Pensions, annuities, and long-term care insurance: On the impact of risk screening

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1 Pensions, annuities, and long-term care insurance: On te impact of risk screening M. Martin Boyer and Franca Glenzer First draft: February 215 Tis draft: February 216 PRELIMINARY DRAFT; PLEASE DO NOT QUOTE Abstract We examine te interaction between te coice of a retirement veicle and te purcase of long-term care insurance in a world were agents learn about teir longevity and long-term care risk over time. In our setting, and absent any long-term care issues, acquiring a retirement product before learning one s risk type would be preferred by risk averse agents. Wen we introduce te possibility of needing long-term care, we sow tat agents may prefer to wait until tey know teir ealt status (i.e., teir risk type) before purcasing a retirement product (asituationakintoavingadefined contribution pension plan). Oters agents will opt to purcase teir retirement product before learning teir ealt status (a situation akin to aving adefined benefit pension plan). Te preference of one retirement veicle over te oter depends, inter alia, on te level of information asymmetry on te market, on an agent s risk aversion, and on te probability of needing long-term care and its potential cost. Wen agents purcase teir retirement veicle before knowing teir teir ealt status, ten agents will coose a contract tat provides tem wit less tan full coverage. Wen agents wait to know teir ealt status before purcasing a retirement product, ten full long-term care insurance is purcased by all agents. We are indebted to Saron Tennyson for discussions on a-many related topic. We also gratefully acknowledge te continuing financial support of Cirano and of te Direction de la recerce at HEC Montréal. Tis paper was written wile te first autor was a visiting scolar at te College of Human Ecology at Cornell University. CEFA Professor of Finance and Insurance and Cirano Fellow, Department of Finance, HEC Montréal (Université de Montréal). 3, cemin de la Côte-Sainte-Caterine, Montréal QC, H3T 2A7 Canada; martin.boyer@ec.ca (corresponding autor). House of Finance, Goete Universität Frankfurt and Cirano; glenzer@finance.uni-frankfurt.de. 1

2 Pensions, annuities, and long-term care insurance: On te impact of risk screening PRELIMINARY DRAFT; PLEASE DO NOT QUOTE Abstract: We examine te interaction between te coice of a retirement veicle and te purcase of long-term care insurance in a world were agents learn about teir longevity and long-term care risk over time. In our setting, and absent any long-term care issues, acquiring a retirement product before learning one s risk type would be preferred by risk averse agents. Wen we introduce te possibility of needing long-term care, we sow tat agents may prefer to wait until tey know teir ealt status (i.e., teir risk type) before purcasing a retirement product (a situation akin to aving a defined contribution pension plan). Oters agents will opt to purcase teir retirement product before learning teir ealt status (a situation akin to aving a defined benefit pension plan). Te preference of one retirement veicle over te oter depends, inter alia, ontelevel of information asymmetry on te market, on an agent s risk aversion, and on te probability of needing long-term care and its potential cost. Wen agents purcase teir retirement veicle before knowing teir teir ealt status, ten agents will coose a contract tat provides tem wit less tan full coverage. Wen agents wait to know teir ealt status before purcasing a retirement product, ten full long-term care insurance is purcased by all agents. JEL Classification: D82, G22, H55. Keywords: Asymmetric information, Information acquisition, Adverse Selection, Longevity risk, Screening, Pooling Equilibrium. 2

3 1 Introduction 1.1 Motivation Planing for retirement is a complex endeavor; individuals are bound to ask temselves many questions. Sould I invest in bonds, stocks, or target retirement date products? Wat will be my actual consumption needs upon retirement in 2 years (or 3 or 4 years)? Do I ave bequest motives or any legacy I want to leave? Wat if I become sick and need elp wit activities of daily living? Sould I acquire information about my genetic pre-disposition to ave some future degenerative disease? Sould I work for an employer tat offers a defined benefit oradefined contribution plan? Do I care? Sould I ask of my employer to offer a defined benefit plan? Even toug most individuals may not be asking all tese questions explicitly, retirement planning is not and sould not be done apazardly. Retirement planning is te more important tat individuals are exposed to two important and potentially catastropic financial risks upon reacing old age: Longevity risk and long-term care risk. Te first risk is represented by te possibility tat one outlives one s savings, wereas te oter is represented by te need of professional elp wit activities of daily living suc as bating, eating and getting dressed, as a result of a debilitating or degenerative disease or illness. Long-term care is an important component of ealt-related expenditures especially for te elderly population tat as to compose wit a relatively small fixed income. To put te problem of long-term care in perspective, te total expenditure on long-term care in te United States in 212 was 22 billion USD. Tis represents 8.7% of total ealt care expenditures according to Konetzka et al. (214). Brown and Finkelstein (29) report tat between one-tird and onealf of retirees in te United States will need some form of nursing ome services, 1% to 2% of tem needing it for over five years. Given tat te annual cost of a nursing ome is approximately four times te average annual gross income of retirees, we see tat long-term care represents a potential catastropic financial event for a ouseold. In te United Kingdom, it is estimated tat 1% of individuals aged 65 and over will incur life-time care costs in excess of 1 GBP (see Wittenberg 214). In Canada, De Donder and Leroux (215) report te results of a 212 study conducted by te Canadian Life and Healt Insurance Association in wic it is predicted tat one-tent of Canadian aged 55 will need long-term care. Te proportion increases of Canadians needing long-term care rises to one in tree wen aged 65, and to one in two wen aged 75. 3

4 Many types of retirement veicles are available to individuals, including defined benefit and defined contribution pension plans. 1 Wile corporate defined benefit pension plan ave been on te decline in te United States, tat is not necessarily te case everywere 2 - and it is not obvious wat are te reason for te decline of corporate defined benefit plans. Does te fault rest wit employers wo welcome te reduction in teir future liability? Are inadequate government rules and regulations to blame for discouraging te use of defined benefit plans? Or are workers temselves responsible for trading a retirement veicle tat offers more in terms of consumption guarantees upon retirement in favor of one tat provides more investment flexibility and encourages mobility (see Aaronson and Coronado 25)? Despite te decline in te use of defined benefit plans in te private sector, te great majority of public employees remain covered by a pletora of different employer-specific defined benefit pension plans. At te end of fiscal year 213, te 99 largest public retirement systems in te United States - representing 85% of all public pension funds - were administering defined benefit pension plans for over 2 million Americans, including million active public employees (see Public Fund Survey 3, and Moan and Zang 214). Figure 1 illustrates te total assets under management in different OECD countries as a function of weter te occupational pension plan is classified as defined benefit ordefined contribution. As we can clearly see, tere is quite a lot of variation in te ways tat countries ave cosen to structure teir occupational pension plan industry. Since few studies so far ave investigated individuals preference over retirement veicles, te motivations beind an individual s coice of a defined benefit versusadefined contribution plan are still not very well understood - or, as Brown and Weisbenner (214) put it, "relatively little is known about wat types of employees coose a DC over a DB plan wen given te option to do so, and even less is known about wy individuals make tese coices" (p.35). Brown and Weisbenner (214) conduct a survey amongst individuals wic, at some point, ad 1 Even toug in many cases, a given employer will not offer a coice of retirement veicles, te pension plan offered is one of te criteria wen coosing an employer. At te margin, te kind of pension plan in place at a company migt ten be te deciding factor for employer coice. Tis is wat we refer to wen we say tat an individual cooses a retirement veicle. 2 In te case of Canada, for instance, Drolet and Morissette (214) find tat te drop in DB coverage from 1977 until 211 is muc less pronounced tan in te United States, and almost negligible for women as 33% of working women in 1977 were covered by a defined benefit plan compared to 3% in ttp:// Last accessed 24 Feb

5 Percentage of total assets in different retirement veicle for selected OECD countries at te end of 213 Only Defined contribution Mostly Defined contribution Mostly or only Defined Benefit DB DC DB DC DB DC Cile 1 Denmark 7 93 United States Czec Rep. 1 Italy 7 93 Turkey Estonia 1 Australia 1 9 Israel 7 3 France 1 Mexico Korea Greece 1 New Zealand 2 8 Luxembourg 8 2 Hungary 1 Iceland Portugal Poland 1 Spain Canada 97 3 Slovak Rep. 1 Finland 1 Slovenia 1 Germany 1 Switzerland 1 Figure 1: Percentage of assets in defined benefit and defined contribution pension plans at year end 213. Occupational pension plans only. Source: OECD Pension Markets in Focus 214 to coose between a defined benefit andadefinedcontributionplanintestate Universities Retirement System (SURS) of te state of Illinois. Based on te respondents personal caracteristics as well as teir answered reasons for te coice of te retirement veicle, te autors igligt te role tat education and financial literacy plays in a ouseold s beavior toward retirement veicles. In particular, more educated individuals are more likely to coose a defined contribution pension plan over a defined benefit pension plan. In contrast, Drolet and Morissette (214) use Canadian pension survey data to not only igligt tat te iger is an individual s level of education, te more likely e or se is to be covered by any type of pension plan in general, but more importantly tat individuals wit iger education are more likely to be covered by a defined benefit pension plan. 4 Te question we address in tis paper, and wic as not been examined before to our knowledge in te insurance economics and finance litterature is related to te interaction between te desire to ave protection against potential catastropic losses associated wit te need to obtain long-term care, and te structure and te individual coices of retirement veicles. More specifically, we 4 For more on te coice between DB and DC see Clark and Pitts (22) and Kanemasu et al. (214). 5

6 examine ow te potential need for long-term care influences an individual s decision between a DB and DC plan. We seek to fill te gap in te economic of risk litterature by developing a model of pension plan coice, taking into account an individual s potential future need of long-term care services as well as information asymmetry wit regards to an individual s ealt conditions. In our model framework, individuals differ wit respect to teir ealt conditions insofar as it affects teir probability of surviving until old age and teir unconditional probability of needing long-term care since suc care is needed only once old age is reaced. Tey only learn about teir survival probability upon entering retirement. We consider te acquisition of a pension plan before learning one s risk type as being very similar to tat of coosing to work for an employer wo offers a defined benefit (DB) pension plan. If an agent cooses to wait to know is risk type before acquiring an appropriate retirement veicle, ten it is as if e was coosing to ave a defined contribution (DC) pension plan. 5 Our results contribute to te understanding of te interaction between longevity risk, long-term care risk, and te retirement financing decision. In particular, te latter result provides a teoretical basis for te self-selection of individuals into different types of retirement veicles based only on te agents possible risk of needing long-term care, tus offering a new reason wy defined benefit and defined contribution pension plan can coexist in te economy. Our results can be summarized as follows: 1. If agents acquire information about teir ealt status before purcasing a retirement veicle (DC plan), ten all agents will purcase full long-term care insurance, if at all; 2. If agents acquire teir retirement veicle before knowing teir ealt status (DB plan), ten all agents will purcase less tan full long-term care insurance, if at all; 3. Low-risk individuals signal teir ealt status by aving less insurance tan te ig-risk individuals; 4. Individuals are more likely to prefer a DB plan to a DC plan, ceteris paribus, wen a) tey are more risk averse, b) tey face greater potential long-term care expenditures, c) tey are 5 Te model we develop can be modified to address te topic of weter one sould get a genetic test before or after purcasing is retirement veicle. Genetic testing is very muc at te forefront of current developments in public policy. For more on te impact of genetic testing on insurance contract, see Doerty and Tistle (1996), Doerty and Posey (1998), Zick et al. (25), Adams et al. (213, 214) and Peter et al. (213). 6

7 more likely to need long-term care, d) tey are more likely to be ig risks (or te proportion of ig risks in te economy is larger), and e) tey are poorer. Te remainder of te paper is organized as follow. Before presenting te full setup of te modelinsection2,weoffer a survey of te literature related to long-term care and te provision of defined benefit and defined contribution pension plans. In Section 3 we solve for te full information retirement and long-term care contracts. We introduce adverse selection in Section 4 and solve first for te optimal contracts wen te two retirement and te long-term care insurance markets are served independently, and ten proceed to looking at te markets togeter. In oter words, we first examine in Section 4 te case were insurers are constrained to make no profit in expectation on eac contract and for eac risk-type, and ten we examine te case were insurers are only constrained to make no profit in expectation for te wole bundled-contract market (i.e., we allow for te subsidization across risk-types and markets). We provide simulation results in Section 5 and alternate model specifications tat yield te same generic results in Section 6. Finally we conclude wit a discussion of potential future avenues of researc in Section Te retirement (DB/DC) decision. Early teoretical researc by Yaari (1965) as suggested tat it would be best for risk averse individuals to annuitize a considerable part of teir wealt upon reacing retirement. Despite tis clear result, wic olds under many oter model specifications including bequest motives, te demand for individual annuities is so low tat some financial economists are referring to it as te "annuity puzzle" (Mitcell et al. 1999; Davidoff et al. 25). One reason tat is often mentioned as to wy te annuity market is so small is tat adverse selection problems are too pronounced. Tat is a well-known result of te Rotscild and Stiglitz (1976) model of adverse selection: A competitive market will collapse provided tat te proportion of ig risks exceeds a certain tresold. Even toug tere is ample empirical evidence for adverse selection in te annuity market, tere is also evidence of advantageous selection in te sense tat more risk-averse individuals take better care of temselves, live longer, and are more likely to buy annuities (see Finkelstein and Poterba 24), and to buy long-term care insurance (Finkelstein and McGarry 26). Akin to Cocco and Lopes (211) wo sow te existence of a link between an individual s earnings caracteristics and is coice of a pension plan structure, we seek to model te interaction between an individual s potential future need for long-term care and is coice of a retirement 7

8 veicle. Altoug we do not use data calibration (see in Cocco and Lopes 211 for suc an approac), we will noneteless draw inferences about te impact of needing long-term care on te coice between defined benefit and defined contribution pension plans based on simulations. An individual s coice of a defined contribution pension plan over a defined benefit pension plan as attracted a fair amount of researc. For instance, Clark and Pitts (22) study an individual worker s decision over te type of retirement plan at one given employer. Tat is also te approaced used in Lacance et al. (23). Using an option-like pricing approac tey examine te situation of pension plans tat allow teir participants enrolled in a defined contribution plan to buy teir waybackintoadefined benefit pension plan (see also Milevsky and Promislow 24). Altoug tey do not examine te caracteristics of te individuals wo exercised tat option to buy back into te defined benefit plan, surely tere are personal caracteristics tat determine one s decision to switc in addition to te pure monetary aspect. More recently, Kanemasu et al (214) examine te response of California public pension plan participants to a potential switc from a defined benefit toadefined contribution plan. Brown and Weisbenner (214) conclude tat, in iger education, over 5% of U.S. states offer some employees te coice between a defined contribution and a defined benefit plan. Our interest in studying te coice of a particular retirement veicle over anoter is based on te fact tat wealt at retirement is an important component of te overall level of poverty in a nation. To see wy, it is interesting to compare te importance of pension plan income relative to a ouseold s total retirement income as in Purcell (28). It appears in tat study tat retirement income tat originates from defined benefit pension plans represents between 15% and 2% of te wealtiest retired ouseolds income, wereas it represents merely 3% of te poorest 2% of ouseolds total retirement income. Defined benefit plansareaveryimportantcontributorto te social welfare of te elderly population according to Porell and Almeida (29) wo estimate tat te number of poor and near poor ouseolds would ave increased by 43% were it not for te ouseolds access a defined benefit pension plan. Te same analysis for social security and for defined contribution plans sows tat te number of poor and of near poor ouseolds would ave increased by approximately 15% and 1% ad tose ouseolds been denied access to social security and a defined contribution plan respectively. Accordingly defined benefit plans play a very important role in keeping retired ouseolds out of poverty. Cocco and Lopes (211) furter point to te role tat pension plans play in te wealt distribution of te nation s retired ouseolds. 8

9 Does tat mean tat coosing a defined contribution plan over a defined benefit planisamistake? Not necessarily for a given individual, but for an entire nation, tat may be problematic (see Brown 214). Gerrans and Clark (213) report tat te welfare debate on te coice of a defined benefit rater tan a defined contribution plan is surrounded by te idea tat people are beaving irrationally: eiter individuals overly discount te future, or tey do not understand te difference between te two types of retirement funding plans (Clark et al. 212). In teir study of Australian pension plan participants Gerrans and Clark (213) find tat individuals tat switced from defined benefit todefined contribution were typically younger and wealtier. 6 Brown and Weisbenner (214) find similar results in te state of Illinois, and add education as a factor increasing te likeliood of picking a defined contribution plan. 1.3 Te markets for long-term care insurance Te market for long-term care insurance is very small despite te potential catastropic losses associated wit long-term care costs and te considerable impact tat te access to long-term care insurance (or te lack tereof) can ave on an individual s ability to smoot is lifetime income. Survey data suggests tat tose wo outlive teir age-gender-coort also need long-term care for a longer-tan-average period of time as compared to teir respective coort so tat longer lived individuals are more likely to use long-term care (Yang et al. 23; Crimmins and Beltrán-Sáncez 211; Hurd et al. 214). In addition to a more likely need for long term care wen an individual gets older, te total cost of long-term care also increases wit age as igligted in Spillman and Lubitz (2, 22), Manton et al. (26) and Hurd et al. (214). According to te American Association of Long-Term Care Insurers, "Most long term care insurance claims begin after a person reaces age % of new claims begin between ages % of all new long-term care insurance claims begin after age 8". 7 6 Recent reforms aimed at introducing fair-value in pension accounting will surely increase te volatility of te financial statements of pension plan sponsors, especially for tose tat ave an important duration mismatc between teir assets and liabilities. As te volatility of a firm s implied debt-to-equity ratio increases because of te (potential) underfunding of pension liabilities, it seems natural to expect firms, tat face some level of solvency risk, to wis to sift te burden of te retirement package tey offer teir workers from a defined benefit (wic is a potential liability for te firm)toadefined contribution plan (wic essentially does not increase te firm s leverage or its risk of financial distress and insolvency). But of course closing DB plans in favour of DC plans does not eliminate te funding risk; it only sifts it from te employer to te employee. 7 ttp:// 27 january

10 Te same message is provided by te National Association of Insurance Commissioners wo claim tat "te longer you live, te more likely it is tat you will need long term care". 8 Te empirical question tat needs anwering terefore seems to be associated wit te individuals apparent disregard for long-term care risk since tey do not seem to buy long-term care insurance. According to Zou-Ricter et al. (21), it may be due to a misperception of te risk; but given te potential catastropic nature of tis risk, it is ard for us to imagine tat individuals would not want to acquire te relevant information about tat risk or do teir proper researc into te nature of long-term care risk. Anoter explanation may be tat individuals are only sligtly risk averse, if not even risk neutral, wit respect to long-term care risk, altoug te results in Sydnor (21) seem to indicate tat individuals are extremely risk averse over small gambles, wic makes it unlikely tat tey would be risk neutral over one as large at tat of long-term care risk. Boyer (215) propose instead tat an individual does not see te cost of long-term care (or of nursing ome) as being borne by imself, but rater by is younger spouse or eirs (tis is equivalent to including a bequest motive in a long-term care framework as in Lockwood 21, and Brunner 212). Put differently, te cost of long-term care only reduces te size of te bequest tat an individual leaves beind so tat it sould be te eirs tat sould be buying long-term care insurance to protect te bequest tat is left beind by teir elderly and rapidly aging parent. Te projected need of long-term care insurance as one grows older olds despite te apparent counter-evidence of compressed morbidity (tat is, te number of years during wic an individual is sick is decreasing). Te debate surrounding te compression of morbidity is best presented in te surveys of Crimmins and Beltrán-Sáncez (21) and Fries et al. (211). Even if morbidity is compressed, te out-of-pocket cost associated wit te last two years of life increases exponentially as te individual gets older. One possible reason is tat individuals prefer to use te professional services of nursing omes instead of te natural services of family and informal care-givers (Robine et al. 27, Tennyson and Yang 214). Tis sift in te demand for geriatric-specific ealt services will necessarily ave a very important impact on te future demand for long-term care insurance. Tis projected rise in te demand for formal long-term care is associated wit te rapid growt in te elderlier population, canges in teir relative independence from teir spouse and cildren (if any), and wit a substitution effect across public and private services of geriatric-specific medical care (see te recent De Donder and Leroux 215 for more on te political economy of long term 8 ttp:// 27 january

11 care, and Brown and Finkelstein 211 and Cremer et al. 29 for more extensive surveys). A longer period spent in need of long-term care is equivalent to a iger present value of future long-term care insurance benefits, ence a iger loss severity in te long-term care segment. Tus, instead of considering different probabilities to eventually need long-term care, it seems straigtforward to model different loss severities instead. Doerty and Jung (1993) put forward a model of adverse selection in loss probabilities wen loss severity is positively correlated wit te loss probability. We built on bot teir and on Webb s (29) model to examine te equilibrium properties of stand-alone annuities and long-term care insurance market equilibria were agents ave private information wit respect to teir survival probability (i.e., teir risk type in te annuity market) and teir severity of loss on te long-term care market. 9 Basedontismodelwe ten examine te situation wen te two markets are integrated at te individual risk level as well as at te entire risk portfolio level. 2 Te Model Setup 2.1 Timing of te model Our model starts from te premise tat individuals enrolled early in a defined benefit pension plan do not yet ave any information about teir longevity risk type to signal to teir potential provider of retirement income. In contrast, individuals enrolled in defined contribution pension plans acquire over time information about suc longevity risk so tat wen te time comes to annuitize teir accumulated wealt, tey ave a more and better information about teir ealt and longevity risk. Tis assumption is reasonable in te sense tat an individual enrolled in a defined benefit pension plan as an income pat upon retirement tat is contracted at te time te individual starts is career and is terefore independent of is future ealt status. Moreover, at te time of is decision to enter a defined benefit pension plan, te individual is so far away from retirement tat e surely as no information watsoever about is future ealt condition. In a defined contribution plan, owever, individuals accumulate a sum of money tat tey will eventually want to transfer into a income-generating veicle upon retirement (say an annuity) 1. Surely at tat 9 In fact, in contrast to Doerty and Jung (1993), our model does not even require a positive correlation between te longevity and te extent to wic long-term care is needed. Assuming tat a ealty individual will live longer and will also live longer in long-term care simply seems to better reflect empirical evidence. 1 Rotscild (215) provides an adverse selection model of annuity purcase in a market were all retirement savings must be annuitized upon retirement. See also Villeneuve 23 and Finkelstein et al. (29). 11

12 point in time individuals ave lived long enoug to accumulate some type of information regarding teir potential ealt status eading into retirement. In suc a world, and absent any oter retirement market caracteristics (see Bodie et al. 1988), adefined benefit pension plan would always dominate a defined contribution pension plan. Te reason is tat te possible distribution of wealt in a defined contribution plan is essentially a meanpreserving spread of te distribution of wealt in a defined benefit pension plan. Consequently opting for a defined contribution pension plan is te equivalent of acquiring a lottery wit a mean payoff of zero. To see wy, note tat agents wo coose a defined benefit plan are typically enrolled in a pool of individuals very early in teir career. In particular, tey are enrolled muc before tey ave any valuable private information about teir ealt status upon retirement and te little information tey may ave is pooled wit tat of all te oter members of te pension plan. And even if tey learn over time teir true ealt status, it does not really affect te benefits tey receive wen tey retire. Wit a DC plan, owever, agents accumulate wealt over time, and acquire information about teir ealt status, suc tat at te time of retirement, wen tey are considering purcasing an annuity contract, tey ave valuable private information, wic affects teir coice of an annuity. Given tis delayed albeit forecasted adverse selection problem, it can be easily sown tat initially uninformed agents are better off purcasing te contract before tey acquireanyinformation. Intecaseoftedefined benefit versusdefined contribution plan, our assumption means tat abstracting from pension portability, autonomy, vesting, investment risk, interest rate risk, inflation risk, incentives, and every oter factor (see Poterba et al. 27 for more on te different risk components associated wit te different retirement veicles), a defined benefit plan is always preferred to a defined contribution plan for risk averse agents. As it as become obvious, we are starting wit a model of pension plan dynamics tat is tilted toward te dominance of defined benefit overdefined contribution pension plans. Our agents will be fully rational and will only care about teir expected utility over final wealt. Tere is no investment risk or wage risk; inflation and interest rates are bot zero. In addition to facing a longevity risk tat can be managed troug eiter a defined benefit or adefined contribution pension plan, we will also assume tat individuals may find temselves in a ealt status tat requires te disbursement of a large amount of money upon reacing retirement age. In oter words, wit a certain probability individuals will be in need of costly long-term care. Tey will ave te possibility of acquiring long-term care insurance upon reacing retirement, wen 12

13 Individuals coose between a DC or a DB plan; If DB, retirement contract is written Individuals learn about teir ealt status Individuals may purcase longterm care insurance; Purcase of annuities for tose wo cose DC Nature cooses between tree states of te world: Die, Live ealty, Live in nursing ome Payoffs are distributed Figure 2: Time line of decision taken by te individuals and te actions of nature in distributing information about types and in coosing te end state of te world. tey privately know teir longevity-risk type. Figure 2 presents te time line more succinctly. As we see, adverse selection of risk types plays a role starting in step 2, tat is after individuals ave privately learned teir ealt status. We terefore coose to simplify te economic model to its simplest expression by concentrating exclusively on te combination of longevity and long-term care risk. Tis means tat we do not model te exact investment coice of bonds versus stock versus ousing (see Davidoff 29 and Cocco 25), nor do we include bequest or legacy motives (see Pauly 199). We concentrate only on te coice between retirement saving veicles (DB versus DC) and on te purcase of long-term care insurance in a world were individuals acquire information about teir ealt status over time. Our results will sow tat one advantage of waiting to ave information about one s ealt status before acquiring a retirement-income contract (i.e., te DC way) is tat full long-term care insurance will be a possibility for all te agents in te economy. If on te oter and individuals acquire teir retirement-income contract before teir ealt status is known, ten full long-term care insurance will not be cosen by any individual upon retirement. Our model differs from te multidimensional screening model of Crocker and Snow (211) in te sense tat our agents private information is does not span te entire dimensionality of te typespace problem. In oter words, Crocker and Snow (211) use an n-dimensional vector screening devices for n-possible types of losses wereas we ave only one screening device for a two-dimensional risk (see also Fluet and Pannequin 1997). 13

14 2.2 Individual decision and variable definition Te initial setup of our model finds its inspiration in Webb (29) wereby individuals care about obtaining a decent income upon retirement and protection against te possibility of needing longterm care. Witout any retirement veicle or long-term care insurance, an agent s vonneumann- Morgenstern expected utility over finalwealtinautarcyisgivenby Ω = ( )+ (1 ) ( )+ ( ) () were and are te agent s initial wealt endowments in period and 1 respectively, wit. is te cost of long-term care; it is distributed according to some function () over te range [ ]. is te probability of needing long-term care and, wic is agent specific, is te probability of living past retirement. If te agent dies, we let is utility be zero. 11 Of course we suppose tat agents are risk averse and prefer to consume more tan less so tat ( ) and ( ). We assume no discounting of future periods. We will assume tat agents can be of two types, { }, wit a proportion of agents of type =. Te unconditional probability tat any individual will live past retirement is terefore given by +(1 ), wereas te unconditional probability tat any individual will need long term care is [ +(1 ) ]. Note tat we assume tat individuals do not ave any private information regarding teir conditional probability of needing long-term care (). Individuals wo coose to acquire a retirement veicle BEFORE knowing teir type will receive apaymentof in all states in period 1. Conditional on teir type, individuals wo coose to acquire a retirement veicle AFTER knowing teir type will receive a payment of in all states in period 1. Irrespective of weter tey coose a retirement contract before or after being informed about teir ealt status and teir longevity risk, individuals also get to acquire some level of longterm care insurance protection, wic is defined as te proportion of te long-term care cost tat is reimbursed by te insurer (or equivalently, tey coose an amount = of protection) AFTER learning about teir risk type so tat is conditional on eac agent s type. We let 11 Alternatively, we could also give te agents in tis model a bequest motive from wic tey extract utility (Γ) for a given amount of wealt bequested Γ. Teir expected utility function would ten be given by Ω = ( Γ)+(1 ) (Γ)+ (1 ) ( )+ ( ) () Te main results of our model are not affected by te inclusion of bequest motives since it will appear only as a reduction in te total wealt. See Pauly (199) for more on bequest motives. 14

15 represent te premium in te initial period of te pre-information (DB) retirement contract, represent te premium in te initial period of te post-information (DC) retirement contract, and represent te premium in te initial period of te long-term care insurance contact. Te expected utility of agents are ten equal to Ω = ( )+ (1 ) ( + )+ ( + (1 ) ) () if te agent cooses te retirement option before knowing is type (DB) and Ω = ( )+ (1 ) ( + )+ ( + (1 ) ) () if te agent cooses is retirement package after knowing is type (DC). We assume tat bot te retirement-provision and te long-term care insurance markets are perfectly competitive in te sense tat neiter te suppliers of retirement income nor te long-term care insurers make any profit in expectation. Even toug markets are assumed to be competitive, insurers face transaction and oter types of costs. We sall account for teses costs troug tree parameters tat beave as fixed loading: sall denote te fixed loading on te long-term care market, wereas (resp. )sallbetefixed loasing on te retirement contracts after (resp. before) te agent as learned about is risk type. Te fixed loadings are tere to represent te fact tat marketing, management, underwriting, te andling of claims are costly, and tat tese costs need to be recouped someow. Given te competitive nature of te markets, te premium tat individuals must pay for te preinformation (DB) retirement income contract is given by =[ +(1 ) ] +,were is te fixed loading factor for te DB contract. Tat is, wit average probability [ +(1 ) ] te agent lives to te next period in wic case s is entitled to receive a payment equal to. Wit average probability 1 [ +(1 ) ] e dies. Wit respect to te premium paid for te retirement contract purcased after te information is learned (DC), it must be set to = +,wit { }. Te premium is of course agent-type specific since agents wo face different probability of living to te next period may not coose te same payment, and surely will not pay te same premium. Finally, in te case of te long-term care insurance contract, te premium is given by = R () +. Again te premium is agent-type specific; wit probability te agent is in te state were e needs long-term care, in wic case te insurer pays a proportion oftelossincurredbyteagent.tislossisrepresentedby and is 15

16 distributed according to some function (). In te case of te long-term care contract, te fixed loading is given by. 3 Full Information Contracts Just as a way to establis our bencmark, let us look at te situation were all information is common knowledge. In suc a situation, tere is obviously no difference between obtaining a retirement contract before or after learning private information since tere is no private information. Witout loss of generality, let us use te notation of te post-information (DC) retirement veicle to distinguis between te contracts for te two types of agents. Te problem tat agent { } faces is given by max Ω = ( )+ (1 ) ( + )+ subject to te following zero-profit constraints for te insurers = + ; = () + ( + (1 ) ) () Substituting te zero-profit constraints in te maximization problem, we can write te entire problem as and max Ω = ( + ) Te first order conditions are : = : = + + (1 ) ( + )+ ( + ) + (1 ) ( + )+ ( + ) ( + (1 ) ) () () + ( + (1 ) ) () () + (1) ( + (1 ) ) () () + () (2) From tese conditions, it can be easily sown tat =1is a solution so tat full insurance on te LTC market is best. Tis takes us to te first proposition of te paper: 16

17 Proposition 1 Under full information and fixed loading factor, te firstbestallocationisobtained suc tat ³ R 1 () =1and = 1+ ³ ³ R Proof. From equations 1 and 2, we isolate ³ ( + ) () + to find tat R Z ( + (1 ) ) () R () =(1 ) ( + )+ ( + (1 ) ) () Tis equality olds only wen =1. From Equation 1, we obtain ( + ) () + = ( + ) wic means tat = ( ()) 1+. Te firstbestallocationgivesteagentperfectincome smooting so tat, watever is te state of te world, is consumption is equal to = + = + ( ()) 1+.Giventat tere is no discounting of future periods, we see tat + is te total expected wealt of te agent (wen te agent is dead, wit probability 1, e does not receive any wealt). Te agent wo purcases annuities and long-term care insurance as to incur te fixed loading factors for bot contracts, +, and must pay te premium of te long-term care insurance contract ³ R () given is coice of =1. All tis must finance is total consumption for te two periods, (1 + ). 4 Information asymmetry We will assume from now on tat tere are two types of individuals tat differ only wit respect to teir probability of reacing retirement age. We sall refer to tese agents as being of type H andoftypel,suctat. Tis means tat ig risk individuals are more likely to live to period 1 tan low risk individuals. Conditional upon reacing period 1, all agents face te same risk of needing long-term care. In oter words te probability of needing long-term care and te potential cost of long-term care to te agents is te same for te two types of agents We make tis assumption regarding te distribution of long-term care losses even toug we are aware teat tere is empirical evidence to suggest tat agents tat are more at risk to live a longer life are also more likely to face ig cost of long-term care (see Spillman and Lubitz 2, Yang et al. 23, and Hurd et al. 214). 17

18 In subsection 4.1, we consider te case tat we associate wit defined contribution retirement plans. In tat situation, individuals ave accumulated wealt tat tey need to use to finance consumption upon retirement. During te wealt-accumulation pase of teir lives, individuals ave also acquired private information about teir ealt status, wic translates into a private information abou teir longevity risk. Providers of retirement scemes, suc as annuity providers in a Yaari-type world, know tat individuals ave acquired suc information but tey do not know exactly wat is te information information tat eac agent as acquired. Tey only know tat a proportion of agents are ig risk and ave a probability of reacing retirement age, wereas a proportion 1 of agents ave a probability of reacing retirement age. In addition to aving to purcase te appropriate annuity contract tat maximizes teir expected utility, individuals must also decide weter tey want to purcase long-term care insurance knowing teir type. Similarly to annuity providers, potential long-term care insurers know tat individuals know teir type and wat is te distribution of types in te economy. In subsection 4.2, we consider te case tat we associate wit defined benefit retirement plans. In tat situation no information about risk types is known by any participant at te time of te signalling of te retirement contract, except for te average probability of living past retirement, +(1 ). Altoug individuals do not know teir type wen coosing teir retirement veicle, tey do know it wen te time comes to purcasing long-term care insurance. As in te defined contribution situation, long-term care insurers know tat individuals know teir type. 4.1 Adverse Selection wit Post-information retirement coices In tis section, we examine te situation were agents purcase retirement veicles and long-term care insurance after tey ave learned about teir risk type. We will start wit te situation were eac contract (retirement or long-term care insurance) must make zero-profit in expectation. We will ten allow for cross-subsidization across agent types and products Zero-profit for eac type, for eac product Given our assumption of zero-profit in expectation for eac product and eac type, we ave tat ³ R for { }, = + and = () +. Te maximization problem 18

19 writes as max Ω = ( )+ (1 ) ( + )+ subject to te zero profit conditions ( + (1 ) ) () = + = + = () + = () + and an incentive compatibility constraint (see below) tat guarantees tat te ig risk individual is better off purcasing te contract tat is designed for im tan purcasing te contract tat is designed for te low risk ( )+ (1 ) ( + )+ ( ) (1 ) ( + )+ Solving tis problem yields te following proposition. ( + (1 ) ) () ( + (1 ) ) () Proposition 2 In a market wen providers of services are restricted to offer menus of contract tat make no profit in expectation, wen tere are two types of agents tat differ wit respect to teir probability of living long, and wen agents privately know tat information wen making teir contract coice, ten all agents will be fully indemnified for losses on te long-term care market. On te retirement market, te ig risk agent receives is firstbestallocation(perfectconsumption smooting) wereas te low risk agent must signal is type by accepting an allocation were e still bears some retirement consumption risk. Proof. See Appendix 9.1 An interesting result ere is tat it is quite possible to ave full insurance on te long-term care market even if agents differ wit respect to teir probability of finding temselves in a situation were long-term care is needed. We may ten wonder wy agents are able to obtain full long-term 19

20 care insurance but not perfect consumption smooting. Te reason is not tat agents prefer to smoot wealt witin a period tan across periods; rater, it is because tere is no conditional information asymmetry on te long-term care market once agents ave signalled teir type on te longevity market. Consequently, conditional on learning te agents longevity risk type or, and () are te same for all te agents in te economy. Te only difference across agents come from te probability of reacing retirement age and not from te probability of needing long-term care conditional on reacing retirement age Zero-profit inteindustry Tere are two ways to look at te problem wen cross-subsidization of types is allowed. Eiter providers on eac market make no profit in expectation (so tat witin a market, cross-subsidization is permitted) or tey make no profit in expectation on te joint markets (so tat cross-subsidization is permitted across agent types and across markets). Altoug providers sould be able to crosssubsidize across agent types (ig and low) and products (long-term care and retirement) wile maintaining a zero-profit condition on teir combined business, it as been more natural to examine eac products separately. Given our very specific setup it sould be clear tat, in terms of Pareto efficiency, te industry structure tat is preferable is te one were subsidization across types and insurance markets is permitted. We will terefore only present and discuss te nature of te contract and te equilibrium in te economy in wic providers are restricted to make zero profit as a wole. Te oter case of cross-subsidization across types but not across products is available from te autors, but te setup is noneteless provided in Appendix 1. Results are similar in teir nature. Assuming a single zero-profit constraint for te entire bundled products market, te problem faced by te providers of retirement and long-term care services is to offer a menu of contracts tat maximizes te low risk agents expected utility over final wealt under te condition tat te ig-risk agents still do not want to acquire te contract designed for te low-risk agents. Let us use to represent te total premium tat agent of type { } must pay to purcase te retirement veicle and long-term care insurance. Recall tat te proportion of ig risk agents is. Te problem can be written as max Ω = ( )+ (1 ) ( + )+ ( + (1 ) ) () 2

21 subject to te unique zero-profit condition +(1 ) = + and te incentive-compatibility constraint +(1 ) + ( )+ (1 ) ( + )+ ( ) (1 ) ( + )+ Te Lagrangian is max B B Ω = ( )+ (1 ) ( + )+ () + + () ( + (1 ) ) () ( + (1 ) ) () ( + (1 ) ) () + R i +(1 ) + () R i (1 ) + () (1 ) ( + ) ( )+ + R + ( + (1 ) ) () (1 ) ( + ) ( ) + R ( + (1 ) ) () Te following proposition igligts te caracteristics of te optimal contract using tis modelling approac. Proposition 3 Wen providers of retirement and long-term care insurance services are allowed to cross-subsidize agents and products, ten te optimal allocation is suc tat agents are fully insured on te long-term care market (i.e., = =1). Furtermore, te ig risk individual is receiving a retirement package tat perfectly smoots is consumption in all states of te world wereas te low risk must signal is type by accepting a contact wereby is consumption is not smooted. Proof: See Appendix 9.2. And of course, it must be tat te allocation under tis contract is preferred to, or at least as good as, te one in te standard Rotscild-Stiglitz case if only because te Rotscild-Stiglitz case is a more constrained problem wit zero expected profit on eac contract type. 21

22 Again, as in te first Rotscild-Stiglitz equilibrium contract we solved, we ave tat agents, no matter wat is teir risk type, purcase full insurance against te potential catastropic expenses associated wit long-term care. As before, te reason is tat tere is no conditional asymmetric information on te long-term care market since agents only differ wit respect to te probability of reacing retirement age ( { }) and not wit respect to teir probability of needing long-term care (). Given tat tere is no conditional asymmetric informationontelong-termcaremarketso tat all agents are fully insured on tat market, it must be tat te signalling and all te crosssubsidization across risk types occur on te retirement product market. We may wonder wat appens if we take tis cross-subsidization to te extreme by forcing ig-risk and low-risk type to acquire a contract tat completely pools teir risk of reacing retirement age altogeter. As we will see in te next section, te complete cross-subsidization troug a pooling contract will not yield te same full insurance results on te long-term care market. 4.2 Adverse Selection wit Pre-information retirement coices In tis section, we examine te situation were agents purcase retirement veicles before knowing teir type, but learn about teir type before deciding to purcase long-term care insurance. Again, we will start wit te situation were eac contract (retirement or long-term care insurance) must make zero-profit in expectation. We will ten allow for cross-subsidization across agent types and products Zero-profit for eac type on te long-term care insurance market Given our assumption of zero-profit in expectation for eac risk type in te long-term care insurance ³ R market, it must be tat for { }, = () +. On te retirement veicle market, te zero profit condition for all is =[ +(1 ) ] +. At te time tey are coosing te structure of teir retirement veicle product, agents are able to forecast wat will be teir long-term care insurance decision once tey learn information about teir type. Given tat agents ave no information about teir type, tey ten face te following retirement veicle 22