Optimal Annuitization with Stochastic Mortality
Probabilities
Felix Reichling
1
2
Kent Smetters
Congressional Budget Oce
2 The Wharton School and NBER
1
May 2013
Disclaimer
This research was supported by the U.S. Social Security Administration through
grant #5 RRC08098400-04-00 to the National Bureau of Economic Research
as part of the SSA Retirement Research Consortium.
The ndings, conclusions, and views expressed are solely those of the author(s)
and do not represent the views of SSA, any agency of the Federal Government,
the CBO, or the NBER.
Outline
Introduction
Yaari Result
Overview of Our Results
Three-Period Model
Model Set-up
Yaari Model
Stochastic Survival Probabilities
Multi-Period Model
Model set-up
Calibration
Results
Sensitivity analysis
Additional Factors that Reduce Annuity Demand
What is a life annuity?
Longevity insurance: protects against the risk of outliving assets
In exchange for one-time premium, receive a stream of payments
with certainty until death
Cannot be bequeathed: you risk this principal at death in exchange
for a mortality credit for living
Examples: Social Security, dened-benet pensions, xed/variable
annuities
Yaari (1965) Result
IF
I Individuals have uncertain life spans (deterministic mortality)
I Individuals do not have bequest motives
I Premiums are approximately fair
THEN
I Full annuitization is optimal
Counterfactual: Annuitization puzzle (Modigliani (1986); many
others)
Our Modications
1. Allow for the survival probabilities themselves to be stochastic.
I Consistent with an investor's health status evolving over her
life in a manner that is not a deterministic function of initial
health status and subsequent age.
I Produces valuation risk where the present value of annuity
payments can rise or fall with health shocks
2. Allow health care shocks to produce correlated reductions in
income (disability) and increase in costs (long-term care).
3. Explore how rate of time preference (even geometric) becomes
important with stochastic mortality probabilities.
Contrast with Previous Literature
Small previous literature examined the role of stochastic mortality
probabilities
I Assumed (i) no correlated costs and (ii) fully patient agents
I Concluded that Yaari's full annuitization result maintains
I Intuition: Agents also want to insure against health
reclassication (valuation) risk by buying annuity early
I Might explain why this literature never really took o
Contrast with Previous Literature
Small previous literature examined the role of stochastic mortality
probabilities
I Assumed (i) no correlated costs and (ii) fully patient agents
I Concluded that Yaari's full annuitization result maintains
I Intuition: Agents also want to insure against health
reclassication (valuation) risk by buying annuity early
I Might explain why this literature never really took o
In contrast we show that relaxing (i) or (ii) can lead to imperfect
annuitization
Overview of Our Results
Theoretical Results (1)
If households are suciently patient and mortality probabilities are
the only source of uncertainty, full annuitization is still optimal
(same intuition as previous literature)
Overview of Our Results
Theoretical Results (1)
If households are suciently patient and mortality probabilities are
the only source of uncertainty, full annuitization is still optimal
(same intuition as previous literature)
If a bad mortality update is correlated with an additional loss
(income or expenses), incomplete annuitization can be optimal,
even without ad-hoc liquidity constraints
Overview of Our Results
Theoretical Results (1)
If households are suciently patient and mortality probabilities are
the only source of uncertainty, full annuitization is still optimal
(same intuition as previous literature)
If a bad mortality update is correlated with an additional loss
(income or expenses), incomplete annuitization can be optimal,
even without ad-hoc liquidity constraints
I Intuition: while the annuity can be rebalanced (equivalently,
borrowed against), the present value of its income stream falls
in states with larger marginal utility
I Aside: We discuss asset mix rebalancing below.
I These constraints can't bind in the Yaari model.
I Introducing them in our model (i.e., asymmetric info with
health shocks) only enhances our dierences
Overview of Our Results
Theoretical Results (2)
If households are suciently impatient then incomplete
annuitization may be optimal even without any additional loss
Overview of Our Results
Theoretical Results (2)
If households are suciently impatient then incomplete
annuitization may be optimal even without any additional loss
I Intuition: A standard lifetime annuity in the traditional of
Yaari and the subsequent literature is not an optimal contract
for impatient households. The optimal annuity is actually a
designer annuity that is a function of the annuitant's rate of
time preference.
I In Yaari model, level of patience aects level of savings but not
the annuitization choice. Hence, issue largely ignored until
now.
Overview of Our Results
Numerical Results (1)
Stochastic lifecycle model to simulate optimal annuitization
(extensive and intensive margins). Even with no bequest motives or
asset-related transaction costs:
Overview of Our Results
Numerical Results (1)
Stochastic lifecycle model to simulate optimal annuitization
(extensive and intensive margins). Even with no bequest motives or
asset-related transaction costs:
I Most households (63%) should not annuitize any wealth
Overview of Our Results
Numerical Results (1)
Stochastic lifecycle model to simulate optimal annuitization
(extensive and intensive margins). Even with no bequest motives or
asset-related transaction costs:
I Most households (63%) should not annuitize any wealth
I Annuitization is mostly found with:
I wealthier households (costs correlated with health shocks are
small relative to their assets)
I households where the expected mortality credit is large relative
to the valuation risk (older and sicker households)
Overview of Our Results
Numerical Results (1)
Stochastic lifecycle model to simulate optimal annuitization
(extensive and intensive margins). Even with no bequest motives or
asset-related transaction costs:
I Most households (63%) should not annuitize any wealth
I Annuitization is mostly found with:
I wealthier households (costs correlated with health shocks are
small relative to their assets)
I households where the expected mortality credit is large relative
to the valuation risk (older and sicker households)
I If households are allowed to sell (short) annuities (i.e., positive
life insurance) then:
I Younger households generally short as a hedge against future
negative health correlated costs.
I
Net aggregate annuitization is negative when summed across
the measure of households
Overview of Our Results
Numerical Results (2)
Results intended to are mostly normative, that is, about optimality
(i.e., we don't claim to explain the annuity puzzle).
But, for kicks, we then throw in various other factors (calibrated
bequests, asset management fees, etc) to see if we can reduce the
extensive and intensive margins even more (of course, these factors
also reduce annuitization in the Yaari model)
Results: extensive margin as low as 10%, intensive margin even
smaller.
Additional realistic modications likely to reduce these amounts
even more
Overview of Our Results
Numerical Results (3)
Hence, the rational expectations model is not necessarily fully down
and out for explaining annuitization patterns, although various
behavioral theories, of course, still likely play an important role.
Overview of Our Results
Numerical Results (3)
Hence, the rational expectations model is not necessarily fully down
and out for explaining annuitization patterns, although various
behavioral theories, of course, still likely play an important role.
Our model is also consistent with another stylized fact from the
literature which indicates that households typically view annuities
as increasing their risk rather than reducing it.
I Brown et al. (2008) interpret this evidence as compatible with
narrow framing.
I In our rational expectations model herein, the presence of
stochastic mortality probabilities implies that annuities deliver
a larger expected return (mortality credit) along with more risk
(valuation risk). A greater level of risk aversion typically
reduces annuitization.
Three-Period Model
Individuals
I Individual can life up to 3 periods, j ,
I Probability of surviving period
depends on health state
h
j
j + 1,
to period
at age
and
j +1
j +2
is
sj (h)
and
j
I Markov transition probability between health state is
0
with h, h ∈ H , where |H| > 1
P (h0 |h),
I Individuals can invest in annuities and bonds
I Annuity: Pays $1 per period (say in real terms) conditional on
survival
I Bonds: WLOG. (Could allow for equities w/ variable annuities)
Three-Period Model
Annuity premium πj and realized net return ρ(h0 |h) for a survivor
sj (h) · 1 sj (h) · ∑h0 P (h0 |h) sj+1 (h0 ) · 1
+
(1 + r )
(1 + r )2
sj (h) · 1
∑h0 P (h0 |h) sj+1 (h0 ) · 1
=
· 1+
(1 + r )
(1 + r )
!
sj (h)
0
0
· 1 + ∑ P h |h πj+1 h
=
(1 + r )
h0
πj (h) =
(1)
Three-Period Model
Annuity premium πj and realized net return ρ(h0 |h) for a survivor
sj (h) · 1 sj (h) · ∑h0 P (h0 |h) sj+1 (h0 ) · 1
+
(1 + r )
(1 + r )2
sj (h) · 1
∑h0 P (h0 |h) sj+1 (h0 ) · 1
=
· 1+
(1 + r )
(1 + r )
!
sj (h)
0
0
· 1 + ∑ P h |h πj+1 h
=
(1 + r )
h0
πj (h) =
1 + πj+1 (h0 )
ρj h0 |h =
− 1.
πj (h)
(1)
(2)
Yaari Model
Subcase: Deterministic mortality probabilities
(
P h0 |h =
h0 = h
0
0, h 6= h
1,
(3)
Yaari Model
Subcase: Deterministic mortality probabilities
(
P h0 |h =
h0 = h
0
0, h 6= h
1,
But standard lifecycle aging eects allowed:
sj+1 (h) < sj (h) < 1
(3)
Yaari Model
Subcase: Deterministic mortality probabilities
(
P h0 |h =
h0 = h
0
0, h 6= h
1,
(3)
But standard lifecycle aging eects allowed:
sj+1 (h) < sj (h) < 1
πj (h) =
sj (h)
· (1 + πj+1 (h))
(1 + r )
(4)
Yaari Model
Subcase: Deterministic mortality probabilities
(
P h0 |h =
h0 = h
0
0, h 6= h
1,
(3)
But standard lifecycle aging eects allowed:
sj+1 (h) < sj (h) < 1
πj (h) =
sj (h)
· (1 + πj+1 (h))
(1 + r )
(4)
(1 + r )
−1
sj (h)
(5)
ρj (h) =
Yaari (1965) Result
Intuition
Denition
We say that annuities statewise dominate bonds if
values of
h.
ρj (h) > r
for all
Yaari (1965) Result
Intuition
Denition
We say that annuities statewise dominate bonds if
values of
ρj (h) > r
h.
Proposition
With deterministic survival probabilities, annuities statewise
dominate bonds for any initial health state at age
j.
Proof.
In equilibrium,
sj (h) (1 + ρj (h)) = (1 + r ) or ρj (h) =
provided
that sj (h) < 1 (i.e. people will die).
(1 + r )
−1 > r
sj (h)
for all
Yaari (1965) Result
Intuition
Denition
We say that annuities statewise dominate bonds if
values of
ρj (h) > r
for all
h.
Proposition
With deterministic survival probabilities, annuities statewise
dominate bonds for any initial health state at age
j.
Proof.
In equilibrium,
sj (h) (1 + ρj (h)) = (1 + r ) or ρj (h) =
provided
(1 + r )
−1 > r
sj (h)
that sj (h) < 1 (i.e. people will die).
Intuition
Annuities are investment wrappers that produce a mortality credit.
Graphical Representation
Annuities statewise dominate bonds
Yaari Result is Very Strong (1)
Literature has not been able to explain the lack of annuitization
observed
I Davido et al. 2005 argues that most deviations from Yaari's
assumptions don't explain incomplete annuitization.
I Brown et al. (2008) conclude: As a whole, however, the
literature has failed to nd a suciently general explanation of
consumer aversion to annuities.
Yaari result is even stronger than commonly appreciated
Yaari Result is Very Strong (2)
Example: Adverse selection does not undermine full annuitization
Yaari Result is Very Strong (3)
Robust to other market imperfections (see paper)
Social Security only reduces saving, not how it is invested
Yaari Result is Very Strong (3)
Robust to other market imperfections (see paper)
Social Security only reduces saving, not how it is invested
Insurance within the marriage, similar
Yaari Result is Very Strong (3)
Robust to other market imperfections (see paper)
Social Security only reduces saving, not how it is invested
Insurance within the marriage, similar
Moral hazard reduces mortality credit but only if annuitization
occurs
Yaari Result is Very Strong (3)
Robust to other market imperfections (see paper)
Social Security only reduces saving, not how it is invested
Insurance within the marriage, similar
Moral hazard reduces mortality credit but only if annuitization
occurs
Liquidity Constraints
I Really an asset rebalancing constraint
I Can't bind in Yaari model since the mortality probability at any
age is just a deterministic function of the initial health
condition when the annuity was written and current age
I Consistently, Sheshinski (2008) writes that no apparent
reason seems to justify these constraints. (p. 33).
Yaari Result is Very Strong (4)
One exception is transaction costs but it's knife-edge in Yaari model
With Stochastic Mortality Probabilities
As in Yaari (1965), we assume rational expectations:
I No ad-hoc liquidity constraints (i.e. we allow full asset mix
rebalancing).
With Stochastic Mortality Probabilities
As in Yaari (1965), we assume rational expectations:
I No ad-hoc liquidity constraints (i.e. we allow full asset mix
rebalancing).
I Realistic? Maybe, maybe not.
With Stochastic Mortality Probabilities
As in Yaari (1965), we assume rational expectations:
I No ad-hoc liquidity constraints (i.e. we allow full asset mix
rebalancing).
I Realistic? Maybe, maybe not.
I But don't confuse models. These constraints make more sense
with stochastic mortality probabilities. The dierence between
the Yaari model and ours would grow even larger if we further
restricted asset mix rebalancing in our model (with one caveat
noted below)
I I.e., we produce incomplete annuitization even without these
additional constraints
Stochastic Rankings
Dominance of Annuities
Proposition
With stochastic survival probabilities ( p (h0 |h) > 0), annuities do
not generically statewise dominate bonds.
Stochastic Rankings
Dominance of Annuities
Proposition
With stochastic survival probabilities ( p (h0 |h) > 0), annuities do
not generically statewise dominate bonds.
I Intuition: Competitive annuity premium at age
j
is equal to
the present value of the expected annuity payments received at
ages
j +1
and
j + 2,
conditional on the health state
h
0
But a suciently negative health realization h at age
produces a capital depreciation at age
the mortality credit received.
j +1
at age
j.
j +1
that is larger than
Stochastic Rankings
Dominance of Annuities
Proposition
With stochastic survival probabilities ( p (h0 |h) > 0), annuities do
not generically statewise dominate bonds.
I Intuition: Competitive annuity premium at age
j
is equal to
the present value of the expected annuity payments received at
ages
j +1
and
j + 2,
conditional on the health state
h
0
But a suciently negative health realization h at age
produces a capital depreciation at age
j +1
at age
j +1
that is larger than
the mortality credit received.
I However, lack of statewise dominance does not rule out
expected utility maximizers (in fact, it's failure is not
surprising)
j.
Stochastic Rankings
Expected return to annuities
Proposition
The expected return to annuities exceeds bonds if the chance of
mortality is positive.
I I.e., Annuities will be preferred by risk neutral agents.
Stochastic Rankings
Expected return to annuities
Proposition
The expected return to annuities exceeds bonds if the chance of
mortality is positive.
I I.e., Annuities will be preferred by risk neutral agents.
Proposition
With stochastic survival probabilities, annuities do not generically
second-order stochastically dominate (SOSD) bonds.
I I.e., Annuities not necessarily preferred by risk averse agents.
Example
Simplied mortality to focus on reclassication risk: Sj (h)=1 (1)
Example
Simplied mortality to focus on reclassication risk: Sj (h)=1 (1)
Assuming that
r = 0,
then for our contract:
πj (h) = 1 + 0.5πj+1 (hG ) + 0.5πj+1 (hB ) = 1.5
Example
Simplied mortality to focus on reclassication risk: Sj (h)=1 (2)
The net return for a survivor in Good health:
ρj (hG |h) =
2
1.5
− 1 = 0.33 > r ,
while the net return for a survivor in Bad health:
ρj (hB |h) =
1
1.5
− 1 = −0.33 < r .
Example
Simplied mortality to focus on reclassication risk: Sj (h)=1 (2)
The net return for a survivor in Good health:
ρj (hG |h) =
2
1.5
− 1 = 0.33 > r ,
while the net return for a survivor in Bad health:
ρj (hB |h) =
1
1.5
− 1 = −0.33 < r .
In other words, statewise dominance fails (but, again, we care
about SOSD for EU maximizers)
Example (Continued)
Failure of SOSD: Assumed Preferences
Example (Continued)
Failure of SOSD: Assumed Preferences
To further simplify, suppose:
I Has $1.5 endowment at age
I Consumes at ages
j +1
j
(for sure) and
j + 2 (if
j +1
survives).
I Recall: Good or Bad health revealed at
I Allowed at age
j
to pool health risk by investing $1.5 in an
Annuity contract (otherwise, a Bond)
Example (Continued)
Failure of SOSD: Assumed Preferences
To further simplify, suppose:
I Has $1.5 endowment at age
I Consumes at ages
j +1
j
(for sure) and
j + 2 (if
j +1
survives).
I Recall: Good or Bad health revealed at
I Allowed at age
j
to pool health risk by investing $1.5 in an
Annuity contract (otherwise, a Bond)
Time-separable conditional expected utility preferences over
consumption:
u (cj+1 |hj+1 ) + β · sj+1 (hj+1 ) u (cj+2 |hj+1 ) ,
(6)
Example (Continued)
Failure of SOSD: Assumed Preferences
To further simplify, suppose:
I Has $1.5 endowment at age
I Consumes at ages
j +1
j
(for sure) and
j + 2 (if
j +1
survives).
I Recall: Good or Bad health revealed at
I Allowed at age
j
to pool health risk by investing $1.5 in an
Annuity contract (otherwise, a Bond)
Time-separable conditional expected utility preferences over
consumption:
u (cj+1 |hj+1 ) + β · sj+1 (hj+1 ) u (cj+2 |hj+1 ) ,
The unconditional expected utility at age
EU =
1
2
where recall
j
is equal to
1
· [u (cj+1 |hG ) + β · u (cj+2 |hG )] + · u (cj+1 |hB ) ,
2
sj+1 (hj+1 = hG ) = 1
and
(6)
sj+1 (hj+1 = hB ) = 0.
(7)
Example (Continued)
Failure of SOSD: High Patience (β = 1)
No Correlated Costs (already a known result)
I Bond:
hG : cj+1 = 0.75, cj+2 = 0.75; hB : cj+1 = 1.5.
I Annuity:
hG : cj+1 = 1.0, cj+2 = 1.0; hB : cj+1 = 1.0
I I.e., Annuities more eective at smoothing consumption (due
to pooling of reclassication risk)
Example (Continued)
Failure of SOSD: High Patience (β = 1)
No Correlated Costs (already a known result)
I Bond:
hG : cj+1 = 0.75, cj+2 = 0.75; hB : cj+1 = 1.5.
I Annuity:
hG : cj+1 = 1.0, cj+2 = 1.0; hB : cj+1 = 1.0
I I.e., Annuities more eective at smoothing consumption (due
to pooling of reclassication risk)
With Correlated Costs: (e.g., $1 lost with Bad health)
I Bond:
hG : cj+1 = 0.75, cj+2 = 0.75; hB : cj+1 = 0.5.
I Annuity:
hG : cj+1 = 1.0, cj+2 = 1.0; hB : cj+1 = 0.0.
I Under the Inada condition (
0
∂ u(c→ )
∂c
→ ∞),
Bond is chosen
I We will refer to this as the correlated cost channel
Example (Continued)
Failure of SOSD: Low Patience (β → 0) (1)
No Correlated Costs
I Bond:
hG : cj+1 → 1.5, cj+2 → 0; hB : cj+1 = 1.5.
I Annuity:
hG : cj+1 → 2.0, cj+2 → 0; hB : cj+1 = 1.0.
Example (Continued)
Failure of SOSD: Low Patience (β → 0) (1)
No Correlated Costs
I Bond:
hG : cj+1 → 1.5, cj+2 → 0; hB : cj+1 = 1.5.
I Annuity:
hG : cj+1 → 2.0, cj+2 → 0; hB : cj+1 = 1.0.
I Annuity increases consumption variation across the two health
states at
j + 1;
the bond investment perfectly smooths it
across allocations actually valued by the agent.
I Intuitively, the traditional competitive annuity fails to shift
consumption across health states in a way that is informed by
the agent's time preferences within each health state.
I See Appendix for discussion on designer annuities
I We'll refer to it as the impatience channel
Example (Continued)
Failure of SOSD: Low Patience (β → 0) (2)
I This channel does not exist in the Yaari model because the
health state is xed; the level of patience only aects the
desired level of saving and not how to invest it.
I Example above is extreme but see the simple illustrative
calculation in the paper with CRRA utility
Extensions
See Appendix if Interested
I Shorter Contracts
I A Richer Space of Mortality-Linked Contracts
I Hybrid and Designer Annuities
Stochastic Survival Probabilities
A gateway mechanism
Multi-Period Model
Individuals
I enter economy at
j = 21
and die with certainty by
I work when young and retire at age
1
j = 120
j = 65
2
3
I can be healthy (h ), disabled (h ), and very sick (h )
I can save into a life annuity and a non-contingent risk-free
bond (initially, no shorting, which we relax later)
I Health transition and mortality probabilities are based on the
actuarial model of Robinson (1996)
I adjusted estimates for working-age population to better match
Social Security disability data
Multi-Period Model
Wages
Wages are product of four factors:
I health status
h
I a predictable age-related productivity
ε
I an individual random productivity
matrix
Qkl ; k, l = 1, ..., Ψ,
where
η with Markov
Ψ ∈ {1, ..., 8}
transition
I the general-equilibrium market wage rate per unit of labor
Age-related and individual productivity states and transition
probabilities are from Nishiyama and Smetters (2005)
w
Multi-Period Model
Correlated Shocks - Workers
2
3
Disabled (h ) and very sick workers (h ) do not work
⇒
loss of
wage income that is not insured by Disability Insurance
I We don't model the lack of full insurance for lost wages,
rather take it as given
Health care loss
L
faced by a very sick worker is assumed to be fully
paid by private insurance
Multi-Period Model
Correlated Shocks - Retirees
Of course, no risk of disability-related wage loss
3
Very sick retirees (h ) face long-term care expenses that are
uninsured
⇒
loss
L,
unless they qualify for Medicaid
Long-term care expenses
L
are equal to 1.2 times the mean wage,
consistent with industry surveys (Genworth Financial (2012),
MetLife (2010))
I Similarly, we don't model lack of insurance market for
long-term care (e.g., Brown-Finkelstein argue that Medicaid
reduces the demand)
I Historically: Private insurance existed, although quite
expensive with low caps.
I Today: Many insurers have pulled out (Met, Pru, Unum,
Hartford, etc).
I We will do sensitivity analysis on this calibration
Multi-Period Model
Social insurance payments
Workers
I Social Security Disability calculated using the U.S. legal bend
point formula
I We assume no waiting periods and all claims accepted (in
reality, most are rejected)
Retirees
I OASI: Social Security retirement benets according the bend
point formula
I Medicaid: Very sick retirees can receive a larger medical
payment if they have minimal assets
Multi-Period Model
Household preferences
U=
"
J
J
∑ β j u(cj ) = ∑
1
j=
1
j=
cj1−σ
1−σ
#
+ ξ Dj Aj+1
Multi-Period Model
Household optimization
Given the state variables (A, η, h, j ) and prices (w ,r ,ρ ), household
solve:
Vj (Aj , ηj , hj , j) = max {u(cj ) + β s(hj , j)E [Vj+1 (Aj+1 , ηj+1 , hj+1 , j + 1)]}
c,α
subject to:
Aj+1 = R(αj , hj , hj+1 )(Aj + Xj − cj )
α ≤1
0
where
≤ cj ≤ Aj + Xj
Xj = I (h = h1 )(1 − T )εj ηj w + Bj + Trj − Lj
R(αj , hj , hj+1 ) = αj ρj (hj , hj+1 ) + (1 − αj )r
Multi-Period Model
Production and payroll taxes
I Output
Y
is generated by
I Utility weight
β
Y = θ K λ L1−λ
chosen to produce 2.8 capital-output ratio
I Social Security, Disability and Medicare transfers are nanced
through a labor income tax, assuming a balanced budget.
Multi-Period Model
General equilibrium
A general equilibrium, therefore, is fairly standard and so a formal
denition will be skipped. In particular:
1. Household Optimization: Households optimize utility, taking as
given the set of factor prices and policy parameters;
2. Asset Market Clearing: The factor prices are derived from the
production technology, with the aggregate levels of saving and
labor properly integrated across the measure of households;
3. Policy Balance: The policy parameters are consistent with
balanced budget constraints (i.e., tax revenue equals
spending); and,
4. Bequest Clearing: Bequests left equal bequests received.
Calibration
Health transition probabilitiesthe healthy
Calibration
Health transition probabilitiesthe disabled
Calibration
Health transition probabilitiesthe very sick
Calibration
Survival probabilities
Calibration
Annuity returns by age and health state transitionthe healthy
Calibration
Annuity returns by age and health state transitionthe disabled
Calibration
Annuity returns by age and health state transitionthe very sick
Calibration
Disability rates: data vs. model
Source: Social Security Administration and authors' calculations.
Calibration
Age structure of the population: data vs. model
Source: U.S. Census Bureau (2011) and authors' calculations. Fractions are based on
the population 21 years and older.
Calibration
Wealth distribution: data vs. model (1)
Sources: (1) Nishiyama (2002), (2) U.S. Census Bureau (2012a).
Calibration
Wealth distribution: data vs. model (2)
I The poverty rate among all workers between the ages of 18 to
64 in our model is 4.2%, which is a bit below the Census value
of 7.2% (U.S. Census Bureau 2012b). Our underestimate is
likely due to the U.S. Census having a broader denition of
working relative to our model's wage data.
Calibration
Wealth distribution: data vs. model (2)
I The poverty rate among all workers between the ages of 18 to
64 in our model is 4.2%, which is a bit below the Census value
of 7.2% (U.S. Census Bureau 2012b). Our underestimate is
likely due to the U.S. Census having a broader denition of
working relative to our model's wage data.
I The poverty rate among all disabled people between the ages
of 18 to 64 is 33.5% in our model, which is pretty close to the
empirical counterpart of 28.8%, as estimated by U.S. Census
Bureau (2012b), although lower than the value of 50%
estimated by Congressional Budget Oce (2012).
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (1)
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (2)
I At CRRA = 2, households with wealth at age 65 below 6
times the national average wage don't annuitize
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (2)
I At CRRA = 2, households with wealth at age 65 below 6
times the national average wage don't annuitize
I At CRRA = 5, that threshold increases to 8 times the national
average
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (2)
I At CRRA = 2, households with wealth at age 65 below 6
times the national average wage don't annuitize
I At CRRA = 5, that threshold increases to 8 times the national
average
I At CRRA = 3, 53% of age-65 retirees hold no annuities and
only 11% fully annuitize
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (2)
I At CRRA = 2, households with wealth at age 65 below 6
times the national average wage don't annuitize
I At CRRA = 5, that threshold increases to 8 times the national
average
I At CRRA = 3, 53% of age-65 retirees hold no annuities and
only 11% fully annuitize
I Annuitization occurs at larger wealth levels because retiree has
enough assets to pay for any potentially correlated long-term
care cost from the annuity stream itself
Results
Annuitized fraction of wealth at Age 65 for healthy person (Black Lines) and population
distribution (Gray) (2)
I At CRRA = 2, households with wealth at age 65 below 6
times the national average wage don't annuitize
I At CRRA = 5, that threshold increases to 8 times the national
average
I At CRRA = 3, 53% of age-65 retirees hold no annuities and
only 11% fully annuitize
I Annuitization occurs at larger wealth levels because retiree has
enough assets to pay for any potentially correlated long-term
care cost from the annuity stream itself
I Recall: Results not driven by ad-hoc liquidity constraints.
Adding those would further decrease annuitization.
Results
Some quick checks
Results
Some quick checks
I We ran a Yaari version of our model where health shocks are
turned o and the mortality rate by age is set equal to the
average mortality rate weighted by the share of households in
each health state at that age.
I 100% of wealth annuitized, as we would expect
Results
Some quick checks
I We ran a Yaari version of our model where health shocks are
turned o and the mortality rate by age is set equal to the
average mortality rate weighted by the share of households in
each health state at that age.
I 100% of wealth annuitized, as we would expect
I We also ran our model but assumed that long-term care costs
were fully insured (but with health shocks)
I Not the same thing as Yaari model due to presence of
valuation risk
I Still, 100% of wealth at retirement fully annuitized
I Hence, the
impatience channel is not driving these results
either.
I Results being driven by the
correlated cost channel.
Results
Annuitized fraction of all wealth, by age (intensive margin). Population density in gray.
(1)
Results
Annuitized fraction of all wealth, by age (intensive margin). Population density in gray.
(2)
I Not monotonic in age due to rate of return to annuities being
a function of the mortality credit (which is monotonic in age)
and valuation risk (and, such heterogeneity across health
groups, the shares of which vary by age).
Results
Fraction of households with annuities, by age (extensive margin) (1)
Results
Fraction of households with annuities, by age (extensive margin) (2)
I At CRRA=3, 37% of all households in the economy hold any
positive level of annuities; 24% of households fully annuitize.
But 57% of wealth is annuitized (due to skewed wealth
distribution)
Sensitivity Analysis
Reducing long-term care costs
Sensitivity Analysis
Allowing short-sales (intensive margin)
Sensitivity Analysis
Allowing short-sales (extensive margin)
Sensitivity Analysis
When healthy, young savers short annuities
I Equivalent to purchasing life insurance
I In other words, they pay (instead of earn) the mortality credit.
I But the mortality credit is cheap when young, and they young
still face a lot of health reclassication risk (and, hence,
correlated expenses)
Then, after a negative health shock, buy (go long in) annuities
I Current life policy is very valuable; annuity is cheap
I Life policy could be sold (life settlement) or borrowed against
to nance annuity (a fully collaterialized swap)
This short-long trade produces a windfall that can be used to pay
for correlated expenses
Additional Factors
Management fees and bequests (1)
How far can our rational model be pushed to produce a low level of
annuitization?
Additional Factors
Management fees and bequests (1)
How far can our rational model be pushed to produce a low level of
annuitization?
Management fees and bequests also reduce annuitization in the
Yaari model. So we typically run both models.
Additional Factors
Management fees and bequests (2)
Management fees: Management fees for annuities are about 1.0%
larger than for bonds (see paper), not including surrender charges
Additional Factors
Management fees and bequests (2)
Management fees: Management fees for annuities are about 1.0%
larger than for bonds (see paper), not including surrender charges
I Yaari model:
I Intensive margin: 73% of all wealth annuitized; 100% of retiree
wealth; 65% of non-retiree wealth
I Extensive margin: 43% of households annuitize; 90% of
retirees (10% have no wealth); 33% non-retirees
I
But knife-edged: most households either 100% or 0%.
Fraction of households that fully annuitize also equal to 43%.
Additional Factors
Management fees and bequests (2)
Management fees: Management fees for annuities are about 1.0%
larger than for bonds (see paper), not including surrender charges
I Yaari model:
I Intensive margin: 73% of all wealth annuitized; 100% of retiree
wealth; 65% of non-retiree wealth
I Extensive margin: 43% of households annuitize; 90% of
retirees (10% have no wealth); 33% non-retirees
I
But knife-edged: most households either 100% or 0%.
Fraction of households that fully annuitize also equal to 43%.
I Our model:
I Intensive margin: 27% of all weath annuitized; 42% of retiree;
20% of non-retiree
I Extensive margin: 22% of all households annuitize; 49% of
retirees; 17% non-retirees
I
Not knife edge: Only 7% fully annuitize
Additional Factors
Management fees and bequests (3)
With positive bequests
management fees)
(ξ > 0)
and a 2.5% bequest-GPD ratio (no
Additional Factors
Management fees and bequests (3)
With positive bequests
(ξ > 0)
and a 2.5% bequest-GPD ratio (no
management fees)
I Yaari model: 67% of wealth annuitized and 90% of households
hold a positive level of annuity
I Our model: 38% of wealth annuitized and 35% of households
hold a positive level of annuity
Additional Factors
Management fees and bequests (3)
With positive bequests
(ξ > 0)
and a 2.5% bequest-GPD ratio (no
management fees)
I Yaari model: 67% of wealth annuitized and 90% of households
hold a positive level of annuity
I Our model: 38% of wealth annuitized and 35% of households
hold a positive level of annuity
Add back in the management fees: 11% of wealth annuitized; 14%
with any annuities in our model
Additional Factors
Management fees and bequests (3)
With positive bequests
(ξ > 0)
and a 2.5% bequest-GPD ratio (no
management fees)
I Yaari model: 67% of wealth annuitized and 90% of households
hold a positive level of annuity
I Our model: 38% of wealth annuitized and 35% of households
hold a positive level of annuity
Add back in the management fees: 11% of wealth annuitized; 14%
with any annuities in our model
Uneven bequests (top 40%): Our model: 8% of wealth annuitized;
10% hold a positive level in our model
Possible Future Extensions
Dierential transaction costs
More Worker Risk
Works Cited in Talk I
Brown, Jerey R., Jerey R. Kling, Sendhil Mullainathan, and
Marian V. Wrobel, Why Don't People Insure Late-Life
Consumption? A Framing Explanation of the
Under-Annuitization Puzzle, American Economic Review, May
2008, 98 (2), 30409.
Congressional Budget Oce, Policy Options for the Social Security
Disability Insurance Program, July 2012.
Davido, Thomas, Jerey R. Brown, and Peter A. Diamond,
Annuities and Individual Welfare, American Economic Review,
December 2005, 95 (5), 15731590.
Genworth Financial, Genworth 2012 Cost of Care
SurveyExecutive Summary, 2012.
MetLife, The 2010 MetLife Market Survey of Nursing Home,
Assisted Living, Adult Day Services, and Home Care Costs,
October 2010.
Works Cited in Talk II
Modigliani, Franco, Life Cycle, Individual Thrift, and the Wealth of
Nations, The American Economic Review, June 1986, 76 (3),
297313.
Nishiyama, Shinichi, Bequests, Inter Vivos Transfers, and Wealth
Distribution, Review of Economic Dynamics, October 2002, 5
(4), 892931.
and Kent Smetters, Consumption Taxes and Economic
Eciency with Idiosyncratic Wage Shocks, Journal of Political
Economy, October 2005, 113 (5), 10881115.
Robinson, Jim, A long-term care status transition model, in The
Old-Age Crisis: Actuarial Opportunities The 1996 Bowles
Symposium Georgia State University, Atlanta 1996, pp. 7279.
Chapter 8.
Sheshinski, Eytan, The economic theory of annuities, Princeton
University Press, 2008.
Works Cited in Talk III
U.S. Census Bureau, Statistical Abstract of the United States:
2012, Washington, DC, September 2011.
, Household Income Inequality Within U.S. Counties:
2006-2010, Washington, DC, February 2012.
, Income, Poverty, and Health Insurance Coverage in the United
States: 2011, Washington, DC, September 2012.
Yaari, Menahem E., Uncertain Lifetime, Life Insurance, and the
Theory of the Consumer, The Review of Economic Studies,
April 1965, 32 (2), 137150.