Early Default Risk and Surrender Risk: Impacts on Participating Life Insurance Policies

Transcription

1 Early Defaul Risk and Surrender Risk: Impacs on Paricipaing Life Insurance Policies Chunli Cheng Jing Li November 2016 Absrac We sudy he risk-neural valuaion of paricipaing life insurance policies wih surrender guaranees when an early defaul mechanism, forcing an insurance company o be liquidaed once a solvency hreshold is reached before mauriy, is imposed by a regulaor. The early defaul regulaion affecs he policies value no only direcly via changing policies paymen sreams bu also indirecly via influencing policyholders surrender behavior. In his paper, we endogenize surrender risk by assuming he surrender inensiy of a represenaive policyholder bounded from below and from above, and uncover impacs of he regulaion on he policyholder s surrender decision making. A parial differenial equaion is derived o characerize he price of a paricipaing policy and solved wih he finie difference mehod. We discuss impacs of he regulaion and he insurance company s reacion o he inervenion in erms of is invesmen sraegy on he policy s value as well as on he policyholder s surrender behavior, which depend on he raionaliy level of he policyholder. Keywords: paricipaing life insurance policies; early defaul risk; surrender risk; parial raionaliy; regulaion of financial markes JEL classificaion: G22; G28 We are graeful o Nadine Gazer, Hao Schmeiser, Judih Schneider, Michael Suchanecki, Alexander Szimayer, and Peer Zweifel for heir valuable commens. This paper has also benefied from he valuable discussions by Carole Bernard, Alexander Braun, and Chrisoph Meinerding. We also hank An Chen and Klaus Sandmann for heir encouragemen and suppor hroughou he projec, as well as he paricipans of he join conference of he 21s Annual Meeing of he German Finance Associaion and he 13h Symposium on Finance, Banking, and Insurance, he 2014 Annual Meeing of he American Risk and Insurance Associaion, he 41s Seminar of he European Group of Risk and Insurance Economiss, he 8h Conference in Acuarial Science and Finance on Samos, and he 18h Inernaional Congress on Insurance: Mahemaics and Economics for heir helpful commens on earlier drafs of his paper. Financial suppor from he German Research Foundaion hrough he Bonn Graduae School of Economics is graefully acknowledged. Corresponding Auhor: Hamburg Universiy, Faculy of Business, Economics and Social Sciences, Office 2002, Von-Melle-Park 5, D Hamburg, Germany. Chunli.Cheng@uni-hamburg.de. PricewaerhouseCoopers AG, Bernhard-Wicki-Sraße 8, Munich, Germany.

2 1 Inroducion A ypical paricipaing life insurance policy provides policyholders wih a minimum ineres rae guaranee and bonus paymens upon deah and upon survival which are linked o he performance of he insurance company. Usually, addiional opions are embedded in he policies o increase heir aracion o he policyholders, among which he mos popular one is a surrender opion. The surrender opion eniles he policyholders o erminae heir conracs premaurely and o obain he surrender benefis promised by he insurance company. The policyholders may no necessarily receive he paymens specified in heir conracs even if hey hold he conracs unil mauriy. If he insurance company does no have enough reserves o pay back is liabiliies a he mauriy dae, he policyholders canno ge more han wha remains in he company. To proec he policyholders from collecing oo few benefis as he insurance company declares bankrupcy a mauriy, regulaory auhoriies impose early defaul mechanisms o monior insurance companies financial saus and close hem before i is oo lae. For example, under Solvency II, he supervisory auhoriy wihdraws he auhorizaion of an insurance company when is capial falls below he minimum capial requiremen and does no recover wihin a shor period of ime, see Solvency II Direcive (2009/138/EC). Also, an insurance company supervised by he Swiss Financial Marke Supervisory Auhoriy (FINMA) can lose is license when is risk-based capial drops below he lowes hreshold specified in he Swiss Solvency Tes (SST), see FINMA Circ. 08/44 SST. Proceeds from liquidaed asses are hen paid o sakeholders. Hence, he policyholders also face early defaul risk of he insurance company accompanied wih he early defaul regulaory inervenion. Boh he surrender and he early defaul inervenion definiely have direc impacs on he fair valuaion of paricipaing life insurance policies since hey change he policies paymen sreams. In he exising lieraure, mos sudies focus on only one of he wo aspecs. For example, Andreaa and Corradin (2003), Bacinello (2003), and Grosen and Jørgensen (2000) sudy he fair value of paricipaing life insurance policies wih an embedded surrender opion bu have no considered early defaul risk riggered by bad performance of he insurance company, while Bernard e al. (2005), Chen and Suchanecki (2007), Grosen and Jørgensen (2002), and Jørgensen (2001) ake ino accoun regulaory inervenion in evaluaing paricipaing policies, bu leave ou surrender risk. The only work ha reas early defaul risk and surrender risk a he same ime is Le Courois and Nakagawa (2013) who model he surrender risk hrough a Cox process of he surrender inensiy which is correlaed o he financial marke bu is independen of he company s liquidaion hreshold. However, since he early erminaion of he insurance company imposed by he regulaor reforms he conracs paymen srucure for he policyholders, which we consider as he direc impacs on he conracs value, as a 1

3 response he policyholders may change heir surrender behaviors. Such an influence of he enforced early bankrupcy on he policyholders surrender behaviors can be considered as a by-produc of he regulaory inervenion, which in urn affecs he conracs paymen sreams and correspondingly, he conracs value. In his paper, we analyze boh he direc and indirec (by-produc) impacs of he early defaul risk on he fair value of he paricipaing policies by endogenizing he policyholders surrender behaviors and uncovering he impacs of he early defaul inervenion on he surrender behaviors. Moreover, when he regulaory rule changes, he insurance company may reac o he change by adoping a differen invesmen sraegy, which again affecs he conracs value direcly and indirecly hrough is influence on he policyholders surrender behaviors. Hence, we also sudy how he insurance company chooses is invesmen sraegy in face of he regulaory inervenion, and he impacs of he insurance company s invesmen sraegy on he policyholders surrender behaviors and heir conracs value. To describe he early defaul inervenion we adop he regulaory framework in Bernard e al. (2005), Grosen and Jørgensen (2002), and Jørgensen (2001), where liquidaion is riggered as he insurance company s asse value drops below a hreshold. Concerning he surrender, in mos lieraure, i is assumed ha policyholders are fully raional, which means hey can erminae heir conracs a an opimal ime so ha he conracs value is maximized, see e.g., Andreaa and Corradin (2003), Bacinello (2005, 2003), and Grosen and Jørgensen (2000, 1997), o jus name a few. However, since here is no an acive marke o monior he conrac values, he surrender opion is hardly exercised a he opimal ime if a policyholder is no capable of valuing he conrac correcly. Also due o he lack of an acive rading marke for he conracs, he policyholders, when in urgen liquidiy needs, have o surrender heir conracs o he insurance company and collec he surrender value, which is usually lower han he conracs fair value. Empirical evidence which confirms he so called emergency hypohesis are found e.g. in Kiesenbauer (2011) and Kuo e al. (2003). Given hese limiaions, i is more reasonable o consider policyholders as parly raional from a purely financial poin of view. We adop he approach of modeling policyholders parial raionaliy in Li and Szimayer (2014). Surrender is considered as a randomized even and he arrival of he even is assumed o follow a Poisson process wih he inensiy bounded from below and from above. The lower and upper bounds refer o he minimum surrender rae due o exogenous reasons and he maximum surrender rae due o limied financial raionaliy, respecively. The maximum conracs value is hen derived by choosing surrender inensiies wihin he wo bounds in he wors case scenario from he perspecive of he insurer. This approach corresponds o he spiri of Solvency II. While valuing opions wrien in he conracs, realisic assumpions concerning he likelihood ha policyholders exercise he opions should be used, see Solvency II Direcive (2009/138/EC). 2

4 Moreover, CEIOPS 1 has poined ou ha policyholders surrender behaviors pose a significan risk o insurance companies and he surrender risk should be reaed differenly for differen policyholders. For example, he surrender risk can be subsanially higher if he policyholders are insiuional invesors since hey end o be beer informed and reac more likely in a financially raional way, see CEIOPS (2009). This indicaes ha he raionaliy level of he policyholders plays an imporan role in analyzing he fair value of heir policies. Therefore, we consider differen bounded values of he policyholders surrender inensiies and analyze how he influence of he early defaul regulaory rule on he policyholders surrender behaviors differs wih respec o heir raionaliy level. Similar o Li and Szimayer (2014), we derive a parial differenial equaion (PDE) o characerize he price of a paricipaing policy. However, his PDE is only valid when he liquidaion hreshold has no been ouched ye. Oherwise, he policy akes immediaely he liquidaion value. In his sense, we are solving a barrier opion pricing problem. We apply he finie difference mehod proposed in Zvan e al. (2000, 1996) o solve his problem numerically. The paper is organized as follows. In Secion 2 we model he insurance company and inroduce he payoff srucure of a paricipaing policy, as well as he early defaul regulaory framework. Besides, boh he financial marke and he insurance marke are modeled wih respec o he sochasic processes of he underlying asse, he moraliy risk inensiy and he surrender risk inensiy. In Secion 3 we derive he PDE for he price of he policy. In Secion 4 we analyze he effecs of he regulaory framework and he invesmen sraegy on he conrac valuaion as well as on he policyholder s surrender behavior. Secion 5 concludes. 2 Model Framework 2.1 Company Overview Inspired by he model framework in Briys and De Varenne (1994), we consider a life insurance company which acquires an asse porfolio wih iniial value A 0 a ime 0 = 0 financed by wo agens, i.e., a policyholder and an equiy holder. The policyholder pays a premium o acquire he iniial liabiliy L 0 = αa 0 wih α (0, 1). The res is levied from he equiy holder who acquires E 0 (1 α)a 0 wih limied liabiliy. The insurance company s balance shee a ime 0 is shown in Table 1. The parameer α is called he wealh disribuion coefficien in Grosen and Jørgensen (2002). 1 CEIOPS refers o he Commiee of European Insurance and Occupaional Pensions Supervisors. I was replaced by he European Insurance and Occupaional Pensions Auhoriy (EIOPA) since

5 Asses Liabiliies & Equiy A 0 L 0 αa 0 E 0 (1 α)a 0 Table 1: Insurance company s balance shee a 0 I is assumed ha he insurance company operaes in a fricionless, complee and arbiragefree financial marke over a ime inerval [0, T ], where he ime T corresponds o he mauriy dae of he insurance conrac. As he insurance conrac maures a T, he insurance company closes and is asses are liquidaed and disribued o sakeholders Paricipaing Life Insurance Policy By invesing in he insurance company a ime 0, he policyholder signs a paricipaing insurance conrac which promises him a share of he insurance company s profis in addiion o he guaraneed minimum ineres rae a he mauriy dae T. If he policyholder dies before ime T, he conrac pays deah benefis. Addiionally, he policyholder can exercise he surrender opion embedded in he conrac before mauriy T and collec surrender benefis from he insurance company. To summarize, he conrac promises survival benefis, deah benefis and surrender benefis, depending on which even happens firs. In any even, he policyholder has a prioriy claim on he company s asses and he equiy holder receives wha is lef. As he conrac maures a mauriy T, he policyholder receives a minimum guaraneed benefi, which is given by compounding he iniial liabiliy L 0 wih a minimum guaraneed ineres rae r g, i.e., L rg T = L 0e rgt, and a bonus condiional on ha he asse value generaed by he conribuion of he policyholder is enough o cover he minimum guaraneed benefi, i.e., αa T L rg T. Suppose δ is he paricipaion rae in he asse surplus. The profis shared wih he policyholder are δ[αa T L rg T ]+. However, i may happen ha a ime T when he company s asses are liquidaed, he asses value is lower han he value of he minimum guaraneed benefi. In his case, based on he assumpions ha he policyholder has a prioriy claim on he company s asses and he equiy holder has limied liabiliy, he policyholder collecs wha is lef, i.e., A T, and he equiy holder walks away wih nohing in his hands. To sum up, when he conrac survives unil mauriy T, he policyholder receives survival benefis which ake he form Φ(A T ) = L rg T + δ[αa T L rg T ]+ [L rg T A T ] +. (1) 2 For simpliciy, we assume he company closes when he conrac ends. I is no a sric assumpion because i can be considered ha asses raised from he policyholder and he equiy holder are pu in a separae fund, as he conrac ends, he fund is closed and asses lef in he fund are liquidaed and disribued o he sakeholders. 4

6 The policyholder may die before he conrac maures. We use τ d o denoe he deah ime of he policyholder aged x a ime 0. A ime τ d < T, he conrac pays deah benefis o he policyholder. We assume ha deah benefis have he same paymen srucure as survival benefis, bu wih all he componens evaluaed a he deah ime τ d. We use r d and δ d o denoe he minimum guaraneed ineres rae and he paricipaion rae for calculaing he promised minimum guaranee, i.e., L r d τd = L 0 e r dτ d, and he asse surplus, respecively. Then, he deah benefis have he following form a ime τ d Ψ(τ d, A τd ) = L r d τd + δ d [αa τd L r d τd ] + [L r d τd A τd ] +. (2) Furhermore, by exercising he surrender opion embedded in he conrac, he policyholder can erminae he conrac before he expiraion dae T. We use τ s o denoe he surrender ime. Once he surrender opion is exercised, he company closes and is asses are liquidaed and paid o he policyholder as specified in he conrac bu no more han he liquidaed asse value. We consider he following surrender paymen form for he policyholder S(τ s, A τs ) = L rs τ s [L rs τ s A τs ] +, (3) where L rs τ s = (1 β τs )L 0 e rsτs is he minimum surrender guaranee when he asse value suffices. Here, r s is he minimum guaraneed ineres rae a surrender and β τs is a penaly parameer which penalizes he policyholder for early erminaing he conrac and is assumed o be a deerminisic decreasing funcion of he ime. Afer he policyholder is paid off, he equiy holder receives he res of he asse value. 2.3 Early Defaul Mechanism Now, we inroduce early defaul risk of he insurance company ino he model. We consider an exernal regulaor who waches on he insurance company s financial saus over is operaing ime horizon. We absrac from cumbersome bankrupcy rules and procedures applied o insurance companies in pracice and assume he insurance company is on-going unil eiher he exernal regulaor inervenes before T or he insurance conrac maures a T. We adop he regulaory mechanism inroduced by Grosen and Jørgensen (2002) and se up a defaulriggering barrier based on he minimum survival guaranee B = θl 0 e rg, where θ is a defaul muliplier. Once he company s asse value drops below he barrier before mauriy T, he company is closed by he regulaor and is asses are liquidaed and disribued o he sakeholders. Accordingly, we define he early defaul ime τ b as he firs ime ha he asse 5

7 value drops below he barrier, τ b = inf { A B }. (4) A ime τ b, he policyholder receives early defaul benefis, denoed by Υ(τ b, A τb ), which have he lower value of he liquidaed asses and he minimum survival guaranee accrued a he guaranee rae r g up o he early defaul ime, Υ(τ b, A τb ) = min{a τb, L rg τ b }, (5) where L rg τ b = L 0 e rgτ b. Accordingly, if he company has he liquidaed asses more han he promised minimum guaranee, he equiy holder obains wha is lef afer paying off he policyholder; oherwise, he equiy holder ges nohing. The defaul muliplier θ is se by he regulaor, which acually reflecs how inensively he regulaor moniors he insurance company and how srongly he regulaor inends o proec he policyholder. If he regulaor believes ha he insurance company is inclined o ake advanage of he policyholder by running a risky business or is no compeen enough o manage is asses, he regulaor may se a higher defaul muliplier o proec he policyholder. This implies ha he insurance company mus bear a higher early defaul risk. Oherwise, he regulaor will se a lower defaul muliplier, which allows he insurance company o recover from is emporary bad performance. In our model, we resric θ o be smaller han 1/α, which ensures A 0 > B 0 so ha he insurance company does no defaul a he iniial ime 0 when he conrac is jus issued o he policyholder. 2.4 Mahemaical Formulaion In his secion we model he financial marke and he insurance marke mahemaically. We fix a filered probabiliy space (Ω, F, F, P), where F = (F ) 0 reflecs he flow of informaion available on he financial marke and he insurance marke. We assume ha he company invess is oal iniial asses in raded (risk-free and/or risky) asses on he financial marke, where he risk-free ineres rae, denoed by r, is assumed o be deerminisic in ime. Under he marke probabiliy measure P, he company s asse price process A is assumed o be governed by he following sochasic process da = a(, A ) A d + σ(, A )A dw, [0, T ]. (6) Here W is a sandard Brownian moion under P and generaes he filraion F W = {F W } 0 T. The funcions a and σ > 0 refer o he expeced rae of reurn and he volailiy of he asse 6

8 process respecively, and boh are regular enough o guaranee a unique soluion of (6). As he payoff of he conrac depends no only on he asse value iself bu also on he occurrence of he deah even or he surrender even, we enlarge he filraion F W in a minimal way o summarize all he informaion relevan o he conrac valuaion. The filraion F W is hus enlarged o G = F W H, where H is joinly generaed by he jump processes H = 1 {τd } and J = 1 {τs }, i.e., he informaion abou wheher he policyholder dies before ime and wheher he surrenders he conrac before ime, respecively. The hazard rae of he random ime τ d, also called moraliy inensiy, is denoed by µ and assumed o be a deerminisic funcion of ime. 3 Similarly, we call he hazard rae of he random ime τ s surrender inensiy and denoe i by γ. Taking ino accoun ha he policyholder has parial raionaliy in surrendering his conrac, we follow he approach in Li and Szimayer (2014) by assuming he surrender inensiy is bounded from below by ρ, in he case ha surrender is no a financially opimal decision for he policyholder, and from above by ρ, in he case ha surrender becomes financially opimal, wih ρ > ρ. Due o personal reasons which urge he policyholder o surrender he conrac premaurely, he lower bound of he surrender inensiy ρ is presen in eiher of he wo cases, i.e., he surrender inensiy a leas akes he value of ρ. The size of he increase in he surrender inensiy as surrendering becomes financial opimal o he policyholder, i.e., ρ ρ, measures how frequenly he policyholder updaes his financial marke informaion and how likely he makes he surrender decision when i is opimal o do so, which is also called endogenous surrender inensiy. The more frequenly he updaes and analyzes financial informaion, he larger he increase in he inensiy is, accordingly, he more raional he is o make his surrender decision. In he case of ρ =, he policyholder surrenders immediaely when i is opimal o do so and ogeher wih a zero exogenous surrender rae ρ = 0, we are back o he case of pricing an American-syle conrac by solving an opimal sopping problem. The decision is made by comparing he coninuaion value of he conrac and he value of surrender benefis, which are denoed by v(, A) and S(, A), respecively. Depending on which decision he policyholder makes, he endogenous surrender inensiy is hus eiher 0 or ρ ρ. To summarize, he surrender inensiy akes he form of ρ, for S(, A) < v(, A) γ = ρ, for S(, A) v(, A). Condiional on he curren informaion available on he financial marke and he insurance 3 In he lieraure, here are many discussions on modeling sochasic moraliy inensiy, see e.g., Bacinello e al. (2010), Biffis e al. (2010), Dahl (2004), Dahl and Møller (2006). However, he sochasic feaure of he moraliy inensiy does no have oo much influence on he conrac value, see Li and Szimayer (2011). Therefore, in his paper, we assume a deerminisic moraliy inensiy funcion for simpliciy and focus more on he early defaul risk and he surrender risk. (7) 7

9 marke, he arrival of he deah even, he arrival of he surrender even, and W are independen. Thus, W is a G-maringale, and µ and γ are G-inensiies of he random deah ime τ d and he random surrender ime τ s, respecively. In he absence of arbirage, we use he risk-neural pricing approach wih a maringale measure Q o price he paricipaing life insurance conrac. Under he maringale measure Q, he company s asse process is described by da = r() A d + σ(, A )A dw Q, [0, T ], (8) where W Q is a sandard Brownian moion. Taking he moraliy risk and he surrender risk, wih µ and γ as G-inensiies of he random imes τ d and τ s, respecively, ino consideraion, pricing he paricipaing life insurance conrac under he maringale measure Q requires addiional jusificaion. Under he condiion ha he moraliy inensiy is deerminisic, if he pool of policyholders is large enough, moraliy risk is diversifiable for he insurer, and hus µ is he (Q, G)-inensiy of moraliy. The surrender inensiy specified in (7) corresponds o he wors-case scenario from he insurance company s perspecive. As long as we assume he insurance company does no ask for an exra risk premium above he wors-case surrender inensiy under he maringale measure Q, 4 he bounds ρ and ρ are sill valid under he measure Q. Then he surrender inensiy specified in (7) corresponds o he (Q, G)-inensiy of surrender on he enlarged marke represened by he filraion G. 5 The conrac value obained under he measure Q wih he wors-case surrender inensiy γ can be inerpreed as he upper price bound of he conrac. We address his issue formally in Remark 1. Consequenly, he choice of he surrender inensiy in (7) is no only moivaed by he observaions of policyholders surrender behaviors on he marke bu also by he wors-case scenario analysis wihin a reasonable range ha is ofen adoped in pracice, see CEIOPS (2009). 3 Conrac Valuaion In his secion we value he conrac by aking boh he early defaul risk and he surrender risk ino consideraion. Since policyholder s surrender behavior is endogenously modeled in his paper, he surrender inensiy can only be deermined endogenously. By applying he PDE approach, we can specify he surrender inensiy and he conrac value a he same ime. Moreover, afer inroducing he early defaul mechanism, he conrac payoff o he policyholder 4 Alernaively, a higher marke price for he surrender risk may be charged by lowering he lower bound ρ and increasing he upper bound ρ under he measure Q. In Proposiion 2 we show formally ha a lower ρ and a higher ρ lead o a higher conrac value. 5 Confer Li and Szimayer (2014) for a formal explanaion of he surrender inensiy afer he change of measure. 8

10 is conneced o he solvency of he company and has a barrier opion feaure. Thus, we need o disinguish he case where he insurance company is ongoing and he case where he regulaor inervenes, which means ha, in order o value he conrac, we differeniae beween he region where A B and he region where A > B for (0, T ). This is similar o he barrier opion pricing. For A B a ime (0, T ), he insurance company mus be liquidaed and he policyholder only obains Υ(, A ). For A > B, we represen he conrac value V on { τ d τ s T } by V = 1 {<τd τ s}v(, A ) + 1 {=τd,τ d <τ s,t }Ψ(τ d, A τd ) + 1 {=τs,τs<τ d,t }S(τ s, A τs ), (9) where v is a suiably differeniable funcion v : [0, T ] R + R + 0, (, A) v(, A), represening he pre-deah/surrender value. Then we apply he no-arbirage pricing condiion on he se { < τ d τ s τ b T }, being r()v (, A )d = E Q [dv G ]. (10) On he se { < τ d τ s τ b T }, we compue he differenial of V as 6 dv = dv(, A ) + (Ψ(, A ) v(, A ))dh +(S(, A ) v(, A ))dj, for 0 < T, (11) where H and J refer o he jump processes wih he Q-inensiies µ and γ, respecively. A jump in H or J leads o a change in he paymen liabiliy eiher of he amoun Ψ(, A ) v(, A ) or S(, A ) v(, A ). Plugging (11) ino (10) and using V = v(, A ) a ime < τ d τ s τ b T, we obain r()v(, A )d = E Q [dv(, A ) G ] + (Ψ(, A ) v(, A ))µ()d By applying Io s Lemma o dv(, A ), we have +(S(, A ) v(, A ))γ d. (12) [ v ] E Q [dv(, A ) G ] = E Q Lv(, A )d + σ(, A )A A (, A )dw Q G = Lv(, A )d, (13) 6 Noice ha in he region A > B, here would no be early defaul afer he insananeous ime period d since he asse process is assumed o be coninuous in our model. 9

11 where Lv(, A) = v v (, A) + r()a (, A) + 1 A 2 σ2 (, A)A 2 2 v (, A). Then, on he se { < A 2 τ d τ s τ b T } we have Lv(, A ) + µ()ψ(, A ) + γ S(, A ) (r() + µ() + γ )v(, A ) = 0. (14) We summarize he pricing PDE in he following proposiion. Proposiion 1. For he conrac value V described by (9), he pre-deah/surrender value v for (, A) [0, T ) R + is he soluion of he parial differenial equaion Lv(, A ) + µ()ψ(, A ) + γ S(, A ) (r() + µ() + γ )v(, A ) = 0, (15) where ρ, for S(, A ) < v(, A ), γ = ρ, for S(, A ) v(, A ) ; (16) subjec o he boundary condiion v(, A ) = Υ(, A ), for [0, T ), A = B = θl 0 e rg, (17) and he erminaion condiion v(t, A T ) = Φ(A T ), for A T R +. (18) The inegral represenaion of he soluion o he above pricing PDE is shown in Corollary 1 and proved in Appendix A. Corollary 1. Suppose he surrender inensiy γ is given. The value of he paricipaing policy V can be represened on { < τ s τ d τ b T } by V =E Q [ τb T + 1 {τb T }Φ(A T )e T e m (r(u)+µ(u)+γ(u,a u))du (µ(m)ψ(m, A m ) + γ(m, A m )S(m, A m ))dm (r(u)+µ(u)+γ(u,au))du + 1 {τb <T }Υ(τ b, A τb )e τ b (r(u)+µ(u)+γ(u,a u))du Remark 1. The pricing problem can be formulaed as looking for he wors case of he riskadjused surrender inensiy γ so ha he conrac value is maximized under he maringale (19) G ]. 10

12 measure Q on { < τ s τ d τ b T }, v(, A) = sup γ Γ(,A) E,A Q [ τb T + 1 {τb T }Φ(A T )e T e m (r(u)+µ(u)+γ(u,a u))du (µ(m)ψ(m, A m ) + γ(m, A m )S(m, A m ))dm (r(u)+µ(u)+γ(u,au))du + 1 {τb <T }Υ(τ b, A τb )e τ b (r(u)+µ(u)+γ(u,a u))du where Γ(, A) = {γ : [, T ] R + R + : ρ γ(u, A) ρ, for all u T and A R + } and E,A Q denoes he expecaion condiional on A = A under he measure Q. This is a sochasic conrol problem, which can be solved, according o he heorem of he Hamilon- Jacobi-Bellman equaion (confer Yong (1997) and Yong and Zhou (1999)), by dealing wih an equivalen problem 0 = sup Lv(, A) + µ()ψ(, A) + γ(, A)S(, A) (r() + µ() + γ(, A))v(, A), γ Γ(,A) subjec o v(, A) = Υ(, A), for A = B = θl 0 e rg, and v(t, A) = Φ(A), for A R +. γ needs o be opimally conrolled: Since in he equaion above he par ha depends on γ is linear in γ, i.e., γ(, A)(S(, A) v(, A)), he soluion o he problem is exacly he same as is presened in equaion (7). Wihin a given regulaory framework and under a given invesmen sraegy, i.e., for given θ and σ, we prove ha a lower value of ρ or a higher value of ρ leads o an increase of he conrac value, see Proposiion 2. This is consisen wih our inuiion, since a lower ρ or a higher ρ indicaes he increase of he raionaliy level of he policyholder in he moneary sense and hus increases he conrac value. The proof is provided in Appendix B. Proposiion 2. Suppose he early defaul mechanism is characerized by he defaul muliplier θ and he insurance company s invesmen sraegy by σ. Furhermore, suppose ha v is he pre-deah/surrender value funcion of he paricipaing policy wih bounds of he surrender inensiy being ρ and ρ, and ha w is he pre-deah/surrender value funcion of he policy wih bounds ζ and ζ. Assume ha ζ ρ and ρ ζ. Then we have w(, A) v(, A), for (, A) [0, τ b T ] R +. (20) (21) ], 4 Numerical Analysis In his secion, we adop he finie difference mehod proposed by Zvan e al. (2000, 1996) o numerically solve he PDE wih a coninuously applied barrier (15) as saed in Proposiion 1 11

13 and sudy he impacs of he early defaul risk and he surrender risk on he fair valuaion of he conrac as well as on he insurance company s invesmen sraegy. The insurance company is se up wih iniial asse value A 0 = 100 and 85% of he asse value is acquired by he policyholder who buys he paricipaing conrac a ime 0 as he iniial liabiliy, which means α = The conrac maures in T = 10 years and promises he same paricipaion rae δ = δ d = 0.9 a mauriy and a deah. 7 The risk-free ineres rae is r = 0.04 and he volailiy of he company s asse process is consan, i.e., σ(, A ) = The volailiy provides informaion abou he riskiness of he insurance company s invesmen sraegy. A higher σ indicaes a higher riskiness of he invesmen sraegy while a lower σ implies a more conservaive invesmen sraegy. 9 The minimum guaraneed ineres raes a survival, a deah and a surrender are r g = r d = r s = As for he moraliy inensiy, we follow he Gomperz- Makeham law by assuming a deerminisic process µ() = A µ + Bc x+ for he policyholder aged x = 40 a 0 = 0 wih A µ = , B = , c = Addiionally, he penaly parameer akes he form 0.05, for 1, 0.04, for 1 < 2, β = 0.02, for 2 < 3, 0.01, for 3 < 4, 0, for > 4. The parameers are summarized in Table 2. Marke Parameers Conrac Parameers A α 0.85 r 0.04 T 10 σ 0.2 δ, δ d 0.9 A µ r g, r d, r s 0.02 B c Table 2: Parameer specificaions The analysis in he following subsecions is conduced for a represenaive policyholder. 7 Regulaors usually require he paricipaion rae o be kep a leas a a cerain level. In Germany, e.g., i lies a 90%. 8 The values of he risk-free ineres rae and he volailiy are chosen purely for he illusraion purpose. 9 On a financial marke wih only one risk-free and one risky asse, a higher asse volailiy is achieved by invesing more ino he risky asse. 10 Source of he parameer values: Delbean (1986). 12

14 Under he assumpion ha he pool of policyholders is large enough, he surrender inensiy of a represenaive policyholder gives an indicaion of he proporion of policyholders who will surrender heir conracs a he porfolio level. The implicaions for a large pool of policyholders will be summarized in Secion 5 o conclude he paper. 4.1 Effecs of Regulaory Framework on Conrac Valuaion In his secion, we analyze he effecs of he early defaul risk on he fair valuaion of he conrac. The magniude of he early defaul risk depends on he sricness of he regulaory framework, which in our model is represened by he defaul muliplier θ ha is specified by he regulaor. I indicaes how he regulaor judges he insurance company s abiliy o manage is asses. If he regulaor has confidence in he experise of he insurance company and abou he financial marke, she will olerae a emporary poor performance of he insurance company and hence choose a lower defaul muliplier giving he company has he chance o recover. Oherwise, she will se a higher value o proec he policyholder from no being able o obain he guaraneed benefis promised by he company. Alhough a lower (higher) defaul muliplier is less (more) effecive o proec he policyholder from a downside developmen of he company, i gives he company a higher (lower) chance o recover from he emporary bad performance and pay ou more benefis (less benefis) o he policyholder when i recovers. Hence, he level of he defaul muliplier has grea influence on he payoff of he conrac and hus on he conrac value. Furhermore,he policyholder akes ino accoun he impacs of he proecion from he regulaor on his conrac s paymens (evenually, on his conrac value) and adjuss his surrender behavior accordingly, which indirecly influences his conrac value. Inuiively, he policyholder makes his surrender decision based no only on benefis ha are promised by he insurance company bu also on he abiliy of he insurance company o mee is promise. The early defaul mechanism ensures he abiliy of he insurance company o mee is promise by imposing a limi on is asse value. A higher defaul muliplier implies ha he policyholder has o worry less abou he second issue because he is beer proeced and will surrender he conrac only when he surrender benefis are very aracive o him. On he conrary, if he defaul muliplier is se lower so ha he policyholder would no be proeced compleely, he mus ake he defaul risk of he insurance company seriously ino accoun when implemening his surrender sraegy. In his case, he policyholder may be willing o surrender his conrac earlier o avoid losing oo much of his iniial invesmen. Table 3 presens he conrac value for differen values of he defaul muliplier θ and differen raionaliy levels represened by (ρ, ρ). In he second column are he conrac values in he case when here is no early defaul mechanism. From he hird o he fifh column are he 13

15 conrac values wih differen levels of regulaory srengh which are represened by he differen values of he defaul muliplier θ. For example, θ = 0.7 means ha he regulaor does no allow he insurance company s asse value o drop below 70% of he minimum guaranee. θ = 1.1 indicaes ha he regulaor is more conservaive and requires he company s asse value o lie above 110% of he minimum guaranee. Comparing he conrac values in columns 2-4 where he early defaul regulaion is inroduced, and hen srenghened, he conrac value increases gradually for all ypes of policyholders. Inroducing he early erminaion rule proecs he policyholder from he downside risk of he insurance company s invesmen and increasing he defaul muliplier enlarges he proecion level. An ineresing feaure is ha he effecs of he early erminaion regulaion no only depend on he defaul muliplier θ bu also on he raionaliy level of he policyholder. For example, a policyholder wih (ρ, ρ) = (0, 0) never surrenders his conrac, which urns o be a European-ype conrac. In his case ha he no early defaul wih early defaul θ = 0.7 θ = 0.9 θ = 1.1 (0, 0) (0, 0.03) (0, 0.3) (0, ) (0.03, 0.03) (0.03, 0.3) (0.03, ) (0.3, 0.3) (0.3, ) Table 3: Conrac values for differen defaul mulipliers θ and differen raionaliy levels represened by (ρ, ρ) policyholder would be beer off if he erminaes his conrac and collecs he surrender benefis when his conrac value drops, imposing an early defaul regulaory rule such as θ = 0.7 helps he policyholder improve his posiion. However, he benefis from he regulaor s proecion become smaller as he policyholder becomes financially more raional. The defaul muliplier does no play a significan role when he policyholder is able o exercise he surrender opion opimally, i.e. (ρ, ρ) = (0, ). 11 Since he fully raional policyholder can find he opimal surrender sraegy anyway, he does no need he proecion of he regulaor. However, he posiive effec of srenghening he early defaul regulaion disappears as an over-regulaion rule is carried ou. As he defaul muliplier increases from 0.9 o 1.1, he conrac value decreases in some cases, e.g., when (ρ, ρ) = (0, ), (ρ, ρ) = (0.03, ), among which we can even observe he disadvanage of inroducing he early defaul regulaion. For he policyholder 11 The conrac values are all around 92 for (ρ, ρ) = (0, ) and θ = {0, 0.7, 0.9}. 14

16 wih (ρ, ρ) = (0, ) and (ρ, ρ) = (0.03, ), he conrac value becomes even lower han when here is no early defaul risk. As we have menioned in Secion 2.4 ha he policyholder wih ρ = may surrender he conrac a any ime when i is opimal o do so, irrespecive of exogenous reasons, he is able o proec himself from he downside risk of he company s invesmen. However, enforcing a erminaion regulaion wih a very large defaul muliplier sops him from obaining more benefis in he favorable developmen of he insurance company, which acually lowers he conrac value. Addiionally, if we ake a look a he column of he conrac values for ρ = 0 and θ = 1.1, he conracs have he same value. Since when he defaul muliplier is so high ha he benefis obained by he policyholder a liquidaion are higher han he surrender benefis, surrendering he conrac becomes unaracive, which means here would also be no endogenous reason for he policyholder o surrender his conrac premaurely. Therefore, if he policyholder does no surrender his conrac for exogenous reasons (ρ = 0), he conrac value says a he same level as he European-syle conrac value , no maer how financially raional he policyholder is. We have discussed in Secion 1 ha due o personal reasons, he policyholder surrenders his conrac even hough he is capable of valuing his conrac and undersands surrender resuls in a financial loss. Such exogenous surrender lowers he conrac s fair value. We isolae he impac of exogenous surrender on he conrac s fair value by calculaing he decrease in he conrac value as ρ deviaes from 0 when ρ =. The decrease in he conrac value acually measures he premium ha he insurance company should no have charged he policyholder due o his personal non-avoidable liquidiy reasons. We call his premium he liquidiy premium. In Figure 1 we presen he liquidiy premium for differen values of he regulaory defaul muliplier and he exogenous surrender inensiy. We observe he following rends. Firs, wihin he same regulaory framework, he liquidiy premium becomes larger as he exogenous surrender inensiy increases. I implies ha as he exogenous surrender inensiy increases, he insurance company needs o compensae he policyholder more in erms of lowering he conrac value in order o make he conrac more aracive o him. Second, he liquidiy premium increases faser a a lower θ-level while more slowly a a higher θ-level, unil he exogenous surrender inensiy ρ also becomes quie large. This indicaes ha he value of he liquidiy premium is more sensiive o he policyholder s exogenous surrender inensiy level ρ a a lower θ-level, where he proecion from he regulaor is low and i is more necessary for he regulaor o urge he insurance company o assess policyholder s exogenous surrender rae more precisely. On he conrary, as he inervenion by he regulaor is enhanced, he probabiliy ha he insurance company is closed increases. Liquidaion may happen before he policyholder exercises he surrender opion due o he exogenous reasons. Since he policyholder is no penalized a liquidaion, he may receive more han he surrender guaranee he may oherwise obain from 15

17 15 liquidiy premium regulaory defaul muliplier exogenous surrender inensiy Figure 1: Liquidiy premium as a funcion of he exogenous surrender inensiy ρ [0, 0.3] and he defaul muliplier θ [0, 1.1] surrendering his conrac. Similar o he above discussion on he impac of he exogenous surrender inensiy on he conrac value, he endogenous surrender inensiy also influences he conrac value. Since he policyholder has limied informaion on he financial marke and limied knowledge o value he conrac on his own, i.e., ρ <, he may fail o surrender he conrac when he should do so, which resuls in a decrease in his conrac value. Similarly, we isolae he impac of endogenous surrender on by calculaing he decrease in he conrac value as he upper bounder surrender inensiy ρ deviaes from infiniy while seing ρ = 0. This decrease in he conrac value measures he premium ha he insurance company should no have charged he policyholder due o his limied informaion and limied valuaion abiliy, which we name raionaliy premium. In Figure 2 we plo he raionaliy premium as a funcion of he upper bound surrender inensiy ρ and he defaul muliplier θ. I is naural o observe ha he raionaliy premium decreases in ρ a a given proecion level θ deermined by he regulaor, which is consisen wih Proposiion 2. Furhermore, for a given value of ρ, he raionaliy premium decreases as he regulaor s inervenion level is enhanced unil he defaul muliplier θ reaches 1. Inuiively, he inervenion by he regulaor helps he policyholder close his conrac premaurely by shuing down he insurance company as is asse value drops below he liquidaion hreshold, which is ofen he iming ha he policyholder should have surrendered his conrac bu failed o do so. In addiion, such an inervenion by he regulaor does no bring penalies o he policyholder, which furher improves his financial posiion. Therefore, wih he 16

18 proecion from he regulaor and as he proecion enlarges, he insurance company charges he policyholder a higher conrac value, which implies a lower a lower value of he raionaliy premium. Bu in a non-sric regulaion environmen, i.e., where he early defaul hreshold is low, i is imporan for he insurance company o assess he endogenous surrender inensiy of he policyholder so ha he conrac price is reduced enough o arac he policyholder. 4 raionaliy premium regulaory defaul muliplier upper bound surrender inensiy Figure 2: Raionaliy premium as a funcion of he upper bound surrender inensiy ρ [0.3, 30] and he defaul muliplier θ [0, 1.1] 4.2 Effecs of Insurance Company s Invesmen Sraegy on Conrac Valuaion In his secion we analyze he effecs of insurance company s invesmen sraegy on he conrac valuaion and discuss insurance company s risk-shifing invesmen sraegy in wo environmens, namely wih and wihou he early defaul inervenion by he regulaor. The invesmen sraegy is represened by he volailiy σ of he underlying asse A. The higher he volailiy σ, he higher he risk ha he insurance company has enered ino. Table 4 presens he conrac value for differen values of σ and differen raionaliy levels represened by (ρ, ρ) wih and wihou he early defaul inervenion. The early defaul muliplier θ is The effecs of he company s invesmen sraegy on he conrac valuaion appear o be differen wihin wo regulaory frameworks. For he no early defaul case, we observe hree endencies, which depend on he policyholder s raionaliy level. When (ρ, ρ) = (0, ) or 12 We have also sudied he cases wih θ = 0.7 and 1.1. However, we have no found any qualiaive differences in he effecs of he volailiy σ and hence do no presen all he resuls here. 17

19 no early defaul wih early defaul σ = 0.1 σ = 0.2 σ = 0.3 σ = 0.1 σ = 0.2 σ = 0.3 (0, 0) (0, 0.03) (0, 0.3) (0, ) (0.03, 0.03) (0.03, 0.3) (0.03, ) (0.3, 0.3) (0.3, ) Table 4: Conrac values for differen invesmen sraegies represened by σ and differen raionaliy levels represened by (ρ, ρ), θ = 0.9 (0.03, ), he policyholder is considered o be financially raional because he policyholder will exercise he surrender opion once his conrac value is higher han he surrender value. An increase in he underlying asse risk also implies a poenially higher expeced rae of reurn, which can be easily capured by a raional policyholder. Hence, he conrac value increases in he underlying asse risk. For (ρ, ρ) = (0, 0), (0, 0.03), (0, 0.3) or (0.03, 0.3), which indicaes eiher zero or very low exogenous surrender inensiy value, and bounded endogenous surrender inensiy, we observe an increase firs and hen a decrease in he conrac value as σ increases. Inuiively, when he underlying asse risk increases bu sill says a a lower level, he downside risk is sill limied and he opimal surrender inensiy during he conrac s life ime says a a lower level anyway. However, he chance o paricipae in he favorable developmen of he asse value increases. Hence, overall he conrac value increases slighly when σ increases from 0.1 o 0.2. However, as he asse risk increases furher, he downside risk could be so high ha i is necessary o check more frequenly wheher o surrender he conrac or no. The bounded endogenous surrender inensiy in his case would hen lead o a lower conrac value for he policyholder. When (ρ, ρ) = (0.03, 0.03) or (0.3, 0.3), he endogenous surrender inensiy is zero and he policyholder surrenders his conrac only for exogenous reasons, which are no relaed o he conrac value a all. Higher invesmen risk requires a more rapid and correc response o changing marke condiions. When he policyholder is no willing o do so or capable of doing so, he conrac value for he policyholder will decrease in he invesmen risk, represened by σ. Now as he early defaul mechanism is implemened by he regulaor, he conrac value increases as he volailiy σ increases in mos cases, excep when he probabiliy ha he policyholder surrenders his conrac due o exogenous reasons is relaively large, i.e., ρ = 0.3. In Table 4, we observe ha he conrac value decreases in σ for (ρ, ρ) = (0.3, 0.3), and says 18

20 consan for (ρ, ρ) = (0.3, ). In he wo cases, i happens ha he policyholder surrenders his conrac even when he asse value increases, which deprives him of he chance o paricipae in he asse appreciaion. Since he policyholder is proeced by he regulaor hrough he early defaul barrier, he poenial downside risk of he insurance company s invesmen is limied while he poenial paricipaion in he favorable asse performance is sill possible. As long as he policyholder is no rushing o liquidae his conrac, he can benefi more from he regulaor s proecion as he riskiness of he invesmen sraegy increases and his conrac value increases accordingly. Similar o Secion 4.1, we presen in Figure 3 he liquidiy premium for differen invesmen sraegies (adoped by he insurance company) boh in he case of no early defaul inervenion by he regulaor, see Figure 3 (a), and he case where here is an early defaul mechanism, see Figure 3 (b). We see ha, for a given raionaliy level (ρ, ρ), liquidiy premium increases in he volailiy of he underlying asse in boh cases. As he invesmen risk of he insurance company increases, he probabiliy ha he policyholder sells his conrac due o exogenous reasons back o he insurance company which has been experiencing financial difficulies becomes higher. This indicaes ha increasing he riskiness of he invesmen generally does harm o he policyholder who is likely o cash ou his conrac due o personal non-avoidable reasons. Hence, he insurance company lowers he conrac price o arac he policyholder as i increases he riskiness of is invesmen. I happens in boh cases wih and wihou he early defaul regulaion. However, wih he proecion of he regulaor, he downside risk of he insurance company s invesmen is limied. Therefore, he increase in he conrac value as he volailiy σ increases is smaller han in he case of no proecion of he regulaor. We presen he raionaliy premium depending on he invesmen sraegy and he endogenous surrender inensiy level in Figure 4. We see ha, given an endogeneous surrender inensiy value ρ, he raionaliy premium increases monoonically wih he riskiness of he invesmen sraegy σ when here is no early defaul inervenion by he regulaor. The raionaliy premium is much higher when he endogenous surrender inensiy level ρ is low. Unlike a fully raional policyholder who can rack he financial performance of he insurance company and ac opimally o maximize his benefis, a parially raional policyholder faces he risk of misakenly holding a conrac whose value is lower han he surrender value. Such risk increases as he company s asse process becomes more volaile and is refleced by he increasing raionaliy premium wih respec o σ. However, when he early defaul mechanism is imposed, he raionaliy premium firs increases and hen decreases in σ. The decreasing effec can be explained by he regulaor s inervenion or proecion, which as a remedy for he policyholder s insufficien surrender inensiy, induces he insurance company o charge he policyholder a higher conrac price and lower he raionaliy premium accordingly. 19

21 15 15 liquidiy premium volailiy exogenous surrender inensiy liquidiy premium volailiy exogenous surrender inensiy (a) no early defaul (b) θ = 0.9 Figure 3: Liquidiy premium as a funcion of he exogenous surrender inensiy ρ [0, 0.3] and he volailiy σ [0.05, 0.5] wih and wihou he early defaul inervenion raionaliy premium raionaliy premium volailiy upper bound surrender inensiy volailiy upper bound surrender inensiy (a) no early defaul (b) θ = 0.9 Figure 4: Raionaliy premium as a funcion of he upper bound surrender inensiy ρ [0.3, 30] and he volailiy σ [0.05, 0.5] wih and wihou he early defaul inervenion Due o he influence of he policyholder s raionaliy level and he regulaory framework on he conrac value, as a response, he insurance company may change is invesmen sraegy. We assume ha he insurance company performs in he ineres of he equiy holder. Since he conrac value can be regarded as he marke value of he insurance company s liabiliies when he insurance company is ongoing, he objecive of he company, maximizing he residual value for he equiy holder, is hus o minimize he value of he policyholder s policy. From Table 4 we can infer which invesmen sraegy he insurance company ends o adop. If here is no early defaul regulaory rule and he raionaliy level of he policyholder is very high, 20

22 he insurance company prefers o ake a low-risk invesmen. This gives us wo implicaions. Firs, if he policyholder is raional enough o surrender his conrac, he regulaor, aiming a inducing he insurance company o avoid oo risky invesmen, does no need o inerfere wih he insurance company s invesmen decision anymore. Second, looking back ino hisory, insurance companies have no always aken conservaive invesmen sraegies. An aspec ha we can infer from our sudy is ha he insurance company has acually assumed ha he policyholder will no always ac opimally. Considering his, i is hen inappropriae o price he surrender opion as a pure American-syle opion as i is ofen assumed in he lieraure, since he policy ends o be overpriced under his assumpion which is unfair for he policyholder. Hence, if he insurance company chooses parameers for pricing he conrac such ha he conrac value is exacly equal o he policyholder s paymen by assuming a high raionaliy level and leading us o hink ha i will adop an invesmen sraegy wih low risk under is pricing assumpion, he company acually has he incenive o increase he riskiness of is invesmen sraegy aferwards. This problem will be avoided mos likely as he early defaul regulaion is inroduced. We can read ou from Table 4 ha he insurance company prefers a low-risk invesmen in all cases bu one when he early defaul regulaion is presen. 4.3 Effecs of Regulaory Frameworks and Invesmen Sraegies on Surrender Behaviors To demonsrae he effecs of he regulaory framework on he policyholder s surrender behavior, we depic in Figure 5 he separaing boundaries which illusrae he regions where he policyholder surrenders he conrac for exogenous reasons and he regions where he policyholder surrenders he conrac for endogenous reasons. When he early defaul regulaion is enforced, par of he surrender region will be replaced by he early defaul region. Based on our consideraion abou he policyholder s raionaliy level as is illusraed in Secion 4.1 and 4.2, we assume ρ = 0.03 and ρ = 0.3. We begin wih he graph for he case where here is no early defaul regulaory rule, see Figure 5 (a). Here we can observe hree regions. When he asse price A is relaively high, he policyholder only surrenders for exogenous reasons, because paricipaion in insurance company s favorable asse performance is very aracive, which is, according o he conrac design, only possible when he policyholder holds he conrac unil deah or unil mauriy. When asse price A is very low, here would also only be exogenous surrender. This is because in his case paricipaion in company s favorable asse performance a deah or a mauriy is hardly possible and early erminaion of he policy carries penaly on he minimum guaranee, he policyholder would raher hold his conrac if he does no have oher exogenous surrender reasons. The region in he middle of he graph 21