where GammaDist(
α
,
β|
.
) denotes the cumulative
distribution function of the gamma distribution (in
Microsoft Excel notation) evaluated at the parame-
ter pair (
α
,
β
). The familiar
μ
and
σ
are the return
and volatility parameters from the investment
portfolio, and
λ
is the mortality rate. The expected
value of the SPV is (
μ
–
σ
2
+
λ
)
–1
.
For example, start with an investment
(endowment, nest egg) of $20 that is expected to
earn a 7 percent real return in any given year with
a volatility (standard deviation) of 20 percent a
year.
5
A 50-year-old (of any gender) with a median
future life span of 28.1 years intends to consume
$1 after inflation a year for the rest of his or her life.
If the median life span is 28.1 years, then by defi-
nition, the probability of survival for 28.1 years is
exactly 50 percent; so, the “implied mortality rate”
parameter is
λ
= ln(2)/28.1 = 0.0247. According to
Equation 8, the probability of retirement ruin,
which is the probability that the stochastic present
value of $1 consumption is greater than $20, is 26.8
percent. In the language of Figure 2, if we evaluate
the SPV at
w
= 20, the area to the right has a mass
of 0.268 units. The area to the left—the probability
of sustainability—has a mass of 0.732 units.
6
In the random life span of
λ
> 0, our result is
approximate, albeit correct to within two moments
of the true SPV density. We will show that this
issue is not significant. In the infinite horizon case
of
λ
= 0, our result is
not an
approximation
. It is a
theorem that the SPV defined by Equation 2b is, in
fact, the reciprocal gamma distributed.
7
Numerical Examples
Our base case is a newly retired 65-year-old who
has a nest egg of $1,000,000 which must last for the
remainder of this individual’s life. In addition to
pensions, the retiree wants $60,000 a year in real
dollars from this nest egg (which is $6 per $100 in
the terms commonly used in practice). The $60,000
is to be created via a systematic withdrawal plan
that sells off the required number of shares/units
each month in a reverse dollar-cost-average strat-
egy. These numbers are prior to any income taxes,
and our results are for pretax consumption needs;
in addition, we are not distinguishing between tax-
sheltered and taxable plans, which is a different
important issue.
The retiree wants to know whether the stochas-
tic present value of the desired $60,000 income a
year is
probabilistically less
than the initial nest egg
of $1,000,000. If it is, the retiree’s standard of living
is sustainable. If the SPV of the consumption plan
is larger than $1,000,000, however, the retirement
plan is unsustainable and the individual will be
“ruined” at some point, unless of course, he or she
reduces consumption.
Table 2
provides an extensive combination of
consumption/withdrawal rates for various ages
based on our model in Equation 8 and based on
exact mortality rates instead of the exponential
approximation. The rates assume an all-equity
portfolio with expected return of 7 percent and
volatility of 20 percent. The time variable is deter-
mined by the first columns in Table 2—retirement