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

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1 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 of Erlangen-Nürnberg Version: Sepember 211

2 ANALYZING SURPLUS APPROPRIATION SCHEMES IN PARTICIPATING LIFE INSURANCE FROM THE INSURER S AND THE POLICYHOLDER S PERSPECTIVE Alexander Bohner, Nadine Gazer This version: Sepember, 211 ABSTRACT This paper examines he impac of hree surplus appropriaion schemes ofen inheren in paricipaing life insurance conracs on he insurer s shorfall risk and he ne presen value from an insured s viewpoin. 1) In case of he bonus sysem, surplus is used o increase he guaraneed deah and survival benefi, leading o higher reserves; 2) he ineres-bearing accumulaion increases only he survival benefi by accumulaing he surplus on a separae accoun; and 3) surplus can also be used o shoren he conrac erm, which resuls in an earlier paymen of he survival benefi and a reduced sum of premium paymens. The pool of paricipaing life insurance conracs wih deah and survival benefi is modeled acuarially wih annual premium paymens; moraliy raes are generaed based on an exension of he Lee-Carer (1992) model, and he asse process follows a geomeric Brownian moion. In a simulaion analysis, we hen compare he influence of differen asse porfolios and shocks o moraliy on he insurer s risk siuaion and he policyholder s ne presen value for he hree surplus schemes. Our findings demonsrae ha, even hough he surplus disribuion and hus he amoun of surplus is calculaed he same way, he ype of surplus appropriaion scheme has a subsanial impac on he insurer s risk exposure and he policyholder s ne presen value. 1. INTRODUCTION Paricipaing life insurance conracs are an imporan produc design in he German insurance marke and comprise various mechanisms of how surplus is disribued o he policyholders. Previous work has shown ha differen surplus disribuion schemes can significanly impac Alexander Bohner and Nadine Gazer are a he Friedrich-Alexander-Universiy of Erlangen- Nürnberg, Chair for Insurance Economics, Lange Gasse 2, 943 Nürnberg, Germany, The auhors would like o hank an anonymous referee for valuable commens and suggesions on an earlier version of his paper.

3 2 he insurer s risk exposure. In his conex, an imporan issue has no been comprehensively analyzed o dae, which is he concree appropriaion of disribued surplus. In paricular, in Germany, policies may feaure differen appropriaion schemes. 1 Surplus appropriaion refers o he way earned surplus, deermined via a given surplus disribuion mechanism, is acually credied o he individual policyholder. In he case of he bonus sysem, surplus is used o increase he guaraneed deah and survival benefi. In conras o his, he ineres-bearing accumulaion emphasizes he survival benefi, which is increased by he surplus, while he deah benefi is kep consan. The hird alernaive uses he surplus o shoren he conrac erm, which resuls in an earlier paymen of he survival benefi and a reduced sum of premium paymens. These hree schemes have no been comparaively examined, even hough heir impac on he insurer s risk siuaion and he policyholders expeced payoff can differ considerably. The aim of his paper is o fill his gap and o analyze his issue in deph. In he lieraure, paricipaing life insurance, along wih is surplus disribuion mechanisms and ineres rae guaranees, have araced widespread aenion. Research on he risk-neural valuaion of paricipaing life insurance conracs includes, for example, Briys and de Varenne (1997), who sudy he fair value of a poin-o-poin guaranee, where he company guaranees only a mauriy paymen and an opional paricipaion in he erminal surplus a mauriy, and deermine a closed-form soluion based on coningen claims heory. Grosen and Jørgensen (22) exend his framework by including a regulaory inervenion rule, which reduces he insolvency probabiliy and can be priced similar o barrier opions. In Grosen and Jørgensen (2), a clique-syle ineres rae guaranee is modeled, where surplus is annually credied o he policy reserves based on a reserve-dependen surplus disribuion mechanism o smooh marke reurns. Once he surplus is credied o he reserves, i becomes par of he guaranee and is hen annually a leas compounded wih he guaraneed ineres rae, hus implying clique-syle effecs. Besides he bonus opion and he minimum ineres rae guaranee, he auhors also include and evaluae a surrender opion by means of American-syle derivaives pricing. Based on he model by Grosen and Jørgensen (2), Jensen, Jørgensen, and Grosen (21) develop and apply a finie difference algorihm in order o numerically evaluae he conracs and furher inegrae moraliy risk. Differen annual surplus smoohing schemes are also examined in Hansen and Milersen (22) for he Danish case and in Balloa, Haberman, and Wang (26), where specific focus is laid on a comparison and radeoff of fair conrac parameers. A comparison of differen surplus disribuion models of paricipaing life insurance wih respec o model risk can also be found in Zemp (211). 1 See, e.g., Schradin, Pohl, and Koch (26, p. 17).

4 3 Oher work ha focuses on he surrender opion embedded in paricipaing life insurance conracs includes Albizzai and Geman (1994), who also accoun for sochasic ineres raes, as well as Bacinello (23) for an Ialian-syle conrac. Siu (25) reas he surrender opion by means of a regime-swiching model for economic saes, including ineres raes, expeced growh raes and volailiy of risky asses, and also presens approximaion mehods for paricipaing American-syle conracs. Schmeiser and Wagner (21) compue fair values of opions o early exercise, including he paid-up opion, he resumpion opion, and he classical surrender opion. Furhermore, several papers have focused on combining risk pricing and risk measuremen. Barbarin and Devolder (25) propose a model o firs assess he risk of a poin-o-poin guaranee and, second, calibrae he erminal bonus paricipaion parameer o obain fair conracs. Graf, Kling, and Russ (211) exend he approach used by Barbarin and Devolder (25) and generalize previous resuls by proving ha he combinaion of acuarial and financial approaches can always be conduced as long as he insurance conracs do no inroduce arbirage opporuniies. Gazer and Kling (27) deermine he real-world risk implied by fair conracs wih he same marke value, and Gazer (28) furher inegraes differen asse managemen and surplus disribuion sraegies in he analysis of paricipaing life insurance conracs wih he aim o assess heir impac on he conracs fair value, while keeping he defaul pu opion value consan. Kleinow and Willder (27) sudy hedging sraegies and calculae fair values for mauriy guaranees, where he surplus paricipaion depends on he insurer s managemen decisions regarding he invesmen porfolio. Wih respec o surplus disribuion schemes and risk measuremen, Gersner e al. (28) provide a general asse-liabiliy managemen framework for life insurance, which incorporaes, iner alia, a reserve-dependen bonus disribuion mechanism based on Grosen and Jørgensen (2). As an applicaion of heir model, hey sudy he impac of differen parameer seings and exemplary producs on he insurer s shorfall risk. Based on a single premium erm-fix insurance and hus focusing purely on financial risks, Kling, Richer, and Russ (27a) analyze he risk exposure of an insurer offering clique-syle ineres rae guaranees for differen conrac characerisics, including he iniial reserve siuaion, asse allocaion, and he acual surplus disribuion. Kling, Richer, and Russ (27b) exend his framework and consider he financial risk inheren in hree surplus disribuion sysems, including surplus appropriaion. The firs sysem incorporaes a clique-syle ineres rae guaranee, where he guaraneed rae also has o be paid on surplus, he second mechanism represens an ineres-

5 4 bearing accumulaion, where surplus canno be reduced once i has been credied o he policyholder s accoun (bu wihou clique-effecs), and hird, a surplus model, where he insurer can reduce surplus o keep he insurance company in business and o avoid insolvencies. As in Kling, Richer, and Russ (27a), moraliy effecs are no included. Hence, wha remains open is a holisic analysis of he financial and moraliy risk of surplus appropriaion schemes on he basis of a ypical life insurance produc, which is modeled acuarially by considering deah and survival benefis. The explici combinaion of acuarial pricing and reserving, as well as financial approaches, wih respec o shorfall risk and valuaion in he analysis of surplus appropriaion schemes has no been done o dae and is inended o offer insigh ino he impac of he ype and characerisics of surplus schemes on an insurance company s risk exposure and he policyholder s ne presen value. Furhermore, he sysem of shorening he conrac erm has no ye been examined. The aim of his paper hus is o fill his gap by analyzing he impac of surplus appropriaion schemes on a life insurer s risk exposure. In addiion and apar from his perspecive on risk, we furher sudy he policyholders ne presen value, i.e., he difference beween he expeced discouned deah or survival benefi and he sum of premium paymens. The model of he life insurance company is based on a paricipaing life insurance conrac 2 wih annual premiums, where moraliy raes are modeled using an exension of he Lee-Carer (1992) model proposed by Brouhns, Denui, and Vermun (22), and he asse base follows a geomeric Brownian moion. In conras o previous lieraure, insurance liabiliies for a pool of policies wih deah and survival benefis are calculaed using acuarial reserving rules, which depend on he surplus mechanism. In paricular, based on he smoohing surplus disribuion scheme of Grosen and Jørgensen (2), we analyze and compare hree companies wih differen appropriaion schemes, including he bonus sysem, 3 he ineres-bearing accumulaion, 4 and shorening he conrac erm. In a numerical simulaion analysis, we sudy he influence of differen asse porfolios and shocks o moraliy on he insurer s risk siuaion and he policyholder s ne presen value. Our findings demonsrae ha, even hough he surplus disribuion and hus he amoun of surplus is calculaed he same way, he ype of surplus appropriaion 2 In his paper, we use he expression paricipaing life insurance analogously o an endowmen conrac. 3 The bonus sysem accouns for clique-syle effecs, and, by including moraliy, surplus leads o higher paymens o he policyholders during he conrac erm due o he increased deah benefis. 4 The ineres-bearing accumulaion has been sudied in a similar form in Kling, Richer, and Russ (27b) bu wihou deah benefis or explici acuarial reserving rules.

6 5 scheme subsanially impacs he insurer s risk exposure and he policyholder s ne presen value. In addiion, he effec of he choice of he asse porfolio as well as shocks o moraliy differ considerably wih respec o he insurer s risk level depending on he respecive surplus appropriaion scheme, which should be aken ino accoun in he conex of underwriing aciviies and in asse managemen. The paper is srucured as follows. Secion 2 inroduces he model framework of he insurance company and he hree surplus appropriaion schemes under consideraion as well as he asse and moraliy model. Numerical resuls are presened in Secion 3, and Secion 4 concludes. 2. MODEL FRAMEWORK 2.1 Overview of he insurance companies We consider hree life insurance companies ha differ only in heir surplus appropriaion scheme, i.e., he way he surplus disribued o he policyholders is acually appropriaed o heir accouns. The schemes are presen in he German insurance marke, bu may as well be exended o similar schemes in oher counries. In he case of he bonus sysem, surplus is used o increase he guaraneed deah benefi as well as he survival benefi and hus increases he policy reserves. In conras o his, he ineres-bearing accumulaion emphasizes he survival benefi, which is increased by he surplus (and guaraneed unil mauriy), while he deah benefi is kep consan. The hird alernaive uses he surplus o shoren he conrac erm, which resuls in an earlier paymen of he survival benefi and a reduced sum of premium paymens. The corresponding balance shee for all hree ypes of companies is exhibied in Table 1. Table 1: Balance shee of a life insurance company a ime Asses Liabiliies A E PR IA RD B A A

7 6 The model is consruced in discree ime, where ime zero indicaes he incepion of he conrac, and = T (in years) mauriy. We furher assume ha he policy period coincides wih he accouning year. Here, A represens he marke value of asses, and E is he book value of equiy, which, similar o he model in Kling, Richer, and Russ (27a, 27b), is assumed o be consan over ime. Furhermore, PR denoes he book value of he policy reserves, IA is he book value of he ineres-bearing accumulaion accoun, RD is he book value of he surplus accoun for he reducion of conrac duraion, and B denoes he buffer accoun, which is deermined residually by subracing E, PR, IA, RD, and dividends paid o equiyholders from he marke value of he asse base, A. To gain insigh ino general effecs of differen surplus disribuion schemes, we assume a run-off scenario wihou new business. 2.2 Modeling he liabiliy side The paricipaing life insurance conracs In he following analysis, we consider a pool of radiional paricipaing life insurance conracs wih a conrac erm of n years, which are acuarially priced based on a moraliy able. Hence, he consan annual (ne) premium for an x-year old policyholder (for all hree surplus schemes) is given by A P S ä x: n = 1, x: n where S1 denoes he iniial guaraneed sum insured in case of deah or survival, paid in arrear, and A and ä represen presen values of an endowmen insurance for n years and a emporary annuiy for n years on he life of x, respecively, given x: n x: n by n n A = v x: n px qx+ + v n px = and n 1 ä = v x: n px =, (1) G where v ( 1 r ) 1 = +, and r G is he one-year calculaory acuarial ineres rae. The probabiliy of an x-year old insured o survive years is denoed by p x, whereas q x + is he probabiliy of an x + -year old insured o die wihin he subsequen year. Figure 1 illusraes he developmen of cash flows resuling from he insurance produc over ime, hereby disinguishing beween December 31 s of year 1 and January 1 s of year, denoed by and +, respecively.

8 7 Figure 1: Developmen of cash flows from he insurance produc over ime ( denoes December 31 s, and + denoes January 1 s for each year) n n + ime x S1 x + 1 S x + x + n - 1 Sn-1 Sn x+n age sum insured P = P P1 = P P = P Pn-1 = P premium D1 D Dn-1 Dn dividend As displayed in Figure 1, while he premium paymen is consan during he conrac erm, he benefi paymen varies depending on he surplus appropriaion scheme of he respecive company. The dividend paymens are based upon he developmen of asses and deah benefis and hus also change over ime. In he following, we consider a cohor of policies wih mauriy n = T. Modeling moraliy probabiliies Regarding he moraliy probabiliies, we disinguish wo cases. For acuarial pricing, he moraliy able by he German Acuarial Associaion ( DAV 28 T ) is used. However, when deermining he acual number of deahs during he conrac erm (relevan for valuaion and shorfall risk as well as he deerminaion of acual policy reserves), we use a furher developmen of he Lee-Carer (1992) model, which consiss of a demographic par and a ime series par. The cenral deah rae or force of moraliy µ x ( τ ) is modeled hrough µ ( τ ) = a + b k τ + ε ( ) ax + bx k + ε x,, τ µ τ = ln x x x x x e τ τ, where ax and b x are ime consan parameers indicaing he general shape of moraliy over age and he sensiiviy of he moraliy rae a age x o changes in k τ, respecively, where k τ is a ime varying index ha reflecs he general developmen of moraliy over ime, and ε x, τ is an error erm wih mean and consan variance. Brouhns, Denui, and Vermun (BDV) (22) propose a modificaion o he model by modeling he realized number of deahs a age x and ime τ, Dx, τ, as Dx, τ ~ Poisson Ex, τ µ x τ ax x ( ( )) wih µ ( τ ) = + b kτ x e

9 8 where Ex, τ is he risk exposure a age x and ime τ. The advanages of he BDV (22) model are ha he resricive assumpion of homoscedasic errors made in he Lee-Carer (1992) model is given up and ha he resuling Poisson disribuion is well suied for a couning variable, such as he number of deahs. The model can be calibraed via he Maximum-Likelihood approach using a uni-dimensional Newon mehod as proposed by Goodman (1979). 5 Lee and Carer (1992) propose o fi an appropriae ARIMA process on he esimaed ime series of k τ, using Box-Jenkins ime series analysis echniques o forecas k τ, k = φ + α k + α k α k + δ ε + δ ε δ ε + ε τ 1 τ 1 2 τ 2 p τ p 1 τ 1 2 τ 2 q τ q τ where p and q are chosen using Box-Jenkins ime series analysis echniques and φ is he drif erm. To model shocks o moraliy, k τ is muliplied by a facor δ. Values of δ less han one resul in moraliy raes greaer han esimaed, and for δ greaer han one, moraliy raes are smaller han esimaed. Policy reserves The acuarial reserve for he considered endowmen insurance for an x+-year old insured a ime (and condiioned on he exisence of he conrac) is denoed by V x, and is prospecive calculaion is given by V = S A P ä, (2) x + 1 x+ : n x+ : n where S+1 is he curren guaraneed sum insured in case of deah or survival payable a he end of year, and P denoes he consan level premium. As before, he presen values are calculaed acuarially as defined in Equaion (1) based on he moraliy able and he calculaory (guaraneed) ineres rae. Hence, he oal porfolio policy reserve a he end of year is deermined by PR = N d V i x, (3) i= 1 5 Sandard Maximum-Likelihood mehods are no feasible due o he presence of he bilinear erm b xk.

10 9 where N is he iniial number of conracs sold and di is he acual number of deahs ha occurred during year i, deermined based on he BDV (22) model. Thus, o obain he policy reserve in he porfolio, he number of policies sill in force is muliplied by he acuarial reserve for one conrac. Buffer accoun As described in Table 1, he buffer accoun a he end of year for all hree companies, i.e. for all hree surplus appropriaion schemes under consideraion, is given residually by B = A PR IA RD E, (4) where IA is se o zero in case of he bonus sysem and in he case of shorening he conrac erm. The accoun RD is used only in he case of shorening he conrac erm and herefore se o zero in he oher wo cases. Furhermore, equiy capial is kep a a consan level as assumed in Kling, Richer, and Russ (27b), i.e. E = 1. E A he end of he las year, i.e. in T, he buffer accoun is paid ou o he policyholders in he sense of a erminal bonus afer subracing dividends. The erminal bonus (TBT) canno become negaive and is given by he residual of he remaining asses and he policyholder accouns as well as dividends and equiy capial, resuling in ( ) ( ) TB = max B D, = max A PR IA RD E D,. T T T T T T T T T Since he buffer accoun has been filled by he excess premiums of he policyholders, his procedure suppors he comparabiliy of he hree companies wih he differen surplus appropriaion schemes. 6 6 We do no consider effecs resuling from reserves, which are passed on o he nex generaion of policyholders. Here, we refer o Døskeland and Nordahl (28) and Faus, Schmeiser, and Zemp (211).

11 1 2.3 Modeling he asse side Developmen of he asse base The oal iniial capial A consiss of equiy capial and he firs premium paymens, which are invesed in a porfolio consising of socks and bonds ha is assumed o follow a geomeric Brownian moion da = µ A d + σ A dw, P where µ is he drif of he asses, σ he asse volailiy, and W P a sandard Brownian moion under he real-world measure P on he probabiliy space ( ΩF,, P), where F is he filraion generaed by he Brownian moion. The soluion of he sochasic differenial equaion is given by (see Björk, 29) ( µ σ 2 2+ σ ε ) r A = A e = A e, ( ) ( ) 1 1 wih ε being a sandard normally disribued random variable and r being he coninuous oneperiod reurn of he porfolio wih expeced reurn ( ) 2 E r = m = µ.5 σ and a sandard deviaion of σ. Under he risk-neural pricing measure Q, he drif of he process changes o he risk-free rae rf. Differen (µ,σ)-combinaions represening differen porfolio composiions are generaed by assuming ha ( 1 ) r = a r + a r, S B where rb and rs sand for he coninuous one-period reurns of bonds and socks, respecively, which follow a normal disribuion wih expeced values of E(rB) = mb and E(rS) = ms, sandard deviaions of σb and σs, and a coefficien of correlaion of ρ. To accoun for decremens in he porfolio of policyholders due o deah, one needs o disinguish beween he end and he beginning of a year wih respec o he evoluion of he ne asses, i.e. asses invesed in he capial marke minus paymens for deahs during year. The erm A hus describes he value of asses a he end of year, which is given by A r = A e S d, wih A =, A = P N + E, (5) ( 1) + +

12 11 where S is he sum insured prevailing in year (ha depends on he surplus schemes), and d is he number of deahs beween ime -1 and, N is he number of conracs sold, and P is he consan premium for each individual conrac (same for all surplus schemes). Furhermore, equiyholders receive annual dividend paymens D ha depend on he developmen of asses and deah benefi paymens: ( ( 1) ) D = max A A,, β + where β denoes he fracion of he increase in asses ha is paid ou as dividends. The insurer is solven if he buffer accoun plus equiy capial is posiive, B + E, implying ha asses are sufficien o cover he liabiliies, i.e., A PR + IA + RD. In his case, he insurer pays ou dividends D o he equiyholders only if B D, leading o B = B D, if B D. + The dividend paymen is se o zero if he insurer is solven bu does no have enough reserves o pay he dividends. Hence, if B + E, bu B < D, hen D =. 7 A he beginning of he subsequen year, premiums are due, which resuls in an asse developmen (see also Equaion (5)) given by r A + = A D + P i. N d = A + e S ( 1) d D + P N d i i= 1 i= 1 If he insurer is insolven, i.e., B + E < and hus A < PR + IA + RD, he equiy capial is no sufficien o cover he losses, he company is closed down, and he remaining funds A e c are disribued o he remaining policyholders in he porfolio, reduced by r + ( 1 ) ( 1) he coss of insolvency c. Noe ha deah benefis are no fully paid ou, bu he beneficiaries receive a remaining fracion of he asses. 7 The insurer also remains solven if he buffer accoun becomes negaive, bu equiy capial is sufficien o cover he losses in his period, i.e., B, bu B E < +. In his case, equiy capial is reduced by he amoun of he loss and B =. In he nex period, we assume ha he amoun of equiy capial is increased again o he original amoun by using gains from he nex period (see Equaion (4)).

13 Surplus appropriaion schemes Afer having defined he general developmen of asses and liabiliies, we now furher specify and disinguish he hree companies under consideraion ha differ only in heir surplus appropriaion scheme. Le r G P denoe he calculaory and guaraneed ineres rae, and r be he acual policy ineres rae credied o he policy reserves for period -1 o, which is o be deermined a ime To smooh marke reurns and o obain less volaile and more sable reurns, he surplus disribuion approach is based on Grosen and Jørgensen (2). Here, a reserve-based sysem is used wih r P B + G ( 1) = max r, α γ, (6) PR IA RD + + ( 1) ( 1) ( 1) where γ indicaes a required proporion of he buffer accoun divided by he policyholder s accouns, which consiue he guaraneed liabiliies, 9 i.e. γ represens he arge buffer raio. The second adjusing parameer for disribuing surplus o he policyholders is he surplus disribuion raio α. I conrols he fracion of he excess amoun of he arge buffer raio, which is acually credied o he policyholders. 1 The policy reserves earn a leas he guaraneed ineres rae r G and serve as he basis for deermining he surplus o be credied o he policyholders. The absolue amoun of surplus generaed in he -h year is hus given by he difference beween he policy ineres rae and he acuarial ineres rae, muliplied by he policy reserve: PR r r. (7) P G ( ) ( 1) While he amoun of surplus is calculaed he same way for all hree companies (and, hence, he surplus disribuion approach is he same), he appropriaion scheme and hus he way he surplus is disribued o policyholders differs and plays an imporan role regarding he insur- 8 This is in line wih he declaraion in advance, see Schradin, Pohl, and Koch (26, p. 14). 9 Since he buffer accoun should provide a cushion o absorb losses wih respec o he guaranees on he balance shee s liabiliies side, all hree policyholder accouns have o be considered in he denominaor. 1 Usually, regulaions, such as hose in Germany, specify a maximum period of ime for he surplus o be kep in he buffer accoun and buffer, respecively, unil i has o be credied o he insureds (see, e.g., Schradin, Pohl, and Koch (26, p. 14)).

14 13 ance payous and hus also for he evoluion of he asse base (see Figure 1). The differen schemes under consideraion are presened in deail in he following subsecions. Company 1: Bonus sysem In he case of he bonus sysem, he surplus is used o increase he iniially guaraneed sum insured S1 (deah and survival benefi) by calculaing a new insurance wih he same ime o mauriy (and he same ype) like he original insurance, using he surplus as a single premium for he new conrac. By using he acuarial equivalence principle, he surplus per insured resuls in an addiional sum insured S of S = P G ( ) ( 1) ( i= 1 i ) PR r r N d A x+ : T which leads o an increased sum insured of S = 1 S + + S. In his seing, he surplus insurance also paricipaes in fuure surplus and hus involves clique-syle ineres rae effecs. In paricular, he increased sum insured impacs he developmen of he policy reserves (see Equaions (2) and (3)) as well as he amoun of surplus ha can be disribued o he policyholders (see Equaion (7)), hus inducing clique-syle effecs. Company 2: Ineres-bearing accumulaion In he case of he ineres-bearing accumulaion, he sum insured is kep consan, i.e., we se S, 1,, = S1 = T. Hence, surplus is no used o increase he sum insured, bu insead is accumulaed on a separae accoun IA. Once funds are credied o his accoun, hey belong o he policyholders and canno be wihdrawn anymore. This implies an ineres rae guaranee of a leas zero percen. The accoun is paid ou a mauriy in case of survival. If he policyholder dies during he conrac period, only he consan sum insured is paid ou, and he remainder is kep by he insurer and in he form of he (non-guaraneed and used for smoohing) buffer accoun is laer evenually paid ou o he remaining policyholders ha are sill alive a mauriy as an opional bonus. Hence, his surplus appropriaion scheme emphasizes he survival benefi as compared o deah and survival benefi in he case of he bonus sysem. The recursive forward projecion of he ineres-bearing accoun is given by

15 14 ( i ) ( ) ( ) 1 ( ) ( ) IA P G = 1+ 1, =. i= IA IA r d N d PR r r IA A ime, he accoun value is calculaed based on is value in he previous period and an ineres rae r IA and is adjused for deahs, i.e. funds ha belonged o policyholders who died wihin he -h year are passed on o he colleciviy of policyholders. Finally, new surplus is added o he accoun. Company 3: Shorening he conrac erm In he hird case, he surplus is used o decremen he remaining years o mauriy, hus resuling in earlier benefi paymens o he policyholders, where S is kep consan. Hence, he oal conrac erm n( ) is considered as a funcion of ime, i.e. n can be reduced from each period o he nex, saring wih n( ) = T. Since he insurer operaes in discree ime wih one year represening one period, he conrac erm is no reduced unil he oal surplus earned is sufficien o finance he gap beween he acuarial reserve for n( 1) and he acuarial reserve for a reduced conrac erm of a leas one year. If he surplus is no sufficien o reduce he conrac erm for one year or surplus remains afer reducing he conrac erm, he remaining surplus amoun RD is ransferred o he nex year. Analogously o he case of he ineres-bearing accumulaion, he fracion of deahs and an ineres rae r RD is accouned for: ( i ) ( ) ( ) = 1+ RD 1 P G, = 1. 1 ( ) ( ) ( i= 1 ) RD RD r d N d PR r r RD Hence, we firs deermine he value of he acuarial reserve for an unchanged conrac period a ime, i.e. for ( 1) ( ) n, denoed by ( 1) V n and deermined analogously o Equaion x (2), where n mus be replaced by n( 1). Nex, we add o his he surplus RD per policyholder, which resuls in ( ( 1) ) ( 1) ( ) ( i= 1 ) V n V n RD N d = +. surplus x x i Third, we calculae he acuarial reserves for a conrac period decremened by h years, where h sars wih zero and is incremened successively o he oal remaining conrac erm a ime, i.e. n( 1). This is given by

16 15 ( ) ( ) surplus { x ( ( )) x ( ( ) ) } hmax = max h : V n 1 V n 1 h h H ( ) { } where Vx n( 1) h is deermined based on Equaion (2) by replacing n wih n( 1) and H ( ) =,, n( 1). By h ( ) max h, we indicae he maximum number of years by which he conrac erm can be reduced in year. The new policy period, saring from year, is given by ( ) ( 1) ( ) n = n h. max ( ) The policy reserve (for one individual conrac) ( ) V n can hen be calculaed, and he surplus accoun is defined by + surplus ( x ( ( 1) ) x ( ( ))) ( i= 1 i ) RD V n V n N d =. x Finally, we can deermine he acuarial reserve for he pool of conracs, i.e., PR, analogously o Equaion (3) by replacing V x wih Vx ( n( )). A mauriy, any remaining amoun RDT is paid ou o he policyholders. 2.5 Evaluaing he surplus appropriaion schemes from differen perspecives To assess he impac of he hree surplus appropriaion schemes from perspecives of he insurer and he policyholder, we calculae he company s shorfall risk and he policyholder s ne presen value of he conrac. Of course, hese wo figures are cerainly relevan o boh paries. For insance, he ne presen value from he policyholder s perspecive can also be inerpreed as he couner value of he conrac o he insurer. 11 Overall, however, boh numbers will be relevan o he insurer and he policyholder and are laid ou in wha follows. A shorfall of he company occurs if he value of he asses A falls below he value of liabiliies, A < PR + IA + RD (or, equivalenly, if B + E < ). Hence, he shorfall probabiliy is defined as 11 Furhermore, policyholders ofen evaluae heir conracs based on individual preferences insead of assuming a risk-neural valuaion approach ha implicily assumes replicabiliy of cash flows. However, he fair value expressed by he ne presen value is sill a relevan figure for policyholders.

17 16 ( ) SP = P T T, s T = inf : A < PR + IA + RD, = 1,..., T. where he ime of defaul is defined as { } s The ne presen value (NPV) of he conrac is calculaed as he expeced value under he riskneural probabiliy measure Q of he difference beween he discouned conrac payoff and he discouned sum of premium paymens aking ino accoun he case of defaul. For an individual policyholder, he NPV is hus given by rf ( + 1) rf ( x x+ + 1 x ) 1{ s } = > = T 1 Q NPV E p q S e p P e T T Q 1 + E T p x ST + IA + RD + TBT e Ts > T T rf ( ) 1 T { } T T N d i= 1 i T 1 1 Q r r 1 f ( + 1 ) rf + + E p x ( A + e ) ( 1 c) e p x P e 1 T = N d i= 1 i { = + 1} s where p x and q x + are he survival and deah probabiliies, respecively, derived hrough he BDV (22) model. Moraliy and marke risks are assumed o be independen, 12 and he insurance company does no demand a risk premium for moraliy risk. 13 In conras o he acuarial pricing, which does no accoun for he surplus disribuion or defaul, he possibiliy of defaul and he surplus disribuion and appropriaion is considered in he calculaion of he fair ne presen value. Thus, he annual premium paymen P is he same for all hree surplus schemes, bu he amoun of surplus differs. If no shorfall occurs, i.e. if Ts > T, he policyholder receives a deah benefi or a survival benefi, which also includes he erminal bonus. If defaul occurs during he conrac erm, he remaining asses are disribued among he policyholders sill alive (and o he heirs of hose who died wihin he year of defaul). 3. NUMERICAL ANALYSIS In his secion, numerical resuls are presened based on he model inroduced in he previous secion wih respec o he insurer s risk exposure and he policyholder s ne presen value for 12 See, e.g., Carriere (1999, p. 34), Gründl, Pos, and Schulze (26), Hanewald (211). 13 See, e.g., Bacinello (23, p. 468), Gründl, Pos, and Schulze (26).

18 17 each of he hree surplus appropriaion schemes. 14 Afer presening he inpu parameers, we nex sudy he exen o which differences in shorfall risk arise for he hree surplus appropriaion schemes wih regard o varying asse porfolios and differen shocks o moraliy. Second, we exend his viewpoin and sudy he effecs on he ne presen value. Numerical resuls are derived using Mone Carlo simulaion mehods based on he same se of 5, asse pahs. Inpu parameer The underlying policies are paricipaing life insurance conracs issued o x = 35 year old males wih a conrac erm of T = 3 years. Wih an iniial sum insured of S1 = 1, he acuarial annual premium is given by P =.247. A oal number of N = 1, conracs are sold. Assumpions abou he evoluion of he asses are based on he hisorical performance (1988 unil 29) of wo represenaive German oal reurn indices. The esimaion for he socks, which is based on monhly daa for he German sock marke index DAX, resuls in an expeced one-period reurn ms = 8.% and a volailiy σs = 21.95%. The esimaion for he bonds, which is based on monhly daa for he German bond marke index REXP, leads o an expeced one-period reurn of bonds mb = 6.2% and a volailiy of bonds σb = 3.3%. The esimaed correlaion coefficien of reurns of he wo indices is ρ = Furhermore, we assume he disribuion raio o be α = 7% and he arge buffer raio o be γ = 1%. Unless saed oherwise, we assume furher relevan parameers o be hose saed in Table 2. Table 2: Parameers for he analysis Expeced one-period reurns of socks ms 8.% Volailiy one-period reurns of socks σs 21.95% Expeced one-period reurns of bonds mb 6.2% Volailiy one-period reurns of bonds σb 3.3% Correlaion beween socks and bonds ρ Sock porion a 1% Guaraneed ineres rae r G 2.25% Rae of ineres for he ineres-bearing accumulaion accoun r IA % Rae of ineres for he accoun RD r RD % 14 For robusness, we also calculaed he mean loss in addiion o he shorfall probabiliy and found ha he general paerns of he resuls were similar. 15 The correlaion coefficien is significan a a level of.1.

19 18 Risk-free rae rf 3% Number of conracs sold N 1, Sum insured in = S1 1 Level premium for T = 3 P.247 Equiy in = E 6 Conrac erm T 3 Age of he policyholders in = x 35 Dividend paymen raio β 5% Disribuion raio α 7% Targe buffer raio γ 1% Reducion coefficien for coss of insolvency c 2% Shock o moraliy δ 1 Incepion dae τ 29 The esimaion of he parameers for he BDV (22) model is conduced on he basis of moraliy daa for Germany for he years 1956 unil 28. Numbers of deahs and exposure o risk are available hrough he Human Moraliy Daabase. For he years 1956 o 199, daa for Eas and Wes Germany are combined, whereas from 199 o 28, daa for he oal of Germany is used. The esimaed values of ax e, which can be inerpreed as a mean cenral deah rae a age x, and bx are given in Figure 2. Figure 2: Esimaed values of he moraliy index k τ and prediced values of k τ for differen shocks o moraliy δ, and esimaed values of he ime consan parameers ax e and b x esimaed and prediced k τ esimaed e a x k τ δ =.7 δ = 1. δ = 1.3 e a x b x age esimaed b x year age

20 19 Furhermore, Figure 2 shows he esimaed ime series k τ as well as is predicion by using Box-Jenkins ime series analysis echniques, illusraing he effec of differen shocks o moraliy. Based on he Bayesian informaion crierion, an ARIMA (3,1,) process is used. Is parameers are given as follows: drif φ = (.5295), α1 = (.1229), α2 =.1124 (.1227), and α3 =.468 (.1246), where he sandard errors are given in parenheses. Residual auocorrelaion can be excluded afer applying he Box-Ljung es (Pormaneau es), ACF and PACF analyses. The impac of surplus appropriaion schemes on shorfall risk Firs, in Figure 3, we sudy he shorfall risk of he hree companies for differen asse allocaions (lef graph) and differen levels of moraliy (righ graph). As expeced, a riskier invesmen leads o subsanially higher shorfall probabiliies. One can see a rapid increase in he shorfall probabiliy as he sock raio grows. For insance, sock raios beween % and 5% resul in a shorfall probabiliy of around.1, while a sock raio of 2% implies a shorfall probabiliy of abou.5, and for sock raios of 2% upwards, he defaul risk increases exponenially. Thus, despie he fac ha gains a he capial marke are smoohed via he buffer accoun (see Equaion (6)), once he surplus is credied o he insureds, i is ransformed o guaranees, which have o be generaed in subsequen periods. Figure 3: Shorfall probabiliy for varying invesmens in socks and varying shocks o moraliy shorfall probabiliy % 5% 1% 15% 2% 25% sock porion a shorfall probabiliy shock o moraliy δ bonus sysem ineres-bearing accumulaion shorening he conrac erm

21 2 Furhermore, he resuls show ha, even hough he amoun of surplus is calculaed in he same way according o he smoohing scheme given in Equaion (6), he specific ype of appropriaion scheme can have a very differen impac on shorfall risk. In paricular, he bonus sysem leads o he highes shorfall probabiliy and hus dominaes he sysem of shorening he conrac erm, which, in urn, dominaes he ineres-bearing accumulaion scheme. This order is mainly due o he differen ypes of guaranees implied by he considered schemes as illusraed in Figures 4 and 5, which is highes in he case of he bonus sysem, since, once he surplus is credied o he policyholder, he guaraneed deah and survival benefi are raised, which in urn increases he policy reserves. This leads o cliquesyle effecs, since a higher reserve resuling from a higher sum insured is subjec o he effec of compound ineres, i.e. he guaraneed ineres rae r G is also paid on he surplus. As illusraed in Figure 4 (lef graph), his implies an increasing guaraneed deah benefi saring from around he 1 h policy year on, which a T = 3 reaches a value ha is more han 7% higher han in case of he oher wo sysems. In conras o he bonus sysem, he guaraneed deah benefi is consan and equal o one in case of he ineres-bearing accumulaion and shorening he conrac erm. The average guaraneed survival benefi a mauriy T = 3 of he ineres-bearing accumulaion is slighly higher han he bonus sysem, bu does no compensae for he considerably higher deah benefis during he conrac erm (righ graph in Figure 4). Figure 4: Average guaraneed sum insureds and average survival benefi including he erminal bonus average guaraneed deah benefi average survival benefi including erminal bonus policy year policy year bonus sysem ineres-bearing accumulaion Noes: Average guaraneed deah benefi E ( S T ) shorening he conrac erm = s >, where T s denoes he ime of defaul; per insured average survival benefi including erminal bonus = E ( S + TB Ts > Tadj = ), where T adj denoes he ime when he survival benefi is paid.

22 21 The sysem wih shoring he conrac erm is more difficul o compare o he oher wo sysems as he survival benefi is paid ou earlier beween he 19 h and he 3 h policy year, on average. However, when considering he lef graph in Figure 5, he developmen of he policy reserves shows ha he bonus sysem implies he highes average policy reserves, followed by shorening he conrac erm and he ineres-bearing accumulaion. Furhermore, Figure 5 (righ graph) shows how he buffer accoun is buil up over ime and ha he ineres-bearing accumulaion feaures he highes value hroughou he conrac erm, followed by he bonus sysem and shorening he conrac erm. 16 Thus, even hough he comparabiliy is sill limied, he order of he hree sysems wih respec o shorfall risk can be generally confirmed by analyzing he policy reserves, he buffer accoun, and guaraneed sums insured in case of deah and survival, implying ha he bonus sysem has he highes risk, followed by shorening he conrac erm and he ineres-bearing accumulaion. Figure 5: Average policy reserves and he average developmen of he buffer accoun average policy reserves average buffer accoun in housand in housand policy year policy year bonus sysem ineres-bearing accumulaion shorening he conrac erm Noes: Average policy reserves = E ( PR T s > Tadj ), where s and T adj he ime when he survival benefi is paid; average buffer accoun E ( B + Ts Tadj ) T denoes he ime of defaul = >. 16 Noe ha he values of he average buffer accoun for he sysem of shorening he conrac erm in he las wo policy years 29 and 3 may vary when using differen ses of random numbers due he low number of scenarios in which he survival benefi is paid in hese las periods. In mos cases, he survival benefi is paid ou unil he 28 h year.

23 22 Thus, depending on he surplus appropriaion scheme, he companies can afford a riskier asse base, implying a higher expeced reurn while keeping he shorfall risk consan, while sill achieving he same shorfall probabiliy. For example, a sock porion of a = 19.8% in he case of Company 1 (bonus sysem), a = 2.5% for Company 3 (shorening he conrac erm), and a = 21.1% in he case of Company 2 (ineres-bearing accumulaion) lead o he same shorfall probabiliy of.5. Moreover, i becomes apparen ha he gap, i.e., he absolue difference in shorfall risk associaed wih he hree surplus appropriaion schemes expands wih an increasing sock porion, even hough gains a he capial marke are smoohed via he buffer accoun. This implies ha i is considerably riskier for Company 1 (bonus sysem) o ceeris paribus hold an asse porfolio conaining more high-risk asses han i is for Company 2 (ineres-bearing accumulaion) and Company 3 (shorening he conrac erm), as a more risky asse sragey implies a higher surplus, which in urn emphasizes he clique-syle effecs and hus he difference beween he hree surplus appropriaion sysems. This behavior is also illusraed in Figure 6, where he number of shorfalls is displayed for differen sock porions, shocks o moraliy, and higher iniial equiy capial. In paricular, a comparison of he case wih a = 1% and a = 25% (for δ = 1) shows ha he bonus sysem and shorening he conrac erm exhibi a much higher number of shorfalls when increasing he sock porion, especially for higher conrac years (saring from he 13 h year), as compared o he ineres-bearing accumulaion. Nex, we focus on he shorfall risk resuling from a change in moraliy as illusraed in he righ graph of Figure 3. We hereby analyze he insurer s shorfall probabiliy as a funcion of a shock o moraliy (δ), for δ [.7, 1.3], where δ =.7 can be considered o represen a pandemic. Since hese shocks o moraliy are modeled by muliplying he negaive ime rend k τ by δ, values of δ less han 1 increase he moraliy raes, while values of δ > 1 decrease he moraliy raes (see also Figure 2, lef graph). The resuls demonsrae ha, even hough he level of shorfall risk is reasonably small for a sock porion of 1% and for he given se of parameers, he relaive changes in risk wih respec o a shock o moraliy are no negligible. For example, a change from δ = 1 (no shock o moraliy) o δ =.7 resuls in abou 25% more deahs wihin he considered conrac period of T = 3 years, and, in he case of he bonus sysem, increases he insurer s risk by abou 5%. In addiion o his general effec of higher shorfall probabiliy for an increasing shock o moraliy, he order of he surplus appropriaion schemes wih respec o he level of shorfall risk remains similar o he case of varying he asse porfolio.

24 23 Figure 6: Number of shorfalls for differen sock porions, shocks o moraliy, and higher iniial equiy capial a = 1%, δ = 1., iniial equiy = 6 a = 1%, δ =.7, iniial equiy = policy year policy year a = 25%, δ = 1., iniial equiy = policy year a = 25%, δ =.7, iniial equiy = policy year a = 25%, δ = 1., iniial equiy = policy year a = 25%, δ =.7, iniial equiy = policy year bonus sysem ineres-bearing accumulaion shorening he conrac erm

25 24 However, in conras o he asse base, for he considered range of parameers, he difference beween he shorfall risk of he hree companies remains fairly sable wihou showing a gap as in he case of high sock porions. Thus, in conras o he sock porion, where paricularly he bonus sysem exhibis an increasing gap compared o he wo oher schemes, all hree companies are affeced in a similar severe way by shocks o moraliy. This can also be seen from Figure 6 when comparing he lef column o he righ column. In paricular, an increase in moraliy by δ =.7 implies a considerably higher number of shorfalls during he firs conrac years, where he difference beween he hree surplus schemes is sill negligible wih respec o guaraneed deah or survival benefi as well as policy reserves (see Figures 4 and 5), while owards he end of he conrac erm, he number of defauls remain overall sable, even hough he number of deahs increase for higher ages (compensaed by a higher buffer accoun owards he end of he conrac erm, see righ graph of Figure 5). This holds rue for a given sock porion, even when increased o 25% and when increasing he amoun of iniial equiy from 6 o 1,2, which implies ha more defauls occur owards he end of he conrac period. In paricular, he increase in E from 6 o 1,2 (for a = 25% and δ = 1., see second and hird row in Figure 6, lef graphs) implies a reducion in he shorfall probabiliy from SP = 16.4% o SP = 1.3% in case of he bonus sysem, from SP = 13.% o SP = 7.2% for shorening he conrac erm, and from SP = 11.5% o SP = 5.9% in case of he ineres-bearing accumulaion, which illusraes he imporance of he iniial buffer siuaion. Hence, while here is a considerable difference beween he hree schemes when comparing differen sock porions (differen rows in Figure 6), heir reacion wih respec o shocks o moraliy is similar and he amoun of he iniial equiy capial (and hus he iniial buffer) has a considerable impac on shorfall risk. Anoher imporan facor in life insurance is he conrac duraion. Figure 7 displays he shorfall risk for differen conrac erms and differen sock porions (a =1%, 25%). All oher parameers being unchanged, including he sum insured, he level premium has o be adjused (e.g., in case of T = 4 o P =.181). Here, hree effecs inerac. Firs, due o he lower premium paymens for a longer conrac erm, reserves build up more slowly, which defers he surplus disribuion mechanism. Second, he prediced ime varying index k ha τ reflecs he general developmen of moraliy over ime is sricly monoonic decreasing as displayed in Figure 2. This enlarges he discrepancy in moraliy beween he projeced moraliy and he moraliy given by he moraliy able (premium calculaion) for he addiional 1 policy years. Third, moraliy raes are higher for people aged 66 o 75 compared o 65 years and younger, which implies ha he addiional conrac period

26 25 pronounces he shorfall risk associaed wih guaraneed deah benefis and, herefore, he company wih he bonus sysem. Figure 7: Shorfall probabiliy as a funcion of he conrac erm T for a sock porion of a = 1% and for a = 25% shorfall probabiliy for a = 1% conrac erm T shorfall probabiliy for a = 25% conrac erm T bonus sysem ineres-bearing accumulaion shorening he conrac erm Generally, he shorfall probabiliy decreases wih an exension of he conrac erm of en years. Figure 7 furher reveals ineracion effecs beween ime o mauriy and sock porion. Here, while he risk level is sill decreasing for all hree surplus schemes for a sock porion of a = 1% (lef graph in Figure 7), for a sock porion of a = 25% (righ graph in Figure 7), he level of shorfall risk for he bonus sysem remains sable, which can o a lesser exen also be observed for he sysem wih shorening he conrac erm, while he shorfall probabiliy of he ineres-bearing accumulaion sill exhibis a clearly decreasing level. This shows ha he gap in he shorfall probabiliy beween he bonus sysem and he ineres-bearing accumulaion as well as shorening he conrac erm increases considerably for riskier asses and an increased conrac erm (see also Figures 4, 5, and 6). This resul is paricularly relevan agains he background of long-erm conracs. Thus, shorfall risk canno be as effecively reduced for a longer erm in he case of higher sock porions when using he bonus sysem and also when shorening he conrac erm, i.e., in he case of he wo oher companies wih emphasis on guaraneed deah and survival benefis (possibly paid ou earlier). Anoher key risk driver in his conex is ypically he guaraneed ineres rae. Resuls are displayed in Figure 8 (lef graph) and show ha a higher conracual guaranee in he form of

27 26 a higher ineres rae guaranee leads o higher shorfall risk for all hree sysems, sressing again ha he bonus sysem is associaed wih he highes shorall risk, he sysem of shorening he conrac erm consiues he second highes shorfall risk, and he ineresbearing accumulaion has he lowes shorfall risk. I can be furher noiced ha hese differences in shorfall risk increase wih an increase in he guaraneed ineres rae. Figure 8: Shorfall probabiliy as a funcion of he guaraneed ineres rae r G (r IA = %, r RD = %, lef graph) and shorfall probabiliy for differen asse allocaions for r G = r IA = r RD = 2.25% (righ graph) shorfall probabiliy shorfall probabiliy % 2.% 2.25% 2.5% 2.75% 3.% guaraneed ineres rae r G % 5% 1% 15% 2% 25% sock porion a bonus sysem ineres-bearing accumulaion shorening he conrac erm The righ graph in Figure 8 shows he shorfall probabiliy as a funcion of asse allocaion, where he ineres raes for he ineres-bearing accumulaion accoun and for shorening he conrac erm are increased and se equal o he guaraneed ineres rae, i.e. r G = r IA = r RD = 2.25%. In his case, he shorfall risk of he ineres-bearing accumulaion is almos equal o he sysem of shorening he conrac erm. Noneheless, hese wo schemes are significanly dominaed by he bonus sysem wih regard o shorfall risk. Furhermore, i is relevan o assess he impac of he parameers for he surplus disribuion mechanism on he shorfall risk wih regard o he differen surplus appropriaion schemes. Figure 9 presens he shorfall probabiliy for varying values of he surplus disribuion raio α and arge buffer raio γ. In he lef graph, γ equals 1%, and in he righ graph, α is se o 7%, while he sock raio is kep consan a a = 1% for boh. As can be seen in Equaion (6), hese parameers conrol he surplus disribuion o he policyholders. In general, he insurer s shorfall risk varies considerably wih a ceeris paribus increase in α and a ceeris

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