Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint

Transcription

1 Dynamic Hybrid Producs in Life Insurance: Assessing he Policyholders Viewpoin Alexander Bohner, Paricia Born, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen Friedrich-Alexander-Universiy (FAU) of Erlangen-Nürnberg Version: Sepember 2014

2 DYNAMIC HYBRID PRODUCTS IN LIFE INSURANCE: ASSESSING THE POLICYHOLDERS VIEWPOINT Alexander Bohner, Paricia Born, Nadine Gazer This version: Sepember 1s, 2014 ABSTRACT Dynamic hybrid life insurance producs are inended o mee new consumer needs regarding sabiliy in erms of guaranees as well as sufficien upside poenial. In conras o radiional paricipaing or classical uni-linked life insurance producs, he guaranee offered o he policyholders is achieved by a periodical rebalancing process beween hree funds: he policy reserves (i.e. he premium reserve sock, hus causing ineracion effecs wih radiional paricipaing life insurance conracs), a guaranee fund, and an equiy fund. In his paper, we consider an insurer offering boh, dynamic hybrid and radiional paricipaing life insurance conracs and focus on he policyholders perspecive. The resuls show ha higher guaranees do no necessarily imply a higher willingness-o-pay, bu ha in case of dynamic hybrid conracs, a minimum guaranee level should be offered in order o ensure ha he willingness-o-pay exceeds he minimum premium he insurer has o charge when selling he conrac. In addiion, srong ineracion effecs can be found beween he wo producs, which paricularly impac he willingness-o-pay of he dynamic hybrids. Keywords: Life insurance, guaraneed ineres raes, dynamic hybrid, consan proporion porfolio insurance, cusomer value, mean-variance preferences, risk-reurn profiles 1. INTRODUCTION Innovaions in he life and pension indusry have become increasingly imporan, especially agains he background of demographic changes and as an alernaive or supplemen o public sae-run pension schemes. However, he currenly low ineres raes and volaile capial markes make providing long-erm guaranees increasingly difficul for insurers. In addiion, he indusry faces increasing regulaion and cos pressure, and consumer preferences for sabiliy, upside poenial and flexibiliy mus be aken ino accoun when developing new conracs. In Alexander Bohner and Nadine Gazer are a he Friedrich-Alexander-Universiy (FAU) of Erlangen- Nürnberg, Deparmen of Insurance Economics and Risk Managemen, Lange Gasse 20, Nuremberg, Germany, alexander.bohner@fau.de, nadine.gazer@fau.de; Paricia Born is a The Florida Sae Universiy, Deparmen of Risk Managemen/Insurance, Real Esae and Legal Sudies, Tallahassee, Florida, pborn@cob.fsu.edu.

3 2 his conex, dynamic hybrid life insurance producs have recenly been inroduced in he German marke. 1 Insead of explicily (exernally or inernally) hedging he guaranees embedded in he conrac or by means of capial, he guaranee is ensured implicily by means of a dynamic reallocaion of he dynamic hybrid accoun value beween hree funds: he policy reserves (i.e. he premium reserve sock), a guaranee fund, and a (risky) equiy fund, following he idea of consan proporion porfolio insurance (see Bohner and Gazer, 2014). In his paper, our aim is o sudy hese producs in deph from he policyholders perspecive by aking ino accoun he preferences and willingness-o-pay of consumers. We hereby also focus on he ineracion effecs ha arise due o he fac ha dynamic hybrid funds are periodically shifed o and from he convenional policy reserves, e.g., in imes of adverse capial markes. Dynamic hybrid producs have firs been modeled in he scienific lieraure by Kochanski and Karnarski (2011), who derive solvency capial requiremens for saic and dynamic hybrids using a rules-based shifing mechanism, bu wihou focusing on possible ineracion effecs wih oher producs. The laer has been sudied in deph by Bohner and Gazer (2014), who presen a comprehensive model framework o assess and demonsrae he (srong) ineracion effecs beween dynamic hybrid producs and radiional paricipaing life insurance policies a he company level, hereby focusing on he insurer s risk siuaion and he policyholders ne presen value. A comprehensive overview of he German marke of dynamic hybrid producs is furher provided in Bohner (2013), who shows he variey of opions embedded in he conracs and he implicaions of differen shifing mechanisms by sudying risk-reurn profiles provided by he indusry. Thus, he scienific lieraure on dynamic hybrid producs is sill raher scarce. In conras, he consumer perspecive on guaranees embedded in life insurance conracs in general has received increasing aenion in he lieraure. Gazer, Holzmüller, and Schmeiser (2012), for insance, use a heoreical model and a simulaion sudy o compare he perspecive of he insurer and he policyholder. They derive he willingness-o-pay for paricipaing life insurance conracs using mean-variance preferences for differen assumpions regarding he diversificaion opporuniies of he policyholder and idenify conrac specificaions ha while keeping he conrac value fixed for he insurer maximize cusomer value. The auhors show ha increasing he guaraneed ineres raes does no necessarily maximize cusomer value. Broeders, Chen, and Koos (2011) use a similar approach based on a power uiliy funcion for he policyholder and sudy wo ypes of annuiy providers (defined benefi pension funds and life insurers) ha differ according o he exen of risk sharing beween beneficiar- 1 Currenly abou 20 life insurance companies in Germany (ou of roughly 100) provide dynamic hybrid producs (see Bohner, 2013). Life insurers in Japan are considering inroducing dynamic hybrid producs as well.

4 3 ies and shareholders, demonsraing he need for regulaion o provide a level playing field for providers. Schmeiser and Wagner (2013) consider he consumers perspecive when deriving minimum solvency capial requiremens, and hereby illusrae how minimum ineres raes should be defined by he regulaor in order o maximize he policyholders uiliy level. While hese papers use heoreical models o sudy he consumers perspecive, Gazer, Huber, and Schmeiser (2011), for insance, also conduc an empirical survey o sudy he willingness-o-pay for ineres rae guaranees in uni-linked life insurance conracs. Their resuls indicae ha cusomers may no be willing o pay he risk-adequae price for he valuable guaranees as, on average, he willingness-o-pay was significanly lower han he minimum prices derived based on opion-pricing heory. A he same ime, however, a subsanial porion of paricipans were willing o pay a considerably higher price, hus indicaing a higher degree of risk-aversion. Furher lieraure also reveals he imporance of such hings as cusomer preferences (e.g., see Doskeland and Nordahl, 2006), demographic characerisics such as income, gender, and educaion (e.g., see Feldman and Schulz, 2004), and insurer characerisics and operaions (e.g., see Marshall, Hardy, and Saunders, 2010) in he deerminaion of willingness-o-pay. In his paper, we explicily focus on he policyholders perspecive, hereby sudying he willingness-o-pay based on risk preferences as well as risk-reurn profiles. We hereby exend he model in Bohner and Gazer (2014) for a life insurer offering dynamic hybrids and paricipaing life insurance conracs by focusing on differen dynamic hybrid guaranee level, varying guaraneed ineres raes (o be credied o he policy reserves). We furher exend he previous seing by inegraing differen shifing mechanisms for he dynamic hybrid funds. This analysis is inended o provide insigh ino he impac of differen ypes of long-erm guaranees as well as feaures and characerisics of hese life insurance financial producs from he policyholders viewpoin. The remainder of he paper is srucured as follows. Secion 2 presens he model framework of he insurance company offering paricipaing life insurance policies and he dynamic hybrid producs including fair valuaion and risk measuremen as well as he derivaion of he willingness-o-pay from he policyholders perspecive. Secion 3 conains a numerical analysis and Secion 4 provides concluding remarks.

5 4 2. MODEL FRAMEWORK Modeling he insurance company - overview In he following, we consider a life insurer offering wo ypes of producs: radiional paricipaing life insurance policies (PLI) and dynamic hybrid producs (). The general model framework for he insurance company is based on he model presened by Bohner and Gazer (2014), which is hen exended by aking he consumers perspecive, which is he focus of he presen analysis. Table 1 shows he simplified balance shee of he insurer. Table 1: Balance shee of he life insurer a ime (see Bohner and Gazer, 2014) Asses A A long erm shor erm GF EF A A A Liabiliies PR PR GF EF B A PLI L L PR AV Regarding he liabiliy side, policyholders of boh conrac ypes are assumed o pay a single up-fron premium P PLI and P, implying iniial policy reserves (PR) of he paricipaing life insurance conracs of PR PLI 0 = P PLI and an iniial accoun value (AV) of he dynamic hybrid producs of AV 0 = P. As exhibied in Table 1, he dynamic hybrid producs accoun value AV is hereby composed of up o hree pars, including a par ha is invesed in he insurer s collecive policy reserves PR, an equiy fund (EF), and a guaranee fund (GF) as described in deail laer. The porion of he oal policy reserves coming from he dynamic hybrid producs is denoed as PR, which, ogeher wih he par coming from he radiional paricipaing life insurance conracs PR PLI, sums up o he oal policy reserves PR = PR + PR. PLI

6 5 The conrac erm T is assumed o coincide wih he lifeime of he considered insurance company. A incepion of he conrac, he buffer B 0, residually given by he difference beween asses and liabiliies, is filled by he iniial conribuion of he company s equiyholders. The conracs are hen calibraed o be fair from he equiyholders perspecive o ensure riskadequae compensaion for heir invesmen. 2 A summary of he various guaranees involved in he following model descripion is given in Table 2. Table 2: Overview of guaranees and guaranee noaions involved in he model Guaranee noaion Ineres rae guaranee Mauriy guaranee Money-back guaranee Guaranee fund GF Minimum dynamic hybrid accoun value G + Descripion Guaraneed ineres rae r G, minimum ineres rae applied o he policy reserves on an annual basis; applies o policies invesed in he policy reserves (especially paricipaing life insurance policies; also relevan for dynamic hybrids for he par invesed in he policy reserves) Mauriy guaranee of dynamic hybrid conracs, only promised a mauriy of he conrac erm; G = x P (x = 1 corresponds o a money-back guaranee) T Guaranees he payback of he single up-fron premium paid ino he conrac a mauriy Equiy fund, which ensures a maximum loss of λ percen wihin one period and hus guaranees ha he fund drops a mos o ( ) 1 λ GF + Accoun value needed a ime o ensure ha he guaranee promised o he dynamic hybrid policyholders G T can be me a mauriy; may vary depending on he concree produc design (especially he guaranee promised o he dynamic hybrids, x) The paricipaing life insurance conrac Paricipaing life insurance conracs feaure an annual guaraneed ineres rae r G and an annual surplus paricipaion rae α. The annual policy ineres rae P r is declared in advance a 2 In he presen seing, ineracion effecs beween he wo conracs are one main reason why he siuaion is no auomaically fair for he policyholders as well. In fac, he value of he policies can considerably depend on he porfolio composiion of he insurer, i.e. he porion of dynamic hybrid conracs in he porfolio (see Bohner and Gazer, 2014).

7 6 he beginning of each year (as is required in he German marke, for insance) and given by he smoohing scheme (see Grosen and Jørgensen, 2000) r B = max r, α γ, PR P G where γ is he arge buffer raio, i.e., he raio of he buffer accoun o policyholder liabiliies PLI ( PR = PR + PR ). The buffer accoun hereby represens he free surplus, which can be used o absorb losses wih respec o he guaraneed posiions on he liabiliy side of he balance shee, i.e. he policy reserves. To ensure ha surplus is smoohed over ime and o reduce volailiy of he policy ineres rae, he proporion beween he buffer accoun and he policy reserves mus amoun o a leas he arge buffer raio γ before surplus is disribued o he policyholders. A second conrol parameer, he surplus disribuion raio α is used o conrol he fracion of he excess amoun of he arge buffer raio ha is acually credied o he policyholders. 3 The policy ineres rae is hen credied o he policy reserves a ime, i.e., PR PR r PLI PLI P ( 1) = ( 1+ + ) The paricipaing life insurance conracs remain invesed in he policy reserves (premium reserve sock) during he whole conrac erm and a mauriy T, he paricipaing life insurance conracs receive heir policy reserves PLI PR T given ha he insurer remains solven. The dynamic hybrid life insurance conrac The policyholders wih he dynamic hybrid produc are promised a fracion x of heir single premium a mauriy T, i.e., he mauriy guaranee amouns o G T = x P (x = 1 corresponds o a money-back guaranee). 4 During he conrac erm, as shown in Table 1, he dynamic hybrid producs accoun value (AV) is dynamically reallocaed beween he policy reserves (PR), an equiy fund (EF), and a guaranee fund (GF), whereby he laer is equivalen o an equiy fund wih a hedge ha ensures a maximum loss of λ percen wihin one period. Similar o a consan proporion porfolio insurance (CPPI) sraegy, his dynamic reallocaion (described in deail below) is inended o ensure he guaranee promised o he policy- 3 In Germany, for insance, regulaors prescribe a maximum period of ime (e.g. hree years), during which he surplus can be kep in he buffer accoun before i has o be credied o he insureds (see, e.g., Schradin, Pohl, and Koch, 2006, p. 14). 4 In general, policyholders can choose a guaranee level up o 100% (see Bohner, 2013).

8 7 holders a mauriy wihou addiional guaranee coss or furher comprehensive hedging aciviies. According o regulaory requiremens in Germany regarding he policy reserves, he par of he dynamic hybrid funds invesed in he policy reserves a ime mus hereby be compounded wih he same policy ineres rae credied o he paricipaing life insurance conracs (even hough he funds may only be invesed shor-erm in he policy reserves), 5 i.e., PR = PR + r. + P ( 1) ( 1 ) A mauriy T, he dynamic hybrid producs receive heir accoun value T T T T L L ( AV = PR + GF + EF ), consising of he oal of he hree funds, given ha he insurer does no defaul during he conrac erm. To secure he mauriy guaranee of = x P promised o he dynamic hybrid policyholders, he accoun value AV is dynamically disribued beween he policy reserves PR G L T, he guaranee fundgf and equiy fund Karnarski (2011) as follows 6 PR GF L ( 1 λ ) G + AV G +, if > 1 G = ( 1+ r ) 1+ λ ( 1 λ ) AV 0, oherwise G + AV PR, if > 1 ( 1 λ) AV = G +, oherwise 1 λ EF = AV PR GF L L. L EF as shown by Kochanski and 5 Subsiuion effecs may arise due o he poenially differen invesmen horizons. In paricular, i can be shown ha for higher porions of dynamic hybrid producs in he porfolio, he guaraneed ineres rae is more difficul o achieve, as i is derived based on an expeced long-erm invesmen (see also Bohner and Gazer, 2014). This also implies ha he porfolio composiion has an impac on he policyholders willingness-o-pay, for insance, as wih an increasing porion of dynamic hybrid producs and hus an increasing share in shor-erm invesmens, he volailiy of he payoff and he defaul risk generally increase (depending on he guaranees promised o he dynamic hybrid producs). 6 This mechanism invess he maximum proporion of he accoun value in he equiy fund (and guaranee fund) along wih ensuring ha he guaranees can sill be me. While his is a common pracice in he marke, an alernaive sraegy would be o balance he radeoff beween he number of shifs (i.e., ransacion coss), and upside poenial (high proporion in equiy funds), which hus imply varied risk profiles.

9 8 The concree shifing mechanism hereby depends on he assumpions regarding G + a he end of each period, for insance, which denoes he accoun value needed a ime o ensure ha he guaranee can be me a mauriy and which may vary depending on he concree produc design. In paricular, in he following, we compare wo ypes of shifing mechanisms, leading o differen risk-reurn profiles from he policyholders perspecive. Firs, in case of he less risky shifing mechanism, we assume ha he accoun value mus be a leas { 0,,, } G + = GT = x P T, (1) i.e., he guaranee mus be ensured a all imes, implying a higher porion of riskless asses in he porfolio of he insurance company (i.e., policy reserves and guaranee fund) and hus generally a lower risk and reurn. Second, we consider a more risky shifing mechanism by discouning he mauriy guaranee o he curren dae (as is usually done in case of consan proporion porfolio insurance (CPPI) sraegy, see Black and Jones (1987), Leland (1980)), implying a higher porion risky asses, i.e., ( 1 ) ( T ) ( 1 ) ( T ) G G G + = GT + r = x P + r. (2) When G + can be fulfilled by he guaranee fund only, he dynamic hybrid producs funds are disribued beween he guaranee fund and equiy fund. Terminal bonus paymens and oal mauriy payoff In addiion o he previously described payoffs, policyholders of boh conracs receive a erminal bonus from he remaining buffer accoun afer he equiyholders have received adequae compensaion for heir iniial conribuion in case he buffer is posiive. The buffer is given by B = A PR PR GF EF, PLI L L and he buffer payback o he equiyholders is deermined by ( ( T 0 ( )) ) BP = max min B, B 1 + b,0, T where b denoes he fair (risk-adequae) buffer ineres rae paid on heir iniial conribuion and represens he dividend for he enire lengh of he invesmen, which is hen calibraed o be fair.

10 9 The policyholders receive he erminal bonus TB max ( 0, B BP ) T T T =, which in case of posiive policy reserves (zero oherwise) is assumed o be disribued beween he wo ypes of conracs as TB TB PR PR PR T T PLI PLI PLI T = T + k k k k = 1 k = 1 ( ) ( ) ( ), TB TB PR PR PR T T PLI T = T + k k k k = 1 k = 1 ( ) ( ) ( ). This disribuion scheme hus uses a discree ime (, laer assumed o be 1/12) weighed average over ime, which akes ino accoun he invesmen in he policy reserves over he whole conrac erm. The inuiion behind his erminal bonus disribuion scheme is ha he erminal accumulaed surplus is generaed by he invesmens in he policy reserves. These funds are collecively invesed in he capial marke by he insurer, while he dynamic hybrids guaranee fund and equiy fund are invesed individually and are direcly credied o he dynamic hybrid policyholders. Hence, he oal conrac payoffs VT are given by ( ) 1{ } 1{ } V = PR + TB T > T + RF T = (3) PLI PLI PLI PLI T T T s s and ( ) 1{ } 1{ } V = AV + TB T > T + RF T =, (4) T T T s s long erm shor erm where TS denoes he ime of defaul wih s { } T = inf : A + A < PR, = 1,..., T, and RF refers o he remaining funds in case he insurer defauls during he conrac erm, i.e. if B ( + ) < 0, which are disribued analogously o he erminal bonus according o he policy reserves over he conrac erm (as is shown in Bohner and Gazer, 2014). The asse side As described before, he policy reserves a ime (PR) are composed of funds semming from he paricipaing life insurance conracs ( PR ) and he dynamic hybrid producs ( PR ). PLI This (synheic) separaion allows us o accoun for he differen asse invesmen mauriies, as funds from he dynamic hybrid producs may be shifed o he policy reserves for a shor period only, and are hen shifed back o he guaranee fund or equiy fund. The funds of paricipaing life insurance policies, in conras, are generally invesed long-erm. This is reflec-

11 10 ed on he asse side in Table 1, where he company s asses are spli ino long-erm invesmens bonds), funcion. A (e.g. long-erm bonds) and shor-erm invesmens (e.g. bills and shor-erm long erm shor erm A, whereby he buffer accoun is also invesed shor-erm due o is smoohing All hree asse invesmen ypes (long-erm asses, shor-erm asses, and equiy fund) are assumed o evolve according o a geomeric Brownian moion di = µ I d + σ I dw, i = 1, 2,3 i i i P i i, i P wih consan drif µi and volailiy σi, P-Brownian moions dw, i defined on he probabiliy P P space ( ΩF,, P) wih correlaions dw, i dw, j = ρi, j, i.e. (see Björk, 2009) i i 1 2 P I = I0 exp µ i σ i + σ i dw, i 2. A he beginning of period, he company s asse invesmens are hus given by A long erm, one obains ( ) = PR, A = PR + B, GF PLI I A =, + 1 long erm long erm + A 1 I I A =, + 2 shor erm shor erm + A 2 I ( ) I EF =, + 3 A A + EF 3 I ( ) GF( ) GF y + shor erm 3 A A + = max 1 λ ; 3 I I, A = GF, EF L A = EF, and a he end of period L where he guaranee fund is given by a fracion y of he equiy fund s reurn, since he periodic downside proecion has o be financed. We hereby assume a pu opion on he equiy fund A A GF ( 0 λ 1) and mauriy, which is purchased a he beginning of each period as proposed by Bohner and Gazer (2014). In his case, Bohner GF wih srike price ( 1 λ) and Gazer (2014) show ha he price of he pu opion depends only on he given se of pa-

12 11 rameers λ, rf, σ3 and and is hus given by a consan fracion y of he guaranee fund, i.e. ( 1 ) P = y GF. 7 The oal asses are hus given by A A A A EF GF long erm shor erm A A = ( + ) ( + ) ( + ) ( + ) ( + ) The equiyholders perspecive To ensure a fair siuaion for he equiyholders, he buffer ineres rae b is calibraed such ha he value of he payou o equiyholders is equal o heir iniial conribuion, i.e. T r ( f T ) Q B0 E BP e =, (5) where E Q denoes he expeced value under he risk-neural pricing measure Q and rf is he consan risk-free ineres rae. Under he risk-neural measure Q, he drif of he invesmen processes changes o he risk-free rae (see Björk, 2009). The policyholder s perspecive willingness-o-pay For fairly calibraed conrac parameers from he equiyholders viewpoin, we nex focus on he policyholders perspecive and deermine he maximum willingness-o-pay for he given conrac design. As he relevan preference funcion, we use mean-variance preferences (Berkei, 1999; Gazer, Holzmüller, and Schmeiser, 2012; Mayers and Smih, 1983), which implies ha he order of preferences is given by he difference beween expeced payoff (wealh) and he variance of he payoff in case insurance is purchased (he paricipaing life insurance policy (PLI) or he dynamic hybrid produc (), respecively) or in he case wihou insurance under he real-world measure P, muliplied by he policyholder s individual risk aversion coefficien a imes one half, i.e., j j a 2 j Φ = E ( VT ) σ ( VT ), j = PLI,, no insurance. 2 To derive he willingness-o-pay, he preference funcion for he case wih and wihou insurance mus be compared, whereby he maximum willingness-o-pay WTP Φ 0, j is he amoun a no insurance PLI, which he cusomer is indifferen beween he wo cases, i.e. where Φ = Φ. The willingness-o-pay mus hen exceed he single premiums P PLI and P ha he insurer mus charge for he conracs, respecively, as oherwise he conrac will no be aken ou. 7 Alernaively, a consan-proporion porfolio insurance sraegy can be used.

13 12 In he following, o keep calculaions simply, we focus on he case of deerminisic wealh, where he policyholder can choose o inves par of his or her wealh in he risk-free asse rf and he respecive insurance policy, and canno furher diversify. Le W0 denoe he iniial wealh of he cusomer. In case no insurance is purchased, one obains no insurance rf T Φ = W0 e and in case a paricipaing life insurance or dynamic hybrid conrac is purchased (willing-, j ness-o-pay for he conrac denoed by WTP Φ ), Φ, rf T a 2 Φ, (( 0 0 ) T ) σ ( 0 0 ) 0 rf T ( T ) j j j j j Φ = E W WTP e + V W WTP e + V, j = PLI,. 2 Equaing he wo condiions implies ha he maximum willingness-o-pay is given by Φ, j rf T j a 2 j WTP0 = e E ( VT ) σ ( VT ), j = PLI,, (6) 2 which does no depend on he policyholder s iniial wealh, W0 (see Gazer, Holzmüller, and Schmeiser, 2012, p. 652). As an alernaive o mean-variance preferences, cerainy equivalens could be derived o obain an impression of he uiliy level in case he premium volume is he same (see Schmeiser and Wagner, 2013; Broeders, Chen, and Koos, 2011). However, since we consider he impac of differen porfolios and hus vary he respecive premium volumes, resuls would no longer be comparable.

14 13 3. NUMERICAL ANALYSIS In he following ables, we provide several numerical examples o illusrae he impac of differen porfolio composiions, conrac designs, and shifing mechanisms on he policyholders willingness-o-pay. The inpu parameers are displayed in Table 3 and are based on hose used by Bohner and Gazer (2014), which were subjec o sensiiviy analyses. Mone Carlo simulaion wih 50,000 lain hypercube samples was used o derive he resuls. Table 3: Inpu parameers for he numerical analyses Parameer Noaion Value Single premiums of paricipaing life insurance conracs P PLI 100 Single premiums of dynamic hybrid producs P 100 Conrac duraion T 10 Guaranee of dynamic hybrid producs x 1 Iniial buffer 8 B 0 6 Guaraneed ineres rae (p.a.) r G Surplus disribuion raio α 0.30 Targe buffer raio γ 0.10 Drif of long-erm invesmens µ Volailiy of long-erm invesmens 9 σ Drif of shor-erm invesmens µ Volailiy of shor-erm invesmens σ Drif of equiy fund µ Volailiy of equiy fund σ Linear correlaion of long-erm and shor-erm invesmens ρ 1,2 0.2 Linear correlaion of long-erm invesmens and equiy fund ρ 1,3 0.2 Linear correlaion of shor-erm invesmens and equiy fund ρ 2,3 0.2 Maximal loss of he guaranee fund per period λ 0.20 Risk-free ineres rae r f 0.03 Lengh of a period 10 1/12 8 Noe ha in his seing, he iniial buffer raio is 6/200 = 0.03, which is below he arge buffer raio of g = 0.10, meaning ha he insurer firs needs o build up he buffer accoun by means of surplus before he surplus can be disribued o he policyholders accouns. 9 Inpu parameers are chosen for illusraion purposes and were subjec o robusness ess o make sure ha he general findings are sable. For he se of invesmen parameers, he Sharpe raios of he hree ypes of invesmens are no equal, bu wih respec o hese parameers, we conduced a sensiiviy analysis by varying he volailiies of he differen invesmens o ensure he sabiliy of he resuls. 10 Funds are shifed once per monh, which resembles he ypical approach in he marke; only a few insurers also shif daily (see Bohner, 2013).

15 14 The impac of differen ypes and levels of guaranees on policyholders willingness-o-pay Figure 1 displays he impac of differen guaranee feaures on he policyholders willingnesso-pay for dynamic hybrids as well as for radiional paricipaing life insurance conracs for differen degrees of risk aversion. In paricular, we sudy he impac of differen levels of he guaraneed ineres rae r G as well as he design wih respec o he shifing mechanisms (i.e., less risky wih a higher safey level and a more risky sraegy wih a higher upside poenial, see Equaions (1) and (2)). The dynamic hybrids are assumed o feaure a full money-back guaranee. 11 In he lef column in Figure 1, he shifing mechanism of he dynamic hybrid produc is se according o Equaion (1), i.e., G (reflecing he minimum accoun value a ime ) is kep consan during he conrac erm, always requiring a leas he mauriy guaranee G = G and hus implying less risk wih a higher safey level. The righ column T shows he case wih a shifing sraegy ha is more risky and wih a higher upside poenial based on Equaion (2), i.e. G is given by he discouned value of he mauriy guaranee in each period. Wih respec o he varying degrees of risk aversion, hree cases are considered. The case of a less risk averse policyholder is shown in he firs row of Figure 1 (wih a = 0.001), while he hird (second) row demonsraes he case for a policyholder ha is more (medium) risk averse wih a = 0.01 (a = 0.005). In all cases, he buffer ineres rae (equiyholders dividend) is calibraed o ensure ha he siuaion is fair from he equiyholders viewpoin, i.e. Equaion (5) is saisfied (see Figure A.1 in he Appendix) Figure A.2 in he Appendix shows he resuls when assuming a 50% money-back guaranee. 12 The fair buffer ineres rae represens he reurn on he equiyholders invesmen over he enire conrac duraion (i.e., 10 years in he example here), given ha he insurer does no defaul during he conrac erm. Since he defaul risk is generally increasing for higher guaraneed ineres raes, for insance, he fair buffer ineres rae increases as well (see Figure A.1).

16 15 Figure 1: Willingness-o-pay for he paricipaing life insurance (PLI) and dynamic hybrid () conracs for differen degrees of risk aversion a = 0.001, 0.005, 0.01 (less / medium / more risk averse) as well as differing shifing mechanisms for a money-back guaranee of he dynamic hybrids for varying guaraneed ineres raes for fair conracs (see Figure A.1) a=0.001 (less risk averse), less risky shifing a =0.001 (less risk averse), more risky shifing willingness-o-pay willingness-o-pay guaraneed ineres rae r G guaraneed ineres rae r G a =0.005 (medium risk averse), less risky shifing a=0.005 (medium risk averse), more risky shifing willingness-o-pay willingness-o-pay guaraneed ineres rae r G guaraneed ineres rae r G a=0.01 (more risk averse), less risky shifing a =0.01 (more risk averse), more risky shifing willingness-o-pay willingness-o-pay guaraneed ineres rae r G guaraneed ineres rae r G W TP P LI WT P D H P P P LI = P D HP

17 16 One can observe ha in he cases of a medium and a more risk averse policyholder considered in Figure 1, he policyholders willingness-o-pay for boh life insurance producs (someimes subsanially) decreases when increasing he guaraneed ineres rae, which mainly sems from an increase in shorfall risk (see Bohner and Gazer, 2014, for his resul) and a higher volailiy of he payoff (see Equaion (6)). However, his increase in he payoff volailiy and he ineracion effecs of he paricipaing life insurance conracs wih he dynamic hybrids cause an increase of he willingness-o-pay for higher guaraneed ineres raes among less risk averse policyholders. When comparing he righ column wih he lef column in Figure 1, i can be seen ha he design of he dynamic hybrids shifing mechanism has a considerable impac on dynamic hybrid policyholders willingness-o-pay. In paricular, a more risky shifing sysem (righ column in Figure 1) leads o a higher volailiy of he dynamic hybrid producs final payoff, which hus resuls in a lower willingness-o-pay for he dynamic hybrids (WTP ) for more risk averse policyholders as compared o he case of he less risky shifing mechanism in he lef column of Figure 1. This is especially pronounced in he las row and also (o a considerable lesser exen) in he second row of Figure 1. while he WTP of dynamic hybrids is affeced in he opposie way by he shifing mechanisms in he firs row, where he willingness-o-pay for dynamic hybrids is even increased for less risk averse policyholders (as compared o he lef graph in he firs row wih a less risky shifing mechanism). The willingness-o-pay for paricipaing life insurance conracs (WTP PLI ) wihin he same porfolio of a life insurance company is also affeced by he differen shifing mechanisms due o ineracion effecs beween he wo ypes of policies. However, in conras o he dynamic hybrid producs, his effec is considerably reduced and he paricipaing life insurance policyholders WTP remains fairly sable in all cases. When using a more risky shifing mechanism for he dynamic hybrids, he WTP even slighly increases (a leas for higher guaraneed ineres raes, see firs row in Figure 1), as he more risky shifing reduces he guaranees for he producs, hus benefiing he paricipaing life insurance policyholders. In addiion o a full money-back guaranee in Figure 1, we furher sudy he case wih a reduced guaranee level of x = 50% of he single up-fron premium (see Figure A.2 in he Appendix). The resuls show ha lowering he dynamic hybrid producs guaranee level from a full (100%) o a 50% money-back guaranee eiher implies a considerable reducion in he dynamic hybrid policyholders willingness-o-pay for more risk averse policyholders, or a considerable increase in he WTP in case of policyholders wih lower risk aversion. These effecs are especially pronounced in case of he less risky shifing mechanism (compare lef columns in Figures 1 and A.2). In he case of he more risky shifing mechanism (righ column, firs and second row), resuls are ambiguous and srongly depend on he guaraneed ineres rae credied on he policy reserves. In paricular, reducing he guaranee level o a

18 17 50% money-back guaranee implies a lower WTP for lower guaraneed ineres raes, bu a higher WTP for guaraneed ineres raes above 1.7% (approximaely). To ensure ha an insurance conrac is acually purchased by cusomers, he policyholders willingness-o-pay for he conrac has o exceed he required premium assumed when calculaing he conracs payoffs, i.e., in our seing a single premium of P PLI = P = 100 for boh producs. While he WTP PLI exceeds he required premium in all considered cases in Figure 1, he siuaion for dynamic hybrids depends heavily on he specific conrac design and especially he guaranee feaures. In paricular, resuls depend on he guaraneed ineres rae, he shifing mechanism and he policyholders risk aversion. For a more risk averse policyholder as given in he las row in Figure 1, he policyholders willingness-o-pay is below he required premium of 100 in case of he risky shifing mechanism (righ graph). This sems from he fac ha only a minor amoun of he dynamic hybrids accoun value is invesed in he policy reserve sock leading o a more volaile final payoff, for which he willingness-opay is low for a more risk averse policyholder. In case of he more conservaive shifing mechanism (lef graph in he las row), he dynamic hybrid policyholders willingness-o-pay is above 100 for guaraneed ineres raes below r G = 1.5% and lower han 100 if he guaranee ineres rae exceeds his value in he presen example. The higher he guaraneed ineres rae, he higher he shorfall risk and he higher he payoff volailiy, which is valued as negaive by a risk averse policyholder. This siuaion changes considerably for a less risk averse policyholder (see firs row in Figure 1). In his case, he willingness-o-pay for dynamic hybrids is sricly above he required premium of 100 and exceeds he willingness-o-pay for he paricipaing life insurance conracs, since he upside poenial and higher volailiy is valued posiively. In conras, he policyholders willingness-o-pay for paricipaing life insurance conracs can be eiher higher or lower han he willingness-o-pay for dynamic hybrids in he case of a medium risk averse policyholder and he less risky shifing mechanism (see lef graph in he second in Figure 1). The impac of porfolio composiion on he policyholders willingness-o-pay We now sudy he impac of he porfolio composiion on he policyholders willingness-opay for dynamic hybrid producs and paricipaing life insurance conracs wihin a porfolio of a life insurance company as exhibied in Figure 2. We hereby fix he oal premium volume o 200. The premium of he dynamic hybrids is hen given on he x-axis, while he single premium for he paricipaing life insurance conracs is residually given by P PLI = 200 P. Figure 2 shows he willingness-o-pay for paricipaing life insurance conracs and dynamic hybrid producs for differen porfolio composiions for a more risk averse policyholder (lef

19 18 graph in Figure 2) and a less risk averse one (righ graph). In line wih he resuls in Figure 1, he policyholders willingness-o-pay for paricipaing life insurance conracs is higher han he corresponding required premium (black doed line) in all considered cases. The willingness-o-pay for dynamic hybrid conracs, in conras, depends grealy on he porfolio composiion and he risk aversion parameer. For a more risk averse policyholder (lef graph in Figure 2), he willingness-o-pay for dynamic hybrids only exceeds he required single premium for a porfolio wih less han abou 50% dynamic hybrid producs in he insurer s porfolio (i.e. P PLI = P = 100). Here, he conracs sabiliy and relaively low volailiy of he payoffs o a large par sems from he paricipaing life insurance conracs in he porfolio, which allows he corresponding asse base o be invesed long-erm wih sable reurns for he policy reserve sock. The upside poenial of he dynamic hybrids fund invesmens are added o his. In urn, for a porfolio composiion wih more han abou 50% dynamic hybrids in he porfolio, he corresponding invesmens of he policy reserve sock canno be invesed longerm, since funds may be shifed o he guaranee fund or equiy fund in a subsequen period. Thus, he policy reserve sock generaes less sable reurns wih fewer surplus (due o higher defaul risk), which resuls in a reduced willingness-o-pay for dynamic hybrids in his seing. Figure 2: Willingness-o-pay for he paricipaing life insurance and dynamic hybrid conracs and he conracs single premiums when varying he porfolio composiion (P PLI = 200 P ) for fair conracs (see Figure A.3) (wih 100% money-back guaranee of he dynamic hybrids as well as he less risky shifing mechanism) a =0.01 (more risk averse) a=0.001 (less risk averse) willingness-o-pay willingness-o-pay single premiums dynamic hybrid produc P D H P single premiums dynamic hybrid produc P D H P WT P P L I P P L I W T P D HP P D H P The righ graph in Figure 2 furher reveals ha for a less risk averse policyholder, he willingness-o-pay for dynamic hybrids is relaively higher, he higher he porion of dynamic hybrid producs is wihin he porfolio. Here, he policyholders prefer he upside poenial of a more volaile payoff, which resuls from a higher porion of dynamic hybrids in he porfolio (see

20 19 Figure 3), insead of he more sable payoff ha resuls from a higher porion of paricipaing life insurance policies in he porfolio. To obain furher insigh, Figure 3 addiionally illusraes he corresponding quariles of he payoff disribuions of he paricipaing life insurance conracs (lef graph) and he dynamic hybrid producs (righ graph) for he various porfolio composiions exhibied in Figure 2, where a money-back guaranee for he dynamic hybrids as well as he less risky shifing mechanism is applied. In paricular, he upper and lower quariles are shown along wih he median indicaed as a black do. When considering he PLI payoff quariles, i can be seen ha he median and he PLI payoffs inerquarile range is increasing wih an increasing porion of paricipaing life insurance conracs in he porfolio, bu a a relaively low level as compared o he payoff quariles. In case of he dynamic hybrid producs, he resuls show a srong increase of he inerquarile range, i.e., an increase in he payoff volailiy for a higher porion of dynamic hybrids in he porfolio. In addiion o his, he payoffs exhibi a posiive skew, which illusraes he much higher upside poenial of dynamic hybrids as compared o paricipaing life insurance conracs, which is preferred by less risk averse policyholders (see Figure 2). Figure 3: Lower quarile, median, and upper quarile (i.e. 25h, 50h, and 75h percenile) of he payoff disribuions for PLIs and s when varying he porfolio composiion (P PLI = 200 P ) for a money-back guaranee of he dynamic hybrids as well as he less risky shifing mechanism (corresponding o Figure 2) PLI payoff quariles D H P payoff quariles single premiums dynamic hybrid produc P D H P single premiums dynamic hybrid produc P D H P

21 20 The impac of differen ypes and levels of guaranees on he conracs payoff disribuion To furher sudy he dynamic hybrid policyholders perspecive, we nex analyze he conracs payoff disribuions, i.e., heir risk-reurn profiles. Toward his end, we derive he real- PLI world disribuions of he PLI and payoffs, i.e. V T and V T (see Equaions (3) and (4)), for varying levels of guaraneed ineres raes and guaranees. Figure 4: Payoff disribuions for PLIs and s (risk-reurn profiles) for differen guaraneed ineres raes and differen guaranee levels (less risky shifing mechanism; corresponding o he lef columns in Figures 1 and A.2) PL I payoff disribuions (money-back guaranee DH P) D HP payoff disribuions (money-back guaranee D HP) 100% < % 14% 13% 12% 10% 9% 7% 100% < % 60% 43% 43% 42% 40% 38% 36% 33% 80% 60% 28% 11% 28% 11% 29% 11% 29% 10% 30% 10% 30% 9% 30% 9% 40% 40% 29% 28% 28% 28% 29% 30% 31% 20% 40% 42% 44% 46% 49% 53% 59% 20% 32% 32% 32% 32% 32% 31% 30% 0% guaraneed ineres rae r G 0% guaraneed ineres rae r G PL I payoff disribuions (lower guaranee DH P, x=0.5) D HP payoff disribuions (lower guaranee D HP, x=0.5) 100% < % 12% 11% 11% 10% 8% 6% 100% < % 60% 44% 44% 43% 42% 40% 38% 34% 80% 60% 50% 50% 49% 49% 48% 48% 47% 40% 20% 40% 41% 42% 45% 47% 52% 60% 40% 20% 10% 11% 11% 10% 11% 11% 10% 12% 12% 10% 12% 12% 10% 12% 12% 10% 12% 13% 10% 13% 13% 18% 18% 18% 17% 17% 17% 17% 0% guaraneed ineres rae r G 0% guaraneed ineres rae r G Figure 4 displays risk-reurn profiles for he less risky shifing mechanism of he dynamic hybrids corresponding o he lef columns of Figures 1 and A.2, which are he basis for deriving he willingness-o-pay. In paricular, he firs and second row show he case of a full

22 21 (100%) and 50% money-back guaranee for he s, respecively. In he lef column of Figure 4, he payoff disribuions for he paricipaing life insurance conracs are exhibied, whereas he case of he dynamic hybrids is shown in he righ column. When comparing he lef column o he righ column, i.e., he case of PLIs o s, he resuls show ha he paricipaing life insurance conracs payoffs are relaively sable wih a low volailiy, whereas he dynamic hybrids payoffs exhibi a considerably higher volailiy, in line wih he previous findings (see also Figure 5). While he paricipaing life insurance payoffs upside poenial (payoffs above 175) remains almos unchanged when reducing he guaranee level of he dynamic hybrids from a full money-back guaranee (firs row) o a 50% money-back guaranee (second row), he dynamic hybrids payoff volailiy considerably increases. In paricular, he upside poenial as well as he downside poenial of he payoffs of he dynamic hybrid producs is considerably higher for a parial money-back guaranee as compared o a full money-back guaranee (compare righ graphs in Figure 4). This is based on he fac ha in case of a 50% money-back guaranee, he dynamic hybrid shifing mechanism invess a higher proporion of he conracs accoun value in he equiy fund (and he guaranee fund). This resuls in higher payoff volailiies compared o a full money-back guaranee, where he available funds are, o a larger exen, invesed in he sable and less volaile policy reserve sock. These findings are in line wih previous resuls showing ha he willingness-opay for dynamic hybrids wih a full money-back guaranee is higher han for a parial moneyback guaranee for more risk averse policyholders, i.e. for a relaively less volaile payoff compared o a more volailiy payoff (see Figures 1 and 5). Furhermore, his resul shows he grea flexibiliy of dynamic hybrids, which can be adjused o various cusomers needs o achieve differen payoff disribuions wihou changing he conracs basic seing. Figure 5 addiionally shows he corresponding quariles (upper and lower quarile, median) for he payoff disribuions of he PLIs and s presened in Figure 4. The lef column in Figure 5 illusraes he relaively sable payoffs of he paricipaing life insurance conracs, while he righ column again shows he righ-skewed and volaile dynamic hybrid producs payoffs. The quariles illusrae he sensiiviy of he dynamic hybrids payoff wih respec o heir guaranee feaures. The resuls also show ha an increase in he guaraneed ineres rae implies a decrease in he median payoff and reducion in he upside poenial of he paricipaing life insurance conracs payoffs. In conras o his, he dynamic hybrids payoff (and hus he willingness-o-pay) is almos no affeced by a change in he guaraneed ineres raes for he less risky shifing mechanism (see also lef column in Figure 1). Resuls for he more risky shifing mechanism are in line wih previous findings and are hus omied here.

23 22 Figure 5: Lower quarile, median, and upper quarile (i.e. 25h, 50h, and 75h percenile) of he payoff disribuions for PLIs and s for differen guaraneed ineres raes and differen guaranee levels (less risky shifing mechanism; corresponding o he lef columns in Figures 1 and A.2) PLI payoff quariles (money-back guaranee D HP) D H P payoff quariles (money-back guaranee DH P) guaraneed ineres rae r G guaraneed ineres rae r G PLI payoff quariles (lower guaranee, x=0.5) D H P payoff quariles (lower guaranee DH P, x=0.5) guaraneed ineres rae r G guaraneed ineres rae r G Furher findings reveal ha he willingness-o-pay for he paricipaing life insurance conracs and dynamic hybrids can be more exreme in a posiive or negaive way depending on he degree of risk aversion and depending on a risk measure ha akes downside risk and skewness ino accoun j We furher sudied he firs-order lower parial momen (LPM 1) wih reference poin = ( T ) z E V, also referred o as lower semi-absolue deviaion measure (LSAD) (see Gusafsson and Salo, 2005) o measure downside risk in a general mean-risk preference model (see Fishburn, 1977) and found he resuls o be robus.

24 23 4. CONCLUDING REMARKS This paper assesses he ineracion effecs when insurers offer dynamic hybrid policies in addiion o paricipaing life insurance conracs and explicily focuses on he policyholders perspecive for he firs ime. We consider a 3-fund dynamic hybrid accoun whose value is periodically reallocaed beween he convenional premium reserve sock (corresponding o he policy reserves), a guaranee fund (which loses a mos a cerain percenage of is value in each period), and a risky equiy fund, following a mahemaical shifing mechanism ha is based on he concep of consan proporion porfolio insurance (CPPI). To assess he policyholders perspecive, mean-variance preferences were used o derive he willingness-o-pay (WTP) for differen degrees of risk aversion of he policyholders. Our resuls show ha higher guaranees (e.g., he guaranee level, riskiness of shifing mechanisms, guaraneed ineres raes) do no necessarily imply an increase in consumers willingness-o-pay. In conras, in he examples considered here, which are based on fair conracs from he equiyholders perspecive, he willingness-o-pay for boh ypes of policyholders clearly decreases for higher guaraneed ineres raes. These findings are consisen wih findings in several oher sudies ha evaluae he willingness-o-pay for guaranees. Higher guaranees wihin insurance producs generally do no necessarily imply an increase in consumers willingness-o-pay (e.g., Gazer, Holzmüller, and Schmeiser (2012). Furhermore, consumers are also willing o pay a subsanially higher price for guaranees when he opion is provided in a simple all else equal conex (e.g., see Gazer, Huber, and Schmeiser, 2011), primarily because hey are risk averse, while on average, he willingnesso-pay is significanly below he insurer s risk-adequae premium. Furher relevan impac facors regarding he deerminaion of willingness-o-pay include cusomer preferences (e.g., see Doskeland and Nordahl, 2006), demographic characerisics such as income, gender, and educaion (e.g., see Feldman and Schulz, 2004), and insurer characerisics and operaions (e.g., see Marshall, Hardy, and Saunders, 2010). The demand for guaranees has imporan implicaions on he supply side, encouraging insurers o consanly look for opporuniies o innovae in heir produc offerings. We show here, for example, ha in he case of he dynamic hybrid producs, a cerain minimum guaranee level mus be offered by he insurer in order o ensure ha he conrac is purchased in he firs place, i.e., ha he willingness-o-pay of he dynamic hybrid policyholders exceeds he required minimum premium ha he insurer mus charge when selling he conrac. For insance, while he willingness-o-pay was sufficien for a full money-back guaranee and guaraneed ineres raes up o around 1.5% in case of a less risky shifing mechanism, he willingness-o-pay considerably decreased when reducing he dynamic hybrid guaranee level o a

25 24 50% money-back guaranee or when using he more risky shifing algorihm o reallocae he dynamic hybrid funds. Thus, he willingness-o-pay by far did no exceed he necessary premium in order o close he conrac. The same holds rue when keeping he guaranee level a a full money-back guaranee bu using a more risky shifing mechanism for disribuing he dynamic hybrid accoun value beween he hree funds. In he laer cases, conracs could only be sold when consumers exhibied a raher lower risk aversion. Thus, he araciveness of hese producs srongly depends on he consumers preferences and varies considerably depending on he conrac design, including he level of guaranee, he shifing mechanism and he degree of risk aversion. A he same ime, his resul also emphasizes he grea flexibiliy of hese producs, which can be easily adjused in order o mee differen consumers needs for sabiliy or upside poenial. Given consumers sensiiviy o paricular conrac feaures, i is imporan ha insurers are ransparen in he markeing of heir producs. Our analysis also emphasizes he need for insurers o regularly (re)evaluae heir mix of producs wih differen ypes of guaranees, recognizing ha he araciveness of any one of produc may be affeced by he insurer s produc mix. Here we found ha he willingness-o-pay for a paricipaing life insurance policy remained almos unchanged when varying he risk aversion parameer, which emphasizes he low volailiy and sable payoff of hese radiional producs. The ineracion effecs ha may arise in he porfolio, which differ depending on he dynamic hybrid conrac feaures, may impac he willingness-o-pay or preference level of he paricipaing life insurances, alhough we do no find his effec o be subsanial for he cases considered. The poenial for ineracion effecs raises concerns ha cusomers may inadverenly choose a life insurance policy ha, subsequenly, is no suiable for heir siuaion, or purchase inadequae amouns of coverage. The presen resuls cerainly depend on he assumed mean-variance preferences used o derive he willingness-o-pay, bu hey are generally consisen wih previous heoreical and empirical work wih respec o he willingness-opay of consumers for guaranees.