On the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes

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1 On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen of Insurance Economics and Risk Managemen Friedrich-Alexander-Universiy (FAU) of Erlangen-Nürnberg Version: November 214

2 ON THE MANAGEMENT OF LIFE INURANCE COMPANY RIK BY TRATEGIC CHOICE OF PRODUCT MIX, INVETMENT TRATEGY AND URPLU APPROPRIATION CHEME Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen This version: November 4, 214 ABTRACT The aim of his paper is o analyze he impac of managemen s sraegic choice of asse and liabiliy composiion in life insurance on shorfall risk and he shareholders fair risk charge. In conras o previous work, we focus on he effeciveness of managemen decisions regarding he produc mix and he riskiness of he asse side under differen surplus appropriaion schemes. We propose a model seing ha comprises emporary life annuiies and endowmen insurance conracs. Our numerical resuls show ha he effeciveness of managemen decisions in regard o risk reducion srongly depends on he surplus appropriaion scheme offered o he cusomer and heir impac on guaraneed benefi paymens, which hus presens an imporan conrol variable for he insurer. Keywords: Paricipaing life insurance, surplus disribuion, risk-neural valuaion, managemen mechanisms Alexander Bohner and Nadine Gazer are a he Friedrich-Alexander-Universiy (FAU) of Erlangen- Nuremberg, Deparmen of Insurance Economics and Risk Managemen, Lange Gasse 2, 943 Nuremberg, Germany, Peer Løche Jørgensen is a Aarhus Universiy, Deparmen of Economics and Business, Fuglesangs Allé 4, Denmark, The auhors would like o hank Michael herris, Helmu Gründl, and he paricipans of he 7h Conference in Acuarial cience & Finance on amos, he 16h Inernaional Congress on Insurance: Mahemaics and Economics in Hong Kong, he 39h seminar of he European Group of Risk and Insurance Economiss, he AFIR/ERM Colloquia 212 in Mexico Ciy as well as he ARIA Annual Meeing 214 in eale for valuable commens on an earlier version of he paper.

3 2 1. INTRODUCTION Managemen decisions regarding asse and liabiliy composiion can considerably impac a life insurer s risk siuaion and also he fair risk-adjused compensaion for he company s shareholders. Decisions can relae o various facors, including a dynamic adjusmen of he porion invesed in high-risk asses, he porfolio composiion on he liabiliy side as well as he ype of surplus appropriaion scheme, which a he same ime influences he exen of he long-erm guaranees ypically embedded in hese conracs. Life insurance conracs in many European counries conain a legally enforced paricipaion mechanism hrough which policyholders paricipae in he company s surplus. This surplus paricipaion represens an imporan facor in compeiion beween insurers and is paid in addiion o a guaraneed ineres rae ha is annually credied o he policyholder s accoun. In addiion, i is no only he absolue amoun of surplus disribued o he policyholders ha has an effec on shorfall risk; he concree way in which disribued surplus is credied o he policyholders also has a considerable influence on he value of he surplus paricipaion par of he conracs (see Bohner and Gazer, 212). These so-called surplus appropriaion schemes also impac he risk profile of he insurance company due o heir varying characerisics of urning surplus ino guaranees. Policies may feaure various appropriaion schemes. In case of an endowmen insurance conrac, for insance, surplus is used o increase he deah as well as he survival benefi, while ineres-bearing accumulaion increases he survival benefi only (and keeps he deah benefi consan). In case of an annuiy conrac, surplus can be used o increase he annual annuiy paymens unil mauriy, or surplus can be direcly paid ou o he policyholders in he corresponding period (direc paymen scheme). Anoher imporan conrol variable besides he surplus appropriaion scheme is he mixure of he produc porfolio, e.g., he percenage of annuiies and life insurance conracs ha a company sells, which impacs liabiliies and asses alike due o he differen iming and amoun of cash in- and ouflows. In addiion, a dynamic pah-dependen asse sraegy can be implemened regarding he riskiness of he asse porfolio o improve he insurer s solvency siuaion, as asses can be more easily adjused over he conrac erm as compared o he liabiliy side. The managemen of asses and liabiliies for a life insurer wih various produc porfolios including a deailed modeling of surplus appropriaion schemes can have an imporan impac on he company s shorfall risk as well as on he risk-adjused compensaion for shareholders. Therefore, he aim of his paper is o examine his issue in more deph, hereby ensuring a fair siuaion for shareholders.

4 3 In he lieraure, various papers examine paricipaing life insurance conracs including surplus disribuion mechanisms and ineres rae guaranees, focusing on differen aspecs. The radiional acuarial surplus managemen focuses on balancing conservaism and fairness (also wih respec o he equiyholders) of surplus disribuion schemes and has been discussed since as early as 1863 by Homans (1863) and by Cox and orr-bes (1963). In he curren lieraure, one aspec of special ineres has been risk-neural valuaion, which has been researched by, amongs ohers, Briys and de Varenne (1997), Dong (211), Grosen and Jørgensen (2, 22), Hansen and Milersen (22), Guillén, Jørgensen, and Nielsen (26), Kling, Ruez, and Russ (211), Tanskanen and Lukkarinen (23), iu (25), chmeiser and Wagner (211), and Goecke (213). In addiion, several papers have focused on combining risk pricing and risk measuremen, including Gazer and Kling (27) Kleinow and Willder (27), and Gazer (28). Kling, Richer, and Russ (27a, 27b) analyze surplus disribuion schemes and heir effec on an insurer s risk siuaion while in Bohner and Gazer (212) differen surplus appropriaion schemes in paricipaing life insurance are analyzed for he firs ime from he policyholders and he insurer s perspecives encompassing moraliy and financial risk, hereby also sudying he impac on defaul risk. Wih respec o managemen discreion, Kleinow and Willder (27) and Kleinow (29) analyze he impac of managemen decisions on hedging and valuaion of paricipaing life insurance conracs, while Gazer (28) examines differen asse managemen and surplus disribuion sraegies for paricipaing life insurance conracs. A general asse-liabiliy managemen framework for life insurance is provided by Gersner e al. (28) ha allows he company o conrol he asse base, he bonus declaraion mechanism and he shareholder paricipaion. Furhermore, Huang and Lee (21) deal wih he opimal asse allocaion for life insurance policies adoping a muli-asse reurn model ha uses approximaion echniques. The opimal porfolio composiion for immunizing a life insurer s risk siuaion agains changes in moraliy has been sudied in Gazer and Wesker (212) wih a focus on endowmen insurance conracs and annuiies, bu wihou including surplus disribuion mechanisms or dynamic asse managemen sraegies. Inspired by he producs on he Danish marke, Guillén e al. (213a, 213b) sudy he performance of Danish wih-profi pension producs and life cycle producs, where hey also accoun for managemen decisions such as asse managemen sraegies. Thus, while asse-liabiliy managemen, porfolio composiion and managemen rules in general have been researched previously, he effeciveness of managemen decisions regarding he asse and liabiliy composiion for differen surplus appropriaion schemes has no been

5 4 examined so far, even hough surplus appropriaion schemes play a cenral role in radiional life insurance and can subsanially impac shorfall risk and shareholder value due o heir consequences for he long-erm guaranees promised o policyholders. One major quesion is, herefore, how surplus appropriaion schemes of differen producs impac he effeciveness of managemen discreion regarding pah-dependen asse managemen sraegies and produc composiions on he liabiliy side. uch an analysis will provide imporan insighs in regard o he managemen of long-erm guaranees induced by surplus appropriaion schemes as well as complex ineracions beween asses and liabiliies in life insurance and heir effec on risk and a fair shareholder posiion. Therefore, in his aricle, we exend previous lieraure by analyzing he effeciveness of managemen decisions regarding he asse and liabiliy composiion for a life insurance company selling endowmen insurance conracs and annuiies under differen surplus appropriaion schemes on he company s shorfall risk and on he fair compensaion of shareholders. Toward his end, we provide a model seing including he wo life insurance producs wih differen surplus appropriaion schemes. The smoohing surplus disribuion scheme considered in he model is hereby similar o he mechanisms ha have been used in, e.g., Denmark for a long ime, implying ha many imporan managemen decisions are now aken on he basis of his ype of models. On he liabiliy side, we consider he impac of he porfolio composiion, hereby always ensuring a fair risk charge for shareholders. On he asse side, he effeciveness of managemen rules ha modify he riskiness of he invesmen is sudied, i.e., where funds are dynamically shifed from socks o bonds o reduce volailiy and vice versa using a consan proporion porfolio insurance (CPPI)-based invesmen sraegy. These asse feedback mechanisms can have an impac on he overall amoun of generaed surplus and hus also on he policyholders surplus paricipaion and he induced increase in guaraneed benefis, which may imply complex ineracions. The remainder of he paper is organized as follows. ecion 2 inroduces he model framework of he insurance company, along wih he managemen decisions and he surplus appropriaion schemes. Numerical resuls are presened in ecion 3 and ecion 4 concludes. 2. MODEL FRAMEWORK In wha follows, we consider a life insurer ha offers wo producs: emporary annuiies and paricipaing life insurance conracs (also referred o as endowmen conracs) wih differen surplus appropriaion schemes. We make use of he model framework inroduced in Bohner

6 5 and Gazer (212) for surplus appropriaion schemes in paricipaing life insurance and expand heir seing in various ways. In paricular, we propose various company seups, where he produc porfolio composiion, surplus appropriaion and asse sraegies can be sudied ha are defined a incepion of he conracs. The insurer s balance shee a ime is laid ou in Table 1. Table 1: Balance shee of a life insurance company a ime Asses A A Liabiliies E R PR PR IA B A A he beginning of he firs conrac year ( = ), equiyholders make an iniial conribuion of E = l A and he colleciviy of policyholders pay single premiums of ( 1 l) A. 1 The book values of he policy reserves for he annuiies and he radiional endowmen insurance conracs are given by R PR and PR, respecively and IA denoes he book value of he endowmen insurance conracs ineres-bearing accumulaion sysem. The buffer accoun B is deermined residually by subracing equiy, he policyholders accouns and dividends paid o he equiyholders from he marke value of he asses (A), where equiy (E) is assumed o be consan over ime (see also Kling, Richer, and Russ (27a, 27b)). 2 Furhermore, a run-off scenario wihou new business is considered. 1 The iniial equiy capial E is se equal in case of annual premium paymens for comparabiliy reasons. 2 Thus, B is a hybrid, since i is he difference beween marke and book values. This is a simplificaion of he acual procedures in an insurance company (see Grosen and Jørgensen, 2).

7 6 2.1 The liabiliy model The insurance conracs The company s range of producs comprises emporary annuiy and endowmen insurance conracs wih a conrac erm of T years. 3 We assume ha a oal number of N conracs is sold, which is disribued among life insurance and annuiy holders, such ha ( ϕ ) R N = ϕ N, N = 1 N, where ϕ is he percenage of annuiy conracs. The annuiies are sold agains single premiums, whereas he endowmen conracs are sold agains single premiums as well as agains annual premiums. We consider pools of conracs ha are acuarially priced based on moraliy ables. Thus, he single (ne) premiums for he emporary annuiy and he endowmen conrac for an individual policyholder are given by P R single = R 1 a, and P x: T single A 1 x : T =, respecively, where R1 denoes he iniially guaraneed annual annuiy paymen in case of survival (wihou any surplus) and 1 denoes he iniially guaraneed sum insured in he case of deah or survival, boh paid in arrears. The corresponding consan annual (ne) premium for he endowmen conrac (paid in advance) is given by annuiizing he single premium, resuling in A P ä x: T = 1. x: T The acuarial presen value of an endowmen insurance wih a sum insured of one and a conrac erm of T years ( A ) for an individual x-year old policyholder and he presen value of x: T 3 In Germany, for insance, endowmen insurance conracs and annuiies ogeher accoun for more han 6% of he premium volume in life insurance and hus represen a major produc design in he life insurance secor (see GDV, 213, Table 34). An endowmen policy is a classical savings produc and ypically feaures a conracually defined duraion afer which a lump sum is paid ou o he policyholder in case of survival. For comparabiliy reasons, we furher consider a emporary annuiy wih a conrac erm ha is idenical o he endowmen policies raher han a lifelong annuiy. The effecs shown for hese conracs ypes are also of relevance for oher ypes of paricipaing life insurance producs wih differen surplus paricipaion schemes.

8 7 an immediae emporary annuiy for T years wih an annual annuiy paymen of one in advance ( ä ) and in arrear ( a x: T : x T ) is given by T T A = v x: T px qx+ + v T px =, T 1 ä = v x: T px = T, and a = x: T v px, (1) = 1 G where v ( 1 r ) 1 = + describes he discoun facor using an acuarial ineres rae of r G. The probabiliy of an x-year old insured person surviving years is given by p x, while q x + saes he probabiliy of dying wihin one year for a policyholder aged x+. The moraliy probabiliies ha we use for pricing and reserving are based on he moraliy ables by he German Acuarial Associaion ha include a safey loading (firs-order moraliy basis). For annuiies, moraliy probabiliies of he able DAV 24 R are used, whereas he able DAV 28 T is applied for he endowmen insurance holders. To simulae he number of deahs in our laer numerical analysis, we use he second-order moraliy basis of he corresponding able, i.e. he ables underlying bes esimaes. 4 An illusraion of he evoluion of cash flows resuling from he insurance producs over ime is given in Figure 1. Figure 1: Developmen of cash flows from he insurance producs over ime ( denoes December 31 s and + denoes January 1 s for each year) T T + ime x R1 x + 1 R x + x + T - 1 RT-1 RT x+t cohor specific age annuiy paymen R P single P single P 1 P P T-1 P T sum insured paymen single premium annuiy single premium endowmen annual premium endowmen D1 D DT-1 DT dividend for equiyholders 4 Noe ha one could alernaively also use moraliy daa from oher counries as he model iself is generic. The DAV ables are he curren moraliy ables provided by he German Acuarial Associaion ( DAV ). The DAV 24 R able is based on German moraliy daa for annuians and includes safey loadings ha accoun for model risk as well as he risk of a long-erm change in he moraliy rend. The DAV 28 T is deermined based on moraliy daa of hose insured on endowmen conracs, erm life insurance and unilinked policies. Here, safey loadings include he risk of random flucuaions, model risk and parameer risk, while he risk of a change in he moraliy rend is negleced. Noe ha he considered ables wihou he safey loadings (bes esimaes) are no idenical in case of he ables DAV 24 R and DAV 28 T, since he corresponding cohors ha are used for consrucing he ables differ. For furher deails, see

9 8 Noe ha he premium paymen(s) are consan, while he benefis from he conracs vary over ime depending on he company s generaed surplus and he seleced surplus appropriaion scheme. Policy reserves The policy reserves for he annuiy and endowmen insurance conracs are calculaed on he same acuarial basis as he premiums. The policy reserves for he pool of annuiy policies (j = R) and he pool of endowmen insurance conracs (j = ) a he end of year are given by j j = j j PR N di V x, j = R,, (2) i= 1 where and j i V represens he acuarial reserve for an individual annuiy or endowmen conrac j x d specifies he number of deahs (of year i) from he cohor of he iniially sold conracs (N j ), which is deermined based on he bes esimaes of he corresponding moraliy able, i.e., he moraliy ables wihou safey loadings. 5 The prospecive calculaion for he acuarial reserve for an x+-year old insured a ime for an individual annuiy conrac is hereby deermined by V = R a, (3) R x + 1 x+ : T and for an endowmen conrac, he individual acuarial reserve is calculaed by V = A P ä, (4) x + 1 x+ : T x+ : T using he acuarial presen values saed in Equaion (1), where R+1 is he guaraneed annuiy ha is paid a he end of year (a ime +1) in case of survival, +1 denoes he guaraneed sum insured (ha is paid ou if deah occurs during year, i.e., from ime unil ime +1, or if he insured survives unil mauriy) and P indicaes he -h premium paymen. In case of a 5 For simplificaion purposes, we refrain from modeling sysemaic longeviy, bu we do ake ino accoun he difference beween anicipaed moraliy including safey loadings (for pricing and reserving) and realized moraliy. This also conribues o he naural generaion of surplus and is ypically conduced in life insurance.

10 9 single premium, P = P and P =, = 1,,T, whereas P = P for a consan annual premium. single Buffer accoun urplus ha has already been generaed, bu no ye been disribued o he policyholders and has hus no been ransformed ino guaranees ye, is saved in he buffer accoun. Funds in his accoun belong o he policyholders bu are no guaraneed, as hey can be used o compensae losses in years of low asse reurns. A he end of year, he buffer accoun is residually deermined by B = A PR PR IA E. (5) R A he end of he las year, he value of he buffer accoun (minus dividend paymens) deermines he erminal bonus (TBT) paid o policyholders, which canno become negaive and is given by 6 R ( ) ( ) TB = max B D, = max A PR PR IA E D,. T T T T T T T T T 2.2 The asse model We assume ha he invesmens on he asse side (I) evolve according o a geomeric Brownian moion, which is given by di = µ I d + σ I dw, P wih an asse drif µ, 7 volailiy σ and a P-Brownian moion W P defined on he probabiliy space ( ΩF,, P). The soluion of his sochasic differenial equaion is given by (see Björk, 29) 6 We hereby ake ino accoun diversificaion effecs beween he pool of endowmen insurance conracs and annuians bu, as in Bohner and Gazer (212), we do no focus on subsiuion effecs across generaions. Analyses on generaional subsiuion effecs can be found in Døskeland and Nordahl (28) and Faus, chmeiser, and Zemp (212), for insance. 7 Under he risk-neural pricing measure Q, he asse drif changes o he risk-free rae r f, and he sochasic Q differenial equaion is hus given by di = r I d + σ I dw wih a Q- Brownian moion W Q. f

11 1 ( µ σ 2 2+ σ ε ) r I = I e = I e, ( ) ( ) 1 1 wih independen sandard normally disribued random variables ε and a coninuous oneperiod reurn r. We furher assume ha he oal asse base is composed of socks and bonds wih a sock raio a. To accoun for differen sock raios in he porfolio, we consider an adjused reurn wih corresponding drif and volailiy for he aggregae asse porfolio, which saisfies ( 1 ) r = a r + a r, B wih coninuous one-year reurns of socks r and bonds rb, corresponding volailiies and 2 drifs σ and σb, expeced values m and mb ( m = µ σ 2, i = B, ) and a correlaion i i i coefficien ρ. A he beginning of he firs year, he iniial asse value is composed of he equiy capial and he (firs) premium paymens, i.e., R R A = P single N Psingle N E, (6) in he case of single premiums for boh conrac ypes and A =. In case of annual premiums for he endowmen insurance, he corresponding single premium ( P single ) has o be replaced by he firs consan level premium P in Equaion (6). During he conrac erm, annual annuiy paymens R have o be paid ou o hose annuians sill alive a ha ime and deah benefis are paid o he heirs of he policyholders who died during he conrac year. Hence, he asse base a he end of year is given by r R R A = A + e R ( 1) N d i d. (7) i= 1 A he end of year, which is assumed o be equal o he accouning dae, wo cases have o be disinguished for he furher developmen of asses and liabiliies. Firs, in case he insurer is solven and asses are sufficien o cover he liabiliies, i.e., R A PR + PR + IA, being equivalen o B + E, a consan fracion β of he equiyholders iniial conribuion is paid ou o as annual dividend paymens D, 8 i.e., 8 If he insurer is solven bu does no have enough reserves o pay he dividends, i.e., if B + E, bu B < D, hen D =. If he buffer accoun becomes negaive, bu equiy capial is sufficienly high o cover he losses in his period, i.e., B <, bu B + E, he insurer sill remains solven. Here, equiy capial

12 11 D = β E if B D., This leads o B + = B D and resuls in an asse base a he beginning of year +1 of (see also Equaions (6) and (7)) + = + A A D P N d i. i= 1 The las summand denoes he annual premium paymens for he endowmen conracs, which are se o zero in he case of single premiums. econd, in he case of an insolvency, liabiliies canno be covered by asses, i.e., A < PR R + PR + IA and equivalenly B + E <, he company is liquidaed by disribuion of he remaining asses less bankrupcy/liquidaion coss c A e c o he policyholders who are sill acive. r + ( 1 ) ( 1) 2.3 urplus disribuion and appropriaion Wih respec o surplus appropriaion, hree schemes are considered. For he annuiy, he direc paymen scheme and he bonus sysem are used, whereas for he endowmen insurance, he bonus sysem and he ineres-bearing accumulaion are applied based on he model in Bohner and Gazer (212). While surplus is used o increase he iniial annuiy and guaraneed sum insured in case of he bonus sysem, surplus is saved on a separae accoun in case of he ineres-bearing accumulaion or direcly paid ou o he annuians in case of he direc paymen scheme. Thus, in addiion o he calculaory ineres rae r G (see Equaion (1)), which has o be credied o he policy reserves annually and which hus consiues a guaraneed ineres rae, he P policy ineres rae r ha includes surplus in addiion o he guaraneed ineres rae is deermined using he reserve-based approach shown in Grosen and Jørgensen (2), 9 is reduced by he amoun of he loss and he buffer accoun is se o zero. In he nex year, he amoun of equiy capial is increased again o he original amoun by using gains from he nex period (see Equaion (5)). 9 Tradiional paricipaing life insurance conracs wih heir surplus disribuion and appropriaion schemes make use of he process of collecive saving (in conras o individual saving as in case of uni-linked policies, for insance). An deailed analysis of reurn smoohing mechanisms is provided in Guillén, Jørgensen, and Nielsen (26), while he meris of collecive saving has been addressed in Goecke (213), for insance.

13 12 r B + G ( 1) = max r, α γ, (8) PR + PR + IA ( 1) ( 1) ( 1) P R wih a arge buffer raio γ, i.e., he raio of he free surplus or buffer divided by he liabiliies belonging o he policyholders and a surplus disribuion raio α, which conrols he exen of surplus ha is disribued o he policyholders. The model proposed by Grosen and Jørgensen (2) has cerain aspecs in common wih an approach suggesed in a repor by he Danish Financial upervisory Auhoriy (1998). In paricular, he idea o deermine bonus (in excess of a guaraneed rae of reurn) as a fracion of an available buffer is a common characerisic of he wo approaches and his is wha has been long-erm pracice in Denmark, for insance. The oal amoun of surplus for an individual conrac in he -h year is derived based on he individual reserves and defined by 1 PR r r, j = R,. j P G ( ) ( ) 1 Based on his surplus disribuion approach, differen appropriaion schemes are applied, which have an impac on he overall dynamics of asses and liabiliies. urplus appropriaion: The bonus sysem for annuiy and endowmen insurance The bonus sysem uses he annually disribued surplus amoun as a single premium o increase he annuiy R1 and he iniially guaraneed sum insured 1 for he res of he conrac erm. For an individual annuiy conrac, he addiional annuiy is calculaed by R = R P G R R ( ) ( 1) ( i= 1 i ) PR r r N d a x+ : T, (9) and for an endowmen insurance, he addiional sum insured is given by 1 This is a ypical approach o model surplus disribuion when guaraneed ineres raes are in place (see, e.g. Grosen and Jørgensen, 2). A model for disribuing surplus o policyholders and equiyholders wih specific characerisics of German regulaion can be found in Maurer, Rogalla, and iegelin (213).

14 13 = P G ( ) ( 1) ( i= 1 i ) PR r r N d A x+ : T, (1) which resuls in an increased annuiy and sum insured, respecively, of R = 1 R + + R and = In his seing, he surplus credi o he policyholders reserves also paricipaes in fuure surplus and is compounded a leas wih he guaraneed ineres rae, hus inducing cliquesyle ineres rae effecs (see Equaions (3)-(6) and (7)). urplus appropriaion: Direc paymen scheme for he annuiy The annuiy s direc paymen direcly pays ou surplus o he policyholders in addiion o heir originally guaraneed annuiy in he subsequen year. In conras o he bonus sysem, he annual surplus amoun per annuian only increases he nex annuiy paymen, i.e., R P G R R ( ) ( 1) ( i ) R R PR r r N d = +, = 1 i while he annuiies afer his addiional paymen are no affeced by he surplus, bu migh again be increased by single surplus paymens in he following years. urplus appropriaion: Ineres-bearing accumulaion for he endowmen conrac In case of he endowmen insurance s ineres-bearing accumulaion, surplus is accumulaed on a separae accoun IA, comparable o a bank accoun ha is paid ou o he policyholder a mauriy in case of survival. In case of deah during he conrac erm, funds are ransferred o he buffer accoun and hus evenually o he colleciviy of policyholders. The policyholders heirs only receive he sum insured of 1, which is consan hroughou he conrac erm. The ineres-bearing accumulaion accoun a he end of year (including new surplus) earns an IA ineres rae r and is given by ( i ) ( ) ( ) 1 ( ) ( ) ( i= 1 ) IA = IA 1+ r 1 d N d + PR r r, IA =. IA P G 1 1

15 Managemen decisions regarding asses and liabiliies The managemen of he insurer has several opions for conrolling he asse and liabiliy side in order o posiively influence he insurer s risk siuaion or shareholder value. The insurance company s risk profile can for insance be alered by means of he produc porfolio composiion by seing he fracion ϕ of annuiies (and endowmen conracs 1 - ϕ). The liabiliy side can furher be conrolled by employing a specific ype of surplus appropriaion scheme for boh producs, which aler he implied guaraneed benefis. Regarding he asse side, a pah-dependen adjusmen rule of risk-relevan conrol variables can be implemened. In he following, we consider a dynamic CPPI-based feedback mechanism, where he sock porion a ime is given by R A + PR PR IA ( 1) ( 1) ( 1) a + = min max m,, a A + max, (11) where m is a muliplier ha conrols he exen o which asses are shifed owards he risky invesmen and amax represens he maximum sock porion allowed. The iniial sock porion is denoed by a. The nominaor hereby represens a buffer beween he liabiliies and he asses available o cover he liabiliies. The lower he buffer becomes, he less is invesed in risky asses and vice versa. A more dynamic approach has recenly been inroduced by Guillén e al. (213b) using an alernaive feedback mechanism, where he sock porion a ime depends on he shorfall probabiliy, which is given by he probabiliy ha he buffer raio falls below a criical level. In paricular, he sock porion is maximized while ensuring ha he shorfall probabiliy does no exceed a cerain hreshold in he subsequen period (e.g. according o a solvency level of 99.9%). This approach could be an ineresing furher developmen for fuure models. Noe ha oher feedback mechanisms are possible as well and ha he focus of his analysis is no on finding an opimal asse allocaion sraegy, bu raher on sudying he general impac and effeciveness of a feedback mechanism ha depends on an insurer s risk siuaion wih respec o differen producs including differen ways of crediing surplus o policyholders. We hus focus on sudying a se of relevan managerial conrol variables and heir fundamenal inerplay in an insurance company and do no sudy how managerial discreion is affeced when facing pressure from cusomer needs or when decisions dynamically depend on shorfall risk as is done in Guillén e al. (213b), for insance, which we leave for fuure research.

16 Risk assessmen and fair valuaion from he shareholders perspecive To deermine he impac of managemen decisions regarding he riskiness of he asse invesmen, he porfolio composiion and various surplus appropriaion schemes, we calculae he life insurer s shorfall risk. A shorfall of he company occurs if he value of he asses R falls below he value of liabiliies, A < PR + IA + PR (or, equivalenly, if B + E < ). Hence, he shorfall probabiliy under he real-world measure P is given by ( ) P = P T T, s R where he ime of defaul is defined as { } T = inf : A < PR + IA + PR, = 1,..., T. s A To ensure a fair siuaion from he shareholders perspecive, he consan dividend rae β is calibraed such ha he value of he paymens o he shareholders (dividends D and final paymen ET) is equal o heir iniial conribuion E, which is calculaed using risk-neural valuaion, i.e., 11 T Q rf T rf E = E e ET + e D = 1 T Q rf T rf = E e min { E, E + B } 1{ T > T} + e β E 1 { T }. T > = 1 (12) If he buffer accoun is nonnegaive a mauriy, i.e., here is no previous defaul, ET = E. However, if he buffer accoun becomes negaive a mauriy bu equiy capial is sufficien o cover hese losses (and, hus, he insurer sill remains solven), equiy capial is reduced by he amoun of he loss, i.e., ET = E + B if B <. T T 3. NUMERICAL ANALYI In wha follows, numerical resuls are presened based on he previously inroduced model wih a focus on analyzing he insurer s risk exposure for fair dividend raes. Afer presening he inpu parameers, we nex sudy he general impac of surplus appropriaion schemes for he wo producs, i.e., he annuiy and he endowmen insurance, on he fair dividend and he corresponding company s shorfall probabiliy. ubsequenly, we analyze o wha exen deci- 11 ince we use moraliy ables ha include safey loadings in order o price he life insurance producs and hus deviaions in moraliy are priced in, Equaion (12) mainly refers o financial risk.

17 16 sions wih regard o managemen rules can reduce he shorfall risk. Numerical resuls are derived using Mone Carlo simulaion based on 1, Lain hypercube samples (see Glasserman, 21). Inpu parameer The underlying policies are annuiies issued o xr = 6 year old males and paricipaing life insurance conracs issued o x = 35 year old males, boh wih a conrac erm of T = 3 years. The iniial annual annuiy is se o one and he acuarial presen values of he benefis for he endowmen insurance and he annuiy (per insured) are equal in order o ensure comparabiliy beween he differen cases considered. Thus he acuarial annual premium for one endowmen conrac is given by P =.88 and he corresponding single premium is R P = P = 18.83, which is equal o he single annuiy premium (due o he calibraion of single single he iniial guaraneed deah benefi). According o his, he iniial sum insured for he endowmen insurance is 1 = The acual daes of deah are simulaed using he inverse ransform mehod based on he moraliy ables of he German acuarial associaion using he DAV 28 T and he DAV 24 R ables of second-order ( bes esimaes wihou safey loadings) for a oal number of N = 1, conracs sold. Assumpions abou he evoluion of he asses are based on he hisorical performance (1988 unil 29) of wo represenaive German oal reurn indices as given in Bohner and Gazer (212). 12 The esimaion for he socks, which is based on monhly daa for he German sock marke index DAX, resuls in an expeced one-year reurn m = 8.% and a volailiy σ = 21.95%. The esimaion for he bonds, which is based on monhly daa for he German bond marke index REXP, leads o an expeced one-year reurn of bonds mb = 6.2% and a volailiy of bonds σb = 3.3%. The esimaed correlaion coefficien of reurns of he wo indices is ρ = To iniiae he CPPI-based feedback mechanism, he iniial sock porion is se o 1%, which is hen immediaely adjused depending on he size of he free buffer (see Equaion (11)). Furhermore, we assume he surplus disribuion raio o be α = 7% and he arge buffer raio o be γ = 1%. The iniial equiy capial E is se o 1% of he oal iniial capial A The parameers are 12 For calibraing parameers, German marke daa is used for illusraion, bu he analysis and he general resuls and ineracion effecs are also relevan o oher insurance markes wih life insurance conracs wih surplus disribuion mechanisms (e.g. Denmark). 13 The correlaion coefficien is significan a a level of This assumpion is based on he equiy capial o balance shee raio of approximaely 1% as in case of he German life insurer Allianz for he year 21 (see and has also been subjec o robusness checks.

18 17 chosen for illusraion purposes only and were subjec o sensiiviy analyses. 15 Unless saed oherwise, we assume furher relevan parameers o be hose saed in Table 2. Table 2: Parameers for he analysis Expeced one-period reurns of socks m 8.% Volailiy one-period reurns of socks σ 21.95% Expeced one-period reurns of bonds m B 6.2% Volailiy one-period reurns of bonds σ B 3.3% Correlaion beween socks and bonds ρ Guaraneed ineres rae r G 1.75% Rae of ineres for he ineres-bearing accumulaion accoun r IA % Risk-free rae r f 3% Number of conracs sold N 1, Annual annuiy paymen in = R 1 1 um insured for he endowmen in = ingle premium for he annuiy P R Alernaive single premium for he endowmen P single Level premium for he endowmen P.88 Equiy in = (1% of oal iniial capial) E 19,2 Conrac erm T 3 Annuiy policyholders age in = x R 6 Endowmen policyholders age in = x 35 Disribuion raio α 7% Targe buffer raio γ 1% Reducion coefficien for coss of insolvency c 2% Muliplier m 1 The impac of surplus appropriaion schemes on he effeciveness of managemen decisions in regard o shorfall risk We firs focus on he impac of he choice of he respecive surplus appropriaion scheme on he effeciveness of asse managemen sraegies in regard o reducing shorfall risk for differen porfolio composiions on he liabiliy side. Resuls are displayed in Figure 2 for a (maximum) sock porion of 25% (lef column) and 1% (righ column). All numerical examples are based on fairly calibraed dividend raes β o ensure an adequae compensaion for shareholders (see Figure A.1 in he Appendix). 15 Noe ha when using a fixed sock porion, we compare wo cases wih 1% and 25% for illusraion purposes, which are realisic assumpions for sock porions (shares held direcly or in funds) for insurers operaing in counries belonging o he OECD (see OECD, 214). In Germany, sock porions are currenly considerably lower and around 3 o 4% (see GDV, 213). However, German insurers also currenly aim o increase heir average sock porions due o he low ineres rae levels and insufficienly available oher invesmen opporuniies.

19 18 The firs row (Figure 2a) conains resuls for porfolios of emporary annuiies and endowmen insurance conracs, boh wih he more risky bonus sysem, whereas he second row shows he case where he annuiies are equipped wih he direc paymen scheme and he endowmen conracs feaure he ineres-bearing accumulaion, i.e. he second row represens he less risky scheme in each case. 16 On he x-axis, a porion ϕ = 1 represens a pool of conracs wih 1% annuiies and for ϕ =, one obains a porfolio enirely composed of endowmen conracs. In addiion o he resuls for varying produc porfolios, he shorfall risk is displayed wihou and wih including he CPPI-based asse mechanism. Is effeciveness for reducing he firm s shorfall risk (in percen) is given on he righ verical axis. Figure 2 illusraes ha he ype of surplus appropriaion scheme plays an imporan role wih respec o he insurer s risk siuaion, 17 especially for higher sock porions and hus a higher asse risk. Figure 2a (lef graph) shows ha he bonus sysem leads o a considerably higher shorfall risk as compared o Figure 2b wih he endowmen conracs ineres-bearing accumulaion and he annuiies direc paymen scheme in he case wihou a feedback mechanism and for higher porions of endowmen conracs in he porfolio due o heir clique-syle ineres rae guaranee effecs. This is paricularly eviden for a maximum sock porion of 25% (compare lef graphs in Figures 2a and 2b), where he bonus sysem increases he defaul risk by more han 2% as compared o he ineres-bearing accumulaion scheme. This is due o he higher expeced reurns associaed wih a higher sock porions ha (on average) increase he surplus amoun and hus he guaraneed deah and survival benefis, while in he case of he ineres-bearing accumulaion, only he guaraneed survival benefi is increased. In he case of an annuiy porfolio (ϕ = 1) and lower sock porions (e.g., 1%, see righhand graphs), he difference beween he wo surplus appropriaion schemes is less disinc. This is in line wih he fac ha reurn guaranees ha vary for differen surplus appropriaion schemes are generally more valuable when he invesmen process is more volaile. This observaion is highly relevan in ha i indicaes ha in he considered seing, surplus appropriaion schemes of differen ypes of producs mainly impac shorfall risk heavily if he asse invesmen is oo risky. Given he curren rend of insurers of invesing more in risky asses 16 We direcly compare he siuaion wih he more and he less risky surplus appropriaion scheme for boh producs, as we are ineresed in he impac of he surplus schemes on he ineracion beween boh producs. The impac of each scheme for each ype of produc individually is sudied in Figure 3 in he following analysis. 17 This is consisen wih he findings in Bohner and Gazer (212), where, however, focus was no laid on annuiies or heir surplus appropriaion schemes nor on he ineracion beween hese producs.

20 19 (socks or credi risky securiies) due o unaracive alernaives, he impac of he surplus appropriaion schemes should be carefully moniored by insurers. Figure 2: horfall probabiliy for various porfolio composiions consising of endowmen conracs and emporary annuiies for differen surplus appropriaion schemes wihou and wih asse feedback mechanism 18 a) Bonus sysem for boh conrac ypes sock porion a = 25% endowmen: bonus sysem, annuiy: bonus sysem sock porion a = 1% endowmen: bonus sysem, annuiy: bonus sysem shorfall probabiliy % -74% -72% -7% -68% -66% risk reducion shorfall probabiliy % -8% -6% -4% -2% % risk reducion % 2% 4% 6% 8% 1% porion of annuiies ϕ % 2% 4% 6% 8% 1% porion of annuiies ϕ b) Endowmen wih ineres-bearing accumulaion and annuiies wih direc paymen sock porion a = 25% endowmen: in.-bear. accum., annuiy: direc paymen sock porion a = 1% endowmen: in.-bear. accum., annuiy: direc paymen shorfall probabiliy % -74% -72% -7% -68% -66% risk reducion shorfall probabiliy % -8% -6% -4% -2% % risk reducion % 2% 4% 6% 8% 1% porion of annuiies ϕ % 2% 4% 6% 8% 1% porion of annuiies ϕ wihou feedback mechanism wih feedback mechanism risk reducion (righ axis) Noes: In he case wihou he feedback mechanism, he sock porion is kep consan a a = 25% (lef graphs) and a = 1% (righ graphs) hroughou he conrac erm. When applying he feedback mechanism, he specified sock porion denoes he maximum sock porion a max given in Equaion (11). 18 Noe he difference in scales when comparing he graphs in he lef and righ column.

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