---
title: "Flexible Life Annuities"
authors:
  - name: "Alexis Direr"
    affiliation: "Paris School of Economics (INRA); Université de Grenoble"
date: "2008-06-04"
doi: "10.1111/j.1467-9779.2009.01446.x"
keywords: [life annuities, adverse selection, longevity risk, expense shocks, flexible contracts, withdrawal option, asymmetric information]
jel_codes: [E21, D82]
language: en
type: research-article
journal: "Journal of Public Economic Theory"
---

# Flexible Life Annuities

**Author**
- Alexis Direr — Paris School of Economics (INRA); Université de Grenoble — *direr@ens.fr*

**DOI**: [10.1111/j.1467-9779.2009.01446.x](https://doi.org/10.1111/j.1467-9779.2009.01446.x)

**Keywords**: life annuities, adverse selection, longevity risk, expense shocks, flexible contracts, withdrawal option.
**JEL codes**: E21, D82.

---

## Abstract

Annuity contracts typically deliver a stream of income at a predetermined level in order to insure against the risk of longevity. This paper explores whether flexible annuities, which give subscribers the possibility to choose between different levels of annuity, are welfare enhancing. In the case where agents gradually discover their actual probability of survival, a predetermined and "one-size-fits-all" annuity plan is optimal. If an expenditure risk is added along with the longevity risk, a flexible annuity plan is better even though the consumption path cannot be isolated from uninsured expenses anymore.

---

## 1. Motivation and contributions

Yaari (1965) and Davidoff, Brown and Diamond (2005) establish that full annuitization is optimal in a wide class of life-cycle environments without bequest motives. Yet observed annuitization is far below this benchmark in the U.S. (401(k) plans rarely offer annuitization options; Brown 2001), in the U.K. (insurers increasingly reluctant to offer mandatory-annuitization products), and in France (the PERP fully-annuitized voluntary plan has had limited commercial success). One frequently cited friction is the loss of liquidity once wealth is annuitized — particularly costly given that out-of-pocket medical expenses are a major retirement risk (Palumbo 1999; Brown 2004; De Nardi, French and Jones 2006).

A natural fix is **contract flexibility**: allow the annuitant to vary the withdrawal level when an expense shock hits. Insurance companies have started offering such products (guaranteed minimum withdrawal benefits in U.S. variable annuities; one-time partial-withdrawal options in fixed life annuities — Brown 2007). The standard objection, voiced by Brown and Warshawsky (2001), is **adverse selection**: if annuitants can update their survival prospects and adjust withdrawals accordingly, the long-lived stay in while the short-lived cash out, undermining the longevity-insurance pool.

The paper makes the following contributions.

1. **Benchmark with longevity risk only: fixed annuities are optimal.** When the only risk is longevity and agents gradually learn their type after annuitization, an unconditional fixed annuity plan strictly dominates any state-contingent plan. The mechanism is the Hirshleifer (1971) effect: the resolution of uncertainty between contract date and information arrival destroys the ex-ante mutuality of insurance. Two propositions formalize this: agents prefer ex-ante commitment (Proposition 1) and do not wish to recontract once their type is revealed (Proposition 2), echoing Brugiavini (1993).

2. **With expense risk added, flexibility becomes optimal.** When out-of-pocket medical-expense shocks are layered on top of longevity uncertainty, the optimal contract menu offers **three options**, corresponding to long-lived without shock, long-lived with expense shock, and short-lived. Agents hit by the expense shock optimally withdraw a higher annuity than the long-lived without a shock — a withdrawal option emerges as part of the second-best contract.

3. **Adverse selection caps but does not eliminate flexibility.** Because short-lived agents will mimic the high-withdrawal option to deplete their account, the high withdrawal level cannot be raised to the first-best amount: the incentive-compatibility constraint pegs the short-lived's annuity to the expense-shock annuity (Lemma 1). The expense-shock type therefore cannot fully smooth consumption (Lemma 2); first-best smoothing is only restored when the short-lived's population share goes to zero (Lemma 3).

4. **Policy reading.** Contrary to the common claim, adverse selection alone does not justify a strictly fixed annuity. A *bounded* withdrawal option captures meaningful consumption-smoothing gains while limiting the damage to the actuarial rate of return — and could plausibly raise demand for annuities in the broader population.

---

## 2. Model A — longevity risk only

**Environment.** Two consumption dates $t=1,2$; gross interest rate $R$; wealth $w$ allocated at $t=1$. At date $0$, agents share a common prior on survival; at date $1$, a fraction $p$ learn they are **long-lived** (survival probability $\beta_h$ to date 2) and the remaining $1-p$ learn they are **short-lived** ($\beta_l < \beta_h$). Utility is time-separable: $v(c_1, c_2; \beta) = u(c_1) + \beta u(c_2)$ with $u'>0$, $u''<0$, $\lim_{c\to 0} u'(c) = \infty$.

**Annuity market.** Insurers cannot observe types ex post and cannot monitor cross-firm holdings. Competition drives the rate of return to the actuarially fair pooled level $R/\bar\beta$, where $\bar\beta$ is the demand-weighted average survival rate.

**Asymmetric-information equilibrium (Eq. 1).** With contracting at date 1 only:

$$
\widehat\beta = \frac{p\,\widehat c_{2h}}{p\,\widehat c_{2h} + (1-p)\,\widehat c_{2l}}\,\beta_h + \frac{(1-p)\,\widehat c_{2l}}{p\,\widehat c_{2h} + (1-p)\,\widehat c_{2l}}\,\beta_l
$$

with each type solving its individual problem at the pooled rate $R/\widehat\beta$.

**Ex-ante contract (Eq. 2–3).** With binding contracts written at date 0 (before information arrives), the optimum equalizes consumption across types — $c^*_{1h} = c^*_{1l}$, $c^*_{2h} = c^*_{2l}$ — and satisfies the first-best Euler equation $u'(c^*_1) = R u'(c^*_2)$.

### Two key results

> **Proposition 1 (Ex-ante commitment dominates waiting).** Date-0 expected utility from subscribing to the ex-ante optimal contract strictly exceeds the expected utility of postponing the contract to date 1.

The proof (Hirshleifer 1971 mechanism) shows the ex-post problem is the ex-ante problem with additional binding constraints — separate per-type budget constraints and uninternalized rate-of-return effects. Information resolution destroys insurance opportunities.

> **Proposition 2 (No date-1 recontracting).** Conditional on having signed the ex-ante contract, agents do not buy or short-sell additional annuities at date 1.

When recontracting offers actuarially fair, type-specific rates $R/\beta_i$, the new-information effect on consumption is exactly offset by the rate-of-return revision. The ex-ante allocation is robust to ex-post side trades — the result echoing Brugiavini (1993).

**Takeaway.** With longevity risk only, the optimal contract is fixed and one-size-fits-all. Flexibility would let the short-lived front-load consumption, contaminating the pool. *This benchmark is the source of the standard adverse-selection argument against flexible annuities.*

---

## 3. Model B — adding an expense shock

**Three types at date 1.**

| Type | Share | Survival to $t=2$ | Date-1 expense |
|:---:|:---:|:---:|:---:|
| $h$ — long-lived, healthy | $p$ | $\beta > 0$ | 0 |
| $m$ — long-lived, expense shock | $q$ | $\beta > 0$ | $m > 0$ |
| $l$ — short-lived | $s = 1 - p - q$ | 0 (dies for sure) | 0 |

The assumption that $\beta_l = 0$ is a simplification; the substantive results extend to $\beta_l < \beta$.

**Contracting structure.** Insurers offer a date-0 menu $\{(c_{1h}, c_{2h}); (c_{1m}, c_{2m}); (c_{1l}, 0)\}$. Agents may **partially recontract upward** at date 1 (buy more annuities) but **cannot short-sell** (cannot consume less than the date-2 annuity in their option).

### 3.1 First best (full information)

With observable types, the contract delivers $c^*_{1m} - m = c^*_{1h} = c^*_{1l}$ and $c^*_{2h} = c^*_{2m}$, with first-best Euler equations $u'(c^*_{1i}) = R u'(c^*_{2i})$ for $i = h, m$. Both longevity and expense shocks are perfectly mutualized.

### 3.2 Second best — incentive compatibility constraints

With private types, two constraints bite:

1. **Type $h$ should not deviate to $(c_{1m}, c_{2m})$.** The reservation value of a deviator who buys $c_{1m} - c_{1h}$ extra annuities is captured by $v_h(z) = \max\{u(c_1) + \beta u(c_2) : c_1 + \beta c_2 / R = z\}$, where $z = c_{1m} + \beta c_{2m}/R$. The constraint $u(c_{1h}) + \beta u(c_{2h}) \geq v_h(z)$ must hold.
2. **Type $l$ chooses the highest available date-1 annuity.** Because the short-lived have no use for date-2 income, $c_{1l} \geq \max(c_{1h}, c_{1m})$.

Using $c_{1m} \geq c_{1h}$ at the optimum (otherwise the planner could redistribute toward type $m$ and raise welfare), the second constraint reduces to $c_{1l} \geq c_{1m}$.

### 3.3 The three central results

> **Lemma 1 (Pooling at the top).** $c_{1l} = c_{1m}$. Short-lived agents and expense-shock agents receive the same date-1 annuity.

The expense shock is a legitimate ground for a higher withdrawal; the short-lived's signal is not. Yet the planner cannot tell them apart, so the high withdrawal is offered to both. The constraint $c_{1l} \geq c_{1m}$ binds.

> **Lemma 2 (Incomplete consumption smoothing).** At the second-best optimum, $u'(c_{1m} - m) > R\,u'(c_{2m})$.

Type-$m$ agents consume too little at date 1 relative to the first best. Lifting their date-1 consumption further would require raising $c_{1l}$ (by Lemma 1), worsening adverse selection by inducing more short-lived agents to deplete the pool.

> **Lemma 3 (First best as $s \to 0$).** If $s = 0$ (no short-lived agents), perfect consumption smoothing is restored: $u'(c_{1m} - m) = R\,u'(c_{2m})$.

The second-best wedge is entirely attributable to the share of short-lived agents.

> **Proposition 3 (Flexibility is optimal).** $c_{1m} > c_{1h}$.

The optimal contract is *strictly* flexible in the form of a withdrawal option: type-$m$ agents withdraw more at date 1 (and accept lower date-2 income) than type-$h$ agents. The adverse-selection argument against flexibility, taken in isolation, does not survive the introduction of an expense risk.

> **Authors' critical reading.** Flexibility in the model is not unbounded freedom — it is a *capped* withdrawal option. Allowing the high-payout level to rise toward the first best would tip more short-lived agents into draining the pool and collapse the longevity insurance. The wedge between the second-best and first-best date-1 consumption of type $m$ is the price paid for keeping the pool actuarially viable.

---

## 4. Conclusion

The paper studies the optimal annuity contract when both longevity risk and an out-of-pocket expense risk (interpretable as health-related) are present. Each risk taken alone delivers a different prescription: pure longevity risk recommends a rigid one-size-fits-all annuity, while an expense risk would be best mutualized with type-contingent payouts. With both risks present, the optimal contract combines elements of each: a withdrawal option of bounded size, allowing expense-shock agents to smooth consumption *partially* without letting short-lived agents fully drain their account.

The headline policy implication is that **adverse selection is not a sufficient argument against contract flexibility**. A bounded-withdrawal annuity can deliver substantial smoothing benefits while keeping rate-of-return reductions modest. This may help reconcile the empirical "annuity puzzle" — low voluntary annuitization despite Yaari-type theoretical benefits — with rational consumer choice, by pointing to the lack of state-contingent withdrawal options in standard contracts.

### Limitations and research extensions

The author flags two main caveats.

1. **Administrative costs of flexibility.** Offering state-contingent withdrawal options entails operational costs not modelled here. How these costs scale with contract sophistication is an empirical question — and could in principle reverse the welfare ranking for sufficiently complex menus.
2. **Stylized risk structure.** The model uses two periods, three discrete types, and a stark $\beta_l = 0$ assumption for the short-lived. Richer dynamic specifications, continuous type distributions, or correlated longevity-expense risk are natural extensions.

A connection is drawn to Brugiavini (1993), who studies *when* in the life cycle individuals should buy annuities; the present paper instead asks *what menu* of annuities should be offered. The two setups become formally close if the date-0 / date-1 information-resolution sequence in this paper is reinterpreted as a working-period / retirement-period sequence.

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## Acknowledgments

The author thanks the participants of the Paris School of Economics "Health and Insurance" workshop and the CESifo Venice Summer Institute Workshop on Longevity and Annuitization. Any errors or omissions are the author's own.

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## Main references

Abel, A. B. (1986). Capital Accumulation and Uncertain Lifetimes with Adverse Selection. *Econometrica* 54(5), 1079–1097.

Brown, J. R. (2001). Private Pensions, Mortality Risk, and the Decision to Annuitize. *Journal of Public Economics* 82(1), 29–62.

Brown, J. R. (2004). Life Annuities and Uncertain Lifetimes. *NBER Reporter: Research Summary*, Spring.

Brown, J. R. (2007). Rational and Behavioral Perspectives on the Role of Annuities in Retirement Planning. *NBER Working Paper* 13537.

Brown, J. R., Warshawsky, M. J. (2001). Longevity-Insured Retirement Distributions from Pension Plans: Market and Regulatory Issues. *NBER Working Paper* 8064.

Brugiavini, A. (1993). Uncertainty Resolution and the Timing of Annuity Purchases. *Journal of Public Economics* 50(1), 31–62.

Davidoff, T., Brown, J. R., Diamond, P. (2005). Annuities and Individual Welfare. *American Economic Review* 95(5), 1573–1590.

De Nardi, M., French, E., Jones, J. B. (2006). Differential Mortality, Uncertain Medical Expenses, and the Saving of Elderly Singles. *NBER Working Paper* 12554.

Hirshleifer, J. (1971). The Private and Social Value of Information and the Reward to Inventive Activity. *American Economic Review* 61(4), 561–574.

Palumbo, M. G. (1999). Uncertain Medical Expenses and Precautionary Saving Near the End of the Life Cycle. *Review of Economic Studies* 66, 395–421.

Yaari, M. E. (1965). Uncertain Lifetime, Life Insurance, and the Theory of the Consumer. *Review of Economic Studies* 32(2), 137–150.

*The full reference list appears in the PDF.*