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

Abstract

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.

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

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

(30)

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

(31)

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 risk³⁵ 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 $d_L^{DB}<d_H^{DB}<1$. $\square$