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

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1 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 share of he reurn surplus as compensaion for he reurn guaranee graned o policyholders. This call-opion-like sake gives he insurance company an incenive o increase he riskiness of is invesmens a he expense of he policyholders. This conflic of ineress can parially be solved by regulaion deerring he insurance company from aking excessive risk. We sugges ha he regulaor implemen a raffic ligh sysem where disressed companies are forced o reduce he riskiness of heir asse allocaion. In a uiliy-based framework, we show ha his approach can increase he benefis of policyholder and insurance company. A he same ime, defaul probabiliies and hus solvency capial requiremens can be reduced. Keywords: regulaion, life insurance, defaul risk, uiliy maximizaion, risk sharing, muliobjecive opimizaion JEL: G11, G23 1. Inroducion Paricipaing life insurance conracs usually provide a yearly or mauriy guaranee for he policyholders. The surplus above his guaraneed amoun is shared beween policyholders and he owners shareholders of he insurance company. In reurn, he policyholders pay insurance premiums ha are invesed by he insurance company. The sakes of policy and shareholders are usually modeled by a coningen-claim approach. The call-opion-like sake of shareholders and limied liabiliy gives he insurance company incenives o inves he premiums as riskily as possible a he expense of he policyholder see, e.g., Dohery and Garven 1986], Cummins 1988]. A very similar conflic arises beween deb and equiy holder of corporaions: Especially if he corporaion is in disress, he equiy holders end Dae. July 13, Insiue of Insurance Science, Universiy of Ulm, Helmholzsr. 20, Ulm, Germany. E Mail: an.chen@uni-ulm.de, peer.hieber@uni-ulm.de. Peer Hieber acknowledges funding by he German Associaion of Insurance Science DVFVW. We wan o hank Jing Li for fruiful discussions and her work on an earlier version of his manuscrip. 1

2 o ake on as much risk as possible, a line of acion called risk shifing. One possibiliy o solve his conflic of ineress is he inroducion of a regulaor ha resrics excessive risk aking of he insurance company see, e.g., Gazer and Schmeiser 2008], Dong e al. 2014], Filipović e al. 2015]. The aim of his paper is o propose ways on how such a regulaory scheme can be implemened in order o solve or a leas alleviae his conflic of ineress. Regulaory supervision is necessary and jusified, as insurance markes are sill raher inransparen: An informaion asymmery beween insurance company and policyholders furher increases he described conflic of ineress see, e.g., Rees e al. 1999]. Regulaory supervision in Europe has ransformed from relaively simple mehods o a comprehensive and very deailed line of acion adequaely reflecing all he risks inheren in life insurance companies. Thereby, a risk-based supervision relying on Value-a-Risk or defaul probabiliy gains more and more imporance see, e.g., Bauer e al. 2005]. There are many differen designs of regulaory supervision: The regulaor may enforce price consrains by inroducing resricions on premium calculaion see, e.g., MacMinn and Wi 1987]. Furhermore, he insurance company may be forced o provide risk-based capial as, for example, specified in he Solvency II accord. The amoun of risk capial needed is usually defined o impose an upper bound on defaul probabiliy see, e.g., McCabe and Wi 1980]. Relaed o his, he regulaor may impose consrains on he riskiness of he insurance company s invesmen decisions. 1 For disressed companies, he las is easier o implemen han he risk-based provision of capial, because a disressed company migh face problems acquiring new capial o fulfill solvency requiremens. Relaed o his sudy, he conflic of ineress beween policyholder and insurance company can be miigaed by a suiable design of he insurance conracs. Several auhors analyzed wha ype of guaranee or securiizaion meachanism bes suis he needs of policy and shareholders see, e.g., Døskeland and Nordahl 2008], Dong e al. 2014], Schlüer 2014], Filipović e al. 2015]. They commonly assume ha he premiums are invesed in a porfolio ha follows Lévy dynamics, i.e. reurns in subsequen periods are independen and idenically disribued ypical asse models used are geomeric Brownian moion or Meron jump-diffusion, 1 This migh include a resricion of he share of socks and oher risky invesmens or some minimum diversificaion requiremen. In Germany his is, for example, regulaed by 3 Anlageverordnung AnlV. An overview of regulaions in oher European counries is given by Davis 2001]. 2

3 see, e.g., Døskeland and Nordahl 2008], Gazer and Schmeiser 2008], Schmeiser and Wagner 2013], Dong e al. 2014], Schlüer 2014], Filipović e al. 2015] and many ohers. However, he opion-like and non-linear payoffs of paricipaing life insurance conracs sugges ha invesing in a consan-risk porfolio is no longer maximizing he benefis of policy and shareholders. This fac was parially recognized in asse-liabiliy managemen 2. We sugges ha he regulaor enforces a non-lévy, regime-dependen invesmen sraegy in order o alleviae he aforemenioned conflic of ineress beween policy and shareholders. The main idea of his approach is o force disressed companies o reduce he riskiness of heir asse allocaion. As menioned earlier, similar conflics of ineress arise beween deb and equiy holders of companies. However, in he lieraure herein, his problem is usually ackled no by a change in he riskiness of he asse allocaion bu by a suiable design of capial srucure see, e.g., Leland 1996], Leland and Tof 1996]. We firs analyze he effec of a defaul consrain on he opimal asse allocaion and assess wheher his helps o a leas parially solve he conflic of ineress regarding he invesmen decision beween insurance company and policyholders. Defaul is modeled coninuously by a srucural approach following he seminal paper of Black and Cox 1976]. In his firs sep, we assume ha he insurance company commis o a consan-mix invesmen sraegy a conrac iniiaion and leaves his sraegy unchanged unil conrac erminaion. In a second sep, we hen analyze a more flexible regulaory scheme: The regulaor inroduces a raffic ligh sysem ha indicaes wheher he life insurance company is in danger of facing solvency problems yellow bulb or even has severe and immediae problems red bulb. If he insurance company is in disress yellow bulb, he regulaor may enforce a decrease in he riskiness of he asse allocaion. This raffic ligh solvency sress es is for example implemened in Denmark and Sweden, see, e.g., Jørgensen 2007]. Similar ideas have been inroduced in oher European counries and in he Solvency II regulaions. One 2 Here, some auhors sugges ha he riskiness of he asse invesmen should depend on he insurance company s funding raio =asses divided by liabiliies. An insurance company migh adap is asse allocaion dependen on he possibiliy ha i is unable o mee is obligaions. Bohner e al. 2015] sugges a CPPI-based sraegy. Graf e al. 2011] and Hieber e al. 2015] change he asse allocaion dependen on risk measures, i.e. he expeced shorfall below he company s invesmen guaranees. Empirically, i is no obvious wheher life insurance companies increase or decrease risk in case of disress: Mohan and Zhang 2014] find ha US public funds increase risk if hey are underfunded, while Rauh 2009] shows ha he asse allocaion is less risky if he company s financial condiion is weaker. 3

4 advanage of he raffic ligh sysem is he fac ha i is easy o implemen and supervise. Furhermore from a mahemaical perspecive he coningen-claim framework is sill analyically racable, even in a coninuous-ime defaul seing. We invesigae he effec of he raffic ligh sysem on he benefis of boh he policyholders and he insurance company. If he regulaor ges he possibiliy o force disressed insurance companies o decrease he riskiness of heir invesmen sraegy, his allows o significanly decrease solvency risk, only marginally changing he benefis of policyholders or he insurance company. If regulaory defaul consrains are he same under boh he sandard and he flexible regulaory framework, we show ha he regulaor migh increase he benefis of policyholders and insurance company. The remainder of he paper is organized as follows. In Secion 2, we describe he model seup and inroduce he payoffs of he policy- and shareholder. We se up heir opimal invesmen problem, aking accoun of he possible defaul of he insurance company. More imporanly, he flexible regulaory inervenion raffic ligh sysem is presened. In he subsequen Secion 3, he expeced uiliy of he policy- and shareholder are compued analyically. We consider fairly-priced insurance conracs only and provide no-arbirage condiions. In Secion 4, we illusrae he advanage of he raffic ligh sysem in a numerical example. Therefore, we show ha he raffic ligh sysem can increase he benefis of policy and shareholder Pareo improvemen. This effec is even more pronounced if he regulaor enforces a defaul consrain. Finally, we provide some concluding remarks and an oulook for fuure research in Secion 5 and deailed proofs in Secion Noaions and Model Seup Our model conains hree paries: an insurance regulaor, a represenaive shareholder also equiy holder and a represenaive policyholder also liabiliy holder. The laer wo consiue a muual life insurance company. We assume ha he represenaive policyholder invess in a paricipaing life insurance conrac wih a mauriy of T years, T <. A he iniiaion of he conrac, he policyholder invess a lump sum L 0 in a single premium conrac; he shareholder provides iniial equiy E 0 > 0. Consequenly, he iniial asse value A 0 of he insurance company is given by he sum of boh conribuions, i.e. A 0 := L 0 E 0. We denoe he share of he policyholder s conribuion or equivalenly he deb raio of our 4

5 insurance company by α := L 0 /A 0, where obviously α 0, 1. Asse model and guaraneed amoun. Le us define a financial marke consising of one risk-free bond B wih risk-free ineres rae r, i.e. db = rb d and B 0 = 1. Furhermore, here is he possibiliy o inves in a risky invesmen ds = µ S d σ S dw, S 0 = 1, 1 where µ > r, σ > 0, and W is a sandard Brownian moion under he real world measure P. To sar wih, we assume ha he insurance company invess he oal proceeds A 0 in a diversified porfolio of risky and non-risky asses. Assume, a consan share θ 1 0, 1] is invesed in he risky asse S and he remainder in he risk-free asse B. Wih he iniial asse invesmen A 0 > 0, his yields he following asse dynamics: da = r θ 1 µ r A d σθ 1 A dw. 2 The asse dynamic remains a log-normal process wih a volailiy of σθ 1. The amoun guaraneed o he policyholder a ime 0, T ] is assumed o be L = L 0 e g, where g r is he guaraneed rae. We wan o draw conclusions regarding he financial risks of paricipaing insurance conracs, herefore, as is usual in his conex, we purely consider financial risks and ignore moraliy risk assume, for insance, ha he fair marke value of he conrac is paid ou immediaely in case of deah. Defaul of he insurance company. We wan o explicily ake he defaul risk of he insurance company ino accoun. Therefore, we make use of a srucural approach and assume ha he insurance company defauls as soon as is asses A hi or drop below a specified percenage η of he guaraneed amoun L. Thus, we inroduce a defaul barrier D := ηl 0 e g whose accrual rae g is he same as for he guaraneed amoun. The ime of defaul is hen he firs hiing ime τ defined by τ := inf { 0 } A D, 3 where we se inf{ } =. The defaul parameer η is assumed o be smaller han A 0 /L 0 such ha he company is solven iniially. Terminal payoff o liabiliy and equiy holder. The insurance payoff o he policyholder is coningen on wheher he insurance company survives he mauriy dae T. If here is no premaure defaul of he insurance company, he policyholder receives he following 5

6 erminal payoff: A T if A T L T Ψ L A T := L T δ ] αa T L T else, = L T δ ] αa T L T LT A T ], 4 where we denoe by ] he maximum max{, 0}. The paricipaion rae δ 0, 1] is he percenage of surpluses ha is credied o he liabiliy holder. If here is no premaure defaul, he erminal conrac payoff is a combinaion of a fixed paymen L T, a bonus call and a shored pu opion on he insurance company s asses. The shored pu opion refers o losses of he liabiliy holder if he company is no defauled premaurely bu asses a mauriy are insufficien o cover he guaraneed amoun. In he case of premaure defaul, a rebae paymen is provided o he policyholder a ime τ. This rebae paymen is given by he minimum of he curren asse value A τ = D τ and he curren liabiliies L τ : minl τ, D τ. If we for ime consisency reasons assume ha he rebae paymen is unil T accumulaed a he risk-free rae r, he policyholder receives he following conrac payoff a ime T : V L A T := 1 {τ>t } Ψ L A T 1 {τ T } e rt τ minl τ, D τ, 5 where 1 B is an indicaor funcion which gives 1 if B occurs and 0 oherwise. The equiy holder always obains he residual asse value. If here is no premaure defaul of he insurance company, he payoff o he equiy holder is 0 if A T L T Ψ E A T := A T L T if L T < A T A 0 e gt A T L T δ ] αa T L T else = A T L T ] δ ] αa T L T. 6 If here is premaure defaul, a rebae payoff D τ minl τ, D τ is provided o he equiy holder. More compacly, he oal payoff of he equiy holder a mauriy T is hus given by V E A T := 1 {τ>t } Ψ E A T 1 {τ T } e rt τ maxd τ L τ, 0. 7 Hereby we have accrued he rebae paymen a τ wih he risk-free rae unil he mauriy dae. 6

7 80% 60% liabiliy and equiy reurn 40% 20% 0% 20% 40% 60% reurn liabiliy holder Ψ L A T /L 0 1 reurn equiy holder Ψ E A T /E 0 1 guaraneed reurn expgt 1 80% 100% 15% 5% 5% 15% 25% 35% 45% asse reurn A T /A 0 1 Figure 1. Reurn of liabiliy and equiy holder dependen on he asse reurn A T /A 0 1 in case ha he insurance company survives unil mauriy T, i.e. τ > T. The parameers are se as A 0 = 1, L 0 = αa 0 = 0.8, D 0 = 0.85, δ = 0.72, g = 1.75 %, T = 10, µ = 6 %, r = 2.5 %, θ 1 = 0.2, and σ = 0.2. If he company survives mauriy T, he payoff o he liabiliy and equiyholder are given in Equaion 4, respecively 6. A numerical example, presening he reurn of liabiliy and equiy holder dependen on he asse reurn A T /A 0 1, is given in Figure 1. The dashed line corresponds o he guaraneed reurn expgt 1. If he asse reurn is greaer han he guaraneed reurn, he liabiliy holder grey line receives a bonus paymen. If he insurance company survives he mauriy, bu he asses a ha ime are insufficien o cover he liabiliies, he liabiliy holder receives a reurn less han he guaraneed one. From Figure 1 one can observe ha he equiy holder s reurn black line is much more volaile han he liabiliy holder s reurn. Risk-neuraliy of he equiy holder. We assume ha he insurance company has possibiliies o diversify is invesmens and hus wans o maximize is expeced payoff a mauriy T. Is invesmen a conrac iniiaion is given by E 0 = 1 αa 0. The insurance company judges insurance conracs using he goal funcion U E Θ := E P VE A T ], 8 where he parameer Θ conains he asse allocion parameers. 3 3 In case of he consan-mix invesmen sraegy 2, we have ha Θ = θ 1. Laer on, he asse allocaion sraegy is allowed o be more flexible and, e.g., Θ = θ 1, θ 2, K. 7

8 Liabiliy holder. The liabiliy holder in conras canno fully diversify he invesmen and is assumed o be risk-averse. She opimizes wih respec o a uiliy funcion u L see Definiion 2.1 and hus evaluaes her paymens according o he goal funcion U L Θ := E P ul VL A T ]. 9 Definiion 2.1 Uiliy funcion. u L is increasing, concave, and wice differeniable on R wih u L > 0, u L < 0 lim x u L x = and lim x u L x =. Laer on, we are exemplarily using power uiliy, see Example 2.2. Example 2.2 Power uiliy. Le γ 1 > 0, γ 1 1 be he relaive risk aversion parameer of he policyholder, i.e. u L V L := V 1 γ 1 L /1 γ 1. Compeiive marke. In our seing, we assume ha he marke is compeiive and insurance conracs are fairly priced see, e.g., Brennan and Schwarz 1976]. Policyholder are fully informed abou available conrac erms from differen insurers. Arbirage-free prices of insurance conracs are obained under he pricing measure Q wih he risk free bond db /B = r d as reference asse. The risky asse evolves as ds = r S d σ S dw Q, 10 where sill B 0 = S 0 = 1 and W Q is a sandard Brownian moion under Q. If an insurance conrac is fairly priced, he iniial invesmen of he shareholder equals is arbirage-free iniial sake E 0 = 1 αa 0, i.e. 1 αa 0 = E Q e rt V E A T ]. 11 Similarly and equivalenly, from he policyholder s viewpoin, one has o ensure ha αa 0 = E Q e rt V L A T ], 12 wih V E A T and V L A T as defined in Equaion 7, respecively 5. We denoe he se of fair conrac erms Θ by X. Regulaory inervenion. The aim of his paper is o examine he impac of regulaion on he opimal asse allocaion deermined by he insurance company see he opimizaion problem 8. The regulaor may force he insurance company o limi is defaul probabiliy Pτ T o an upper limi ɛ. If he reference porfolio has dynamics 2, we will laer prove ha his defaul consrain is equivalen o a bound on he share of risky asses θ 1, 8

9 i.e. 0 θ 1 θ max for a given consan θ max 1. Boh policy and equiy holder value heir conrac erms using 9, respecively 8. In a second sep, he regulaor allows for more flexibiliy in he invesmen sraegy, while sill keeping he defaul probabiliy consrain Pτ T ɛ. The concep is in analogy o Solvency II regulaions in Europe where he regulaor has he possibiliy o inervene, as soon as he asses drop below some criical level {K } 0 yellow bulb o avoid a defaul even. If he company s asses neverheless drop below he defaul barrier {D } 0 red bulb, he insurance company defauls. The possible ineracion in case of he yellow bulb gives more freedom o ac in he ineress of boh liabiliy and equiy holder. The second upper hreshold K is se as K := K 0 e g, 13 where D 0 = ηl 0 < K 0 < A 0. The hiing ime of his barrier is denoed ˆτ := inf { 0 } A K, 14 where we again se inf{ } =. In case his barrier is hi, he regulaor may once force he insurance company o change is invesmen sraegy from θ 1 o θ 2 0, 1]. Then, he asse value process is for 0 given by da = r θ Z µ r A d θ Z σ A dw, A 0 > 0, 15 where Z = 1 for ˆτ and Z = 2 for > ˆτ. The effec of his more flexible design on he benefis of equiy and liabiliy holder is analyzed in he remainder of his paper. For reasons of analyical racabiliy, we do no consider a sraegy recovery of he insurance company, i.e. i is no possible o reurn o he original asse sraegy θ 1. Under his more flexible regulaion, he defaul-riggering even remains unchanged. A defaul occurs when he asse process A his he lower hreshold D i.e. if {τ T }. Since he asse process is coninuous and he regulaory barrier K by definiion greaer han D, he even {τ T } implies ha {ˆτ T }, i.e. he upper hreshold is hi before he lower one. 4 4 The even {ˆτ > T } delineaes he siuaion ha he asses perform well unil mauriy T and all he ime exceed he upper regulaory hreshold. The even {ˆτ T, τ > T } describes he siuaion ha he asses perform moderaely unil mauriy T. The asses have hi he regulaory barrier bu he insurance company has no defauled premaurely. The even {τ T } describes he siuaion ha he company has defauled. 9

10 3. Theoreical resuls In order o deermine he opimal invesmen sraegy and examine he regulaory effecs on i, we need o compue he expeced payoff of he equiy holder 8 and he expeced uiliy of he policyholder 9. 3.A. Tradiional regulaory framework. In his firs case, we assume ha here is no regulaory barrier {K } 0 and hus he invesmen sraegy says consan a θ 1 0, 1]. Theorem 3.1 gives analyical expressions for he expeced payoff o he equiy holder and he expeced uiliy of erminal payoffs o he policyholder. Theorem 3.1 Expeced uiliy: No inervenion. Assume he model seup as described in Secion 2 wih asse process 2. Then, he desired expecaions are given by where κ 1 L κ i U L Θ =: κ 1 L A 0, D 0, L 0, 0, T, U E Θ =: κ 1 E A 0, D 0, L 0, 0, T, and κ1 E can be compued via L A, D, L,, T = κ i lnd /A E A, D, L,, T = lnd /A T u L e rt g rτ minl, D f i, τ, A, D dτ u L LT δ αa e ygt L T ] LT A e ygt ] g i y,, T, A, D dy, T e rt g rτ maxd L, 0 f i, τ, A, D dτ A e ygt L T ] δ αa e ygt L T ] g i y,, T, A, D dy. The densiies g, respecively f, are defined as g i 1 y µ y,, T, A, D := i T ϕ 1 e 2 lnd /A σθ i T σθ i T f i, τ, A, D := lnd /A σθ i τ 3 2 ϕ 2 y lnd /A σ 2 θ i, 2T lnd /A µ i τ, µ i := r θ i µ r g σ 2 θ 2 σθ i τ i /2, where ϕ denoes he densiy of he sandard normal disribuion. Proof: See he Appendix. 10

11 In he case of power uiliies, mos of he inegrals in Theorem 3.1 can be derived analyically. The defaul probabiliy on he ime inerval 0, T ] is given by lnd 0 /A 0 µ 1 T Pτ T = Φ σθ 1 T D0 A 0 2 µ 1 σ 2 θ 2 1 lnd 0 /A 0 µ 1 T Φ, 16 σθ 1 T where Φ is he sandard normal disribuion funcion and µ 1 is defined as in Theorem 3.1, see also he Appendix. From Theorem 3.1, we can conclude ha he fair pricing condiion 11 is valid if and only if we solve he equaion κ 1 E A 0, D 0, L 0, 0, T = 1 αa 0 e rt for δ. 3.B. Flexible regulaory framework. Now, we are going o derive he same resuls as in Theorem 3.1 under he assumpion ha he invesmen sraegy is changed from θ 1 o θ 2 as soon as he regulaory barrier {K } 0 is hi. This leads o he asse process given by 15. Technically, his seup is sill analyically racable: Unil firs hiing he regulaory hreshold K a ime ˆτ, he asse process behaves as a geomeric Brownian moion one of he rare cases where he firs-hiing ime densiy is known analyically he hiing ime is disribued according o an inverse Gaussian law, see, for example, Folks and Chhikara 1978]. A ime ˆτ, he asses equal he barrier Kˆτ. Afer his hiing ime, he asses are again a geomeric Brownian moion now wih a differen mean and volailiy parameer due o he changed invesmen sraegy θ 2. Thus, he ime o defaul follows again an inverse Gaussian law. To sum up, he defaul ime τ is given by he convoluion of wo inverse Gaussian random variables. The defaul probabiliy can be evaluaed via Pτ T = = T P τ T Aˆτ = Kˆτ f 1 0, ˆτ, A 0, K 0 dˆτ 0 T T τ 0 0 f 2 ˆτ, τ, Kˆτ, Dˆτ f 1 0, ˆτ, A 0, K 0 dˆτ dτ, 17 wih f as defined in Theorem 3.1. Equaion 16 resuls as he special case θ 1 = θ 2. Similarly o Theorem 3.1, one can derive he expeced erminal payoff o he equiy holder E P V E A T ] and he expeced uiliy E P u L V L A T ] of he liabiliy holder, see Theorem 3.2. Theorem 3.2 Expeced uiliy: Regulaory inervenion. Assume he model seup as described in Secion 2 wih asse process 15. The regulaor may inervene a ime ˆτ he firs hiing ime of he insurance company s asses A breaching he regulaory barrier K = K 0 e g. A ime ˆτ, he insurance company is forced o change is invesmen sraegy from θ 1 o θ 2. 11

12 Then, he desired expecaions are given by where U L Θ =: ζ L A 0, D 0, K 0, L 0, 0, T, U E Θ =: ζ E A 0, D 0, K 0, L 0, 0, T, ζ L A, D, K, L,, T = T T ˆτ lnk /A ζ E A, D, K, L,, T = T T ˆτ lnk /A T lnd /K u L LT δ αkˆτ e ygt ˆτ L T ] LT Kˆτ e ygt ˆτ] f 1, ˆτ, A, K g 2 y, ˆτ, T, Kˆτ, Dˆτ dy dˆτ u L e rt g rˆτ minl, D f 1, ˆτ, A, K f 2 ˆτ, τ, Kˆτ, Dˆτ dτ dˆτ u L LT δ αa e ygt L T ] LT A e ygt ] g 1 y,, T, A, K dy T lnd /K Kˆτ e ygt ˆτ L T ] δ αkˆτ e ygt ˆτ L T ] f 1, ˆτ, A, K g 2 y, ˆτ, T, Kˆτ, Dˆτ dy dˆτ e rt g rˆτ maxd L, 0 f 1, ˆτ, A, K f 2 ˆτ, τ, Kˆτ, Dˆτ dτ dˆτ A e ygt L T ] δ αa e ygt L T ] g 1 y,, T, A, K dy, wih f and g as defined in Theorem 3.1. Proof: See he Appendix. Again, i is sraighforward o rephrase he fair pricing condiion 11 in erms of he paricipaion rae δ, i.e. he insurance conrac is fairly priced if and only if we solve he equaion ζ E A 0, D 0, K 0, L 0, 0, T = 1 αa 0 e rt for δ. Remark 3.3 Implemenaion of Theorems 3.1 and 3.2. The expecaions presened in Theorems 3.1 and 3.2 are inegrals over normal densiies. Thus, hey can easily be implemened a high precision. Compuaion ime is wihin fracions of seconds. Tha is why i does no make sense o furher simplify he given expressions and solve he 12

13 inegrals analyically, alhough i is, for example, possible o presen κ i E A, D, L,, T in Theorem 3.1 in a lenghy closed-form expression. 3.C. Asse allocaion decision. So far, we have shown how o compue he goal funcion pairs U L Θ, U E Θ of policyholder and equiy holder. The goal funcion pairs are deermined under he assumpion of a compeiive marke, i.e. we have shown how he paricipaion rae can be se in order o fulfill he fair pricing condiion 11. In he remainder of he paper we wan o focus on he following wo quesions: 1 Is here some opimaliy crierium for he insurance conracs in our model seup? Can we choose he asse allocaion parameers Θ = θ 1, θ 2, K 0 in order o maximize he pair U L Θ, U E Θ? 2 How does a regulaory defaul consrain affec he consideraions in 1? A general answer o 1 is impossible as his requires o find a reasonable way o somehow weigh he goals of liabiliy and equiy holder. However, here should be broad consensus ha if a given conrac wih uiliy pair U L Θ, U E Θ sricly dominaes anoher pair U L Θ, U E Θ, i.e. if U L Θ > U L Θ and U E Θ > U E Θ, hen his second conrac is in some sense sub-opimal. Tha is why, following, e.g., Bokranz and Fredriksson 2014], Filipović e al. 2015], we define Pareo-efficien conracs via Definiion 3.4. Definiion 3.4 Pareo-efficien conrac erms. Conrac erms Θ = θ 1, θ 2, K 0 X are Pareo-efficien if here does no exis a conrac erm Θ X such ha Θ Θ, meaning ha a leas one of he following holds: U L Θ U L Θ and U E Θ > U E Θ, or U L Θ > U L Θ and U E Θ U E Θ. If he se of feasible conrac erms X i.e. he conrac erms ha fulfill he fair pricing condiion 11 is non-empy and compac, Theorem 3.5 gives condiions for he exisence of Pareo-efficien conrac erms. Theorem 3.5 Condiions for Pareo-efficiency. Assume ha he se of feasible conrac erms X is non-empy and compac and ha boh U L Θ and U E Θ are lower semiconinuous in Θ. Then, here exiss a se of Pareo-efficien conracs. Proof: See, e.g., Bokranz and Fredriksson 2014] and he references herein. 13

14 We can apply Theorem 3.5 o our seing. Therefore, firs define he se of feasible conrac erms by conracs ha fulfill he fair pricing condiion 11: X := { Θ 1 αa0 = E Q e rt V E A T ]}, 18 where he disribuion of A T depends on he asse allocaion parameers Θ = θ 1, θ 2, K 0. As 0 θ 1, θ 2 1 and D 0 K 0 A 0, his se is obviously compac. Furher, as we can see from he smooh inegrands in Theorem 3.2, he funcions U L Θ and U E Θ are coninuous in Θ. Thus, if we assume ha here exiss a leas one fairly priced conrac, hen, according o Theorem 3.5, here exiss a se of Pareo-efficien conracs. If a regulaor is inroduced, he se of feasible conracs addiionally has o fulfill he defaul consrain Pτ T ɛ. Then X := { Θ 1 αa0 = E Q e rt V E A T ] ; Pτ T ɛ }. 19 As Pτ T is coninuous in he asse allocaion parameers, he compacness of X implies he compacness of X. Thus, again, we conclude from Theorem 3.5: If here is a fair conrac ha fulfills he defaul consrain, hen here exiss a se of Pareo-efficien conracs. 4. Numerical illusraions We now illusrae he noion of Pareo-efficiency in a numerical example. We show ha he flexible regulaory scheme can be beneficial o boh he liabiliy and he equiy holder Pareo improvemen. Furher, due o he non-lineariy of he conrac payoff, we show ha i is no he Meron-ype consan-mix sraegy ha maximizes he liabiliy holder s uiliy. Therefore, we firs fix he conrac parameers ha are he same in boh he radiional and he flexible regulaory framework. In conras o he asse allocaion sraegy, hose parameers are eiher se or resriced by naional law insolvency condiion η or changing hem migh be difficul for he insurance company equiy share 1 α: In line wih balance shee daa from big German insurance companies where equiy is mainly composed of socks and undisribued reserves, we se he equiy share o 1 α = 0.1. Form his, he iniial asses are wihou loss of generaliy A 0 = 1; he iniial single premium is L 0 =

15 We assume ha he insurance company defauls as soon as he asses drop below he guaraneed amoun L, i.e. D 0 = L 0 = 0.9, η = 1, an assumpion ha is common in he conex of srucural models see, e.g., Brennan and Schwarz 1976]. The uiliy funcion of he policyholder is assumed o be power uiliy wih a relaive risk aversion parameer γ 1 = 3. The paricipaion rae δ is se according o he fair pricing condiion 11. Noe ha in some counries, legal consrains on δ are imposed. If his is he case, we can incorporae his fac in our analysis by including his consrain in our definiion of feasible conrac erms X. Furher, we se he ime horizon o T = 10, and adap he financial marke parameers o he curren siuaion in Germany: µ = 6 %, r = 2.5 %, and σ = 0.2. The guaraneed rae is se o 1.25%, which corresponds o he curren maximum echnical ineres rae in Germany. The qualiaive resuls in his secion are consisen if he parameerd α, η, γ 1, T, µ, r, σ are changed in a reasonable way e.g. if an underfunding of he insurance company is allowed by law, and hus η < 1. In he following, we wan o opimally deermine he asse allocaion parameers in case of he radiional Θ = θ 1, θ 1, and he flexible regulaory scheme Θ = θ 1, θ 2, K 0. Therefore, we firs consider he case wihou defaul consrain by he regulaor feasible se X ; hen we assume ha his defaul consrain is imposed feasible se X. Tradiional regulaory framework No. sraegy θ 1 U L Θ, U E Θ P D 1a θ 1 = , % 2a θ 1 = , % 3a θ 1 = , % Table I. Differen asse allocaion sraegies in a radiional regulaory framework consan-mix, θ 1 = θ 2. For each sraegy, we comupue he uiliy pair U L Θ, U E Θ and an annualized defaul probabiliy P D := 1 1 Pτ T 1/T. Before we furher examine he efficiency condiion in Theorem 3.5, we wan o look a he effec of he invesmen sraegy θ 1, θ 2 on he goal funcion pairs U L Θ, U E Θ. Firs, we deal wih he radiional consan-mix regulaory framework. Table I presens he opimal 15

16 θ θ θ θ Figure 2. Goal funcion of he liabiliy holder U L Θ op and equiy holder U E Θ below in a numerical example dependen on he invesmen sraegy θ 1, θ 2. We se K 0 = 0.94, A 0 = 1, L 0 = D 0 = 0.9, g = 1.25 %, T = 10, µ = 6 %, r = 2.5 %, and σ = 0.2. The conracs are iniially fairly priced, i.e. Θ X. 16

17 asse allocaion of he liabiliy holder sraegy 1a and he equiy holder sraegy 2a. Furhermore, i presens he riskies sraegy ha is in line wih a Solvency II annualized defaul probabiliy of 0.5%. 5 Noe ha, in our example, he opimal sraegy 1a of he liabiliy holder is a feasible sraegy even if a defaul probabiliy consrain is imposed P D 0.50%. If his defaul consrain is furher ighened or he parameer se changed, i migh happen ha boh he liabiliy and he equiy holder s opimal asse allocaion sraegy leads o an annualized defaul probabiliy P D ha is greaer han 0.50%. In his case, sraegy 3a Pareo-dominaes sraegy 1a. Nex, we consider he flexible regulaory scheme. For illusraory purposes, we for he momen fix he regulaory hreshold o K 0 = Figure 2 presens he goal funcion pairs U L Θ op and U E Θ below for differen asse allocaion sraegies θ 1, θ 2. Therefore, we use a grid of lengh 0.01 for boh variables θ 1 0, 1] and θ 2 0, 1]. The whie line corresponds o he case of no regulaory inervenion, i.e. θ 1 = θ 2. We firs observe ha he equiy holder below seems o prefer more risky asse allocaions his goal funcion is increasing in boh θ 1 and θ 2. He obains a global opimum for θ 1 = θ 2 = 1 an opimum ha is unchanged if he flexible regulaory scheme is implemened. The liabiliy holder s preferences are differen: If here is no regulaory inervenion whie line, he maximum uiliy U L Θ = is reached for θ 1 = θ 2 = In he flexible regulaory scheme, however, he global opimum for he liabiliy holderis a uiliy of U L Θ = if θ 1, θ 2, K 0 = 0.22, 0.07, This already indicaes ha he liabiliy holder would prefer a scheme ha decreases risk in disress i.e. θ 2 < θ 1. To reurn o our definiion of Pareo efficiency in Theorem 3.5, i makes sense o no longer se K 0 = 0.94 bu o addiionally opimize over he parameer K 0. Then, we can presen goal funcion pairs U L Θ, U E Θ for each asse allocaion sraegy θ 1, θ 2, K 0, see Figure 3 lef hand side. Again, he case wihou any regulaory inervenion is represened by he whie line. Each Pareo-efficien porfolio, i.e. a porfolio ha is no dominaed by anoher uiliy pair U L Θ, U E Θ, lies on he upper edge of he black area. We observe ha each 5 In order o inerpre he resuls, we anualize defaul probabiliies o P D := 1 1 Pτ T 1/T. Noe, however, ha he srucural model implies a erm srucure of defaul probabiliies ha canno be summarized by jus one number. The value presened can hus be seen as some average yearly defaul probabiliy of he insurance company. 17

18 Figure 3. Goal funcion pairs U L Θ, U E Θ. Each poin in his graph corresponds o one asse allocaion sraegy θ 1, θ 2, K 0. The lef hand side presens all possible pairs; he righ hand side only conracs whose annualized defaul probabiliy P D := 1 1 Pτ T 1/T is smaller han 0.5%. The consan-mix sraegy θ 1 = θ 2 is given by he whie line. conrac from he radiional regulaory framework whie line is always Pareo-dominaed by several conracs from he flexible regulaory framework black area. The righ hand side of Figure 3 presens he subse of conracs whose annualized defaul probabiliy is lower han 0.5% feasible se X. To demonsrae his fac more clearly, Table II akes some specific asse allocaion sraegies. For each sraegy, he uiliy pair U L Θ, U E Θ and annualized defaul probabiliies are compued. 6 Firs, he able presens he opimal asse allocaion sraegy of he liabiliy holder sraegy 1b and he equiy holder sraegy 2b under he flexible regulaory scheme. As he equiy holder prefers he highes risk possible, his resuls are unchanged, i.e. 2a and 2b coincide. 7 For he liabiliy holder, however, resuls are differen: His opimal uiliy 6 In our example, he defaul probabiliy is increasing in boh θ 1 and θ 2. Inuiively, a higher equiy holding leading o a higher volailiy resuls in a higher probabiliy of hiing he defaul barrier D. If here is no regulaory barrier K, his can easily be proved analyically, i.e. from Equaion 16 we obain Pτ T = 2 lnd 0/A 0 lnd 0 /A 0 µ 1 T ϕ θ 1 σθ1 2 T σθ 1 T 4r gt lnd 0 /A 0 σ 2 θ1 3 µ r D0 2 µ 1 σ 2 θ 1 2 lnd 0 /A 0 µ 1 T σ 2 θ1 2 Φ > 0, A 0 σθ 1 T since D 0 < A 0 and g r < µ. 7 Noe ha for θ 1 = θ 2, he asse allocaion sraegy does no depend on he parameer K 0. 18

19 Flexible regulaory framework No. sraegy θ 1, θ 2, K 0 U L Θ, U E Θ P D 1 θ 1 = , % 1b 0.230, 0.040, , % 2b 1.000, 1.000, , % 3b 0.240, 0.110, , % Table II. Differen asse allocaion sraegy in a flexible regulaory framework asse allocaion is changed from θ 1 o θ 2 if regulaory barrier K = K 0 e g is hi. For each sraegy, he uiliy pair U L Θ, U E Θ and an annualized defaul probabiliy P D := 1 1 Pτ T 1/T is compued. is no obained under a consan-mix invesmen sraegy. Insead, he prefers he sraegy θ 1, θ 2, K 0 = 0.23, 0.04, 0.91, a sraegy wih a slighly higher iniial share in he risky asse and a sronger downside proecion. Sraegy 3b is Pareo-opimal if conracs have o fulfill he solvency requiremen P D 0.5% in his case he se of feasible conracs is X ; he resuls in Theorem 3.5 sill apply. Noe ha conrac 3b dominaes conrac 3a from Table I. To summarize our numerical illusraions, noe ha he flexible regulaory framework allows us o find fair conrac erms ha increase he goal funcion of boh he liabiliy holder and he equiy holder Pareo improvemen. An asse allocaion wih iniially θ 1 higher equiy raios and a reducion in case of disress θ 2 < θ 1 is benefical for liabiliy and equiy holder. 5. Conclusion The presen paper discusses flexible regulaory supervision o parly solve he conflic of ineress ha arises by he opion-like sakes of he insurance company. Typically, he regulaor imposes a Value-a-Risk-ype consrain defaul probabiliy consrain on he invesmen sraegy. In view of he opion-like sakes and non-linear sharing rules of a paricipaion life insurance conrac, we show ha he regulaor should no only impose a defaul consrain. We show ha an inervenion of he regulaor in case of disress forcing he insurance company o decrease risk in disress, migh miigae he conflic of ineress beween policy- and shareholder and lead o Pareo-improvemens for he wo involved paries. 19

20 The advanage of his flexible scheme is ha i is raher simple and in conras o, for example, a more flexible and dynamic asse allocaion sraegy like CPPI easy o implemen and supervise for a regulaory auhoriy. A leas from an academic perspecive, i is, however, desirable o generalize he resuls o oher dynamic asse allocaion sraegies. In our opinion, his is an imporan bu challenging furher research direcion. I is a leas possible o allow for more regulaory barriers o enlarge he se of possible invesmen sraegies. An exension o general dynamic asse allocaion sraegy seems o be difficul he problem lies in he non-lineariy of he conrac payoff and he fac ha he asse allocaion is opimized subjec o a fair pricing consrain. 6. Appendix Proof of Theorem 3.1. Firs, we recall resuls on he firs-hiing ime τ of a geomeric Brownian moion, i.e. he process A as defined in 2. The law of τ is known o be inverse Gaussian see, e.g., Folks and Chhikara 1978]. Lemma 6.1 recalls some resuls on he firshiing ime in his special case. Lemma 6.1 Firs-hiing ime disribuion. Consider he process A from 2. Then, he survival probabiliy wihin he inerval, T ] is given by P τ > T τ > µ 1 T lnd /A = Φ σθ 1 T D A 2 µ 1 σ 2 θ 2 1 µ 1 T lnd /A Φ, σθ 1 T where D < A, µ 1 := r θ 1 µ r g σ 2 θ 2 1/2, and Φ denoes he sandard normal cumulaive disribuion funcion. The densiy of τ can be obained from f i, τ, A, D := lnd /A lnd /A µ i τ ϕ. 20 σθ i τ 3 2 σθ i τ For y := lne gt A T /A, we define g 1 y,, T, A, D := P y dy, τ > T, 21 which is known o be g 1 y,, T, A, D = 0 for y lnd /A ϕ y µi T σθ 1 T σθ 1 T 1 e 2 lnd 2 /A y lnd /A σ 2 θ 1 2T else 20 22

21 where ϕ denoes he densiy of he sandard normal disribuion. Proof: See, e.g., Folks and Chhikara 1978], He e al. 1998], and Shreve 2004]. Noe ha he same resuls hold, if we replace he asse sraegy θ 1 by θ 2 and similarly µ 1 by µ 2 := r θ 2 µ r g σ 2 θ 2 2/2. We denoe he densiies ha resul from his parameer change by f 2, τ, A τ, D τ, respecively g 2 y,, T, A, D. We are now using Lemma 6.1 o prove Theorem 3.1. Noe firs ha if he barrier D is no hi in he inerval, T ], 22 helps us o obain he disribuion of he asses A a mauriy T. To compue he expeced uiliy of he erminal payoff Ψ L A T from 4, one simply has o inegrae is uiliy over 22 on he se lne gt D T /A, = lnd /A,. If he barrier is hi, i.e. τ T, he erminal payoff depends solely on he defaul ime τ whose disribuion can be obained from 20. This hen leads o E P ul V L ] = E P ul 1{τ>T } Ψ L A T 1 {τ T } e rt τ minl τ, D τ ] = E P 1{τ>T } u L ΨL A T ] E P 1{τ T } u L e rt τ minl τ, D τ ] = lnd /A T u L LT δ αa e ygt L T ] LT A e ygt ] g 1 y,, T, A, D dy u L e rt g rτ minl, D f 1, τ, A, D dτ. 23 In he case of power uiliy see Example 2.2, he laer inegrals can be furher simplified. Proof of Theorem 3.2. Theorem 3.2 can raher sraighforwardly be derived using he previous resuls. Noe ha he regulaory barrier K = K 0 e g is always hi prior o defaul due o he coninuiy of he process A. Up o ime ˆτ he process A is a geomeric Brownian moion wih sraegy θ 1 allowing us o use he densiy f 1, ˆτ, A, K from Lemma 6.1 for ˆτ. A ime ˆτ, we are back in he siuaion ha is already solved in Theorem 3.1: One has o adap he iniial values for A, D, and L. Furhermore, he ime o mauriy is now T ˆτ insead of T and he invesmen sraegy is now θ 2. If he regulaory hreshold K is never hi, we can in analogy o he proof of Theorem 3.1 compue he expeced uiliy of he erminal 21

22 payoffs o ge he firs erms of ζ L A, D, K, L, T, respecively ζ E A, D, K, L, T : ζ L A, D, K, L,, T = = T lnk /A lnd /K T T ˆτ lnk /A ζ E A, D, K, L,, T = = T lnk /A lnd /K T T ˆτ lnk /A T κ 2 L Kˆτ, Dˆτ, Lˆτ, ˆτ, T f 1, ˆτ, A, K dˆτ u L LT δ αa e ygt L T ] LT A e ygt ] g 1 y,, T, A, K dy, u L LT δ αkˆτ e ygt ˆτ L T ] LT Kˆτ e ygt ˆτ] f 1, ˆτ, A 0, K 0 g 2 y, ˆτ, T, Kˆτ, Dˆτ dy dˆτ u L e rt g rˆτ minl, D f 1, ˆτ, A, K f 2 ˆτ, τ, Kˆτ, Dˆτ dτ dˆτ u L LT δ αa e ygt L T ] LT A e ygt ] g 1 y,, T, A, K dy T κ 2 E Kˆτ, Dˆτ, Lˆτ, ˆτ, T f 1, ˆτ, A, K dˆτ A e ygt L T ] δ αa e ygt L T ] g 1 y,, T, A, K dy Kˆτ e ygt ˆτ L T ] δ αkˆτ e ygt ˆτ L T ] f 1, ˆτ, A, K g 2 y, ˆτ, T, Kˆτ, Dˆτ dy dˆτ e rt g rˆτ maxd L, 0 f 1, ˆτ, A, K f 2 ˆτ, τ, Kˆτ, Dˆτ dτ dˆτ A e ygt L T ] δ αa e ygt L T ] g 1 y,, T, A, K dy, wih κ i L, κi E, f, and g as defined in Theorem 3.1. Again, power uiliy simplifies he given expressions. 22

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