Article from:
Risk Management
December 2011 – Issue 23
RISK RESPONSE
Spending Retirement on Planet Vulcan: The Impact of
Longevity Risk Aversion on Optimal Withdrawal Rates
By Moshe A. Milevsky and Huaxiong Huang
Copyright 2011, CFA Institute. Reproduced and republished from the Financial Analysts Journal, March/April 2011 with permission from CFA
Institute. All rights reserved.
RECOMMENDATIONS FROM THE MEDIA AND
FINANCIAL PLANNERS REGARDING RETIREMENT SPENDING RATES DEVIATE CONSIDERABLY FROM UTILITY MAXIMIZATION MODELS.
This study argues that wealth managers should advocate
dynamic spending in proportion to survival probabilities, adjusted up for exogenous pension income and
down for longevity risk aversion.
In our study, we
attempted to derive,
associate professor at the Schulich
analyze, and explain
School of Business and executive
the optimal retirement
director of the IFID Centre at York
spending policy for a
University in Toronto. He can be
utility-maximizing conreached at [email protected].
sumer facing (only) a
stochastic lifetime. We
deliberately ignored
financial market risk
by assuming that all
Huaxiong Huang, PhD, is
investment assets are
professor of mathematics at York
allocated to risk-free
University in Toronto. He can be
bonds (e.g., Treasury
reached at [email protected]
Inflation-Protected
Securities [TIPS]). We
made this simplifying
assumption in order to
focus attention on the
role of longevity risk aversion in determining optimal
consumption and spending rates during a retirement
period of stochastic length.
Moshe A. Milevsky, PhD, is an
By longevity risk aversion, we mean that different
people might have different attitudes toward the “fear”
of living longer than anticipated and possibly depleting
their financial resources. Some might respond to this
economic risk by spending less early on in retirement,
whereas others might be willing to take their chances
and enjoy a higher standard of living while they are still
able to do so.
Indeed, the impact of financial risk aversion on optimal
asset allocation has been the subject of many studies
and is intuitively well understood by practitioners. On
the one hand, investors who are particularly concerned
about losing money (i.e., risk averse) invest conservatively and thus sacrifice the potential upside, leading to
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a reduced lifetime standard of living. On the other hand,
financially risk-tolerant investors accept more risk in
their portfolios in exchange for the potential—never a
guarantee—of a higher standard of living in retirement.
But the impact of longevity risk aversion on retirement
spending behavior has not received as much attention,
and most practitioners are unfamiliar with the concept.
Although neither our framework nor our mathematical
solution is original—they can be traced back almost 80
years—we believe that the insights from a normative
life-cycle model (LCM) are worth emphasizing in the
current environment, which has grown jaded by economic models and their prescriptions. Our pedagogical
objective was to contrast the optimal (i.e., utilitymaximizing) retirement spending policy with popular
recommendations offered by the investment media and
financial planners.
The main results of our investigation are as follows:
Counseling retirees to set initial spending from investable wealth at a constant inflation-adjusted rate (e.g.,
the widely popular 4 percent rule) is consistent with
life-cycle consumption smoothing only under a very
limited set of implausible preference parameters—that
is, there is no universally optimal or safe retirement
spending rate. Rather, the optimal forward-looking
behavior in the face of personal longevity risk is to consume in proportion to survival probabilities—adjusted
upward for pension income and downward for longevity risk aversion—as opposed to blindly withdrawing
constant income for life.
HISTORY OF THE PROBLEM
The first problem I propose to tackle is this: How much
of its income should a nation save?
With those words, the 24-year-old Cambridge University
economist Frank R. Ramsey began a celebrated paper
published two years before his tragic death, in 1930.
The so-called Ramsey (1928) model and the resultant
Keynes–Ramsey rule, implicitly adopted by thousands
of economists in the last 80 years (including Fisher
1930; Modigliani and Brumberg 1954; Phelps 1962;
Yaari 1965; Modigliani 1986), form the foundation for
life-cycle utility optimization. They are also the workhorse supporting the original asset allocation models of
Samuelson (1969) and Merton (1971).
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In its basic form, the normative LCM assumes a rational individual who seeks to maximize the discounted
additive utility of consumption over his entire life.
Despite its macroeconomic origins, the Ramsey model
has been extended by scores of economists. Indeed, ask
a first-year graduate student in economics how a consumer should be “spending” over some deterministic
time horizon T, and she will most likely respond with
a Ramsey-type model that spreads human capital and
financial capital (i.e., total wealth) between time zero
and the terminal time, T.
The pertinent finance literature has advanced since
1928 and now falls under the rubric “portfolio choice”
or extensions of the Merton model. We counted more
than 50 scholarly articles on this topic published in
the top finance journals over the last decade alone.
Unfortunately, much of the financial planning community has ignored these dynamic optimization models,
and nowhere is this ignorance more evident than in the
world of “retirement income planning.”
Lamentably, the financial crisis, coupled with general skepticism toward financial models, has moved the
practice of personal finance even further away from a
dynamic optimization approach. In fact, many popular
and widely advocated strategies are at odds with the
prescriptions of financial economics. For examples of
how economists “think about” problems in personal
finance and how their thinking differs from conventional wisdom, see Bodie and Treussard (2007); Kotlikoff
(2008); Bodie, McLeavey, and Siegel (2008); Ayres and
Nalebuff (2010).
LITERATURE ON RETIREMENT
SPENDING RATES
Within the community of retirement income planners,
a frequently cited study is Bengen (1994), in which he
used historical equity and bond returns to search for the
highest allowable spending rate that would sustain a
portfolio for 30 years of retirement. Using a 50/50 equity/bond mix, Bengen settled on a rate between 4 percent
and 5 percent. In fact, this rate has become known as
the Bengen or 4 percent rule among retirement income
planners and has caught on like wildfire. The rule simply states that for every $100 in the retirement nest egg,
the retiree should withdraw $4 adjusted for inflation
each year—forever, or at least until the portfolio runs
dry or the retiree dies, whichever occurs first.
Indeed, it is hard to overestimate the influence of the
Bengen (1994) study and its embedded “rule” on the
community of retirement income planners. Other studies in the same vein include Cooley, Hubbard, and Walz
(1998), often referred to as the Trinity Study. In the last
two decades, these and related studies have been quoted
and cited thousands of times in the popular media (e.g.,
Money Magazine, USA Today, Wall Street Journal).2
The 4 percent spending rule now seems destined for the
same immortality enjoyed by other (unduly simplistic)
rules of thumb, such as “buy term and invest the difference” and dollar cost averaging. And although numer-
Along the same lines, we attempted to narrow the gap
between the advice of the financial planning community regarding retirement spending policies and the
“advice” of financial economists who use a rational
utility-maximizing model of consumer choice.1
In particular, we focused exclusively on the impact of
life span uncertainty—longevity risk—on the optimal
consumption and retirement spending policy. To isolate
the impact of longevity risk on optimal portfolio retirement withdrawal rates, we placed our deliberations on
Planet Vulcan, where investment returns are known
and unvarying, the inhabitants are rational and utilitymaximizing consumption smoothers, and only life
spans are random.
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ous authors have extended, refined, and recalibrated
these spending rules, the spirit of each rule remains
intact across all versions.3
We are not the first to point out that this “start by
spending x percent” strategy has no basis in economic theory. For example, Sharpe, Scott, and Watson
(2007) and Scott, Sharpe, and Watson (2008) raised
similar concerns and alluded to the need for a life-cycle
approach, but they never actually solved or calibrated
such a model. The goal of our study was to illustrate
the solution to the lifecycle problem and demonstrate
how longevity risk aversion—in contrast to financial
risk aversion, so familiar to financial analysts—affects
retirement spending rates.
Other researchers have recently teased out the implications of mortality and longevity risk for portfolio choice
and asset allocation (see, e.g., Bodie, Detemple, Ortuba,
and Walter 2004; Dybvig and Liu 2005; Babbel and
Merrill 2006; Chen, Ibbotson, Milevsky, and Zhu 2006;
Jiménez-Martín and Sánchez Martín 2007; Lachance,
forthcoming 2011). Likewise, Milevsky and Robinson
(2005) argued that retirement spending rates should be
reduced because the embedded equity risk premium
(ERP) assumption is too high. In our study, however,
we used an economic LCM approach to retirement
income planning.
NUMERICAL EXAMPLES AND CASES
The model we used is fully described in Appendix A so
that readers can select their own parameter values and
derive optimal values under any assumptions. Using
our equations, readers can obtain values quite easily
in Microsoft Excel. We selected one (plausible) set to
illustrate the main qualitative insights, which are rather
insensitive to assumed parameter values.
Note that we use the following terms (somewhat
loosely and interchangeably, depending on the context)
throughout the article: (1) The consumption rate is an
annualized dollar amount that includes withdrawals
from the portfolio, as well as pension income, and is
scaled to reflect an initial portfolio value of $100. The
retirement consumption rate is synonymous with the
retirement spending rate. (2) The portfolio withdrawal
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rate (PWR) is the annualized ratio of the amount withdrawn from the portfolio divided by the value of the
portfolio at that time. (3) The initial PWR is the annualized ratio of the initial amount withdrawn from the
portfolio divided by the initial value of the portfolio.
Recall that we are spending our retirement on Vulcan,
where only life spans are random. Our approach forced
us to specify a real (inflation-adjusted) investment
return. So, after carefully examining the real yield from
U.S. TIPS over the last 10 years on the basis of data
from the Fed, we found that the maximum real yield
over the period was 3.15 percent for the 10-year bond
and 4.24 percent for the 5-year bond. The average yield
was 1.95 percent and 1.50 percent, respectively.
The longer-maturity TIPS exhibited higher yields but
obviously entailed some duration risk. After much
deliberation, we decided to assume a real interest rate
of 2.5 percent for most of the numerical examples, even
though current (fall 2010) TIPS rates were substantially
lower. Readers can code the formulas in Appendix A—
with their favorite riskfree-return estimate—to obtain
their own rational spending rates. Our values are
consistent with the view expressed by Arnott (2004)
regarding the future of the lower ERP.
As for longevity risk, we exercised a great deal of modeling caution because it was the impetus for our investigation. We assumed that the retiree’s remaining lifetime
obeys a (unisex) biological law of mortality under
which the hazard rate increases exponentially over
time. This notion is known as the Gompertz assumption
in the actuarial literature, and we calibrated this model
to common pension annuitant mortality tables. (See
Appendix A for a full description of the mortality law.)
In most of our numerical examples, therefore, we
assumed an 86.6 percent probability that a 65-year-old
will survive to the age of 75, a 57.3 percent probability
of surviving to 85, a 36.9 percent probability of reaching 90, a 17.6 percent probability of reaching 95, and
a 5 percent probability of attaining 100. Again, note
that we do not plan for a life expectancy or an ad hoc
30-year retirement. Rather, we account for the entire
term structure of mortality.
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Our main objective was to focus attention on the impact
of risk aversion on the optimal PWR and especially the
initial PWR. Therefore, we display results for a range
of values—for example, for a retiree with a very low
( = 1) and a relatively high ( = 8) coefficient of relative risk aversion (CRRA).
To aid a clear understanding of mortality risk aversion, we offer the following analogy to classical asset
allocation models. An investor with a CRRA value of
( = 4) would invest 40 percent of her assets in an
equity portfolio and 60 percent in a bond portfolio,
assuming an equity risk premium of 5 percent and volatility of 18 percent. This analogy comes from the famed
Merton ratio. Our model does not have a risky asset and
does not require an ERP, but the idea is that the CRRA
can be mapped onto more easily understood risk attitudes. Along the same lines, the very low risk-aversion
value of ( = 1), which is often labeled the Bernoulli
utility specification, would lead to an equity allocation
of 150 percent, and a high risk-aversion value of ( = 8)
implies an equity allocation of 20 percent (all rounded
to the nearest 5 percent).
Finally, to complete the parameter values required for
our model, we assume that the subjective discount rate
, which is a proxy for personal impatience, is equal
to the risk-free rate (mostly 2.5 percent in our numerical
examples). To those familiar with the basic LCM without lifetime uncertainty, this assumption suggests that
the optimal consumption rates would be constant over
time in the absence of longevity risk considerations.
Again, our motivation for all these assumptions is to
tease out the impact of pure longevity risk aversion.
In the language of economics, when the subjective discount rate (SDR) in an LCM is set equal to the constant
and risk-free interest rate, a rational consumer will
spend his total (human plus financial) capital evenly
and in equal amounts over time. In other words, in a
model with no horizon uncertainty, consumption rates
and spending amounts are, in fact, constant, regardless
of the consumer’s elasticity of intertemporal substitution (EIS).4
The question is, what happens when lifetimes are stochastic?
Table 1. Optimal Rate (pre-$100)
under Medium Risk Aversion (γ=4)
Let us now take a look at some results. We will assume
a 65-year-old with a (standardized) $100 nest egg.
Initially, we allow for no pension annuity income,
and therefore, all consumption must be sourced to the
investment portfolio that is earning a deterministic
interest rate. On Planet Vulcan, financial wealth must
be depleted at the very end of the life cycle (say, age
120) and bequest motives are nonexistent. So, according to Equation A5 (see Appendix A), the optimal consumption rate at retirement age 65 is $4.605 when the
risk-aversion parameter is set to ( = 4) (see Table 1),
and the optimal consumption rate is $4.121 when the
risk-aversion parameter is set to ( = 8).
Note that these rates—perhaps surprisingly—are within
the range of numbers quoted by the popular press for
optimal portfolio withdrawal (spending) rates. Thus, at
first glance, these numbers seem to suggest that simple
4 percent rules of thumb are consistent with an LCM.
Unfortunately, the euphoria is short-lived. The numbers
(may) coincide only in the first year of withdrawals
(at age 65) and for a limited range of risk-aversion
coefficients (most importantly, no pension income). As
retirees age, they rationally consume less each year—in
proportion to their survival probability adjusted for risk
aversion. For example, at our baseline intermediate
level of risk aversion ( = 4), the optimal consumption
rate drops from $4.605 at age 65 to $4.544 at age 70,
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and then to $4.442 at age 75, $3.591 at 90, and $2.177
at 100, assuming the retiree is still alive. All these values are derived from Equation A5.
Note how a lower real interest rate (e.g., 0.5 percent
in Table 1) leads to a reduced optimal retirement consumption/spending rate. Indeed, in the yield curve and
TIPS environment of fall 2010, our model offered an
important message for Baby Boomers: Your parents’
retirement plans might not be sustainable anymore.
The first insight in our model is that a fully rational plan
is for retirees to spend less as they progress through
retirement. Life-cycle optimizers (i.e., “consumption
smoothers” on Vulcan) spend more at earlier ages and
reduce spending as they age, even if their SDR is equal
to the real interest rate in the economy.
Intuitively, they deal with longevity risk by setting
aside a financial reserve and by planning to reduce
consumption (if that risk materializes) in proportion to
their survival probability adjusted for risk aversion—all
without any pension annuity income.
As Irving Fisher (1930) observed in The Theory
of Interest,
The shortness of life thus tends powerfully to
increase the degree of impatience or rate of
time preference beyond what it would otherwise be. . . . Everyone at some time in his life
doubtless changes his degree of impatience for
income. . . . When he gets a little older . . . he
expects to die and he thinks: instead of piling
up for the remote future, why shouldn’t I enjoy
myself during the few years that remain? (pp.
85, 90)
Table 2. Initial PWR at age 65 with Pension Income,
as a Function of Risk Aversion
Including Pension Annuities.
Let us now use the same model to examine what happens when the retiree has access to a defined benefit
(DB) pension income annuity, which provides a guaranteed lifetime cash flow. In the United States, the
maximum benefit from Social Security, which is the
ultimate real pension annuity, is approximately $25,000
per annuitant. Let us examine the behavior of a retiree
with 100, 50, and 20 times this amount in her nest
egg—that is, $2,500,000, $1,250,000, and $500,000 in
investable retirement assets.
Alternatively, one can interpret Table 2 as displaying the optimal policy for four different retirees with
varying degrees of longevity risk aversion, each with
$1,000,000 in investable retirement assets. The first
retiree has no pension (
= $0), the second has an
annual pension of $10,000 ( = $1), the third has an
annual pension of $20,000 ( = $2), and the fourth has
a pension of $50,000 ( = $5).
Table 2 shows the net initial PWR (i.e., the optimal
amount withdrawn from the investment portfolio)
as a function of the risk-aversion values and preexisting pension income. Thus, for example, when the
( = 4) retiree (medium risk aversion) has $1,000,000
in investable assets and is entitled to a real lifetime pension of $50,000—which, in our language, is a scaled
nest egg of $100 and a pension ( = $5)—the optimal
total consumption rate is $10.551 in the first year. Of
that amount, $5.00 obviously comes from the pension
and $5.551 is withdrawn from the portfolio. The net
initial PWR is thus 5.551 percent.
In contrast, if the retiree has the same $1,000,000 in
assets but is entitled to only $10,000 in lifetime pension
income, the optimal total consumption rate is $5.873
per $100 of assets at age 65, of which $1.00 comes from
the pension and $4.873 is withdrawn from the portfolio.
Hence, the initial PWR is 4.873 percent. All these numbers are derived directly from Equation A5.
The main point of our study can be summarized in
one sentence: The optimal portfolio withdrawal rate
depends on longevity risk aversion and the level of
pre-existing pension income. The larger the amount of
the pre-existing pension income, the greater the optimal
consumption rate and the greater the PWR.
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The pension acts primarily as a buffer and allows the
retiree to consume more from discretionary wealth.
Even at high levels of longevity risk aversion, the
risk of living a long life does not “worry” retirees too
much because they have pension income to fall back
on should that chance (i.e., a long life) materialize. We
believe that this insight is absent from most of the popular media discussion (and practitioner implementation)
of optimal spending rates. If a potential client has substantial income from a DB pension or Social Security,
she can afford to withdraw more—percentagewise—
than her neighbor, who is relying entirely on his investment portfolio to finance his retirement income needs.
Table 3. Impact of Pensionization on Retirement Consumption Rates
Table 2 confirms a number of other important results.
Note that the optimal PWR—for a range of risk-aversion and pension income levels—is between 8 percent
and 4 percent, but only when the inflation-adjusted
interest rate is assumed to be a rather generous 2.5
percent. Adding another 100 bps to the investment
return assumption raises the initial PWR by 60–80 bps.
Reducing interest assumptions, however, will have the
opposite effect. Readers can input their own assumptions into Equation A5 to obtain suitable consumption/
spending rates.
The impact of longevity risk aversion can be described
in another way. If the remaining future lifetime has a
modal value of (m = 89.335) and a dispersion (volatility) value of (b = 9.5), then a consumer averse to longevity risk behaves (consumes) as if the modal value
were [m* = m + bln( )] but with the same dispersion
parameter, b.
Longevity risk aversion manifests itself by (essentially)
assuming that retirees will live longer than the biological/medical estimate. Only extremely risk-tolerant retirees ( = 1) behave as if their modal life spans were the
true (biological) modal value. Note that this behavior is
not risk neutrality, which would ignore longevity risk
altogether.
In the asset allocation literature, the closest analogy
to these risk-adjusted mortality rates is the concept of
risk-adjusted investment returns. A risk-averse investor
observes a 10 percent expected portfolio return and
adjusts it downward on the basis of the volatility of the
return and her risk aversion. If the (subjectively) adjust-
ed investment return is less than the risk-free rate, the
investor shuns the risky asset. Of course, this analogy is
not quite correct because retirees cannot shun longevity
risk, but the spirit is the same. The longevity probability
they see is not the longevity probability they feel.
Again, an important take-away is the impact of pension annuities on retirement consumption. Although
the point of our study was not to advocate for pension annuities or examine the market for longevity
protection—already well achieved in a recent book by
Sheshinski (2008), as well as the excellent collection of
studies by Brown, Mitchell, Poterba, and Warshawsky
(2001)—we present yet another way to use Equations
A5 and A6.
Table 3 reports the optimal consumption rate at various
ages, assuming that a fixed percentage of the retirement
nest egg is used to purchase a pension annuity (“pensionized”). The cost of each lifetime dollar of income
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Figure 1. Optimal Consumption: $5 Pension Income with
Investment
Rate =
2.5%
Op#mal Consump#on: $5 Pension Income with Investment Rate = 2.5% Results are reported for a retirement age of 65 and
planned consumption 15 years later (assuming the
retiree is still alive), at age 80. We illustrate with a
variety of scenarios in which 0 percent, 20 percent 40
percent, 60 percent, or 100 percent of initial wealth is
pensionized—that is, a nonreversible pension annuity
(priced by Equation A2) is purchased on the basis of the
going market rate.5
Figure #1 $14.00 CRRA = 1, Spend 8.01% @65 $13.00 CRRA = 2, Spend 6.55% @65 $12.00 CRRA = 4, Spend 5.55% @65 $11.00 CRRA = 8, Spend 4.84% @65 $10.00 $9.00 $8.00 $7.00 $6.00 $5.00 $4.00 65 Figure #2 68 70 73 75 78 80 83 85 88 90 93 95 98 100 Figure 2. Financial Capital: $5 Pension Income with
Investment Rate = 2.5%
Financial Capital: $5 Pension Income and Investment Rate= 2.5% $100.00 CRRA = 1, WDT = 24.6 years $90.00 In contrast, the retiree with a high degree of longevity
risk aversion ( = 8) will receive the same $1.261 from
the $20 that has been pensionized but will optimally
spend only $3.535 from the portfolio (a withdrawal
rate of 4.419 percent), for a total consumption rate of
$4.801 at age 65.
CRRA = 2, WDT = 29.6 years $80.00 CRRA = 4, WDT = 34.9 years $70.00 CRRA = 8, WDT = 40.5 years $60.00 $50.00 $40.00 is displayed in Equation A2, which is the expression
for the pension annuity factor. So, if 30 percent of $100
is pensionized, the corresponding value of F0 is $70 and
the resulting pension annuity income is
If the entire nest egg is pensionized at 65, leading to
$6.3303 of lifetime income, the consumption rate is
constant for life—and independent of risk aversion—
because there is no financial capital from which to draw
down any income. This example is yet another way
to illustrate the benefit of converting financial wealth
into a pension income flow. The $6.3303 of annual
consumption is the largest of all the consumption plans.
Thus, most financial economists are strong advocates of
pensionizing (or at least annuitizing) a portion of one’s
retirement nest egg.
We note that the pricing of pension (income) annuities
by private sector insurance companies usually involves
mortality rates that differ from population rates owing
to anti-selection concerns. This factor could be easily
incorporated by using different mortality parameters,
but we will keep things simple to illustrate the impact of
lifetime income on optimal total spending rates.
Visualizing the Results. Figure 1 depicts the optimal
consumption path from retirement to the maximum
length of life as a function of the retiree’s level of longevity risk aversion ( in our model). This figure provides yet another perspective on the rational approach
and attitude toward longevity risk management. It uses
Equation A5 to trace the entire consumption path, from
retirement at age 65 to age 100.
$30.00 $20.00 $10.00 $-­‐ 65 30 |
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Table 3 shows total dollar consumption rates, including
the corresponding pension annuity income. These rates
are not (only) the PWRs that are reported in percentages in Table 2. For example, if the retiree with medium
risk aversion allocates $20 (from the $100 available) to
purchase a pension annuity that pays $1.261 for life,
optimal consumption will be $1.261 + $3.997 = $5.258
at age 65. Note that the $3.997 withdrawn from the
remaining portfolio of $80 is equivalent to an initial
PWR of 4.996 percent.
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“Using our methodology, one can examine
the optimal reaction to financial shocks over the
retirement horizon.
“
Note that the optimal consumption rate declines with
age and in relation to the retiree’s attitude toward longevity risk as measured by the CRRA.
Figure 1 plots four cases that correspond to various
levels of the CRRA. Note that the consumption rate
eventually hits $5, which is the pension income flow.
For example, the consumer with a CRRA of 2 (i.e., very
low aversion to longevity risk) will start retirement by
withdrawing 6.55 percent from his nest egg plus his
pension income of $5. The withdrawals from the portfolio will continue until the retiree rationally exhausts
his wealth at age 95. From the wealth depletion time
(WDT) onward, he consumes only his pension.6
Figure 2 shows the corresponding trajectory for financial
capital. At all levels of longevity risk aversion, the curve
begins at $100 and then declines. The rate of decline is
higher and faster for lower levels of longevity risk aversion
because the retiree is “unafraid” of living to an advanced
age. She will deplete her wealth after 24.6 years (at age
90), after which she will live on her pension ($5).
In contrast, the retiree with a longevity risk aversion
of CRRA = 8 does not (plan to) deplete wealth until
age 105 and draws down wealth at a much slower
rate. When there is no pension annuity income at all,
the WDT is exactly at the end of the terminal horizon,
which is the last possible age on the mortality table. In
other words, wealth is never completely exhausted. This
result can also be seen from Equation A8, in which the
only way to obtain zero (on the right-hand side) is for
the survival probability to be zero, which can happen
only when equals the maximum length of life.
Reacting to Financial Shocks. Using our methodology, one can examine the optimal reaction to financial
shocks over the retirement horizon. Take someone who
experiences a 30 percent loss in his investment portfolio
and wants to rationally reduce spending to account for
the depleted nest egg. The rule of thumb suggesting that
retirees spend 4–5 percent says nothing about how to
update the rule in response to a shock to wealth.
The rational reaction to a financial shock at time s,
which results in a new (reduced) portfolio value, would
be to follow these steps:
1. Using Equation A8, recalibrate the model from time
zero but with the shocked level of wealth and compute the new WDT.
2. Use Equation A7 to compute the new level of initial
consumption, which will be different from the old
consumption level because of the financial shock.
3. Continue retirement consumption from time s onward
on the basis of Equation A5.
To understand how this approach would work in practice, let us begin with a (CRRA = 4) retiree who has
$100 in investable assets and is entitled to $2 of lifetime
pension income. With a real interest rate of r = 2.5 percent, the optimal policy is to consume a total of $7.078
at age 65 ($2 from the pension and $5.078 from the
portfolio) and adjust withdrawals downward over time
in proportion to the survival probability to the power of
the risk-aversion coefficient. The WDT is at age 105.
Under this dynamic policy, the expectation is that at age
70, the financial capital trajectory will be $86.668 and
total consumption will be $6.984 if the retiree follows
the optimal consumption path for the next five years.
Now let us assume that the retiree survives the next five
years and experiences a financial shock that reduces the
portfolio value from the expected $86.668 to $60 at age
70, which is 31 percent less than planned. In this case,
the optimal plan is to reduce consumption to $5.583,
which is obtained by solving the problem from the
beginning but with a starting age of 70. This result is a
reduction of approximately 20 percent compared with
the original plan.
Of course, this scenario is a bit of an apples-to-oranges
comparison because (1) a shock is not allowed in our
model and (2) the time zero consumption plan is based
on a conditional probability of survival that could
change on the basis of realized health status. The problem of stochastic versus hazard rates obviously takes us
far beyond the simple agenda of our study.
In sum, a rational response to an x percent drop in one’s
retirement portfolio is not to reduce consumption and
spending by the same x percent. Consumption smooth-
CONTINUED ON PAGE 32
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DECEMBER 2011
|
31
RISK RESPONSE
Spending Retirement on Planet Vulcan | from Page 31
Figure 3. Economic Tradeoffs at Retirement:
Stocks
Bonds
Pensions
ing in the LCM is about amortizing unexpected losses
and gains over the remaining lifetime horizon, adjusted
for survival probabilities.
SUMMARY
To a financial economist, the optimal retirement consumption rate, asset allocation (investments), and product allocation (insurance) are a complicated function of mortality expectations, economic forecasts,
and the trade-off between the preference for retirement sustainability and the desire to leave a financial
legacy (bequest motive). Although it is not an easy
problem to solve even under some very simplifying
assumptions, the qualitative trade-off can be illustrated
(see Figure 3).
Retirees can afford to spend more if they are willing to
leave a smaller financial legacy and risk early depletion
times. They should spend less if they desire a larger legacy and greater sustainability. Optimization of investments and insurance products occurs on this retirement
income frontier. Ergo, a simple rule that advises all
retirees to spend x percent of their nest egg adjusted up
or down in some ad hoc manner is akin to the broken
clock that tells time correctly only twice a day.
We are not the first authors—and will certainly not be
the last—to criticize the “spend x percent” approach to
retirement income planning. For example, as noted by
Scott, Sharpe, and Watson (2008),
32 |
DECEMBER 2011
| Risk management
The 4 percent rule and its variants finance a constant, non-volatile spending plan using a risky,
volatile investment strategy. Two of the rule’s inefficiencies—the price paid for funding its unspent
surpluses and the overpayments for its spending
distribution—apply to all retirees, independent of
their preferences. (p. 18)
Although we obviously concur, the focus of our study
was to illustrate what exactly a life-cycle model says
about optimal consumption rates. Our intention was to
contrast ad hoc recommendations with “advice” that a
financial economist might give to a utility-maximizing
consumer and see whether the two approaches have
any overlap and how much they differ. In particular,
we shined a light on aversion to longevity risk—uncertainty about the human life span—and examined how
this aversion affects optimal spending rates.7
Computationally, we solved an analytic LCM that was
calibrated to actuarial mortality rates (see Appendix A).
Our model can easily be used by anyone with access to
an Excel spreadsheet. Our main insights are as follows:
1. The optimal initial PWR, which the “planning literature” says should be an exogenous percentage of
(only) one’s retirement nest egg, critically depends on
both the consumer’s risk aversion—where risk concerns longevity and not just financial markets—and
any preexisting pension annuity income. For example, if the portfolio’s assumed annual real investment
return is 2.5 percent, the optimal initial PWR can be
as low as 3 percent for highly risk-averse retirees and
as high as 7 percent for those who are less risk averse.
The same approach applies to any pension annuity
income. The greater the amount of pre-existing pension income, the larger the initial PWR, all else being
equal. Of course, if one assumes a healthier retiree
and/or lower inflation-adjusted returns, the optimal
initial PWR is lower.
*
2. The optimal consumption rate ( ct )—which is the
total amount of money consumed by the retiree in
any given year, including all pension income—is a
declining function of age. In other words, retirees
(on Vulcan) should consume less at older ages. The
consumption rate for discretionary wealth is proportional to the survival probability
and is a func-
ON
N’SSE C O R N E R
C HRAI ISRKS PREERS SPO
“You might live a very long time, so you better make
sure to own a lot of stocks and equity. “
tion of risk aversion, even when the subjective rate
of time preferences
is equal to the interest rate.
The rational consumer—planning at age 65—is willing to sacrifice some income at 100 in exchange the
age of 100 the same preference weight as the age of
80 can be explained within an LCM only if the SDR
is a time-dependent function that exactly offsets
the declining survival probability. That people might
have such preferences is highly unrealistic.
3. The interaction between (longevity) risk aversion and
survival probability is quite important. In particular,
risk aversion tends to increase the effective probability of survival. So, imagine two retirees with the
same amount of initial retirement wealth and pension
income (and the same SDR) but with different levels
of risk aversion (γ). The retiree with greater risk aversion behaves as if her modal value of life were higher. Specifically, she behaves as if it were increased
by an amount proportional to ln(γ) and spends less
in anticipation of a longer life. Observers will never
know whether such retirees are averse to longevity
risk or simply believe they are much healthier than
the population.
4. The optimal trajectory of financial capital also
declines with age. Moreover, for retirees with preexisting pension income, spending down wealth
by some advanced age, and thereafter living exclusively on pension income, is rational. The WDT
can be at age 90—or even 80 if the pension income
is sufficiently large. Greater (longevity) risk aversion, which is associated with lower consumption,
induces greater financial capital at all ages. Planning
to deplete wealth by some advanced age is neither
wrong nor irrational.8
5. The rational reaction to portfolio shocks (i.e., losses)
is nonlinear and dependent on when the shock occurs
and the amount of pre-existing pension income. One
does not reduce portfolio withdrawals by the exact
amount of a financial shock unless the risk aversion is (γ = 1), known as the Bernoulli utility. For
example, if the portfolio suffers an unexpected loss
of 30 percent, the retiree might reduce consumption
by only 30 percentage points.
6. Converting some of the initial nest egg into a stream
of lifetime income increases consumption at all ages
regardless of the cost of the pension annuity. Even
when interest rates are low and the cost of $1 of
lifetime income is (relatively) high, the net effect
is that pensionization increases consumption. Note
that we are careful to distinguish between real-world
pension annuities—in which the buyer hands over a
nonrefundable sum in exchange for a constant real
stream—and tontine annuities, which are the foundation of most economic models but are completely
unavailable in the marketplace.
7. Although not pursued in the numerical examples,
one result that follows from our analysis is counterintuitive and perhaps even controversial: Borrowing
against pension income might be optimal at advanced
ages. For retirees with relatively large pre-existing
(DB) pension income, preconsuming and enjoying
their pensions while they are still able to do so might
make sense. The lower the longevity risk aversion,
the more optimal this path becomes.
The “cost” of our deriving a simple analytic expression—described by Equations A1–A8—is that we had
to assume a deterministic investment return. Although
we assumed a safe and conservative return for most of
CONTINUED ON PAGE 34
Risk management |
DECEMBER 2011
|
33
RISK RESPONSE
Spending Retirement on Planet Vulcan | from Page 33
our numerical examples, we essentially ignored the last
50 years of portfolio modeling theory. Recall, however,
that our goal was to shed light on the oft-quoted rules of
thumb and how they relate to longevity risk, as opposed
to developing a full-scale dynamic optimization model.
CONCLUSION: BACK TO PLANET
EARTH
How might a full stochastic model—with possible
shocks to health and their related expenses— change
optimal consumption policies? Assuming agreement on
a reasonable model and parameters for long-term portfolio returns, the risk-averse retiree would be exposed
to the risk of a negative (early) shock and would plan
for this risk by consuming less. With a full menu of
investment assets and products available, however, the
retiree would be free to optimize around pension annuities and other downside-protected products, in addition to long-term-care insurance and other retirement
products. In other words, even the formulation of the
problem itself becomes much more complex.
More importantly, the optimal allocation depends on
the retiree’s preference for personal consumption versus bequest, as illustrated in Figure 3. A product and
asset allocation suitable for a consumer with no bequest
or legacy motives— those in the lower left-hand corner
of the figure—is quite different from the optimal portfolio for someone with strong legacy preferences. In
our study, we assumed that the retiree’s objective is to
maximize utility of lifetime consumption without any
consideration for the value of bequest or legacy.
Although some have argued that a behavioral explanation is needed to rationalize the desire for a constant
consumption pattern in retirement, we note that very
high longevity risk aversion leads to relatively constant
spending rates and might “explain” these fixed rules.
In other words, we do not need a behavioral model to
justify constant 4 percent spending. Extreme risk aversion does that for us.
That said, we believe that another important take-away
from our study is that offering the following advice to
retirees is internally inconsistent: “You might live a
very long time, so you better make sure to own a lot of
34 |
DECEMBER 2011
| Risk management
stocks and equity.” The first part of the sentence implies
longevity risk aversion, while the second part is suitable
only for risk-tolerant retirees. Risk is risk.
To make this sort of statement more precise, we are
working on a follow-up study in which we derive the
optimal portfolio withdrawal rate for both pension and
tontine annuities in a robust capital market environment à la Richard (1975) and Merton (1971) but with
a model that breaks the reciprocal link between the
elasticity of intertemporal substitution and general risk
aversion. Another fruitful line of research would be to
explore the optimal time to retire in the context of a
mortalityonly LCM, which would take us far beyond
the current literature.9
One thing seems clear: Longevity risk aversion and
pension annuities remain very important factors to consider when giving advice regarding optimal portfolio
withdrawal rates. That is the main message of our study,
a message that does not change here on Planet Earth.
We thank Zvi Bodie, Larry Kotlikoff, Peng Chen,
François Gaddene, Mike Zwecher, David Macchia,
Barry Nalebuff, Glenn Harrison, Sherman Hanna,
We thank Zvi Bodie, Larry Kotlikoff, Peng Chen, François
and Bill Mike
Bengen—as
as Macchia,
participants
atNalebuff,
the 2010
Gaddene,
Zwecher,well
David
Barry
Retirement
Income
Industry
Association
conference
Glenn Harrison, Sherman Hanna, and Bill Bengen—as
in Chicago,
seminar
at Income
GeorgiaIndusState
well
as participants
at theparticipants
2010 Retirement
try
Association
in Chicago,
seminar particiUniversity,
andconference
participants
at the QMF2010
conferpants
at Georgia
State University,
and participants
the
ence in
Sydney—for
helpful comments.
We alsoatoffer
QMF2010
conference in Sydney—for
helpful comments.
a special acknowledgment
to our colleagues
at York
We
also offer a special acknowledgment
to our colleagues
University—Pauline
Shum, Tom Salisbury,
Nabil
at York University—Pauline Shum, Tom Salisbury, Nabil
Tahani, Chris Robinson, and David Promislow—
Tahani, Chris Robinson, and David Promislow—for helpfor discussions
helpful discussions
the of
many
years of
ful
during the during
many years
this research
this
research
program.
Finally,
we
thank
Alexandra
program. Finally, we thank Alexandra Macqueen and
Macqueen
Faisal
Habib
at the
QWeMA
Group
Faisal
Habiband
at the
QWeMA
Group
(Toronto)
for assistance
with for
editing
and analytics.
(Toronto)
assistance
with editing and analytics.
APPENDIX A. A.
LIFE-CYCLE
MODEL
IN
Appendix
Life-Cycle
Model
RETIREMENT
in
Retirement
The value function in the LCM during retirement
The
in the
during retirement
yearsvalue
whenfunction
labor income
is LCM
zero, assuming
no bequest
years when labor income is zero, assuming no
motive, can be written as follows:
bequest motive, can be written as follows:
max c V ( c ) = ∫0D e−ρt ( t p x ) u ( ct )dt ,
(A1)
where
x = the age of the retiree when the consumption/spending plan is formulated (e.g.,
60 or 65)
D = the maximum possible life span years in
retirement (the upper bound of the utility integration, which is currently 122 on
u(c) = c1 –
(longevit
from Ber
The
by aTx ( v
probabil
is define
aTx
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which is
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sion ann
or time T
not inclu
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rate towa
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Spending Retirement on Planet Vulcan
Spending
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onon
Planet
Vulcan
Spending
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on Planet
Planet
Vulcan
Spending
Retirement
Planet
Vulcan
Spending
Retirement
on
Vulcan
u(c)
= c1 1– –/(1
– ), where  is the coefficient of relative
 where  is the coefficientCof
u(c)
c1c–1=–=/(1
relative
RArelative
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SRK
ON
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c1 –– ),
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is the
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),
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which
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can take
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values
We thank Zvi Bodie, Larry Kotlikoff, Peng Chen, François (longevity)
risk
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can
take
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values
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risk
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take
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values
We
Zvi
Larry
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Chen,
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riskrisk
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which
cancan
taketake
on values
(longevity)
on values
from
Bernoulli
(aversion,
= 1) up
towhich
infinity.
We
thank
Zvi
Bodie,
Larry
Kotlikoff,
Peng
Chen,
François (longevity)
We thank
thank
Zvi Bodie,
Bodie,
Larry
Kotlikoff,
Peng
Chen,
François
Gaddene,
Mike
Zwecher,
David
Macchia,
Barry
Nalebuff,
We
thank
Zvi
Bodie,
Larry
Kotlikoff,
Peng
Chen,
François
from
Bernoulli
((= =1)(
up
toto
infinity.
from
Bernoulli
=up
1)
up
to
infinity.
Gaddene,
Mike
Zwecher,
David
Macchia,
Barry
Nalebuff,
from
Bernoulli
1)
infinity.
from
Bernoulli
(
=
1)
up
to
infinity.
Gaddene,
Mike
Zwecher,
David
Macchia,
Barry
Nalebuff,
The
actuarial
present
value
function,
denoted
Gaddene,
Zwecher,
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Macchia,
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Nalebuff,
GlennMike
Harrison,
Sherman
Hanna,
and
Bill
Bengen—as
Gaddene,
Mike
Zwecher,
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Macchia,
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Nalebuff,
The actuarial
present
valuevalue
function,
denoted
The
actuarial
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function,
denoted
TThe
Glenn
Harrison,
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and Bill
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andBengen—as
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The
actuarial
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depends
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+ ,t)/b
denotes
thethe
income
(in
real
dollars)
from
any
preexistrch
denotes
the
income
(in
dollars)
from
any
preexponentially
with
the
denotes
income
(in
real
dollars)
from
any
prehazard
rate(e.g.,
isthat
is
=pays
(1/b)e
grows
hazard
rate
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,spective
which
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plying
wealth
itself
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defined
bythe
sion
annuity
aage—m
real
amt)/b
year
until
death
Retirement
onwhich
Planet
Vulcan
t Spending
of
life
80
years),
and$1
b denotes
the
dispersion
t Spending
Retirement
on
Planet
Vulcan
is
possible
in
terms
of
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incomplete
gamma
funcplying
wealth
itself
is
defined
by
(1–γ),
where
γ
is
the
coefficient
of
relative
(longevity)
model,
one
could
include
it
by
tilting
the
survival
of
life
(e.g.,
80
years),
and
b
denotes
the
dispersion
plying
wealth
itself
is
defined
by
and
existing
pension
annuities,
andthe
thefunction
function
multiing
pension
annuities,
and
multiplying
of T,
lifewhichever
(e.g.,
80
years),
and
bthe
denotes
the
dispersion
existing
pension
annuities,
and
the
function
multiexponentially
with
age—m
denotes
the
modal
value
exponentially
with
age—m
the
modal
value
or
time
comes
first.
Although
we
do
coefficient
(e.g.,
10
years)
ofdenotes
future
lifetime
ran- retirement
The
value
function
inlifetime
the
LCM
during
tion
(A,B),
which
is available analytically:
r
,
F
≥
0
rate
toward
a
longer
life.
⎧
siscoefficient
(e.g.,
10
years)
of
the
future
ranrisk
aversion,
which
can
take
on
values
from
Bernoulli
t
plying
wealth
itself
is
defined
by
coefficient
(e.g.,
10
years)
of
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future
lifetime
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wealth
itself
is
defined
by
–variable.
/(1
wealthv (itself
by0F ≥ 0,
–mortality
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life
(e.g.,
years),
and
denotes
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not
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risk
from
the
perof
(e.g.,
80
years),
b coefficient
denotes
the
dispersion
dom
Both
numbers
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calibrated
to
U.S.
, ⎧ rF, t ≥
t , F⎧) is
=r defined
=life
c1 =
–/(1
),
and
isbpremium
the
of
relative
(A3a)
u(c)
c1a80
–where
),
where
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is
the
ofzero,
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years
when
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income
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assuming
dom
variable.
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numbers
calibrated
to
AU.S.
closed-form
representation
v ( t , no
Ftv)(=t , ⎨tFt ) =⎨⎩ ⎨Rof+ Equation
(γ
= 1)
up tables
to
infinity.
(A3a)
λt , Ft, t< 0A2,
dom
Both
numbers
are
calibrated
toranU.S.
spective
of variable.
the
insurance
company
in
this
valuation
coefficient
(e.g.,
10
years)
of
the
future
lifetime
rancoefficient
(e.g.,
10
years)
of
the
future
lifetime
x − (A3a)
m ⎞⎤
mortality
to
fit
advanced-age
survival
rates.
(longevity)
risk
aversion,
which
can
take
on
values
⎡
⎛
(longevity)
risk
aversion,
which
can
take
on
values
bequest
motive,
can
be
written
as
follows:
R
+
λ
,
F
<
0
,
≥
r
F
0
r
,
F
≥
0
⎧
R
+
λ
,
F
<
0
⎧
t
t
⎩
nçois
t
François model,
b
−
vb
,
exp
Γ
is
possible
in terms ofvthe
ttfuncmortality
tables
to include
fitnumbers
advanced-age
survival
rates.
t
mortality
tables
to
fitassumed
advanced-age
survival
⎜ (A3a)
⎟ ⎥)
one
could
it
by
tilting
the
survival
dom
variable.
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are
calibrated
to U.S.
dom
variable.
Both
numbers
are
calibrated
torates.
U.S.
t ,vFincomplete
=t ⎨) r.
, , ⎢⎣ function
tR
,F
= ⎨⎩Thegamma
(
)
(A3a)
(
In
our
study,
we
that
the
utility
funcwhere
discontinuous
v(t,F
b
from
Bernoulli
(
=
1)
up
to
infinity.
from
Bernoulli
(
=
1)
up
to
infinity.
t
⎝ t)v(t,F
⎠t⎦ )
buff,
lue function
in our
the
LCM
during
retirement
Dthe
t tion
−ρutility
Nalebuff,
(A,B),
which
is where
available
analytically:
R
λThe
+⎩(discontinuous
T
R
+,mλ,tF
,t) <=F0t < 0 function
In
study,
we
assumed
that
funcwhere
R  r.R The
v(t,F
In
our
study,
we
assumed
that
the
utility
func
r.
discontinuous
function
t
⎩
rate
toward
a
longer
life.
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actuarial
present
value
function,
denoted
by
a
v
,
b
(A1)
max
V
c
=
e
p
u
c
dt
,
mortality
tables
to
fit
advanced-age
survival
rates.
mortality
tables
to
fit
advanced-age
survival
rates.
(
)
(
)
(
)
t
∫
tion
of
consumption
exhibits
constant
elasticity
of
denotes
the
interest
rate
on
financial
capital
and
c
t
x
t
x
0
The
actuarial
present
value
function,
denoted
The
actuarial
present
value
function,
denoted
—as
when labor
income
is
zero,representation
assuming
noof
ngen—as
x and
− m ⎞⎤
tion A
of
consumption
exhibits
constant
elasticity
offunc-ofdenotes
theRinterest
rate
on
financial
capital
and
⎛
⎡
tion
of
consumption
exhibits
constant
elasticity
denotes
the
interest
rate
on
financial
capital
closed-form
Equation
A2
T
depends
implicitly
on
the
survival
probability
In
our
study,
we
assumed
that
the
utility
funcwhere
R
r.
The
discontinuous
function
v(t,F
)
T
In
our
study,
we
assumed
that
the
utility
where

r.
The
discontinuous
function
v(t,F
intertemporal
substitution,
which
is
synonymous
allows
F
to
be
negative.
For
credit
cards
and
other
where
R≥
exp function
− exp
t
( m − xcards
) v (t,F
by
depends
implicitly
the
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t)
xThe
− mdiscontinuous
t r.
where
depends
implicitly
on the
survivalallows
aby
va,xm(,vb, )m
, ,as
⎜ other
b )follows:
, substitution,
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⎛to
⎞ ⎤ For credit
dust) denotes
me
Indus- is
st
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intertemporal
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ison
synonymous
Fvb
be
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which
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be
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and
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t to
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−
,
exp
Γ
possible
in
terms
of
the
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gamma
func⎝
tion
of
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exhibits
constant
elasticity
of
denotes
the
interest
rate
on
financial
capital
and
⎜
⎟
tion
of
consumption
exhibits
constant
elasticity
of
denotes
interest
rate
on
financial
capital
and
with
(and
the
reciprocal
of)
constant
relative
risk
unsecured
lines
of
credit,
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)
=
R
+

.
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borcurve
(tpx)
via
the
parameters
(m,b).
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defined
and
probability
curve
(txpx)(t=
via
the
parameters
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t and allows
t
the
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financial
capital
Ft to be
probability
curve
pof)
via
the
parameters
(m,b).
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⎥⎦ credit,
age
of
the
retiree
when
b on
ticirent
partici- tion
x)the
with
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reciprocal
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lines
of
credit,
v(t,Fcredit
=R
. The
bor⎝rate
⎠of
(A2a)
with
(and
the
reciprocal
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constant
relative
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lines
)+=(to
R
+and
t.and
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borDintertemporal
t
−ρ
t)v(t,F
tcards
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which
isunder
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tcards
isthe
allows
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negative.
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ofsynonymous
perfect
rower
pays
R
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insurance
protect
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isfollowing:
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(A1)
ax
V (the
c ) =intertemporal
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is∫0defined
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using
tbe
isaversion
defined
and
by
using
the
following:
−
+the
x
mother
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negative.
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credit
cards
and
other
unsecured
lines
tion/spending
plan
(e.g.,
t the
c at
t pand
x )by
t )dtcomputed
nts
⎡
⎛
aversion
under
conditions
of
perfect
certainty
rower
pays
R
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the
(RRA)
under
conditions
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perfect
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rower
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no
x
−
m
−
b
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vb
,
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⎞
⎡
⎤
with
(and
the
reciprocal
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constant
risk
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lines
of
credit,
v(t,F
)
=
R
+

.
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borand
time-separable
utility.
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exact
specification
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lender
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rower
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⎡died
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or
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utilexp
m
−
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v
−
exp
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)
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b
⎣
⎠
⎦
longest-lived
= company
thecompany
SDR,inorthis
personal
time preference
spective
of the
valuation
The wealth trajectory
(financial capital during
spective
of insurance
the insurance
in
this
valuation
of the
Milevsky
(2006,
p. 61) for returns.
instructions
on A1, A2,
. See
−world’s
s in the basisinsurance
company
in this
valuation
one
could0 percent
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stochastic
Equations
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(which
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value
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x −tilting
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model,
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it byitin
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model,
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by
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⎞
⎡
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person,
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t, and the dynamic conhow to code the gamma function in Microsoft
tilexp ( m − x ) vas− high
exp ⎜ as 20 percent
⎟
in
some
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rate 1997)
toward
a longer
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straint in our model—linked to the objective functoward
a⎢⎣longer
⎝ b Excel.
⎠ ⎥⎦
on France in rate
CONTINUED ON PAGE 36
studies) of Equation
tion capital
in Equation
A
closed-form
representation
A2
A
closed-form
representation
of
Equation
A2
=
the
SDR,
or
personal
time
preference
The
wealth
trajectory
(financial
duringA1—can now be expressed as
ved
See Milevsky (2006, p. 61) for instructions on
follows:
p
=
the
conditional
probability
of
survival
is
possible
in
terms
of
the
incomplete
gamma
funcis
possible
in
terms
of
the
incomplete
gamma
funcranges
in value
to in retirement)
t x 0 percent
in (which
how
to code
the from
gamma
function
Microsoft is denoted by Ft, and the dynamic confrom analytically:
retirement
age xintoour
agemodel—linked
x + t, which to the objective
ment
tion
(A,B),
which
is available
irement
(A,B),
which
isempirical
available
analytically:
as high
astion
20 percent
in
some
Risk management | DECEMBER 2011 | 35
straint
 = v ( t , FfuncExcel.
(A3)
t
t ) Ft − ct + π0 ,
is based on antion
actuarial
mortality
table now be Fexpressed
nostudies)
ming
no
in
Equation
A1—can
as
nce
The wealth trajectory
(financial
capital
during
⎡ −of
⎛ x −⎛ mx −⎞(t⎤pmfollows:
⎡,−survival
⎞⎤
where the dot is shorthand notation for a derivative
parameterize
basis of the Gom= the conditional
bΓWe
bby
, ⎜exp
Γvb
t to
⎟ x)⎟on the
retirement) probability
is denoted
Fexp
,vb
and
the
con⎜ dynamic
Appendix
A. Life-Cycle
Model
Appendix
A. Life-Cycle
Model
Appendix
A.
Life-Cycle
Model
Appendix
A.A.
Life-Cycle
Model
Appendix
Life-Cycle
Model
in
Retirement
in Retirement
inRetirement
Retirement
in in
Retirement
Appendix A. Life-Cycle Model
in Retirement
pendix A. Life-Cycle Model
etirement
el
el
odel
0
0
can show that when R > , borrowing is not optimal
0
the W
life-contingent
pension
under
a shifted
stant
a pivotal
to the
model.
We
reiterate
denotes
theannuity
investable
assets
atand
retirement.
The
pension
income
is
consumed.
Leunginput
(1994,
The
optimal
financial
capita
ofannuity
follows:
The
Euler–Lagrange
theorem
from
th
c0* as
and  < D under certain
conditions.
For our
numersump
modal
value
ofvalid
m
+ only
bln()
and
aonly
shifted
valuation
that
Equation
A4
is
until
the
wealth
terminal
condition
is
F
=
0,
where

denotes
the
2007)
explored
the
existence
of
a
WDT
in
a
series
of
defined
as
until
time
t
<
),
w
lus
of
variations
leads
to
the
following.
The
ical results, we assume a high-enough valueof R.
π0 Equation
rate
ofcan
r.– But
(r
–atone
)/
instead
of
r. Plugging
⎛ a wealth
⎞ rt be expres
RISK RESPONSE
depletion
time,
can
always
force
wealth
depletion
time,
at
which
point
only
the
theoretical
papers.
In
theory,
the
WDT
be
the
tion
to
Equation
A4,
can
trajectory, Ft, in the
Ft =region
+ over
ewhich it is
The Euler–Lagrange theorem from the calcu⎜ W however,
into the
Equation
A4,
con- c
r ⎟⎠
depletion
time
income
< differential
Dassuming
by is
assuming
a follows:
minimal
pen⎝
annuity income
is D)
consumed.
Leung
(1994,
horizon
time
( =
if the A6
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abe
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)
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t
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(
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athe
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e
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sumption
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Salisbury
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A7
into
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A5
Salisbury
if
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lus zero
of variations
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the following.
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value
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factor
firms
that
the
solution
is
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and
over
the
values
of
the
function.
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the
defini56
www.cfapubs.org
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CFA
Institute
*
factor
is
m*
=
m
+
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mu
Alexandra Macqueen the
and
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stages.
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the
optimal
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permanently.
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term A5
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Ftt, in
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tion
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Mathematically,
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not that
include
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from
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pert
assuming
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)
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these
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areinpension
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und
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ics.
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spective
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ar shift
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ical
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them,
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rate
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*
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when
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)/
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tory
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A5
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the
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to
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e initial
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) of
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ain
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tionF
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sumption
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that Equation
A4
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isreiterate
valid
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be
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wealth
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one
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wealth
xthe
−the
m to
solution
to
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=
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itten as follows:
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= (r –  time,
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is
introduced
simplify
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)
In other words,
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function
in Equation
A1— utility—is
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−vbdepletion
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Γ second-order
life-cycle
⎟ ⎥τa <minimal
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c0* = life-cycle
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from
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A6
andPut
theanother
of (A7)
force
abwealth
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by assuming
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<
by
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geneous
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is⎠nonhomoimportant
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thus
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⎦a very
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τ utility—is
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thus
life-cycle
maximized
when
the
T
when
the
rate
and the w
−
,
*,
a
r
k
m
b
e
56
www.cfapubs.org
a
v
,
m
,
b
=
(
)
(A1)
p x ) u ( ct )dt ,
(
)
geneous
differential
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annuity,
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
when
the consumption
and
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τ =trajectory
f ( γ, π0 trajecW
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observable
from the
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to rate
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EquationsEquations
A5 and A6,
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course,
these
bthe⎠⎟ wealth
−πonly
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rktt valid
F⎢⎣ttt)=on
tt − ( ktt + r ) A4
00 ktt , until
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detailed
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asecond
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| DECEMBER 2011 | Risk
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them, which
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A6 and
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Note that when  = 1, 0 = 0, and  = r, Equation
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r
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arate:
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sumption
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and
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A7 collapses
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.
resulting nonlinear equation over the range (0,D) for
ax
solution
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A4 is
. differential Equation
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(A7)
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A7
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.
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Finally,
the
WDT,
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by
substituting
Equation
x = 1,Ft A5
Equation
into
Equation
the
if a WDT exists, then for consumption
to remain
tion
rate
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can
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to satisfy the
andsearching
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Finally,
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and
Chang
(1995)
and
Andersen,
Harrison,
Lau,
and
A7
into
Equation
A5
and
searching
the
resulting
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resulting
nonlinear
equation
over
the
range
(0,D)
for
smooth
at
that
point—which
is
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A7 collapses
to W/
.
ax
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A5
and
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Rutstrom (2008).
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the value
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the
value
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over
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to
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Mathematically,
satisfies
equation
1/ γthethe
outcome
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a very
important
(and
tionMathematically,
of life-cycle
theory—it
to 
(A8)
= π0 . implication
e τmust
p xconverge
(
)
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τ
ter
assumes
a
pool
in
which
survivors
inherit
the
rτ
r τb ) ethe
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τ = f ( γ, π0 W , ρ, r , x, m, b ) .
a xτ+( π
robservable
− /kr, )me*,
result)
from
the the
LCM.
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to of the deceased,
Mathematically,
WDT,
,
satisfies
equation
π
−
/
r
(W
whereas
the
former
requires
/
1
γ
0
0
ek τ ( τ p x ) = π0 . (A8)
rτ
Puta τanother
an insurance company or pension fund to guarantee
*,
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(Wx (+rπ−0k/,rm)way,
1/ γ
(A8)
the lifetime payments. See Huang, Milevsky, and
=
π
e
p
.
( )
www.cfapubs.org
©2011 CFA Institute
0 (A8a)
τ(56
τPut
= afanother
γ( ,rπ−0kW
,*,
ρ,br), exr, τm, b ) . τ x
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©2011 CFA Institute
τ = f ( γ, π0 W , ρ, r , x, m, b ) .
(A8a)
©2011 CFA Institute
©2011 CFA Institute
The optimal consumption policy (described by Equation A5) and the optimal trajectory of wealth (described
by Equation A6) are now available explicitly. Practically speaking, the WDT (τ ≤ D) is extracted from
Equation A8, and the initial consumption rate is then
obtained from Equation A7. Everything else follows.
These expressions can be coded in Excel in a few
minutes.
NOTES
1. Thus, our use of “Planet Vulcan” in the title of
our study, inspired by Thaler and Sunstein (2008),
who distinguished “humans” from perfectly rational
“econs,” much like the Star Trek character Spock,
who is from Vulcan.
2. See, for example, Walter Updegrave, “Retirement:
The 4 Percent Solution,” Money Magazine (16
August 2007): http:/ /money.cnn.com/2007/08/13/pf/
expert/expert.moneymag/ index.htm.
3. In fact, to some extent, Milevsky and Robinson
(2005) encouraged this approach by deriving and
publishing an analytic expression for the lifetime
ruin probability that assumes a constant consumption
spending rate.
between the two and their impact on optimal retirement planning in a stochastic versus deterministic
mortality model.
6. The consumption function is concave until the WDT,
at which point it is nondifferentiable and set equal to
the pension annuity income.
7. A (tongue-in-cheek) rule of thumb that could be substituted for the static 4 percent algorithm is to counsel
retirees to pick any initial spending rate between
2 percent and 5 percent but to reduce the actual
spending amount each year by the proportion of
their friends and acquaintances who have died. This
approach would roughly approximate the optimal
decline based on anticipated survival rates.
8. Thus, one could say that there are bag ladies on
Vulcan. 9. See Stock and Wise (1990) for an example of this burgeoning literature.
REFERENCES
Andersen, S., G.W. Harrison, M.I. Lau, and E.E. Rutstrom.
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Arnott, R.D. 2004. “Sustainable Spending in a LowerReturn World.” Financial Analysts Journal, vol. 60, no. 5
(September/ October):6–9.
Ayres, I., and B. Nalebuff. 2010. Lifecycle Investing: A New,
Safe, and Audacious Way to Improve the Performance of
Your Retirement Portfolio. New York: Basic Books.
Babbel, D.F., and C.B. Merrill. 2006. “Rational Decumulation.”
Working paper, Wharton Financial Institutions Center.
Bengen, W.P. 1994. “Determining Withdrawal Rates Using
Historical Data.” Journal of Financial Planning, vol. 7, no. 4
(October):171–181.
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Risk management |
DECEMBER 2011
|
37
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Bodie, Z., and J. Treussard. 2007. “Making Investment
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