This post picks up from a line of thought
started in “Variable Withdrawals in Retirement.” I’m going to present a
simplified scenario for two different withdrawal rates and explain about some
quantitative ways to answer the question: How do you decide which spending path
is more desirable?

The classic 4% withdrawal rate rule is
about

**constant inflation-adjusted withdrawal amounts**. The benefit of this approach is that it provides a smooth and predictable income stream for as long as wealth remains. But wealth can run out.
Another rule is to spend a

**constant percentage**of the remaining portfolio each year. This allows for spending to increase when wealth grows. As well, though spending amounts may drop to uncomfortably low levels, this spending rule prevents wealth from ever running out. The main disadvantages of using this rule are that the spending path is unpredictable and spending fluctuates widely over time.
The simplified scenario I will discuss consists
of one set of paths for the real income and real remaining wealth provided by
these two strategies. This is one simulation out of 10,000 or so that might
typically be part of a Monte Carlo simulation study on retirement income. I
will consider the case for a 65-year old female. In this simplified case, she
can see the evolving path from these two outcomes when she is ready to retire. She
wants to know which is better.

She retires at 65. A market boom in the
first year of retirement allows her spending to rise from the initial level of
5. Her wealth grew from a retirement date value of 100 to nearly 140 in the first
year. After year 8, her wealth begins to decline. Though the constant
inflation-adjusted amount stays the same at 5, the constant percentage spending
amount fluctuates quite a bit. In the 34

^{th}year of retirement (her 98^{th}year), she uses the last of her wealth with the constant amount strategy. Should she make it to 99, nothing is left. With the constant percentage strategy, she still has income through at least 105 (40 years after retiring) that is somewhere around 30% of the initial retirement value of 5.
The big uncertainty she faces is that she doesn’t
know how long she will live. She has to choose her income strategy in advance
without knowing this. Also, though the constant inflation-adjusted withdrawals
can result in no income at some point in the future, we can suppose that while
this is a very undesirable outcome, there is a safety net in place such that
she does not necessarily need to absolutely prevent this outcome. How can she
decide in a systematic way about which of these strategies to use?

**Failure Rates**

Traditional safe withdrawal rate studies
focus on failure rates, often over a fixed retirement duration. For the
constant amount strategy, failure is represented by wealth depletion. In this
case, if the retirement duration is 30 years, her failure rate is 0%, but with
a 40 year horizon her failure rate is 100%.
Trying to apply this same concept to the constant percentage strategy doesn’t
really work though, since the strategy never runs out of wealth. It fails in
the sense that spending is below the desired real amount of 5, but it succeeds
in the sense that it always provides some spending power and some remaining
wealth. Basically, we need to throw the failure rate concept out the window
when considering retirement income strategies with variable spending amounts.

**Average Lifetime Spending and Average Bequest Value**

Retirees may like a strategy that provides
the highest average lifetime income and the highest amount of remaining wealth
to leave as an inheritance. To calculate these, we need to incorporate
mortality data.

Average lifetime income is the average of
income in each year of retirement multiplied by the probability of surviving to
that age.

Average bequest is the wealth remaining in
each year of retirement multiplied by the probability of that being the final
year of the 65 year old’s life.

These survival and death age distribution
probabilities, derived from the Social Security Administration Periodic Life Table from 2007, are shown in Columns G (survival probabilities) and Column H
(death probabilities) in the spreadsheet
at the end of this post.

For the constant amounts strategy, average
income is the sum of Column C times Column G, divided by the sum of Column G.
It is 4.97. Though wealth runs out in year 34 of retirement, the probability of
the 65 year old female surviving to her 99

^{th}birthday is 3.9%. Earlier income receives a greater weight because it has a higher probability of being achieved. Likewise, for the constant percentage strategy, the average lifetime income is 5.23 (sum of Column C times G, divided by sum of Column G). The extra spending early in retirement accounts for a lot.
As for expected bequests, for constant
amounts, this is the sum of Column D times H, the wealth remaining in each year
times the probability that it ends up being the final bequest (i.e. the
probability of death). It is 73.97. As for constant percentages, it is the sum
of Column F times H. That is 69.75.

The constant amounts strategy has a
slightly lower average income, but a slightly higher bequest.

**Spending Shortfall Below a Desired Floor Level**

Average bequest may be helpful since it
gives some indication about the surplus provided by the strategy, but average
lifetime income may not be completely satisfactory.

A risk averse retiree may instead choose to
focus on how often and by how much spending power may fall under a minimally
acceptable flooring level. This floor may vary from person to person, and it
represents the spending level that you really hope to avoid broaching because
it may mean a serious reduction in your quality of life, such as a need to move
to a cheaper home or give up some important spending.

To demonstrate this, I will use an income
floor of 3. It is arbitrary, but our hypothetical retiree really hopes she can
avoid experiencing spending shortfalls below the level of 3. Surely, she would
like to be able to spend more than 3, and 3 is just her basic level of necessities.
With this measure, we focus on downside and forget about the upside. Over the
40 year period, the total amount of spending shortfall below the floor of 3
with constant amounts is -18. There are 6 years with income of 0, which
represents a shortfall of 3, and 6 x 3 = 18. The corresponding sum for the
constant percentage strategy is -20.83.
If we add up the shortfalls weighted by the probability of surviving to
the age of the shortfalls, we get a deviation with constant amounts of -.016,
and the deviation for constant percentages is -.1051. Constant percentages had more shortfalls
earlier in retirement (the first shortfall happened in the 24

^{th}year), and there are higher probabilities for experiencing these shortfalls.
While this measure tells us about the
downside, and in a sense it provides a modified form of failure rates that can
be applied to variable withdrawal strategies, it does ignore any upside
potential.

**The Value of Spending (a.k.a. utility maximization)**

A final measure I will look at provides an
attempt to translate spending amounts into something that may more closely
represent the value of that spending to the retiree.

A couple of issues will be at work here.
First, more spending is better than less, but the additional value provided by
more and more spending diminishes as spending increases. The additional value
of increasing spending from 1 to 2 is very high, because this is helping you to
fulfill same basic needs that vastly improve your quality of life. But the same
1 unit increase in spending from 9 to 10 may not provide much more value. That
high spending might mean living in a home with 1800 sq. ft. instead of 1600, or
it might mean 50 bottles of champagne per year instead of 30. The additional life-improving
value provided by spending will surely be less as spending grows.

Another important aspect is that I will
include the floor described in the previous section. Again, I will consider a
floor of 3 as a reference point. Spending may fall below 3, but the value
plummets for spending below 3. A spending drop from 4 to 3 is not nearly as bad
as a drop from 3 to 2.

To discuss the value of spending, I will
adopt the same basic function used by Joe Tomlinson in his February 2012

*Journal of Financial Planning*article, except I will reconfigure it to a new context. He calculated one value based on the lifetime outcome of how much wealth or total shortfall remained at death. I will translate each year of spending into a utility value and then add these up after multiplying them by the probability of surviving to that age.
In other words, this measure is similar to
average lifetime spending, except that I don’t use spending from each year
directly. Instead, I translate each year’s spending into a corresponding value based
on the relationships seen in the following figure for two different loss
aversion ratios (which measure the impact of how bad it is to fall under the
floor of 3).

A few things to note about this: First, it
will be hard to identify the precise shape of the function for real retirees.
The shape depends on several parameters including one which explains the
steepness of the curve, and also one which defines the appropriate floor level,
and the loss aversion ratio that defines how bad it is to fall below the floor.
More research is needed on this aspect. But we can still gain some insights by
looking at outcomes. Also, note that this is only about spending power, and
bequests are not incorporated into the calculations at all. But an important
aspect here is that the measure does incorporate both downside risk and upside
potential. Unfortunately, there is no connection between the value of spending
in successive time periods. Real people may grow accustomized to a particular
spending level, so that their frame of reference changes as spending changes.
More research can eventually incorporate this aspect, as well as the
possibility that the minimum acceptable floor could change over time. Finally,
please keep in mind that the numbers defining the utility value are abstract
and don’t have any direct meaning. What matters is the relative value of
different spending levels, and not their absolute value. We don’t have to worry
about utility being negative for levels below the income floor, it’s all
relative.

This being said, we can calculate the
“expected utility” of each spending path using this function which translates
spending into a corresponding value, and then weighs these values by the
probability of surviving to that age. With loss aversion of 2:1, the constant
amount strategy provides expected lifetime utility of 0.387, while the
corresponding value for the constant percentage strategy is 0.361. Again, these numbers by themselves have no
meaning. What matters is that 0.387>0.361 and the constant amount strategy
provides higher expected utility in this case. With loss aversion of 10:1, a
value function that more strongly punishes spending deviations below 3, the
constant amount strategy provides utility of 0.364, which is more than the
constant percentage’s strategy of 0.161.

As a final point about utility maximization,
Michael Kitces recently wrote:

The reality is that utility functions are not new to the analysis of economic and financial problems, but these research papers are some of the first instances that we have seen in trying to apply utility functions in the context of financial planning, and potentially represent a significant step forward in determining effective solutions for clients.

**Putting it all together**

For these two spending paths, we’ve
considered 8 ways to quantify which strategy provides a more satisfying
outcome. Failure rates were inconclusive, and the constant percentage strategy
provided a higher average lifetime income. For the other six measures, the
constant amount strategy performed better.

But understand that this is not the whole
story. This was only one simulation. It doesn’t provide a final answer. We need
to consider the outcomes for many more simulations, such as 10,000. Then we can
look at the distributions of these measures and think more systematically about
which strategy gives the best chance for the most satisfactory outcomes.

We can also include more asset allocation
and withdrawal rate choices, and more retirement income strategies such as
partial annuitization with SPIAs or GLWBs, strategies which use buckets for
different time horizons, the impacts of delaying annuitization, the use of bond
ladders, the decision about when to begin Social Security, and so on.

Please stay tuned and keep reading, because
all of this will be considered here in the coming weeks and months.