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

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1 Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen Friedrich-lexander-Universiy of Erlangen-Nürnberg Version: February 2013

2 FIR VUTION ND RISK SSESSMENT OF DYNMIC HYBRID PRODUCTS IN IFE INSURNCE: PORTFOIO CONSIDERTION lexander Bohner, Nadine Gazer This version: February, 2013 BSTRCT Dynamic hybrid producs are innovaive life insurance producs paricularly offered in he German marke and inended o mee new consumer needs regarding sabiliy and upside poenial. These producs are characerized by a periodical rebalancing process beween he policy reserves (i.e. he premium reserve sock), a guaranee fund and an equiy fund. The policy reserve hereby corresponds o he one also valid for radiional paricipaing life insurance producs. Hence, funds of dynamic hybrids ha are allocaed o he policy reserves in imes of adverse capial marke environmens earn he same policy ineres rae deermined for he paricipaing life insurance policyholders and, hence, a leas a guaraneed ineres rae. In his paper, we sudy he fair valuaion and risk siuaion of an insurer offering boh, dynamic hybrid and radiional paricipaing life insurance conracs. The resuls reveal considerable ineracion effecs beween he wo conrac ypes wihin he porfolio ha srongly depend on he porfolio composiion, hereby emphasizing meris as well as risks associaed wih offering dynamic hybrids. Keywords: ife insurance, risk measuremen, risk-neural valuaion, dynamic hybrid, consan proporion porfolio insurance 1. INTRODUCTION Dynamic hybrid producs are innovaive life insurance and deferred annuiy producs paricularly offered in he German marke. The aim of hese producs is o mee new consumer needs by combining he sabiliy of radiional life insurance (by means of he convenional policy reserves) wih he upside poenial of uni-linked policies (hrough invesing in a guaranee and / or equiy fund). While he firs inroduced version of his class of conracs (referred o as saic hybrid ) used a decomposiion of he premiums o ensure he guaranee promised o he policyholders, dynamic hybrid producs use a periodical dynamic rebalancing of he accoun value ino wo o hree differen invesmens based on a consan proporion porfolio lexander Bohner and Nadine Gazer are a he Friedrich-lexander-Universiy (FU) of Erlangen- Nürnberg, Deparmen of Insurance Economics and Risk Managemen, ange Gasse 20, Nuremberg, Germany,

3 2 insurance-ype sraegy (CPPI). 1 The hree ypical invesmens of 3-fund dynamic hybrid producs include he policy reserves, a guaranee fund (ha loses a mos a cerain percenage in each period), and a risky equiy fund. In paricular, in imes of adverse capial marke environmens, for insance, funds may be shifed shorerm from he guaranee fund and he equiy fund o he convenional policy reserves (and hence he premium reserve sock), hus earning he same policy ineres rae also credied o he radiional life insurance conracs, which is a leas he guaraneed acuarial ineres rae. Hence, considerable ineracion effecs can be expeced beween he radiional paricipaing life insurance policies and he dynamic hybrid producs when considering a mixed porfolio, which may have a srong impac on an insurer s risk siuaion and also on he fair valuaion of boh conrac ypes. The aim of his paper is o comprehensively sudy hese ineracion effecs wih focus on fair valuaion and risk measuremen for an insurer wih a produc porfolio consising of radiional life insurance conracs and dynamic hybrid producs for a specific rebalancing mechanism in deph. In he early versions of hybrid producs, premiums were spli in order o mee a cerain (e.g. money-back) guaranee, such ha one premium par is invesed in he policy reserves and he remaining par is invesed in a risky equiy fund, whereby he policy reserves are assumed o earn a leas he minimum acuarial ineres rae. 2 The drawback of his approach is he fac ha only a minor porion of he premiums is available for he invesmen in risky funds wih upside poenial, as he equiy fund s value is assumed o drop o zero in he wors case in one period, e.g. one monh or even one day, which can be considered raher unrealisic. Hence, newer versions of he producs used a guaranee fund, which is equal o he equiy fund bu ensures ha he guaranee fund value does no lose more han a cerain percenage wihin one period. Thus, he premium par invesed in he policy reserves can be relaively lower, such ha policy reserves are parly relieved from having o cover he full guaranee promised o he hybrid conracs. However, o mee a final guaranee afer more han one period, funds have o be shifed from he guaranee fund o he policy reserves in case he guaranee fund has dropped, which is he saring poin for dynamically reallocaing he invesmen as done in he case of dynamic hybrid producs. Unil recenly, he lieraure has paid only lile aenion o dynamic hybrid producs, which are mainly discussed in he non-academic lieraure and here wihou providing a model framework or numerical sudies. Beels, Grosner, and eischkis (2011), for insance, indicae ha here is a need o analyze ineracions of dynamic hybrid producs wih he exising porfolio of policies o adequaely assess he risk for insurers ha are conneced o dynamic hybrid producs. In addiion o he risk for insurers, hey poin ou ha since he guaranees in 1 2 See eland (1980), Rubinsein and eland (1981), and Black and Jones (1987) regarding CPPI mechanisms. See, e.g., Kochanski and Karnarski (2011) for a more deailed and formal descripion of hese producs.

4 3 he guaranee fund are provided by exernal invesmen companies, repuaional risk may arise in case hese invesmen companies fail o keep heir guaranees. Oher qualiaive discussions by acuaries are provided in Menzel (2008) and Sieber (2008). Menzel (2008) highlighs he poenial risk an insurer would face wih dynamic hybrid producs in he porfolio, as he porfolio of radiional polices sells an opion o he policyholders of dynamic hybrid producs, whose value may no be negligible. Sieber (2008) opposes ha here are no only risks conneced o dynamic hybrid producs bu also benefis for an insurance company selling dynamic hybrid producs in addiion o radiional policies, as here are posiive subsiuion effecs for boh producs. Thus, while boh aricles address poenial ineracion effecs of a porfolio of dynamic hybrid producs wih a radiional insurance porfolio, hey do no provide a model or numerical examples. model framework for hybrid conracs is presened in Kochanski and Karnarski (2011), who calculae solvency capial requiremens according o Solvency II for saic as well as dynamic hybrid producs. They implemen a parial inernal model for a porfolio of 3-fund dynamic hybrid producs including a rules-based shifing process o reallocae he conracs accoun value each monh. Since focus is laid on deermining solvency capial requiremens in accordance o Solvency II, hey do no analyze ineracion effecs wihin a porfolio of dynamic hybrid producs and radiional conracs. In his paper, we conribue o he lieraure by analyzing he impac of dynamic hybrid producs on he fair valuaion and risk assessmen of an insurer wih a porfolio consising of radiional paricipaing life insurance conracs and dynamic hybrid producs. We hereby provide a model framework for he developmen of a pool of paricipaing life insurance conracs and he accumulaion phase of a pool of dynamic hybrid producs ha allows us o sudy ineracion effecs beween he wo ypes of producs in deph. The dynamic reallocaion of funds of he dynamic hybrid producs over ime is mainly based on Kochanski and Karnarski (2011), whereas he surplus disribuion mechanism for he policy reserves of he radiional conracs is based on he smoohing scheme in Grosen and Jørgensen (2000). Our findings show srong inerdependencies beween radiional paricipaing life insurance conracs and innovaive dynamic hybrid producs, which can considerably affec he fair value of he wo producs as well as he insurer s risk siuaion. Policyholders of he wo producs can eiher profi or lose from he porfolio composiion and he arising subsiuion effecs, depending on he conrac parameers and especially he guaranee level offered o he dynamic hybrid policyholders, which srongly impacs he dynamic reallocaion of funds. In addiion, he findings show ha even hough he siuaion is fair from he equiyholders perspecive, his is no necessarily he case for he wo life insurance producs, whose presen values vary considerably depending on he porfolio composiion and he choice of inpu parameers.

5 4 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. Secion 3 conains a numerical analysis and Secion 4 concludes. 2. MODE FRMEWORK Modeling he insurance company We consider a life insurance company wih a produc porfolio consising of radiional paricipaing life insurance (PI) and dynamic hybrid producs (DHP). Tradiional paricipaing life insurance (radiional guaraneed wih-profis policies) place he conracs available funds enirely and hroughou he whole conrac erm in he acuarial policy reserves (PR). In he case of dynamic hybrid producs, he policyholder s funds are dynamically allocaed beween up o hree pos, he policy reserves, a guaranee fund (GF) and an equiy fund (EF), in order o ensure he guaranee promised o he policyholder. Table 1 shows he balance shee of he life insurance company a ime. The liabiliies side comprises accouns for he policy reserves (PR ), (synheically) pariioned in policy reserves for he paricipaing life insurance conracs ( PR ) and for he dynamic hybrid producs ( PR ), accouns for he dynamic hybrid producs guaranee fund GF and equiy fund EF, plus an accoun B ha serves as buffer. Thus, B is residually given by he company s DHP oal asses ( ) minus he policyholders accouns, i.e. PR, GF and EF. For simplificaion, B is no furher subdivided ino a buffer accoun and equiy. The policy reserves for par- PI DHP icipaing life insurance producs as well as dynamic hybrid producs, i.e. PR and PR, are reaed as one accoun, as heir separaion ino wo accouns is conduced o keep rack of he corresponding invesmen on he asse side (he premium reserve sock). PI Table 1: Balance shee of he life insurer a ime sses longerm shorerm GF EF iabiliies PI PR DHP PR PR GF V EF B

6 5 The asse side of he balance shee is srucured as follows. The company s oal asses are allocaed o wo differen groups of invesmens, he fund invesmens for he dynamic hybrid produc policyholders (he guaranee fund GF and he equiy fund EF ), which are he offseing iems o GF and EF on he liabiliies side, 3 and he remaining asses, hereafer referred o as company s asses. Even hough policy reserves of he paricipaing life insurance and dynamic hybrid producs on he liabiliies side are reaed equally, he corresponding asses canno be handled alike, since mauriies differ considerably. Unlike he policy reserves of paricipaing life insurance conracs ha have o and can be invesed for a relaively long period of ime, funds in he policy reserves of he dynamic hybrid produc migh only be available in his accoun for a shor period, unil hey are shifed o he guaranee fund or he equiy fund. Therefore, we assume ha he company s asses are spli ino longerm invesmens invesmens shorerm longerm and shorerm, whereby he buffer accoun is also assumed o be invesed shorerm due o is smoohing characerisics over ime. To be able o disinguish beween he beginning and he end of a period, we use and - o denoe he period s beginning and end, respecively. Hence, a = 0, i.e. a he beginning of he firs period, B 0 is filled up by an iniial conribuion of he company s owners. nalogously o he policyholders, we do no assume furher paymens from he equiyholders during he lifeime of he company. Furhermore, equiyholders are no able o wihdraw funds from he company unil liquidaion a ime T, a which hey receive an ineres-bearing payback of heir iniial conribuion, in case he company s funds are sufficien o cover he policyholders liabiliies. he end of he considered ime horizon, i.e. a ime T, policyholders holding a radiional paricipaing life insurance conrac receive heir final sum insured including heir surplus paricipaion, while policyholders wih a dynamic hybrid produc are paid ou he sum of heir hree pos, i.e. heir final accoun value a he end of year T ( DHP DHP T T T T V = PR GF EF ). In addiion, policyholders of boh conracs receive a erminal bonus. In he presen analysis, focus is only laid on he accumulaion phase, while he payou is assumed o be a single lump sum paymen insead of a lifelong annuiy. Furhermore, policyholders pay heir single premiums P PI and P DHP resuling o iniial policy reserves of he paricipaing life insurance conracs of PR PI PI = P, 0 3 For he insurer, hese accouns are no risk-bearing, as marke risks are fully carried by he policyholder. However, he defaul of a guaranee fund would represen a repuaional risk for he insurance company.

7 6 and an iniial accoun value of he dynamic hybrid producs of V DHP DHP = P. 0 DHP DHP The disribuion of funds in V 0 o PR 0, GF 0 and EF 0 is laid ou in he subsequen secion. To focus on ineracions of he wo produc ypes wihin a life insurer, he conrac erm T is assumed o coincide wih he lifeime of he considered insurance company. Developmen of asses he beginning of period, he company s asse invesmens are hus given by = PR, longerm PI shorerm DHP = PR B, GF EF = GF, EF =. We assume ha he guaranee fund evolves analogously o he equiy fund excep for he downside proecion (see also Kochanski and Karnarski, 2011). Hence, here are hree ypes of invesmens, including longerm invesmens corresponding o he policy reserves of he radiional paricipaing life insurance conracs, shorerm invesmens ha correspond o he par of policy reserves ha arise from he dynamic hybrid producs and he buffer, and he equiy fund. These hree invesmens are all assumed o evolve according o a geomeric Brownian moion given by he following sochasic differenial equaion, di = µ I d σ I dw, i = 1, 2, 3, i i i P i i, i P wih consan drif µ i and volailiy σ i, P-Brownian moions dw, i defined on he probabiliy P P space ( Ω, F, P) wih he linear correlaions dw, i dw, j = ρi, j. The soluion of he equaion resuls o (see Björk, 2009) i i 1 2 P I = I0 exp µ i σ i σ i dw, i 2. he end of period, he asse iems are hus given by I =, 1 longerm longerm ( ) 1 I

8 7 I =, 2 shorerm shorerm ( ) 2 I EF I =, 3 EF ( ) 3 I GF GF y 3 = max 1 λ; ( ) 3 I I, where he developmen of he guaranee fund is deermined by a fracion y of he performance of he equiy fund, since he downside proecion has o be financed. In paricular, we can show ha in he presen seing, he fracion y is consan over ime for a given se of parameers λ, r f, σ 3 and and hus independen of he curren value of he guaranee fund, as we assume ha a pu opion on he equiy fund wih price P and a srike price ( 1 ) λ GF wih 0 λ 1 is used for he downside proecion (alernaively, a guaranee fund can also be secured by a consan proporion porfolio insurance (CPPI)-based sraegy). In paricular, only a fracion of he guaranee fund he pu opion price ( 1 ) y GF wih 0 y 1 can be invesed in he equiy fund, as P = y GF has o be paid for hedging he downside risk. The pu opion price P can be calculaed via he Black-Scholes formula, which is given by ( 1 ) exp( ) ( 1) ( 1 3 ) λ f σ P = GF r Φ d y GF Φ d, ( 1 λ ) GF λ 1 2 ln rf σ 3 ln rf σ 3 y GF 2 y 2 d1 = =. σ σ 3 3 Hence, he available funds in he guaranee fund a ime can be spli ino an invesmen in he equiy fund and he paymen for he pu opion, resuling in ( 1 λ ) exp ( f ) ( 1) ( 1 σ 3 ) GF = y GF P = y GF GF r Φ d y GF Φ d, which can be rewrien as ( λ ) ( ) ( 1) ( 1 σ 3 ) 1 = y 1 exp rf Φ d y Φ d. The laer equaion is hus independen of GF for a given se of parameers and mus only be solved once for y using a roo-finding algorihm (as d 1 depends on y as well), which considerably simplifies he furher simulaion analysis.

9 8 The oal asses amoun of he balance shee hus resuls o = EF GF. longerm shorerm ( ) ( ) ( ) ( ) ( ) Developmen of liabiliies On he liabiliy side, all policy reserves are annually compounded wih a policy ineres rae P r ha in he following is assumed o be based on Grosen and Jørgensen (2000) (bu can as well be replaced by oher smoohing mechanisms depending on he regulaory rules of he respecive counry), 4 r B = max r, α γ, PR PR P G PI DHP where r G is he guaraneed ineres rae, α is he annual surplus paricipaion rae and γ is he buffer raio, which mus be exceeded in order o allow surplus paricipaion. The policy ineres rae mus no only be ensured for he radiional paricipaing life insurance conracs, bu also for he par of funds of he dynamic hybrid producs ha is allocaed o he insurer s policy reserves a ime. In case of he paricipaing life insurance conracs, he policy reserves in each period are hus increased by PR = PR r. PI PI P ( ) ( 1 ) 1 3-fund dynamic hybrid produc invess no only in policy reserves, bu also in an equiy fund and a guaranee fund, whereby he laer is equivalen o an equiy fund wih a hedge ha ensures a maximum loss of λ percen wihin one period. In conras o he paricipaing life insurance policies, where he policy ineres rae is guaraneed, we assume ha he dynamic hybrid policyholders are guaraneed a fracion x of heir single up-fron premium, DHP DHP DHP 1 G = G = x P 0,, K, T, 12 where G DHP denoes he accoun value needed a ime o ensure ha he guaranee can be me and which may vary depending on he concree produc design. Hence, for x = 1, he cusomer 4 Noe ha he amoun of surplus paricipaion ypically depends on how pruden he insurer calculaes premiums when aking ino accoun moraliy risk and coss, surplus is increased by means of he moraliy and he cos resul, which comes in addiion o he invesmen resul (see, e.g. Bohner and Gazer (2012) for sudies regarding differen surplus appropriaion schemes and heir impac on an insurer s risk siuaion).

10 9 DHP obains a money-back guaranee. In case he guaranee a he end of each period G can be fulfilled by he guaranee fund only, he dynamic hybrid producs funds are disribued beween he guaranee fund and equiy fund. The disribuion of he accoun value DHP policy reserves PR, he guaranee fundgf and equiy fund and Karnarski (2011) and given by 5 PR DHP GF ( 1 λ ) DHP DHP DHP G V G, if > 1 G DHP = ( 1 r ) 1 λ ( 1 λ) V 0, oherwise DHP DHP DHP G V PR, if > 1 DHP ( 1 λ) V = DHP G, oherwise 1 λ EF = V PR GF DHP DHP. DHP V o he EF is based on Kochanski In wha follows, we consider a family of idenical 3-fund dynamic hybrid producs where funds are reallocaed every period. In sum, liabiliies a he end of period are hus given by PI PI P ( ) ( 1 ) DHP DHP P ( ) ( 1 ) PR = PR r, PR = PR r, GF EF = GF, ( ) ( ) = EF. ( ) ( ) The buffer a he end of a he period is calculaed by B = PR PR GF EF. PI DHP ( ) ( ) ( ) ( ) ( ) ( ) The company is insolven, if he asses are no sufficien o cover he liabiliies, i.e. if B ( ) < 0. 5 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 sysem in he marke, here are also differen approaches aiming 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.

11 10 In his case, he remaining funds RF are assumed o be paid ou o he policyholders as follows by aking ino accoun he invesmen in he policy reserves over he conrac erm (as surplus is generaed by hese means and since dynamic hybrid policyholders sill receive heir invesmen in he guaranee fund and he equiy fund), i.e. ( ) ( ) PI longerm shorerm PI PI DHP RF = ( 1 c) PR PR PR ( ) ( ) ( ) ( k 1 ) (( k 1) ) (( k 1) ) k = 0 k= 0 and GF EF ( ) ( ) DHP longerm shorerm DHP PI DHP RF = ( 1 c) PR PR PR ( 1) ( ) ( ) ( k 1 ) (( k 1) ) (( k 1) ) k = 0 k = 0 ( 1) ( 1), respecively, if he policy reserves are posiive, and where c represens he coss of insolvency. 6 mauriy T, he remaining buffer is disribued among policyholders and equiyholders, whereby he equiyholders firs receive a buffer payback of ( ( ( ) T 0 ) ) BP = max min B, B 1 b,0, T which includes a buffer ineres rae b paid on heir iniial invesmen. The policyholders receive he remainder as an (opional) erminal bonus TB max ( 0, B BP ) T T T =, which analogously o he remaining funds in case of defaul is assumed o be disribued beween PI and DHP conracs as follows: TB TB PR PR PR T T PI PI PI DHP = T T ( k ) ( k ) ( k ) k = 1 k = 1, TB TB PR PR PR T T DHP DHP PI DHP = T T ( k ) ( k ) ( k ) k = 1 k = 1, if he policy reserves are posiive, and zero else. Hence, he oal payous o he paricipaing life insurance and dynamic hybrid policyholders are given by ( ) 1{ } 1{ } V = PR TB T > T RF T = PI PI PI PI T T T s s 6 See also Grosen and Jørgensen (2002) for an analysis of early defaul using barrier opions.

12 11 and ( ) 1{ } 1{ } V = V TB T > T RF T =, DHP DHP DHP DHP T T T s s respecively, where he accoun value if given by DHP DHP T T T T longerm shorerm s = inf : <, = 1,...,. V = PR GF EF and T S denoes he ime of defaul wih { } T PR T Fair valuaion and risk measuremen 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. Q 0 T r ( f T ) B = E BP e, where E Q denoes he expeced value under he risk-neural pricing measure Q and r f 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). For he fairly calibraed b, he presen values from he paricipaing life insurance and dynamic hybrid policyholders perspecive are given by T rf rf (( ) { s }) { s } T T ( ) PI Q PI PI 1 Q PI PV = E PR TB e T > T E RF e 1 T = 0 and T rf rf (( ) { s }) { s } T T ( ) DHP Q DHP DHP 1 Q DHP PV = E V TB e T > T E RF e 1 T =, 0 respecively, which in case of a fair siuaion for policyholders should be equal o heir iniially paid single premiums. We furher calculae he shorfall probabiliy under he real-world measure P as ( ) SP = P T T, s longerm shorerm where { } T = inf : < PR, = 1,..., T. s

13 12 3. NUMERIC NYSIS Descripion of inpu parameers The following secion presens numerical resuls based on he model laid ou in he previous secion. The inpu parameers are summarized in Table 2 and serve o illusrae cenral effecs. They were furher subjec o sensiiviy analyses. In paricular, we assume one period o be one monh, i.e. he ime horizon of T years is subdivided wih = 1/12. Resuls are generaed based on Mone Carlo simulaion using 50,000 simulaion runs and lain hypercube sampling o reduce variance. In he following, we paricularly sudy he fair valuaion and risk measuremen for differen inpu parameers and varying porfolio composiions (in erms of he single upfron premium) in order o idenify possible porfolio subsiuion and risk ransfer effecs. Table 2: Inpu parameers for he numerical analyses Single premiums of paricipaing life insurance conracs P PI 100 Single premiums of dynamic hybrid producs P DHP 100 Conrac duraion T 10 Guaranee of dynamic hybrid producs x 1 Iniial buffer B 0 6 Guaraneed ineres rae (p.a.) r G Surplus disribuion raio α 0.3 Targe buffer raio γ 0.1 Coss of insolvency c 0 Drif of longerm invesmens µ Volailiy of longerm invesmens σ Drif of shorerm invesmens µ Volailiy of shorerm invesmens σ Drif of equiy fund µ Volailiy of equiy fund σ inear correlaion of longerm and shorerm invesmens ρ 1,2 0.2 inear correlaion of longerm invesmens and equiy fund ρ 1,3 0.2 inear correlaion of shorerm 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 engh of a period 1/12

14 13 The impac of he guaraneed ineres rae on fair values and shorfall risk The impac of he guaraneed ineres rae r G is exhibied in Figure 1 for differen porfolios of paricipaing life insurance conracs and dynamic hybrids (only paricipaing life insurance conracs in he firs row; equally weighed in erms of he single up-fron premium in he second and hird row) and differen guaranees for he dynamic hybrid (money-back guaranee in second row; guaranee equals 50% of he single premium in he hird row). In he righ column of Figure 1, he fairly calibraed buffer ineres raes are shown (from he equiyholders perspecive), and he lef column exhibis he corresponding presen values of he paricipaing life insurance and dynamic hybrid conracs (lef y-axis) along wih he corresponding shorfall probabiliy of he insurer (righ y-axis). The righ graphs in Figure 1 show ha he fair buffer ineres rae b (o be paid o he equiyholders) increases (along wih an increasing shorfall risk) if he guaraneed ineres rae is raised. However, even hough all conracs in Figure 1 are fair from he equiyholders perspecive, hey are no necessarily fair from he policyholders viewpoin as shown in he lef graphs. While he paricipaing life insurance conracs are approximaely fair if dynamic hybrid conracs are no sold by he insurer (firs row in Figure 1 where he premium by he dynamic hybrid policyholders is P DHP = 0), hey are no longer fair in case of a mixed porfolio (second and hird row), i.e. as soon as dynamic hybrid conracs are offered in addiion o he paricipaing life insurances. In paricular, in case of a fair siuaion for boh policyholders and equiyholders, he presen value of each conrac ype should be equal o he corresponding iniial single up-fron premium, which in he second row is 100 in case of boh producs, for insance. In his seing, he wo ypes of conracs would be fair from he policyholders perspecive (i.e. presen value = 100 for boh conracs) for a guaraneed rae of approximaely 2.4%, i.e. where boh curves inersec. Therefore, o ensure fair conracs for shareholders and boh groups of policyholders, a corresponding opimizaion problem would comprise hree objecive funcions and requires a sufficien number of variable inpu parameers o solve his problem.

15 14 Figure 1: Presen values of he paricipaing life insurance and dynamic hybrid conracs wih corresponding insurer shorfall risk (lef column) for differen porfolio composiions and dynamic hybrid guaranees when varying he guaraneed ineres rae and given fair buffer ineres raes (righ column, equiyholder perspecive) P D H P =0 P D H P =0 presen value shorfall probabiliy fair buffer ineres rae b guaraneed ineres rae r G guaraneed ineres rae r G P D H P =100, G D H P =1.0 P D H P P D H P =100, G D H P =1.0 P D H P presen value shorfall probabiliy fair buffer ineres rae b guaraneed ineres rae r G guaraneed ineres rae r G P D H P =100, G D H P =0.5 P D H P P D H P =100, G D H P =0.5 P D H P presen value shorfall probabiliy fair buffer ineres rae b guaraneed ineres rae r G guaraneed ineres rae r G paricipaing life insurance dynamic hybrid produc shorfall probabiliy (righ axis)

16 15 Figure 1 hus demonsraes he srong impac of including dynamic hybrid producs in he insurance porfolio on he presen value of paricipaing life insurance conracs and on shorfall risk for he considered rebalancing mechanism, whereby he effec also depends on he guaranee G DHP promised o he dynamic hybrid policyholders. In paricular, one can observe in he middle lef graph (where a money-back-guaranee is embedded in he dynamic hybrid conracs) ha for an increasing guaraneed ineres rae, he value of he dynamic hybrid produc increases, while he presen value of he paricipaing life insurance policies decreases. However, he presen value of he PIs lies above heir iniial premium of 100, which is mos pronounced for low guaraneed ineres raes, while DHPs exhibi a presen value well below heir iniial paymen. he same ime, he shorfall probabiliy is higher in he second row (especially for higher guaraneed ineres raes) as compared o he case wihou DHPs (firs row). Hence, he decrease in he presen value of PIs (from a higher level han 100) is caused by he inclusion of dynamic hybrid producs. This can be explained when looking a Figure 2, which shows he average monhly pariion of he accoun value (V) of he dynamic hybrid produc over he conrac erm, i.e. he amoun of financial resources invesed in he hree funds (given fair conracs from he equiyholders perspecive). The lef and righ column shows he pariioning for a guaraneed ineres rae of r G = 1.0% and 2.5%, respecively, while from op o he boom, differen guaranees for he dynamic hybrids G DHP are displayed. s can be seen in he firs row of Figure 2, where G DHP = P DHP, a higher guaraneed ineres rae as shown in he righ graph implies ha afer he firs periods, fewer funds need o be allocaed o he policy reserves o ensure he guaranee (see also he formula for he dynamic reallocaion, which depends on he guaraneed rae). he same ime, a larger par can be allocaed o he guaranee fund and he equiy fund, which in urn increases he upside poenial regarding he average expeced payoff for he dynamic hybrid policyholders. However, he shorfall probabiliy of he company as a whole increases due o he higher guaranee also promised o he paricipaing life insurance policyholders. This is also influenced by he fac ha he allocaion of funds of dynamic hybrids o he policy reserves ypically happens in imes of low marke reurns, which makes i difficul for he insurer o generae he necessary guaraneed ineres raes by invesing in he capial marke. Hence, in hese imes, more money is shifed o he policy reserves and needs o earn a leas he guaraneed rae, which is paricularly difficul wih shorerm invesmens. In addiion, as soon as he guaranee is credied o he policy reserves, i becomes par of he guaranee, which also increases he shorfall risk.

17 16 Figure 2: verage pariion of he dynamic hybrid produc accoun value ino he hree funds (equiy fund, guaranee fund, policy reserve) over he conrac erm for differen guaraneed ineres raes and differen dynamic hybrid guaranees given a fair siuaion from he equiyholders perspecive G D H P =1.0 P D H P, r G =0.01 G D H P =1.0 P D H P, r G =0.025 average value average value ime ime G DH P =0.75 P D H P, r G =0.01 G D H P =0.75 P D H P, r G =0.025 average value average value ime ime G D H P =0.5 P D H P, r G =0.01 G D H P =0.5 P D H P, r G =0.025 average value average value ime ime PR D H P G F EF PR P I

18 17 s Figure.1 in he ppendix shows, shorfall mainly occurs in he firs conrac periods, where no only paricipaing life insurance, bu also dynamic hybrid policyholders are sill heavily invesed in he policy reserves (see Figure 2, firs row), which means ha in case of defaul, he remaining funds of he company are disribued almos evenly beween paricipaing life insurance and dynamic hybrid policyholders, whereby he laer group addiionally obains he guaranee fund and equiy fund (which are no subjec o defaul in he presen seing). Hence, on average and in erms of he presen value, dynamic hybrid policyholders profi from he radeoff of a higher reurn in he sense of an increasing presen value (which, however, is sill no fair and sill below heir iniial premium paymen) and higher shorfall risk in case of a higher guaraneed ineres rae, while he presen value of paricipaing life insurance policyholders is decreasing in he radeoff (which would no be he case wihou dynamic hybrids in he porfolio, see firs row in Figure 1), bu exhibis an overall higher value as compared o he case wihou dynamic hybrids. Thus, he siuaion for paricipaing life insurance policyholders improves in he considered examples, bu is deerioraing for increasing guaraneed ineres raes. However, his picure changes when reducing he guaraneed sum insured of he dynamic hybrids from x = 1 o x = 0.5 (of he iniially paid up-fron premium P DHP ) as shown in he hird row in Figure 1 and he hird row in Figure 2. In his case, he presen value of he paricipaing policies even slighly increases, while he dynamic hybrid producs exhibi a decrease when increasing he guaraneed ineres rae. This is rue even hough only a very small amoun of he dynamic hybrid funds is allocaed o he policy reserves (since he guaraneed sum insured of he DHPs for he mos par can be covered by he guaranee fund GF wihou need of he policy reserves). While he policy reserves are hus hardly affeced by he DHP producs and, hus, here is almos no effec on shorfall risk as compared o he case where no dynamic hybrids are sold (firs row in Figure 1), he shorfall probabiliy sill increases for higher guaranee ineres raes. Hence, in case of a defaul (which is riggered by he paricipaing life insurance conracs), paricipaing life insurance policyholders obain he vas majoriy of he remaining funds due o having been invesed in he policy reserves, while dynamic hybrid policyholders only receive heir curren values of he guaranee fund and he equiy fund. Hence, for lower dynamic hybrid guaranees, i is he paricipaing life insurance policyholders who profi from he radeoff beween risk and reurn in case of defaul in erms of an increasing guaraneed ineres rae, which is opposed o wha we observed in he second row of Figure 2. However, paricipaing life insurance conracs sill exhibi a higher presen value han heir premium paymen of 100, while dynamic hybrid policyholders are again below ha value for he considered inpu parameers.

19 18 The impac of he surplus paricipaion rae on fair values and shorfall risk ooking a he impac of differen surplus paricipaion raes α in Figure 3, one can see ha he fair buffer ineres raes and he shorfall probabiliy are increasing for increasing surplus paricipaion raes, bu ha his effec is much less pronounced as in case of he guaraneed ineres rae. Furhermore, in conras o he case of increasing he guaraneed ineres rae, he presen value of he paricipaing life insurance conrac is higher when including dynamic hybrid conracs in he porfolio (second and hird row in Figure 3) and increasing for higher surplus paricipaion raes, while he presen value of he dynamic hybrid produc is decreasing (paricularly in he second row in Figure 3, G DHP = 1.0 P DHP ). This can be explained by he fac ha he mahemaical algorihm for he reallocaion of funds in case of he dynamic hybrid producs is based on he guaraneed ineres rae only, which is consan, while he surplus paricipaion rae (only) conribues o a higher reurn earned on invesmens in he policy reserves. Hence, he pariion of he dynamic hybrid funds and he oal dynamic hybrid accoun value remains almos unchanged when increasing he surplus paricipaion rae for a given dynamic hybrid guaranee (see Figure 4). he same ime, he average policy reserves of he paricipaing life insurance policies increase considerably due o he increase in α (see solid black line wih sars in Figure 4, compare lef and righ column), which, associaed wih he lower shorfall risk as compared o he higher guaraneed ineres rae (see Figure.2 in he ppendix), conribues o an advanage of he paricipaing life insurance conrac as opposed o he dynamic hybrid. Thus, while paricipaing life insurance policies are a a disadvanage in he radeoff beween higher reurn and higher risk when increasing he guaraneed ineres rae (bu sill exhibi a posiive ne presen value), hey profi in case of he annual surplus paricipaion rae. When reducing he guaranee level promised o he DHP policyholders o G DHP = 0.5 P DHP, Figure 4 shows ha as in Figure 2, almos no funds are disribued o he policy reserves, implying ha he siuaion remains almos unchanged and only slighly improved for he paricipaing life insurance policyholders (see hird row in Figure 3).

20 19 Figure 3: Presen values of he paricipaing life insurance and dynamic hybrid conracs wih corresponding insurer shorfall risk (lef column) for differen porfolio composiions and dynamic hybrid guaranees when varying he annual surplus paricipaion rae and given fair buffer ineres raes (righ column, equiyholder perspecive) P D H P =0 P D H P =0 presen value shorfall probabiliy fair buffer ineres rae b surplus paricipaion rae α surplus paricipaion rae α P D H P =100, G D H P =1.0 P D H P P D H P =100, G D HP =1.0 P D H P presen value shorfall probabiliy fair buffer ineres rae b surplus paricipaion rae α surplus paricipaion rae α P D H P =100, G D H P =0.5 P D H P P D H P =100, G D HP =0.5 P D H P presen value shorfall probabiliy fair buffer ineres rae b surplus paricipaion rae α surplus paricipaion rae α paricipaing life insurance dynamic hybrid produc shorfall probabiliy (righ axis)

21 20 Figure 4: verage pariion of he dynamic hybrid produc accoun value ino he hree funds (equiy fund, guaranee fund, policy reserve sock) over he conrac erm for differen annual surplus paricipaion raes α and differen dynamic hybrid guaranees given a fair siuaion from he equiyholders perspecive G D H P =1.0 P D HP, α=0.1 G D H P =1.0 P D H P, α=0.7 average value average value ime ime G D H P =0.75 P D H P, α=0.1 G D H P =0.75 P D H P, α=0.7 average value average value ime ime G D H P =0.5 P D HP, α=0.1 G D H P =0.5 P D H P, α=0.7 average value average value ime ime PR D H P G F EF PR P I

22 21 The impac of he porfolio composiion on fair values and shorfall risk We nex focus on he impac of differen porfolio composiions by varying he premium volume of paricipaing life insurance and dynamic hybrid conrac as shown in Figure 5 in order o idenify furher porfolio subsiuion and risk ransfer effecs. We hereby fix he oal premium volume o 200 and only vary he single premium of he dynamic hybrid conracs P DHP, such ha he premium of he paricipaing life insurances is given by P PI = 200-P DHP. Figure 5: Presen values of he paricipaing life insurance and dynamic hybrid conracs wih corresponding insurer shorfall risk (lef column) when varying he porfolio composiion (P PI = 200-P DHP ) and given fair buffer ineres raes (righ column, equiyholder perspecive) G D H P =1.0 P D H P G D H P =1.0 P D HP presen value shorfall probabiliy fair buffer ineres rae b single premiums dynamic hybrid produc P D H P single premiums dynamic hybrid produc P D H P paricipaing life insurance dynamic hybrid produc shorfall probabiliy (righ axis) Figure 5 shows ha boh conrac ypes are approximaely fair, as he presen value of he payoff approximaely equals he respecive premium. 7 Despie his fac, he shorfall probabiliy varies subsanially for differen porfolio composiions. Hence, increasing he porion of he dynamic hybrid conracs in he porfolio firs implies a decrease in he shorfall risk, unil he porion of dynamic hybrids is P DHP = 125, which consiues a minimum. Increasing he porion of DHPs above his level leads o an increase in shorfall risk. Hence, here is an opimum regarding he shorfall probabiliy in he given seing, which is due o subsiuion effecs. In paricular, he fac ha he funds of he dynamic hybrids invesed in he policy reserves (which depend on he size of he guaranee promised o he dynamic hybrid policyholders, see e.g. Figure 2) are always a leas compounded wih he same policy ineres rae credied o he paricipaing life insurance policyholders accoun 7 Noe ha in case of fair conracs, he lines of he presen values of boh producs should be exacly linear in he premium volume.

23 22 implies ha dynamic hybrid producs profi from he longerm invesmens ha are possible due o he longerm commimen of paricipaing life insurances, even hough hey are only invesed shorerm in he policy reserve. In addiion, following he mahemaical algorihm, funds are ypically shifed o he policy reserves in imes of an adverse capial marke environmen in order o ensure he guaranee promised o he dynamic hybrid conracs. Hence, as he policy ineres rae paid o he policy reserves is he same for boh conrac ypes, a leas he guaraneed rae mus be covered by he insurer, which may be especially difficul o achieve in imes of low marke ineres raes, where he guaraneed rae may even be higher han he risk-free rae (see Menzel, 2008; Sieber, 2008). This represens an opion graned o he policyholders of he DHPs and he corresponding risk should be accouned for by an insurer, whereby he value of he opion also depends on he amoun shifed o he policy reserves (and hus he guaranee promised o he dynamic hybrid policyholders in he firs place). In addiion, in case funds in he policy reserves from dynamic hybrids are no invesed shorerm bu longerm, hidden reserves may have o be realized if funds are shifed from he policy reserves o he guaranee fund or equiy fund (see Sieber, 2008). Furhermore, liquidiy risk may arise if he CPPI-based sraegy requires a frequen rebalancing, especially in imes of marke urbulences, which may imply ha asses are no as liquid as assumed (see, e.g., Rubinsein and eland, 1981). The differen guaranees and opions as well as associaed risks can be reduced by he insurer by adjusing he dynamic reallocaion procedure, inroducing, e.g., limis in regard o he amoun shifed beween guaranee fund o policy reserves and vice versa or dependen on he sock marke environmen (see Sieber, 2008). 4. SUMMRY This paper assesses he fair valuaion and risk assessmen associaed wih an insurer s porfolio ha consiss of dynamic hybrid policies and paricipaing life insurance conracs. The paper hus conribues o he lieraure by aking he insurer s perspecive in ha he porfolio ineracion effecs by he wo producs are comprehensively sudied. Toward his end, we presen a model of a life insurer who offers boh ypes of conracs. The considered 3-fund dynamic hybrid accoun value is hereby 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 algorihm ha is based on he concep of consan proporion porfolio insurance (CPPI). Our resuls emphasize ha here are srong ineracion effecs beween he wo produc ypes, especially as funds allocaed o he policy reserves earn he policy ineres rae acually deer-

24 23 mined for he paricipaing life insurance policyholders and, hence, a leas a guaraneed ineres rae. Even hough conracs are calibraed o be fair from he equiyholders perspecive, he conracs are no necessary fair for he wo life insurance producs, whose presen values srongly depend on he porfolio composiion and he choice of inpu parameers. In paricular, one main finding is ha increasing he guaraneed ineres rae on average (in erms of he presen value) implies an increase in he presen value for he dynamic hybrid policyholders, as hey profi from he radeoff beween a higher reurn and a higher shorfall risk. However, in he considered examples, heir presen value sill generally says below heir iniial premium paymen. Paricipaing life insurance policyholders, in conras, lose in he described radeoff as for increasing guaraneed raes, he presen value decreases, which is no he case wihou dynamic hybrids in he porfolio, where he presen value remains approximaely consan in case of a fairly calibraed equiyholder ineres rae. However, hey can also gain from including dynamic hybrid conracs in he porfolio in ha he presen value of heir conracs may say above heir single up-fron premium, even if i is decreasing for higher guaraneed ineres raes. Our findings also emphasize ha he ineracion effecs depend on he guaranee promised o he dynamic hybrid policyholders, which may be a money-back guaranee or less (or more). In addiion, while paricipaing life insurance policies are a a disadvanage in regard o he radeoff beween a higher reurn and a higher risk in case of increasing he guaraneed ineres rae, hey profi in case of increasing he annual surplus paricipaion rae in he sense of an increasing presen value, while dynamic hybrid policies show a reducion in he presen value of fuures payoffs. Furhermore, as for a lower fracion of paricipaing life insurance conracs in he porfolio, less longerm invesmens can be made since dynamic hybrid funds are in general no invesed longerm, he invesmen reurn generaed by he insurer s asses decreases and he shorfall risk increases, as he guaraneed rae is more difficul o be covered. However, his is only rue for low fracions of radiional life insurances in he porfolio, as including dynamic hybrid conracs can also help reducing shorfall risk, in paricular since dynamic hybrids feaure lower overall guaranees. In he examples considered in he numerical analysis, for insance, he minimum shorfall risk is approximaely achieved for an approximaely equally weighed porfolio of paricipaing life insurance and dynamic hybrid producs (in erms of he single up-fron premium). In fuure research, furher analysis is necessary regarding he inerdependencies observed in a porfolio wih differen conrac ypes, e.g. wih respec o he capial marke environmen and ineres rae dynamics, he impac of ransacion coss as well as variaions of he rebalancing mechanism, which can be used o adjus he risk-reurn profile of he dynamic hybrids. In addiion, he impac of moraliy risk and managemen rules regarding asses and profi paricipaion on porfolio effecs could be sudied o idenify furher ineracion effecs.

25 24 REFERENCES Beels, C., T. Grosner, and M. eischkis, 2011, Vorsorge: Dynamische Hybride: Chancen und Risiken für ebensversicherer, Versicherungswirschaf, 66(20): Björk, T., 2009, rbirage Theory in Coninuous Time, 3rd ed. (New York: Oxford Universiy Press). Black, F., and R. Jones, 1987, Simplifying Porfolio Insurance, Journal of Porfolio Managemen, 13(3): Bohner,., and N. Gazer, 2012, nalyzing Surplus ppropriaion Schemes in Paricipaing ife Insurance from he Insurer s and he Policyholder s Perspecive, Insurance: Mahemaics and Economics, 50(1): Grosen,., and P.. Jørgensen, 2000, Fair Valuaion of ife Insurance iabiliies: The Impac of Ineres Rae Guaranees, Surrender Opions and Bonus Policies, Insurance: Mahemaics and Economics, 26(1): Grosen,., and P.. Jørgensen, 2002, ife Insurance iabiliies a Marke Value: n nalysis of Insolvency Risk, Bonus Policy, and Regulaory Inervenion Rules in a Barrier Opion Framework, Journal of Risk and Insurance, 69(1): Kochanski, M., and B. Karnarski, 2011, Solvency Capial Requiremen for Hybrid Producs, European cuarial Journal, 1(2): eland, H. E., 1980, Who Should Buy Porfolio Insurance?, Journal of Finance, 35(2): Menzel, P., 2008, Opionen in Dynamischen Hybridproduken, Der kuar, 14(1): Rubinsein, M., and H. E. eland, 1981, Replicaing Opions wih Posiions in Sock and Cash, Financial nalyss Journal, 37(4): Sieber,., 2008, kuarielle Fragen zu dynamischen Hybridproduken, Der kuar, 14(2):

26 25 Figure.1: Number of shorfalls per period corresponding o he siuaion in Figure 2 G D H P =1.0 P D H P, r G =0.01 G DH P =1.0 P D H P, r G =0.025 number of shorfalls number of shorfalls ime ime G DH P =0.75 P D H P, r G =0.01 G D H P =0.75 P D H P, r G =0.025 number of shorfalls number of shorfalls ime ime G D H P =0.5 P D H P, r G =0.01 G DH P =0.5 P D H P, r G =0.025 number of shorfalls number of shorfalls ime ime

27 26 Figure.2: Number of shorfalls per period corresponding o Figure 4 G D H P =1.0 P D H P, α=0.1 G D H P =1.0 P D H P, α=0.7 number of shorfalls number of shorfalls ime ime G D H P =0.75 P D H P, α=0.1 G D H P =0.75 P D H P, α=0.7 number of shorfalls number of shorfalls ime ime G D H P =0.5 P D H P, α=0.1 G D H P =0.5 P D H P, α=0.7 number of shorfalls number of shorfalls ime ime

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