A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul"

Transcription

1 universiy of copenhagen Universiy of Copenhagen A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI: 1.339/risks32183 Publicaion dae: 215 Documen Version Publisher's PDF, also known as Version of record Ciaion for published version (APA): Jensen, N. R., & Schomacker, K. J. (215). A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk. Risks, 3(2), /risks32183 Download dae: 19. Jun. 216

2 Risks 215, 3, ; doi:1.339/risks32183 OPEN ACCESS risks ISSN Aricle A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Ninna Reizel Jensen 1, * and Krisian Juul Schomacker 2 1 Deparmen of Mahemaical Sciences, Universiy of Copenhagen, Universiesparken 5, DK-21 København Ø, Denmark 2 Edlund A/S, Bjerregårds Sidevej 4, DK-25 Valby, Denmark; * Auhor o whom correspondence should be addressed; Tel.: Academic Edior: Nadine Gazer Received: 3 November 214 / Acceped: 27 May 215 / Published: 4 June 215 Absrac: Using a wo-accoun model wih even risk, we model life insurance conracs aking ino accoun boh guaraneed and non-guaraneed paymens in paricipaing life insurance as well as in uni-linked insurance. Here, even risk is used as a generic erm for life insurance evens, such as deah, disabiliy, ec. In our reamen of paricipaing life insurance, we have special focus on he bonus schemes consolidaion and addiional benefis, and one goal is o formalize how hese work and inerac. Anoher goal is o describe similariies and differences beween paricipaing life insurance and uni-linked insurance. By use of a wo-accoun model, we are able o illusrae general conceps wihou making he model oo absrac. To allow for complicaed financial markes wihou dramaically increasing he mahemaical complexiy, we focus on economic scenarios. We illusrae he use of our model by conducing scenario analysis based on Mone Carlo simulaion, bu he model applies o scenarios in general and o wors-case and bes-esimae scenarios in paricular. In addiion o easy compuaions, our model offers a common framework for he valuaion of life insurance paymens across produc ypes. This enables comparison of paricipaing life insurance producs and uni-linked insurance producs, hus building a bridge beween he wo differen ways of formalizing life insurance producs. Finally, our model disinguishes iself from he exising lieraure by aking ino accoun he Markov model for he sae of he policyholder and, hereby, faciliaing even risk.

3 Risks 215, Keywords: wo-accoun model; economic scenarios; paricipaing life insurance; uni-linked insurance; sochasic differenial equaions; guaranees; bonus; fairness; marke valuaion 1. Inroducion Classical life insurance mahemaics deals wih he compuaion of reserves and cash flows for guaraneed paymens in paricipaing life insurance. Non-guaraneed paymens in paricipaing life insurance and guaraneed and non-guaraneed paymens in uni-linked insurance depend on he evoluion of he financial marke, and his makes hem difficul o model, in paricular on op of he sae model of he policyholder. Noe ha by non-guaraneed paymens, we mean all fuure paymens ha are no guaraneed, wih bonus in paricipaing life insurance as he leading example. The paper [1] offers examples of his advanced combined modeling in he case of a Black Scholes financial marke. To lower he mahemaical complexiy and allow for more complicaed financial markes while mainaining a general biomeric sae model, we focus on economic scenarios. An economic scenario could, for example, consis of a sample pah for he shor ineres rae and/or a sock index. The scenarios may be wors-case scenarios, sress scenarios from Solvency II, scenarios generaed via Mone Carlo simulaion or bes-esimae scenarios. For a given scenario, he balance of he policy is projeced ino he fuure. For scenarios generaed via Mone Carlo simulaion, one obains a valid approximaion of he expeced fuure paymens, guaraneed as well as non-guaraneed, by averaging over sufficienly many projecions (as is common pracice wih Mone Carlo simulaion). For wors-case or bes-esimae scenarios, a single projecion is enough o obain he corresponding wors-case or bes-esimae approximaion of he fuure paymens. We will no go ino deails abou he generaion of sochasic scenarios; we simply ake hem as financial inpu o our model. For Mone Carlo simulaion, we refer o [2]. For he generaion of wors-case scenarios, we refer o [3]. Scenario-based calculaions have he advanage ha he overall projecion approach does no change wih he financial model, because he sochasic scenarios are he only financial inpu. Economic scenarios are widely used in he insurance indusry; see, for example, [4] and Chaper 1 A Simulaion-Based ALM Model in Pracical Use by a Norwegian Life Insurance Company in [5]. Considering non-guaraneed paymens when valuing he liabiliies has many applicaions, such as risk managemen, produc developmen and solvency. In [6], Paragraph 79, i is saed ha he value of financial guaranees and conracual opions should ake ino accoun non-guaraneed as well as guaraneed paymens. Scenario-based calculaions allow for marke valuaion, solvency calculaions, hedging and pricing of guaraneed and non-guaraneed paymens in paricipaing life and uni-linked insurance. Hence, scenario-based calculaions, which is exacly wha we propose, are useful for complying wih curren Solvency II regulaion. We model life insurance conracs using wo ineracing accouns described by sochasic differenial equaions. One accoun measures he asses, and he oher accoun is a echnical accoun. For each scenario, he sochasic differenial equaions simplify o deerminisic differenial equaions ha can be solved numerically. A numerical soluion can, for example, be obained by applying a simple numerical

4 Risks 215, discreizaion. Thereby, our model is simple o implemen. Furhermore, our model allows us o model paricipaing life and uni-linked insurance in he same framework. By doing so, we are able o compare he wo. In heir naure, uni-linked and paricipaing life insurance seem differen, bu hey are really no. The producs may vary in riskiness, bu projecion-wise, hey are almos he same. The main difference lies in he specificaion of how non-guaraneed paymens arise, saed in he conrac from he beginning (uni-linked) or deermined fairly by he company along he way (paricipaing life). By use of a wo-accoun model, we are able o illusrae general conceps wihou making he model oo absrac. Our wo-accoun model is based on he wo-accoun model in [7], and boh models offer a common framework for he valuaion of guaraneed and non-guaraneed paymens in paricipaing life and uni-linked insurance. Our model disinguishes iself from he model in [7] by aking ino accoun he Markov model for he sae of he policyholder, hereby including even risk. Here, even risk is used as a generic erm for life insurance evens, such as deah, disabiliy, ec. The exising lieraure considers he valuaion of guaraneed and non-guaraneed paymens in paricipaing life insurance or uni-linked insurance wihou even risk, whereas a common framework and inclusion of even risk are rare. The papers [8 11] are examples of recen lieraure ha considers valuaion in paricipaing life insurance wihou (or wih very limied) even risk. The papers [12 14] are oher examples wihin he same area, bu heir focus is more risk-relaed. On he oher hand, [15] models paricipaing life insurance, aking ino accoun he Markov model for he sae of he policyholder, bu he model is only racable for a very simple financial environmen, and i does no apply o uni-linked insurance. The work in [1] and, more specifically, [16] cover paricipaing life and uni-linked insurance wih broad even risk, bu only in a Black Scholes marke, and he resuls involve non-rivial parial differenial equaions. In our reamen of paricipaing life insurance, we have special focus on bonus allocaion and on he bonus schemes consolidaion and addiional benefis. These bonus schemes are he mos common in he Danish life insurance and pensions indusry, bu o our knowledge, consolidaion is barely menioned in he lieraure. An imporan goal of his paper is o formalize exacly how hese bonus schemes work and inerac. Oher papers wih a similar focus on bonus include [17 2], bu again, none of hem include even risk. In our reamen of uni-linked insurance, we have special focus on he implemenaion of guaranees and on he similariies and differences in relaion o paricipaing life insurance. For boh produc ypes, we include numerical examples o demonsrae he possible applicaions of our wo-accoun model. In Secion 2, we discuss scenario-based projecion in general. Our main focus is on projecion level and which measure o projec under (physical or pricing measure). In Secion 3, we discuss valuaion bases in life insurance and formalize a common model for he sae-wise evoluion of he policies under consideraion. In Secion 4, we consider paricipaing life insurance. We briefly ouch upon differen bonus schemes, and we presen our wo-accoun model for a general paricipaing life insurance policy, alhough no allowing for policyholder behavior opions. We include simple survival model examples o illusrae formulas and provide inuiion. We end he secion wih a numerical example building on he survival model. The example illusraes a fair bonus sraegy and he risk of unfair redisribuion beween policies in a porfolio. I also highlighs some of he many possible applicaions of scenario-based calculaions. In Secion 5, we consider uni-linked insurance. We ouch upon differen aspecs of uni-linked insurance, and we presen our wo-accoun model for a general uni-linked

5 Risks 215, insurance policy. Again, we include simple and illusraive survival model examples. We end he secion wih a numerical example ha is a uni-linked version of he numerical example in he previous secion. The example illusraes a fair guaranee fee sraegy, and we compare he uni-linked insurance policy o is paricipaing life insurance counerpar, making good use of our common modeling framework. 2. Projecion in General In paricipaing life insurance, we inroduce sochasic scenarios o allow for marke valuaion of non-guaraneed paymens, pricing and hedging of guaranees, bonus and benefi prognoses and solvency calculaions. In uni-linked insurance, we inroduce sochasic scenarios o uilize reiremen savings and benefi prognoses, solvency calculaions and hedging and pricing of uni-linked guaranees. In boh cases, each scenario consiss of wo sample pahs: one for he shor ineres rae, r, and one for he reurn on he fund ha he policyholder and/or he insurance company has chosen o inves in, R X. We assume ha he sochasic scenarios arise from a financial model equipped wih a physical measure P and a risk-neural pricing measure Q. For each scenario, we projec he accouns ha deermine he financial progress of a given policy. When making he projecions in paricipaing life insurance, as well as in uni-linked insurance, i is imporan o bear in mind he requesed oucome. For pricing, hedging, marke valuaion and solvency assessmens of guaraneed and non-guaraneed paymens (in paricipaing life) or of a provided guaranee (in uni-linked) or for examining a bonus allocaion sraegy (in paricipaing life), i is he expeced evoluion of he policy, boh financially and across saes, ha is relevan. Hence, he evoluion of he policy is considered on an average porfolio level. For pricing, hedging and marke valuaion, he projecions are carried ou under he pricing measure (Q), since he focus is on pricing and valuaion. For solvency assessmens, he projecions are carried ou under he physical measure (P ) up o some relevan ime poin, and from hen on, hey are carried ou under he pricing measure (Q). For examining a bonus allocaion sraegy and quanifying he oal expeced fuure bonus, he projecions are carried ou under he physical measure (P ), since he focus is on he acual oucome. For reiremen savings, benefi and bonus prognoses, i is he expeced financial evoluion of he policy ha is relevan. The policyholder wans o know wha o expec in each sae, no he average expecaion. Hence, he evoluion of he policy is considered on an individual policy level. However, in paricipaing life insurance, he amoun of bonus allocaed o he policy depends on he financial evoluion and he average evoluion of he policy. Hence, for he purpose of prognoses in paricipaing life insurance, he asses and he reserves mus, firs, be projeced on porfolio level o produce a sample pah for he bonus allocaion. Second, he sample pah for he bonus allocaion is used o projec he reserves on an individual pah-wise policy level. In eiher case, he projecions are carried ou under he physical measure (P ), since he focus is on he acual bonus, reiremen savings and benefis. In his paper, we limi our focus o projecion on porfolio level and leave projecion on policy level for fuure research.

6 Risks 215, Valuaion Bases and Insurance Model A cornersone in life insurance mahemaics is he principle of equivalence, which saes ha he expeced presen values of premiums and benefis should be equal. The principle relies on he law of large numbers ha will, hen, on average, make premiums and benefis balance in a large insurance porfolio. To apply he equivalence principle, one needs assumpions abou ineres, moraliy and oher relevan economic-demographic elemens. The uncerain developmen of hese elemens subjecs he insurance company o a risk ha is independen of he size of he porfolio. In paricipaing life insurance, he insurance company can neiher raise he premiums nor reduce he benefis along he way, so he only way for he insurance company o miigae his risk is o build a safey loading ino he premiums. This is done by performing he equivalence principle under conservaive assumpions abou ineres, moraliy, ec. These assumpions make up he so-called echnical basis, and i represens a provisional wors-case scenario for is elemens. Below, we mark elemens of he echnical basis by superscrip. For marke-consisen valuaion of fuure paymens, he echnical basis does no apply due o is wors-case naure. Insead, valuaion is performed under he so-called marke basis, which is made up of bes-esimae assumpions abou he various elemens. Below, we mark elemens of he marke basis by superscrip m. In uni-linked insurance, he benefis are ypically allowed o flucuae wih he marke, hereby making he echnical basis superfluous. In paricipaing life insurance, as well as in uni-linked insurance, we consider a policy whose sae-wise evoluion is governed by a coninuous-ime Markov process Z wih a finie sae space J, saring a zero. For a deailed descripion of he Markov model, see [21] or [22]. For k, j J, j k, we define he couning process N jk and he indicaor process I k by N jk () = # {s : Z (s ) = j, Z (s) = k}, I k () = 1 {Z()=k}. Wih his definiion, N jk () couns he number of jumps from sae j o sae k unil ime, and I k () indicaes a sojourn in sae k a ime. Under he echnical basis, we model he evoluion of Z by he ransiion inensiies µ jk (), j, k J, j k, and under he marke basis, we model he evoluion of Z by he ransiion inensiies µ m jk (), j, k J, j k. The corresponding echnical and marke ransiion probabiliies from sae j o sae k over he ime inerval [, s] are denoed by p jk (, s) and p m jk (, s), and wih =, m indicaing he basis, we have p µ jk (, + h) jk () = lim. h h The ransiion probabiliies can be calculaed numerically from he ransiion inensiies by use of he Kolmogorov equaions; see, for example, [21]. We assume ha he process Z governing he sae of he policy is independen of he financial marke, and under boh P and Q, he evoluion of Z is described by he ransiion inensiies from he marke basis. In addiion o he marke ransiion inensiies, he marke basis consiss of a marke ineres rae. The marke basis has no more elemens, as we do no ake expenses or any oher economic-demographic elemens ino accoun. Similarly, he echnical basis consiss of he echnical ransiion inensiies and a echnical ineres rae. In uni-linked insurance, only he marke basis comes ino play.

7 Risks 215, Paricipaing Life Insurance In his secion, we consider paricipaing life insurance. We ouch upon differen bonus schemes, and we presen our wo-accoun model for a general paricipaing life insurance policy. We include simple survival model examples o illusrae formulas and provide inuiion. We end he secion wih a numerical example building on he survival model Non-Guaraneed Paymens (Bonus) In paricipaing life insurance, he guaraneed paymens are based on he echnical basis. The conservaive echnical basis gives rise o a sysemaic surplus ha is o be paid back o he policyholders in erms of bonus. There are many possible ways o do so. For a shor survey, see [1]. We consider a bonus scheme consising of wo seps: firs, consolidaion, and hen, when he policy is consolidaed on a sufficienly low echnical ineres rae (if ever), addiional benefis. The bonus scheme consolidaion (in Danish syrkelse ) is much used in he Danish marke, bu i can easily be skipped below, heading sraigh for he bonus scheme addiional benefis. The bonus scheme consolidaion is primarily used for policies wih a echnical ineres rae ha is oo high compared o he marke ineres rae. Bonus is used o consolidae he policy on a lower echnical ineres rae. By consolidae, we mean ha he echnical ineres rae is lowered wihou changing he guaraneed paymens. This may seem o be less favorable for he policyholder, bu since he guaraneed paymens are no changed, he policyholder is no worse off. When a sufficienly low echnical ineres rae has been reached, he remaining bonus is used for addiional benefis. Hence, consolidaion does no benefi he policyholder in erms of more favorable paymens immediaely afer bonus paymens, bu i helps o ensure ha he liabiliies of he policy can be me. Furhermore, he lower echnical ineres rae gives rise o a higher sysemaic surplus in he fuure, which will evenually be redisribued and refleced in he paymens. The bonus scheme addiional benefis is primarily used for policies wih a low echnical ineres rae compared o he marke ineres rae. Bonus is used o increase pars of he guaraneed benefis proporionally, whereas he remaining benefis, he premiums and he echnical ineres rae are mainained. I is usually he reiremen par of he benefis (such as a pure endowmen or a life annuiy) ha is increased and he insurance par of he benefis (such as a erm insurance or disabiliy coverage) ha is no. There is good reason o increase he reiremen par of he benefis insead of decreasing he premiums or increasing all of he benefis, since he reiremen benefis are ypically se according o which premiums he policyholder can afford and which insurance coverage he/she needs, and no he oher way around. Furhermore, here is good reason o increase he reiremen benefis proporionally, as he benefi profile reflecs he policyholder s preferences Produc Specificaion We consider a paricipaing life insurance policy wih guaraneed paymens based on a echnical basis whose elemens are marked by superscrip. The sae-wise evoluion of he policy is described in Secion 3. We le r () denoe he echnical ineres rae a ime. By B u, B f and C, we denoe he

8 Risks 215, guaraneed paymen sreams a ime. Here, C is he premium sream ( C for conribuions), B u is he benefi sream for he benefis ha are increased ( B for benefis and superscrip u for upscaled) and B f is he benefi sream for he benefis ha are kep fixed (superscrip f for fixed). The paymens sreams are given by dc = j J db i = j J I j dc j, I j db i j + j,k J :k j b i jk dn jk, i = f, u, where c j, b f j and bu j are deerminisic, sae-wise paymen sreams and b f jk and bu jk are deerminisic lump sum paymens upon jumps. We noe ha we, hereby, exclude policyholder behavior opions, such as surrender and free policy, since hey imply non-deerminisic paymens. However, for surrender modeling, see he remark on Page 197. Examples of deerminisic lump sum paymens upon jumps include insurance coverage, such as a deah sum, a disabiliy sum or a sum upon criical illness. The policy erminaes a ime T. Thereafer, here are no paymens Two-Accoun Model We denoe by X he asses of he policy, including is share of he collecive bonus poenial, and by Y, we denoe he marke expeced echnical reserve for he policy. By marke expeced echnical reserve, we mean he expecaion of fuure sae-wise echnical reserves where he expecaion across saes is aken under he marke basis. Thus, he marke expeced echnical reserve is no a sae-wise reserve, bu a probabiliy weighed sum of sae-wise reserves. The accouns X and Y are he backbone of our wo-accoun model. The policy is issued before or a ime, and he wo accouns amoun o X ( ) = x and Y ( ) = y jus before ime. For a policy issued a ime, y and x are boh zero. For a policy issued before ime, y is equal o he echnical reserve for he policy jus before ime, and x is equal o he asses of he policy jus before ime. Boh are assumed o be known when iniiaing he projecion. The asses X are invesed in a fund wih sochasic reurn R X. The marke expeced echnical reserve Y accumulaes according o he echnical ineres rae. In good imes, he reurn rae on he asses exceeds he echnical ineres rae. Pars of he excess reurn are allocaed o he policy in erms of bonus, which adds o he marke expeced echnical reserve, bu pars are saved for imes where he reurn rae on he asses is less favorable. In really bad imes, he asses may be insufficien o cover he guaraneed paymens of he policy. In ha case, he equiy holders of he insurance company sep in wih a capial injecion aken from he company s equiy. We speak of he possible capial injecion as a guaranee injecion, and is role is o raise he asses in case of unfavorable developmens in he financial marke. The policyholder pays for he company s risk aking by having a guaranee fee deduced from he asses and paid o he equiy holders of he insurance company in good imes. We assume ha he insurance company s equiy is always sufficien o cover he guaranee injecions and ha all guaranee injecions and guaranee fees are seled via he equiy. In Denmark, he guaranee fee used o be known as he drifsherreillæg (ranslaes o echnical yield ). All of he above does no happen coninuously, bu

9 Risks 215, 3 19 a pre-specified, deerminisic ime poins < 1 <... < n = T (for example, once a year) where he wo accouns X and Y are updaed. We le ɛ () = # {i = 1,..., n : i } coun he number of updaes prior o ime. The updaes consis of bonus allocaion d (if funds are sufficien), guaranee injecion g from he equiy holders of he insurance company (if needed) and deducion of he guaranee fee π g in reurn for he possible guaranee injecion. All hree are non-negaive. For echnical convenience, we assume ha he sochasic reurn on he asses, R X, does no jump a he ime poins < 1 <... < n = T wih accoun updaes. Furhermore, for all wih dɛ () = 1, i.e., for all ime poins wih an accoun updae, we assume ha d () and π g () are known a ime and ha g () is calculaed a ime. This is o ensure predicabiliy and, hereby, sochasic inegrabiliy Bonus Mechanisms As menioned, we consider a bonus scheme where bonus allocaed o he policy is, firs, used o lower he echnical ineres rae unil i his a pre-described level r. Typically, his level coincides wih he echnical ineres rae for new policies. Thereafer, bonus is used o increase he benefis B u. The addiional benefis are priced using he echnical ransiion inensiies µ jk and he echnical ineres rae r. This means ha he minimum echnical ineres rae for consolidaion and he pricing ineres rae for addiional benefis are assumed o coincide. One could have chosen anoher echnical ineres rae for he pricing of addiional benefis, bu ha would require a division of he echnical reserve on wo differen echnical bases, so we insis on using r. We le r (n) denoe he echnical ineres rae afer he n-h bonus accrual and k (n) denoe he upscaling of he benefis B u afer he n-h bonus accrual. We noe ha he upscaling facor sars a one, i.e., k () = 1. Afer he n-h bonus accrual, he guaraneed benefi sream for he policy is given by B (n) = k (n) B u + B f. We poin ou ha r (n) and k (n) depend on he developmen of he financial marke and are herefore sochasic. However, for each economic scenario, we have a procedure for calculaing hem according o he equivalence principle. The procedure is presened in Secion 4.8. We noe ha we have k (n) = 1 for all n wih r (n) > r, and if k (n) > 1, hen necessarily r (n) = r. This is because we do no increase he guaraneed benefis unil he echnical ineres rae has been lowered o r. Finally, we noe ha he echnical ineres rae and he upscaling facor amoun o r (ɛ()) and k (ɛ()) a ime, since here has been ɛ () accoun updaes a ime. For all wih dɛ () = 1, ha is for all ime poins wih an accoun updae, we assume ha he echnical ineres rae r (ɛ()) and he upscaling facor k (ɛ()) are calculaed a ime. Again, his is o ensure predicabiliy. Furhermore, addiional benefis are in effec from ime, such ha benefis paid ou a ime include he upscaling k (ɛ()). The laer ensures ha a policyholder wih a final lump sum paymen acually benefis from he las bonus updae.

10 Risks 215, Technical Reserves We denoe by V f,,+ j (, ρ) and V u,,+ j (, ρ) he sae-wise echnical benefi reserves for he benefi sreams B f and B u given ha he policy is in sae j and ha he echnical ineres rae is ρ. Similarly, we denoe by V, j (, ρ) he sae-wise echnical premium reserves for he premium sream C. Noe ha we use superscrip + o indicae he benefi reserves and superscrip o indicae he premium reserve. Furhermore, we use superscrip o indicae ha he reserve is evaluaed under he echnical basis. Finally, we use he generic consan ρ in place of he echnical ineres rae, because we need o evaluae he echnical reserves for differen echnical ineres raes in connecion wih he bonus scheme consolidaion. We have [ T V, j (, ρ) = E T ] e ρ(s ) dc (s) Z () = j = e ρ(s ) p jl (, s) dc l (s), l J [ T ] (1) V i,,+ j (, ρ) = E e ρ(s ) db i (s) Z () = j T = e { ρ(s ) p jl (, s) db i l (s) + } µ lk (s) b i lk (s) ds l J k J :k l for i = f, u. Here, E denoes echnical expecaion and p jl is he echnical probabiliy of ransiion from sae o j. Boh are deermined by he ransiion inensiies from he echnical basis. The sae-wise echnical reserves can be calculaed numerically by use of Thiele s differenial equaions; see [23]. We denoe by Vi (, ρ, k) he sae-wise echnical reserve for he (parly upscaled by k) paymen sream B f + kb u C, given ha he policy is in sae i and ha he echnical ineres rae is ρ, i.e., V i (, ρ, k) = kv u,,+ i (, ρ) + V f,,+ i (, ρ) V, i (, ρ), i J. (2) Here, V f,,+ j, V u,,+ j, and V, j are he sae-wise echnical benefi and premium reserves defined in Equaion (1). Wih he inroducion of Vi, we can wrie he marke expeced echnical reserve as Y () = j J p m j (, ) V j (, r (ɛ()), k (ɛ())). We recall ha p m j is he marke probabiliy of ransiion from sae o j, which is deermined by he ransiion inensiies from he marke basis. We noe ha he sochasiciy in Y () comes from he sochasic developmen of he echnical ineres rae r (ɛ()) and he upscaling facor k (ɛ()). However, for each, he echnical ineres rae r (ɛ()) is deermined as a consan ineres rae over [, T ], so we never plug a non-consan echnical ineres rae ino he reserves in Equaion (1) when calculaing V j (, r (ɛ()), k (ɛ())). Example 1 (Survival model). We consider a simple example ha provides he basis for numerical illusraions laer on. The sae of he policy is described by he classical survival model wih wo saes, (alive) and 1 (dead). The paymens of he policy consis of a consan coninuous premium paymen π while alive, a erm insurance sum b ad upon deah before expiraion T and a pure endowmen

11 Risks 215, sum b a upon survival unil expiraion T. Under he bonus scheme addiional benefis, bonus is used o increase he endowmen sum. There are no paymens in he deah sae. For simpliciy, we wrie I = I 1, N = N 1, µ = µ 1 and p = p for =, m, and we have The paymen sreams of he policy read p (s, ) = e s µ (v) dv, s. dc () = πi d, T, db f () = b ad dn (), T, db u () = b a I () dɛ T (), T, where ɛ T is he Dirac measure in T, i.e., for a measurable se A R { 1 for T A, ɛ T (A) = 1 {T } (A) = for T / A. We noe ha T db u () = b a I (T ). The echnical premium and benefi reserves are zero in he sae dead, and in he sae alive, hey read 4.6. Cash Flows V, (, ρ) = V f,,+ (, ρ) = V u,,+ (, ρ) = = π T T T = b ad T T e ρ(s ) p (, s) π ds e ρ(s ) e s µ (v) dv ds, T, e ρ(s ) p (, s) b ad µ (s) ds e ρ(s ) e s µ (v) dv µ (s) ds, T, e ρ(s ) p (, s) b a dɛ T (s) = b a e ρ(t ) e T µ (v) dv, T. For projecion on porfolio level, i i useful o consider marke cash flows of he policy. Here, we use he erm marke cash flows for he expecaion of he sochasic paymen sreams aken under he marke

12 Risks 215, basis. By ς, β f and β u, we denoe he ime marke cash flows for he premium sream C and he benefi sreams B f and B u, i.e., [ ] ς () = E m dc (s) = j J β i () = E m [ = j J p m j (, s) dc j (s), ] db i (s) { p m j (, s) db i j (s) + k J :k j µ m jk (s) b i jk (s) ds }, i = f, u, where he expecaion E m is aken under he marke basis. Furhermore, we need he marke expeced marke reserve. By V (), we denoe he marke expeced marke reserve a ime for he mos recenly guaraneed paymen sream B (ɛ()) C = B f + k (ɛ()) B u C, i.e., V () = E m k (ɛ()) [ E m k (ɛ()) [ T = T e s r(v) dv d ( B (ɛ()) C ) ]] (s) Z () e s r(v) dv ( k (ɛ()) dβ u (s) + dβ f (s) dς (s) ). Here, E m denoes marke expecaion given k (ɛ()), r is he sochasic shor ineres rae and r k (ɛ()) is he yield curve seen from ime. Similar o he marke expeced echnical reserve, he marke expeced marke reserve is no a sae-wise reserve, bu a marke probabiliy weighed sum of sae-wise marke reserves. This is no eviden from he formula above, since he reserve simplifies due o he ower propery. We emphasize ha only addiional benefis, and no consolidaion, raise he guaranee. However, consolidaion has an effec on he non-guaraneed benefis as he echnical reserve increases. Example 2 (Survival model coninued). For he simple policy in Example 1, he ime marke premium and benefi cash flows read ς () = β f () = β u () = p m (, s) π ds = π p m (, s) b ad µ m (s) ds = b ad The marke expeced marke reserve V reads e s µm (v) dv ds, T, e s µm (v) dv µ m (s) ds, T, p m (, s) b a dɛ T (s) = b a e T µm (v) dv I { T }, T. V () = e T + T r (v) dv e T µm (v) dv k (ɛ()) b a e s r(v) dv e s µm (v) dv ( b ad µ m (s) π ) ds. (3)

13 Risks 215, Two-Accoun Projecion On porfolio level, he asses X and he marke expeced echnical reserve Y of he policy evolve according o he sochasic differenial equaions (SDEs) dx () = X ( ) dr X () dβ f () k (ɛ()) dβ u () + dς () + [g () π g ()] dɛ (), T, X ( ) = x, dy () = Y () r (ɛ()) d dβ f () k (ɛ()) dβ u () + dς () (4) Y ( ) = y. Here, α is an adjusmen erm given by α (, ρ, k) = j J + d () dɛ () + α (, r (ɛ()), k (ɛ())) d, T, l J :l j p m j (, ) ( µ jl () µ m jl () ) ( ) Vj (, ρ, k) kb u jl () b f jl () V l (, ρ, k) where Vi (, ρ, k) are he sae-wise echnical reserves defined in Equaion (2). The adjusmen erm accouns for he marke expeced surplus arising from he conservaive echnical ransiion inensiies. See, for example, [15]. We recall ha R X is he sochasic reurn on he asses, g is he guaranee injecion provided by he equiy holders of he insurance company, π g is he guaranee fee deduced from he asses and paid o he equiy holders, d is he allocaed bonus and ɛ couns he number of updaes of guaranee injecion, guaranee fee and bonus (ypically annual). The bonus d and he guaranee fee π g are specified by he company, whereas he guaranee injecion g is designed o ensure ha he asses are a leas equal o he guaraneed liabiliies. We define he guaraneed liabiliies L as he maximum of he marke expeced marke reserve and he marke expeced echnical reserve for he guaraneed paymens, i.e., L () = max {V (), Y ()}. (5) This definiion has been common pracice in Denmark since he inroducion of marke values. However, he guaraneed liabiliies can easily be defined differenly, for example L = V. The guaranee injecion g () is calculaed according o he formula g () = (L ( ) (X ( ) π g ())) +. (6) This guaranee design ensures ha he asses X are sufficien o cover he guaraneed liabiliies L afer he guaranee fee π g has been paid o he equiy holders of he insurance company. The guaraneed liabiliies L represen he lowes amoun ha he insurance company can se aside for he guaraneed paymens. Hence, he asses should always exceed he guaraneed liabiliies, and by design of he guaranee injecion, his will always be he case afer adding he guaranee injecion. The inclusion of he guaranee fee is a echnicaliy ha ensures ha he asses are no drained by guaranee fee paymens,

14 Risks 215, o he equiy holders of he insurance company in bad imes where he liabiliies exceed he asses. By he design of he guaranee injecion, no guaranee fee is deduced from he asses in hose imes. In Secion 4.9, we ge ino deails abou how he bonus allocaion and guaranee fee are deermined. The sochasic elemen R X eners via a sample pah for he asse reurns. Furhermore, he size of he guaranee injecion g depends on he sample pah for he shor ineres rae. In pracice, one will ofen work wih a discreized version of he sochasic differenial equaions in Equaion (4). For an example, see Secion Example 3 (Survival model coninued). For he simple policy in Example 1 2, he adjusmen erm α reads α (, ρ, k) = p m (, ) (µ () µ m ()) ( V (, ρ, k) b ad), where he oal echnical reserve V in he sae alive is given by V (, ρ, k) = kv u,,+ (, ρ) + V f,,+ (, ρ) V, (, ρ) = kb a e ρ(t ) e T + T µ (v) dv e ρ(s ) e s µ (v) dv ( b ad µ (s) π ) ds, T Procedure for Deermining he Technical Ineres Rae and he Upscaling Facor Assume ha dɛ () = 1, meaning ha here is an updae a ime. In deermining he echnical ineres rae r (ɛ()) and he upscaling facor k (ɛ()), he disribuion of he policy across saes a ime eners. The disribuion depends on he choice of basis; in our case, he echnical basis or he marke basis. The marke basis reflecs he rue disribuion of he policy across saes. Therefore, we srongly sugges o work under he marke basis. Working under he echnical basis has he advanage ha he ower propery applies (see below), which limis he number of compuaions. However, aking he shor cu and using he arificial echnical basis leads o a wised picure of he evoluion of he policy, so we discourage i. For compleeness, we include boh opions and model hem by below. Assume ha r (ɛ( )) > r, so ha he policy is sill in he consolidaion phase of he bonus scheme. Then, necessarily, k (ɛ( )) = 1 (since we consolidae firs), and he echnical ineres rae r (ɛ()) is deermined as he soluion o he equaion Y ( ) + d () = V, (, r (ɛ())), (7) where V, (, ρ) is he marke or echnical (indicaed by he ) expeced echnical reserve for he paymen sream B () C = B f + B u C, given ha he echnical ineres rae is ρ. Tha is V, (, ρ) = E [ E [ T = E [ V Z() (, ρ, 1) ], e ρ(s ) d ( B f + B u C ) ]] (s) Z () where he sae-wise echnical reserves V j, j J, are given in Equaion (2), and E denoes marke or echnical expecaion. Hence, r (ɛ()) is he echnical ineres rae ha complies wih he equivalence

15 Risks 215, principle on porfolio level. simplifies o Under he echnical basis, he ower propery applies, and he reserve V, (, ρ) = j J = T p j (, ) V j (, ρ, 1) e ρ(s ) d ( β f, + β u, ς ) (s), where ς, β f, and β u, are he ime echnical cash flows for he premium sream C and he benefi sreams B f and B u. This means ha he reserve can be calculaed using only he echnical cash flows. Under he marke basis, he reserve reads V,m (, ρ) = j J p m j (, ) V j (, ρ, 1). Hence, using he marke basis, boh ransiion probabiliies and sae-wise echnical reserves are needed in order o solve Equaion (7). This is a drawback, bu in our opinion, i is no enough o swich o he arificial echnical basis. If he soluion r (ɛ()) is sricly smaller han r, hen r (ɛ()) is se o r, and he remaining bonus Y ( ) + d () V, (, r ) is used o raise he upscaling facor k (ɛ()) as below. Oherwise, we se k (ɛ()) = 1. Now, assume ha r (ɛ( )) = r. Then, he policy is in he addiional benefis phase of he bonus scheme, and we se r (ɛ()) = r. The upscaling facor k (ɛ()) is deermined as he soluion o he equaion i.e., d () = ( k (ɛ()) k (ɛ( ))) V u,,,+ ( ), k (ɛ()) = k (ɛ( )) + d () V u,,,+ ( ). Here, V u,,,+ is he marke or echnical (indicaed by he ) expeced echnical reserve for he benefi sream B u, given ha he ineres rae is r, i.e., V u,,,+ () = E [ E [ T = E [ V u,,+ Z() (, r ) ]] e r (s ) db u (s) Z () ], where he sae-wise echnical benefi reserves V u,,+ j, j J, are given in Equaion (1), and E denoes marke or echnical expecaion. Hence, k (ɛ()) is he upscaling facor ha saisfies he equivalence principle on porfolio level. Under he echnical basis, he ower propery applies, and he reserve simplifies o V u,,,+ () = j J = T p j (, ) V u,,+ j (, r ) e r (s ) dβ u, (s).

16 Risks 215, Under he marke basis, i reads V u,,m,+ () = j J p m j (, ) V u,,+ j (, r ). Again, we see ha, using he marke basis, boh ransiion probabiliies and sae-wise echnical reserves are needed. We emphasize ha here is no reason o consider he case r (ɛ( )) < r. For he explanaion, recall ha consolidaion serves o lower he echnical ineres rae, so if he echnical ineres rae is already low, here is no need for consolidaion. If he iniial echnical ineres rae r (ɛ()) is high compared o he pre-described level r, he allocaed bonus is used for consolidaion unil r (ɛ()) = r for some. Thereafer, he bonus is used for addiional benefis, and he echnical ineres rae is kep fixed. If he iniial echnical ineres rae r (ɛ()) is equal o r, he consolidaion phase is skipped, he allocaed bonus is used for addiional benefis and he echnical ineres rae is kep fixed from he beginning. In neiher case, we arrive a r (ɛ( )) < r. In he hird and las case where he iniial echnical ineres rae r (ɛ()) is low compared o r, here is clearly no need for consolidaion. Now, one has wo opions. Eiher, one can lower r o r (ɛ()) and proceed as in he case where r (ɛ()) is equal o r ; or, one can raise r (ɛ()) o r, use he decline in he echnical reserve for addiional benefis and hen proceed as in he case where r (ɛ()) is equal o r. Boh soluions will aver he case r (ɛ( )) < r. We noe ha he case r (ɛ()) < r represens a siuaion wih increasing echnical ineres rae. This has no been observed in Denmark in recen years, which is why we exclude he case from our paper. However, as argued above, our model can easily handle he case. Example 4 (Survival model coninued). For he simple policy in Example 1 3, he expeced echnical reserve V, (, ρ) reads ( V, (, ρ) = e T µ (v) dv The expeced echnical benefi reserve V u,,,+ ( ) reads e ρ(s ) e s µ (v) dv ( b ad µ (s) π ) ds + b a e ρ(t ) e T V u,,,+ () = b a e µ (v) dv e r (T ) e T µ (v) dv. ) µ (v) dv, T. Remark 1. In Secion 4.2, we menioned ha our seup does no allow for policyholder behavior opions, such as surrender or free policy. However, i is no paricularly complicaed o include surrender, since i is an absorbing sae. For he sake of clariy, we will no go ino deails on how. We jus menion ha, under he bonus scheme addiional benefis, he surrender cash flow needs o be spli ino an upscaled and non-upscaled par. Furhermore, under he bonus scheme consolidaion, he bonus suddenly raises he guaranee hrough a higher surrender value (ypically equal o he echnical reserve), and he marke cash flows need o be recalculaed every ime he policy is consolidaed o accoun for he higher surrender value.

17 Risks 215, Bonus Allocaion and Guaranee Fee In Secion 4.7, we ook he bonus d and he guaranee fee π g as exogenously given. This is imprecise for a leas hree reasons. Firsly, he oal bonus allocaed o he policies in a (homogeneous) porfolio ypically depends on he collecive bonus poenial of he porfolio. The collecive bonus poenial K is defined as he maximum of zero and asses less guaraneed liabiliies, i.e., K () = (X () L ()) +, where he asses X and he guaraneed liabiliies L are calculaed on porfolio level. Wih his definiion, he balance shee can be represened as in Figure 1. Collecive bonus poenial K Asses X Guaraneed liabliies L = max (Y, V ) Figure 1. Porfolio balance shee. The collecive bonus poenial is a resul of he sysemaic surplus o which he conservaive echnical basis gives rise. The sysemaic surplus of he policy is o be paid back o he policyholder in erms of bonus, bu he collecive bonus poenial serves as a buffer in years wih poor financial reurns and/or poor risk resuls, so mos ofen, he sysemaic surplus is no paid ou righ away. Therefore, o avoid redisribuion across policies via he collecive bonus poenial, he porfolio mus be homogeneous wih respec o ineres rae and risk (and coss, bu in his paper, we leave ha ou). Furhermore, in order o avoid redisribuion across generaions, he sysemaic surplus should be paid ou as soon as possible. Secondly, he policy s share of he porfolio bonus depends on how much he policy has conribued o he porfolio s sysemaic surplus. As menioned, he adjusmen erm α in he projecion SDEs in Equaion (4) is he marke expeced surplus arising from he conservaive echnical ransiion inensiies. We choose o pay ou he adjusmen erm immediaely as risk bonus, such ha he collecive bonus poenial collecs surplus from capial gains only. The surplus colleced in he collecive bonus poenial is hen paid ou, bu no immediaely, as ineres rae bonus, i.e., proporional o he marke expeced

18 Risks 215, echnical reserve Y. If he echnical ransiion inensiies are no chosen carefully enough (which can be difficul for varying producs), he adjusmen erm can be negaive for some ages. In ha case, no risk bonus is paid ou. Thirdly, for he conrac o be fair, he bonus d and he guaranee fee π g mus be chosen in such a way ha he equivalence principle is saisfied for he oal paymens under he marke basis, i.e., E Q [ T e s r(v) dv d ( B (ɛ(s)) C ) ] (s) = x. (8) In a muli-policy porfolio, he fairness consrain can be difficul o honor. I is possible o have fairness on porfolio level, bu no on policy level, implying an unfair redisribuion of sysemaic surplus across policies. Ofen, he guaranee fee is a fracion of eiher he asses or he asse reurns. The bonus allocaion akes on more forms, bu is ulimaely a funcion of he collecive bonus poenial, he marke reserve and he echnical reserve. In Secion 4.11, we presen a numerical example wih a one-policy and a wo-policy porfolio. We show how o find a fair bonus and guaranee fee sraegy, and we exemplify he challenges of fairness in a wo-policy porfolio Applicaion of Projecions We recall ha he sae process Z is independen of he financial marke, and ha, under boh P and Q, he evoluion of Z is described by he marke basis. Mos imporanly, he projecions of X and Y can be used o calculae he oal ime marke cash flow CF and marke value W for he guaraneed and non-guaraneed paymens, i.e., o calculae dcf () = E Q [ k (ɛ()) db u () + db f () dc () ] = E Q [ k (ɛ())] dβ u () + dβ f () dς () (9) and W () = E Q [ T = E Q [ T e s r(v) dv d ( B (ɛ(s)) C ) ] (s) e s r(v) dv ( k (ɛ(s)) dβ u (s) + dβ f (s) dς (s) )]. (1) Here, r is he sochasic shor ineres rae. We emphasize ha he cash flow and marke value disinguish hemselves from he usual cash flows and marke values by including non-guaraneed paymens as well as guaraneed paymens. In paricular, we have W () = V () + E Q [ T e ( s r(v) dv k (ɛ(s)) 1 ) ] dβ u (s) where V is he usual marke value from Equaion (3). The addiional erm is he marke value of he non-guaraneed benefis. If he projecions are based on scenarios generaed via Mone Carlo simulaion, hen for each, he expecaion E Q [ k (ɛ())] in Equaion (9) is approximaed by averaging over a sufficien number of Q-projecions up o ime. If, insead, he projecions are of he wors-case or bes-esimae,

19 Risks 215, 3 2 ype (and, hence, singular), hen E [ Q k (ɛ())] is approximaed by he single projeced value. If he shor ineres rae is deerminisic, hen Equaion (1) simplifies o W () = T e s r(v) dv dcf (s). (11) Oherwise, Equaion (1) is approximaed by averaging over a sufficien number of inegraed sample pahs k (ɛ()) dβ u () + dβ f () dς (), discouned by he shor ineres rae. The marke value is useful for deermining he bonus allocaion d and guaranee fee π g according o he fairness crierion in Equaion (8), which can be wrien as W () = x. So far, we have suppressed he influence of he invesmen sraegy, bu i eners hrough he sochasic reurn on he asses. Hence, he ask of deermining d and π g is he classical rade-off beween he aggressiveness of dividend allocaion (expressed by d) and he opion price (expressed by π g ) given he aggressiveness of he invesmen sraegy (ypically expressed by he volailiy). The projecions of X and Y are also useful for calculaing he ime P -expeced cash flow for he guaraneed and non-guaraneed paymens, i.e., for calculaing dcf P () = E P [ k (ɛ()) db u () + db f () dc () ] = E P [ k (ɛ())] dβ u () + dβ f () dς (). If he projecions are based on scenarios generaed via Mone Carlo simulaion, hen for each, he expecaion E [ P k (ɛ())] is approximaed by averaging over a sufficien number of P -projecions up o ime. If insead, he projecions are of he wors-case or bes-esimae ype (and, hence, singular), hen E [ P k (ɛ())] is approximaed by he single projeced value. The P -expeced cash flow is an esimae of he money ou flow from he insurance company a fuure ime poins, and i is, herefore, useful for liquidiy consideraions. Again, he cash flow disinguishes iself by including non-guaraneed paymens as well as guaraneed paymens, hereby, providing a more complee picure. Finally, for solvency purposes, one can use scenarios generaed via Mone Carlo simulaion o calculae P -quaniles for he capial requiremen a ime T 1. [ T T1 E Q e s T r(v) dv 1 d ( B (ɛ(s)) C ) ( (s) k (ɛ(s)), r (s) ) ] s T 1 = T1 e s T 1 r(v) dv ( k (ɛ(s)) dβ u (s) + dβ f (s) dς (s) ) + E Q [ T T1 T 1 e s T r(v) dv ( 1 k (ɛ(s)) dβ u (s) + dβ f (s) dς (s) ) k (ɛ(t1)), r (T 1 )]. The capial requiremen is expressed in erms of he capial needed up o ime T 1 plus he marke value of fuure liabiliies a ime T 1. The condiional Q-expecaion appears in he capial requiremen, because he capial requiremen concerns fuure paymens and balance shees. The quaniles are obained by projecing up o ime T 1 under he physical measure P. However, for each projecion, he Q-expecaion is approximaed by projecing from ime T 1 o ime T under he pricing measure Q. Hence, if N sample pahs are needed for approximaing cash flows and marke values, hen N 2 pahs are needed for he quaniles. The quaniles can be used for solvency assessmens of he provided guaranee.

20 Risks 215, Numerical Examples In his secion, we go hrough wo numerical examples wih a one-policy porfolio and a wo-policy porfolio. A larger porfolio would, of course, be more realisic, bu a large number of policies could easily drown he key insighs from he examples. Going from one o wo policies is by far he bigges sep, and concepually, here is no impedimen o exending he heory o larger porfolios. Working in a discree projecion seup, we show how o find a fair bonus and guaranee fee sraegy for he one-policy porfolio, and we exemplify he fairness challenges in a wo-policy porfolio. The examples are based on 5 scenarios generaed via Mone Carlo simulaion. We have made sure ha he number of simulaed scenarios is sufficienly high for our numerical resuls and graphs no o change beween simulaions, bu we do no go ino deails abou he robusness of he simulaions, since he examples only serve o demonsrae he possible applicaions of our model One-Policy Porfolio We consider a porfolio consising of a single policy. The policy is he one from Examples 1 4. The policyholder is a female aged 25 a ime, where he policy is issued. We fix r =.2, and we assume ha r () = r, which is naural for a newly-issued policy. Thereby, we only consider he bonus scheme addiional benefis. We recall ha bonus is used o increase he endowmen sum and no he erm insurance sum. The deah of he policyholder is governed by he echnical moraliy inensiy µ () = e.87498(25+). For he las hree decades, his has served as a sandard moraliy inensiy for adul women in Denmark. I is par of he so-called G82 echnical basis ha was se forh as a Danish indusry sandard in The marke moraliy inensiy is given by µ m () =.8µ (). Wih his choice of moraliy inensiies and wih he produc choices below, he echnical basis is on he safe side, excep for low ages, where he deah sum exceeds he savings, resuling in a negaive conribuion from moraliy risk. However, due o he low moraliy for low ages, he negaive conribuion is insignificanly small. The policy expires a ime T = 4 when he policyholder is 65. We fix he erm insurance sum a b ad = 1 and he pure endowmen sum a b a = 3. The equivalence premium is deermined via he equivalence relaion V, (, r ) = V f,,+ (, r ) + V u,,+ (, r ), i.e., π = ba e T (r +µ (v)) dv + b ad T T e s e s (r +µ (v)) dv ds (r +µ (v)) dv µ (s) ds. Using numerical mehods, we obain π = The bonus d is allocaed and he guaranee fee π g is paid once a year. Hence, we have ɛ () = # {i = 1,..., 4 : i }.

21 Risks 215, 3 22 We noe ha ɛ () = for = 1,..., 4. We projec he wo accouns X and Y using seps of a size of one year by applying a discreized version of he sochasic differenial equaions for X and Y. For he discreizaion, we recall from Example 2 ha β u is a pure jump funcion and ha ς and β f are coninuous funcions. Hence, we ge he sochasic difference equaions ( X ( ) = X ( 1) (1 + R X ()) dβ f (s) dς (s) ), ( 1,) X () = X ( ) k () β u () + g () π g (), = 1,..., 4, X () =, Y ( ) = Y ( 1) e r ( 1,) ( dβ f (s) dς (s) ) + α (, r, k ( 1)), Y () = Y ( ) k () β u () + d (), = 1,..., 4, Y () =. We assume a deerminisic marke ineres rae r =.4, and he asses of he porfolio (in his case, he asses of he policy) are invesed in a fund wih log-normal reurns ha are paid ou once a year, i.e., R X () = S () S ( 1) S ( 1), = 1,..., 4, where S is a geomeric Brownian moion. We basically consider a simple Black Scholes financial marke. We assume ha he fund size S has drif.7 and volailiy.2 under he physical measure P (and, consequenly, drif r =.4 and volailiy.2 under he pricing measure Q). The bonus d is deermined as a fracion θ 1 of he excess collecive bonus poenial K jus before he bonus allocaion over a hreshold K if his fracion exceeds he posiive par of he naural risk bonus α (see Secion 4.9 for more on risk bonus), i.e., { (α ( d () = max, r, k (ɛ( )))) + ( ) } +, θ1 K ( ) K ( ), where K and K are given by K () = (X () L ()) +, K () = θ 2 L (), = 1,..., 4, wih he guaraneed liabiliies L defined in Equaion (5). The hreshold K can be seen as a preferred minimum collecive bonus poenial. We fix θ 1 =.2 and θ 2 =.1. As menioned, he chosen echnical ransiion inensiy is no on he safe side for low ages. Therefore, we need o ake he posiive par of α in he expression above o exclude negaive risk bonus. Finally, we choose he guaranee fee π g o be a fracion θ 3 of he posiive par of he reurns on he asses, i.e., π g () = θ 3 (R X () X ( 1)) +. In addiion o he yearly guaranee fee, he equiy holders of he insurance company receives he remaining collecive bonus poenial a expiraion as par of he final guaranee fee. We deermine he fracion θ 3 according o he fairness crierion in Equaion (8). Furhermore, using his guaranee fee, we consider: (12)

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Longevity 11 Lyon 7-9 September 2015

Longevity 11 Lyon 7-9 September 2015 Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Life insurance cash flows with policyholder behaviour

Life insurance cash flows with policyholder behaviour Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of Erlangen-Nuremberg Lange Gasse

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees. The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees 1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

More information

Fair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration

Fair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen Friedrich-lexander-Universiy

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies 1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz- und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective

Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics Friedrich-Alexander-Universiy

More information

Differential Equations in Finance and Life Insurance

Differential Equations in Finance and Life Insurance Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

More information

A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *

A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities * A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:

More information

The Grantor Retained Annuity Trust (GRAT)

The Grantor Retained Annuity Trust (GRAT) WEALTH ADVISORY Esae Planning Sraegies for closely-held, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach Opimal Consumpion and Insurance: A Coninuous-Time Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems

More information

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed

More information

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

On the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes

On the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Double Entry System of Accounting

Double Entry System of Accounting CHAPTER 2 Double Enry Sysem of Accouning Sysem of Accouning \ The following are he main sysem of accouning for recording he business ransacions: (a) Cash Sysem of Accouning. (b) Mercanile or Accrual Sysem

More information

Fifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance

Fifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of bes-esimae provisions... 3 2.1

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

CVA calculation for CDS on super senior ABS CDO

CVA calculation for CDS on super senior ABS CDO MPRA Munich Personal RePEc Archive CVA calculaion for CDS on super senior AS CDO Hui Li Augus 28 Online a hp://mpra.ub.uni-muenchen.de/17945/ MPRA Paper No. 17945, posed 19. Ocober 29 13:33 UC CVA calculaion

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear

More information

LEASING VERSUSBUYING

LEASING VERSUSBUYING LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss

More information

This page intentionally left blank

This page intentionally left blank This page inenionally lef blank Marke-Valuaion Mehods in Life and Pension Insurance In classical life insurance mahemaics, he obligaions of he insurance company owards he policy holders were calculaed

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN

More information

Equities: Positions and Portfolio Returns

Equities: Positions and Portfolio Returns Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi

More information

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

More information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Annuity Decisions with Systematic Longevity Risk

Annuity Decisions with Systematic Longevity Risk Annuiy Decisions wih Sysemaic Longeviy Risk Ralph Sevens This draf: November, 2009 ABSTRACT In his paper we invesigae he effec of sysemaic longeviy risk, i.e., he risk arising from uncerain fuure survival

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß ** IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

Markit Excess Return Credit Indices Guide for price based indices

Markit Excess Return Credit Indices Guide for price based indices Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semi-annual

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

LIFE INSURANCE MATHEMATICS 2002

LIFE INSURANCE MATHEMATICS 2002 LIFE INSURANCE MATHEMATICS 22 Ragnar Norberg London School of Economics Absrac Since he pioneering days of Black, Meron and Scholes financial mahemaics has developed rapidly ino a flourishing area of science.

More information

ABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION

ABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

THE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS

THE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

Life insurance liabilities with policyholder behaviour and stochastic rates

Life insurance liabilities with policyholder behaviour and stochastic rates Life insurance liabiliies wih policyholder behaviour and sochasic raes Krisian Buchard Deparmen of Mahemaical Sciences, Faculy of Science, Universiy of Copenhagen PFA Pension Indusrial PhD Thesis by: Krisian

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

Chapter Four: Methodology

Chapter Four: Methodology Chaper Four: Mehodology 1 Assessmen of isk Managemen Sraegy Comparing Is Cos of isks 1.1 Inroducion If we wan o choose a appropriae risk managemen sraegy, no only we should idenify he influence ha risks

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

Research Article Optimal Geometric Mean Returns of Stocks and Their Options

Research Article Optimal Geometric Mean Returns of Stocks and Their Options Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics

More information

PRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES

PRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely

More information

Appendix D Flexibility Factor/Margin of Choice Desktop Research

Appendix D Flexibility Factor/Margin of Choice Desktop Research Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\223489-00\4

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Agnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010-085

Agnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010-085 Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/2010-085 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint

Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint Dynamic Hybrid Producs in Life Insurance: Assessing he Policyholders Viewpoin Alexander Bohner, Paricia Born, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen Friedrich-Alexander-Universiy

More information

NASDAQ-100 Futures Index SM Methodology

NASDAQ-100 Futures Index SM Methodology NASDAQ-100 Fuures Index SM Mehodology Index Descripion The NASDAQ-100 Fuures Index (The Fuures Index ) is designed o rack he performance of a hypoheical porfolio holding he CME NASDAQ-100 E-mini Index

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

PREMIUM INDEXING IN LIFELONG HEALTH INSURANCE

PREMIUM INDEXING IN LIFELONG HEALTH INSURANCE Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages

More information

Markov Models and Hidden Markov Models (HMMs)

Markov Models and Hidden Markov Models (HMMs) Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook. Markov Model

More information

Optimal Life Insurance Purchase, Consumption and Investment

Optimal Life Insurance Purchase, Consumption and Investment Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.

More information

Time variant processes in failure probability calculations

Time variant processes in failure probability calculations Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TU-Delf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,

More information

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.

More information

Renewal processes and Poisson process

Renewal processes and Poisson process CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

More information

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a

More information