How to Maximize Wealth in Retirement

Imagine you’re recently retired and living off your savings and investments. You feel fairly comfortable that you have enough invested to live on for the rest of your life, but you would like to be able to pass on some inheritance to future generations as well.  You’re deciding between two portfolios:

• Portfolio A, which you expect to have 8% volatility and a real geometric return of 3%
• Portfolio B, which you expect to have 12% volatility and a real geometric return of 3.3%

Which portfolio will provide a higher growth rate, leaving more wealth for future generations?

You’re probably thinking, “I’ve read this blog enough to know the geometric return projects long term wealth. So since 3.3% is higher than 3%, I should expect to have more with portfolio B”. 

Let’s say you can see the future and know the above assumptions are true, is the answer definitive?

Well, the correct answer is probably not the obvious one, and it’s certainly not definitive. To understand why we first need to dive into what causes sequence risk. 

Sequence Risk

Sequence risk is a phenomenon where the order of returns greatly impacts future wealth. It is usually discussed in terms of withdrawals in retirement, as very bad investment returns at the beginning of retirement can be catastrophic for the retiree. Sequence risk can also exist in investing while saving, but that’s not the focus of this post.

Sequence risk arises from the ways that multiplication and addition/subtraction interact with each other.  Let’s see how different sequences lead to different results.

Here is a list of random numbers.  They have a standard deviation of 22 and an average of 0.¹ This will be our imaginary return stream through time.

Different Sequences

Now let’s shift the order around and randomly make different sequences with the same set of numbers. The original sequence is in the first row.

Addition

Ok, so we have our same set of numbers in different sequences. Do we find any differences if we start at 500 and then add those different returns up to create a total for each sequence?

Well, as we know from the commutative property of addition, nothing changes.  The end total is still the same in all of them.

The sequence doesn’t matter with addition.

Multiplication

Ok, let’s make our returns products now.  Here is the same set of numbers but instead of adding them, we’re going to multiply them . So where we saw 20 before, its now +20% (or as I like to think of it, X 1.2).

Now let’s take the same re-sequenced numbers as before, use them as products, and see where the different sequences end up.

Well, multiplication is commutative too, so the end total is all the same for each sequence.

Now the total is different with multiplication than addition since multiplication and addition have different properties, but the sequence doesn’t matter when the returns are multiplied.

There is no such thing as “sequence risk” in either of these two worlds. The order doesn’t effect the end result.

Let’s Add Withdrawals

Now let’s add withdrawals to the mix.  We’ll start with the first example using addition between each round and subtract 5 after each round as a consistent withdrawal.

Addition with Withdrawals

Notice again, still no difference in any of the end values.  There isn’t any “risk” in the sequence of the data as the final value in each path is always 400. This test was still just addition and subtraction, so this isn’t surprising.

Multiplication with Withdrawals

Now we try multiplication, subtracting $5 after each round.

Ah, there it is.  Now we have differences in all of them.  And you can see that row 21, with the downward moves clustered together in rounds 4 through 11 end up the worst² and row 22 with the upward moves early ends up the best (those two aren’t random I purposefully set them up to be the best and worst, the others sequences are random).

Essentially, sequence risk comes from mixing multiplication and addition/subtraction in the same string of numbers.  If you keep only to multiplication, or keep only to addition/subtraction, then the series is commutative, and the sequence doesn’t matter.³  But once you mix the two together, the order starts to matter deeply.

So it should be clear now that the answer to the original question can’t be definitive. If the higher returning portfolio has a bad sequence to it, it could easily perform worse than the lower returning portfolio. But there is still more to this story.

Impact to Portfolio Growth Rate

Now I know the above was obvious to some of you, but I wanted to make it very clear that sequence risk exists because of the mixture of multiplication and addition/subtraction in a series. When you mix multiplication and addition/subtraction strange things start happening.

Row 21 is a typical sequence risk example. Grouping the bad years up front and then taking withdrawals greatly reduces the overall growth of the portfolio. But look at the other sequences as well. Many returns seem low.

With no withdrawals, the compound growth of the multiplicative world for all paths is -2.46%. It’s negative because of volatility drag. A 5 withdrawal after each round is 1% of the total at the start (5/500= 0.01). At the end of multiplication with no withdrawal, we end up with 304, so a 5 withdrawal is about 1.6% or so at the end (5/304 = 0.016).

Therefore we’d expect the average growth rate of the portfolio with withdrawals to be between -2.46%-1% = -3.46%, and -2.46%-1.6% = -4.06% . Maybe we split the difference and expect numbers around -3.75%. However that’s not what we see in the actual returns.

The average compound growth rate is -4.24% in our sample. The median is -4.35%. Both well below the target and even below our “upper range” of -4.06%. Only 5 out of 20 of our random sequences are “better” than the-3.75% target. That’s not expected is it? Why do most portfolios seem to end up doing so poorly?

Well, the driver of sequence risk (mixing multiplication with subtraction) affects most of those portfolios, not just the extreme ones.

Withdrawals Hurt Compound Growth

This effect isn’t random. The average compound growth rate under withdrawals will always be lower than you would expect from simply combining the withdrawal rate and the growth rate of the portfolio. This happens because of the same math that creates volatility drag in the geometric return.

Sequence risk is amplified volatility drag.

The true wisdom in the geometric return: downward moves require a larger move up to return back to even. A 20% down move requires a 25% move back up to return to flat.  You need a markedly better gain to offset the loss. The same concept underlies sequence risk, but importantly it impacts most portfolios not just the unlucky portfolios.

If my portfolio falls 20%, and then I subtract another 10% (of my original wealth) for a withdrawal to cover my retirement expenses, my portfolio is down 30%.  So I would need 90%/70% = 28.6% to get back to where I would have been with out the negative return.⁴

But what if the bad return was only half as volatile–10%–and I then subtracted 10% for my living expenses?  That portfolio is down 80% and I’d need a 12.5% gain to get back to where it would be without any loss (90%/80%).

See the difference here?  Twice the loss (20% vs 10%) requires much more than twice the gain (28.6% to 12.5%) to get back to where the portfolio would be without any loss. 

Now let’s compare to a world without any withdrawals. With no withdrawal, the “back to even percentage” for the same two bad returns would be 25% and 11.11% respectively⁵. This gives us a 25% / 11.11% = 2.25 “back to even” ratio between the two drawdowns.  This ratio is kind of like a volatility drag co-efficient.  With a withdrawal, the ratios were 28.6% / 12.5% = 2.29.  So the withdrawal amplifies the same volatility drag effect, making it worse.

A drawdown — any drawdown — hurts a portfolio more when a withdrawal is taken on top of it.

When you are taking withdrawals from a portfolio, higher volatility strategies always have “a higher threat of sequence risk” because there is a higher probability of a deep pullback early on. But very importantly, higher volatility strategies will also likely suffer even higher volatility drag because of the withdrawals.

Amplifying Volatility Drag

Let’s dig into this relationship a bit more, first by evaluating what happens when we adjust the portfolio volatility.

Using the same data as above, I’ve scaled the returns to shrink the standard deviation, but leave the arithmetic return the same. The bottom row at 22% volatility is the numbers from above.

You can see that the base growth rate with no volatility is -1.11%. This is because of the withdrawal. As volatility grows, the amount of drag increases faster than we would expect from the volatility of the portfolio alone.

The withdrawals seem to be amplifying the volatility drag. The more volatility in the portfolio, the greater the effect. The higher the volatility, the more the withdrawal negatively effects the portfolio growth rate.

Combining Volatility and Withdrawal Rates

Now let’s look at a chart showing the combined effect of volatility and withdrawals on average portfolio growth rate. I’ve used the exact same data as above (excluding rows 21 and 22 as those were not randomly created). The 5 withdrawal and 22% volatility on this cart match the the data point in our earlier example above.

The column with zero withdrawal on the right shows the standard volatility drag that I’ve discussed many times before. More volatility, lower compound growth. Total drag here is 2.46% and the implied standard deviation is 22% just like the actual return’s standard deviation.

But notice how as the withdrawal rate grows, the size of the volatility drag goes up.⁶ The implied standard deviation climbs with it, rising above the true standard deviation of the returns. This change is not a linear increase either. At the same volatility, an increase in withdrawal amount causes the portfolio’s average growth rate to drop further and further.⁷

Plotting this chart in 3D shows how the surface twists, and returns fall off convexly as the volatility and withdrawals increase (click to rotate).

You can literally see the withdrawals amplifying the volatility drag as the chart curves downward.

Retirees May Benefit From Lower Volatility Portfolios

Now what does that mean for the question at the start of the blog? Which portfolio should provide the retiree with the most wealth to pass on to the next generation? The low volatility portfolio with slightly less return or the high volatility portfolio with slightly more return ?

The correct answer depends on the size of the withdrawals taken from the investments. But I hope you can see now that at a certain level of withdrawals, the lower volatility portfolio will perform better. The low volatility portfolio will suffer less amplified volatility drag than the high volatility portfolio. Therefore it might provide a higher compound growth rate and more wealth at the end to the retiree, even though the investment itself has a lower compound growth rate.

It’s a strange paradox.

Keep this in mind if you are retired and trying to plan out an investment process. A portfolio with lower volatility and lower return can potentially provide a retiree with more wealth than a portfolio with a higher return.

Now for the rest of us who aren’t retired quite yet, don’t worry we’re just at the beginning of this thought process. This revelation leads down a very interesting path on how to invest when a portfolio is in withdrawals or contributions, which we will tackle in the next post.

Footnotes:

1-The random generator used an average of 0, and a standard deviation of 20. I tweaked one number to return the true average back to zero to make the scaling later on in this post clearer.

2-Sequence risk for someone living off their investments is often thought of as being about avoiding the early market crash. The early drawdown is bad, but if you notice, the worse outcome (line 21, the second to last line which wasn’t random), goes up the first 3 periods and then the downward years are clustered together. So it’s clearly more complicated than just avoiding the drawdowns the first few years. You also want to avoid having the bad years cluster together later on.

3-I could have done another set of charts showing that withdrawals that are a percent of wealth do not produce any sequence risk when the investment returns are multiplicative. That case is entirely multiplication so the the end values are all the same leading to no sequence risk.

4-90% is the numerator because it’s where the portfolio would have been with no risky gamble and the subsequent withdrawal.

5-100%/80% = 25%. 100%/90% = 11.11%. The fact that 25%/11.11% here is greater than 2–the ratio of the losses–is the foundational concept of standard volatility drag.

6-I went back and forth on how to scale the returns. Keep the arithmetic return stable or the geometric return stable? Each has its own benefits in communicating this idea. I went with the arithmetic, but the best part of the scaling around the geometric was it kept the zero withdrawal column at constant zeros, so with a withdrawal, you can clearly see that higher volatility makes the compound growth of the portfolios worse.

7-I specifically stopped this chart where I did to ensure none of our sequences fell to zero. I wanted to show that this curvature effect didn’t come from the sequences going bankrupt. It is inherent in the nature of a portfolio with withdrawals. If you do push volatility or withdrawals out further to bankrupt some of the sequences, the chart shows even more convexity, and starts to look a bit like a waterfall.