# The Perfect Withdrawal Amount

In 2014 Suarez, Suarez, and Waltz wrote “The Perfect Withdrawal Amount” which introduces an interesting system. I mentioned it in passing in another post on “Perfect Withdrawal Amount and Modifying the Initial Rate”. Their core concept, the “perfect withdrawal amount” isn’t quite as nifty as they seem to think. It is basically the same as Blanchett et al’s “Sustainable Spending Rate” or gummy’s “Magic Number”: given a sequence of returns you can calculate what initial withdrawal rate would leave you with exactly $0 at the end.

In other words, if you have the returns +10%, 0, -10% then the perfect withdrawal amount — or sustainable spending rate, or magic number — is 35.48%. And this is how it would play out:

You end up with exactly $0 left.

Suarez et al make a generalisation of the technique that allows you to finish with any portfolio value, not just $0. This is handy if you want to try to deal with bequests for heirs. Maybe you want to leave $100,000 to your cats.

To be honest, the first part of the paper, where they are introducing all of this, left me underwhelmed. Their next step is more interesting though: they realise that this “perfect withdrawal amount” is a way of a translating a series of numbers into a single number. You take 30-years of returns, feed it into the algorithm, and out pops a single number: 0.070465. That gives us a building block.

If there were a way to generate lots of sequences, we’d have lots of numbers, and could build a probability distribution. We can combine Monte Carlo simulations of stock market returns with the “perfect withdrawal amount” calculation and then plot the results as a histogram.

Which is exactly what they do.

The way to read this histogram is:

- $70,465 is the 50th-percentile “perfect withdrawal amount”. If you were to retire and withdraw $70,465 (adjusting for inflation every year) then in 50% of all possible scenarios you will run out of money. (But in 50% of them you’ll never run out of money!)
- $65,000 is the 40th-percentile “perfect withdrawal amount”. So 40% of the time you’ll succeed and 60% of the time you’ll fail.
- And so on…

This is interesting because it gives us a bit more insight and transparency into the probabilities. Say that you really, really want to retire and withdraw 5.2% instead of the traditional 4%. Are you totally doomed? Not exactly, you still have an 80% chance of success.

The authors go on to point that, in the real world, no one blindly follows a plan that is going to leave them broke. So the same technique can be reapplied every single year. So it could look something like this:

- You retire and withdraw $52,000 — the “perfect withdrawal amount” analysis for a 30-year retirement tells you that you are in the 20th percentile. It’s not perfectly safe but you’re willing to take a little bit of a risk, especially while you are still relatively young.
- The market does really well, so the next year the “perfect withdrawal amount” analysis for a 29-year retirement says you are now in the 18th percentile. You don’t increase your spending but you feel a tad more confident you won’t need to cut your spending either.

You can use this to as a “warning indicator”, something that is sorely lacking from much of retirement spending research, which is a pretty nifty application.

The authors show that you can even use it to build a variable spending strategy. Say you set up some guardrails: you don’t want to be too safe or too risky: so if you drop below the 15th percentile (too safe) or above the 85th percentile (too risky) you reset your spending to the 50th percentile at that point in time.

In practice it might look something like this:

Or this

The authors conclusion is a bit surprising (all emphasis is mine):

But one that seems to take shape intuitively is a major departure from conventional wisdom: the

best level of failure risk is 50%. To be more precise, we should use the mode of the PWA distribution — because that is the most likely value that the PWA will end up taking in our particular case. For symmetric or moderately skewed distributions, failure risk at the mode will be close to 50%. If the assumptions used for the distribution of return rates are sound, thenclose to half of the retirees who follow this policy will be able to keep withdrawing this amount (or more!)for the entire retirement period. For the other half, the procedure outlined here will steer them clear of funds’ depletion or bequest collapse throughtimely warningsthat the amount needs to be decreased — and probably not by much. If ours is indeed the case when the sequence of returns awaiting in our future is a terrible one, then the procedure will keep signalling for downward adjustments, taking the withdrawal path asymptotically to a very low value, but probably not too far below what it would have been under perfect foresight.

I’m not totally convinced that nearly 50% of retirees would be willing to tolerate cuts in their annual withdrawals, even if the cuts are not large. I think the authors overestimate the risk tolerance of most people. Still, it is an interesting thought that, with the right timely warnings, we have the flexibility to be a bit more aggressive.

To me, that’s more palatable than dying with millions going to my heirs.

## Problems

The authors introduce a new tool but I’m not convinced it is actually that novel or useful.

First, it is based on Monte Carlo analysis to generate the probability distributions. That means you are at the mercy of the Monte Carlo engine. One of the perennial problems is how much the future will resemble the past. There seems to be a growing consensus that — at least for the next decade or two — assuming historical US growth with be unwise.

It also means that, unless you’re using a pretty sophisticated Monte Carlo engine, the simulated portfolio has only a passing resemblance to your actual portfolio. For instance, in the simulation I might use a portfolio that is 70/30–70% S&P 500 and 30% intermediate Treasuries. My actual portfolio is 70% stocks…but not 70% S&P 500. It has some international and some small-cap value. And my 30% bonds aren’t all intermediate duration and aren’t all Treasuries.

When you realise that a Monte Carlo analysis is underpinning the whole thing…there are already quite a few things out there showing you how to use Monte Carlo analysis to dynamically adjust a portfolio over time.

In 2008 (i.e. 6 years before the Perfect Withdrawal Amount paper) Spitzer published “Retirement Withdrawals: An Analysis on the Benefits of Periodic ‘Midcourse’ Adjustments”. In it he performs a very similar analysis — though he doesn’t need to generate “perfect withdrawal amounts”; he just uses traditional “risk of ruin”. But you’re able to build up essentially the same kinds of tables, albeit with a bit more computational effort.

You can use this in a very similar way — pick your risk-tolerance and the remaining length of retirement and use that as your withdrawal amount. Say you only want a 5% chance of failure and have 15 years of retirement left: you can withdraw 6.6% of your portfolio.

Ultimately, I struggle to see either of these approaches as being an improvement over a PMT-based scheme. With a PMT-based scheme I don’t need to rely on historical data and its applicability to the unknown future.

That said, I think the “perfect withdrawal amount” does provide an interesting risk metric, even if you’re not using it to drive your withdrawals. Using it, I can see that withdrawing $75,000 would put me in the 13th percentile (that is, only a 13% chance of needing to ever lower my withdrawals).

The “perfect withdrawal amount” lends itself to being put in a table so I’ve generated lots of tables if you’re curious.

- I used 6 different inputs to the Monte Carlo simulator, so you can get a feel for how things vary.
- I generated the “perfect withdrawal amount” for 10- to 60-year retirements. So you can see how things would change every year.
- I show the “perfect withdrawal amount” for every decile from 10th to 90th. You can see how different risk preferences look.

Here’s a snippet of what a “conservative” 50/50 portfolio looks like. (Where “conservative” means adjusted down the historical numbers somewhat.)

Say you wanted to variable spending strategy where you adjust things every year. And you want something in the 30th percentile. So there’s only a 1-in-3 chance that you’ll need to lower spending.

When you are 65 (and have 30 years of retirement left), you would withdraw 4.62% of your portfolio. The next year, when you have 29 years of retirement left, you would withdraw 4.74% of your portfolio.

As I said, I’m not convinced this is an improvement over just using PMT but it is fun to play with the numbers. It feels like there the “perfect withdrawal amount” offers a potential building block that hasn’t been found yet.

The authors themselves seem to realise this since they close their paper with

The obvious next step for the framework’s future development is to use it to derive new “standard” rules — à la Bengen’s 4% — for different portfolio compositions and retirement objectives, and to look at the rules currently in favor through the lens of PWA.