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

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We examine the interaction between an individual’s pension scheme and her purchase of long-term care insurance in a context where individuals learn their longevity risk type over time. We show that the structure of an individual’s retirement pension scheme is an important component of her selection of long-term care insurance coverage. When individuals purchase their retirement product and long-term care insurance after learning their risk type, low-risk individuals signal their type solely on the retirement product market, which allows all individuals, irrespective of their risk type, to perfectly insure against the incidence of long-term care shocks. When individuals purchase their retirement product before learning their risk type, then the retirement product will pool all risk types, which prevents any signaling in that market. If individuals still learn their type before purchasing long-term care insurance, then having to signal their type in the long-term care insurance market considerably reduces the take-up rate for such protection for all risk types.

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  1. Fluet and Pannequin (1997), Crocker and Snow (2011) and Picard (2019) also examine multidimensional adverse selection problems in insurance markets, as well as Rochet and Choné (1996) and Armstrong and Rochet (1999) in more general setting.

  2. We will abstract from other differences between DB and DC plans, and tax-financed pension plans, such as Social Security in the USA and the Canadian (and Québec) Pension Plan, which are similar in nature to DB schemes in that contributions are not based on risk type.

  3. See for example Hurd et al. (2014) who calculate the prevalence of nursing home stays by age for HRS respondents. The average number of nights respondents spent in a nursing home stay in the two years prior to the survey increases steadily from 0.64 nights for respondents under the age of 55, to 202.97 nights for respondents aged 95 and older.

  4. From a practical point of view, there is a subtle difference between DB and DC schemes and the timing of annuitization since private annuities always come with risk underwriting, even when the contract is written at a young age. One could argue that if annuitization occurs when an individual is young, it is the insurer that is better informed about an individual’s longevity risk because it has underwriting experience and access to data that allow a risk classification even in the absence of concrete information about an individual’s health. The individual, who has no concrete evidence for her health type nor the data and statistical tools allowing a personalized prediction of her risk type becomes the less informed party. To abstract from these complications, we choose to frame early annuitization as a DB scheme, which does not entail any underwriting, and late annuitization as a DC scheme with underwriting.

  5. To lighten the text, we will let the agents be feminine (she) and the principals (the LTC insurers and the annuity providers) be neutral (it).

  6. For a more detailed examination of this puzzle, please see Brown and Finkelstein (2004, 2011), Cremer et al. (2009), Grignon and Bernier (2012), and Pestieau and Ponthiere (2012).

  7. As Brown and Weisbenner (2014) point out, it is increasingly common for U.S. public sector employees to be given the choice between a DB and a DC scheme. In the private sector, it is rare that an employer offers a choice between different retirement vehicles. Nonetheless, in a global compensation framework, the type of pension is one of the criteria when choosing an employer. At the margin, a firm’s pension plan might become the deciding factor. It is in this perspective that we say that an individual “chooses” a retirement vehicle.

  8. See Brown et al. (2017) for an overview of how jurisdictions vary internationally in how much leeway individuals are given with respect to annuitization of their retirement wealth.

  9. The 99 largest public retirement systems in the USA in 2013, representing 85% of all public pension funds, covered over 20 million Americans, including 12.65 million active employees (Mohan and Zhang 2014).

  10. See the OECD report for more details regarding the pension arrangements in those countries that are reported as not having occupational pension plans. In some countries, individuals are obliged to join a personal pension plan, sometimes with voluntary (e.g., in Peru) or obligatory (e.g., in Colombia) contributions from the employer. These are not occupational pension plans according to the OECD since the employer did not set up the plan and is not responsible for its operations.

  11., last accessed on 12 March 2019.

  12.; last accessed on 21 November 2019.

  13. See Webb and Zhivan (2010) for a more conservative estimate.

  14., last accessed on 19 August 2019.

  15. This means that all savings occur through the employer in the case of a DB scheme. We further assume, following Villeneuve (2003), Finkelstein and Poterba (2004), and Finkelstein et al. (2009), that all DC scheme accumulated sums must be annuitized with a single provider, so that annuity contracts are exclusive. See Rothschild (2015) for a discussion of the impact of removing the exclusivity aspect of annuity purchases. In Sect. 6.2 we examine the impact of introducing savings outside of the DB and DC schemes.

  16. By assuming that \(\pi\) is the same for all agents, we are diverging from Murtaugh et al. (2001), Warshawsky (2007) and Webb (2009) who suppose that \(\pi\) is correlated with \(p_i\). While this may seem restrictive, what is relevant for our model is that obtaining information about an individual’s risk on one market is informative about the individual’s risk on the other market. Therefore, our results would hold for a type-specific \(\pi _i\) if it is public knowledge that type i always has a specific combination of longevity and LTC risk.

  17. The impact of introducing bequest and legacy motives in our model is not obvious as it would depend on the way such motives are presented. If bequest motives exist only in the event the policyholder is dead in the last period (which occurs with average probability \((1-\bar{p})=1- (1-x) p_L - x p_H\)), then having a product that pays in the event of death—say, a life insurance policy—would be valuable. This bequest product would have a cost paid for in the initial period, which would necessarily reduce the amount annuitized because total wealth is reduced. An alternative, we could say that all bequest motives are embedded in the last period’s wealth, \(W_2\), such that \(U(W_2)=0\). Another alternative would be to let bequest motives be present in all states of the world in the final period, which would then increase the amount annuitized because policyholders want to leave a legacy after death, whenever death happens (i.e., either at the beginning or at the end of the final period). Including bequest and legacy motives would surely change the nature of the equilibrium demand for retirement and/or LTC insurance products. Nonetheless, the demand for LTC insurance would remain smaller under a DB scheme than under a DC scheme, whichever way we model bequest motives.

  18. Even though in reality insurers must incur expenses for marketing, management, underwriting, and claims-handling, even in competitive markets, we will assume that there are no loading factors, so that in each market, the premium paid is simply equal to the expected loss.

  19. Equivalently, we could also assume that suppliers are restricted to offer a single contract designed for either type of agents, so that cross-subsidization becomes irrelevant. Moreover, as shown by Sandroni and Squintani (2007), if insurers compete locally in single-contract offers in the Nash sense, then, irrespective of the share of high risks in the market, the Rothschild-Stiglitz allocation is feasible, sustainable, and stable. See Sect. 6 for further discussion.

  20. The two simplified first-order conditions are \(U^\prime ( W_1-\eta ^{FB})=( 1-\pi ) U^\prime ( W_2+B^{FB}) + \pi U^\prime ( W_2+B^{FB}-( 1-d^{FB})\lambda )\) and \(U^\prime ( W_1-\eta ^{FB}) = U^\prime ( W_2+B^{FB}-( 1-d^{FB})\lambda )\).

  21. Put differently, \(\Delta =U^\prime (W_1- \bar{p} B^*- p_i \pi d_i^{DB-sym} \lambda )-p_i [(1-\pi ) U^\prime ( W_2+B^*) +\pi U^\prime ( W_2+B^*-( 1-d_i^{DB-sym}) \lambda )]\) is the difference between the marginal utility of an agent of type \(i \in \{L,H\}\) in the initial period and her expected marginal utility in the final period. Knowing that under symmetric information, an agent’s intertemporal expected utility is maximized when \(\Delta =0\), we can show that as \(d_i^{DB-sym} \longrightarrow 0\), then \(\Delta <0\) and \(\frac{\partial ^2 \Delta }{\partial d_i \partial p_i}>0\).

  22. Note that when the average survival probability is 0.60, and for a share of high-risk agents of \(x=0.3\), the high-risk agent’s survival probability would have to be higher than 1 for a survival probability of the low-risk agent that is smaller than 43%. While for the purpose of consistency with the other numerical example, the graph is shown for the entire domain of survival probabilities, the results for a survival probability lower than 43% should be disregarded.

  23. In principle, there are three possible contract combinations that high-risk agents could choose instead of the one designed for them: (1) Both the annuity and the LTC insurance contract designed for the low-risk agent; (2) the annuity contract designed for the high-risk agent and the LTC insurance contract designed for the low-risk agent; or (3) the annuity contract designed for the low-risk agent and the LTC insurance contract designed for the high-risk agent. For simplicity, we will assume that only the first incentive constraint binds, so that the high-risk agent can only choose the low-risk agent’s annuity contract if she also chooses the low-risk agent’s LTC insurance contract and vice versa. This is tantamount to assuming there is information sharing between insurance providers as described by Fluet and Pannequin (1997)

  24. All remaining figures are in the paper’s Appendix.

  25. Or that their saving choice is made before the accepting the DB/DC contract, and that this choice is observable.

  26. The argument is similar with respect to insuring the deductible in a “simple” Rothschild-Stiglitz equilibrium: If the deductible is insured (through a non-exclusive contract), then there is no signal.

  27. In other words, by saving outside of the DC scheme agents would reduce the validity of the signal they send about their type, which means that all information gathered on the annuity market would have no value for LTC insurers.

  28. We thank an anonymous referee and the editor for suggesting that we discuss the implications of allowing savings outside of the DB / DC schemes.

  29. An interesting side issue is that our model does not need to be of a repeated nature for the market insurance equilibrium to collapse, as in Allen (1985) and Chiappori et al. (1994).

  30.; last accessed on 15 July 2020

  31. The report can be found at; last accessed on 15 July 2020. See also Chien and Morris (2018).

  32. In a DB system, individuals are grouped together with their fellow workers in such a way that each individual’s specific risk type is irrelevant. In a DC scheme, however, agents can convert their retirement savings into an annuity at the time of their retirement—at which point they will have more information about their risk type.

  33. An individual’s insurance decision and preference over pension schemes are affected by many factors that were left our of our model. One such important factor is risk aversion (Finkelstein and McGarry 2006; Webb 2009). Another important factor is the investment risk that individuals assume in a DC scheme, but not in a DB scheme (see Davidoff (2009) for more on the investment choices through which retirement wealth is accumulated). And even though investment returns can be considerably higher in a DC scheme than in a DB scheme (OECD 2017), the investment risk (and necessary investment in financial literacy) that accompanies higher expected investment returns might deter individuals from choosing a DC scheme (Clark et al. 2006; Gerrans and Clark 2013). Other considerations, such as bequest and legacy motives (Pauly 1990; Courbage and Montoliu-Montes 2018), as well as inter-generational moral hazard (Courbage and Zweifel 2011; Ko 2017), were left out to keep our model tractable.

  34. For more on the impact of genetic testing on insurance contracts, see Doherty and Thistle (1996), Doherty and Posey (1998), Zick et al. (2005), Adams et al. (2015a, b), Hoy et al. (2003), Hoy and Ruse (2005), Barigozzi and Henriet (2011), and Peter et al. (2017).

  35. This cannot possibly be correct since it would imply no signalling whatsoever. The exercise is done to see whether signalling makes the low-risk agent’s annuity contract more or less generous than the high-risk agent’s annuity contract, \(A_H^{sym}\).


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Correspondence to M. Martin Boyer.

Additional information

This paper was written while the first author was a visiting scholar at the College of Human Ecology at Cornell University, and the second was finishing her doctoral studies at the House of Finance, Goethe Universität Frankfurt. We are indebted to Sharon Tennyson for discussions on a-many related topic, and for very insightful comments by the Journal’s editor, Michael Hoy, and the Journal’s referees. We both gratefully acknowledge the continuing financial support of the Direction de la recherche at HEC Montréal and Boyer acknowledges the financial support of the Social Science and Humanities Research Council of Canada (435-2016-1109). We would also like to thank seminar and conference participants at Université Laval, Universität St-Gallen, St. John’s University, the Société canadienne de science économique, the Southern Finance Association, the Southern Risk and Insurance Association, and the American Risk and Insurance Association.

Appendix: Proofs

Appendix: Proofs

1.1 Proof of Proposition 2

For each agent \(i\in \{L,H\}\), the second stage problem is

$$\begin{aligned} \max _{d_i^{DB}}\Psi _{i}^{DB-sym} = &\; U\left( W_{0}-\bar{p} B^{*}-p_{i}\pi d_i^{DB}\lambda \right) \\&+p_i\left[ \left( 1-\pi \right) U\left( W_2+B^{*}\right) +\pi U\left( W_2+B^{*}-\left( 1-d_i^{DB}\right) \lambda \right) \right] . \end{aligned}$$

The first-order conditions, one for each risk type, can be written as:

$$\begin{aligned} U^{\prime }\left( W_{0}-\bar{p} B^{*}-p_{L}\pi d_L^{DB-sym}\lambda \right)= & {} U^{\prime }\left( W_2+B^{*}-\left( 1-d_L^{DB-sym}\right) \lambda \right) \\ U^{\prime }\left( W_{0}-\bar{p} B^{*}-p_{H}\pi d_H^{DB-sym}\lambda \right)= & {} U^{\prime }\left( W_2+B^{*}-\left( 1-d_H^{DB-sym}\right) \lambda \right) \end{aligned}$$

so that

$$\begin{aligned} W_1-\bar{p}B-p_L\pi d_L^{DB-sym} \lambda= & {} W_2+B-(1-d_L^{DB-sym})\lambda \end{aligned}$$
$$\begin{aligned} W_1-\bar{p}B-p_H\pi d_H^{DB-sym} \lambda= & {} W_2+B-(1-d_H^{DB-sym})\lambda \end{aligned}$$

Subtracting (31) from (30) and rearranging the terms, we find

$$\begin{aligned} d_L^{DB-sym}=d_H^{DB-sym} \left[ \frac{1+p_H \pi }{1+p_L \pi }\right] . \end{aligned}$$

Since \(p_H > p_L\), it follows that \(d_L^{DB-sym} > d_H^{DB-sym}\).

We thank the editor, Michael Hoy, for proposing a simpler proof of this Proposition. \(\square\)

1.2 Proof of Proposition 3

The first two parts of the proof are a simple restatement of Proposition 1: 1- Once agents have signaled their type on the annuity market, no residual adverse selection is left so that perfect consumption smoothing in the final period becomes possible for all agents (i.e., \(d_L^{DC}=d_H^{DC}=1\)); 2- High-risk agents do not need to signal their type and thus receive the contract allocation they would have received under symmetric information (i.e., \(A_H=A_H^{sym}\)). For the third case, let us use the incentive compatibility constraint of (22) in which we let \(d_L^{DC}=d_H^{DC}=1\) and \(A_H=A_H^{sym}\), and find that the incentive compatibility constraint holds if and only if

$$\begin{aligned} U ( W_1- p_H A_H^{sym} -p_H \pi \lambda ) +p_H U( W_2+A_H^{sym}) \ge U( W_1- p_L A_L- p_L \pi \lambda ) + p_H U(W_2+A_L) \end{aligned}$$

Suppose \(A_L=A_H^{sym}\) (i.e., the low-risk agent receives the annuity contract of the high riskFootnote 35 agent), then the above equation simplifies to \(U ( W_1- p_H A_H^{sym} -p_H \pi \lambda ) \ge U( W_1- p_L A_H^{sym} - p_L \pi \lambda )\), which cannot be true unless \(p_L=p_H\). It therefore follows that

$$\begin{aligned} U( W_1- p_L A_L- p_L \pi \lambda ) + p_H U(W_2+A_L) < U( W_1- p_L A_H^{sym}- p_L \pi \lambda )+ p_H U(W_2+A_H^{sym}) \end{aligned}$$

This inequality holds only if \(A_L<A_H^{sym}<A_L^{sym}\). \(\square\)

1.3 Proof of Proposition 4

In a DB scheme, agents choose their LTC insurance in the second stage of the initial period after they have already chosen their pension benefit (\(B^*,\beta ^*\)). This means that agents are maximizing only with respect to the LTC insurance coverage. The problem in its Langrangian form is:

$$\begin{aligned} \begin{aligned} \max _{d_L,d_H,\gamma }\Psi _L =&U ( W_1-\beta ^*- p_L\pi d_L\lambda ) \\&+p_L [( 1-\pi ) U( W_2+B^*) +\pi U( W_2+B^*-( 1-d_L) \lambda )] \\&+\gamma \left[ \begin{array}{c} U( W_1-\beta ^*-p_H\pi d_H\lambda ) +\pi U ( W_2+B^*-( 1-d_H) \lambda ) \\ -U( W_1-\beta ^*- p_L\pi d_L\lambda ) -\pi U ( W_2+B^*-( 1-d_{L}) \lambda ) \end{array}. \right] \end{aligned} \end{aligned}$$

The first-order conditions are

$$\begin{aligned} \frac{\partial \Psi _L}{\partial d_L} :0= & {} -p_L\pi \lambda [U^\prime ( W_1-\beta ^*- p_L\pi d_L\lambda ) - U^\prime ( W_2+B^*-(1-d_L) \lambda ) \\&\quad +\gamma [ U^\prime ( W_1-\beta ^*-p_L\pi d_L\lambda ) p_L\pi \lambda -p_H\pi \lambda U^\prime ( W_2+B^*-(1-d_L) \lambda )]\\ \frac{\partial \Psi _L}{\partial d_H}:0= & {} -\gamma p_H\pi \lambda \left[ U^\prime ( W_1-\beta ^*-p_H\pi d_H\lambda ) - U^\prime ( W_2+B^*-(1-d_H) \lambda ) \right] \\ \frac{\partial \Psi _L }{\partial \gamma }:0= & {} \left[ \begin{array}{c} U( W_1-\beta ^*-p_H\pi d_H\lambda ) +\pi U ( W_2+B^*-( 1-d_H) \lambda ) \\ -U(W_1-\beta ^*-p_L\pi d_L\lambda )-\pi U( W_2+B^*-( 1-d_L) \lambda ) \end{array} \right] . \end{aligned}$$

To first show that \(d_H^{DB}<1\), consider the \((\frac{\partial \Omega _L}{\partial d_H})\) constraint from (27), and suppose that the second stage solution yields \(d_H^{DB}=1\). It would then follow that

$$\begin{aligned} U^\prime \left( W_1-[ xp_H+( 1-x) p_L] \widehat{B}- p_H\pi \lambda \right) =U^\prime \left( W_2+\widehat{B}\right) , \end{aligned}$$

which would be a solution provided that

$$\begin{aligned} \widehat{B}=\frac{W_1-W_2-p_H\pi \lambda }{1+ xp_H+( 1-x) p_L }. \end{aligned}$$

Retirement amount \(\widehat{B}\) assumes, however, that agents already know their risk type since the premium paid by high-risk agents is included in the calculation of \(\widehat{B}\). That cannot be the case since agents do not know their type at the time the DB contract is purchased. It must therefore be that \(B^*>\widehat{B}\). Because there is a negative relationship between the choice of annuities and the choice of the LTC indemnity, knowing that \(B^*>\widehat{B}\) means that \(d_H^{DB}<1\).

Lastly, given the presence of information asymmetry, low-risk agents, who cannot signal their type on the annuity market under a DB scheme, must signal their type on the LTC insurance market by accepting a contract that is less generous than what is offered to high-risk agents. We therefore have that \(d_L^{DB}<d_H^{DB}<1\). \(\square\)

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Boyer, M.M., Glenzer, F. Pensions, annuities, and long-term care insurance: on the impact of risk screening. Geneva Risk Insur Rev 46, 133–174 (2021).

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