a ield curve ranging from 1 monh o 30 ear erm o mauri. The principle ued for rediribuion cah flow were ha; a) marke value hould be preerved, bu ignoring credi pread, (b) marke rik hould be preerved and (c) cah flow ign hould be unchanged. In relaion o (3.3) onl crieria (c) i aified. We will herefore ugge a mapping which approximael aifie a) and b). The idea i impl o pli he cah flow beween he wo neare verice. Once again we confine he dicuion o a zero-coupon bond mauring in > 1 ear. The neare verice are and auming here are verice for ever ear. In order o aif a) and b) imulaneoul we acuall need o ue hree verice in he mapping algorihm. To keep hing imple we are, however, aified if boh condiion are approximael rue. Equaion (3.5) ue principle a), while equaion (3.6) ue principle b) and ield idenical approximae mapping rule. d 0 (1 ) d d ( 1 ( 1 d d ) ( 0 d 0 d0 d ) d 1 1) 1 d d (3.5) 1 r (1 ) (3.6) 1 r 1 r 1 r 1 r 1 r 1 r 11 RikMeric i oda commercialized and merged wih MSCI in

31 r In (3.6) we have aumed ha r r. Thi hould be a fairl cloe approximaion excep poeniall in he horer end of he ield curve in period where high or low polic rae lead o invered or eep ield curve, repecivel. In (3.5) we make ue of wo approximaion. Firl, (1 r) (1 r) 1 1 for [0,1], econdl (1 r) 1 r which i a fir order Maclaurin erie. Thu, we have eablihed he following imple mapping rule: A. B. C. If of he cah flow o verex of he cah flow o verex : Allocae he cah flow o verex Allocae Allocae (3.7) Convenienl, i he duraion of he zero-coupon bond. We ma approximae a bond cah flow (no necearil a zero-coupon bond) for rik meauring purpoe b uing onl he informaion abou duraion, ear, and he marke value b formula (3.8). The cah flow ma hen be rediribued o verice b uing he mapping rule in (3.7). Marke value Cah flow (3.8) d 0 We dicued life annui produc in chaper 2 auming a coninuou pamen ream. In ecion (2.3) we defined he expeced cah flow of hee produc and formula (3.3) give he ne ingle premium marke value. We can improve he approximaion (3.3) b appling (3.7). For implici, we aume ha he urvival funcion i conan beween each verex and ha iniiaion or erminaion of a life annui onl happen a a verex. We can hen impl pli he cah flow equall beween he wo neare verice a defined b mapping algorihm (3.9). Thi follow from (3.7) ince he duraion i equal o he midpoin beween wo neighboring verice. The approximaion i le exac for he higher age. Bu he midpoin approximaion ma anwa be a ignifican improvemen ince he duraion for higher age are horened relaive o he midpoin. A. B. C. E E E 1 E 1 E E 2 2 E 2 E 2 1, 2,, 1 (3.9) 19

32 We round off he dicuion abou cah flow b referring o he echnical documenaion of RikMeric (J.P. Morgan, 1996). Thi documen conain mehod o repreen man common financial inrumen a nheical cah flow which can be ueful for repreening inere rae expoure a cah flow. 3.5 Swapion We will briefl decribe wapion a he are ueful for pricing embedded inere rae guaranee in inurance policie. We will pecificall addre he opion premium of a receiver wapion. The owner ha he righ o ener an inere rae wap, receiving he fixed leg and paing he floaing leg. The inere rae on he fixed leg i deermined b he rike of he wapion, while he inere rae on floaing leg will be deermined b he po inere rae a he wapion expire. Thu a policholder having a earl inere guaran ma be hough of a owning a erie of receiver wapion wih one ear enor enuring ha he invemen reurn each ear are no le han he rike when he porfolio (premium reerve) i inveed imilarl o he floaing leg. r( 1) L e i N d 2 F N d1 2 ln F / / 2 1 i d d 2 d1 (3.10) F 1 1d 1 The upper formula in (3.10) give he premium of a receiver wapion, having; rike level i (he echnical rae), one ear enor, principal L, expiring in ear, he rik free inere rae r, he forward one ear inere rae F, volaili, and N i he andard cumulaive normal diribuion. Thi i he Black model applied o wapion (Hull, 1993). If we aume ha r equal F, all parameer expec he volaili are known. If he price of a wapion i quoed in he marke we ma calculae he implied volaili ielding b finding he volaili ha ield a heoreical premium equal o he quoed premium. Furhermore, if we do hi for all expirie and rike (holding he one ear enor conan), we will end up wih a volaili urface (e.g. 20

33 ee figure 9.5). Eeniall, he volaili urface expree he quoed price a volailiie, uing he Black formula a a ranlaion rule. We are hen able o calculae he price of he embedded inere rae guaranee, uing he relevan one ear forward inere rae and volaili urface from he marke. The premium are calculaed for each ear uing he Black formula. Hence, he heoreical wapion premium are marke conien ince hee equal he quoed price. Uing hi approach we don need o aume ha he Black model i correc ince i i onl ued o ranlae he quoaion ino implied volailie. In fac, he marke price are ofen quoed in hi wa. We herefore refrain from dicuing he underling aumpion or deriving he Black model which ma be found in mo andard finance ex book, ee for inance (Hull, 1993). However, we will need o make ome aumpion when uing he Black model in chaper 9, in he re cenario. The implied volaili urface i readil available for he preen ield curve. Thi i no he cae for he reed ield curve. We will herefore aume ha he volaili urface i unchanged. Oher poibiliie exi, e.g. we ma cale he volailiie o preerve he abolue volailie (relaive o inere rae level). Thi ha alo i weaknee. Preumabl here are more realiic model han he Black model which ma improve hi iue. Inernal value-a-rik model ma alo be more appropriae enabling ochaic volaili model. 21

34 4 Counerpar rik For implici, we inerpre counerpar rik generall for he purpoe expoiion in hi chaper. In QIS5 he erm i limied o he counerpar rik module. The reaon for he general inerpreaion i ha boh he marke rik module and he counerpar defaul module include ome elemen of credi rik. Thi chaper eek o give an inuiive background covering he rik calculaion which are baed explicil or implicil on raing. We herefore compile he pread rik ub module, he concenraion rik ub module, and he counerpar (defaul) rik module in hi chaper. The wo previou chaper have dicued applicable heor and mehodolog, alhough no direcl conained in he QIS5 echnical documenaion. Thi chaper differ in hi repec, ince all formula are explicil aed in he echnical documenaion, and herefore he andard formula in hee cae are explici formula. Thi i no he cae for all (ub) module. Raing for each ecuri or counerpar ma be found b uing one of hree mehod; a) official credi raing from raing agencie, b) implicil b olvenc capial raio uing a QIS5 ranlaion able, and c) unraed, which reul in he lowe credi raing. 4.1 Spread rik We confine he dicuion o fixed income ecuriie. Thi include among oher corporae bond, ubordinaed deb, hbrid deb, morgage backed ecuriie, municipal bond, and governmen bond. In QIS5 he pread rik onl capure he widening of he pread. The pread are calculaed relaive o heir repecive rik free ield curve. The andard formula (4.1) ue he relaionhip in expreion (3.4) and a pecified increae in he pread for each raing cla in QIS5. i up MV duraion F (raing ) (4.1) i Spread ma alo narrow. Thi i no accouned for in (4.1). Furhermore, raing migraion i non-exien. Thi would picall be accouned for in a credi rik model which i no olel confined o pread, e.g. ee CrediMeric (J.P. Morgan, 1997). i i 22

35 4.2 Concenraion rik Concenraion rik ma ake everal form. On he aggregae level i could for inance appl o counr and ecor expoure. Anoher poibili i correlaed invemen heme. For inance, expoure o energ, maerial, indurial, agriculure and emerging marke, which have been driven b he ame economic uper-ccle he la decade. However, he concenraion rik in QIS5 i defined a he concenraion of expoure again he ame counerpar. 2 E E i i i g i (raingi) Max CT(raing i), 0 (4.2) i E i i The proporion above a hrehold level depending on each counerpar raing i caled b a concenraion facor alo depending on he raing. Thi i quared o ield a variance-like proper. Uing an aumpion of being non-correlaed, he formula aggregae all erm and ake he quare roo. The concenraion rik i hen found b mulipling previou reul b he ae applicable o he concenraion rik ub module. Thu, expoure o everal counerparie above he hrehold level will ill gain diverificaion. 4.3 Counerpar defaul rik There are wo model ued in he QIS5 counerpar defaul rik module, baed on Tpe 1 or Tpe 2 expoure. Tpe 1 i mainl credi defaul rik from rik miigaing conrac, eiher from reinurance or finance. The expoure i ofen undiverified bu he counerparie are uuall raed. Tpe 2 i he remaining expoure capured b he counerpar rik module (e.g. morgage loan and depoi if ufficienl diverified). Tpe 2 i imple o calculae uing formula (4.3), aking a 15 per cen charge of he expoure ha haven been due for more han hree monh, and 90 per cen of he expoure which have been due for more han hree monh. 15 % E 90% E po due (4.3) The counerpar defaul rik of Tpe 1 i aociaed wih more andard credi rik model, for inance CrediMeric (J.P. Morgan, 1997). However, i relie on more complicaed 23

36 aumpion. We will herefore no dicu hee in deail and will onl give he necear background for calculaing he rik. For an explanaion we refer o he conuled advice for he counerpar rik module (EIOPA, 2009). Formula (4.4) calculae he variance of he loe. The variance formula ma a fir look imilar o a normal variance formula. However, he la erm i non-andard. The j and k indexe are ued for indexing raing caegorie, while he i index i ued for accumulaing expoure wihin a raing caegor. Formula (4.5) i calculaed b eing equal o The probabili of defaul for raing caegor j i p j. 2 u jk j k v j z j (4.4) j k j u jk p j (1 p j ) pk (1 pk ) (1 )( p j pk ) p j pk and v j (1 2 ) p j (1 p j ) (2 2 ) p j (4.5) Formula (4.6) accumulae he lo given defaul (LGD) wihin each raing caegor and he quared LGD. j LGD and z LGD i i j i i 2 (4.6) Formula (4.7) calculae he lo given defaul for each counerpar. The lo given defaul i calculaed a a hare, x, of he be eimae and he rik miigaion ha i no covered b collaeral. The rik miigaion of a conrac i calculaed according o he conrac rik miigaion effec in he QIS5 re cenario. The hare x i eiher 50 percen or 90 percen. LGD i Max x Recoverable RM Collaeral 0 (4.7) i i i, To compue he capial charge for Tpe 1 rik in ecion (7.8), he andard deviaion i caled o roughl a 99.5 percen confidence level wih ome addiional afe adjumen, depending on he relaive ize of he andard deviaion o he accumulaed LGD a hown in formula (4.8). 3 if 5% LGD i, or oherwie min5, LGDi (4.8) i i 24

37 5 The Norwegian legilaion In hi chaper we will give a hor overview of he mo relevan par of Norwegian legilaion for Solvenc II. The dicuion will be limied o occupaional defined benefi cheme managed b life inurance companie or penion fund. In he expoiion we will aume ha he curren Norwegian legilaion will coninue in he preen form. However, Finanilne 12 ha propoed o modif exiing buffer fund ino a ingle buffer fund wih increaed flexibili in ligh of Solvenc II (Finanilne, 2011a), which ma impoe evere conrain for he underaking under he curren framework. We bae he dicuion on (Banklovkommijonen, 2010). The Norwegian em i in general characerized b; 1) full funded cheme, 2) linearl earned benefi b emploee, 3) he righ o ranfer paid up policie 13 4) well defined profi haring principle, 5) earl capializaion guaranee, and 6) pricing of rik. We will dicu hi in more deail below. 5.1 Ae and liabiliie The premium reerve are proviion covering he acuarial value of he benefi capialized a a fixed rae, which i called he echnical rae. The auhoriie define he maximum echnical rae, which normall ha a ignifican margin of afe, o he acual bond ield for longer ime o mauriie. The echnical rae ma no be a ingle rae a an given poin in ime, ince lower rae omeime are phaed in graduall for exiing conrac. An underaking ma however chooe o ue a lower echnical rae han he maximum rae and can even operae wih everal echnical rae. The addiional reerve i a buffer fund which can be ued o cover par or all of he earl required accrual of he premium reerve b he underaking choen echnical rae(). For hi reaon he echnical rae i alo known a an inere rae guaranee and i iued b he inurance compan or penion fund o he inured. Boh of hee fund are broken down o he individual level of each inured and ma be ranferred o anoher underaking if a polic ha been convered o a paid up polic. 12 The Norwegian regulaor auhori for banking and inurance 13 Fripolier 25

38 The premium fund can be ued for proviion ino he premium reerve. A ponor ma ue he fund o cover premium during a ear (e.g. he linearl earned benefi or due o wage increae). Anoher poible opion i o revalue he premium reerve b a change of ariff, e.g. reducing he echnical rae or increaing life expecanc or accoun for poible increaing rend in diablemen. In hi repec he fund belong o he inured, bu he ponor decide how he proviion are ued. The reiree urplu fund on he oher hand belong o he penioner and i ued o regulae benefi which oherwie will a unchanged. The inere rae guaranee i applicable o all four fund dicued in hi ecion hihero. The price adjumen fund i a buffer fund which coniue he unrealized profi for ome par of he liquid ae (uuall, lied equiie, and bond ha are no claified a hold-omauri). I i impl he difference beween he marke value and he book value of hee ae, and one ma ap from he fund ju b realizing he profi. Thu, he price adjumen fund ma be ued o cover negaive reurn, earl lawful accrual and reurn above he echnical rae. Anoher frequenl ued fund i he rik adjumen fund which work imilarl o a buffer fund, bu i claified a equi. I ma be ued o cover loe in he acuarial profi in he echnical accoun. Turning o he ae ide, regulaion require ha he ae are pli ino a collecive porfolio() and a compan porfolio. The collecive porfolio() coniue eligible ae ha mu amoun o he um of premium reerve, addiional reerve, premium fund, reiree urplu fund and he price adjumen fund (no necearil a complee li). The book reurn on he collecive porfolio() coniue he financial profi(). The difference beween acual reurn baed on marke value, excep for bond claified a hold o mauri, and he book reurn equal he change in he price adjumen fund. Bond claified a hold o mauri are amorized a book value over he ime o mauri. The accouning effec i imilar o a buffer if inere rae rie, bu i hen in reali a negaive reerve in conra o he oher buffer fund which ac a rue rik miigaion. The ponor() decide on he raegic ae allocaion, direcl or indirecl, and accordingl pa a reurn guaranee premium which we will ouch on in ecion (5.3). In conra o he collecive porfolio(), he underaking ha full conrol of he raegic ae allocaion in he compan porfolio and he invemen reurn are direcl linked o he underaking equi. For penion fund hi diviion ma be le clear a he ponor() ma 26

39 define he invemen guideline for boh porfolio alhough no necearil being equal, alo repreening he owner. 5.2 Profi haring Unil 2008 he inere rae guaranee wa paid for implicil b he paid premium and a profi haring model ielding 35 percen o he underaking and 65 percen o he clien. From 2008 hi wa changed b new legilaion o make rik pricing more ranparen. Rik i now priced explicil baed on hree pe of profi cener; 1) financial profi, 2) acuarial profi, and 3) managemen profi. An negaive profi in a given ear are covered full b he underaking own fund, while an profi are credied in full o he clien (normall he premium fund, reiree urplu fund 14 ). Thu, he clien receive an profi while he underaking bear an loe. Thi rik i priced individuall and charged each clien. The book reurn on a given collecive porfolio coniue he financial earning, while financial expene amoun o he earl accrual b he echnical rae for he applicable fund. The acuarial reul i he difference beween he expecaion and he acual oucome in a given ear wih repec o he defining biomeric rik caegorie, e.g. morali, diabili or longevi for he defined benefi cheme conidered here. Over ime premium loading (afe margin) will maerialize a acuarial profi if he acuarial expecaion come rue. Lal, he managemen profi i he difference beween charged managemen expene for he inurance cheme ubraced b acual managemen co. The dicuion above aume ha he policie are no paid up policie, and herefore have a ponor backing he cheme o which he underaking can charge inurance and rik premium. Thi i obvioul no an opion for paid up policie ince here i no ponor backing he cheme. Paid up policie are herefore on a modified profi haring model. For hee conrac 80 percen of he combined profi i credied he clien while he underaking receive he remaining 20 percen of he combined profi o compenae for he underaken rik. 14 Proviion for he addiional reerve are charged a an expene in he financial profi cener. 27

40 5.3 Rik pricing Since 2008 underaking have been obliged o price and charge he rik aociaed wih he clien a dicued in he previou ecion, auming ha a polic i no a paid up polic. The rik premium are priced a he beginning of each ear, while on he oher hand an compenaion for underaking rik on paid up polic, poeniall onl will maerialize afer he accoun for he ear are eled. Thi eemingl difference in regime i however no he full or. Sponor in he privae ecor ma erminae he defined benefi cheme 15. In hi cae he inurance policie will be convered o paid up policie and he modified profi haring model will appl o hee conrac a well. Thu, if he penion cheme become oo expenive for ponor (or for oher reaon) hi ma be a probable oucome leaving he underaking bearing he rik. We will follow hi line of reaoning in chaper 6. Generall, he price of rik depend on he available buffer which can ac a rik miigaion, in paricular dampening he effec of hor o medium erm flucuaion in marke value. In he long run rend in inere rae, equi reurn, morali rae and diabili rae ma in an cae have a ignifican effec on he underaking olvenc poiion, if rik i no priced accordingl or addiional proviion are e aide o cover he changing environmen. Thi become epeciall criical if inurance cheme are erminaed, in hee even. The price of rik hould reflec an underaking rik of covering loe from i own fund. Thi i a complex maer and will depend on fuure acion. For inance if he addiional reerve i empied in a ingle ear, hi will no be available he following ear preumabl increaing he rik of needing o cover ubequen loe wih own fund. Even wore, if he ield curve fall below he echnical rae he inere rae guaranee can onl be me b aking marke rik, realizing unrealized profi from he price adjumen fund, or uing own fund. In he laer cenario an underaking ma be inclined o increae he bond duraion in he compan porfolio o hedge again lower inere rae. 15 And a a minimum eablih a defined benefi cheme in accordance wih OTP dicued in chaper 2. 28

41 6 Solvenc II: The rucure of QIS5 Hihero we have been concerned wih explaining he analical ool from life inurance mahemaic and financial heor ha are necear ai he calculaion of SCR and MCR. In he nex hree chaper we will decribe he formal compuaion uing he andard formula in he QIS5 echnical documenaion (European Commiion, 2010a) and he annexe (European Commiion, 2010b). The documenaion cover boh life and non-life inurance. Here we will onl dicu he relevan par for Norwegian life and penion inurance, focuing on he Norwegian occupaional defined benefi cheme. I ma alo be helpful o conul he manual for compleion of he preadhee for olo underaking (EIOPA, 2010a) and he QIS5 queion and anwer (EIOPA, 2010b). The wo laer documen are, however, no par of he formal documenaion and are publihed b EIOPA. The inerpreaion of he QIS5 pecificaion i no unique, in he ene ha addiional aumpion will be necear which we will ake in due coure. Conequenl, he andard formula doe no ield a unique olvenc capial requiremen for a given underaking. Thi chaper ouline he calculaion a he aggregae level, while he wo nex chaper decribe he calculaion for each rik caegor uing he andard formula. Each rik caegor ma be inerpreed a a parial olvenc capial requiremen, which ubequenl i aggregaed o a oal olvenc requiremen. Thi i generall lower due o diverificaion effec (in he modular approach). 6.1 Inroducion Figure 6.1 illurae he general approach in Solvenc II requiremen, which i borrowed from (Finanilne, 2010). The lef ide repreen an underaking oal ae a marke value excluding inangible, unle inangible have a documened ranacion value. The righ hand ide how he echnical proviion for he inurance liabiliie, which i baed on he following Solvenc II principle (European Commiion, 2010a): Solvenc 2 require underaking o e up echnical proviion which correpond o he curren amoun underaking would have o pa if he were o ranfer heir (re)inurance obligaion immediael o anoher underaking. 29

42 The echnical proviion coni of he Be Eimae and he Rik Margin. In addiion he underaking need ae o cover he Solvenc Capial Requiremen (SCR). The Minimum Capial Requiremen (MCR) i he abolue minimal level of olvenc capial for avoiding pervaive regulaor acion. Figure 6.1: Solvenc and minimum capial requiremen The Solvenc II balance hee above doen ake accoun of which liabiliie he ae belong o. The ae ma no floa a freel on he underaking acual balance hee and Solvenc II herefore require ha SCR and MCR mu be covered b eligible own fund. Thi i briefl decribed in ecion (6.6). The re cenario for he andard SCR formula are explicil given, bu in heor he hould correpond o a 99.5 percen confidence level over a ear. In accordance wih Solvenc II principle he ma follow a couner cclical approach dampening he effec of he buine ccle which ofen reul in large flucuaion in ae value. Correpondingl, QIS5 i omewha adaped o he ae of buine ccle a he end of 2009 wih ae value having recovered onl pariall afer he downurn in Underaking ma chooe o ue inernal model, full or pariall, baed on a Value-a-Rik approach wih a 99.5 percen confidence level over a one ear horizon. However, he inernal model() mu be approved b he naional auhoriie. We noe ha an underaking reling on dnamic rik managemen 30

43 ma onl accoun for hi in inernal model, which ma encourage ome underaking o develop uch model. 6.2 The Be Eimae The conrac boundar i a concep deermining he exen of he liabiliie ha ma be incurred from exiing conrac. The conrac boundar, full or parl, i defined a he line where he underaking unilaerall can erminae he conrac or regulae premium wihou limiaion. In relaion o he Norwegian defined benefi cheme, one mu conider o which exen i i poible o change he ariff or erminae he conrac wih ponor, which in hi even will reul in paid up policie. Thi hould define he line of he conrac boundar. The be eimae concep correpond o he dicouned value of fuure expeced cah flow unil he conrac boundar. Obvioul one need o ake ino accoun expeced cah flow from he inurance liabiliie, bu he be eimae alo include fuure ervicing expene, managemen expene, and axe. On he oher hand he underaking ha he benefi of including he expeced fuure premium and cerain receivable, bu he policholder behavior need o be accouned for. The expeced cah flow hould be calculaed on a ne expeced bai excluding an afe loading. The rik margin cover hi par and andardize he afe loading o he are comparable. In addiion he cah flow hould be calculaed gro of reinurance and SPV in order o calculae he olvenc capial requiremen for defaul rik, which i alo aumed o carr over o he reference underaking when conidering he rik margin. The cah flow calculaion hould alo be compued a he policholder level and hould cover he full lifeime. Furhermore, underaking hould ake ino accoun inflaion appropriael, bu no invemen reurn. Simplificaion are allowed for, bu he echnical documen ree he proporionali aumpion repeaedl. To enure homogenei and realiic implificaion, he be eimae hould be calculaed b each line of buine. For life inurance here are ixeen line of buinee reuling from wo level of egmenaion; life inurance wih profi paricipaion, index-linked and uni-linked life inurance, oher life inurance, and acceped reinurance. The econd level of egmenaion divide hi furher ino primar rik driver; deah, urvival, diabili and morbidi, and aving conrac (he rik i born b he policholder). 31

44 The defined benefi cheme dicued in hi hei relae onl o life inurance conrac wih profi paricipaion having longevi rik a he main rik driver. Unbundling of policie beween ub egmen i no necear. Uing he above guideline life and penion underaking need o ae a lea even cah flow elemen. I.e. cah flow reuling from; a) inurance liabiliie, b) reinurance conrac, c) premium, d) ervicing expene, e) inere rae guaranee, f) fuure dicreionar benefi, and g) axe. In dicuion we will aume ha policie are paid up o we can diregard fuure premium. Neiher will we conider reinurance and axe, a our main focu i he Norwegian occupaional defined benefi cheme. We will herefore in hi ecion calculae he be eimae a he um of he dicouned value of inurance liabiliie (guaraneed benefi), he earl inere rae guaranee, fuure dicreionar benefi and ervicing expene. We dicu each elemen below Inurance liabiliie We defined he ne ingle premium and he ne expeced cah flow in chaper 2. Thu, for benefi relaed o he Norwegian occupaional defined benefi cheme we ma compue he dicouned value of he expeced cah flow raighforward. We do hi b exending formula (3.3). The andardized cah flow are caled b he policholder earl benefi for each liabili, i, from he e of available liabiliie, P. Le S be he ock of policholder, hen (6.1) i he dicouned value of he guaraneed benefi. ip ( i) V ( c, p) Y d ( c) E ( p) (6.1) 1 0 ( i) The formula obvioul applie o oher life and penion inurance conrac alo, bu he expeced cah flow need o be eimaed uing mehod aifing he guideline dicued above. The funcion V depend on everal facor, pecificall he ield curve (c) and he urvival funcion (p) ued o calculae he expeced cah flow. We condiion on hee wo explicil ince he capure he re cenario relaed o he liabiliie. We ma alo add he premium fund o he liabiliie following a preenaion b Finanilne (Finanilne, 2010). However, he premium fund ma alo be ued o change he cheme ariff, which ma increae he premium reerve fund wihou regulaing he 32

45 benefi. We herefore chooe o conider he premium fund a omehing beween premium reerve and he addiional reerve fund. The QIS5 package include he necear ield curve, which hould be ued o dicoun he cah flow. There are everal ield curve for each currenc aking differen level of illiquidi premium ino accoun. The Norwegian ield curve including he 75 percen liquidi premium i preumabl he mo relevan ield curve for mo life inurance conrac wih profi haring in Norwa. The crieria for uing a 100 percen illiquidi premium i; a) underwriing rik i onl longevi and expene rik, b) no rik of urrender, and c) i a paid up polic. Crieria c) ma qualif, bu b) conflic wih he righ o ranfer he liabiliie, and c) conflic ha wih he occupaional defined benefi cheme picall include diabili inurance and inurance relaed o deah The inere rae guaranee There are poeniall a wide range of differen policholder opion ha hould be aeed when calculaing he be eimae. However, we will onl conider he o-called inere rae guaranee dicued in chaper 5. The guaranee relae o he book reurn o he policholder each ear. We will onl conider he financial rik in hi cae. Ne income from acuarial and managemen accoun will alo affec he book reurn. Therefore, we will aume ha he are independen from financial reurn. Underaking ma value he opion uing hree mehod; a) ochaic approach, eiher cloed form or b imulaion, b) cenario baed wih aigned probabiliie, and c) a deerminiic valuaion baed on he expeced cah flow if i i marke conien. We will ue mehod c), alhough i ma alo be inerpreed a mehod a). We will price he rik inherenl from he ield curve. Deviaing ae allocaion raegie ma be priced eparael or viewed a he owner rik. In ome ene, hi i parallel o he mehodolog ued for calculaing he rik margin (where underaking onl hould include unavoidable marke rik). Auming he book reurn equal he one ear forward ield curve for each ear, one ma price he inere guaranee a a receiver wapion a dicued in chaper 3. Thee are quoed a over-he-couner conrac, bu price are available for mauriie up o 30 ear. Price 33

46 ma be quoed a implici volailiie, e.g. b uing he Black formula, ielding a volaili urface acro rike and expirie for one ear enor. We ma herefore find he relevan volaili (i.e. price) for a cerain combinaion of expirie and rike. Thi mehod i marke conien and correpond o mehod c). However, i ma alo be viewed a a ochaic model. Having obained he price in chaper 3, we onl need o calculae he principal. We do hi b uing formula (6.1) and a fla dicoun rae equal o he echnical rae, i. We alo cale he expeced cah flow accordingl o he afe loading ued for calculaing he premium reerve. We aume ha all benefi are covered b he inere rae guaranee. L n (1 loading ) ip Y ( i) n d ( echnical rae) E 0 ( i) ( p) (6.2) Now, uing formula (3.10) we can calculae he value of he inere rae guaranee a in (6.3), where d 1, d 2,r, F are given a in (3.10). 1 IRG ( c, p) L exp( r ( 1)) i N ( d2) F N ( d1) 0 (6.3) We alo ue he condiional denoaion for IRG a he forward ield curve i implied from he ield curve and he cah flow depend on he urvival funcion for he re cenario. Formula (6.3) ake ino accoun of he hape of he ield curve which i relevan for underaking conidering maching ae wih liabiliie. One hould alo noe ha realiicall here are no marke conien price for long mauriie. Bu, in reali hi i alo he cae for he exrapolaed ield curve. A long a he echnical rae i ufficienl low compared o he long erm macroeconomic aumpion ued in he exrapolaion, hi hould be a minor iue. We alo noe ha in he cae of acive policie (which we have aumed do no exi) he ponor are charged earl rik premium. The rik repreened b (6.3) hould be refleced in rik pricing charged he ponor. However, a dicued earlier, he underaking ma end up bearing he whole rik if he rik premium become expenive. Evenuall, riking ha all policie ma be convered o paid up policie. In hi conex, our aumpion ha all policie are paid up, ma no be compleel unrealiic in a re cenario. 34

47 In he nex ub ecion we dicu fuure dicreionar benefi (FDB). The value of he inere rae guaranee for he policholder will depend on hee, a he managemen ma draw from buffer fund when needed o ield he guaraneed inere rae. In hi repec he pecificaion choen above for modeling he inere rae guaranee i conien wih he principle ha he fuure dicreionar benefi hould be calculaed eparael. For inance if we price he opion uing imulaion and ake ino accoun fuure managemen acion of drawing from buffer fund, hi would be inconien wih adjuing for FBD eparael. In hi cae i would alo be necear perform imulaion holding FDB unchanged, which we inuiivel believe hould ield imilar reul o (6.3) Fuure dicreionar benefi Fuure dicreionar benefi are poenial benefi belonging o he policholder, and depend on being realized. Thi ma happen b managemen acion, e.g. b realizing profi, or indirecl hrough change in he inere rae level. We will aume ha he FDB i alwa poiive in he dicuion. Thi i no he cae for under-funded penion cheme relaive o he marke rae. For he Norwegian occupaional defined benefi cheme, he FDB i given b expreion (6.4). There are, however, oher poible buffer which we haven aken accoun. Expreion (6.4) ma hen be appropriael modified 16. FDB br PR V ( c, p) Max[ PAF ARF IRG ( c, p), 0] (1 br) ARF (6.4) The fir par i he difference beween he premium reerve and he dicouned value of he guaraneed benefi. Thi i he exce value from he bond marke when ield are higher han he echnical rae, and he value of afe loading. Over ime hi hould accumulae a profi if he ne expeced cah flow are reaonabl correc. The econd par i he unrealized profi on he collecive porfolio, and he addiional reerve which ma be ued o cover he earl accumulaion b he echnical rae. We ubrac IRG, ince he buffer ma be ued o cover he echnical rae in period where inere rae are oo low. However, if he buffer are oo low, hi miigaion i no poible. The final erm make ure ha he policholder receive he whole addiional reerve fund. 16 PR = premium reerve, PAF = price adjumen fund, ARF = addiional reerve fund, and br = bonu rae. 35

48 6.2.4 Expene We won dicu expene, bu give a implified formula for calculaing he dicouned value baed on he implificaion for expene rik in (8.5). In hi formula, following he noaion in he echnical documenaion, i i he expeced inflaion rae, n i he average number of ear he rik run off weighed b renewal expene, and E i a repreenaive expene figure. n n E 1 i d0 n n 1 i n d0 (6.5) Thi end our dicuion of he calculaion of he be eimae for life and penion inurance. The rik premium will laer be dicued briefl in order o calculae he full echnical proviion. However, for hi we will need he olvenc capial requiremen. 6.3 The Solvenc Capial Requiremen Thi i he main objecive of he hei. In hi par we decribe he aggregae level, and he overlaing rucure. Thi i necear o aggregae he capial charge from each ub module uing he andard formula. We will alo onl conider he andard formula. Simplificaion ma be poible condiional on qualifing aumpion. The andard formula i baed on a modular re cenario approach. Thi i illuraed in figure 6.2 borrowed from he QIS5 echnical documenaion (European Commiion, 2010a). The approach aken i o calculae he change in he underaking ne ae value for each ub module reuling from he re cenario. The ne ae value i he difference beween he ae and liabiliie, which i denoed b SCR i for (ub) module i 17. Thee calculaion will be addreed horoughl in chaper 7 and chaper 8, covering repecivel financial rik and life underwriing rik. For all SCR i calculaion (and nscr i calculaion below) he rik margin in he echnical proviion are lef ou o avoid circulari ince he SCR i ued o calculae he rik margin. In addiion he fuure dicreionar benefi are kep unchanged when calculaing each SCR i, i.e. he lo aborbing capaci of he echnical proviion i no accouned for. Furhermore, a poiive change in ne ae value i defined a a lo of ne ae value. 17 The echnical documenaion ue differen noaion for capial requiremen a he ub module level. In hi chaper we implif for improving he clari in he expoiion. 36

49 Figure 6.2: Rik module in he andard formula Auming ha each individual SCR i i calculaed, one ma hen aggregae hee o he module level. The module level i depiced b level hree in figure 6.2. The aggregaion ue he QIS5 correlaion marice and he formula for calculaing a porfolio andard deviaion wih all weigh equal o one, a hown in (6.6). The aggregaion i performed for each module reuling in a SCR i for each module i. For life and penion inurance, primaril marke, defaul and life rik module will be relevan. In addiion, he diabili and morbidi (SLT) ub module of he Healh module ma be necear, if conrac having hi rik a he primar rik driver, are unbundled. i j SCR SCR Corr (6.6) i j ij The Baic Solvenc Capial Requiremen (BSCR) i imilarl compued b aggregaing from module o he op level. For illuraive purpoe we have included he op level correlaion marix in figure 6.3. We noe ha here are wo inere rae cenario (higher and lower inere rae). The correlaion marix aociaed wih he marke rik module depend on which cenario i choen, i.e. he cenario reuling in he large lo. 37

50 Figure 6.3: Top level correlaion marix Formula (6.7) define he SCR. We have o far calculaed he BSCR, and need in addiion he adjumen erm and he capial charge for operaional rik. SCR BSCR Adj SCR Op (6.7) SCR Op i he olvenc capial requiremen for operaional rik. We onl conider defined benefi cheme and ae he formula for hee below: Earn TP echnical proviion SCR Op Min 0.3 BSCR, Max 0.04 earned premium he pa 12 monh pearnearned premium over he prior 12 monh period Earn Max Earn pearn,0,0.0045tp (6.8) The Adj(umen) erm i he lo aborbenc of he echnical proviion and deferred axe. Thee hould alwa be negaive. We will aume ha axe have a minimal lo aborbing effec on he dicued inurance cheme and diregard deferred axe in he expoiion. The calculaion of he adjumen erm for lo aborbenc of he echnical proviion mu follow wo mehod in QIS5. Thee are explained in ub ecion (6.3.1) and (6.3.2). The reaon for uing wo mehod i ha he poliical level ha no e decided which mehod will be adoped in Solvenc II. However, he equivalen cenario approach hould be ued for calculaion which depend on he SCR (e.g. poibl when calculaing he rik margin). Coninuing below we will (or ma) need he nscr i for each ub module. Thi i he ne SCR i aking accoun of he lo aborbing of he echnical proviion in he re cenario. The difference beween SCR i and nscr i a he ub module level i impl he reduced value of fuure dicreionar benefi reuling from he re cenario (if boh erm are ricl poiive). Underaking mu in each cenario calculae he lo aborbing effec cauioul o enure ha he lo aborbing capaci i no ued in oher cenario. Thi i o avoid poenial double couning. 38

51 6.3.1 The equivalen cenario approach Thi approach require he underaking o compue a ingle re cenario where all rik are accouned for imulaneoul. Thi cenario i called he equivalen cenario and he reducion in ne ae value for hi cenario i denoed b nbscr (alhough hi alo being he cae for he modular approach). In he calculaion of he change of echnical proviion one hould ake accoun of he relevan managemen acion ha would be applied. Having calculaed BSCR above, he adjumen erm for echnical proviion i calculaed b he formula (6.9). FDB i he oal value of fuure dicreionar benefi from he be eimae. The formula enure ha he aumed lo aborbing in he cenario i no larger han he maximum poible lo aborbing, which i equal o FDB. Adj TP min( BSCR nbscr, FDB) (6.9) The mehod for calculaing he equivalen cenario i baed on he Componen VaR pariion which indicae he change in VaR if a componen i aken awa (i.e. he rik i immunized). We hall herefore label hi he Componen BSCR where componen i, denoed b ΔSCR i, approximae he change in he BSCR if he rik reuling from ub module i i immunized. The principle rule i o ue he SCR i in he calculaion, bu here i an opion o ue he nscr i if he are more appropriae for he buine and don ield ignificanl differen reul. The Componen BSCR pariion can be compued direcl b aking he calar produc beween he vecor of he SCR i and he gradien of he BSCR. The ih pariion of he Componen BSCR i calculaed in (6.10). The feauring proper of he Componen BSCR pariion i ha he um of he pariion equal he BSCR a hown in (6.11). Corrij SCR j BSCR j SCRi SCRi SCRi (6.10) SCR BSCR i i SCR i i SCR i j Corr SCR BSCR ij j BSCR BSCR 2 BSCR (6.11) Thu, one ma calculae he SCR i ne of porfolio diverificaion raio b dividing ΔSCR i b BSCR (bu will onl be rue for marginal change and uing a linear model). The equivalen cenario approach ue hee raio o cale he original re cenario in QIS5 a hown in expreion (6.12). 39

52 Equivalen cenario i SCR BSCR i re cenarioi (6.12) For inance if ub module i raio i 0.8, he equivalen cenario reduce he rik for ub module i o 80% of he original re cenario. The Componen BSCR aume a linear relaionhip which i onl rue for ome of he ub module. However, we ma view hi a a fir order approximaion for he non-linear cae. Chaper 9 give a numerical example uing he cae ud. The advanage of he equivalen cenario i ha he hock are calculaed imulaneoul. Thu, i i more raighforward o compue he lo aborbing effec of he echnical proviion (and deferred axe) and appropriae managemen acion, while avoiding double couning he lo aborbing capaci. The diadvanage i, however, ha he equivalen cenario i pecific for an underaking and depend on he parial olvenc capial requiremen reuling from each ub module. An underaking ma herefore opimize he equivalen cenario b juggling he rik on he balance hee, and ulimael alo opimize he emploed mehod in each ub module ince here are opening for uing implificaion The modular approach The modular approach alo equae he adjumen erm b uing formula (6.9). However, for hi approach he nbscr i calculaed b aggregaing up he nscr i from he ub module level b uing he correlaion marice in he ame wa a when calculaing he BSCR, a explained above. If an underaking whihe o implif he calculaion i ma chooe o ubiue one or everal nscr i b he repecive SCR i which ma reul in a higher SCR. The advanage of he modular approach i ha i ue he original re cenario wihou modificaion. The diadvanage i however ha i ma be complicaed o calculae he lo aborbing capaci of he echnical proviion (and he deferred axe) ince he hock are calculaed one-b-one and no imulaneoul. If hi i no carefull calculaed, hi could lead o double couning of lo aborbing capaci of he echnical proviion. Thi i no allowed for ince he andard formula, uing he modular approach, i eeniall a capial charge for each re cenario, while accouning for ome rik reducion due o diverificaion. To a leer exen i ma be poible o opimize he olvenc capial requiremen b being elecive where he fuure dicreionar benefi are ued. 40

53 6.4 The Minimum Capial Requiremen The Minimum Capial Requiremen (MCR) for life inurance underaking i calculaed a a combinaion of a linear formula depending on componen of he echnical proviion and a minimum floor of he guaraneed benefi. Poibl overruling he linear formula, MCR mu be a a minimum 25 percen of he SCR bu no more han 45 percen of SCR, if he Abolue Minimum Capial Requiremen (AMCR) i aified. The AMCR i expreed a an abolue amoun in Euro. For life inurance underaking he AMCR i EUR The linear MCR formula for life inurance underaking i given in (6.13). However, we have excluded conrac where he policholder bear he invemen rik and conrac wihou profi paricipaion. TP1 TP2 MCR L Max 5% TP1 8.8% TP2, Technical proviion for guaraneed benefi 1.6% TP1 Technical proviion for fuure dicreionar benefi (6.13) 6.5 The Rik Margin The rik margin i par of he echnical proviion o enure ha he echnical proviion are equivalen o he amoun ha anoher underaking would require o aume he liabiliie. If he underaking onl receive he be eimae, i can onl expec o earn he rik free rae on he rik capial and hould herefore no be able o arac rik capial, being unable o mee he olvenc capial requiremen. Since rik capial i expenive, corporaion have a naural incenive o opimize he ue of rik capial. The required rik capial for an inurance underaking i he SCR. The echnical proviion hould herefore include a loading for ervicing he appropriae level of SCR when underaking he liabiliie. Thi i he rik margin. In QIS5 he Co-of-Capial i 6% above he rik free rae 18. We will onl ouline he principle for calculaing he rik margin briefl. Firl, i i aumed he underaking a all ime onl hold he required olvenc capial a he rik run off he balance hee. Secondl, he rik margin onl accoun for he necear SCR o aume he liabiliie. Thu, he rik margin onl cover he unavoidable marke rik, life underwriing rik, he defaul rik due o reinurance conrac or SPV and operaional rik. Thirdl, he fuure dicreionar benefi carr over o he new underaking. Finall, he SCR 18 Underaking replicaing he liabiliie hould onl be charged for operaional rik. 41

54 hould be dicouned wih he rik free rae, and no include an illiquidi premium. The rik margin i 6% of he dicouned SCR (in QIS5). A dicued in ecion (6.3) he rik margin hould be baed on he equivalen cenario. Thi i convenien ince one ma impl add he capial charge reuling from he relevan ree. However, one mu accoun for unavoidable marke rik which i uuall no par of he calculaion, unle an underaking eek o minimize rik. Unavoidable marke rik i aumed o come from he cah flow where he mauriie are longer han he available mauriie from he rik-free financial inrumen. In order o calculae he rik margin we need o calculae he appropriae SCR for ever ear a he rik run off. Thi ma be cumberome for ome ub module. For inance if he re cenario he fir ear i differen from he ubequen ear, underaking will need o run he SCR calculaion for all relevan fuure ear (e.g. he diabili hock). To faciliae hi, here are everal poible implificaion. The imple formula for life underaking i he level four of he hierarch, where one roughl ake he produc of he appropriae curren SCR, he modified duraion of he obligaion, and he Co-of-Capial dicouned one ear. In order o ue hi implificaion he underaking alo need o eimae he unavoidable marke rik. There i alo a implified formula for hi. I i he produc of he following erm; he par of modified duraion of obligaion exending beond he marke, he inere rae hock for he mauri of he longe financial inrumen, he average number of ear he unavoidable marke rik exi, and he be eimae. 6.6 Own fund Solvenc II define he own fund which ma be ued o cover he olvenc capial requiremen. Own fund are divided ino hree caegorie depending on he characeriic of he capial. The characeriic are decribed along hree dimenion; availabili, durabili, and he abili o aborb loe during normal buine acivi. We will onl give a implified decripion. Tier 1 capial comprie of equi wih full lo aborbing capabili and cerain hbrid capial wihin a maximum limi. The equi i in he conex of Solvenc II roughl defined a he difference beween he ae and liabiliie. Tier 2 capial coni of ub ordinae deb of a cerain quali and equi wih limied abili o aborb loe. Tier 3 capial include eeniall capial elemen no qualifing for Tier 2 capial. 42

55 The SCR mu be covered b minimum 50 percen Tier 1 capial and maximum 15 percen Tier 3 capial. The MCR normall being le han SCR, ha ronger requiremen. A lea 80 percen mu be covered b Tier 1 capial, and he remaining ma onl be covered b Tier 2 capial. 43

56 7 Financial Rik Life and penion underaking are generall expoed o financial rik in wo wa. Eiher hrough invemen and counerparie, or indirecl hrough he ime value of mone ued for dicouning he expeced cah flow of he inurance liabiliie. The invemen and counerpar rik reul in acual loe on he financial accoun when being realized. Loe from he laer ofen pla ou over ime being realized hrough lower income on he financial accoun. Bu, when income doen cover he charge b he echnical rae, underaking will need o ue own fund o cover he difference. In more exreme cae underaking ma alo need o ue own fund o cover increae in he premium reerve. Thi epeciall applie when here are no ponor backing he cheme. If long erm inere rae are cloe o or below he echnical rae, he maximum allowed echnical rae ma be lowered b he auhoriie. Thi will increae he premium reerve b he dicouning mechanim. Similarl for financial rik, if he marke value of he ae fall below he echnical reerve, underaking will need o ue own fund o cover he difference. Thi chaper cover he marke rik module and defaul rik module, which we dicu a he end. We decribe each ub module which ma be aggregaed o BSCR level uing he relevan correlaion marice and formula (6.6). We noe ha he defaul module i no a ub module and herefore hould be ued direcl a inpu a he op level. Furhermore, here are wo mehod o aggregae o he nbscr level. Boh are dicued in chaper 6. Before dicuing each rik we will inroduce he necear formula for calculaing he gro and ne olvenc capial charge reuling from he ree. We denoe he oal marke value of he collecive porfolio and compan porfolio repecivel a AK and AS. We ma hen compue and underaking lo or profi from a change in marke value uing expreion (7.1). The bonu rae (br) in he profi haring model dicued in chaper 5, are eiher 80 percen or 100 percen of he profi received b he clien on he collecive porfolio, while an loe are covered b he underaking. The compan porfolio i he underaking porfolio and conequenl receive all profi and bear all loe. A ( 1 br) Max[ AK, 0] Min[ AK, 0] AS (7.1) 44

57 Equaion (7.2) and (7.3) will onl be affeced b inere rae in hi chaper. Hence, we ma define hem equal o zero when he ield curve i no reed. Thi i done in order o generalize expreion (7.4) (7.7) below o all (ub) module. Expreion (7.2) give he change in he dicouned value of he guaraneed benefi b uing (6.1) for boh he reed and he normal ield curve. Similarl, (7.3) ield he change in marke value of he embedded inere rae guaranee uing (6.3). L V ( cre, p) V ( c, p) (7.2) IRG IRG ( cre, p) IRG ( c, p) (7.3) The gro capial charge ma hen be compued b formula (7.4). When ae and liabiliie are mached, he expreion inide he bracke hould ield value cloe o zero 19. Capial charge ma no be negaive, o we enure ha i i poiive b uing he Max operaor. Mk i Max[ L IRG A,0] (7.4) We calculae he lo aborbing of he echnical proviion in (7.5) and (7.6). The fir expreion how he lo aborbing of he premium reerve. Thi canno be larger han he capaci, which i he difference he premium reerve and he dicouned value of he obligaion before he re cenario ha occurred 20. The expreion oherwie follow he ame reaoning a in ub ecion (6.2.3). FDB br Min[ L, PR V ( c, )] (7.5) 1 p Expreion (7.6) i lighl more complicaed. An underaking ma draw from he price adjumen fund (PAF) and/or if qualified from he addiional reerve fund (ARF) o cover loe in he collecive porfolio and/or an increae in value of he embedded inere guaranee. The la par of he equaion ake ino accoun ha he addiional reerve fund aborb he covered lo compleel independen of he bonu rae. Min[ PAF ARF, IRG AK] (1 br) Min[ ARF, IRG ] FDB2 br i i i AK (7.6) 19 In hi cae we aume ha he compan porfolio i alo ued o mach liabiliie. 20 The illiquidi cenario urn ou be inignifican for Norwegian ield curve (ee chaper 9), o we have implified he equaion b auming ha we in pracice onl need o bound he lo aborbing in he inere rae re cenario. 45

58 We can finall calculae he ne capial charge afer accouning for he lo aborbing of he echnical proviion b uing (7.7). The ne capial charge hould no be larger han he gro charge ince he gro charge i bounded b zero, i.e. negaive capial charge are no allowed for in profiable cenario. We herefore enure ha he ne figure i equal or le han he gro figure. nmk i Min[ Mk FDB FDB, Mk ] (7.7) i 1 2 i Underaking mu alo enure ha he accumulaed ue of fuure dicreionar benefi in each cenario i le han he capaci in he modular approach. We herefore impoe rericion (7.8) and (7.9) o guaranee hi. i i PAF PAF (7.8) i ARF ARF (7.9) i Having oulined he necear equaion we now decribe each (ub) module. We have explicil addreed he change in value of guaraneed benefi and he embedded inere rae guaranee in expreion b uing formula in (7.2) and (7.3). However, we have no addreed he formula for calculaing he change in ae value. Thi will depend on each (ub) module, and we will addre hi in each ub ecion below. 7.1 Inere rae rik Purpoe: To accoun for inere rae rik from all ae and liabiliie eniive o ield curve and/or inere rae volailiie. Boh nominal and real ield curve hould be accouned for. The ub module doe no exend o ae indirecl eniive o inere rae (e.g. equi and proper). Definiion: There are wo (inananeou) cenario ha hould be evaluaed; a) higher inere rae, and b) lower inere rae. Underaking expoed o ield curve in everal currencie hould calculae he capial charge reuling from all ield curve in a combined cenario. The hocked ield curve are conruced b caling he appropriae ield curve b facor depending on he ime o mauri. In cenario a), he facor range from 1.70 in he 46

59 hor end o 1.25 for 30 ear and longer mauriie. In cenario b), he facor range from 0.25 o The abolue change in inere rae hould a a minimum be one percenage poin. Nominal inere rae are however bounded below b zero (hi doe no appl o real inere rae). We refer o he QIS5 echnical documenaion of a complee pecificaion of he ield curve hif (European Commiion, 2010a). Calculaion: The expeced cah flow from he inurance obligaion are alread accouned for b uing (7.2) and (7.3). We herefore onl need o eimae he cah flow from he inere rae eniive ae. We will onl conider bond and refer o chaper 3 for a dicuion. If he cah flow are readil available, one ma ue he mapping algorihm (3.7) o rediribue he cah flow o he neare inere rae verice. Oherwie one ma ue (3.8) o approximae cah flow and proceed wih (3.7) hereafer. Coninuing one ma ue (3.3) o calculae (7.1), and he lo aborbing of he echnical proviion i found b uing (7.5) and (7.6). Finall, he gro and ne capial charge ma be eimaed uing (7.4) and (7.7). We have no explicil addreed he muli-currenc cae. Thi ma impl be performed b ieraing he ep above for one currenc a a ime, while accumulaing. The liabili equaion however onl reflec he Norwegian penion em covering onl liabiliie in local currenc, o (7.2) and (7.3) hould be zero for all oher currencie. The ep above need o be calculaed for boh inere rae cenario. The inere rae cenario ha ield he highe ne capial charge i choen for calculaing he olvenc capial requiremen. Thi will alo define which correlaion hould for he marke rik module. 7.2 Equi rik Purpoe: To accoun for equi rik reuling from he price level or price volaili in he equi marke. The ub module exend o all ae and liabiliie which are expoed o equi rik. Definiion: Equiie are caegorized ino wo group; a) global, and b) oher. The QIS5 re cenario are an immediae fall of 30 percen in he global caegor, and 40 percen in he oher caegor. All lied equiie on ock exchange in EEA and/or OECD member counrie belong o he global caegor. So for he purpoe of calculaing he capial charge, Norwegian lied equiie are global. 47

60 However, here are ome excepion. Equi paricipaion in financial and credi iniuion ge a capial charge of zero, while he ame rae for oher raegic paricipaion i 22 percen. The laer i no confined o EEA and/or OECD counrie. Calculaion: For each caegor we ake he produc of he price fall and he marke value or he equi expoure. We ma hen proceed b uing formula (7.1) and (7.4) o calculae he gro capial charge, and coninuing wih (7.5) and (7.7) o find he ne capial charge. The equi caegorie ma hen be aggregaed uing he equi correlaion marix and formula (6.6). We have implicil aumed ha here i no equi expoure in he liabiliie. Furhermore, we have alo aumed ha he price eniivi of he equi inrumen i conan. Thi ma eail be refined uing appropriae pricing model, which we will no cover in he hei. Finall, we noe ha underaking ma no ake ino accoun dnamic hedging raegie when uing he andard formula in QIS5, i.e. underaking can onl ake ino accoun poiion ha are in place a he valuaion dae. 7.3 Proper rik Purpoe: To accoun for proper rik reuling from he price level or he price volaili in he proper marke. The ub module exend o all ae and liabiliie which are expoed o proper rik. Definiion: The change in he ne ae value from an inananeou fall of 25 percen in he proper price. Calculaion: We ma calculae he ne and gro capial charge uing he ame ep a above. In addiion, lied real eae companie are covered b he equi rik ub module and hould no be included. Furhermore, proper fund are picall leveraged. Underaking will in hee cae need o calculae he proper expoure inead of uing he marke value of he proper fund. 7.4 Spread rik Purpoe: To capure he rik from widening pread or volaili of he pread relaive o he rik free ield curve in boh ae and liabiliie. Thi rik i for inance applicable o 48

61 governmen bond iued b non-eea member, corporae bond, covered bond, ubordinaed deb, hbrid capial, and credi derivaive. Definiion: The change in ne ae value uing formula (4.1) for bond and he QIS5 defined pread rik facor for each caegor. An addiional marix of pread rik facor i provided for non-eea governmen bond. There are alo pecific duraion floor and cap which hould accouned for in he formula. Formula for rucured produc and credi derivaive depending on pread rik are given in he QIS5 echnical documenaion. Calculaion: We ma calculae (7.1) b uing (4.1). We have aumed ha (7.2) and (7.3) do no depend on pread rik. So, we ma carr on b uing (7.4), (7.6) and (7.7) o calculae he ne and gro capial charge. 7.5 Currenc rik Purpoe: To accoun for foreign exchange rik from all ource wihin an underaking reuling from he change in price level or he volaili of he price. Boh ae and liabiliie are included in he ub module. Definiion: The change in ne ae value, reuling from wo inananeou cenario uing he underaking local currenc a bai 21 ; a) a 25 percen appreciaion of each currenc, and b) a 25 percen depreciaion of each currenc. Calculaion: We ma hen calculae he ne and gro charge imilarl o he calculaion for he equi rik, for each currenc rik cenario eparael. The cenario wih he highe ne capial charge i ued for calculaing he olvenc capial requiremen. We don addre foreign exchange rik on he inurance liabiliie for he ame reaon a in ecion (7.1). 7.6 Concenraion rik Purpoe: To capure concenraion rik ariing from having expoure o he ame counerpar. The ub module onl accoun for expoure conidered in he equi, proper and pread rik ub module. In paricular, he expoure accouned for in he credi defaul module hould no be included. 21 The local currenc i defined a he currenc he underaking prepare i financial aemen in. 49

62 Definiion: Formula (4.2) define he concenraion rik. Threhold level and concenraion facor depending he raing i pecified in he QIS5 echnical documenaion. Calculaion: We ma calculae (7.1) uing (4.2). Oherwie he calculaion i idenical a for he pread rik in ecion (7.4). 7.7 Illiquidi rik Purpoe: The illiquidi premium i incorporaed in he QIS5 ield curve and reul in lower echnical proviion. When he illiquidi premium fall he echnical proviion will rie. Thi ub module herefore accoun for a fall in he illiquidi premium, bu no a rie. The echnical documenaion (European Commiion, 2010a) ae ha: The effec of an increae of he illiquidi premium i capured in he calibraion of he pread rik module 22. Definiion: The change in he ne ae value reuling from an inananeou 65 percen fall in he illiquidi premium. Calculaion: The capial charge ma be calculaed b following he ep for calculaing he inere rae rik, while uing he reed ield curve from he illiquidi premium hock. The muli-currenc ma alo be relevan. 7.8 Counerpar rik (module) Purpoe: Thi module capure counerpar defaul rik no accouned for in he pread rik ub module. Rik miigaing conrac, uch a derivaive and reinurance arrangemen, depoi, receivable and loan belong o hi module. The counerpar defaul rik i defined a poible loe due o unexpeced defaul or deerioraing credi anding. Definiion: There are wo pe of counerpar rik expoure, Tpe 1 and Tpe 2. Thee are dicued in ecion (4.3). Calculaion: Formula (4.3) define he capial charge for Tpe 2 rik. The capial charge for Tpe 1 rik ma be calculaed uing formula (4.8) and he pecified defaul probabiliie in he QIS5 echnical documenaion. The capial charge for he counerpar defaul module 22 We are however no convinced abou hi argumen ince he pread rik module i primaril relaed o bond, ubordinaed deb and credi derivaive. Inurance obligaion are uuall unaffeced b higher pread in QIS5. 50

63 ma hen be calculaed b uing formula (6.6) on boh Tpe 1 and Tpe 2 rik, uing a correlaion coefficien of We can hen proceed b calculaing (7.1). The ep hereafer are idenical o he pread rik in ecion (7.4). 51

64 8 Life Underwriing Rik In hi chaper we will decribe he andard formula for he life underwriing module. All ub module are relevan for life and penion inurance underwriing rik. We end hi chaper b briefl referring o he diabili and morbidi ub module in he SLT 23 Healh Module for he purpoe of compleene. Thi ma be relevan for life inurance, bu no for penion fund in Norwa which onl cover diabili benefi a par of an occupaional defined benefi cheme. Unbundling i in hi cae unnecear and he rik i repreened b he diabili and morbidi ub module of he life underwriing module. Finall, we noe ha he ub module in hi chaper ma be relevan for non-life inurance conrac which can give rie o life annuiie. Unbundling of inurance policie involve pliing conrac ino i baic par or group of baic par. Le P denoe he e of available inurance produc. A polic i a ube U P wih coverage Y (i) > 0 for iu. Thi i a ligh abue of noaion ince we ue o repreen boh he age and he life ielf. We aume ha an inurance polic onl conain one principall inured. Oherwie, each inurance polic need o be pli ino conrac of ingle principal live. We ma unbundle he inurance produc haring one or everal common characeriic, e.g. a primar rik driver, b forming he ube U P of inurance produc haring he common characeriic() and aking he dijuncion U U. The dicouned expeced value of he unbundled par of a life inurance polic can hen be compued b formula (8.1). iu U ( i) V ( U ) ( c, p) Y d ( c) E ( p) (8.1) 1 0 ( i) Furhermore, le S repreen he ock of inurance policie. We ma hen calculae he change in he dicouned expeced value of he liabiliie reuling from a re cenario b uing (8.2). pδ indicae ha he hocked urvival funcion() hould be ued for calculaing he reed cah flow, while p indicae uing he expeced urvival funcion() a before. 23 Similar o Life Technique 52

65 L( U ) ( c, p, p) Max[ V ( U ) ( c, p) V ( U ) ( c, p), 0] (8.2) S Noe ha (8.2) aggregae on a polic-b-polic level. Thi allow for no diverificaion beween differen principall inured live ince onl increae in liabiliie are accouned for. On he oher hand, if U > 1, (8.2) allow for diverificaion effec for rik depending on he ame life. Thi repreenaion follow he principle e forward b he QIS5 echnical documenaion. Le U be he relevan level of unbundling for Life ub module i. (8.3) i hen he gro olvenc capial requiremen for hi rik wihou accouning for lo aborbenc of he echnical proviion. If he profi haring i on he acuarial accoun level, hen hi i alo he ne requiremen (8.4). Thi i he cae for acive policie in he Norwegian defined benefi cheme a dicued in chaper 5. Life i L( U ) ( c, p, p) (8.3) nlife i L( U ) ( c, p, p) (8.4) In he Norwegian em paid up policie have profi haring on he combined profi cener a alo dicued in chaper 5. Auming ha he combined profi i non-poiive, an underaking ma draw on he price adjumen fund (PAF i ) o cover he lo full or pariall, depending on he bonu rae. Thi equae o he ne olvenc charge in (8.5), and ma be viewed a a managemen acion in line wih he following formulaion in he QIS5 echnical documen (European Commiion, 2010a): Addiionall, he reul of he cenario hould be deermined under he condiion ha he value of fuure dicreionar benefi can change and he underaking i able o var i aumpion in fuure bonu rae in repone o he hock being eed. Below he bonu rae i no allowed o change, bu he underaking ma realize unrealized profi from he price adjumen fund appropriael. nlife i 1 br) L( U ) ( c, p, p) br Max[ L( U ) ( c, p, p) PAF, 0] (8.5) ( i Addiionall he underaking need o check ha he um of he draw for all ub module, no limied o he ub module conidered in hi chaper, i le han he capaci reuling in he 53

66 rericion repreened b (8.6). An underaking ma alo conider uing he addiional reerve fund (ADR), bu he fund i ignificanl more rericed a dicued in chaper 5. i PAF PAF (8.6) i We now urn o dicu each relevan ub module, having dicued ome principle and defined he necear formula for he hree fir conidered ub module. We decribe each ub module baed on purpoe, definiion of he andard formula, unbundling, if implificaion i allowed and dicu he Norwegian occupaional defined benefi cheme (NODBS) oulined in chaper 2 in he conex of each ub module. The definiion are encloed in quoaion mark being aken direcl from he QIS5 echnical documenaion (European Commiion, 2010a). The capial requiremen for each ub module i he change in ne ae value reuling from he applied mehod. 8.1 Morali rik Purpoe: To accoun for morali rik aociaed wih (re)inurance obligaion reuling from deah of a policholder. Definiion: A permanen 15 percen increae in morali rae for each age and each polic where he benefi are coningen on morali rik. Unbundling: Inurance policie covering benefi in he even of boh deah and urvival of he ame life don need no o be unbundled. The naural diverificaion of morali and longevi rik aociaed wih he ame peron i allowed for. Simplificaion: I poible a he aggregae level if he andard mehod i an undue burden for he underaking and he proporionali crieria i qualified. NODBS: Morali rik doe appl ince widow and orphan benefi depend on deah of he policholder. There i however no need for unbundling he conrac ince all erm, repreened b (2.9) (2.15), depend on he life au of he policholder, i.e. U = P. Furhermore, widow benefi and orphan benefi are picall ufficienl low compared o he policholder benefi o miigae he morali hock compleel for paid up police. However, hi ma no be he cae for acive policie. Expreion (8.3) and (8.4) or (8.5) ma 54

67 be ued o calculae he capial charge for morali rik according o he andard formula. The reed morali urvival funcion(), pδ, ma be compued b uing he mehod in appendix A and eing F = +15 percen. 8.2 Longevi rik Purpoe: To accoun for longevi (or urvival) rik aociaed wih (re)inurance obligaion reuling from a decreae in morali rae, picall emming from life annuiie or erm inurance. Definiion: A 20 percen (permanen) decreae in morali rae for each age and each polic where he benefi are coningen on longevi rik. Unbundling: Same iue a for morali rik. Simplificaion: I poible a he aggregae level if he andard mehod i an undue burden for he underaking and he proporionali crieria i qualified. NODBS: Longevi rik i he primar rik driver and doe appl ince he benefi o he policholder depend on urvival. No need for unbundling equivalenl o he cae for morali rik, i.e. U = P. We ue (8.3) and (8.4) or (8.5) o calculae he gro and ne capial charge longevi rik b he andard formula. We compue he reed morali urvival funcion(), pδ, b uing he mehod in appendix A and eing F = -20 percen. 8.3 Diabili and Morbidi rik Purpoe: To accoun for he diabili or morbidi rik reuling from (re)inurance obligaion coningen on a definiion of diabili. The rik i characerized b change in level, rend or volaili of diabili rae. Definiion: An increae of 35 percen in diabili rae for he nex ear, ogeher wih a (permanen) 25 percen increae in diabili rae a each age in following ear. Plu, where applicable, a permanen decreae of 20 percen in morbidi/diabili recover rae. 55

68 Unbundling: Thi ub module i onl applicable where i i no appropriae o unbundle conrac. Oherwie he rik hould be handled in he Healh SLT diabili and morbidi ub module. Simplificaion: I poible a he aggregae level, if he andard mehod i an undue burden for he underaking and he proporionali crieria i qualified. NODBS: The ub module i applicable when NODBS include diabili penion wih earned righ. Oherwie he rik, if exien, hould be handled b he Healh SLT underwriing module. In he calculaion we e U = {diabili inurance 24 } ince hi i he onl par depending on diabili rae. However, in a model where morali rae depend on diabili rae here would likel be ome diverificaion and we would ue U = P. Now, coninuing a above, we ue (8.3) and (8.4) or (8.5) o calculae he gro and ne capial charge for diabili and morbidi rik b uing he andard formula. We compue he diabili urvival funcion(), pδ, b uing he mehod in appendix A and eing F 1 = 35 percen and F 2 = 25 percen. We implicil ue a recover rae of 0 percen, o a permanen decreae of 20 percen in recover rae i no feaible and herefore no applicable. 8.4 Lape rik Purpoe: Accoun for lape rik which i defined a: he rik of lo or change in liabiliie due o a change in he expeced exercie rae of policholder opion. The aeed policholder opion are he legal and/or conracual policholder opion uch a, erminaion, renewal, exenion, ha would ignificanl aler he fuure cah flow. Definiion: The definiion i comprehenive. In broad erm, i i baicall he maximum change in ne ae value reuling from one of hree poible cenario; 1) exercie rae increae b 50 percen, bu no o a higher abolue level han 100 percen, 2) exercie rae decreae b 50 percen, bu no b more han 20 percenage poin reducion in abolue level, and 3) a ma lape of 30 percen of policie wih (poiive) urrender rain for reail buine and 70 percen for non-reail buine. Unbundling: Necear, following he policholder legal and conracual opion. 24 Diabili inurance i underood a no receiving (full) diabili penion a he valuaion dae. 56

69 Simplificaion: I poible, mu be bu calculaed a an appropriae granulari, and onl if he andard mehod i an undue burden for he underaking and he proporionali crieria i qualified. NODBS: Thi can be applicable, bu we have no empirical daa covering hi iue. If an emploee leave a compan he emploee ha an opion o coninue in NODBS b paing premium privael. In hi cae married emploee wih children ma have bigger incenive o exend ince NODBS dicued in chaper 2 i baed on populaion mean. Likewie could here poeniall be a elecion effec occurring from individual having peronal healh informaion. We will however neglec hi rik ince he ax incenive i on he emploer hand which hould moderae he incenive. Thi i no relevan in regard o penion fund, ince hi pe of inurance mu be conraced hrough a life inurance compan. 8.5 Expene rik Purpoe: To accoun for he rik of increae in ervicing expene for (re)inurance conrac in he fuure. Expene pamen ha are fixed a he valuaion dae hould be excluded from he anali, and one hould alo ake ino conideraion realiic managemen acion for policie wih adjuable expene loading. Definiion: Increae of 10 percen in fuure expene compared o be eimae anicipaion, and increae b 1 percen per annum of he expene inflaion rae compared o anicipaion. Unbundling: Condiional on expene loading rucure. Simplificaion: Poible condiional ha he andard mehod i an undue burden for he underaking and he proporionali crieria qualifie. We will ue he propoed implificaion in chaper 9 and ae he formula in (8.7). The inpu variable are; 1) E = ervicing expene during he la ear for life inurance conrac, 2) n = average number of ear he rik run off weighed b renewal expene, and 3) i = he expeced inflaion rae. The expeced inflaion rae ued in he exrapolaion of he Norwegian rik free ield curve i 2 percen. n 1 i i n Life n E 1 exp 10% E (8.7) i i

70 NODBS: Applicable. The formula above ield a gro capial charge. We ma arrive a a ne figure b uing (8.4) or (8.5), where he gro change in liabiliie i defined a in hi ecion raher han (8.3). 8.6 Reviion rik Purpoe: To accoun for he reviion rik aociaed wih he ae of healh of a policholder or poible change in legal environmen affecing an annui. Definiion: Increae of 3 percen in he annual amoun paable for annuiie expoed o reviion rik. The impac hould be aeed conidering he remaining run-off period of he annuiie. Unbundling: Ye, he ub module hould onl be applied o annuiie where he paable benefi ma increae a a reul of reviion rik. Simplificaion: No poible. NODBS: Applicable for earned diabili penion benefi no called for. Thu we ake U = {diabili inurance}. In addiion we need o modif (8.1). Inead of uing he expeced cah flow relevan for hi benefi, we will calculae he dicouned value a if he policholder are diabled. Thi i done b hifing he expeced cah flow index o {receiving diabili penion} in (8.1). The capial charge i 3 percen of he oal dicouned value in (8.1). Noe ha he echnical documenaion doen ae a ne figure, o for calculaion purpoe in he coninuaion we will ue ne capial requiremen = gro capial requiremen. 8.7 Caarophe rik (CAT) Purpoe: To accoun for he rik of major irregular even reuling in he deah of policholder, e.g. pandemic even, nuclear exploion and earh quake. Definiion: Abolue increae in he rae of policholder ding over he following ear of 0.15 percenage poin (onl applicable o policie which are coningen on morali). Unbundling: Ye, he ub module i rericed o (re)inurance obligaion coningen of morali. The echnical documenaion doen explicil allow for he naural diverificaion 58

71 from benefi depending on boh deah and longevi of he policholder in he decripion of he ub module. We will herefore aume ha hi diverificaion i no allowed for. Simplificaion: I poible a he aggregae level if he andard mehod i an undue burden for he underaking and he proporionali crieria i qualified. NODBS: Applicable for earned widow and orphan benefi. Thu we ake U = {widow inurance, orphan inurance} 25. Addiionall we alo need o modif (8.1) analogoul o he modificaion in ecion (8.6), bu hi ime for widow and orphan benefi. The gro capial charge i 0.15 percen of he oal dicouned value. We ma arrive a a ne figure b uing (8.4) or (8.5), where he gro change in liabiliie i defined a in ecion raher han (8.3). We noe we ma alo calculae hi figure b uing (8.3) and hocking he morali urvival funcion appropriael. However, we have no deigned he algorihm in appendix A wih repec o abolue change in morali rae, o we refrain from hi ince i eeniall will ield he ame reul. 8.8 SLT Healh Rik: Diabili and morbidi Purpoe: To cover rik of diabili or morbidi rik from (re)inurance liabiliie reuling from hif and variabili in claim frequencie or claim amoun from diabili, ickne and morbidi rae, and medical inflaion. Thi i noabl more comprehenive han he diabili and morbidi ub module of he life underwriing module in ecion (8.3). The ub module diinguihe beween medical expene inurance and income proecion inurance obligaion. Definiion: We will kip a general definiion and refer o he QIS5 echnical documenaion for more informaion. The purpoe of hi ecion i onl o bring aenion o hi ub module. We alo noe ha he income proecion inurance obligaion are defined imilarl o he diabili and morbidi ub module in ecion (8.3). NODBS: No applicable if NODBS i of he form dicued in chaper 2. However, i hould anwa no ield ignificanl differen reul if he medical expene inurance componen menioned above i no covered. On he oher hand, here ma be mall diverificaion 25 Widow and orphan inurance are underood a no receiving widow and orphan benefi a he valuaion dae. 59

72 difference ince he SLT Healh module ue a lighl differen correlaion marix. Furhermore, underaking would alo loe he diverificaion in combinaion wih he Life underwriing module. 60

73 9 Cae ud: QIS5 for a penion fund We have o far decribed and oulined relevan heor, mehod, principle, regulaion and legilaion in relaion o he forhcoming Solvenc II direcive and QIS5. Ulimael, he objecive of he aignmen i o calculae he olvenc capial requiremen for a real life inurance or penion underaking. Thi i he ubjec in hi chaper where we will ue he mehod dicued in he previou chaper. Pål Lillevold ha provided daa for a Norwegian penion fund. We will refer o hi penion fund a PF in hi chaper. 9.1 Overview We will bae our calculaion on he penion fund au a he end of We noe ha he formal QIS5 reporing aumed underaking filing he au a he end of In hi repec he calculaion ma no be compleel repreenaive in a formal Solvenc II filing, ince Solvenc II ue a couner cclical approach. However, he poenial ighening for a uppoed 2010 Solvenc II requiremen i raher uncerain ince he economic recover afer 2008 ha been low for he developed counrie. The direed iuaion among ome indebed counrie in Europe preumabl ma dela or dampen he couner cclical ighening. Furhermore, governmen bond iued b direed EU counrie are raed a prime quali in QIS5 even if ield on ome of hee bond rade above 10 percen. The deail of QIS5 were finalized a he direed iuaion wa emerging. One ma aume ha regulaor wih o avoid ecalaing hee iue furher a ome financial iniuion are heavil expoed and ma need governmen aid if reconidered. The QIS5 filing form are comprehenive Excel preadhee, bu wih implificaion for olo underaking. Neverhele, he are unuiable for he expoiion in hi hei. We have herefore prepared our own preadhee performing he necear calculaion from a ub module level imilar o he formal pread hee. Compuaion compriing or depending on (expeced) cah flow for he penion fund are calculaed in Mahemaica. The code i lied in appendix C. We will briefl kech he logic behind he algorihm in ub ecion (9.3.1), bu will refer o he previou chaper for a heoreical and principle dicuion. 61

74 The Norwegian legilaion for life and penion inurance proec he policholder holder from flucuaion in he dicouned value of he guaranee benefi. Thi ma moivae underaking o chooe cauiou raegic ae allocaion raegie ring o preerve he hareholder equi in he hor run. Solvenc II ma amplif hi effec ince i explicil reveal he inheren rik of he earl inere rae guaranee. Conrar o hi, PF i a well-managed and well poiioned penion fund wih ver comforable buffer and own fund. Thi enable he penion fund o run a long erm focued invemen raeg which i more comparable o inernaional counerpar operaing wihou hor erm reurn guaranee. A we will ee, PF i ufficienl capialized o purue i long erm invemen raeg, even under Solvenc II uing he andard formula in QIS5. Table (9.1) depic PF balance hee. I hold 1.85 billion (NOK) in own fund and 2.41 billion (NOK) in buffer fund. Thi amoun o 4.27 billion (NOK) in rik miigaing capial, and compare o 6.97 billion (NOK) in fund ha are covered b he embedded inere rae guaranee. PF ue a comforabl low echnical rae of 2.6 percen for all policholder. Table 9.1: Ae and liabiliie The financial informaion we have ued hroughou he chaper i he annual repor and accoun for The aemen conain he mo imporan daa for compleing QIS5. However, ome deail are no dicloed and we will herefore make appropriae aumpion when necear. We believe he undicloed deail won have a major impac on he concluion ince he main rik em from equi rik and inere rik from guaraneed benefi (while no accouning for change in fuure dicreionar benefi). To ae he 62

75 liabiliie we are provided wih a complee daae covering benefi and premium reerve for each individual policholder. Technicall, he addiional reerve fund i alo allocaed o each policholder, bu we will rea i a an aggregae level ince hi informaion i no dicloed. The penion fund cover defined benefi for reiremen and diabili, widow or parner, and orphan. Reiremen benefi principall ar when he policholder urn 67 ear and la for he remaining lifeime. We will aume ha all policholder receive reiremen benefi when urning 67 ear, alhough here are ome cae where he policholder mu wai unil urning 68 ear. Diabili inurance la unil reiremen in he even of diablemen, while widow and parner benefi la for he whole lifeime in he even of policholder deah. In he laer cae an orphan will receive benefi unil urning 18 and/or 21 ear. Quie commonl he orphan benefi double in ize afer urning 18 ear, while laing unil urning 21 ear. Thee are eeniall he benefi dicued in deail in chaper Technical proviion A dicued previoul he echnical proviion in he Solvenc II for inurance liabiliie hould be equal o he (marke) value counerpar are willing o underake he inurance liabiliie. The valuaion i pli ino wo par; a) calculaing he expeced value of he liabiliie uing no afe loading, and b) calculaing he rik premium o cover he co of capial for he necear olvenc capial requiremen. We will purue par a) in hi ecion. We will ue he K2005 parameer for calculaing he expeced cah flow. K2005 give Gomperz-Makeham parameer for urvival funcion and addiional funcion decribing populaion mean and age difference necear o calculae ne expeced cah flow for all benefi covered b PF, excep for diabili benefi. Lillevold & Parner AS ha addiionall provided relevan diabili Gomperz-Makeham parameer. Parameer for calculaing ne expeced cah flow are herefore readil available. However, i i imporan o be criical ince hee are baed on populaion mean and he inurance liabiliie of he underaking ma no be repreenaive for he enire populaion. We hall, however, aume ha hi i aifacor. PF ha more han member which are geographicall dipered over variou par of Norwa. The conrac boundar i cenral for deermining a penion fund liabiliie a dicued in chaper 6. Abou 35 percen of he policholder are emploed and acquire benefi linearl 63

76 b eniori. The remaining ock are paid up policie and/or receiving benefi. The ponor, alo repreening he owner ake in hi cae, ma decide o change he penion cheme or enirel cloe i. In hi cae all policie will be convered o paid up policie. Thi i ulimael he conrac boundar for PF, and we will ue hi a a bai for calculaing he olvenc capial requiremen. Wih hi aumpion we can rea all conrac equall and par of he ame modified profi haring model dicued in chaper 5. We will include earned benefi over he nex ear for increaed eniori, bu no poenial wage increae. We will aume ha he premium for he earned benefi i charged o he premium fund and credied he premium reerve fund. The one ear imeframe i choen accordingl o he VaR pecificaion of he olvenc capial requiremen (i.e. one ear ime horizon wih 99.5 percen probabili). To dicoun he expeced benefi we ue he Norwegian ield curve from he QIS5 dicouning helper ab, including he 75 percen illiquidi premium which i relevan for hee conrac. In addiion we will ue he implied volaili urface calculaed from Norwegian wapion marke (a of 31 of December 2009) which i illuraed in figure 9.5. We dicu boh in more deail below. Uing hee aumpion we are able o compue he echnical proviion excluding he rik margin a hown in able 9.2. Table 9.2: Technical proviion ex. rik margin The value of he guaraneed benefi, inere rae guaranee and expene in able 9.2 i calculaed uing he Mahemaica algorihm lied in appendix C. The value of he embedded inere rae guaranee reflec onl ime value ince he inrinic value i zero. The expene are calculaed uing formula (6.5) baed on ervicing co (oal managemen expene wa million NOK la ear), he expeced inflaion rae (2%) and he duraion of he expeced cah flow. The oher inurance fund ma be obained from able 9.1 while 64

77 deducing he eimaed gro premium for he earned benefi for nex ear (abou 120 million NOK). The premium reerve are credied wih he ame amoun. The fuure dicreionar benefi in he able are hen raighforward o calculae. Before proceeding o he olvenc capial requiremen, we will briefl look a he Norwegian ield curve and volaili urface ued in he calculaion. The zero-coupon ield curve are diplaed in figure 9.3 up o mauriie of 40 ear, while figure 9.4 how he implied forward (one) ear ield curve. Yield Time o mauri Figure 9.3: Zero-coupon ield curve (NOK) Yield Time o mauri Figure 9.4: Forward one ear ield curve (NOK) The mooher curve in each graph depic he rik free rae, while he oher curve in each graph include he relevan illiquidi premium (75 percen of 20 bai poin). A can be een from figure 9.4, he effec of he illiquidi premium run compleel off from 10 o 15 ear mauriie having a negaive pread o he rik free ield curve. Thi i omewha irregular and effecivel mean ha (expeced) cah flow wih long mauriie are no expoed o and 65

78 neiher are dicouned b he illiquidi premium obainable in he horer end of he curve. Conequenl, he illiquidi premium i irrelevan for mauriie longer han 15 ear. When he illiquidi premium narrow in he re cenario, forward inere rae will rie for mauriie beween 10 and 15 ear, and fall for mauriie le han 10 ear. Thu, he effec of he illiquidi premium i no obviou for an underaking. Figure 9.5 depic he Norwegian volaili urface for wapion wih one ear enor baed on implied volailiie uing he Black model and marke price a of 31 of December The ime cale follow from righ o lef o improve viualizaion. The rike in he graph var from -200 o +200 bai poin on op of he relevan implied forward curve which hould be approximael imilar o he forward curve in figure 9.4. Price are onl available for expirie up o 30 ear. We have herefore exrapolaed he volaili urface uing 30 ear volailiie alo for longer mauriie. The volaili urface converge o abou percen volaili for rike above he forward ield curve. Thi i however fairl low compared o he inere rae level. A he end of la ear, he ame par of he urface wa 2-3 percenage poin higher han hi. To be conien wih dae we hall, however, ue he volaili urface below a inpu. The hape of he volaili urface will acuall reul in larger abolue price change of he embedded opion in he inere rae down cenario. Figure 9.5: Volaili urface for Norwegian wapion wih one ear enor 66

79 9.3 Solvenc capial requiremen Calculaing he olvenc capial requiremen i he main objecive in hi hei. We ar fir b briefl oulining he algorihm in appendix C below. The algorihm compue he life underwriing rik in he fir ub ecion, and he inere rae rik and he illiquidi rik in he econd ub ecion. The remaining par of he marke rik and he defaul rik i alo covered in he econd ub ecion. We keep he fuure dicreionar benefi unchanged, o we can calculae he baic olvenc capial requiremen in (9.3.3). Change in he fuure dicreionar benefi are accouned for in he wo nex ub ecion. Finall ub ecion (9.3.6) calculae he operaional charge and he adjumen facor obained from he wo previou ub ecion. Thi ield he Solvenc Capial Requiremen (SCR) Life underwriing rik The algorihm coni of hree par. The fir par calculae vecor of ne expeced cah flow for andardized benefi (100 NOK per ear) for each andardized age beween 0 and 120 ear and each gender, looking 120 ear ino he fuure. The algorihm produce boh he ne expeced cah flow and he reed cah flow aking he relevan re parameer a inpu. For age where cerain benefi are no applicable, he ne expeced cah flow i impl zero. For inance we aume ha emploee ma ener he defined benefi cheme a age 20. Thu, he ne expeced cah flow are zero for emploee under 20 ear old. On he oher hand, orphan receiving benefi loe hee when urning 18 or 21, and o he ne expeced cah flow for orphan receiving benefi are zero for age above 18 or 21 repecivel. The ne expeced cah flow for each gender and benefi are aved ino able ranging from 0 o 120 ear, and likewie for he reed cah flow. Widow and parner benefi depend on wo live, and we re boh live imulaneoul o be conien 26. The econd par compue he expeced cah flow and he reed cah flow for each polic uing he able from par one. For non-ineger live we ue linear inerpolaion o approximae he relevan cah flow. Thi i a fair approximaion peeding up he algorihm coniderabl. Thi i done for each covered benefi and caled appropriael according o he erm in each polic. All expeced and reed cah flow for each policholder are 26 And effecivel reduce he diverificaion when conidered in combinaion wih benefi having longevi a he primar rik driver. 67

80 accumulaed 27. Addiionall expeced and reed cah flow relaing o diabili benefi (bu no currenl receiving diabili benefi) are alo accumulaed. Similarl, expeced cah flow auming ha all policholder have deceaed, are accumulaed for he purpoe of calculaing CAT rik a oulined in ecion 8.7. Thi ield all he necear cah flow from he liabiliie. Bu, we will need o rerun hee algorihm when compuing he ree in he equivalen cenario. Par hree compue he rik ub module. We don give an rik figure here a he are diplaed in able 9.10 and dicued in ub ecion (9.3.3). The relevan ield curve are loaded. We alo appl he rediribuion algorihm o he cah flow a dicued in chaper 3. Morali rik i conidered equal o zero a dicued in ecion (8.1). Longevi rik i compued b calculaing he dicouned value of he difference beween he reed and ne expeced cah flow. Similarl, he diabili rik i calculaed a he dicouned value of he difference beween he reed and ne expeced cah flow. Thee were accumulaed addiionall. We aume ha lape rik i zero according o he dicuion in ecion (8.4). Expene rik i compued b formula (8.7) needing hree inpu. The la ear ervicing expene, he inflaion rae and he average number of ear he rik run off weighed b he renewal expene. Here we have ued he duraion of he expeced cah flow. Reviion rik i 3 percen of he dicouned value of he diabili benefi (no including hoe ha are currenl receiving diabili benefi). Finall, CAT rik i 0.15 percen of he dicouned value of he CAT cah flow Financial rik In he dicuion we follow he ame order a in chaper 7, aring wih inere rae ub module. We coninue uing he Mahemaica algorihm from he previou ecion, and addiionall load he appropriae reed ield curve, uuall denoed b up and down, and he volaili urface. We will aume ha laer alo applie in reed iuaion. Realiicall however i i likel ha he implied volailiie will rie omewha in he even of lower inere rae. We alo compue he cah flow for he fixed income porfolio, and keep he collecive and compan porfolio eparael. 27 Thi i a ligh implificaion ince we hould eparae cah flow b longevi or morali a he primar rik driver. However, we aume ha longevi rik i he primar rik driver for all polic holder ince hi i he naural rik driver for hi defined benefi cheme. Thi i even more appropriae when all policie are conidered paid up a here. 68

81 Par hree of he algorihm ield he change in value reuling from reing he ield curve, upward or downward. We calculae he change in value individuall for he guaraneed benefi, he embedded inere rae guaranee (wih rike equal o he echnical rae of 2.6 percen), price change of he collecive porfolio and he compan porfolio. The financial aemen conain informaion abou he marke value and duraion acro rik caegorie for he bond porfolio. We ue formula (3.8) o eimae cah flow and mapping algorihm (3.7) o diribue cah flow o he appropriae verice. The average duraion in boh porfolio i fairl hor (2.05 in oal). Higher inere rae will herefore onl have a moderae impac on he marke value of he bond porfolio. The dicouned value of he guaraneed benefi and he embedded inere rae guaranee will fall. Thu, i i highl likel ha he cenario wih lower inere rae will be relevan for PF (which i verified b he calculaion in appendix C). We will aume ha all bond are iued in NOK, having no informaion ha conradic hi. We alo eimae he inere income for he nex 12 monh in able 9.6. The effecive ield i 6.7 percen according o he financial aemen. Thi i raher high compared o hor erm Norwegian governmen bond, o we upec here are ome floaing rae noe in he porfolio. We have ubraced 0.5 percenage poin o ake accoun of he moderae ield curve eepne. In able 9.6 we have charged he inere income accoun for he collecive porfolio for he earl inere rae guaranee b he echnical rae. Thu, he inere income hould cover he inere rae guaranee. However, no financial reurn hould be included, and we herefore aume ha he ne income i zero. There are no bond claified a hold-omauri. Table 9.6: Inere pamen accoun 69

82 The financial aemen give all equi holding percen are lied on he Olo Sock Exchange, 1.3 percen on Olo Axe, 2.8 percen on foreign ock exchange, while 8.5 percen are unlied equiie. Having looked hrough he holding, we find ha mo of he unlied equiie are in he compan porfolio. For implici we will aume ha he collecive porfolio hold onl lied equiie. All ock exchange are in EEA and/or OECD counrie. All lied equiie herefore qualif for he global caegor. Uing he marke value we caegorize he equiie a in he upper par of able 9.7. The oher equiie are he unlied par which i 8.5 percen of he oal marke value. We have alo added 7.9 million NOK of call opion o he compan porfolio. When doing hi we need o aume ha he opion i ver deep in-he-mone o ha he opion eeniall behave a a ock (i.e. dela equal o one). The lower par of he able how he re cenario. Table 9.7: Equi expoure and rik We ma kip he proper ub module ince PF hold no proper. When conidering foreign exchange rik, he financial aemen ae ha PF eek o reduce foreign exchange rik, i.e. here are no acive be bu here ma be paive be. The marke value of he currenc hedge in he collecive porfolio i 2 million NOK, while zero for he compan porfolio. The laer ma indicae ha he compan porfolio i unhedged. Having aumed ha all bond are iued in NOK, FX rik can onl em from equiie. Table 9.8 how he equi expoure for each currenc uing he holding in he financial aemen. We will aume ha none of hee currencie are hedged ince he compan porfolio hold he larger rik and preumabl i unhedged. We herefore ue a 25 percen capial charge on he expoure for all currencie. 70 Table 9.8: FX expoure from he equiie and FX rik

83 The capial charge from he pread rik module i difficul o calculae wihou more informaion abou he fixed income porfolio. We will herefore ue he following aumpion; a) financial iniuion are raed A, b) ecuriie wih he highe collaeral level are raed A, c) morgage backed ecuriie are raed AAA, and d) he re i unraed. Table 9.9 how he capial charge, which i copied from he QIS5 pread rik helper ab. Table 9.9: Helper ab preadhee for pread rik In order o calculae concenraion rik we will need o aume ha he bond porfolio are well diverified and do no hold iuer from he four large equi ake. From he lied equi holding we have idenified he four large equi holding a of 5.82, 2.92, 2.16 and 1.66 percen of he oal ae. Thee are unraed and herefore have a concenraion rik hrehold of onl 1.5 percen. We ma compue he concenraion rik b uing he QIS5 concenraion rik helper ab. Anoher opion i impl o mulipl he porion exceeding he hrehold b he relevan concenraion rik facor (which i 0.73), quaring and adding all erm, aking he quare roo of he um, and finall mulipl b he oal ae value excluding he loan porfolio conidered below (abou 11 billion NOK). The Mahemaica code in appendix C calculae he illiquidi rik uing he ame mehod a for he inere rae down cenario, bu uing he relevan reed illiquidi premium ield curve. We have obained hi from he QIS5 dicouning helper ab. Finall, we will conider he counerpar defaul rik. There are hree poenial ource we need o ae. The counerpar rik relaed o derivae. We will diregard hi ince we have aumed ha all equiie are unhedged. The ame applie o he reinurance conrac (which i relaed o caarophe inurance) ince we implicil have aumed ield no rik 71

84 miigaing effec in he re cenario. Furhermore, PF ha a porfolio of loan collaeralized b proper (of abou 96 million NOK). The loan are primaril morgage loan o policholder. We herefore caegorize hee a Tpe 2 expoure. According o he financial aemen none of hee are a rik, o we will aume ha none of he downpamen are overdue. The capial charge i herefore impl 15 percen of he loan The Baic Solvenc Capial Requiremen (BSCR) Having worked hrough all relevan par of he rik ub module, we can calculae he baic olvenc requiremen. Table 9.10 how he relevan informaion from he rik ub module (and ub-ub module in he cae of equiie) holding he value of fuure dicreionar benefi unchanged. We ue he following column heading; ΔL = change in marke value of guaraneed benefi, ΔIRG = change in marke value of embedded opion, ΔAK = change in ae value on he collecive porfolio, and ΔAS = change in ae value on he compan porfolio. Table 9.10: BSCR calculaion able The row in he SCR i column ma now be calculaed b impl adding he ΔL and ΔIRG column, ubracing he ΔAS column, and ubracing 20 percen of he ΔAK column if 72

85 poiive and 100 percen if negaive. The laer follow from he modified profi haring model which hare he profi (20/80) bu no he loe (100/0). The loe ma hen be aggregaed o a higher level uing formula (6.6) and he relevan correlaion marice (e.g. uing he correlaion marice for he cenario where inere rae fall). We have lied he correlaion marice in appendix B. The ma alo be found in he QIS5 echnical documenaion. Thi add up o a BSCR of 2.83 billion NOK. To arrive a an SCR we need o calculae he adjumen facor and operaional rik capial charge. We calculae he adjumen according o he wo precribed mehod in he nex wo ub ecion. We hall aume ha lo aborbing of deferred axe i inignifican. A a noe, in he dicuion above we have implicil aumed ha he ne income from he acuarial and managemen accoun i nil before he re cenario appear nbscr: The modular approach In hi approach we calculae a ne Baic Solvenc Capial Requiremen (nbscr) b accouning for he lo aborbing of echnical proviion for each cenario eparael. There are hree poible ource a hown in able 9.11 under he ource heading; he change in he difference beween he premium reerve and he value of he guaraneed benefi, poenial ue of he adjumen reerve fund, and he (poibl poenial) ue of he price adjumen fund. Poenial in hi conex mean (poibl fuure) managemen acion. The ne change in FDB ake accoun of he profi haring, while he change in gro FDB impl add he hree column under he ource heading. For each ub module we calculae he ne capial charge under he column headed nscri, found b adding he ub module amoun from he SCRi column wih he change in ne FDB. The SCRi column follow from he previou ecion. Alhough he modular approach calculae each cenario individuall, we hould be careful no o double coun he lo aborbing capaci. We have herefore racked he uilizaion in he la row of able A can be een boh he price adjumen fund and addiional reerve fund are depleed. The difference beween he premium reerve and he value of he benefi i ill large, bu uilizaion will mol depend on he inere rae level. We have choen o ue he adjumen reerve fund in he equi re cenario ince hi ield he 73

86 highe effec being he mo imporan rik driver (he addiional reerve fund can onl be ued o cover he earl capializaion b he echnical rae). We have alo aumed ha he managemen appropriael draw from he price adjumen fund in each cae of he life re cenario. Thi i poible ince we have aumed ha all policie are convered o paid up policie, enabling profi haring on he combined ne income of he financial, acuarial and managemen accoun. Table 9.11: nbscr calculaion modular approach Finall, he nbscr calculaed b he modular approach ma be found b uing formula (6.6) appropriael in combinaion wih he ame correlaion marice ued above. A he able how, hi ield an nbscr of million NOK. The adjumen erm ma hen be compued a BSCR nbscr = million NOK, ince he ource are no depleed uing formula (6.9). We will make ue hi nbscr in ub ecion and will now urn o calculaing he adjumen erm uing he equivalen approach. 74

87 9.3.5 nbscr: The equivalen cenario The equivalen cenario i pecific for each underaking a dicued in chaper 6. We herefore need o calculae hi fir, b uing he mehod decribed in chaper 6. Table 9.12 ouline he calculaion. We noe ha hi i done a hree level 28 aring a he boom level and working upward o he op level. The principle rule i o ue he SCRi and no he nscri. The SCRi column can be recognized from he andard cenario in ub ecion The muliplicaion of SCRi b Y and diviion b Capial correpond o formula (6.10). We have choen hi eup following he example in Annex J in (European Commiion, 2010b). We noe ha formula (6.10) aume onl one level, o BSCR in he formula (6.10) correpond wih Capial in he able, while pscri i he SCRi a he ub-ub or ub module level. Y repreen he remaining par of formula (6.10). Table 9.12: Calculaing he equivalen re cenario The diverificaion facor raio in he % Top column ield he underaking pecific equivalen cenario. Each cenario i quoed a a percenage of he original re. For inance he global equi hock i onl 91 percen of he original cenario where marke value fall 30 percen. Thu, equiie fall 27.3 percen in he equivalen cenario. All gro 28 Effecivel hi mean ha correlaion beween ub module from differen module can change ince he correlaion a he module level are conan, depending on each pecific underaking poiion. 75

88 charge are in fac calculaed b able 9.12 if he rik ub module are linear. Thi i no he cae for he inere rae, illiquidi premium, longevi, diabili and expene rik ub module. We herefore need o recalculae hee ub module uing he new parameer and he modified ield curve. To do hi, we reue par hree of he Mahemaica code, adjuing parameer and imporing appropriael caled reed ield curve for he inere rae down cenario and he illiquidi premium cenario. The ield curve ma be appropriael caled uing he QIS5 dicouning helper ab. The oher parameer follow he ame line of reaoning a decribed for equiie. For inance he longevi hock ma be found b mulipling he original re (-20 percen) b he diverificaion facor raio for he longevi ub module from he op level (29 percen), which ield a reduced re cenario of -5.8 percen. We proceed in a imilar wa for he oher rik ub module. Having done hi, able 9.13 how he equivalen cenario uing a eup imilar o able 9.10 for he modular approach. However, in hi cae we don ue correlaion marice o aggregae he individual capial charge o he op level. The equivalen cenario i a ingle cenario where all ree happen imulaneoul, o we impl add he cenario. The gro charge i 2.65 billion NOK, auming fuure dicreionar benefi are unchanged. Table 9.13: The gro change in ne ae value under he equivalen cenario 76

89 One ma compare he non-linear ub module in 9.13 wih he linear approximaion in able 9.12 reuling from he Componen BSCR pariion. We ee ha inere rae rik i abou 20 percen off he exac calculaion, while longevi rik i abou 13 percen lower. Diabili rik i on he oher hand roughl 6 percen higher. We can now calculae he change in fuure dicreionar benefi. Thi i illuraed in able 9.14 which i imilarl rucured a able However, we now caegorize he row b liabiliie, collecive porfolio and compan porfolio which are he relevan level for profi haring when conidering a combined imulaneou re cenario. We ue he ame aumpion a in he modular cae regarding poible fuure managemen acion. Table 9.14: nbscr under he equivalen cenario Thi reul in an nbscr of million NOK. The adjumen erm i again found b formula (6.9) which ield million NOK. We noe ha hi i differen from he change in ne FDB in able 9.14 which i onl million NOK. We have calculaed he laer o keep rack of he ource. When uing formula (6.9) we mu enure ha he lo aborbing i no greaer han he capaci. The lo aborbing capaci i 3.86 billion NOK, and 1.98 billion i herefore well wihin he limi. 77

90 9.3.6 The olvenc capial requiremen Having worked hrough he BSCR and nbscr calculaion, we are almo able o calculae he olvenc requiremen. We will alo need he capial charge for operaional rik. Thi i raighforward uing he expreion (6.8). We noe ha (6.8) i incomplee ince i diregard ome apec, bu, i i exac for PF. The necear inpu i eiher previoul calculaed or can be found in he financial aemen. Table 9.15 how he calculaion. I i impl he maximum of he charge for eiher premium or echnical proviion, bu bounded a above b 30 percen of he BSCR. We ee ha he 0.45 percen charge for he echnical proviion i binding. Technical proviion in hi cae don include he rik margin o avoid circulari. Table 9.15: Solvenc capial charge for operaional rik We are finall able o compue he olvenc capial requiremen which i hown in able We ee ha he equivalen cenario approach require approximael 5 percen lower SCR han he modular approach. In addiion he equivalen cenario approach ha ignificanl more lo aborbing capaci lef in he echnical proviion. The difference i abou 764 million NOK when comparing he BSCR nbscr for each approach. Table 9.16: Solvenc capial requiremen The penion fund i well capialized and ha ufficien fund o cover he required olvenc requiremen uing boh mehod. For compleene we will addiionall compue he minimum capial requiremen in able 9.17, uing formula (6.13). 78

91 Table 9.17: Minimum capial requiremen The olvenc capial requiremen under he curren regulaion i 283 million NOK according o he financial aemen. The MCR i ju 18 and 22 percen lower, while he SCR i 229 and 214 percen higher han under he curren regime, under he modular and equivalen cenario approach repecivel. Thi ma involve a ignifican ighening which i ma be difficul o operae under for underaking wih weaker balance hee. One hould alo ake ino conideraion ha Solvenc II ue a couner cclical approach. Thi ma reul in even more demanding olvenc capial requiremen when he developed economie have recovered ufficienl. For inance, he bae level equi ree are 39 percen and 49 percen for he global and he oher caegor, compared o 30 percen and 40 percen in QIS5 repecivel. 9.4 The penion fund olvenc II balance hee We have worked hrough an exenive number of calculaion, and round up b howing he penion fund liabili ide of he Solvenc II balance hee. In order o do o we alo need o calculae he rik margin. Thi i hown in able We have ued he crude approximaion ha i allowed for (i.e. level 4). Thi ma lead o an unnecear high rik margin. However, he lo aborbing capaci of PF i abundan and we ma no gain oo much uing a more refined mehod. In addiion, he life re cenario where draicall reduced in he equivalen cenario which benefi he calculaion of he rik margin. Thi follow ince we onl need o ue he capial charge from he life module, he operaional rik and a capial charge for unavoidable marke rik from he equivalen cenario approach. The gro capial charge for he unavoidable marke rik i, however, large a can be een from able We can probabl improve hi ignificanl if needed, e.g. if he penion fund 79

92 experience a large lo of aborbing capaci. Bu, in ha cae he unavoidable marke rik ma alo decreae ome ince he fuure dicreionar benefi are included in he be eimae. Table 9.18: Calculaion of rik margin Finall, in able 9.19 we ee ha he echnical proviion (8.83 billion NOK) are 4.3 percen lower han he inurance liabiliie (9.23 billion NOK) from able 9.1. The echnical proviion are herefore full covered b he inurance liabiliie. 80 Table 9.19: Solvenc II balance hee

93 We have aumed ha he penion fund own fund are Tier 1 o ha all exce ae over liabiliie have Tier 1 characeriic. The olvenc capial i ignificanl higher han under he curren regime. Thi i parl due o he liabiliie having a lower value, and ha he equi i alo including he unrealized earning. Thi counerac a ignifican par of he higher olvenc capial requiremen. The olvenc raio onl fall from 324 percen under he curren regulaion o 251 percen or 239 percen in QIS5 depending on which approach i adoped when Solvenc II i finalized. The Norwegian regulaor auhori ha publihed he reul from he Norwegian QIS5 reporing (Finanilne, 2011b). The preenaion doe no cover penion fund, bu we ma compare wih he en life companie ha have repored. The average be eimae and rik margin i abou 1.6 percen lower han he gro reerve fund. Thi compare wih 4.3 percen lower for PF. Onl hree have olvenc raio above 150 percen, while four are below 100 percen. PF i coniderabl beer capialized han he average, while probabl alo running a much higher invemen rik. We ma alo compare he rik decompoiion. The life module repreen on average abou 40 percen of he BSCR, while 75.6 percen of he BSCR come from he marke rik module. The rik figure are aed in undiverified erm in he repor, and herefore accumulae o more han 100 percen. PF i ignificanl more kewed oward invemen rik wih onl 9 percen of he BSCR coming from he life module, and 97 percen from he marke rik module. We end hi chaper b concluding ha he penion fund i exremel well capialized and i well prepared o handle he forhcoming olvenc requiremen. 81

94 10 Concluion The Solvenc II direcive i a fundamenal review of he capial adequac requiremen for he European inurance indur cheduled o be implemened Januar I will have a major impac on life and penion inurance underaking. Thi i in paricular rough for he Norwegian indur illuraed b he reul from he Norwegian QIS5 reporing (Finanilne, 2011b) and in he previou chaper. The Norwegian QIS5 reporing indicae ha he iuaion i challenging. The average olvenc capial requiremen i 240 percen of he olvenc margin under he curren regime, while own fund are increaed b onl 5 percen. The penion fund, analzed in chaper 9, i on he oher hand well capialized. QIS5 i no real concern for hi penion fund. The olvenc capial requiremen i increaed dramaicall, however, being offe o a large degree b he increae in own fund. We will in hi chaper ummarize ome of our finding and reflecion hroughou he projec. The calculaion and aumpion behind QIS5 are non-rivial. I aume horough inigh ino an underaking inurance buine a well a financial poiion. Thi e a high andard for all paricipan needing qualified experie combining acuarial cience and finance. Thi applie boh o underaking and regulaor auhoriie. Furhermore, he echnical documenaion ouline he calculaion for he olvenc capial requiremen baed on he o-called andard formula. The andard formula i however in everal cae onl a decripion of a re cenario and doe no necearil ae an explici formula. In ome of hee cae implificaion are given, bu ree repeaedl ha he proporionali aumpion mu be reaonable and ha a full calculaion mu be an undue burden. Furhermore, when underaking op o ue inernal model, an even higher degree of complexi i inroduced. The complexi of QIS5 i alo illuraed b he fac he paricipan in he QIS5 reporing have ued varing aumpion leading o ha he reul ma no be compleel comparable. Thi reveal ha he QIS5 mehodolog probabl need addiional uning before implemened in Solvenc II. The iue of he conrac boundar i non-rivial for Norwegian life and penion inurance. We have ued he aumpion ha all policie are convered o paid up policie. Thi i a real poibili for defined benefi cheme in he privae ecor, bu ma 82

95 no be relevan for he public ecor. Under he curren re e for life inurance underaking, Finanilne balance hi oucome for he privae ecor, bu no for he public ecor (Finanilne, 2009). Furhermore, i i unclear if one raher hould ue he lape ub module o accoun for he poibili of ma converion o paid up policie. Thi ma be relevan for he life inurance underaking ha have man ponor backing he defined benefi cheme. Penion fund uuall have onl one or a few ponor, and differ from life inurance companie ince he ponor alo picall repreen he owner. Coninuing hi reaoning one ma alo queion how large rik premium he ponor are willing o pa for he inere rae guaranee, if long erm inere rae fall ignificanl below he echnical rae. We noe ha i onl favourable for underaking o aume buine a uual in he QIS5 reporing, if he aumed rik premium paid b he ponor cover hi mimach ufficienl in he ubequen ear afer he re cenario occur. A reaonable aumpion ma however be ha lower inere rae ma amplif he rend oward replacing defined benefi cheme wih defined conribuion cheme, alread being under preure from he ongoing penion reform. Thu, underaking ma in he end bear he rik of lower inere rae anwa. Claical value inveing raegie eeniall boil down o buing cheap cah flow and elling expenive cah flow having a long erm view, while avoiding momenum raegie. Solvenc II combined wih he Norwegian inere rae guaranee ma moivae he revere. If he buffer beween he premium reerve and he dicouned value of guaraneed benefi become low, underaking will be hreaened b he rik of bearing he co of lower inere rae, and ma bu bond wih long duraion o hedge hi rik. Shor erm hi ma lead o a elf-reinforcing effec of even lower inere rae due o he increaed demand for bond wih long duraion. Momenum raegie are alread rongl moivaed b he earl inere rae guaranee 29. Solvenc II ma amplif hi addiionall and can reul in lower invemen reurn o he clien. However, we noe ha Solvenc II oherwie o ome degree have adoped a couner-cclical approach in he calibraion of he re e. We alo oberved ha he exploied capaci of he buffer fund were ignificanl lower in he equivalen cenario approach compared o he modular approach, alhough he repecive olvenc capial requiremen were imilar. Which mehod ha finall will be adoped in 29 Dnamic rik managemen and conan proporion porfolio inurance (CPPI) are frequenl ued echnique in financial marke. 83

96 Solvenc II, ma be a concern for he Norwegian underaking. The buffer fund are necear for enduring marke flucuaion when operaing under he earl inere rae guaranee. Furhermore, more flexible buffer fund ma eae he burden omewha. In he equivalen cenario where all ree occur imulaneoul he addiional reerve fund can onl cover capializaion b he echnical rae. Thu, addiional reerve fund beond hi will a unued in hi approach. We noe ha Finanilne ha uggeed amendmen o Norwegian legilaion in relaion o Solvenc II, among oher propoing a ingle more flexible buffer fund (Finanilne, 2011a). The reul from he Norwegian QIS5, and in paricular for he penion fund analzed in chaper 9, how ha marke rik coniue he larger par of he olvenc capial requiremen. Thi will preumabl bring aenion o developing inernal marke rik model in order o reduce he olvenc requiremen. Furhermore, i i onl poible o ake accoun of dnamic hedging program in inernal model, diplaing he behavior of an underaking in urbulen marke. The andard formula can onl include exiing poiion (excep for rolling hedging program, which will onl accoun for he rolling of exiing conrac ino new conrac a he expire). The illiquidi premium ued for Norwegian QIS5 ield curve i mall (maximum 20 bai poin). However, we oberved irregular effec on he ield curve a he illiquidi premium i phaed ou beween 10 and 15 ear mauriie in he Norwegian QIS5 ield curve. Thi ma be improved in he calibraion (b EIOPA) uing he forward ield curve inead of he zero-coupon ield curve. A a final poin, we ma alo queion he o-called macroeconomic mehod for exrapolaing ield curve beond he longe bond mauriie in he marke 30. The argumen i baed on ome of he ame principle of he inflaion argeing adoped b man cenral bank, which became a cenral banking paradigm in he 1990 ie. The globalizaion in he following ear pu rong downward preure on conumer good reuling in low impored inflaion in developed economie. The afermah of he Grea Receion in 2008 ma change he inflaion argeing paradigm, a he dfuncional global rade model ha been uncovered. I i herefore reauring o oberve ha Finanilne i uncomforable wih increaing he incenive o hold long bond duraion a low inere rae level. 30 Alhough, probabl no ver differen from he prevailing ulra long bond ield for he ime being. 84

97 Appendix A Sreing urvival funcion In hi appendix we decribe how we compue he reed urvival funcion. We aume a Gomperz-Makeham hazard rae funcion defined in ub ecion (2.1.3), alhough no needed before (B.8). The hock in (European Commiion, 2010a) are defined wih repec o nonurvival rae (i.e. morali or diabili rae) for each age. The are picall formulaed a; a) a permanen F percen increae/decreae for each age, b) F 1 percen increae/decreae nex ear, and c) a F 2 percen increae/decreae permanenl in he following ear afer he fir ear. In each cae we repecivel define; a) F = F, b) F 1 = F 1 and F = 0 for > 1, and c) F 1 =0 and F = F 2 for > 1. We ma combine b) and c) o obain an individual hock rae for he fir ear and equal hock rae in he ubequen ear (bu leaving ou he zero aignmen in b) and c)). Each hock ma hen be expreed b (B.1) for a given age x. q hock x ( 1 F 1) q, 0,1,, x (B.1) x B uing relaion (2.3) we find he hocked urvival funcion in (B.2). p hock x 1 (1 F 1) qx 1 (1 F 1) (1 px ), 0,1,, x (B.2) The idea i now o find coefficien c 1, c which aifie expreion (B.3) o ha we can decribe he hocked urvival funcion a a produc he ordinar urvival funcion and he exponenial of he accumulaed c. Noe ha he coefficien depend on x when pecified hi wa. Thi allow for a general hock pecificaion of F. p hock x exp j1 c j p x, 1,2,, x (B.3) We coninue b finding an analical oluion for c b inering he hocked urvival funcion ino (2.4) and leing = - 1 and = 1. Thi reul in (B.4) which i olved for c in (B.5). 85

98 hock hock ( 1) px exp( ) px x c exp( c ) hock ( 1) px ( 1) px p p (B.4) x( 1) hock px( 1) 1 (1 F ) 1 px( 1) c ln ln (B.5) p x( 1) px( 1) In order o ue he formula in chaper 2 we need an appropriae coninuou hocked urvival funcion. We herefore define a coninuou funcion f x+ in (B.6). The hocked urvival funcion (B.7) aifie (B.1) exacl for,, x 1. fx c j c, [0, x] (B.6) j1 hock f p, [0, x] px exp x x (B.7) The derivaive of f x+ i epwie coninuou. If he coefficien are ufficienl bounded he inegral of (B.7) doe exi. Numericall, we ma need o be cauiou when chooing o enure ha he coefficien, c, don change oo rapidl a he urvival funcion approache zero. For inance, a morali hock i more challenging ince i i difficul o decreae he urvival funcion furher when he urvival funcion i cloe o zero. Converel, a longevi hock reul in a urvival probabili of approximael F for he nex ear, auming he ordinar urvival funcion i cloe o zero. Thu, he hocked urvival funcion run off imilar o a geomeric erie which i numericall able. Finall, combining (2.6) (2.7) and (B.7) ield he hocked hazard rae funcion in (B.8). x x c c (B.8) 86

99 Appendix B QIS5 correlaion marice We have included he relevan correlaion marice from he QIS5 Technical Specificaion for compleene. Thee are par of he formal QIS5 documenaion iued b he European Commiion (European Commiion, 2010a). Figure B.1: Top level correlaion marix Figure B.2: MarkeDown correlaion marix (lower inere rae cenario) Figure B.3: MarkeUp correlaion marix (higher inere rae cenario) 87

100 Figure B.4: Equi ub module correlaion marix Figure B.5: Life module correlaion marix 88

101 Appendix C Mahemaica code Par I - See ub ecion 9.3.1: 89

102 90

103 91

104 92

105 93

106 94

107 Par II See ub ecion 9.3.1: 95

108 96

109 97

110 98

111 99

112 Par III ee ub ecion and 9.3.2: The Calculaion a he end i baed on he andard formula. The reul from he equivalen cenario i no hown. 100

113 101

114 102

115 103

116 104

117 Reference Aalen, O. O., Borgan, Ø., & Gjeing, H. K. (2010). Survival and Even Hior Anali. Springer. Banklovkommijonen. (2010). Penjonlovene og folkergdreformen I. Norge offenlige uredninger. Bølviken, E., & Moe, A. (2008). Gabler ne beregninggrunnlag GAP07. Cannon, E., & Tonk, I. (2008). A hor hior of annuiie. Annui Marke, Dow, J. B. ( ). Earl acuarial work in eigheenh-cenur Scoland. Tranacion of he Facul of Acuarie, 33: EIOPA. (2009). CEIOPS' Advice for Level 2 Implemening Meaure on Solvenc II: SCR andard formula - Counerpar defaul rik module. EIOPA. (2010a). Manual for he compleion of he QIS5 preadhee for olo underaking. EIOPA. (2010b). Quaniaive Impac Sud 5 - Queion & Anwer. European Commiion. (2010b). Annexe o he QIS5 Technical Specificaion. European Commiion. (2010a). QIS5 Technical Specificaion. Finanilne. (2011b). Erfaringer fra QIS5 - og veien videre. Finanilne. (2010). QIS5 Informajonmøe for livforikringelkap augu Finanilne. (2009). Veiledning for ree for livforikring - verjon 6c. Finanilne. (2011a). Virkomheregler i livforikring. Leer o he Minir of Finance, March 8h Gerber, H. U. (1997). Life Inurance Mahemaic. Springer-Verlag. Hull, J. C. (1993). Opion, fuure, and oher derivaive ecuriie. Prenice Hall. J.P. Morgan. (1997). CrediMeric - Technical Documen. J.P. Morgan. (1996). RikMeric - Technical Documen. Lillevold & Parner AS. (2010). Referaneplan. The European Parliamen and of he Council. (2009). Direcive 2009/138/EC. Official Journal of he European Union,