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W
'DENE*
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
ANNUITIES AND INDIVIDUAL WELFARE
Thomas
Davidoff
Brown
Peter Diamond
Jeffrey
Working Paper 03-1
May 7, 2003
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Annuities and Individual Welfare
Thomas
Davidoff, Jeffrey Brown, Peter
May
*Davidoff, Haas School of Business,
UC
at
2003
Berkeley; Diamond,
Champaign and NBER. Davidoff and Diamond
ment Research
7,
Diamond
MIT; Brown, University of Illinois
at
Urbana-
are grateful for financial support from the Center for Retire-
Boston College pursuant to a grant from the U.S. Social Security Administration funded
as part of the Retirement Research Consortium.
The opinions and conclusions
are those of the authors and
should not be construed as representing the opinions or policy of the Social Security Administration or any
agency of the Federal Government, or the Center
support from the National Science Foundation.
for
Retirement Research. Diamond
is
grateful for financial
Abstract
This paper advances the theory of annuity demand. First, we derive
ditions
under which complete annuitization
result holds true in
is
optimal, showing that this well-known
a more general setting than in Yaari (1965).
markets are complete,
sufficient conditions
sufficient con-
Specifically,
when
need not impose exponential discounting,
intertemporal separability or the expected utility axioms; nor need annuities be actuarially fair,
is
required
nor longevity risk be the only source of consumption uncertainty. All that
is
that consumers have no bequest motive
and that annuities pay a
rate of
return for survivors greater than those of otherwise matching conventional assets, net
of administrative costs. Second,
when markets
asset
are incomplete.
markets are complete.
itization
we show that
Some
full
annuitization
The incompleteness
annuitization
is
may
not be optimal
optimal as long as conventional
of markets can lead to zero annu-
but the conditions on both annuity and bond markets are stringent. Third,
we extend the simulation
literature that calculates the utility gains from annuitization
by considering consumers whose
utility
"standard-of-living" to which they have
tion hinges critically
Key Words:
on the
depends both on present consumption and a
become accustomed. The value of
annuitiza-
size of the initial standard-of-living relative to wealth.
Annuities, annuitization, Social Security, pensions, longevity
surance, standard-of-living, habit.
risk, in-
Introduction
1
Providing a secure source of retirement income
States today
is
62 years
an issue of increasing importance to
The most common
viduals and policy-makers alike.
1
is
retirement age for a male in the United
and, thanks to the substantial reduction in mortality risk at older
ages witnessed over the past century, expected remaining
nearly 19 years - almost to age 81.
2
There
is,
span
life
about how long one
retirees.
live to
will live
for a 62 year old
will die before
age 90 or beyond. As a result, longevity risk
- is
male
is
however, substantial uncertainty around this
expected value. Approximately 16 percent of 62 year old males
another 16 percent will
indi-
age 70, while
-
uncertainty
a substantial source of financial uncertainty facing today's
Consideration of couples extends the upper
expectancy outcomes.
tail of life
Since the seminal contribution of Yaari (1965) on the theory of a life-cycle consumer
with an unknown date of death, annuities have played a central role in economic theory. His
widely cited result
is
that certain consumers should annuitize
all
of their savings. However,
these consumers were assumed to satisfy several very restrictive assumptions: they were von
Neumann-Morgenstern expected
utility
maximizers with intertemporally separable
utility,
they faced no uncertainty other than time of death and they had no bequest motive.
addition, the annuities available for purchase
arially fair.
While the subsequent
literature
In
by these individuals were assumed to be actu-
on annuities has occasionally relaxed one or two
of these assumptions, the "industry standard" is to maintain
most of these conditions. In
particular, the literature has universally retained expected utility
and additive
separability,
the latter dubbed "not a very happy assumption" by Yaari.
This paper advances the theory of annuity demand in several directions. Section 2 derives
sufficient conditions for
known
complete annuitization to be optimal, demonstrating that this well-
result holds true in
a much more general setting than that in Yaari (1965). Specifically,
we show that when markets
discounters, for utility to
obey expected
annuities to be actuarially
optimal
is
are complete,
fair.
Rather,
utility
all
matching financial
risk.
consumers to be exponential
axioms or be intertemporally separable, or
that
is
for
required for complete annuitization to be
long as there
is
is
greater than the return on conventional assets
Section 2.1 considers a two period setting with no uncertainty
other than date of death, in which
all
trade occurs at once. Here,
no bequest and annuities have a higher return
'Gustman and Steinmeier (2002)
2002 Annual Report of the Board
2
ability
for
that consumers have no bequest motive and that annuities pay a rate of return to
survivors, net of administrative costs, that
of
not necessary
it is
Insurance Trust Funds (2002)
all
savings are annuitized so
for survivors
than conventional
of Trustees of the Federal Old- Age and Survivors Insurance and Dis-
assets. Section 2.2
many
extends this result to the Arrow-Debreu case with arbitrarily
future
periods with aggregate uncertainty, as long as conventional asset and annuity markets are
complete.
Despite this strong theoretical prediction, few people voluntarily annuitize outside of
and defined benefit
Social Security
To
plans.
called "annuity puzzle" might exist, in section 3
can break down when markets
Section
3. 1
why
provide theoretical guidance on
we show how the
full
this so
annuitization result
for either annuities or conventional assets are incomplete.
examines the case where conventional markets are complete but annuity markets
We
are incomplete.
derive the weaker result that as long as trade occurs
all
at once
and
preferences are such that consumers avoid zero consumption in every state of nature, then
consumers
we
once,
will
always annuitize at least part of their wealth.
if
trade occurs
for
consumers with no bequest motive, even
every state of nature.
off in
An
securities such as stocks or
the asset does
if
important consequence of this result
finding that "annuities dominate conventional assets" extends past riskless
A
all at
derive the result that an annuitized version of any conventional asset will always
dominate the underlying asset
not pay
Also,
is
that the
bonds to risky
mutual funds.
practical implication of these results
that "variable
is
life
annuities"
may dominate
mutual funds, provided that the higher expenses associated with variable annuities are not
too high. 3 For example, suppose the provider of a mutual fund family doubles the number of
available funds
of investors
The
by
who
offering a
die
matching annuitized fund that periodically takes the accounts
and distributes the proceeds across the accounts of surviving
investors.
4
returns to this annuitized fund will strictly exceed the returns of the underlying fund
for surviving investors.
Section 3.2 considers situations in which
all.
it
can be optimal not to annuitize any wealth at
A key finding of this section is that under plausible conditions on returns, 5
incompleteness
of conventional asset markets as well as incompleteness of annuity markets themselves,
required for zero annuitization to be optimal. This highlights the
part of the solution to the annuity puzzle
may
lie in
common
is
observation that
the lack of complete insurance against
other types of risk. Section 3.2.3 sharpens this observation by showing an example where a
critical role is
3
We
played by a decrease in the possible maximal date of death.
refer to true life annuities, not
the variable annuities widely marketed that contain only an annuiti-
zation option.
4
TIAA-CREF
5
Milevsky and Young (2002) considers a violation of the return condition that
currently provides annuities with such a structure.
may
render zero annuiti-
zation optimal. In particular, in the absence of variable annuities they demonstrate that
to defer annuitization, with deferral
more
attractive as risk tolerance grows.
it
can be optimal
6
Section 4 extends the simulation literature, that calculates the utility gains from an-
whose
nuitization by considering consumers
depends both on present consumption
utility
and a "standard-of-living" to which they have become accustomed. 7 In our
whether annuities are more or
less valuable
under this standard-of-living model than under
the conventional model hinges on whether the
retirement resources.
is
8
In particular,
specification,
initial
standard-of-living
is
large relative to
the initial standard-of-living at the start of retirement
if
large relative to the individuals stock of resources, complete annuitization in the
a constant real annuity
not optimal, since
is
it
form of
does not allow the individual to optimally
"phase down" from the pre-retirement level of consumption to which she had become accus-
tomed.
If,
however, the stock of retirement wealth
annuities are even
more valuable than
in the usual
is
large relative to the standard-of-living,
model
of separability.
Section 5 concludes and proposes directions for future research.
When
2
The
literature
is
Complete Annuitization Optimal?
on annuities has long been concerned with the "annuity puzzle." This puzzle
consists of the combination of Yaari's finding that, under certain assumptions, complete
annuitization
is
optimal with the fact that outside of Social Security and defined benefit
pension plans, very few U.S. consumers voluntarily annuitize any of their private savings. 9
This issue
how
to
is
of interest from a theoretical perspective because
model consumer behavior
because of the gradual
as
shift in
it
bears
in the presence of uncertainty. It
the
US from
upon the
issue of
also of policy interest
is
defined benefit plans, which typically pay out
an an annuity, to defined contribution plans, that often do not require, or even
retirees the opportunity to annuitize.
The
important
role of annuitization is also
defined contribution plans, which have been growing in importance.
offer,
in national
This section of the
paper adds to the annuity puzzle by deriving much more general conditions under which
annuitization
6
is
optimal. Section 3 will then shed light on potential resolutions to the puzzle
See for example Kotlikoff and Spivak (1981), Friedman and Warshawsky (1990), Yagi and Nishigaki
(1993) and Mitchell, Poterba,
7
We
Warshawsky and Brown (1999)
Diamond and Mirrlees (2000). This formulation
use the formulation in
referred to as an "internal habit." Different
for
full
involves
models of intertemporal dependence
what
is
sometimes
in utility are discussed in,
example, Dusenberry (1949), Abel (1990), Constantinides (1990), Deaton (1991), Campbell and Cochrane
(1999),
Campbell (2002) and Gomes and Michaelides
(2003).
8
This might occur due to myopic failure to save, or due to adverse health or financial shocks.
9
This assertion
is
consistent with the large market for
what are
called variable annuities since these
insurance products do not include a commitment to annuitize accumulations, nor does there appear to be
much voluntary
annuitization. See for example
Brown and Warshawsky
(2001).
by examining market incompleteness.
Annuity
2.1
Demand
Two
in a
No
Period Model with
Aggregate
Uncertainty
Analysis of intertemporal choice
all
Consumers
at once.
of time
will
greatly simplified
is
time and
all
is
first
if
Alternatively,
consumers
start
condition, standard in the complete market
consumers are able to trade goods across
that, at the start of time,
states of nature.
across states of nature
made
resource allocation decisions are
be willing to commit to a fixed plan of expenditures at the
under either of two conditions. The
Arrow-Debreu model
if
first
period asset trade obviates future trade
only two periods.
live for
Yaari considered annuitization in a continuous time setting where consumers are uncertain only about the date of death.
Some
however, can be seen more simply by
results,
dividing time into two discrete periods: the present, period
and period
nitely alive
assumption that there
period 2
2,
is
when the consumer
no bequest motive and
and planned consumption
is
1
— q. We
moment assume
defined over
consumer
in the event that the
U=
we
for the
uncertain. In this case, lifetime utility
is
when the consumer
with probability
alive
is
1,
is
first
is defi-
maintain the
that only survival to
period consumption
alive in period 2, c 2
By
.
C\
writing
U( Cl ,c2 )
allow for the possibility that the effect of second-period consumption on utility depends
on the
level of first
axioms
satisfy the
period consumption. This formulation does not require that preferences
for
U
to
be an expected
value.
We approach both optimal decisions and the welfare evaluation of the availability of annuities
by taking a dual approach. That
is,
we analyze consumer
expenditures subject to attaining at least a given level of
in units of first period
consumption. Assume that there
units of consumption in period
2,
whether the consumer
unit of the consumption good in period
which returns
RA
in period 2
Whereas the bond
for
10
Ra =
y^-.
paying bonds
1.
Assume,
the consumer
is
if
the saver
is alive.
If
We
utility.
Rb
is
alive or not, in
and nothing
exchange for each
if
the consumer
whether or not the saver
the annuity were actuarially
drive returns below this level.
Rb
a bond available which returns
Adverse selection and higher transaction costs
may
measure expenditures
in addition, the availability of
alive
requires the supplier to pay
annuity pays out only
have
if
is
choice in terms of minimizing
for
fair,
an annuity
not
is
is alive,
then
alive.
10
the
we would
paying annuities than
However, because any consumer
These values should be interpreted as net of the transaction cost of a consumer buying these
will
assets.
have a positive probability of dying between now and any future period, thereby relieving
borrowers' obligation,
Assumption
if
there
is
the following as a
weak assumption: 11
Ra > Rb
1
Denoting by
we regard
A
no other income
in period 2 (e.g. retirees),
c2
and expenditures
B
savings in the form of annuities and by
savings in the form of bonds,
then
= RA A + RB B,
(1)
consumption are
for lifetime
E=
+A + B.
Cl
(2)
The expenditure minimization problem can thus be
defined as a choice over
period
first
consumption and bond and annuity holdings:
min
ci
+A+B
s.t.U( Cl ,R A A
By Assumption
1,
(3)
+ R B B) > U
purchasing annuities and selling bonds in equal numbers would cost
nothing and yield positive consumption
when
alive in period 2
but leave a debt
However, such an arbitrage would imply that lenders would be faced with losses
that such a trader failed to live to period
planned consumption
is
in the
if
dead.
in the
event
The standard Arrow-Debreu assumption
2.
consumption
possibility space. For
would require that the consumer not be in debt.
someone who
is
is
that
dead, this
In this simple setting the restriction
is
therefore that
B>0.
This setup yields two important results.
The
first
considers improving an arbitrary
allocation while the second refers to the optimal plan.
Result
1
(i) If
and holding
the
Proof. For
and
B>
0,
then
(i)
annuitization can be increased while reducing expenditures
consumption vector constant,
(i)
a sale of -g^ of the bond
the definition of C2-
For
(ii),
(ii)
The
solution to problem (3) sets
violated, annuities
by(i), a solution
yr^: is
with
B
supported empirically by Mitchell
would be dominated by bonds.
0.
and purchase of 1 annuity works by Assumption
Solutions with the inequality reversed are not permitted,
n That Rb < Ra <
B=
>
fails to
1
minimize expenditures.
u
et
al.
(1999).
If
the
first
inequality were
In this two period setting, Part
of Result
(ii)
1 is
an extension of Yaari's result of complete
annuitization to conditions of intertemporal dependence in
satisfy
there
expected
is
utility,
preferences that
axioms and actuarially unfair annuities. All that
utility
is
may
required
is
not
that
no bequest motive and that the payout of annuities dominates that of conventional
assets for a survivor.
Part
constant
(i)
of Result
utility,
implies that the introduction of annuities reduces expenditures for
1
thereby generating increased welfare (a positive equivalent variation or a
negative compensating variation).
We
might be interested
in
two related calculations: the
reduction in expenditures associated with allowing consumers to annuitize a larger fraction
of their savings (particularly from a level of zero)
consumers to annuitize
all of
That
their savings.
is,
and the
benefit associated with allowing
we want
to
know
the effect on the expen-
diture minimization problem of loosening or removing an additional constraint
(3) that limits
annuity opportunities.
To examine
this issue,
we
restate the expenditure
minimization problem with a constraint on the availability of annuities
min
:
Cj
on problem
as:
+A+B
(4)
v
'
ci,A,B
s.t.
:U{c u
R A A + R B B) > U
A <
A.
(5)
B >
We know
that utility maximizing consumers will take advantage of an opportunity to
annuitize as long as second-period consumption
is
(6)
is
positive.
ensured by the plausible condition that zero consumption
Positive planned consumption
is
extremely bad:
Assumption 2
lim
dU
—
—=
oo for
t
=
ct^o dc t
We can see from the
itize
optimization
(4)
1
,
that allowing consumers previously unable to annu-
any wealth to place a small amount of their savings into annuities (incrementing
zero) leaves second period consumption
period consumption
is
unchanged and so
A
from
unchanged (since the cost of the marginal secondtoo, therefore,
is
the optimal level of consumption
in
By
both periods).
Result
a small increase in
in this case,
1,
substitution of the annuity for the
bond proportional to the
A
generates a very small
prices
dA _
RA
~d~A~~ ~R~
B
= RA dA + RsdB = R A — Raj^ =
dB
_
'
leaving consumption unchanged:
The
effect
on expenditures
dc<i
is
equal to (1
-
4
j^
-)
<
0.
This
is
0.
the welfare gain from
increasing the limit on available annuities for an optimizing consumer with positive
bond
holdings.
If
of
constraint (5)
first
is
removed altogether, the price of second period consumption
period consumption
falls
second-period consumption,
parts.
One
part
is
its level will
to £-.
We
is
With a change
Thus the
adjust.
the savings while financing the
no annuitization and the second
change in prices.
from £-
in the cost of
cost savings
is
same consumption bundle
in units
marginal
made up
as
of two
when there
is
the savings from adapting the consumption bundle to the
can measure the welfare gain in going from no annuities to potentially
unlimited annuities by integrating the derivative of the expenditure function between the
two
prices:
rRg 1
E\a=o ~ £U=oo
where
c 2 is
compensated demand
arising
=~
/ _,
JR
A
c 2(P2)dp 2
from minimization
(7)
,
of expenditures equal to cj
+ c 2p2
subject to the utility constraint without a distinction between asset types.
Equation
tion) benefit
(7) implies that
more from the
Result 2 The
greater for
consumers who save more (have larger second-period consump-
ability to annuitize completely:
benefit of allowing complete annuitization (rather than
consumer
i
than for consumer j
if
consumer
i 's
no annuitization)
is
compensated demand for second
period consumption (equivalently, compensated savings) exceeds consumer j
's
for any price
of second period consumption.
2.2
Extending the Model to
Many
Periods and States with
Com-
plete Markets
While a two-period model with no aggregate uncertainty
consumers face a more complicated decision
setting.
offers the virtue of simplicity, real
In particular, they face
many
periods
consumption and each period may have several possible states of nature. For
of potential
example, a 65 year old consumer has some probability of surviving to be a healthy and active
some chance of
80 year old,
and
finding herself sick
chance of not being alive at
at age 80.
all
home
in a nursing
at age
80 and some
Moreover, rates of return on some assets are
stochastic.
The
complete annuitization survives subdivision of the ag-
result of the optimality of
gregated future defined by
many
c 2 into
and
future periods
A
states.
particularly simple
subdivision would be to add a third period, so that survival to period 2 occurs with probability
1
—
and to period 3 with probability
q2
(1
—
g 2 )(l
annuities which pay out separately in period 2 with rates
rates
R A3
Rb3 and
in period 1
E=
Assumption
that
we have
set
1
is
Rb2 and Ra2, and period
That
3.
with the appropriate subscript,
ci
C2
=
+
^42^2,
c3
= RB3B3 +
RA3A3.
will call
states of nature that differ in no other
is,
3 with
defining bonds
12
+ A2 + A 3 + B2 + B 3
RbiB-I
modified to hold period by period, Result
up what we
In this case, bonds and
<?3)-
are sufficient to obviate trade in periods 2 or
and annuities purchased
If
~
"Arrow bonds" (here
B2
way except whether
this
annuities" which also recognize whether this consumer
is
alive
extends
1
and
53
)
trivially.
Note
by combining two
consumer
complete the
is
alive.
"Arrow
set of true
Arrow
securities of standard theory.
In order to take the next logical step,
c2
B2
,
,
and
A2
future periods
we can continue
to treat C\ as a scalar
as vectors with entries corresponding to arbitrarily
(t
<
T), within arbitrarily
many
states of nature
many
(a;
<
and
interpret
(possibly infinity)
Q).
Ra2 {Rbz)
is
then a matrix with columns corresponding to annuities (bonds) and rows corresponding to
payouts by period and state of nature. Thus, the assumption of no aggregate uncertainty
can be dropped. Multiple states of nature might
refer to uncertainty
about aggregate
issues
such as output, or individual specific issues beyond mortality such as health. 13 In order to
extend the analysis, we need to assume that the consumer
each state of nature where the consumer
1
implicitly,
we
are assuming that
if
is alive,
is
sufficiently "small" that for
there exists a state where the consumer
is
markets reopened, the relative prices would be the same as are
available in the initial trading period
13
For a discussion of annuity payments that are partially dependent on health status, see Warshawsky,
Spillman and Murtaugh (forthcoming).
dead and the equilibrium prices are otherwise
Completeness of markets
identical.
construction of Arrow bonds which represent the combination of two
Arrow
still
allows
securities.
Annuities with payoffs in only one event state are contrary to our conventional perception of (and
name
for) annuities as
paying out in every year until death.
complete markets, separate annuities with payouts
such securities.
in
each year can be combined to create
clear that the analysis of the two-period
It is
However, with
model extends to
this setting,
provided we maintain the standard Arrow-Debreu market structure and assumptions that
do not allow an individual to die in debt. In addition to the description of the optimum, the
formula for the gain from allowing more annuitization holds
for state-by-state increases in
the level of allowed level of annuitization. Moreover, by choosing any particular price path
from the prices inherent
in
bonds to the prices inherent
gain in going from no annuitization to
full
we have extended the Yaari
In this section,
we can measure the
annuitization. This parallels the evaluation of the
by a lumpy investment
price changes brought about
in annuities,
(see
result of
Diamond and McFadden
of aggregate uncertainty, actuarially unfair (but positive) annuity
premiums and intertem-
porally dependent utility that need not satisfy the expected utility axioms.
shown that increasing the extent
who
hold conventional bonds.
that complete annuitization
14
is
We
have also
of available annuitization increases welfare for individuals
These
results
deepen the annuity puzzle by demonstrating
optimal under a wider range of assumptions about individual
Thus, given available empirical evidence about the small
preferences.
(1974)).
complete annuitization to conditions
annuity market, a natural question
is:
when might
size of the private
individuals not fully annuitize? This
is
explored in the next section.
When
3
In Section 2,
Is Partial
Annuitization Optimal?
we explored annuity demand
Arrow bonds and Arrow
plete markets
annuities were
in a setting with complete
assumed
and without a bequest motive, the
were very weak
just that the
-
value of security payments not
added
Arrow
to exist for every event.
securities
-
both
With such com-
sufficient conditions for full optimization
costs of administering annuities were less than the
made because
of the deaths of investors.
The
full
annuitiza-
tion result depends on market completeness. In settings without market completeness,
The
generalization of Result 2 to this case requires the very strong condition that after the present,
consumption
for agent
i
exceeds that of agent j state of nature by state of nature. That
grows at a greater rate than
different prices
in
we
j's
is
not sufficient: allowing complete annuitization
by increasing any of many ratios A
time past period
1.
A
'
—
B
W
.
may yield
i's
consumption
reduction
In general, these price changes are
in
many
non-monotonic
consider sufficient conditions for partial annuitization
in
We
optimized demand.
we
First,
Then we
consider two alternative tpes of annuity market incompleteness.
consider a setting with complete
consider a setting with complete
many
involve payoffs in
some annuitization
the inclusion of
-
Arrow bonds but only some Arrow
Arrow bonds and compound annuities
events rather than being
Arrow
annuities.
The
first
annuities.
-
ones that
setting relates
the annuity puzzle to the circumstance that insurance firms provide limited opportunities for
annuitization.
The second
products that do
setting explores the puzzle in annuity
demand given
the annuity
exist.
3.1
Incomplete Annuity Markets (When Trade Occurs Once)
3.1.1
Incomplete Arrow Annuities
The
logic of the
of an
argument
Arrow bond, the
in Section 2
cost of
was straightforward. Whenever there was a purchase
meeting a given
utility level
purchase of an Arrow annuity for an Arrow bond.
row bonds and Arrow
annuities,
savings was invested in
annuitization
if
consumption
in
the set of
the
Incomplete
3.1.2
Most
annuities. This line of
Arrow annuities
is
is
argument
exist, the
optimum
consumption
will include
for
securities.
both Ar-
the only
(since
all
of
complete
way
to get
no Arrow annuity
that event will be part of the
in that event. Conversely, as long as
some
annuitization.
Annuities
markets require that a consumer purchase a particular time path
of payouts, thereby combining in a single security a particular
Arrow
is, if
by purchasing an Arrow bond
positive
Compound
real world annuity
sets of
will not result in
not complete. That
then some purchase of Arrow bonds
optimum has
any Arrow annuities
Thus with complete
no Arrow bond would be purchased, implying that
some future event
exists for that event),
optimum when
Arrow
could be reduced by substituting
"compound" combination of
For example, the U.S. Social Security system provides annuities that are
indexed to the Consumer Price Index and thus offer a constant real payout (ignoring the
role of the earnings test). Privately
nominal terms, or
offer
purchased immediate
life
annuities are usually fixed in
a predetermined nominal slope such as a 5 percent nominal increase
per year. Variable annuities link the payout to the performance of a particular underlying
portfolio of assets
and combine Arrow
participating, which
securities in that way.
means that the payout
CREF
annuities are also
also varies with the actual mortality experience
for the class of investors.
Numerous simulation
studies have examined the utility gains from annuities with these
10
types of payouts that combine Arrow securities in a particular way.
we
annuities in this setup,
in
complete set of Arrow bonds and consider the
We
is alive.
effect of
We
continue to assume a
the availability of particular types
need to consider whether the return from annuities and bonds can be
also
reinvested (markets are open) or
must be consumed (markets
complete annuitization
lose the result that
consider such lifetime
continue to assume a double set of states of nature, differing only
whether the particular consumer we are analyzing
of annuities.
To
is
are closed) In general,
optimal. Nevertheless,
we
we
will
will get optimality
of complete annuitization of initial savings in real annuities satisfying the return condition
provided that optimal consumption
more general
of annuities
To
setting
is
we examine
is
rising over
time and markets for bonds are open. In a
sufficient conditions for
the result that the optimal holding
not zero.
illustrate these points,
and a complete
set of
bonds.
we consider a three-period model with no aggregate uncertainty
Then we
will
show how the
annuities, then the expenditure minimization
min
:ci
problem
results generalize. If there are
is:
+ B2 + B3
(8)
ci,A,B
s.t.:U(c 1 ,R B2 B 2 ,R B3 B3
That
With
B3
is,
we
have:
are positive.
in the
Now assume
C2
=
RB2B2,
c3
=
RB3B3.
that there
is
The minimization problem
min
:ci
c\,A,B
s.t.
Before proceeding,
>
Rbiui
Viw.
:
U(c 1 ,Rb 2 B 2
we must
A
all
three of
cj,
B2
and
a single available annuity, A, that pays given
two periods. Assume further that there
initial contracting.
RAtu
)>U
the assumption of infinite marginal utility at zero consumption,
amounts
no
is
is
no opportunity
for trade after
the
now
+ B 2 + B3 + A
(9)
+ R A 2A, R B3 B 3 + R A3 A) > U
c2
= Rb2B2 +
c3
= RbsBs + R A3 A.
RA2A,
revise the superior return condition for
more appropriate formulation
that combines Arrow securities to exceed bond returns
stream provided by the annuity, the cost
is
less if
11
for the return
is
Arrow annuities that
on a complex security
that for any quantity of the payout
bought with the annuity than
if
the
same
stream
Define by £ a row vector of ones with length equal to
bought through bonds.
is
the number of states of nature distinguished by bonds,
let
the set of bonds continue to
be represented by a vector with elements corresponding to the columns of the matrix of
RB
returns
and
let
RA
be a vector of annuity payouts multiplying the scalar
A
to define
state-by-state payouts.
Assumption 3 For any
RB B
A
and any
RA3
if
there
is
an annuity that pays
through annuities
when purchased by
RaA
is less
expensive
when
< (^ + jf )- By
that may be purchased
we would have
any consumption vector
period
41
az
1
financed strictly through annuities than
15
a set of bonds with matching payoffs.
Given the return assumption and the presence of positive consumption in
is
B,
R A2 per unit of annuity in the second
per unit of annuity in the third period, then
linearity of expenditures, this implies that
strictly
collection of conventional assets
A<£B.
=>
For example,
and
annuitized asset
clear that the cost goes
down from the introduction
of the
first
small
all
amount
periods,
it
of annuity,
which can always be done without changing consumption. Thus we can also conclude that
the
optimum
purchase.
It is
(including the constraint of not dying in debt) always includes
also clear that full annuitization
tion pattern with complete annuitization
denoting partial derivatives of the
consumption given
full
is
may
if
the implied consump-
worth changing by purchasing a bond. That
utility function
annuitization,
not be optimal
some annuity
with subscripts, optimizing
we would have the
first
first
is,
period
order condition:
Ul (c1 ,RA2 A,RA3 A) = RA2 U2 (c ,R A2 A,RA3 A) + Ra 3 U3 {c 1i R A2 A,Ra 3 A).
1
Purchasing a bond would be worthwhile
tfifo,
if
we
satisfy either of the conditions:
RA2 A,RA3 A) < RB2 U2
,RA2 A,RA3 A)
(10)
RB3 U3 (c u RA2 A,RA3 A)
(11)
{ Ci
or
Uiia,
By
our return assumption,
A2 A,R A3 A)
we can not
but we might satisfy one of them. That
annuitized asset and
15
may
involve
<
satisfy
is,
both of these conditions at the same time,
the optimum
some bonds, but not
all
will involve
holding some of the
of them.
This assumption leaves open the possibility considered below that both bond and annuity markets are
incomplete and some-consumption plans can be financed only through annuities.
12
It is clear
compound
that these results generalize to a setting with complete Arrow bonds and some
many
annuities with
many
periods and
states of nature.
We
show below that
expenditure minimization requires that there must be positive purchases of at least one
annuity.
Lemma
plan
Consider an asset
1
A*
Any consumption
with finite, non-negative payouts Ra*-
with positive consumption in every state of nature can be financed by a combina-
\c\ C2}'
tion of first period consumption, a positive holding of
A* and another
strictly
non-negative
consumption plan.
•>>"* =
Proof. Define Ra*
Now
C2
—
Ra*
We now
Result 3
+ Z,
[-5-^
l
—
,
-fM»2l'
where
Z
—
••, -5-^
K-A*tu
is
-5-^
,
'
marginal
]'
i
and define
J
utilities are infinite at zero
constant.
held, a
Also, then
(ii)
Ra*)'
•
1
consumption (Assumption 2 holds) and
there exist annuities with non-negative payouts which satisfy
no annuities are
a = minfco
V
the scalar
weakly positive.
have a weaker version of Result
If
—
ttA*TO.
Assumption
3,
then
(i)
small increase in annuitization reduces expenditures, holding
when
utility
expenditure minimization implies positive holdings of at least one
annuity.
Proof. Suppose
sibilities: first,
that the optimal plan (ci,A,
consumption might
is
and fails
positive in every state of nature, then
positive linear combinations of the
some
be zero in
this implies infinitely negative utility
B)
features
0.
Then
future state of nature.
there are
bonds.
two pos-
By Assumption 2
to satisfy the utility constraint. If
consumption
Arrow
A=
consumption
a linear combination of all strictly
is
But then since some
Assumption 3 and
strictly positive con-
Lemma
sumption plan can
be financed by annuities, by
be reduced holding
consumption constant by a trade of some linear combination of the bonds
for
some combination of annuities with
strictly positive payouts.
1,
expenditures can
This contradicts optimality
of the proposed solution.
Part
all
(i)
of Result 3 states that
at once, then starting
if
consumers are willing to commit to lifetime expenditures
from a position of zero annuitization, a small purchase of any
annuity (with a good return) increases welfare. This applies to any annuity with returns
excess of the underlying nonannuitized assets, no matter
Part
(ii) is
the corollary that optimal annuity holding
that
up
some
to
effect is to
point, annuity purchases
do not
how
is
distort
distasteful the payout stream.
always positive.
Lemma
1
is
annuitized,
if
the supply of annuitized assets
13
shows
consumption, so that their only
reduce expenditures, as in the case where annuities markets are complete.
a large fraction of savings
in
fails
to
When
match
demand, annuitization
Prom the proof
distorts
of Result 3,
it
consumption and some conventional assets may be
follows that the annuitized version of
preferred.
any conventional asset
(with higher returns) that might be part of an optimal portfolio dominates the underlying
asset.
Incomplete Annuity Markets With Trade More than Once
3.2
The setup
so far has not allowed a second period of trade. However,
payout trajectories are unattractive, households
may wish
if
the existing annuities'
to modify the consumption plan
yielded by the dividend flows purchased at retirement through trade at later dates.
find in this case that positive annuitization remains optimal as long as conventional
are complete
and a
We
markets
revised form of the superior returns to annuitization condition holds.
With incomplete conventional markets,
it is
possible for liquidity concerns to render zero
annuitization optimal.
Trade in
3.2.1
Many
Suppose that trade
in
Periods with Complete Conventional Markets
bonds
is
allowed after the
first
with the returns that were present for trade before the
there
is
period, with
period.
first
bond
To
prices consistent
begin,
we assume that
not an annuity available at any future trading time and that the consumer can save
out of annuity receipts but can not
sell
the remaining portion of the annuity. Since there
would be no further trade without an annuity purchase out of
without any annuity
is
wealth, the
optimum
unchanged. Utility at the optimum, assuming some annuity purchase
and consumption of the annuity return,
the result that
initial
some annuity purchase
is
raises welfare as above.
Thus we conclude that
optimal (Result 3) carries over to the setting with
complete bond markets at the start and further trading opportunities in bonds that involve
no change in the terms of bond transactions. The possibility of reinvesting annuity returns
can further enhance the value of annuity purchases and
may
result in the optimality of full
annuitization.
Returning to the three period example with no uncertainty beyond individual mortality,
a sufficient condition for complete annuitization at the start
associated with complete annuitization at the
first
would wish to save, rather than dissave. This
(11)
is
violated.
To examine
this issue,
we now
is
that the consumption stream
trading point was such that the individual
true even
set
is
if
one of the inequalities (10) or
up the expenditure minimization problem
with retrading, denoting saving at the end of the second period by
14
Z
min
:c x
c u A,B,Z
s.t.
The
:
U( Ci ,Rb2B2
zero:
is
(12)
y
'
+ RaiA -
restriction of not dying in debt
+ £2 + £ 3 + ,4
Z,
R B3 B 3 + R A3 A +
(R B3 /R B2 ) Z) > U.
the nonnegativity of consumption
if
A
set equal to
is
16
B S Z>
2
3
,
,
>
RB2B2
RB3 B 3 +
The assumption
( Cl
,RA2 A,R A3 A) <
This condition can be readily
future periods, as long as trade
T—
is
3 (c ly
and
is
RA2 A,RA3 A)
interest rates
allowed in each.
future periods
1
R B3 U
annuitization
full
To show
and the
this,
(13)
a suitable relationship
satisfied for preferences satisfying
(implicit) utility discount rates
a world with
Z >
that dissaving would not be attractive given
R B2 U2
between
{RB3/RB2)
result extends with
we consider
and no uncertainty except individual
many
as a special case
mortality, so that
future consumption conditional on survival can be described by a vector c 2 with one element
each period up to T, beyond which no individual survives:
for
c2
=
[c 2
,
Consumers
c 3 ...cx}'.
have access to "Arrow" bonds and a single annuity product which pays out a constant real
amount
of
RA A
per period, where
A
is
assume that no annuities are available
allowed.
by
T—
1
By completeness
the
amount
of the annuity purchased in period
after the first period,
1.
We
but that future bond trades are
bond markets, we can
consider the set of bonds to be described
_1
securities, each of which pays out at a rate of (1 + r)'
at date t only. We assume
further that there
is
of
a constant real interest rate of r on bonds and that the rate of return
condition (Assumption 3)
is satisfied.
That
is,
the internal rate of return on the annuity,
with periodic payouts multiplied by survival probabilities, exceeds
Because Assumptions 2
(infinite disutility
r.
from zero consumption
and 3 (any consumption plan that can be financed by annuities alone
is
in
any future period)
financed most cheaply
by annuities alone) are met:
16
B3
can be negative
if
Z is
positive.
However, a budget-neutral reduction
A
constant, then yields equivalent consumption, so there
is
non-negative, then
Z
must be zero as long as
B2
is
is
no restriction
15
Z
B2
and increase
in
B3
,
holding
disallowing negative B3.
positive, or else constant
expenditures could be obtained at a lower price by reducing
savings out of bonds.
in
in
If
S3
consumption with reduced
and increasing A. That
is,
there are no
Result 4 The
above features
solution to the expenditure minimization problem with markets as described
A>
0.
Proof. Follows immediately from Result
By
3.
the no bankruptcy constraint, consumers
itization renders
may undo
annuitization by saving
if
annu-
consumption too weighted towards early periods, but not by borrowing
With bonds
annuitization renders consumption too weighted to later periods.
liquidity constraint given a constant real annuity requires that expenditures
up to any date r must be
plus expenditures on
liquid, the
on consumption
than total bond holdings plus annuity receipts up to that date,
less
first
if
period consumption. This constraint can be written
JZct{l+r)
l
-t
YB
<c +
l
/
t
+ RA AJ2{l + r)
t=2
t=l
This induces one constraint
-t
Vr.
(14)
t=2
for every period in
by the required annuity. Annuities are costly
l
as:
which consumption
in optimization
is
bound from above
terms because they contribute
to these constraints.
The expenditure minimization problem becomes:
min
+A+B
cl
(15)
'
v
ci,A,B
> U
s.t.U(c 1 ,c2 {A,B))
s.t.
Result 5
// optimal
nuitization
Proof.
is
consumption
optimal. That
is,
equation (14)
weakly increasing over time, then complete
is
initial net
That
is satisfied.
purchased from future savings. Hence,
can be reduced and
>
e
bond purchases are
units of
B2
utility
initial
an-
zero.
With non- decreasing consumption, constraint 14
budget constraint
e-ji
is satisfied.
is satisfied
when
the lifetime
is,
bonds maturing as needed to satisfy (17) can be
if
net bond holdings are greater than zero, expenditures
increased by an additional purchase of
e
units of
A
and
sale of
.
Without the annuity, expenditures are given by
T
E(c,0)
=
Cl
T
~
+ Y, c R B \ =
t
ci
+
(=2
With
annuities, the cost of a
E
c t(l
1
+
"'-
(16)
t=2
consumption plan
is
equal to the cost of annuitized con-
sumption plus the difference between annuitized consumption and actual consumption in
every period:
E(c, A)
=
Cl
+A+
j^ict
16
~ RaA)(1 +
r)
1
"4
,
(17)
where
Ra
is
the per-period annuity payout. For
>
t
consumption
1, if
is less
than the annuity
payout, the difference can be used to purchase consumption at later dates, with the relative
prices given
by bond returns.
bond maturing
at date
of additively separable preferences over consumption, exponen-
discounting and access to an actuarially
results.
If 1
—
m
=
t
greater than the annuity payout, then a
is
must be purchased.
t
Adding the assumptions
tial
consumption
If
n' =2 (l
—
g s ) is
constant real annuity generates additional
fair
the probability of survival to period
t,
then actuarial
fairness implies that the cost per unit of the annuity is equal to the survival-adjusted present
discounted value of bond purchases yielding the same unit per period:
-
1
l~rn
— Ra
£ -"'(Tn^
(1
t )
t=2
Ra -
—
^
M
E^l-^Xl+r)
1
Wl
Assumption 3 applies as long as there
is
—
This
.
RaA per period past
—-— —
than =^r
is less
•
a positive probability of death by the end of
periods because the cost of consuming any plan
is -=-r
(18)
-''
_m_,
1
,
-
,
period
the cost of purchasing
A
1
T
with annuities
per period with
conventional securities.
Here,
we assume that
utility is given by:
U(c 1 ,c 2 )
= Y,S - (l-rn
1
t
t
)u(c t ),
(19)
t=i
Where
>
u'
0,
Result 6 For
u"
<
lim C( ^
0;
u'
=
oo,
and
6
is
the rate of time preference.
the dual utility maximization problem with fixed expenditures, if the optimal
level of annuitization
A
is less
than initial wealth savings, so that there are positive initial
expenditures on bonds, an increase in S yields an increase in optimal
Proof. With an increase in
initial
5,
for any periods
period consumption and investment
marginal
utilities increases
must
>
s,
the ratio of
increase.
relative to savings.
consumption induced by
This follows since the ratio of
with 5 and the ratio can be increased with a small budget-neutral
exchange of
Bs
the original
consumption plan plus a weakly increasing sequence with negative elements for
all
dates up to
for
B
—
s'
A
s >.
some date
minimal expenditures by
Result 7
Hence, planned consumption with the increase in 5 must be equal to
If 5(1
+ r) >
t.
By
the result above, the old
consumption plan
selling bonds with maturity less than
1,
complete initial annuitization
17
is
t
is
revised with
and increasing A.
optimal.
By
Proof.
result
6,
is
it
complete annuitization to be suboptimal,
it
purchasing a bond with maturity at date
t
5
then
1,
t
-\l
If
+ r)
t
-l
and
t
bond purchases so that
it is
some
For
1.
for which
t
hand
>
receipt, or:
^*'\l -
£f=2 (l-m
because the
the right
=
r)
provides greater marginal utility than purchase of
(l-m )>
this is impossible,
(by non-negative mortality)
later
be the case that there exists
^(l + rrV(*^)(l-mO
3t>l:
+r) =
must
+
this is true for 5(1
consumption of each period's annuity
the real annuity with
If 6(1
show that
sufficient to
left
t
t )u>
(RA A)
)(l+r)i-<"
hand side
side equals one.
is less
than or equal
Note that
to
one
this applies to
any
concluded optimal to have constant consumption.
uncertainty were introduced, for complete annuitization to remain optimal,
we would
require that marginal utility in every state of nature not be so large to justify the cost of
adding consumption
in that period
through a bond rather than adding consumption in every
period through the constant real annuity (which
amount across
we might assume would pay out a constant
states of nature as well as periods).
Future Purchase of Annuities and the Possibility of Zero
3.2.2
Initial
Annuiti-
zation
As we have
seen, the possibility of future trade in
bonds can increase the demand
Conversely, the possibility of future trades in annuities can decrease the
nuities.
for initial annuities, replacing it
demand
with a later
for annuities.
Young
(2002)).
If
is
it is
possible that
survival probability for the first period
it is
is
zero,
an annuity purchased
in period
one pays $0.55
annuity purchase in period two pays $1.50 in period three,
are
more cheaply purchased by placing
two and investing
17
all
is
larger for the
if
the
large enough.
annuity and suppose an individual lives for at most three periods.
is
period one
worthwhile to delay annuity purchase,
Consider the case considered above where the only annuity available
bonds
in
addressed in Milevsky and
the internal rate of return (unadjusted for mortality)
delayed annuity, then
demand
Continuing to assume
complete bond markets, assume that real annuities can be purchased starting
and, in a reopened market, also in period two (this possibility
for an-
all
If
in periods
17
is
a constant real
the interest rate on
two and three and an
then some consumption plans
period one savings in a bond maturing in period
period two savings in the annuity available in period two.
Such an unrealistic payout scenario could
in principle
annuitizers are longer lived than late annuitizers.
18
be a product of a selection process whereby early
Incomplete Markets for Nonannuitized Assets and the Possibility of Zero
3.2.3
Annuitization
In the original Yaari model, stochastic length of
ical
life
was the only source
expenses and nursing home costs represent large uncertainties
insurance for these events
less liquid
than bonds or
is
if,
incomplete, this will affect the
for
some
The
guaranteeing positive annuity purchases
which pays out in
many
for
demand
Med-
consumers.
for annuities
if
If
they are
reason, the available annuities' payouts are relatively
large in low marginal utility states.
nuities available
of uncertainty.
general incomplete markets sufficient condition
same
the
all
that there
is
an annuity or combination of an-
is
states of nature as a nonannuitized asset,
with payouts that are weakly greater state-by-state. In the real world, this seemingly strong
met by an annuitized
condition could be
version of an underlying asset such as shares in
a particular stock or mutual fund. However, with complete Arrow pure bond markets and
given survival probabilities, such that price- weighted marginal utility
states, as long as the optimal
as the return condition
Basically, the
times
is
plan involves some consumption throughout
and as long
life
remains the case that some annuitization
is satisfied, it
argument above that the minimal consumption over
expectancy as the only
life
expectancy that
bonds would
if
equated across future
all
affect
is
risk,
possible states
not recognized in the market structure. Again, a greater liquidity for
annuity demand. In this case, there can be zero
the news implies that the maximal possible length of
life
is
demand
has decreased
-
for annuities
that
Conversely,
zero.
if
changes the probabilities of survival, without shortening the possible maximal
some annuitization remains optimal, by the same argument as above.
model with
life
expectancy news, we derive a necessary condition
Suppose that
— q 2 and
in
period
1,
to period 3 with probability (1
health news") with probability
compound annuity
three, respectively.
consumer
will sell
If
is
is,
that
the news
life,
then
In a three period
for zero annuitization.
a consumer expects to survive to period 2 with probability
—
g 2 )(l
—
93)-
However, the consumer knows that
in period 2, the conditional probability of survival to period 3 will
single
and
individuals can receive information about remaining
the minimal consumption over the initially possible ages
1
optimal.
is
best financed by an annuity continues to hold.
With
life
is
a
or to
j^
("good health news") with probability
available in period one, paying
the bonds
fail
R&2 and Ras
to distinguish between the
whatever bonds pay
be updated to zero ("bad
off in
in periods
1
—
a.
A
two and
two health conditions, the
period three on obtaining bad health news in
period two, but will be unable to cash out the illiquid third period annuity claim.
Suppose that without annuitization, the consumer divides period one savings between
the bonds maturing in periods two and three such that no trade
19
is
undertaken in period
two
if
Consumption
the consumer obtains good health news.
by RB2B2
if
there
is
good health news and R.B2B2
Assume that the consumer's
marginal utility of savings
by
utility is given
in either
bond
is
optimal
if
j^-RbzB?,
f/(c 1; c 2 c3 )
,
=
if
u(ci)
two
thus given
is
the health news
+ 6u(c 2 ) +
is
bad.
6 u(cs).
The
2
thus:
is
5R B2 {au'(R B 2(B2 + B 3 )) +
Zero annuity purchase
+
in period
and only
(1
if
-
a)u'{R B2 B 2 ).
expression (20)
to the marginal utility of a small purchase of the annuity.
This
is
(20)
greater than or equal
latter value
is
simplified
by noting that optimal allocation across periods two and three conditional on good health
news imply
R B 2u' (R B 2B2 = 5R B sW'(RbsB3
the annuity
is:
)
6(aR A2 u'(R B 2(B2
+ B 3 )) +
(1
)
.
The marginal
utility of a small
- a)(RA2 + RAS ^-)u' (RB2 B 2
purchase of
(21)
)
Expression (20) can exceed expression (21) and hence zero annuitization can be optimal
without violating the superior return condition for annuities, here
can occur
if
bad health news a
>
1.
This
is
sufficiently large
and u
is
not too
Hence, in this particular incomplete markets setting, zero annuitization, partial
annuitization and complete annuitization are
further assumptions.
4
-j^
the annuities' payouts are sufficiently graded towards future payouts relative
to the bonds, the probability of
concave.
-^ +
all
consistent with utility maximization without
18
.
Special Cases:
The Welfare Gains from Annuitiza-
tion with Additive and Standard-of-Living Utility
Much
of the hterature
on annuities has focused on the welfare gains that can be generated
by providing access to annuity markets.
These simulations have typically assumed that
individuals have intertemporally additive utility that exhibits constant relative risk aversion.
The
gains from annuitization have been
Mitchell
shown to be quite
and Poterba (2002) show that a consumer with
actuarially fair real annuity
substantial. For example,
Brown,
log utility would find access to an
market equivalent to nearly a 50 percent increase
in unannuitized
wealth.
18
2.
In this
example zero annuitization cannot be optimal unless the support
for being alive
changes
in
period
For example, uncertainty about medical expenses might change the extent of annuitization, but would
not eliminate annuitization. With psychic or monetary costs to annuitization, demand sufficiently close to
zero could result in an
optimum no
annuitization at
all
20
We saw in Section
a weaker
effect
under these conditions, a consumer
3.2.1 that
than interest rates
whom
for
will annuitize completely. In this section,
welfare consequences of annuitization in this "industry standard" case.
the prior literature by examining annuity valuations
of-living,
we consider a case
In particular,
separable.
i.e.,
any period
utility in
is
in
which
the initial standard-of-living
make
is
utility is
we
then expand on
no longer intertemporally
dependent on a standard-
utility is
both
more or
annuities
sets of
is
discuss the
a function of current and past consumption.
calculate the welfare gains from annuitization under
a standard-of-living effect can
when
We
discounting
We
assumptions and show how
less valuable,
depending on how large
This relationship
relative to available retirement resources.
plays a major role in the level of savings as well as the attractiveness of constant consumption.
We
consider as in Section 3.2.1 a world with
than time of death.
units of
We
an actuarially
T—
1
future periods and no uncertainty other
evaluate the welfare consequences of the required purchase of
fair
annuity with constant real return
RA
in each future period
there are no future opportunities to purchase annuities, but bonds
in the present
and
CV
when
may be purchased both
in the future.
As discussed above, a small
the
A
increase in
A from zero has no effect on consumption,
from incremental annuitization from
to a small
number
e is
so that
equal to the difference
between E(c,0) and E(c,e):
^
(JF
T
= l-E(^(Hr)
w )<0-
The
inequality follows from equations (16)
The
welfare effects of larger increases in annuitization are
they
may
tions.
constrain consumption. Below,
We
4.1
we
and (18) as long as
(22)
ttlt
more
>
0.
difficult to sign
because
consider the effects for particular utility func-
also consider the value of a complete annuity market.
The Gains from Annuitization under "Usual Assumptions"
If utility is
additively separable and features exponential discounting, as in specification (19),
then the extension to Result 7 follows from the proof above:
Result 8
If 6(1
+ r) >
1,
then any increase in annuitization in the range
A
G
[0,
E — c{\
is
welfare enhancing.
For
more impatient consumers
(lower S),
we
solve for the optimal fraction of savings put
into annuities numerically. Results are detailed below.
21
Beyond the
results
we have above, making statements about the
from complete annuitization to zero annuitization
lation
in
must take
which should push consumption
5,
where optimal consumption
increase valuation. Hence, for 6(1
parameter of
risk aversion,
liquidity constraints
is
a move
summarized
is
that valuation will
later in
life.
Further,
decreasing over time, increased smoothing should
+ r) <
we should expect valuation
1,
to increase with any
because the desire for decreasing consumption, which makes the
brought on by annuitization bind, would then be tempered by a desire
consumption smoothing.
We
confirm these intuitions below with numerical examples.
The Gains from Annuitization when
4.2
for
because in general, this calcu-
equation (14). That said, a plausible conjecture, based on Result 6
in cases
EV
into account the period-by-period positive wealth constraints
increase in the patience parameter
for
is difficult,
size of
Depends on a
Utility
standard-of-living
Additive separability of utility does not
no car
dio apartment with
apartment without a car than
abandon a
section,
we
four
more
surely
is
for
sit
well with intuition.
tolerable for
someone who was
bedroom house and an Escalade
For example,
someone used to
in a stu-
living in a studio
forced by a negative
for a studio
life
income shock to
apartment and no
car. In this
consider an extreme and hence illustrative, example of intertemporal dependence
in the utility function,
formulation
is
that
taken from
it is
consumers with such
Mirrlees (2000).
The
intuition behind this
not the level of present consumption, but the level relative to past
consumption that matters.
difference could also
Diamond and
We
consider the ratio of present to past consumption, but the
be considered. In choosing how to allocate resources across periods,
utility trade off
immediate
gratification
from consumption not only
against a lifetime budget constraint, but also against the effects of consumption early in
on the standard-of-living later
in
life
life.
U(c1 ,c2 )
= jr6 - (l-m
t
1
t
)ufr,
(23)
s<
t=i
where
St
If
=
+ act_!
1 + Q
St-i
individuals' subjective standard of living
the additively separable case.
A
is
positive value of
•
constant
a
(i.e.
if
a =
indicates that past
0)
we
are back in
consumption makes
individuals less satisfied with a given level of present consumption.
In the absence of the positive wealth constraints (14), the marginal utility of consumption
in
any period incorporates two
effects
not present in the additively separable case:
22
(i)
the
effect of the present standard-of-living
consumption on future periods'
on present marginal
utility
1 ,, c t n
^-1 -„(-(l-
We
note that
finite si, so
To do
if
consumption
some annuitization
calculations,
u
mt ))-g<5
v^xfc-i
=
lim£t^o u '( c t)
>
=
°°>
0) case
is
7 <
Effect
(ii)
will
2,
this
a
u \n
(^^(-1(1-^
°k
Ck
\
for
m.) -
= -^—
and that 7
>
1.
Hence:
g^.^-^cJ-^-d - ™,).
tend to push consumption towards later periods relative to the
if
decreasing over time and 7
periods. For
Under
living.
given by:
the standard-of-living
is
increasing over time since a higher
standard-of-living increases the marginal utility of consumption.
is
the effect of present
tnen Assumption 2 holds and Result 4 applies
we assume that u(-)
1, effect (i) will
no standard (a
(ii)
optimal.
is
g . *- W'd For 7
and
through subsequent standards of
specification, the marginal benefit of present
dU
=*
utility
the effect
>
is
then effect
2,
(i)
will
If
the standard-of-living
tend to push consumption to earlier
ambiguous.
tend to push consumption towards later periods in
life
since later
consump-
tion raises the standard-of-living in fewer periods. Hence, the result of complete annuitization
when
the discount rate
is less
than the interest
rate,
Result
stant or decreasing over the period of annuitization.
is
small and the required level of
utility,
U,
is
large.
7,
continues to hold
This occurs
If
may undo
s
con-
is
the initial value of s
the initial value 5j
large relative to the expenditures required to attain U, then the
aversion
if
if
is
sufficiently
smoothing implied by
risk
the result by rendering optimal consumption relatively decreasing over
time.
With the
constraint that the only annuity available pays out a constant real amount, rel-
ative valuations are particularly difficult to calculate with standard-of-living effects, because
the intertemporal effects
compound
the difficulty of the multiple positive wealth constraints.
However, we conjecture that parameter changes that tend to defer optimal consumption
will
tend to increase valuation. Hence, simulated valuations should tend to be increasing in
Further, large
both
4.3
effects
Sj
should yield decreasing valuation and small
magnified by
with
7.
Numerically Estimated Magnitudes of Welfare Effects
To estimate numerically the value
that an individual places on annuitization,
we
—
= ^37
1
uix)
sj increasing valuation,
5.
for
both the additively separable and standard-of-living
23
effect cases.
specify that
We
assume
exponential discounting and a
In the separable case, this gives constant
flat yield curve.
relative risk aversion utility, with a relative risk aversion of 7 and an intertemporal rate of
au
t "'
)' T }l"' In the standard-of-living effect case, both risk aversion
^
substitution of ^fr
=
:
(f
•
and intertemporal substitution are complicated by the intertemporal
We
utility linkage.
We
calculate the utility gains from annuitization for a single, 65 year old male.
use
survival probabilities from the U.S. Social Security Administration for the cohort turning
age 65
We
in 1999,
modified (to ease computation) so that death occurs for sure by age 100.
use a real interest rate r of 0.03 and vary
100 in
all
cases.
We
find the
7 and
6.
We
normalize wealth at age 65 to be
consumption vector that solves the expenditure minimization
problem numerically using standard optimization techniques. 19
In Table
1,
we report on nine
simulations.
The
first
three simulations, in the top panel of
the table, report results for a consumer with the usual additively separable utility function.
The middle panel contains
function. In this case, the
three simulations for an individual with a standard-of-living utility
consumer
retires
with a stock of wealth that
is
20 times larger than
the standard-of-living to which they are accustomed at age 65. Specifically, the consumer
has a starting wealth of 100 and standard-of-living
(23)) equal to
The
s\
equal to
We
5.
set
a
(from equation
1.
last three simulations, in
the bottom panel, are also for a consumer with preferences
that depend on their standard-of-living. In this case, however, the stock of wealth
is
only
twice as large as the standard-of-living to which they are accustomed. Specifically, we set s\
equal to 50, while we continue to hold wealth at 100 and
a
equal to
1.
Within each panel, we examine three cases to show how results are affected by 7 and
The
first
(and thus
case in each panel
is
is
the discount rate
our "base case"
is
for
which
7=1
equal to the real interest rate).
(log utility)
We
third case returns 6 to
its
value of 1.03
for the separable utility cases,
we
and explores change
7 represents the
of
7
annuity receipts, but
may
We
—
to a value of
1.10
1.03
2.
-1
.
it
The
Note that
coefficient of relative risk aversion.
use the same values of 7 for the standard-of-living effect cases,
as the risk aversion coefficient.
=
then explore how results
change when the individual discounts the future more heavily by setting 5
-1
and 5
5.
-1
While
cannot be interpreted
assume that the consumer cannot borrow against future
save annuity payments in bonds with the interest rate of
For each of the nine simulations, we calculate four values. In the
first
.03.
column, we report
the equivalent variation (EV) for fully annuitizing in a constant real annuity. In other words,
19
Inspection of case two shows suboptimally increasing consumption in the last few years of
life.
The
solutions are approximations with only very small deviations from equalized marginal utility to price ratios
tolerated for years in which consumption
is
not equal to the real annuity.
24
the numbers in column (1) represent the increase in wealth required to hold utility constant
while moving
all
wealth from a constant real annuity to conventional bonds. In the second
column, we report the fraction of wealth that
bonds
if
is
optimally placed in the real annuity instead of
a continuous choice over annuitization levels
the equivalent variation associated with the optimal
in
column
Thus, column
(2).
permitted. In column
is
amount
(3) represents the increase in
(3),
we
report
of annuitization as reported
wealth required to hold
utility
constant while moving from having the optimal amount annuitized in a real annuity, to
of wealth in bonds.
The
having
all
in the
form of an equivalent variation)
final
column reports the gains from annuitization (again
to choose an optimal payout trajectory,
which the individual
for the case in
i.e.,
is
permitted
they are no longer constrained to purchase a
constant real annuity.
In addition to the four welfare measures presented in table
through
profiles for each of the nine cases in figures 1
consumption with
and types
different levels
9.
we graph the consumption
1,
Each graph
plots the optimal
of annuitization: the series plotted with circles
is
optimal consumption without annuitization; the series plotted in squares represents optimal
consumption with an equivalent
utility level,
but with 100 percent of wealth (100 units) put
into a constant real annuity; the series plotted in triangles represents optimal
consumption
with the same level of expenditures as in the complete annuitization case (rather than the
same
level of
consumption or
utility)
but with the consumer free to place an optimal fraction
of wealth in the constant real annuity
represents optimal consumption
when
and the remainder in bonds. The
all initial
A
which are allowed any desired time shape.
wealth (again 100 units)
series plotted in
is
xs
placed in annuities
rough estimate of the magnitude of EVs can be
obtained by observing the difference in trajectories between the circled consumption plan and
the other, annuitized consumption plans.
When
optimal consumption
is
sharply decreasing,
the constraints implied by (18) bind consumption away from the optimal path.
cases, the price benefit of annuitization
is
by the
largely offset
unconstrained (zero annuitization) consumption
is
constraints.
hump shaped and
the constraints impose less costs, so the net benefit to annuitization
Turning our attention to the
results,
we
see that the first case
In these
When
optimal
less steeply decreasing,
is
greater.
is
for a
consumer with
intertemporally separable preferences, log utility and a discount rate equal to the interest
rate.
For this individual, a constant real annuity provides an optimal consumption path.
Therefore,
all
Specifically,
be made
wealth
we
is
annuitized and the
find that the individual
as well off with
EV
is
the
for
columns
(1),
would require a 44 percent increase
no annuities as he would be
purchase a constant real annuity.
same
This result
25
is
if
permitted to use
(3)
and
(4).
in wealth to
his full
wealth to
very close to those found in the existing
literature, despite the truncation of the
constant real annuity
no benefit to
The second
now
optimal given actuarially
is
a
annuitization are
more
real annuity
much
This consumer would
heavily.
than to invest entirely
lower, with an
arises because the individual
is
Figure
1
demon-
consumption. Hence, there
fair pricing of
case considers a different discount factor of 1.10
of her wealth in
20
annuity payout trajectory.
flexibility in
discounts the future
present, but
lifespan at age 100.
from annuitization graphically, as the consumption path provided by the
strates the gains
is
maximum
EV of only
in
-1
still
,
such that the consumer
prefer to place 100 percent
bonds. However, the gains from
full
This decline in the value of the annuity
19.
would prefer to reallocate consumption from the future to the
essentially liquidity constrained
payments, as can be seen in figure
2.
Were
by the constant
real
nature of the annuity
the individual permitted to annuitize any amount,
he would optimally choose to place 72 percent of his assets in the real annuity and retain
28 percent in bonds.
column
as indicated in
he pursued this strategy, the consumer would have an
If
(3).
Column
(4)
shows the
EV
for
to choose any annuitized payout trajectory that he wishes.
complete annuitization
is
a consumer
We know
who
is
EV
of 19,
permitted
from Result
that
1
optimal when any consumption stream that can be purchased by
bonds can be mimicked by annuities. This number must be weakly greater than the
EV
associated with complete real annuitization, or equal in the knife-edge case where optimal
consumption
to place
is
constant with actuarially
of his wealth in
all
an annuity with a downward sloping payout trajectory and
would give him an even larger
The
final case in the
In this case, the consumer would choose
fair prices.
EV
this
of 24.
top panel shows the effect of increasing risk aversion from
1
to
2.
As
has been found elsewhere, this increases the value of annuitization. With a discount factor of
1.03
-1
,
the
EV
of complete annuitization rises to 56.
Complete
real annuitization is
optimal
for this individual.
The
three cases in the middle panel consider a standard-of-living effect case, where the
individual has a large
amount
of wealth to standard-of-living
of wealth relative to his standard-of-living. This large ratio
means that the endowment
more consumption per year than the consumer
middle panel to the upper panel
of annuitization
is
much
(i.e.,
This
greater.
20
the
used
to.
no standard-of-living
is
enough to sustain
Comparing the
effect),
we
results in the
see that the value
is
backloaded
in the first panel. For the case of log utility
interest rate,
EV
is
64 for a real annuity and 82 for an
For example, Brown, Mitchell and Poterba (2002) found that the
maximum
is
not surprising since consumption
compared with the additively separable cases
and a discount rate equal to the
is
of wealth
lifespan to run to age 115.
26
EV for this case was 0.50 when allowing
Even when the individual discounts the future more
optimally chosen payout trajectory.
case,
where the individual
wealth in a real annuity.
Consistent with the
highly, annuities are quite valuable, as indicated
would choose to place 99 percent of
case in which there
is
is
their
no standard-of-living
by the middle
we
effect,
increasing with the concavity of the utility function
and
because of the standard-of-living
arises
and determined by
Figures
7.
4,
5
show the consumption paths with and without annuitization. The hump
6 graphically
shape
see that the value of annuitization
them
stock of wealth that allows
to
consume
effect.
At retirement, the individual has a
in excess of their standard-of-living. Therefore,
the individual gradually increases consumption and raises their standard-of-living to a point
that
it
can be sustained given the wealth endowment. The fourth and sixth figures show
a considerable difference between optimal consumption with choice over annuity trajectory
and given the constant
real annuity;
hence we see a considerable benefit to
flexibility in
annuity payout in these cases.
In the
bottom
we explore
panel,
—
is
rapidly decreasing over time, as indicated
Such a consumption path
is
inconsistent with a constant real annuity
and
9.
as a result the standard-of-living effect
where 7=1 and 6
—
1.03
-1
,
now
reduces the value of the annuity. In the case
the value of the annuity
falls
from 44 percent of wealth without
the standard-of-living effect to 36 percent with a standard-of-living
discount rate, complete, mandatory real annuitization
EV
of only 3.
When
risk aversion increases to 2,
is
even
Even
in the latter case,
fraction of wealth
is
which
is
in
a real annuity, this
Mitchell and
and
Moore
Social Security
smoothing the
is
if
ratio
higher
providing an
a becomes an
even
utility.
a constant real annuity
the individual
if
is
is
the only form
permitted to annuitize 60 percent of
equivalent to a 27 percent increase in wealth. For perspective,
(2000) find that the
median household nearing retirement has pensions
making up 60 percent
holds have annuities that
simulations
is
With a
the worst case for annuitization analyzed here, a large
optimally annuitized even
of annuity available. In particular,
effect.
less attractive,
greater priority and complete real annuitization actually reduces
wealth
large
requires large initial consumption that
in figures 7, 8
and
is
smoothing the ratio of consumption to the standard-of-
relative to resources. In this setting,
living
the case in which the initial standard-of-living
make up a
of its retirement wealth. Thus, while
many
house-
substantial fraction of wealth, the implication of these
that preferences alone
may have a
difficult
time explaining the absence of
annuitization for households with substantial asset holdings.
27
Conclusions and Future Directions
5
With complete markets,
the result of complete annuitization survives the relaxation of
several standard, but restrictive assumptions.
Utility
need not satisfy the von Neumann-
Morgenstern axioms and need not be additively separable. Further, annuities must only
positive net premia over conventional assets; they need not be actuarially
Even with
fair.
a positive premium to annuitizing wealth
incomplete annuities markets, as long as there
is
and conventional markets
some
are complete, at least
offer
positive fraction of wealth
is
optimally
annuitized.
Even without bequest motives, we
find that a lack of complete insurance markets
can
render even a small amount of annuitization suboptimal. This suggests that an increase in
the use of other forms of insurance might encourage annuitization from a
This
is
interesting in light of the suggestion
by Warshawsky
et
al.
demand
perspective.
(forthcoming) that linking
annuities and long term care insurance might improve the problem of adverse selection in
both markets.
In the much-studied case of a world
find that there
However, even
may be
available. It
that
we
is
which render a constant
uncertain,
we
optimally annuitized even
would be interesting to consider
for
if
real annuity relatively unattractive, a
this
is
what fraction
the only form of annuitization
of the
and pensions amount to more than the lowest optimal
security
is
considerable individual heterogeneity in the value of annuitization.
for preferences
large fraction of wealth
where only individual mortality
American
elderly social
fraction of wealth (60
%)
find.
In our simulations,
we have retained the
abstractions of no bequest motive,
no
risks other
than longevity and no learning about health status or other liquidity concerns. Exploring
the consequences of dropping these assumptions in the context of non-separable preferences
and
unfair annuity pricing
would be an important generalization, but obtaining
require strong assumptions both
and
of bequest preferences
liquidity needs.
The near absence
life
on annuity returns and on the nature
results will
of voluntary annuitization
and the absence of annuitization early in
are puzzling in the face of theoretical results suggesting large benefits to annuitization.
Our
analysis extends the puzzle by demonstrating that annuitization of
all
financial assets
optimal more generally than previously thought. In general, incomplete annuity markets
is
may
render annuitization of a large fraction of wealth suboptimal; our simulation results show
that this
is
not the case for some special cases of preferences and
when annuity markets
are
incomplete only in that they impose a single payout trajectory across time.
It is
sometimes argued that the lack of annuity purchase
28
is
evidence for bequests. This
raises the question of
there
is
what
sort of bequest motive
no annuitization, then a bequest
PDV. Assuming one
is
would
random
in
sum
at a fixed time
purchases
is
both timing and
and annuitizing the
annuitization can reduce the variation in the bequest.
factors;
an absence of annuities.
cares about the risk aversion of recipients, this
giving the heirs a fixed
on load
call for
21
The
with a bequest motive, the load factor that
lower, because
we expect that
rest.
size,
If
measured as a
may
be dominated by
More
generally, partial
extent of dominance depends
is
sufficient to cut off annuity
sharing the outcome with someone else reduces
risk aversion.
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This
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31
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