Surrender Risk in the Context of the Quantitative Assessment of Participating Life Insurance Contracts under Solvency II

Transcription

1 risks Aricle Surrender Risk in he Conex of he Quaniaive Assessmen of Paricipaing Life Insurance Conracs under Solvency II Tobias Burkhar ifa (Insiue for Finance and Acuarial Sciences), Lise-Meiner-Srasse 14, 8981 Ulm, Received: 18 May 218 ; Acceped: 22 June 218; Published: 27 June 218 Absrac: Paricipaing life insurance conracs enile he policyholder o paricipae in he company s annual surplus. Typically, hey are also equipped wih a surrender opion ha allows he policyholder o erminae he conrac prior o mauriy, receiving a predeermined surrender value. The opion ineracs wih (ofen clique-syle) ineres guaranees ha are a key feaure of radiional paricipaing conracs. Surrender opions can considerably affec an insurer s liabiliies and bear maerial risks. This paper addresses he recogniion of hose risks in he quaniaive assessmen of a heerogeneous insurance porfolio under Solvency II, aking ino accoun he complex inerrelaion beween minimum ineres guaranees, reserving requiremens, and profi sharing. The lapse risk module of he Solvency II sandard formula requires he idenificaion of porfolio segmens ha are exposed o a specific change of surrender raes (long-erm increase/decrease, one-off increase). We provide a heurisic ha idenifies homogeneous risk groups in he sense ha he respecive sress would increase he insurer s liabiliies. Our approach can be used o derive an appropriae segmenaion in pracical applicaions. We furher analyze implicaions of he segmenaion on he Risk Margin (as par of he Technical Provisions under Solvency II) and discuss consequences of policyholder opions on he calculaion of Going Concern Reserve and Surplus Funds. To illusrae our findings, we se up a sochasic balance shee and cash flow projecion model for a sylized life insurance company. We conclude ha curren mehods used for pracical applicaions underesimae surrender risk under Solvency II and ha he proposed modeling refinemens may improve he appropriaeness of solvency raios for paricipaing business. Keywords: paricipaing life insurance; surrender risk; Solvency II; Going Concern Reserve; Surplus Funds 1. Inroducion Life insurance policies ofen come wih several policyholder opions. In paricular, his includes he surrender opion which allows he policyholder o erminae he conrac prior o mauriy in combinaion wih he payou of a (predeermined) surrender value. Hence, surrender opions may significanly affec a company s fuure cash flow profile and have maerial impac on he insurer s liabiliies (Bauer e al. 26; Gazer 29; Kling e al. 214). This especially holds for radiional paricipaing life insurance business, which makes up for a significan par of he life insurance marke in Germany (and oher member saes of he European Union). Such conracs are characerized by wo key feaures: Firs, hey come wih an (ofen clique-syle) minimum ineres guaranee (ha may differ wihin he insurance porfolio). In addiion, hey are eniled o an appropriae paricipaion in he company s annual (posiive) surplus. In fac, a subsanial par of an insurer s fuure obligaions agains policyholders is relaed o fuure surplus paricipaion. Risks 218, 6, 66; doi:1.339/risks6366

2 Risks 218, 6, 66 2 of 38 The surplus paricipaion process iself is highly complex and resuls in ineracion effecs wihin he insurance porfolio (Burkhar e al. 215). Policyholders surplus paricipaion is no deermined on a single conrac basis, bu he annual surplus is joinly deermined for he whole porfolio and allocaed o individual conracs in a muli-sage process. The process includes a collecive bonus reserve, he so-called Reserve for Bonuses and Rebaes (RfB, Rücksellung für Beiragsrückersaung), in which surplus is pooled firs o buffer flucuaions in he annual surplus o achieve a sable surplus paricipaion for all policyholders. As a resul, he insurer s fuure obligaions arising from a specific conrac canno be isolaed from he remaining porfolio, bu hey depend on he guaranees and he developmen of he enire porfolio. Due o hese inerdependencies, he impac of a policyholder exercising her surrender opion is no limied o her own conrac, bu i affecs he insurer s oal obligaions. In fac, he profi sharing mechanisms lead o diverse inerrelaions beween surrender opions, differen levels and ypes of minimum ineres guaranees, and policyholders surplus paricipaion wihin a heerogeneous insurance porfolio. Surrender opions also bear maerial risks for a life insurance company (Kuo e al. 23): Massive surrenders may cause liquidiy issues and force he selling of asses. Surrenders may furher lead o a loss of poenial fuure profis. In case of early surrenders, he insurer may no be able o refinance high up-fron invesmens made in he course of acquiring new business. The opion can add o adverse selecion, reducing he effeciveness of risk diversificaion and balancing over ime. High surrender raes can also have a negaive effec on he insurer s repuaion, which may resul in even more policyholders surrendering or harm new business. On he oher side, low surrender raes bear he risk ha he insurer may no be able o mee he obligaions resuling from high guaranees given in he pas. Clearly, he risks due o a policyholder s surrender opion depend on he conrac and in paricular, on he conracual minimum ineres guaranee. The high economic relevance of surrender opions has drawn he aenion no only of academics bu also of he indusry and regulaors, especially in connecion wih he inroducion of increasingly sophisicaed financial reporing sandards and risk-based solvency requiremens such as Solvency II. The laer is he risk-based solvency sysem ha applies o insurance companies in he European Union. Various auhors have done work on he valuaion of surrender opions. They find ha surrender opions can be quie valuable (e.g., Albizzai and Geman 1994; Bacinello 23; Grosen and Jørgensen 2). Anoher branch of he exising lieraure sudies he deerminans and modeling of surrender behavior, wih parly conradicory resuls. 1 e.g., Geneva Associaion (212) do so concenraing on poenial illiquidiy risks in he life insurance secor due o an increase of surrender raes. Surrender risks are also addressed by oher auhors. In a recen sudy, Feodoria and Försemann (215) sressed he poenial risk of a policyholder run in case of a sharp increase of ineres raes. Similarly, Berdin e al. (217) examine he impac of rising ineres raes in combinaion wih he poenial increase of lapse raes on he insurer s liquidiy and solvency. In fac, regulaors have idenified lapse risk as he mos imporan risk among he life underwriing risks (EIOPA 211). Lapse risk under Solvency II reflecs all risks due o changes in he level or volailiy of opion exercise raes. Hence, i appears paricularly imporan for an insurer o appropriaely assess he surrender risks he is exposed o. In his paper, we analyze he recogniion of surrender risk in he quaniaive assessmen under Solvency II. The Solvency II sandard formula recognizes surrender risk in is lapse risk module. The module includes hree differen sress scenarios: an immediae mass lapse even as well as a permanen increase and decrease of he expeced lapse raes. I requires he idenificaion of porfolio segmens ha are exposed o a specific even in he sense ha he respecive sress would increase he insurer s liabiliies. We provide a heurisic ha can be used o derive an appropriae segmenaion in pracical applicaions. We furher compare his heurisic wih he segmenaion approach currenly 1 For an overview of he exising lieraure e.g., see Eling and Kochanski (213) or Campbell e al. (214).

3 Risks 218, 6, 66 3 of 38 used by mos insurance companies in Germany. For his purpose, we se up a sochasic balance shee and cash flow projecion model for a sylized life insurance company. The model covers he relevan feaures of he German marke, considering sauory requiremens as well as ypical managemen rules. We furher discuss surrender risk in he conex of he Risk Margin (RM) which is an imporan par of he Technical Provisions (TP) under Solvency II. To analyze he surrender risk profile of he insurance company, we explicily projec he Solvency Capial Requiremen (SCR) for lapse risk over he lifeime of he insurance porfolio. Based on ha projecion, differen mehods o approximae he risk profile are compared o an alernaive approximaion mehod inroduced in his paper. In paricular, our comparison includes he mehod currenly applied in he German sandard valuaion model for Solvency II, developed by he German Insurance Associaion (GDV 217a). Finally, we exend our analysis o he poenial impac of a lapse sress on he mechanisms inheren in he German life insurance business ha resul in wo special Basic Own Funds (BOF) iems: Going Concern Reserve (GCR) and Surplus Funds (SF). Regarding he former, we sudy how inheriance effecs beween he business in-force and expeced fuure new business change in a lapse sress and how his affecs he GCR. Regarding he laer, we discuss he impac of a lapse sress on he economic value of SF. We paricularly focus on he risk reducing capaciy of collecive bonus reserves available a he valuaion dae, which is he basis for he calculaion of SF. The remainder of his paper is organized as follows. Secion 2 covers he requiremens of Solvency II relevan for subsequen analyses regarding surrender risk. Secion 3 inroduces he valuaion framework underlying he analysis. The resuls of our analysis are presened and discussed in Secion 4. Secion 5 summarizes our findings and concludes. 2. Regulaory Requiremens on he Recogniion of Policyholder Behavior The following secion provides aspecs of he Solvency II framework relevan for subsequen analyses. We furher discuss regulaory requiremens regarding he reflecion of policyholder behavior (risk) for he calculaion of solvency capial requiremens. In paricular, we rely on he Solvency II Direcive (European Union 29) and he laes version of he Solvency II Delegaed Ac (European Commission 215). The main focus is on he reflecion of surrender risk in he lapse risk module of he Solvency II sandard formula General Definiions of Solvency II The main arge of capial requiremens under he Solvency II framework is o ensure ha he insurance underaking will be able o mee is obligaions o policyholders over he following 12 monhs wih a leas 99.5% probabiliy (cf. recial 64 of he Solvency II Direcive (European Union 29)). More precisely, ar. 11 of he Solvency II Direcive (European Union 29) defines he SCR as he Value-a-Risk of he BOF wih a confidence level of 99.5 % over a one-year period. 2 The amoun of economic capial available o cover he SCR is denoed by BOF. 3 By definiion, he BOF are derived from he excess of he underaking s asses over is liabiliies. The marke consisen valuaion of hose iems is based on he concep of a ransfer value (cf. ar of he Solvency II Direcive (European Union 29)). For mos ypes of insurance conracs, he ransfer value of so-called TP for he obligaions owards policyholders is deermined as he sum of he Bes Esimae of Liabiliies (BEL) and he RM (cf. ar. 77 of he Solvency II Direcive (European Union 29)). In general, he BEL equals he expeced presen value of all fuure cash in- and ouflows required o sele he insurance obligaions. I is ypically derived by applying a sochasic balance shee 2 For more deails concerning he definiion of SCR see Chrisiansen and Niemeyer (214). 3 We do no address he allowance of ancillary own funds and eligibiliy resricions under he Solvency II framework. Therefore, BOF coincide wih Eligible Own Funds.

4 Risks 218, 6, 66 4 of 38 and cash flow projecion model. In case of paricipaing life insurance conracs, he BEL includes boh he conracually guaraneed benefis and he expeced Fuure Discreionary Benefis (FDB) from policyholders fuure surplus paricipaion. However, regulaors allow for wo special iems o reflec balancing effecs beween (differen generaions of) policyholders and risk sharing beween policyholders and he insurer, inheren in radiional German life insurance business: GCR and SF. 4 Those iems do no increase he TP, bu hey are par of he BOF insead. The GCR is relaed o he usage of going concern assumpions for he valuaion of he business in-force (cf. ar. 7 of he Solvency II Delegaed Ac (European Commission 215)). I quanifies he inheriance effecs caused by he pre-financing of acquisiion coss of new business (no included in he valuaion) via he business in-force a he valuaion dae. The par of surplus ha (under going concern assumpions) is no expeced o resul in FDB for he business in-force, bu o be used for he pre-financing, is recognized in he GCR. SF are defined as accumulaed profis which have no been made available for disribuion o policyholders, ye. In Germany, his is closely relaed o RfB funds under sauory accouning. To he exen hose funds fulfill he crieria for classificaion as Tier 1 capial, hey shall no be considered as insurance liabiliies (cf. ar. 91 of he Solvency II Direcive (European Union 29)). The economic value of SF is given by he expeced presen value of all cash flows o policyholders resuling from hose iniial funds. The RM represens an add-on o he BEL o ensure ha he TP correspond o he acual ransfer value. Assuming an immediae ransfer of he company s enire porfolio of insurance obligaions, he RM equals he expeced cos of capial for a reference underaking o provide he solvency capial (SCR RU ) required o suppor he insurance obligaions over he lifeime of he ransferred porfolio (cf. ar. 77 of he Solvency II Direcive (European Union 29)). Following he cos-of-capial approach saed in ar. 37 of he Solvency II Delegaed Ac (European Commission 215), he RM is given by: SCR RM = CoC RU, (1) (1 + r ( + 1)) +1 where CoC = 6 % is he Cos-of-Capial rae prescribed in ar. 39 of he Solvency II Delegaed Ac (European Commission 215), r ( + 1) denoes he basic risk-free ineres rae for he mauriy of + 1 years (cf. Secion 3.1), and SCR RU denoes he reference underaking s SCR afer years. In paricular, SCR RU includes all underwriing risks. Under he Solvency II sandard formula, he SCR for a single risk module is derived based on prespecified sresses which are assumed o occur a he valuaion dae. Applying he same valuaion model under sressed assumpions, BEL and BOF afer sress are deermined (assuming he RM remains unchanged). Hence, he SCR for a specific risk (sress) equals he loss of BOF due o he sress, i.e., ( ) SCR (sress) = max BOF BOF (sress) ; Reflecion of Policyholder Behavior in Quaniaive Assessmens under Solvency II Policyholder behavior is explicily addressed in he Solvency II framework. I prescribes ha boh value and risks of embedded policyholder opions mus be aken ino accoun for he quaniaive assessmen of life insurance business under Solvency II. In paricular, ar. 79 of he Solvency II Direcive (European Union 29) saes ha he value of TP mus include he Time Value of Financial Opions and Guaranees (TVFOG). Hence, he impac 4 For deails regarding GCR and SF see Burkhar e al. (215) and Burkhar e al. (217).

5 Risks 218, 6, 66 5 of 38 of policyholder opions mus be refleced in he cash flow projecion for he BEL. Assumpions on fuure policyholder behavior have o be realisic and based on curren informaion (cf. ar. 26 of he Solvency II Delegaed Ac (European Commission 215)). For his purpose, companies have o ake ino accoun how fuure changes in financial and non-financial condiions may affec he exercise of hose opions. In paricular, when analyzing pas policyholder behavior, he impac and he change of such condiions over ime have o be refleced in a prospecive view. The insurer s maerial risks resuling from policyholder behavior are recognized in a separae lapse risk module of he Solvency II sandard formula. Ar. 15 of he Solvency II Direcive (European Union 29) defines lapse risk as he risk of loss, or of adverse change in he value of insurance liabiliies ha resuls from changes in he level or volailiy of opion exercise raes. Please noe ha as par of he underwriing risks, lapse risk mus be aken ino accoun for he calculaion of boh SCR and RM. According o he sandard formula, he SCR for lapse risk equals he maximum capial requiremen based on he following hree lapse sress scenarios (cf. ar. 142 of he Solvency II Delegaed Ac (European Commission 215)): lapse up: a permanen increase of opion exercise raes by 5%; lapse down: a permanen decrease of opion exercise raes by 5%, wih he decrease no exceeding 2 percenage poins; mass lapse: he insananeous disconinuance of 4% of he insurance policies. The lapse risk module does no only cover he risk arising from he surrender opion bu from all policyholder opions ha significanly affec he presen value of fuure cash flows. In deail, he lapse up and lapse down sress mus ake ino accoun all opions ha give a policyholder he righ o fully or parly erminae, surrender, decrease, resric or suspend he insurance cover, bu also he righ o fully or parially esablish, renew, increase, exend or resume he insurance cover. Regarding he laer, he change in he opion exercise rae mus be applied o he rae reflecing ha he relevan opion is no exercised. Under he sandard formula, he lapse sresses are only applied o hose conracs and opions exercise raes ha would resul in an increase of he BEL and a respecive loss of BOF (RM is ignored for his purpose). In some sense, his represens a wors-case scenario. I assumes policyholders o behave sricly adverse from he insurer s perspecive. However, conrary o oher underwriing risks like he moraliy or longeviy risk, i is no per se clear wha lapse risks a cerain conrac bears for he insurance company. In fac, a conrac s relevance for a lapse sress depends on several facors and his has o be refleced in he segmenaion of he insurance porfolio for he hree lapse sresses. Due o he complex profi sharing mechanisms, he idenificaion of porfolio segmens relevan for a cerain lapse sress is paricularly challenging. In fac, a segmenaion on an individual conrac basis is unsuiable for pracical applicaions. Insead, simplified segmenaion approaches are applied. 5 We will discuss his furher in Secion Analysis Framework This secion describes he asse-liabiliy framework used o illusrae our findings. We apply a sochasic balance shee and cash flow projecion model for a sylized life insurance company ha covers he relevan feaures of life insurance in Germany. We exend he model of Burkhar e al. (217), mainly by he policyholder opion o surrender he conrac. Subsequen analyses include boh SF and GCR. The mehod o quanify hose iems is adoped from Burkhar e al. (215) and Burkhar e al. (217), respecively. 5 See commens given by he German Insurance Associaion regarding he required segmenaion (GDV 215). 6 Please noe ha alhough his paper concenraes on he opion o surrender a conrac prior o mauriy, hose challenges apply o all ypes of opions, e.g., annuiizaion or premium indexaion opions.

6 Risks 218, 6, 66 6 of 38 To se up he economic balance shee, asses and liabiliies are projeced unil complee run off of he iniial business in-force. 7 Due o he complexiy of he sochasic fuure cash flows, he BEL canno be deermined using closed-form soluions. Insead, he valuaion is based on Mone Carlo simulaions, using a risk neural valuaion approach. Please noe ha he following model descripion addresses he valuaion a he curren valuaion dae T Val =. Analyses in Secion 4 include SCR and RM, wih he laer requiring a projecion of fuure SCRs. Accordingly, he economic balance shee is recalculaed under differen sress scenarios and for valuaion daes T Val > Financial Marke Model The insurance company invess in wo ypes of risky asses: coupon bonds and socks. Therefore, we inroduce a fricionless and coninuous financial marke under he risk neural measure Q, wih a shor rae process (r ) > following he Vasicek model and a sock price (S ) > process following a geomeric Brownian moion: dr = κ (θ r ) d + σ r dw (1), ds = S ( r d + ρσ S dw (1) + 1 ρ 2 σ S dw (2) The iniial values r and S are deerminisic. The parameers κ, θ, σ r, σ S and ρ are also deerminisic and consan. W (1) and W (2) are wo uncorrelaed sandard Wiener processes adaped o a filraion F on some probabiliy space (Ω, F, Q), which saisfies he usual condiions. Par yields ha deermine he coupon raes of he considered coupon bonds can be derived from he discreely compounded yield curve a ime (e.g., Branger and ( Schlag 24). ) The bank accoun a ime is given by B = exp r udu. I also serves as he discoun facor in he risk-neural framework such ha he marke value a he valuaion dae T Val = of a sochasic cash flow Y occurring a ime is given by PV (Y ) = E Q ( Y B ). Thereby, E Q ( ) denoes he expeced value under he risk neural measure Q. In addiion o he sochasic valuaion, we also consider he so-called Cerainy Equivalen (CE) scenario which represens he expeced developmen of he financial marke under Q (Oechslin e al. 27). In each ime period ( ; + 1], all asses are assumed o earn he forward rae f (, + 1) implied by he iniial yield curve (r (s)) s> wih ). r (s) = ( s 1 1s (1 + f (, + 1))) 1. = The presen value is hen given by PV [CE] (Y ) = Y[CE] (1+r ()), where Y [CE] denoes he respecive cash flow in he CE scenario Liabiliy Model The considered company sells radiional paricipaing endowmen policies agains annual premium paymens, wih a conrac duraion of n years and policyholder s age x a incepion of he conrac. The policies come wih an annual minimum ineres rae guaranee i and provide a guaraneed benefi G (sum insured) a mauriy or deah. Policyholders have he opion o surrender heir conrac, receiving a predeermined surrender value. The premium also covers iniial acquisiion 7 The acual valuaion does no consider new business. However, assumpions regarding expeced new business are required o deermine he GCR.

7 Risks 218, 6, 66 7 of 38 charges α (as percenage of premium sum), amorizaion charges α γ (as percenage of sum insured) and adminisraion charges β (as percenage of premium) Premium Calculaion and Reserving The company s insurance porfolio consiss of several cohors k of idenical conracs, all concluded a he beginning of year = k + 1. Cohors may differ in he calculaion assumpions applied, in paricular regarding he guaraneed ineres rae and charges, which is denoed by (k) ( ). Applying he acuarial principle of equivalence, he annual premium (k) P of cohor k is given by (k) G ( (k) A P = x:n + (k) α γ (k) ä x:n ). (1 (k) β) (k) ä x:n (k) α n Thereby, (k) A x:n and (k) ä x:n represen he presen values of an n year endowmen insurance and annuiy-due of an x year old wih guaraneed ineres (k) i. Following he so-called Zillmerisaion procedure, he acuarial reserve (k) AR a he end of year = k + 1,..., k + n evolves according o (k) AR = ((k) AR 1 + (k) P (1 (k) β) G (k) α γ ) (1 + (k) i) G q x+( k) 1 1 q x+( k) 1, where (k) AR k = (k) α n (k) P and q x denoes he firs-order probabiliy rae of an x year old o die wihin one year (e.g., Führer and Grimmer 21). The minimum surrender value (k) SV, policyholders receive in case of surrender, is calculaed in line wih 169 of he Insurance Conrac Law (VVG) which requires he spreading of iniial acquisiion charges over five years. Ignoring surrender charges, his implies ha (k) SV = max ( ; (k) ÃR ) wih (k)ä (k) AR + (k)ãr (k) α n (k) x+( k):5 ( k) P = (k)ä for k k + 4, x:5 (k) AR for k + 5 k + n. The difference (k) ZR = (k) SV (k) AR is shown as a Zillmer receivable on he asse side of he sauory balance shee; i is amorized over he firs five years of he conrac Surplus Paricipaion and Benefi Paymens Besides he guaraneed benefis, policyholder paricipae in he company s annual surplus. The bonuses (k) bon of year for policyholders of cohor k are credied in wo ways: ongoing bonuses (k) bon acc ha are accumulaed in an ineres-bearing bonus reserve (k) BR, i.e., (k) BR = (k) BR 1 (1 + (k) i) + (k) bon acc (2) wih (k) BR k = ; erminal bonuses (k) bon erm, allocaed o he erminal bonus fund (k) TBF, ha evolves according o (k) TBF = (k) TBF 1 + (k) bon erm (k) 14, TBF CF Policyholders do no have a claim on he Terminal Bonus Funds (TBF) unil hey are declared for payou in he nex year. Funds may even be wihdrawn from he TBF according o 14 Insurance 14, TBF Supervision Law (VAG, Versicherungsaufsichsgesez) in case of adverse evens ( (k) CF ).

8 Risks 218, 6, 66 8 of 38 In case of surrender, he policyholder s deah, or a mauriy of he conrac, bonus reserve and TBF are paid ou on op of he conracually guaraneed benefis. In oal, he benefis (liabiliies) a he end of year add up o (k) L = (k) BR + (k) L erm + { G in case of deah or a mauriy of he conrac, (k) SV in case of surrender, wih (k) L erm = (k) TBF Liabiliy Porfolio Developmen Before mauriy of he conrac, policyholders may leave he insurance porfolio eiher due o deah or if hey surrender he conrac. 8 Second-order (bes esimae) moraliy raes are assumed o equal he firs-order moraliy raes q x adjused by a facor q. Hence, (k) l = (k) l 1 (1 q q x+( k) 1 (k) s ) policyholders of cohor k remain in he porfolio a he end of year = k + 1,..., k + n 1, wih (k) lk+n =. Thereby, (k) s denoes he rae of a policyholder of cohor k o surrender his conrac in year. The overall size of he company s insurance porfolio a ime is given by l n 1 ( k) = l. k= In he same way, respecive figures for he enire insurance porfolio are derived by summing up relevan figures deermined for each cohor Surrender Model As menioned in Secion 2.2, TP under Solvency II mus accoun for he value of surrender opions. In academic lieraure, here is a broad range of approaches for modeling and valuaion of surrender opions. The assumpion of a solely financially raional policyholder is of raher heoreical naure. For pracical applicaions, he valuaion models used o deermine he BEL ypically apply wha Eling and Kochanski (213) call dynamic lapse rae models ha allow for subopimal policyholder behavior. In hese models, policyholder behavior is no modeled as he opimal sopping ime problem for a raional risk-neural or risk-adverse invesor. Insead, deerminisic lapse raes are adjused by dynamic, (financial marke) scenario specific facors, so-called dynamic lapse mulipliers. A benchmark ha is used in academic lieraure as well as by praciioners is he spread beween a marke rae and he oal yield of he policy (Clark e al. 213). Following he ineres rae hypohesis, lapse raes are assumed o increase if policyholders receive a low yield compared o he marke rae and vice versa (e.g., Kuo e al. 23). In line wih his approach, he dynamic lapse muliplier in our model is based on he spread (k) i surr = r (T surr ) (k) i oal 1 beween he curren T surr -year spo rae r (T surr ) and he conrac s oal yield credied in he previous year (k) i oal 1 = (k) i + (k) i 1. I is he sum of he echnical ineres rae and he bonus rae declared a he end of he previous year for a policyholder of cohor k. We assume ha policyholders do no change heir behavior before he spread exceeds a cerain olerance hreshold i surr >, i.e., (k) i surr > i surr. 8 For echnical reasons, boh deah and surrender occur a he end of he year.

9 Risks 218, 6, 66 9 of 38 In his case, he deerminisic lapse rae (k) ŝ for a policyholder of cohor k o surrender his conrac in year is adjused as follows: (k) s = (k) ŝ 1 ± ( (k) i surr i surr ) q surr 1, if (k) i surr i surr 1, else, where q surr denoes he policyholders ineres rae sensiiviy Cos Model The coss C arising a he beginning of each year can be decomposed ino one-off acquisiion coss AcC for new business and ongoing adminisraion coss AdC = var AdC l, derived by a per policy adminisraion cos rae var AdC. 1 In addiion, claims selemen coss CC +1 = C claims (l l +1 ) occur a he end of year in case of deah, surrender, or a mauriy of he conrac. The model also reflecs commission refunds from inermediaries. In case of surrender wihin he cancellaion liabiliy period T cancel, he inermediary has o repay a par of he commission α (as a percenage of he premium sum) received for his conrac. In our model, he par of he commission ha is refunded is linearly decreasing wihin he cancellaion liabiliy period. Hence, he oal commission refunds a he end of he year sum up o: Asse Model CRe f T cancel 2 ( ) +1 = ( k) s +1 ( k) l α n ( k) P Tcancel (k + 1) T k= cancel The company invess in socks and coupon bonds yielding a par wih fixed iniial erm T B. A he end of each year, he asse porfolio has o be adjused based on he cash flows CF + a he beginning of he year (cash flow o shareholders X, premium paymens P, and coss incurred AcC + AdC ) and he cash flows a he end of he year (coupon paymens CP +1, nominal repaymen of bonds a mauriy N (+1 T B) +1, commission refunds CRe f +1, benefi paymens L +1, and claims selemen coss CC +1 ). Over he year, CF+ is invesed in a risk-less bank accoun earning he ineres rae r (1). The asse porfolio is furher rebalanced o achieve a consan arge sock raio q sock (in erms of marke values). If necessary, bonds are sold proporionally o heir marke values. The company may realize Unrealized Gains or Losses (UGL) due o he rebalancing. If he remaining UGL exceed a limi q+ UGL, a cerain porion d of he unrealized gains on socks is realized o sabilize he invesmen reurn. Unrealized gains are furher realized o avoid/reduce losses for he insurer due o a negaive invesmen surplus. Please noe ha unless hey exceed he limi q UGL, unrealized losses on socks are no realized. The overall invesmen reurn is given by R +1 = CF+ r (1) + CP +1 + UGL real +1, where UGL real +1 denoes he realized porion of he UGL. 9 Our surrender model is similar o he approach used in he German sandard valuaion model for Solvency II. Furher surrender funcions can e.g., be found in Kochanski (21). 1 The valuaion does no include new business and acquisiion coss are only relevan for he calculaion of he GCR (see Secion 3.8.2). 11 Please noe ha he par of he commission ha has o be repaid in case of surrender is similar bu no idenical o he Zillmer receivable.

10 Risks 218, 6, 66 1 of Surplus Disribuion The surplus disribuion process applied in our model reflecs he corresponding regulaory consrains. Each year, he company has o decide how o spli raw surplus beween shareholders and policyholders and o which exen bonuses are allocaed o policyholders in he nex period Sources of Surplus Annual raw surplus Sp +1 is derived based on sauory accouning rules. I consiss of invesmen surplus, risk surplus, cos surplus, and surrender surplus. The sources of surplus are derived from he difference beween pruden assumpions applied for premium calculaion and acual realizaions experienced over he year, i.e.,: invesmen surplus, represening he difference beween invesmen reurn R +1 and guaraneed reurn R gar +1 : Sp+1 I n 1 ( ( k)l = R +1 Rgar +1 = R +1 ( k) AV i) ( k), where he policyholders accoun value a he beginning of year (afer premium paymen) is given by ( k) AV = ( k) AR + ( k) BR + ( k) P (1 β) G ( k) α γ ; risk surplus, represening he difference regarding moraliy: k= Sp+1 R n 1 ( )) = (1 q ) q x+( k) ( k) l (G ( k) AR +1 ; k= cos surplus, represening he difference beween charges included in he premium and acual coss incurred: Sp C n 1 (( ) ) +1 = β ( k) P + ( k) α γ G ( k) l C CC +1 ; k= surrender surplus, represening he surplus due o acual surrender, consising of he ne benefis o be paid o policyholders and he possible refunds received from he inermediary (wih no surrender raes being included in he premium calculaion): Sp S n 1 ( ( k)s ( +1 = +1 ( k) l ( k)ar+1 )) ( k) SV +1 + CRe f +1. k= The sum of he laer wo is denoed by oher surplus Sp O +1 = SpC +1 + SpS Spliing of Surplus In a firs sep, he annual raw surplus is spli beween he insurance company and he policyholders based on a arge annual reurn i Eq on shareholders equiy (from he sauory balance shee). 12 Policyholders receive he remaining par of surplus PS +1 = max (Sp +1 i Eq Eq loc ; PS min +1 ) bu a leas a minimum share of { [ ( )] PS min +1 = max ; min R +1 Rgar +1 ; max 9 % R +1 Rgar +1 ; ) )} + max (9 % Sp+1 R ; + max (5 % Sp O +1 ;. 12 This managemen rule is in line wih common pracice bu differen from Burkhar e al. (217).

11 Risks 218, 6, of 38 In case of losses originaing from invesmen surplus, profis from oher surplus sources can be used o offse hose losses o he exen ha policyholders share of surplus remains non-negaive. This reflecs he Execuive Guidance Order on Minimum Surplus Paricipaion (MindZV, Mindeszuführungsverordnung). If minimum surplus paricipaion rules are no me, he insurer s share of surplus is reduced accordingly. In case of a negaive raw surplus, losses may parly be passed o policyholders by applying 14 VAG. Hereby, funds are wihdrawn from he free RfB firs and, if necessary, also from he TBF. Losses are spli in he same proporion q+1 PH as surplus has been spli in he pas T 14 years. Of course, wihdrawals are limied o he funds available. Hence, he oal wihdrawal CF from he undeclared RfB is given by he sum of 13 wih q PH ( +1 = min CF 14, f R f B CF 14,TBF +1 = min T = s=1 PS +1 s T 14 s=1 max (Sp +1 s ; ) f R f B +1 + PS +1 ; q PH +1 max ( Sp +1 ; ) ), ( TBF L erm +1 ; qph +1 max ( Sp +1 ; ) CF 14, f R f B +1. Please noe ha for echnical reasons, erminal bonuses bindingly declared in he previous year o be paid ou a he end of he curren year only remain par of he TBF unil he paymen L+1 erm is made. Hence, hose funds are no available o cover losses. The shareholder cash flow X +1 resuls in a respecive cash ou-/inflow a he beginning of he nex year. Besides he share of surplus (including emergency wihdrawals from he RfB), i also includes he developmen of shareholders equiy over ime. Hence, i is given by Declaraion of Surplus X +1 = ( ) ( ) Sp +1 PS CF +1 + Eq loc Eq+1 loc. (3) A he end of he year, he insurance company deermines he amoun o be wihdrawn from he RfB and credied o policyholders accouns in he nex period via ongoing and erminal bonuses. The oal amoun bon +1 = bon+1 I + bon+1 R + bon O +1 is derived from he allocaion of invesmen, risk, and oher surplus o he RfB in he previous T bon years. Neverheless, he reserve raio beween he free RfB (afer he [ declaraion) ] and he accoun value (a he beginning of he year) mus remain in a cerain corridor q f R f B min ; qmax f R f B. Oherwise, he planned declaraion is adjused accordingly. A fixed porion q TBF of he declared bonuses is passed on o policyholders via erminal bonuses bon+1 erm. Please noe ha for simplificaion, declared ongoing bonuses bon+1 acc are immediaely allocaed o he policyholders bonus reserves Allocaion of Surplus o Individual Policyholders In he las sep of surplus disribuion, declared bonuses are allocaed o individual conracs. We apply a so-called naural allocaion sysem (Wolfsdorf 1997). This implies ha all conracs of a cerain cohor k receive he same bonus (k) bon +1, bu he amoun may differ beween conracs from differen cohors. Toal bonuses are allocaed o individual conracs as follows: 14 Invesmen bonuses bon+1 I are disribued such ha all policyholders receive he same oal yield (sum of he guaraneed ineres rae (k) i and he bonus rae (k) i+1 ) on heir accoun values. However, if invesmen bonuses are no sufficien for all policyholders o receive a oal yield ), 13 Please noe ha since he offseing of differen surplus sources is limied o he case of negaive invesmen surplus, i is generally possible o have a posiive surplus paricipaion PS +1 despie a negaive raw surplus Sp For a more deailed descripion of he surplus allocaion process see Burkhar e al. (215).

12 Risks 218, 6, of 38 above heir minimum guaraneed ineres rae, bonus raes of cohors wih a lower guaraneed ineres rae have o be reduced accordingly. Risk bonuses bon R +1 are allocaed based on he capial a risk (G (k) AR +1 ). Oher bonuses bon O +1 are allocaed based on he premium (k) P Sauory Balance Shee A he end of each year, he sauory balance shee can be se up as presened in Table 1. The asse side consiss of he book value of he asses (socks and bonds) and he Zillmer receivables. On he liabiliy side, he shareholders equiy equals a fixed percenage q loc of he acuarial reserves AR +1. TBF and free RfB a he end of he year evolve according o TBF +1 = TBF L erm +1 14,TBF CF +1 + bon+1 erm f R f B +1 = f R f B + PS +1 CF 14, f R f B +1 bon +1. Table 1. Sauory balance shee a ime + 1. Asses Liabiliies BV+1 A Eq+1 loc X +1 ZR +1 SV +1 BR +1 TBF +1 f R f B Economic Balance Shee Based on he sochasic projecion of he company unil complee run off of he business in-force, he economic balance shee a he valuaion dae T Val = can be se up for he base case. We also describe he adjusmens in case of allowance for GCR or SF Base Case The asse side of he balance shee conains he marke value MV A of he insurer s invesmens in bonds and socks, which is derived from he financial marke daa a he valuaion dae. Before consideraion of he RM, he liabiliy side of he economic balance shee conains he following iems: he BOF, which can be decomposed ino he Presen Value of Fuure Profis (PVFP) and he shareholders equiy Eq loc, i.e., BOF = PVFP + Eq loc; he BEL, represening he insurer s fuure obligaions from he business in-force. Based on J realizaions of he sochasic capial marke, PVFP and BEL are esimaed by PVFP = 1 J BEL = 1 J J j=1 J j=1 T 1 X [j] +1 = B [j] +1 T 1 = AdC [j] Eq loc, (4) B [j] P [j] + L[j] +1 + CC [j] [j] +1 CRe f +1 B [j] +1, (5) where ( ) [j] denoes he respecive values/cash flows in scenario j. Thereby, he PVFP also includes he presen value of expeced 14 wihdrawals from he RfB (cf. Equaion (3)). The economic balance shee can also be se up for he CE scenario as inroduced in Secion 3.1. The resuling BEL [CE] will be used o quanify he asymmery of he insurer s liabiliies due o he

13 Risks 218, 6, of 38 financial opions and guaranees included in he insurance conracs. From a policyholders perspecive, he TVFOG can be defined as 15 TVFOG = BEL BEL [CE]. Please noe ha he fuure benefis (k) L o be paid o a policyholder of cohor k leaving he company a ime can be decomposed ino he (guaraneed) benefis already locked-in a he valuaion dae, (k) L Gar = (k) BR (1 + i) + { G, in case of deah or a mauriy of he conrac (k) SV, in case of surrender and he remaining benefis (k) L FDB = (k) L (k) L Gar resuling from he policyholders fuure surplus paricipaion. In oal, he BEL can be divided ino he Bes Esimae of Guaraneed Obligaions (BE Gar ), which covers he coss incurred, premiums, and guaraneed benefis, he FDB [CE] from policyholder surplus paricipaion (boh calculaed under he CE scenario), and he TVFOG. Hence, he BEL a he valuaion dae is given by BEL = BE Gar + FDB [CE] + TVFOG. The TVFOG can furher be decomposed ino he Time Value of Financial Opions (TVO) due o he policyholders dynamic behavior and he Time Value of Financial Guaranees (TVG) due o policyholders minimum ineres and surplus paricipaion guaranees, wih he laer being given by TVG = TVFOG TVO. The former is derived as TVO = ( ) ( BEL BEL BE Gar BE Gar) (FDB [CE] FDB [CE]) where ( ) denoes he respecive values based on a cash flow projecion ha does no allow for dynamic policyholder behavior. Please noe ha he ime value of he policyholder opions is parly included in he deerminisic CE values BE Gar and FDB [CE] Allowance for Going Concern Reserve The mehod used o quanify he GCR in our model is adaped from Burkhar e al. (215). The basic idea is o compare he projeced fuure cash flows relaed o he business in-force beween a siuaion wih fuure new business (going concern) and wihou new business (run off). For his purpose, he projecion model is run under wo differen ses of cos assumpions. Ongoing adminisraion coss are he same for boh runs. The going concern cos assumpions reflec he addiional cos burden of he business in-force due o he pre-financing of uncovered acquisiion coss by including arificial acquisiion coss AcC > in he projecion. The addiional coss are derived from a hird run ha includes new business. In his run, he amoun of uncovered acquisiion coss o be pre-financed is deermined. This amoun is spli beween he business in-force and he expeced new business based on he premium income from he amorizaion charges α γ of each sub-porfolio. The former deermines he arificial acquisiion coss relevan for our going concern cos assumpions. Please noe ha he arificial acquisiion coss also consider he poenial impac of he expeced new business on he average ongoing adminisraion cos per policy. 15 Please noe ha he Marke Consisen Embedded Value framework (CFO Forum 216) defines he TVFOG from he shareholders perspecive (based on he PVFP), bu in our model framework boh approaches resul in he same value.

14 Risks 218, 6, of 38 In oal, GCR equals he valuaion differences beween he run off (RO), ha is equal o he base case, and he going concern (GC) run, given by GCR = ( ) ( ) PVFP RO + FDB RO PVFP GC + FDB GC. The allowance for a GCR increases BOF and reduces TP such ha Allowance for Surplus Funds BOF GCR = PVFP GC + Eq loc + GCR, BEL GCR = BEL GC. The economic value of SF is deermined according o he mehod developed in Burkhar e al. (217). The wihdrawals from he RfB are racked unil he oal of hose wihdrawals (undiscouned) exceeds he undeclared RfB a he valuaion dae, i.e., f R f B + TBF. Afer he developmen of he iniial RfB and he respecive wihdrawals have been idenified, he economic value of SF is deermined as he presen value of benefi paymens L SF resuling from hose wihdrawals. Compared o he wihdrawals, he acual benefi paymens are delayed by several years and also include he ineres earned on hose wihdrawals unil paid ou. Please noe ha he erminal bonus paymens L1 erm paid ou a he end of he firs year afer he valuaion dae should no be considered in he calculaion of SF, since hey have already been bindingly declared a he end of he previous period and are no available o cover fuure losses. Applying an ex-pos approach, he preliminary value of BEL is derived from he base case (wihou consideraion of SF) and subsequenly he amoun of SF is deduced. 17 Hence, BOF SF = PVFP + Eq loc + SF, BEL SF = BEL SF. Thereby, he oal value of SF is limied o he nominal value of he relevan par of he RfB a he valuaion dae, such ha SF = min 4. Numerical Resuls and Discussion [ f R f B + TBF L erm 1, 1 J J j=1 T =1 [j] L SF B [j] Focusing on surrender opions, he following secion analyzes differen aspecs of lapse risk under he Solvency II sandard formula. We discuss he regulaory requiremens regarding lapse risk and implicaions on he quaniaive assessmen of paricipaing life insurance under Solvency II. To illusrae our findings, we presen numerical resuls, based on he model inroduced in Secion 3. The assumpions underlying our model are presened nex Model Assumpions The valuaion dae is 31 December 216. The iniial insurance porfolio is buil in line wih he hisoric developmen in Germany (cf. Table A1 in Appendix A). We assume ha in he pas (k) l k = 1 idenical endowmen policies were sold a he beginning of each year k + 1. ]. 16 Please noe ha AcC are no included in he cash flows used o deermine BEL (cf. Equaion (5)). 17 SF can easily be combined wih he GCR via he ex-pos approach.

15 Risks 218, 6, of 38 A he valuaion dae, he porfolio conains 24 cohors of conracs from ariff generaions o 6 presened in Table 2, wih ime o mauriy from 1 o 24 years. The ariff generaions are se up in line wih sauory provisions regarding he maximum echnical ineres rae i and he maximum Zillmer rae α. Tariff generaions 6 and 7 include increased α γ -charges o compensae for he reducion of he sauory Zillmer rae. The ariff generaions furher reflec he coninuous decline of he average adminisraion cos rae of he German life insurance marke in pas years (cf. Table A1 in Appendix A). The ime horizon for he projecion is T = 24 years. Table 2. Insurance conrac parameers. Tariff Years of Sale x n G i α α γ β Moraliy Table AC 2, 3.5 % 4. %.1 % 6. % DAV 28 T a AC 2, 4. % 4. %.1 % 4.5 % DAV 28 T AC 2, 3.25 % 4. %.1 % 3.5 % DAV 28 T AC 2, 2.75 % 4. %.1 % 3.5 % DAV 28 T AC 2, 2.25 % 4. %.1 % 3. % DAV 28 T AC 2, 1.75 % 4. %.1 % 2.5 % DAV 28 T AC 2, 1.25 % 2.5 %.2 % 2.5 % DAV 28 T 7 since AC 2,.9 % 2.5 %.2 % 2.5 % DAV 28 T a This is he German sandard moraliy able. The acuarial reserve AR, he bonus reserve BR, and TBF are derived from a projecion in a deerminisic scenario which is based on hisoric daa from he German life insurance marke concerning ne invesmen reurn and cos parameers (cf. Table A1 in Appendix A). The iniial free RfB equals 2.5 % of he sum of acuarial and bonus reserve. The book value of he asse porfolio coincides wih he book value of liabiliies (including shareholders equiy). We assume a sock raio of q = 1 % wih unrealized gains on socks of 25 % of he book value of socks. The coupon bond porfolio a = consiss of bonds wih coupons based on hisoric yields, derived from he erm srucure of ineres raes on covered bonds wih annual coupon paymens and iniial erm of 12 years (Bundesbank 217). The ime o mauriy is equally spli beween 1 and T B = 12 years. The resuling sauory balance shee a ime = is given in Table 3. Table 3. Sauory balance shee a ime = (in AC 1). Asses Liabiliies BV A 18,98 Eq loc 1895 ZR 988 SV 95,746 BR 767 TBF f R f B 2546 The bes esimae moraliy raes are assumed o equal q = 7 % of he firs-order raes q x. For he bes esimae surrender raes, we assume deerminisic surrender raes ha only depend on he curren year of he conrac and are he same for all cohors k. As done in Kling e al. (214), we assume declining raes over he firs five years of he conrac and a consan base rae hereafer (cf. Table 4). The reference value for he dynamic lapse mulipliers is he T surr = 5-year spo rae. If he spread (k) i surr exceeds i surr = 1.5 %, he deerminisic surrender rae changes by q surr = 15 % per 1 basis poins. Under going concern assumpions, he surrender model calibraion leads o a oal surrender rae of 4.2 % of he gross wrien premium income in he firs projecion year, which is in line wih he average rae of he German life insurance marke in 216 (GDV 217b).

16 Risks 218, 6, of 38 Table 4. Deerminisic lapse raes. Conrac Year m Deerminisic surrender rae (k) ŝ k+m 8.3 % 7.3 % 6.3 % 5.3 % 4.3 % 3.3 % Regarding he bes esimae cos parameers, he adminisraion coss var AdC per policy are chosen such ha under going concern assumpions he oal annual adminisraion cos rae including claims selemen coss of Cclaims = AC 5 equals 2.3 % of he gross wrien premium income. Similarly, he GCR calculaion assumes one-off acquisiion coss AcC ha resul in an acquisiion cos rae of 4.8 % of he new business premium sum. The one-off acquisiion coss include a commission paymen of α = 4. %. Boh adminisraion and acquisiion cos rae are in line wih he average of he German life insurance marke in 216 (cf. GDV 217b). Since Solvency II requires a run-off valuaion, we do no consider any new business. However, our analysis regarding he GCR assumes ha under bes esimae assumpions (k) lk = 1 new conracs, belonging o ariff generaion 7, are concluded a he beginning of each year. The parameers for he managemen rules are presened in Table 5. The managemen rules are consisen wih curren regulaion. The parameers are adoped from Burkhar e al. (217). Minor adjusmens were made o beer fi he curren siuaion in he German life insurance marke. Based on he hisoric developmen of he reurn on equiy in he German life insurance marke, we assume a arge annual reurn of i Eq = 1 % (cf. BaFin 213). Table 5. Parameers for managemen rules. q sock T B d q UGL + q UGL T 14 T bon q f R f B min q f R f B max q TBF q loc i Eq 1 % 12 years 5 % 15 % 15 % 1 years 5 years 1 % 4 % % 1 % The financial marke parameers for he projecion are shown in Table 6. The parameers are adoped from Reuß e al. (215). Adjusmens were made o beer fi he Solvency II ineres rae erm srucure a he end of 216. Besides he base case, we also conduc a sensiiviy analysis wih respec o he ineres rae level, o illusrae furher aspecs of lapse risk under Solvency II. In fac, we reduce boh he iniial shor rae r and he mean reversion level θ by 5 basis poins. Table 6. Financial marke parameers. Scenario r θ κ σ r σ S ρ Base case.5 % 4.2 % 2. % 2. % 2. % 15. % Ineres rae sensiiviy 1. % 3.7 % 2. % 2. % 2. % 15. % The sochasic projecion is performed for 5 scenarios of he financial marke. Furher analysis showed ha his allows for a precise esimaion of he relevan figures (Glasserman 21). Based on he bes esimae parameers, he valuaion resuls in he economic balance shee (before RM) a ime = as presened in Table 7. In case of he reduced ineres rae level, he marke value of he iniial asse porfolio increases by On he liabiliy side, i resuls in reduced surplus paricipaion for policyholders FDB [CE] ( 328). A he same ime, he insurer s guaraneed obligaions BE Gar (+6745) as well as he TVFOG (+877) increase. In oal, he BEL increases by 4342, reducing he insurer s BOF close o zero ( 1215).

17 Risks 218, 6, of 38 The able furher shows ha for boh ineres rae levels dynamic policyholder behavior plays a minor role. The TVO makes only up for a small porion of he TVFOG, wih he mos par resuling from he policyholders guaranees TVG. 18 Table 7. Economic balance shee a ime = (in AC 1). Base Case Ineres Rae Sensiiviy Asses Liabiliies Asses Liabiliies MV A 116,337 BOF 1236 MV A 119,464 BOF 21 BEL 115,11 BEL 119,443 BE Gar 98,785 BE Gar 15,53 FDB [CE] 13,934 FDB [CE] 1,654 TVFOG 2382 TVFOG 3259 TVO 12 TVO 211 TVG 2262 TVG 348 ( ) ( ) TVFOG TVFOG FDB [CE] (17 %) FDB [CE] (31 %) Following he sandard formula, we ake ino accoun all hree lapse sresses. To calculae he SCR for lapse risk, he cash flow projecion and valuaion is performed for each sress, wih bes esimae surrender raes being adjused accordingly (cf. Secion 2.2). Please noe ha for echnical reasons, he mass lapse sress in our model occurs a he end of he year, coinciding wih he regular surrenders of our model Surrender Risk in he Conex of Solvency Capial Requiremens As described in Secion 2.2, he lapse sresses are only o be applied o hose conracs and opion exercise raes ha resul in an increase of he insurer s liabiliies. Concenraing on surrender opions, we analyze and compare wo differen segmenaion approaches o idenify relevan porfolio segmens Segmenaion Alernaive 1: Change of BE Gar The firs segmenaion mehod only akes ino accoun he insurer s guaraneed obligaions owards policyholders. I represens he segmenaion approach currenly used by mos insurance companies in Germany. The relevance of a conrac for a cerain lapse sress scenario is evaluaed based on he change of guaraneed obligaions owards policyholders in he considered sress scenario. This means conracs are sressed if and only if he insurer s guaraneed liabiliies increase under he considered sress scenario. For a conrac of cohor k he change of (k) BE Gar due o a sress a ime equals (k) Gar (sress) = (k) BE Gar (sress) (k) BE Gar. Thereby, (k) Gar (sress) > implies ha he sress increases he insurer s guaraneed liabiliies for conracs of cohor k and hese conracs have o be sressed for he calculaion of SCR (sress). Please noe ha (k) Gar (sress) is linked o he change of expeced profiabiliy, measured in erms of he Presen Value of Margins (Mrg) included in he (pruden) acuarial assumpions applied for pricing and reserving of he conrac. In general, he pruden assumpions resul in sysemaic profis over he lifeime of he conrac, since he calculaed premiums exceed expendiures for 18 The raher small value of TVO is in line wih curren observaions from pracical applicaions.

18 Risks 218, 6, of 38 benefi paymens and coss incurred. However, hose profis depend on he developmen of he sochasic environmen. If safey margins included in he iniially deermined premiums urn ou o be insufficien, he insurance company may incur losses over ime. Technically speaking, he profis or losses expeced o be realized in he fuure can be characerized by he difference beween he presen value of expeced fuure premiums minus guaraneed benefi paymens and coss, using pruden (sauory accouning) and realisic (Solvency II) acuarial assumpions. Hence, he presen value of margins (k) Mrg for a conrac of cohor k a ime can be calculaed as ( (k) (k)ar ) Mrg = + (k) BR (k) BE Gar. Since he sauory reserves do no chance due o a sress a ime, he difference beween he margins expeced under bes esimae assumpions and in case of a lapse sress scenario is given by (k) Mrg (sress) = (k) Mrg (sress) (k) Mrg = (k) BE Gar (k) BE Gar (sress) (k) (sress) = Gar. Hence, he increase (decrease) of he insurer s guaraneed liabiliies due o a sress corresponds o a respecive decrease (increase) of expeced fuure margins of his conrac. Al. 1 only depends on paymens already guaraneed a he ime of he sress. I does no ake ino accoun any fuure inerdependencies beween differen insurance conracs, i.e., (k) Gar (sress) of cohor k is independen of he remaining insurance porfolio. Therefore, he mehod does no require a sochasic simulaion. Insead, wo projecions under he CE scenario are sufficien, using bes esimae or sressed assumpions. In fac, he mehod allows a simulaneous analysis of all conracs wihin a single projecion of he insurance porfolio. To avoid inerdependencies beween conracs, he projecion does no allow for dynamic policyholder behavior (which in urn depends on porfolio level facors such as a conrac s annual oal yield). Table 8 provides he margins (k) Mrg per conrac for each cohor under bes esimae assumpions and he change of guaraneed obligaions (k) Gar (sress) in he considered lapse sress scenarios. The cohors belonging o differen ariff generaions are separaed by horizonal lines. We observe maerial differences beween margins of differen ariff generaions. This implies a srong dependence of a cohor s profiabiliy on is echnical ineres rae. A higher guaraneed ineres rae leads o higher annual ineres obligaions and herefore lower margins from invesmen income. Overall, margins are posiive for cohors 1 o 11 ( (k) i 2.75 %) and negaive for cohors 12 o 24 ( (k) i 2.75 %). Table 8 also shows a dependence of a conrac s profiabiliy on is remaining duraion. For cohors 6 o 16, he profiabiliy of conracs wih he same calculaion basis decreases wih decreasing remaining duraion. 19 Due o he raher low echnical ineres raes of hose cohors, younger conracs creae posiive margins during addiional conrac years. Conversely, margins are increasing from cohors 19 o 22 and cohors 23 o 24 (alhough he remaining duraion decreases). Apparenly, hese conracs creae negaive margins during addiional conrac years, which is linked o he high echnical ineres rae. Margins also increase from cohor 1 o 2 and cohor 3 o 4 (alhough he remaining duraion decreases). This is linked o ineracions beween Zillmer receivables and commission refunds wihin he firs 5 years afer incepion of he conrac. In line wih he margin before sress, a decrease of margins ( (k) Gar (sress) > ) in he mass lapse and he lapse up scenario can be observed for younger cohors wih a raher low echnical ineres rae (k) i 2.75 %. However, due o heir differen naures, he resuling segmenaions (prined in 19 Please noe ha margins increase from cohors 16 o 14, 13 o 11, and 1 o 6.

19 Risks 218, 6, of 38 bold ype) beween hese wo lapse sresses slighly differ (mass lapse: cohors 1 o 12; lapse up: cohors 1 o 13). Furhermore, excep for cohors 11 o 13, he change in margins is considerably more pronounced for he insananeous 4 per cen mass lapse sress. For he lapse down sress, decreasing margins are observed for cohors 14 o 23 (wih a echnical ineres rae (k) i 3.25 %). Overall, he absolue change in margins of he wo permanen lapse sresses are fairly similar, wih he change being slighly more pronounced for he lapse down sress. Table 8. Resul of segmenaion alernaives 1 and 2 (in AC 1). k (k) Mrg Al. 1: (k) Gar (sress) Al. 2: BEL (sress) (Sand-Alone Sress) Mass Lapse Lapse Up Lapse Down Mass Lapse Lapse Up Lapse Down (-) 42 (-) - (-) ( 16.1) 42 (.5) 1 (.5) ( 41.5) 43 ( 2.) 2 (1.8) ( 61.9) 45 ( 4.) 6 (3.7) ( 76.4) 49 ( 6.1) 12 (5.9) ( 95.7) 56 ( 9.) 22 (9.1) ( 11.2) 65 ( 12.1) 34 (12.2) ( 123.3) 77 ( 15.2) 5 (15.8) ( 91.3) 93 ( 12.7) 64 (13.4) ( 95.8) 15 ( 14.5) 79 (15.4) ( 98.7) 119 ( 15.6) 97 (16.9) ( 57.6) 135 ( 7.9) 16 (8.9) ( 53.) 142 ( 6.7) 114 (7.1) ( 47.8) 149 ( 5.2) 119 (5.3) ( 5.1) 154 (5.) 113 ( 5.9) ( 3.6) 149 (5.1) 16 ( 6.1) ( 1.6) 144 (5.4) 1 ( 6.2) (.5) 138 (5.2) 94 ( 5.4) (2.4) 134 (5.8) 88 ( 5.8) (37.9) 129 (15.7) 71 ( 16.8) (37.7) 114 (15.8) 55 ( 17.2) (39.9) 98 (17.4) 35 ( 19.2) (87.5) 81 (38.) 8 ( 43.9) (84.6) 42 (42.) 57 ( 49.3) Please noe ha he lapse sresses do no have any impac on he margins of he oldes cohor 24 wih only one year of conrac remaining. This is due o he fac ha hose conracs will erminae a he end of he year, receiving he same benefi paymen regardless of he cause of he erminaion (deah, surrender, or mauriy of he conrac). In resul, (24) Gar (sress) = for all hree sresses. Furher noe ha he segmenaion of al. 1 does no depend on he algebraic sign of (k) Mrg bu on he algebraic sign of he change in margins due o he sress and ha hose wo can differ. For example, cohor 12 already has negaive margins of.12 under bes esimae assumpions, bu he margins furher decrease by.5 in he mass lapse and by.6 in he lapse up sress Segmenaion Alernaive 2: Change of BEL In conras o al. 1, he second segmenaion approach considers he insurer s oal expeced liabiliies agains policyholders, including FDB and TVFOG. Sricly following he Solvency II requiremens as presened in Secion 2.2, he sresses are only applied o hose conracs ha would increase he BEL, i.e., BEL (sress) = BEL (sress) BEL >. The sochasic valuaion underlying he BEL reflecs he complex profi sharing mechanisms wihin he insurance porfolio, including ineracions wih he asse porfolio and assumed managemen

20 Risks 218, 6, 66 2 of 38 acions. As a consequence, he impac of a sress applied o a cerain conrac also depends on he remaining porfolio. For al. 2, he conracs are classified ino homogeneous risk groups (HRGs) and sochasic valuaions are performed for cerain subses of HRGs. The arge is o idenify he subse ha resuls in he highes SCR. A brue-force approach would apply he sress o all possible subses in order o deermine an adequae segmenaion. Even hough he number of possible combinaions is finie, compuaional resricions make such an approach unsuiable for pracical applicaions. To reduce he number of possible combinaions, he following heurisic is inroduced: In he firs sep, he insurance porfolio is classified ino HRGs ha are expeced o bear similar lapse risk for he insurer. For his purpose, relevan risk facors have o be idenified ha characerize how a lapse sress affecs he insurer s liabiliies. In he second sep, he HRGs are ranked based on he (sand-alone) exposure of each HRG o he considered ype of lapse risk. In he hird sep, he sressed subse is successively adjused based on he ranking as long as SCR is increasing. Hereby, he fundamenal assumpion is ha if an HRG wih a higher exposure does no increase he SCR neiher will HRGs wih a lower exposure o ha risk. Please noe ha his reduces he maximum number of possible combinaions o he number of HRGs. In paricular, we can eiher increase he sressed subse, saring wih a sress of he HRG wih he highes exposure and successively adding he HRG wih he nex highes exposure or sar wih a sress of he enire porfolio and successively eliminae he HRG wih he lowes exposure. Of course, he more risk facors are used for he classificaion, he more HRGs are obained in sep 1. On he one hand, his increases he homogeneiy of each HRG wih respec o is risk exposure. On he oher hand, due o he (combined) impac of he risk facors, i also increases he complexiy of he ranking in sep 2. Hence, one has o find he righ balance beween he homogeneiy or number of HRGs and he pracicabiliy of he heurisic. Naurally, a firs level of classificaion ino HRGs would be based on he ype of he conrac (e.g., endowmen, erm life, disabiliy). Wihin hose groups, furher risk facors, depending on he ype of conrac, can be idenified. For radiional paricipaing life insurance conracs, characerized by a savings process wih a minimum ineres guaranee, liabiliies highly depend on he echnical ineres rae. A lower echnical ineres rae resuls in higher margins (cf. Secion 4.2.1) and a he same ime, reduces he value of he policyholder s financial guaranees. Accordingly, he exposure of an HRG o he mass lapse and lapse up risk ypically increases wih he echnical ineres rae decreasing. Vice versa, a higher echnical ineres rae resuls in a higher exposure o he lapse down risk. Hence, i seems naural o use he echnical ineres rae as he firs risk facor. Boh fuure margins and he value of he guaranees (in relaion o he BEL) also depend on he remaining duraion of he conrac. Therefore, his could be used as an addiional risk facor for he classificaion. For our model, he righ-hand side of Table 8 (values in brackes) conains he change of BEL resuling from a sand-alone sress of each HRG. 2 Obviously, he echnical ineres rae is he crucial facor for he change of BEL whereas he remaining duraion only has a minor impac. For example, in he lapse up sress he change of BEL varies considerably beween cohors 5 o 6 (from wo differen ariff generaions), whereas he difference is much smaller beween cohors 4 and 5 (which belong o he same ariff generaion). Therefore, we choose each cohor as a separae HRG and deermine he subse of cohors o be sressed along wih he respecive SCR as follows: 2 Please noe ha similar o al. 1, applying he lapse sresses o he oldes cohor 24 does no change he BEL a all.

21 Risks 218, 6, of 38 mass lapse/lapse up: we increase he number of sressed HRGs as long as SCR increases, saring wih he younges cohor (lowes echnical ineres rae and longes remaining duraion); lapse down: we reduce he number of sressed HRGs as long as SCR increases, also saring wih he younges cohor. The resuls of his heurisic are presened in Table 8. I shows he cumulaive BEL (sress) for each lapse sress (which corresponds o he SCR before maximizaion wih zero), increasing he number of cohors being sressed from 1 o 24 for he mass lapse and lapse up sress and decreasing i from 24 o 1 for he lapse down sress. The SCR for he mass lapse risk increases unil cohor 7, reaching a maximum of 36, and decreases hereafer. Hence, al. 2 resuls in cohors 1 o 7 o be sressed in he mass lapse scenario. Nowihsanding, he maximum SCR of 154 for he lapse up risk is obained for addiionally sressing cohors 8 o 1, which implies ha cohors 1 o 1 are relevan for he lapse up sress according o al. 2. The riskies sub-porfolio regarding he lapse down risk includes cohors 11 o 24 and resuls in a SCR of 119. Comparing he combined sresses wih he sand-alone resuls, Table 8 shows ha he SCRs resuling from he sand-alone sresses do no add up o he SCRs resuling from a combined sress of he respecive HRGs. For example, he combined mass lapse sress of cohors 1 and 2 resuls in a oal SCR of 178, which exceeds he sum of he respecive sandalone SCRs by approximaely 6. This confirms ha he impac of a sress depends on he oal insurance porfolio. The effec of an HRG being sressed can even change in a simulaneous sress of several HRGs, as is he case for he mass lapse sress in our model: The sand-alone sress indicaes ha a sress of cohor 7 reduces SCR (.3), bu he maximum SCR is obained for a sress of cohors 1 o 7. Again, his is he resul of he inerdependencies beween cohors and demonsraes ha a segmenaion on a sand-alone basis is no sufficien Comparison and Conclusions We now compare he resuls of he wo alernaive segmenaion approaches. Table 9 provides he change of he BEL if he lapse sresses are applied o he enire insurance porfolio wihou any segmenaion (all) as well as for al. 1 and The decomposiion of he change of BEL reveals he maerial impac of boh he risk miigaion via FDB [CE] and he value of he guaranees (TVFOG). As Table 8 shows, boh mehods resul in similar segmenaions. Cohors wih low echnical ineres raes are included in he mass lapse and lapse up scenario. For hose cohors we observe ha (k) Gar (sress) >, which resuls in an increase of BE Gar as shown in Table 9. This loss of margins is largely compensaed by a reducion of policyholders surplus paricipaion (change of FDB [CE] < ). In addiion, he increase of surrenders reduces he value of he guaranees (change of TVFOG < ). Despie he decrease in absolue erms, we observe an increase of he TVFOG when expressed as a percenage of FDB [CE]. For boh sresses, he relaive increase is larger for al. 2 (mass lapse: 3.5 % vs. 4.9 %; lapse up: 2.3 % vs. 2.6 %). In conras, cohors wih higher echnical ineres rae are subjec o he lapse down sress. If less of hose policyholders erminae heir conracs prior o mauriy, he insurer s guaraneed obligaions increase (change of BE Gar > ), bu also he value of he policyholders guaranees increases (change of TVFOG > ). The financing of he addiional liabiliies reduces he company s annual raw surplus such ha policyholders oal surplus paricipaion decreases (change of FDB [CE] < ), alhough he projeced number of conracs in-force is higher. 21 Please noe ha he change in BEL for all hree cases presened in Table 9 can also be found in he righ-hand side of Table 8.

22 Risks 218, 6, of 38 For he base case, boh alernaives resul in a disjoin segmenaion regarding he lapse up and lapse down sress. However, alhough he wo sresses represen conrary evens, a disjoin segmenaion is neiher required nor expeced. Change of Table 9. Change of BEL ( BEL (sress) ) and is componens (in AC 1). Mass Lapse Lapse Up Lapse Down All Al. 1 Al. 2 All Al. 1 Al. 2 All Al. 1 Al. 2 BEL BE Gar FDB [CE] TVFOG ( ) TVFOG (3.5%) (4.9%) (2.3%) (2.6%) (.8%) (1.1%) FDB [CE] Wihou any segmenaion, he SCRs for each lapse risk are severly underesimaed. The mass lapse and lapse down sresses even resul in a reducion of he BEL (mass lapse: 576; lapse down: 57), whereas SCR (lapse-up) = 42 is above zero. The resuling SCRs for he alernaive segmenaion approaches imply ha al. 1 is no conservaive. In paricular, he SCR for mass lapse risk based on al. 2 exceeds he al. 1 value by 65 % (36 vs. 185). Alhough he differences for lapse up and lapse down risk are smaller, al. 2 SCRs sill exceed he respecive al. 1 values by 14 % and 23 % (lapse up: 154 vs. 135; lapse down: 119 vs. 97). For he mass lapse sress, he addiional sress of cohors 8 o 12 in al. 1 resuls in a higher loss of margins (533 vs. 427). Also afer consideraion of he risk miigaion due o policyholders surplus paricipaion (FDB [CE] ), he increase of he insurer s liabiliies is sill higher for al. 1 (658 vs. 452). However, he sress of he addiional cohors leads o a sronger decrease of he TVFOG ( 473 vs. 146) such ha he resuling SCR is lower for al The same can be observed for he lapse up sress. Before consideraion of he TVFOG, he addiional sress of cohors 11 o 13 in case of al. 1 resuls in a higher increase of he insurer s expeced liabiliies (BE Gar + FDB [CE] ) compared o al. 2 (257 vs. 23). However, he addiional reducion of he TVFOG for al. 2 by = 46 leads o a higher overall SCR for al. 2. Regarding he lapse down scenario, he increase in guaraneed obligaions (BE Gar ) for al. 1 is almos hree imes as large as for al. 2 (183 vs. 64). For boh alernaives, he increase is largely offse by a reducion of policyholders FDB. However, he sress of addiional cohors in al. 2, which herefore remain longer in he insurance porfolio, bears a higher risk of losses for he insurer due o higher minimum ineres rae guaranees. The laer is refleced in a higher (increase of) TVFOG (83 vs. 135) such ha he overall SCR is higher for al. 2. The resuls illusrae he differen risk exposures refleced in he hree lapse sresses. For he mass lapse and lapse up sress, he risk largely originaes from he loss of fuure margins due o higher surrender raes, and in resul, from a loss of fuure profis for he insurer. In conras, he lapse down sress reflecs he insurer s risk resuling from unprofiable insurance conracs wih high guaranees ha were given in he pas and urn ou o be quie expensive for he insurer due o he decrease in ineres rae levels over he pas years. If he holders of such policies remain in he insurance porfolio longer han expeced due o reduced surrender raes, he insurance company may incur addiional losses. 22 The laer can be raced back o he asymmery of he profi sharing process in Germany which is refleced in our Mone Carlo valuaion framework. Addiional surrenders of conracs wih fairly high echnical ineres raes in al. 1 may reduce losses relaed o invesmen reurns below he echnical ineres rae in downside scenarios, bu has a smaller impac in upside scenarios. Overall, his has a posiive impac on he PVFP ha exceeds he loss of fuure margins. Hence, despie losing addiional margins in al. 1, he insurer is beer off due o reduced volume of minimum ineres rae guaranees.

23 Risks 218, 6, of 38 We conclude ha a segmenaion based on a deerminisic projecion (al. 1) of guaraneed benefis considerably underesimaes SCR. I only akes ino accoun he change in margins (BE Gar ) of he sressed cohors, alhough he SCR is also significanly affeced by changes of FDB and TVFOG. Due o he inerdependencies beween cohors, he laer wo can only be deermined on porfolio level. Therefore, a sand-alone assessmen of each conrac is no sufficien, bu a sochasic valuaion is required (al. 2) Surrender Risk in he Conex of he Risk Margin In he following, we discuss how lapse risk is refleced in he RM which is he second componen of he TP on op of he BEL. By definiion, he calculaion of RM requires an explici projecion of SCR RU over he lifeime of he insurance porfolio. However, i is also allowed (and common pracice) o deermine he RM based on a simplified mehod ha approximaes SCR RU (cf. ar. 58 of he Solvency II Delegaed Ac (European Commission 215)). Guideline 61 of he Level 3 Guidelines (EIOPA 215) specifies ha appropriae risk drivers may be applied o approximae fuure SCRs eiher individually for each risk (mehod 1) or for SCR RU as a whole (mehod 2). We apply hree differen risk drivers for he approximaion of fuure SCRs for lapse risk based on he SCR a ime =. The appropriaeness of such risk drivers is analyzed and compared o an explici projecion of fuure SCRs Explici Projecion of Fuure Solvency Capial Requiremens The explici projecion of he SCR requires an annual valuaion of he projeced iniial insurance porfolio over he projecion horizon. In accordance wih Equaion (1), he insurance company is projeced under he CE scenario unil ime (wihou new business). Subsequenly, a sochasic valuaion of he company s liabiliies a ime is performed (under bes esimae and sressed assumpions) o derive SCR (sress) ( = max BOF BOF (sress) ) ;. Previous resuls showed ha he segmenaion for he hree lapse sresses depends on several facors, which are subjec o change over ime. Hence, he explici projecion of he SCR for lapse risk requires a separae segmenaion for each valuaion dae. For his purpose, we apply segmenaion al. 2 (cf. Secion 4.2.2). Since he SCR projecion has o be in line wih he iniial yield curve, he risk-neural scenarios used for he valuaion are required o fi he yield curve (r (s)) s> a ime which is derived from he iniial forward raes f (, + s): r (s) = (1 + f (, + s)) 1 s 1. Please noe ha in general, he Vasicek model canno be calibraed o an arbirarily given yield curve. Hence, we approximae he argeed yield curve (r (s)) s> by performing a goal seek for he shor rae r a ime ha resuls in a yield curve ( r (s)) s> wih r (y) = f (, + y) and y = min (T B, T ) being he minimum of he iniial mauriy T B of coupon bonds he company invess in and he remaining projecion period (T ) Approximaion of Fuure Solvency Capial Requiremens For he approximaion of fuure SCRs, we se SCR (sress) following (sress depending) risk drivers rd (sress). = SCR (sress) rd(sress) rd (sress), based on he

24 Risks 218, 6, of 38 Risk Driver A: Change of BE Gar Following segmenaion al. 1, he firs risk driver is based on he change of he insurer s guaraneed obligaions (k) Gar (sress) owards policyholders caused by a cerain lapse sress occurring a ime, i.e., rd (sress) n 1 ( ( k) Gar ) (sress) = max ( k) l ; =: + Gar (sress). k= Analogous o he explici SCR projecion, risk driver A sars wih a CE projecion of he iniial insurance porfolio unil ime. Based on ha, he insurance porfolio is projeced deerminisically under bes esimae or sressed assumpions from ime on o deermine + Gar (sress) for he lapse sress considered (cf. Secion 4.2.1). Please noe ha (k) Gar also includes he insurer s fuure obligaions due o policyholders (deerminisic) surplus paricipaion unil ime. Furher noe ha o allow for a simulaneous calculaion of + Gar (sress) for each conrac wihin a single projecion, inerdependencies wihin he insurance porfolio mus be avoided. Hence, he projecion does no allow for dynamic policyholder behavior, bu deerminisic lapse raes are applied (cf. Secion 4.2.2). Alhough risk driver A is based on deerminisic projecions, he exac calculaion of + Gar (sress) sill requires hree addiional sress projecions for each projecion year. To reduce he compuaional effor for pracical applicaions, he change of he insurer s guaraneed obligaions in projecion years > may be approximaed based on + Gar (sress). The sress-depending risk driver A includes a specific segmenaion for each projecion year ha is based on segmenaion al. 1. Risk Driver B: Liabiliies agains Policyholders The second risk driver is based on a forward projecion of he insurer s expeced obligaions agains policyholders, derived from he sochasic valuaion of he BEL a ime =. Therefore, i only considers cash flows occurring afer ime and in line wih Equaion (5) i is given by rd (sress) = (1 + r ()) 1 J J j=1 T 1 s= AdC [j] s B [j] s P [j] s + L[j] s+1 + CC [j] [j] s+1 CRe f s+1 I assumes a consan raio beween he lapse risk and expeced fuure obligaions agains policyholders. Please noe ha his is he approximaion mehod applied in he curren version of he German sandard valuaion model for Solvency II. Risk Driver C: Premium Income Similar o risk driver B, risk driver C assumes a consan raio beween he insurer s risk and he expeced fuure premium income. Again, i is derived from he bes esimae run as follows: rd (sress) = (1 + r ()) 1 J J j=1 ( ) T 1 P s s= B [j]. s Please noe ha risk driver A depends on he lapse sress considered, i.e., in accordance wih mehod 1, i approximaes each lapse risk sub-module individually. In conras, risk drivers B and C are he same for each lapse sress, since hey are derived under bes esimae assumpions. Hence hey boh represen examples of RM approximaion mehod 2. B [j] s+1.

25 Risks 218, 6, of Comparison For he purpose of our analysis, we define he presen value of cos-of-capial RM (sress) for a cerain lapse risk sub-module as RM (sress) = CoC T = SCR (sress) (1 + r ( + 1)) rd(sress) +1 rd (sress) wih rd (sress) = SCR (sress) in case of he explici SCR projecion. Please noe ha he Solvency II RM is based on he maximum capial requiremen from he hree lapse sress scenarios a each poin in ime. Any change of he relevan lapse sress scenario over ime has an impac on he presen value of cos-of-capial RM (lapse) for he lapse risk sub-module such ha RM (lapse) max (SCR (sress) ) ( = CoC (1 + r ( + 1)) +1 = max RM (sress)). (6) Figure 1 shows projeced and approximaed SCRs for he hree lapse sresses for he base case and in case of he reduced ineres rae level. In addiion, i also presens he corresponding lapse sress segmenaion applied for he explici SCR projecion (grey highlighed area) and for risk driver A (paerned area). 23, 23 For he base case, SCRs and he respecive segmenaions a ime = correspond o he resuls presened in Secion 4.2.

26 Risks 218, 6, of Mass lapse Al. 2 Al. 1 SCR Risk driver A Risk driver B Risk driver C Lapse-up lapse-down Mass-lapse Lapse-up lapse-down (a) (b) Figure 1. Developmen of SCR over ime (in AC 1). (a) Base case; (b) Ineres rae sensiiviy. Please noe ha in he following, he analysis focuses on he base case. However, he basic saemens also hold if he ineres rae level is reduced. Hence, we do no analyze he ineres rae sensiiviy in deail, bu we only address i in case i provides addiional insigh. We also discuss specific implicaions of an ineres rae increase on he lapse risk under Solvency II. For he explici SCR projecion, he number of cohors included in he mass lapse and lapse up scenario increases over ime, such ha he whole insurance porfolio is finally relevan for he mass lapse sress (ime = 6) and he lapse up sress (ime = 1). Vice versa, he number of cohors included in he lapse down sress is already down o zero a ime = 9. This is parially due o he increasing yield curve (cf. Figure 2). I causes more and more cohors o be profiable for he insurer and hence, o be included in he mass lapse and lapse up sress. In addiion, noe ha he echnical ineres rae in he German life insurance marke was coninuously reduced for more han 15 years. Since he insurance porfolio of our sylized company is buil in line wih ha, cohors wih a higher echnical ineres rae also have a raher shor remaining duraion. Hence, he conracs ha migh sill be relevan for he lapse down sress are he firs o be winded up.

27 Risks 218, 6, of % 2.5% 1.5%.5% -.5% forward rae f_(-1,) spo raes r_() Figure 2. Iniial spo raes r () and forward raes f ( 1, ). As a resul, he poenial loss due o a lapse down even drops fases over ime, whereas he risk due o a lapse up even decreases more slowly. For he mass lapse even, even an increase of SCR can be observed unil year 7 (from 36 o 465), followed by a gradual decrease ha reflecs he run-off of he porfolio. On he one hand, his is due o he increasing yield curve ha leads o increasing margins. Hence, despie he run off, he poenial loss in case of a mass lapse even increases in early years. On he oher hand, he risk-miigaing capaciy of he FDB decreases over his ime span, as can be seen in Figure 3 (increasing projeced raio beween SCR and change of margins), and his leads o an overall increase of he SCR. Figure 1 shows ha in early years, he segmenaion based on al. 1 (used for risk driver A) differs from al. 2 (applied for he explici SCR projecion). However, boh segmenaions show he same behavior, wih he number of cohors relevan for he mass lapse and lapse up scenario increasing, and respecively decreasing for he lapse down scenario. Finally, he segmenaions mach (mass lapse: = 8; lapse up/down: = 9). The reduced ineres rae level confirms he dependency of boh he segmenaion and he respecive lapse risk on he yield curve. In line wih he lower yield curve, more cohors are iniially included in he lapse down sress (al. 1: +2; al. 2: +5), whereas he number of cohors included in he mass lapse (al. 1: 2; al. 2: 2) and lapse up (al. 1: 2; al. 2: 5) sress are reduced. Accordingly, Figure 1 shows higher SCRs for he lapse down risk and reduced SCRs for he lapse up and mass lapse risk compared o he base case. However, he segmenaion over he projecion horizon evolves similar o he base case due o increasing ineres raes over ime. In conras o he reduced ineres rae level, rising ineres raes may diminish he lapse down risk, bu furher increase he mass lapse and lapse up risk (and he number of cohors o be sressed). On he one hand, his is due o more insurance conracs becoming profiable for he insurer, which increases he poenial loss in case of higher surrender raes in he mass lapse and lapse up even. On he oher hand, high lapse raes, especially in case of a mass lapse even, rigger immediae surrender paymens which impair he insurer s liquidiy. The laer requires he insurer o sell asses a due o rising ineres raes depreciaed marke values. This leads o a realizaion of unrealized losses, which reduces he raw surplus and he insurer s profis. 24 Risk drivers B and C, derived from he bes esimae run, are independen of he sress and do no include any segmenaion of he insurance porfolio. In fac, he shape of he approximaed SCR curves are he same for each lapse risk and only differ due o he varying saring poins a ime =. Comparing he differen SCR curves, risk driver A bes fis he explici SCR projecion for all hree lapse sresses. The figure shows ha wih excepion of he SCR approximaion for he lapse up risk based on risk driver C, risk drivers B and C do no reflec he acual risk profile derived from he explici SCR projecion. 24 For furher deails regarding he join impac of increasing ineres raes and surrender raes on he insurer s solvency, see Berdin e al. (217).

28 Risks 218, 6, of 38 In paricular, risk driver A works quie good for boh he lapse up and lapse down risk. However, larger deviaions from he projeced SCRs can be observed for he mass lapse risk. Alhough i replicaes he general shape of he SCR projecion, he curvaure of he approximaion is less pronounced. Table 1 conains he resuling cos-of-capial RM (sress) for each lapse sress. The cos-of-capial based on risk driver A differs from he SCR projecion by less han 2 % for all hree sresses (mass lapse: 18 %; lapse up: +12 %; lapse down: 11 %). In conras, risk drivers B and C are a leas 2 % off he mark, wih errors being up o 422 %. Table 1. Cos-of-capial RM ( ) resuling from projeced and approximaed SCRs (in AC 1). RM ( ) SCR Projecion SCR Approximaion (In % of Explici SCR Projecion) Risk Driver A Risk Driver B Risk Driver C (Base case) Mass lapse (82 %) 233 (72 %) 161 (5 %) Lapse up (112 %) 117 (177 %) 81 (123 %) Lapse down (89 %) 91 (522 %) 63 (361 %) Lapse (82 %) 233 (72 %) 161 (5 %) (Ineres rae sensiiviy) Mass lapse (71 %) 169 (6 %) 115 (41 %) Lapse up (17 %) 9 (168 %) 61 (114 %) Lapse down (92 %) 183 (478 %) 124 (325 %) Lapse (71 %) 183 (65 %) 124 (44 %) For he reduced ineres rae level, risk driver A performs even beer for he lapse up (+5 %) and lapse down (+3 %) sress. However, he approximaion error for he mass lapse sress rises o 29 %. Risk drivers B and C sill exhibi errors of more han 4 % (excep for risk driver C in he lapse up sress). Please noe ha he wo sress-independen (mehod 2) risk drivers B and C ignore ha he lapse risk relevan for he calculaion of he overall RM may change over ime. This means ha he resuling overall RM only reflecs he lapse risk relevan a ime =. Figure 1 shows ha his has an impac on he overall RM in case of he reduced ineres rae level: The lapse down sress is relevan a ime =, bu he mass lapse sress is he relevan risk hereafer. Due o is sress-dependency, risk driver A reflecs his change in a oal RM (lapse) = 22 ha is larger han he sand-alone RM (mass lapse) = 21 (cf. Table 1). For a beer undersanding of he approximaion error for mass lapse risk, we consider wo ypes of errors: Error 1: The approximaion assumes a consan raio beween he (posiive) change of margins and he resuling SCR (sress) over ime. Error 2: The posiive change of margins used for he risk driver is based on segmenaion al. 1 and differs from he acual change of margins for he SCR projecion which is based on segmenaion al. 2 (cf. Figure 1, grey highlighed area vs. paerned area). Figure 3 shows ha only a minor par of he deviaions are due o error Error 2 even parly offses error 1, which causes he main par of he deviaions. The approximaion assumes a consan raio of 6.1 % or, vice versa, a consan risk miigaion of 93.9 % by FDB (including TVFOG). 25 Alhough he segmenaion beween al. 1 and 2 only differs unil year 7, error 2 can be observed hroughou he projecion. This is due o he fac ha risk driver A does no allow for dynamic policyholder behavior which slighly affecs he change of margin.

29 Approximaion error Raio beween SCR and change of margins Risks 218, 6, of 38 In conras, he projeced raio shows a differen paern, which reflecs he overall siuaion of he company over ime. Firs, he projeced raio increases from 7.4 % in year 1 o a maximum raio of 9.5 % a ime = 7, which means ha he risk miigaion declines. This is primarily due o low invesmen income, which even under he CE scenario is no sufficien o cover he minimum ineres rae obligaions. Hence, an addiional loss of margins in hose years due o a mass lapse even can less and less be compensaed by a reducion of policyholders surplus paricipaion, bu i resuls in a loss for he insurer. From year 8 on, he siuaion regarding he company s ineres rae guaranees improves and resuls in increasing annual surpluses. In line wih ha, he projeced raio decreases unil he end of he projecion. However, he approximaed SCR is sill underesimaed unil year 12, when he acual raio falls below 6.1 %. 2 1.% % 1 5.% 5 2.5% Error 1 Error 2 Toal error Projeced raio Assumed raio.% -2.5% -5.% Figure 3. Error due o approximaion of SCR (mass lapse) using posiive change of margins (in AC 1) Conclusions Our analysis shows ha he hree considered risk drivers differ no only in heir appropriaeness o approximae fuure SCRs for lapse risk bu also in he compuaional effor required o derive hem. The wo sress-independen risk drivers B and C can be derived from he sochasic projecion already used for he economic valuaion a ime =, wihou any major addiional overhead. However, hey do no fi he risk profile of a porfolio of radiional life insurance conracs wih respec o lapse risk. Boh risk drivers lead o maerial approximaion errors regarding he RM and in consequence, o an inaccurae valuaion of TP. In conras, an exac calculaion of risk driver A based on segmenaion al. 1 requires hree addiional projecions under sressed assumpions a each valuaion dae of he projecion horizon. However, he addiional compuaional effor may be reduced if he risk driver is approximaed based on he CE projecion used for he valuaion a ime =. The sress-dependen risk driver A improves he resuls considerably, bu approximaion errors are sill maerial. Finally, a complee SCR projecion based on segmenaion al. 2 ha would allow for an exac RM calculaion is quie complex. I requires a sochasic valuaion under bes esimae and sressed assumpions as well as a respecive segmenaion for each lapse sress and each valuaion dae of he projecion horizon. Due o he maerial compuaional overhead, his approach appears unsuiable for pracical applicaions. A possible compromise may be an approximaion based on grid poins. This means ha he acual SCR calculaion is limied o cerain poins in ime and he remaining SCRs are inerpolaed based on hose grid poins. As an illusraive example, Table 11 shows he resuls of a linear inerpolaion wih grid poins a imes =, 4, 12, 23. Despie he raher simple inerpolaion mehod, he approximaion

30 Risks 218, 6, 66 3 of 38 error is already less han 1 % for he mass lapse and lapse up scenario. A more sophisicaed inerpolaion or revised choice of grid poins may furher reduce he approximaion error, in paricular for he lapse down sress. Table 11. Cos-of-capial RM ( ) (in % of SCR projecion) resuling from inerpolaion of SCRs (in AC 1). Scenario Mass Lapse Lapse Up Lapse Down Lapse Base case 33 (94 %) 71 (17 %) 22 (128 %) 33 (94 %) Ineres rae sensiiviy 283 (11 %) 57 (17 %) 49 (128 %) 284 (11 %) 4.4. The Impac of Lapse Sresses on Going Concern Reserve and Surplus Funds As analyzed in Burkhar e al. (215, 217), he allowance for GCR and SF and heir recogniion as par of he company s BOF is in line wih regulaory requiremens under Solvency II. However, hose papers furher menion ha in reurn, he risks resuling from hose addiional BOF iems also have o be refleced in he SCR calculaion. In paricular, i has o be considered o which exen funds are sill available for loss coverage in a lapse sress scenario. Given he maerial impac on an insurer s solvency raio, we analyze how GCR and SF change in such scenarios Going Concern Reserve The share of cos surplus of he business in-force used o pre-finance uncovered acquisiion coss of fuure new business depends on several facors (Burkhar e al. 215): Firs, he amoun of uncovered acquisiion coss ha needs o be pre-financed depends on he volume of new business. New business volume may be characerized by he number of prospecive policyholders exercising heir opion o sign a life insurance conrac. Second, he uncovered acquisiion coss in year are no only pre-financed by he iniial business in-force bu also by new business expeced o be wrien unil year. Since only he share allocaed o he iniial business in-force a ime = resuls in GCR funds, i is highly relevan how he pre-financing is spli beween hose wo sub-porfolios. In general, he higher he raio beween iniial business in-force and fuure new business, he higher he porion resuling in GCR funds. This raio is subjec o he opion exercise raes of boh new business and business in-force. Whereas he Solvency II sandard formula explicily prescribes how and which conracs of he business in-force are o be sressed in he lapse sresses (cf. Secion 2.2), i does no give furher guidance on how a sress may affec new business assumpions relevan for he valuaion. However, o ensure an appropriae assessmen of he insurer s lapse risk, sound and consisen assumpions regarding volume and developmen of expeced new business are required for each lapse sress scenario. In he following, we analyze he possible impac of a lapse sress on he pre-financing of uncovered acquisiion coss and is impac on he SCR. We assume ha all new conracs belong o ariff generaion 7 wih a echnical ineres rae of.9 %. Based on he increasing yield curve (cf. Figure 2), his implies ha new business expeced o be wrien in he fuure is profiable from an insurer s poin of view. Hence, in line wih our previous resuls, such conracs are subjec o he mass lapse and he lapse up sress. For he mass lapse sress, we assume ha such an even also affecs he volume of new business o be wrien in he year of he occurrence of he sress. In line wih he insananeous 4 per cen surrender even we consider a 4 per cen drop of he new business volume in year 1 and no change of new business volume from year 2 on. For he lapse up sress, we assume a permanen decrease of he volume of fuure new business. In line wih ar. 142 of he Solvency II Delegaed Ac (European Commission 215), he 5 % increase is applied on he rae reflecing ha he opion o conclude a conrac is no exercised. Saring wih a bes esimae conclusion rae of 9 %, his resuls in a decrease of his rae in he lapse up sress o

31 Risks 218, 6, of 38 1 (1 9 %) 1.5 = 85 %. Hence, he volume of new business decreases by % % = 5.6 % for each projecion year. We also analyze he impac of a (simulaneous) permanen increase of he surrender raes for fuure new business. We assume ha he lapse down sress does no affec he expeced new business. Hence, he impac of he sress on he GCR is limied o he change in he policyholder behavior wih respec o he business in-force. Resuls are presened in Table 12. Wih respec o he business in-force, he analysis is based on cohors 1 hrough 1 being sressed in he mass lapse and lapse up scenario and cohors 11 hrough 24 being sressed in he lapse down scenario. Applying segmenaion al. 2, his resuls in he maximum SCRs for each of he following varians. Hence, allowing for a GCR, he segmenaion for he mass-lapse scenario differs from he base case (cohors 1 o 7). Figure 4 shows how a mass lapse even affecs he pre-financing of uncovered acquisiion coss by he business in-force. The medium grey bars show he expeced amoun pre-financed by he iniial business in-force in each projecion year under bes esimae assumpions. Assuming 1 new conracs each year, he annual pre-financing need equals AC 425,. The share pre-financed by he iniial business in-force gradually drops from 1 % in year 1 o zero a he end of he projecion. The ligh grey bars show how he pre-financing of he business in-force evolves if cohors 1 hrough 1 are sressed in a mass lapse even (and new business volume is no affeced by he sress). The sress resuls in a reduced pre-financing share compared o he bes esimae, since a higher porion of fuure uncovered acquisiion coss is borne by fuure new business. In oal, his resuls in a decrease of GCR by 556 ( 16 %) and ulimaely o a maerial increase of SCR by = 332 (+68 %) compared o an SCR based on bes esimae pre-financing assumpions. If he volume of firs year s new business is also reduced, we observe a corresponding reducion in pre-financing by 4 % in he firs year (dark grey bars). A he same ime, he reduced new business of he firs year resuls in an increased pre-financing share of he business in-force in subsequen years compared o he mere sress of he business in-force. Wih he former effec dominaing, GCR decreases by anoher 76 resuling in a oal SCR of 878 (+53) Bes Esimae Sress of business in-force Combined sress Figure 4. Pre-financing of acquisiion coss over ime for bes esimae and mass lapse sress (in AC 1). Table 12. BOF and SCR under differen sress assumpions regarding he GCR (in AC 1). Assumpions Regarding GCR BOF Change of BOF in Case of Bes Esimae Mass Lapse Lapse Up Lapse Down GCR Toal GCR Toal GCR Toal GCR Toal Bes esimae Sress of business in-force Combined sress n/a n/a For he lapse up sress, he mere sress of he business in-force reduces is share of he uncovered acquisiion coss, as illusraed in Figure 5 (ligh grey bars). The reducion gradually increases unil reaching is maximum afer 15 years, when only cohors underlying he lapse up sress are sill par of he porfolio. I gradually decreases hereafer. The addiional sress of he expeced new business

32 Risks 218, 6, of 38 resuls in wo conrary effecs. On he one hand, he pre-financing need permanenly drops due o he reduced volume of new business. A he same ime, he pre-financing share of he business in-force is higher due o he reduced volume in combinaion wih higher surrender raes of new business. In oal, he reducion (dark grey bars) is less pronounced compared o he mere sress of business in-force. In boh cases, GCR considerably decreases and SCR increases, respecively. However, he oal effec is smaller for he combined sress (GCR: 118 vs. 184; SCR: +74 vs. +134) Sress of business in-force Combined sress Sress of business in-force (a) (b) Figure 5. Change of pre-financing of uncovered acquisiion coss over ime for he lapse up and lapse down sress compared o he bes esimae (in AC 1). (a) Lapse up sress; (b) Lapse down sress. For he lapse down sress, he reduced surrender raes of he business in-force resul in a higher pre-financing share. As shown in Figure 5, a subsanial effec can only be observed unil year 11, i.e., as long as sressed cohors are sill par of he porfolio. The minor effecs hereafer are due o he dynamic policyholder behavior. In oal, GCR increases by 32, reducing SCR by 17. We conclude ha he lapse sresses have maerial impac on he pre-financing of uncovered acquisiion coss, i.e., on he size of he GCR afer sress. If he impac of a lapse sress on he expeced pre-financing of uncovered acquisiion coss is aken ino accoun, he allowance for a GCR increases he poenial loss of BOF in a mass lapse and lapse up sress. Vice versa, he lapse down risk is reduced by an increasing GCR. Since he GCR is par of he insurer s BOF, his direcly affecs he SCR. Of course, his has o be aken ino accoun when assessing he relevance of a conrac for a cerain lapse sress. Please noe ha he pre-financing and hus he size of he GCR in he lapse sresses highly depends on he assumpions regarding expeced volume and developmen of fuure new business. To ensure an adequae and consisen assessmen of he insurer s lapse risk, furher guidance from he regulaor on relevan assumpions regarding he expeced new business in lapse sresses is desirable Surplus Funds Alhough he iniial value of he RfB is he same for he bes esimae and he sress scenarios, he economic value of SF recognized as par of he BOF migh differ. If he same valuaion mehodology is applied under sressed assumpions, relevan cash flows (respecive benefi paymens o policyholders) will change in a (lapse) sress scenario and so will he value of SF in he sress scenario (Burkhar e al. 217). However, also he porion of he iniial RfB which is acually available for loss coverage and herefore qualifying as Tier 1 capial may change compared o he bes esimae. The RfB accumulaes profis of previous periods unil allocaed o policyholders. Each year, he insurer has o declare in advance, o which exen RfB funds are o be allocaed/paid ou o policyholders in he nex year, eiher via ongoing or erminal bonuses. Hence, he RfB can be divided ino a declared par of funds bindingly reserved for policyholders surplus paricipaion in he curren declaraion period and he remaining funds of he undeclared par.

33 Risks 218, 6, of 38 In general, he funds of he undeclared par qualify as a candidae for classificaion as Tier 1 capial, since i is no before he binding declaraion ha he insurer is deprived of he righ o use hose funds in case of emergency (cf. 14 VAG). Wihou furher consideraions, SF would herefore equal he presen value of all cash flows o policyholders resuling from he undeclared RfB a he valuaion dae. However, siuaions may arise in which funds of he iniial undeclared RfB may no be available for fuure loss coverage. E.g., he Federal Financial Supervisory Auhoriy saes ha his is he case if raw surplus is insufficien o finance he immediae credi of he curren year and funds of he undeclared RfB may (parly) be used o do so insead (BaFin 215). Even hough i is no explicily menioned, he same mus hold if he amoun reserved in he declared RfB for policyholders expeced surplus paricipaion is no sufficien. In his case, addiional funds mus be wihdrawn from he undeclared RfB and are no available for loss coverage in fuure periods. Therefore, he iniial funds used for he calculaion of he SF have o be reduced accordingly. Such a siuaion may be riggered by an insananeous sress like a mass lapse even. Due o he maerial increase of policyholders erminaing heir conracs righ away, he amoun of erminal bonuses ha mus be paid ou may exceed he expeced amoun reserved in he declared RfB. The exceeding porion is aken from he undeclared RfB and clearly hese funds are no eligible for loss coverage any more, i.e., hey do no qualify as BOF and he iniial RfB funds relevan for he valuaion of SF have o be reduced accordingly. Oherwise, he risk reducing capaciy of SF in case of a mass lapse even is overraed and he SCR is oo low. In he following, we analyze he impac of he valuaion mehod used o deermine he value of SF for he mass lapse scenario and he respecive impac on he SCR. In paricular, we consider hree varians. The firs varian does no include a separae valuaion of SF for he sress scenario. The second varian includes a separae valuaion of SF for he sress scenario. However, i does no ake ino accoun he reducion of he iniial RfB funds available for loss coverage due o an increase of he erminal bonuses L1 erm paid ou in year 1. Finally, he laer is aken ino accoun for he hird varian. Resuls are presened in Table 13. For his analysis, cohors 1 hrough 1 are sressed in he mass lapse scenario, since his resuls in he maximum SCR for varian 3. Hence, he segmenaion for he mass lapse scenario differs from he base case (cohors 1 o 7) due o he allowance for SF. Table 13. Change of relevan BOF componens in he mass lapse scenario (in AC 1). Calculaion of Surplus Funds Bes Esimae BOF Mass Lapse Change of BOF SF Toal SF Toal SF Toal based on bes esimae valuaion wihou change of L1 erm due o sress wih change of L1 erm due o sress (Terminal bonuses L erm 1 in year 1) (411) (633) (+222) If he change of L erm 1 due o he sress is ignored (varian 2), he value of SF for he sress scenario even slighly increases by 3 compared o is bes esimae value (varian 1), which resuls in a respecive decrease of SCR by 1 %. However, erminal bonuses paid ou in year 1 acually increase by 222 for he mass lapse even. If he iniial RfB funds available for loss coverage are reduced accordingly (varian 3), he value of SF for he sress scenario decreases by 222 (6 %) compared o varian 2. In oal, his leads o a maerial increase of SCR by 65 % compared o varian 1 and by 83 % compared o varian 2. Our analysis shows he maerial impac of SF on he SCR for mass lapse risk. The insananeous mass lapse sress no only affecs he expeced cash flow profile, bu i also reduces he iniial RfB funds eligible for loss coverage. Boh effecs mus be aken ino accoun for he quanificaion of he lapse risk and hey may also affec he required segmenaion of he insurance porfolio.

34 Risks 218, 6, of Conclusions Surrender risk ranks among he bigges risks a life insurance company faces oday. Therefore, i is imporan for an insurer o properly evaluae hose risks in order o be sufficienly capialized. According o he Solvency II sandard formula, he SCR for lapse risk is calculaed based on hree lapse sresses: an immediae mass lapse even as well as a permanen increase and decrease of he expeced lapse raes. Under he sandard formula, he lapse sresses are only applied o hose conracs and opions exercise raes ha would resul in an increase of he BEL and a respecive loss of BOF. A conrac s relevance for a cerain ype of lapse risk depends on several facors and his has o be refleced in he segmenaion of he insurance porfolio for he hree lapse sresses. The idenificaion of relevan porfolio segmens is no sraigh forward due o he inerdependencies beween conracs (wih differen levels of minimum ineres guaranees and differen remaining duraions). In a firs sep, we show ha he applied segmenaion mehod can have maerial impac on he SCR for lapse risk. We analyze and compare wo differen segmenaion mehods: Al. 1 approximaes he impac of a lapse sress on he BEL by he change of he guaraneed obligaions (BE Gar ). The mehod allows for a simulaneous analysis of all conracs wihin a single projecion. I is he mehods currenly applied by mos life insurance companies in Germany. Al. 2 represens a segmenaion heurisic based on sochasic valuaions. Conracs are classified ino HRGs based on heir expeced lapse risk exposure. The HRGs are ranked based on a sand-alone assessmen of heir lapse risk. Based on his ranking, he acual cash flow projecion model is repeaedly applied in order o idenify he subse of sressed HRGs ha resuls in he highes SCR. Despie showing similariies, boh mehods lead o differen segmenaions and resuling SCRs for an illusraive insurance porfolio of endowmen policies. Conracs wih low echnical ineres raes are included in he mass lapse and lapse up sress. The risk in hose wo sresses mainly resuls from a loss of fuure margins due o higher surrender raes of profiable conracs. In conras, cohors wih higher echnical ineres rae are subjec o he lapse down sress. I reflecs he risk of low surrender raes in combinaion wih high guaranees given in he pas. In he curren low ineres environmen, hose guaranees urn ou o be quie expensive and risky for he insurer. Our analysis shows a srong impac of FDB and TVFOG on he SCR. In fac, a segmenaion based on a deerminisic projecion (al. 1) of guaraneed benefis considerably underesimaes SCR. I ignores he complex inerrelaions of he surplus paricipaion process. Hence, we conclude ha a sand-alone assessmen of each conrac is no sufficien. A sochasic valuaion as implemened in al. 2 is required for an appropriae assessmen of an insurer s lapse risk. In a second sep, we analyze he surrender risk profile of a life insurance company in he ligh of he RM calculaion. We explicily projec he SCR for lapse risk over he lifeime of he iniial insurance porfolio, applying segmenaion al. 2. The projecion confirms our previous resuls regarding he dependency of a conrac s surrender risk exposure on he relaion beween a conrac s echnical ineres rae and he respecive yield curve. We find ha ypical risk drivers (presen value of fuure insurance obligaions and presen value of fuure premiums), currenly used in pracical applicaions o approximae he RM, do no fi he surrender risk profile of a life insurance company. This leads o maerial approximaion errors regarding he RM and in consequence, o an inaccurae valuaion of TP. A crucial drawback of hose risk drivers is ha hey do no ake ino accoun he impac of ime-depending facors on a conrac s risk exposure. Hence, we propose an alernaive mehod based on segmenaion al. 1 ha reflecs hose dependencies. Our analysis shows ha he new mehod improves he approximaion error of he RM calculaion compared o he risk drivers currenly used. However, approximaion errors may sill be

35 Risks 218, 6, of 38 subsanial. To avoid he complexiy of a complee SCR projecion for he RM calculaion, a possible soluion may be an approximaion based on grid poins. In a las sep, we discuss lapse risk in he conex of an allowance for GCR and SF. We find ha he lapse sresses affec he value of boh GCR and SF. Wih hose iems being par of he insurer s BOF, his direcly ranslaes ino a change of he SCR for lapse risk. The effecs also have o be aken ino accoun for he segmenaion of he insurance porfolio for he lapse sresses. Regarding he GCR, he pre-financing of uncovered acquisiion coss by he business in-force may considerably change in a lapse sress scenario. Our analysis also reflecs possible implicaions of a lapse sress on volume and surrender raes of he expeced new business and heir impac on he GCR. We conclude ha o ensure an appropriae assessmen of he insurer s lapse risk, sound and consisen assumpions regarding he expeced new business are required and furher guidance from he regulaor seems necessary. Regarding he SF, we noe ha each conrac ha is sressed in a sress scenario affecs he surplus paricipaion process and wih i he cash flows (benefi paymens o policyholders from he iniial RfB) relevan for he valuaion of SF. Thereby, one also has o ake ino accoun ha due o sauory requiremens he iniial RfB funds acually eligible for loss coverage may change due o he sress, which is paricularly imporan for he insananeous mass lapse sress. The curren paper focuses on he policyholder opion o surrender he conrac. Alhough covering he main characerisics of he German marke, considering sauory requiremens as well as ypical managemen rules, illusraive resuls presened are based on a simplified model. We herefore feel ha here is room for addiional research. Since he lapse risk module of he Solvency II sandard formula covers he risk arising from all policyholder opions, i would be valuable o exend our analysis o oher opions like he annuiizaion opion, which has paricular impac on an insurer s long-erm guaranees. Fuure sudies may also ake ino accoun furher ineracion effecs due o sauory provisions which, under cerain circumsances, require he insurer o build up addiional reserves for he policyholders minimum ineres guaranees. In oal, our analysis shows ha he recogniion of lapse risk under he Solvency II sandard formula is highly complex. The segmenaion of he insurance porfolio for he hree lapse sresses consiues a major challenge in pracical applicaions. A he same ime, curren provisions regarding he required segmenaion are quesionable wih respec o he sricly adverse policyholder behavior assumed for he lapse sress scenarios. We conclude ha given he maerial impac on solvency raios, a criical review of he approximaion mehods and assumpions applied in pracical applicaions is required, also including respecive regulaory requiremens. Abbreviaions The following abbreviaions are used in his manuscrip: BE Gar BEL BOF CE CoC ExC FDB GCR HRG Mrg PVFP RfB RM SCR Bes Esimae of Guaraneed Obligaions Bes Esimae of Liabiliies Basic Own Funds Cerainy Equivalen Cos-of-Capial rae Excess Capial Fuure Discreionary Benefis Going Concern Reserve Homogeneous risk group Presen Value of Margins Presen Value of Fuure Profis Reserve for Bonuses and Rebaes (Rücksellung für Beiragsrückersaung) Risk Margin Solvency Capial Requiremen

36 Risks 218, 6, of 38 SF TBF TP TVG TVFOG TVO UGL VAG Surplus Funds Terminal Bonus Funds Technical Provisions Time Value of Financial Guaranees Time Value of Financial Opions and Guaranees Time Value of Financial Opions Unrealized Gains or Losses Insurance Supervision Law (Versicherungsaufsichsgesez) Appendix A. Daa from he German Life Insurance Marke Table A1. Hisoric daa from he German life insurance marke used for deriving he porfolio of insurance conracs. Year Tariff Cohor Ne Invesmen Adminisraion Acquisiion Generaion k Reurn a Cos Rae a,c Cos Rae a,d % 5.8 % 5.5 % % 5.7 % 5.5 % % 5.5 % 5.5 % % 5.4 % 5.5 % % 5.16 % 5.5 % % 4.92 % 5.5 % % 4.68 % 5.5 % % 4.44 % 5.5 % % 4.2 % 5.5 % % 4.6 % 5.52 % % 3.92 % 5.54 % % 3.78 % 5.56 % % 3.64 % 5.58 % % 3.5 % 5.6 % % 3.44 % 5.6 % % 3.38 % 5.6 % % 3.32 % 5.6 % % 3.26 % 5.6 % % 3.2 % 5.6 % % 3. % 4.9 % % 2.9 % 5.2 % % 2.8 % 4.9 % % 2.7 % 5.2 % % 2.4 % 5.1 % % 2.4 % 5. % % b 2.4 % 5. % % b 2.3 % 5.1 % % b 2.2 % 5. % % b 2.3 % 4.9 % % b 2.3 % 4.8 % a Daa is aken from GDV (217b). b Side effecs from he increased realizaion of valuaion reserves for he financing of he ZZR have been eliminaed. c The adminisraion cos rae is given as a percenage of gross wrien premium income. No daa is given for years 1987 o 1994; he missing values are derived by inerpolaion. d The acquisiion cos rae is given as a percenage of new business premium sum. No daa is given for 1994 and previous years; we use he acquisiion cos rae for 1995 also for hose years. References Albizzai, Marie-Odile, and Hélyee Geman Ineres rae risk managemen and valuaion of he surrender opion in life insurance policies. Journal of Risk and Insurance 61: [CrossRef] Bacinello, Anna Ria. 23. Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. Journal of Risk and Insurance 7: [CrossRef]

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38 Risks 218, 6, of 38 Kling, Alexander, Frederik Ruez, and Jochen Ruß The impac of policyholder behavior on pricing, hedging, and hedge efficiency of wihdrawal benefi guaranees in variable annuiies. European Acuarial Journal 4: [CrossRef] Kochanski, Michael. 21. Solvenzkapial für flv uner berücksichigung von dynamischem sorno. Zeischrif für Die Gesame Versicherungswissenschaf 99: [CrossRef] Kuo, Weiyu, Chenghsien Tsai, and Wei-Kuang Chen. 23. An empirical sudy on he lapse rae: The coinegraion approach. Journal of Risk and Insurance 7: [CrossRef] Oechslin, Joachim, Olivier Aubry, Mahias Aellig, Alexander Kappeli, Daniel Bronnimann, Arnaud Tandonne, and Guillaume Valois. 27. Replicaing embedded opions. Life & Pension Risk February 3: Reuß, Andreas, Jochen Ruß, and Jochen Wieland Paricipaing life insurance conracs under risk based solvency frameworks: How o increase capial efficiency by produc design. In Innovaions in Quaniaive Risk Managemen. Springer Proceedings in Mahemaics & Saisics. Cham: Springer, vol. 99, pp Wolfsdorf, Kur Versicherungsmahemaik: Teil 1: Personenversicherung. Sugar: Vieweg + Teubner Verlag. c 218 by he auhor. Licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion (CC BY) license (hp://creaivecommons.org/licenses/by/4./).