A TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul


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1 universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased 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 TwoAccoun Life Insurance Model for ScenarioBased 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 TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Ninna Reizel Jensen 1, * and Krisian Juul Schomacker 2 1 Deparmen of Mahemaical Sciences, Universiy of Copenhagen, Universiesparken 5, DK21 København Ø, Denmark 2 Edlund A/S, Bjerregårds Sidevej 4, DK25 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 woaccoun model wih even risk, we model life insurance conracs aking ino accoun boh guaraneed and nonguaraneed paymens in paricipaing life insurance as well as in unilinked 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 unilinked insurance. By use of a woaccoun 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 worscase and besesimae 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 unilinked 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: woaccoun model; economic scenarios; paricipaing life insurance; unilinked 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. Nonguaraneed paymens in paricipaing life insurance and guaraneed and nonguaraneed paymens in unilinked 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 nonguaraneed 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 worscase scenarios, sress scenarios from Solvency II, scenarios generaed via Mone Carlo simulaion or besesimae 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 nonguaraneed, by averaging over sufficienly many projecions (as is common pracice wih Mone Carlo simulaion). For worscase or besesimae scenarios, a single projecion is enough o obain he corresponding worscase or besesimae 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 worscase scenarios, we refer o [3]. Scenariobased 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 SimulaionBased ALM Model in Pracical Use by a Norwegian Life Insurance Company in [5]. Considering nonguaraneed 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 nonguaraneed as well as guaraneed paymens. Scenariobased calculaions allow for marke valuaion, solvency calculaions, hedging and pricing of guaraneed and nonguaraneed paymens in paricipaing life and unilinked insurance. Hence, scenariobased 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 unilinked insurance in he same framework. By doing so, we are able o compare he wo. In heir naure, unilinked and paricipaing life insurance seem differen, bu hey are really no. The producs may vary in riskiness, bu projecionwise, hey are almos he same. The main difference lies in he specificaion of how nonguaraneed paymens arise, saed in he conrac from he beginning (unilinked) or deermined fairly by he company along he way (paricipaing life). By use of a woaccoun model, we are able o illusrae general conceps wihou making he model oo absrac. Our woaccoun model is based on he woaccoun model in [7], and boh models offer a common framework for he valuaion of guaraneed and nonguaraneed paymens in paricipaing life and unilinked 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 nonguaraneed paymens in paricipaing life insurance or unilinked 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 riskrelaed. 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 unilinked insurance. The work in [1] and, more specifically, [16] cover paricipaing life and unilinked insurance wih broad even risk, bu only in a Black Scholes marke, and he resuls involve nonrivial 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 unilinked 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 woaccoun model. In Secion 2, we discuss scenariobased 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 saewise 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 woaccoun 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 scenariobased calculaions. In Secion 5, we consider unilinked insurance. We ouch upon differen aspecs of unilinked insurance, and we presen our woaccoun model for a general unilinked
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 unilinked version of he numerical example in he previous secion. The example illusraes a fair guaranee fee sraegy, and we compare he unilinked 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 nonguaraneed paymens, pricing and hedging of guaranees, bonus and benefi prognoses and solvency calculaions. In unilinked insurance, we inroduce sochasic scenarios o uilize reiremen savings and benefi prognoses, solvency calculaions and hedging and pricing of unilinked 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 riskneural 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 unilinked insurance, i is imporan o bear in mind he requesed oucome. For pricing, hedging, marke valuaion and solvency assessmens of guaraneed and nonguaraneed paymens (in paricipaing life) or of a provided guaranee (in unilinked) 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 pahwise 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 economicdemographic 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 socalled echnical basis, and i represens a provisional worscase scenario for is elemens. Below, we mark elemens of he echnical basis by superscrip. For markeconsisen valuaion of fuure paymens, he echnical basis does no apply due o is worscase naure. Insead, valuaion is performed under he socalled marke basis, which is made up of besesimae assumpions abou he various elemens. Below, we mark elemens of he marke basis by superscrip m. In unilinked 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 unilinked insurance, we consider a policy whose saewise evoluion is governed by a coninuousime 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 economicdemographic elemens ino accoun. Similarly, he echnical basis consiss of he echnical ransiion inensiies and a echnical ineres rae. In unilinked 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 woaccoun 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 NonGuaraneed 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 saewise 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, saewise 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 nondeerminisic 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 TwoAccoun 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 saewise echnical reserves where he expecaion across saes is aken under he marke basis. Thus, he marke expeced echnical reserve is no a saewise reserve, bu a probabiliy weighed sum of saewise reserves. The accouns X and Y are he backbone of our woaccoun 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 prespecified, 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 nonnegaive. 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 predescribed 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 nh bonus accrual and k (n) denoe he upscaling of he benefis B u afer he nh bonus accrual. We noe ha he upscaling facor sars a one, i.e., k () = 1. Afer he nh 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 saewise 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 saewise 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 saewise echnical reserves can be calculaed numerically by use of Thiele s differenial equaions; see [23]. We denoe by Vi (, ρ, k) he saewise 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 saewise 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 nonconsan 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 saewise reserve, bu a marke probabiliy weighed sum of saewise 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 nonguaraneed 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, TwoAccoun 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 saewise 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 saewise 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 saewise 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 saewise 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 saewise 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 predescribed 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 nonupscaled 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 mulipolicy 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 onepolicy and a wopolicy porfolio. We show how o find a fair bonus and guaranee fee sraegy, and we exemplify he challenges of fairness in a wopolicy 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 nonguaraneed 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 nonguaraneed 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 nonguaraneed 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 Qprojecions up o ime. If, insead, he projecions are of he worscase or besesimae,
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 radeoff 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 nonguaraneed 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 worscase or besesimae 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 nonguaraneed 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 Qexpecaion 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 Qexpecaion 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 onepolicy porfolio and a wopolicy 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 onepolicy porfolio, and we exemplify he fairness challenges in a wopolicy 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 OnePolicy 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 newlyissued 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 socalled 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 lognormal 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)
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