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

2 II

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

4 IV

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

6 VI

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

11 XI

12 XII

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