# Mathtastic Advice on Sequence of Returns Risk

Folks we need to sit down and talk about sequence of returns risk. What is it, how can we measure it, and how it can bite you on the ass! So stay with me as we dive into this a bit more.

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# Calculating Sequence of Returns Risk

Sequence of return risk (SORR) is rarely mentioned in the mainstream finance and actuarial literature. I have never come across it in discussions with employers about their 401k plans, and to my knowledge it’s not discussed in any depth in the actuarial and CFA exams. But it might just be the most important factor standing between you and an early retirement spent sipping Pina Coladas on the beach and a life of penury.

### What is it?

On average a basketball game is played in in the center of the court, but if you put all your focus on that section you would miss all the important plays.

It’s the same with investments.

On average the market might go up 7% a year over decades, but this disguises the volatile nature of that path. The ride might be extremely bumpy and when you are spending-down your savings, you might not have the luxury of being able to wait for decades of recovery after experiencing significant market loss.

### Show me!

You’re right – we need some pictures to understand this thing! And lucky for you I have something prepared. Let’s look at US stocks from 1984 to 2016. I happen to have this data lying around, and as will become clear in a moment, the time period is not that important in what we are going to measure.

From the chart below you can see that if you had invested \$100 in US stocks in 1984 it would have growth to a whopping \$1,224.

Note that I have adjusted all my returns to strip-out inflation, and these returns are net of the rising cost of living – called “real returns”. The figures \$100 and \$1,244 are in current dollar terms.

But what if we change the sequence of these returns? What if we run the returns backward? We first apply the 2016 return to our \$100 then the 2015 return and so on… You get the following growth of \$100 shown below.

The blue line shows the growth of \$100 if we reverse the returns and the red line growth is as before. Note that both the blue and red lines start at \$100 but crucially they both end at \$1,244!

It doesn’t matter what order the returns are, the \$100 will grow to the same end point over the period. It’s kinda obvious if you think about it.

We can take this one step further and scramble the returns up even more than simply running them backwards. We can take the returns in any order we like. Suppose we take 1,000 different permutations of these returns from 1984 to 2016. We then end up with some crazy paths, but they all start and end at the same point. Look at the far right of the chart, all the colored paths come together at the same point (cool!).

[Incidentally how many different permutations do you think there are for 32 years? It’s actually far more than there are teaspoons of water in the Atlantic Ocean. Which is why I restricted myself to 1,000, since I didn’t want to blow up my laptop.]

### Ok – Very Nice, But What’s the Point?

Let’s recap. We’ve taken the real stock returns from 1984-2016 and scrambled them up in 1,000 different ways. All these 1,000 scenarios crucially provide exactly the same growth rate over this period. The only thing that is different about them is the sequence of returns. We’ve isolated the sequence of returns.

It’s like in Jurassic Park where they take a mosquito frozen in amber and isolate the DNA of the dinosaurs. We’ve taken one set of historical returns and from that we’ve been able to isolate the sequence of returns. We can now do experiments on them without the bother of worrying about whether ancient man-eating carnivores will rip us to shreds.

Note that we are not looking at simulating different historical periods. That has been done to death and there are some very good pieces of work out there (e.g. here) that look at what historical safe withdrawal rates could have been. Instead we are taking one period and re-simulating it with different sequences of returns (I know crazy huh?).

### How Big a Deal is SORR?

Now that we have isolated the sequence of returns we can measure the impact SORR has on withdrawals.

Instead of letting the \$100 grow unimpeded we will withdraw \$4 each year. Here we will simulate a simple 4% withdrawal strategy and examining the impact of SORR. We end up with the following simulations.

The first thing to notice is that at the end of the projection period the values are spread out. They no longer converge at one single point. As soon as you introduce cashflows SORR rears it’s ugly head!

The median of the end point is now \$640 and the 95th percentile is \$260. In other words:

• At the end of the projection, half the scenarios were above \$640 and half below this amount.
• At the end of the projection, 1 in 20 scenarios resulted in less than \$260 at the end of the projection, and 19 in 20 were more than that.

So this gives us an idea of the risk of SORR; the difference between having a lucky sequence of returns, and an unlucky sequence of returns is hundreds of dollars at the end of our analysis period. So I think we can safely say that SORR is a significant risk, but as an actuary I need to quantify it. So let’s press on….

### Quantifying SORR

What we’re going to do now is vary the withdrawal rate. in addition we will look at the difference between the median amount of money you have left at the end of the period, with the amount of money you have left in only 1 in 20 of scenarios (5th percentile). But note that I’m not really interested in the amount of money, since our \$100 starting point was arbitrary, I want to convert this into a difference in return.

You can think of SORR as being like a “return drag”. Remember that the total return over the whole period of all 1,000 scenarios is exactly the same, but as soon as you introduce spending over the period then if you are unlucky with a particular sequence of returns you will end up with less money. This is like a “drag” on your return.

Conversely, you could be really lucky with your sequence, in which case you get a return boost. But actuaries are pessimistic, so let’s stick with a drag.

[Technical pro’ note: I will calculate the IRR of the cashflow streams for the 50th and 5th percentiles for the 1,000 simulations.]

We end up with the following results. I have taken withdrawal rates from 0% to 5% going up in 0.5% increments. For each withdrawal rate I have found the return drag due to SORR.

### Give me an example!

For example you can see that for a 4% withdrawal rate the return drag is -2% per year. In other words if you are unlucky with your sequence of returns and end up at the 5th percentile this is equivalent to a drag on your return of 2% a year. That is a huge number!

Bonus observations: you can see that the return drag for a 0% withdrawal rate is zero. Remember, when we don’t have any cashflows coming in or out then all our paths converge to the same point, and so there is no return drag since SORR has no impact.

Also note that the drag on return accelerates as the withdrawal rate increases. As you increase your withdrawal rate you create more sensitivity to SORR. So a high withdrawal rate dramatically increases the impact SORR can have on your savings.

### What should I do?

This drag on return from SORR gives you a rule of thumb for your own calculations. If you are planning a 4% withdrawal rate then you should factor in a provision for the risk of SORR. Simply reduce all your asset projections by 2% a year and re-run your calculations. Things are going to look pretty ugly but that reflects the fact that SORR can be a significant problem if left unchecked. in fact I’m sure that if you run any FIRE numbers with a 2% headwind to your returns you are going to be pretty dismayed, and your goals might even look unobtainable.

So you really need a plan to cope with SORR. The main routes to investigate are:

• Look at alternative spend-down strategies. I’ve just looked at a naïve fixed percentage withdrawal rate, and you would be correct to point out that no sane investor would keep withdrawing the same amount when the markets go south. There is considerable advice around on different withdrawal strategies that we will not cover here.
• Increase your income with side hustles.
• Geo-arbitrage – live somewhere with a cheaper cost of living.

Remember SORR is a big risk with a big potential adverse impact on your solvency, but also remember things could go in your favor…

Does SORR keep you up at night? Are you going to build any provisions for SORR in your projections? What spend-down strategy will be using to mitigate this risk?

### 14 thoughts on “Mathtastic Advice on Sequence of Returns Risk”

1. I thoroughly enjoy your posts, especially your narrative style. Keep up the awesome work. I found your analysis of SORR to be very illustrative. Great idea to look at it the way you did. Best wishes on your path to FI.

1. Thanks dr mom! So kind of you to stop and say that. You are welcome back any time and help yourself to the free coffee!

2. A great post. Thank you for the analysis – it is both insightful and concerning. A couple of thoughts I would like your thoughts on 1) The use of a diversified portfolios between stock funds, bonds/gold and international funds would smooth the extremes out 2) The concept of turning a portion of your portfolio ( my thought is 33%) would provide some insurance.

Also regarding good posts on variable withdraw strategies, can you point us to some ?
THANK YOU

1. Sean, 1) yes I need to show how that would mitigate the impact to some extent.
Not sure I understand 2).
On variable withdrawal that is key. But I want to deep dive on that before coming back to you. That’s for commenting!

1. Sorry , what I meant by item 2) was the concept of using part of the invested capital to purchase a fixed income annuity to help cover the most basic of retirement needs. For example, one could possibly convert a \$700K lump sum into a \$40K per year income stream for the remainder of one’s life time. This might not be the most efficient use of capital but it would provide a level of comfort against the prospect of prolonged negative returns.

3. I agree that it is important to manage sequence of returns risk, especially for those that are planning to retire in this high valuation environment. I plan to simply target a 3% withdrawal rate and adjust up or down depending upon market performance.

1. sounds sensible to me. As long as you don’t feel that you are leaving money on the table. 3% might be a bit lean for me, and I feel that could be a little cautious. But I don’t know anything more about the future market direction than anyone else! Thanks for dropping by.

4. Is that Matlab? Sure looks like its chart formatting. I’m curious what method you used to run the simulations. Is there a built in function for scrambling all the returns?

Your last graph is awesome and very telling. This is a helpful planning tool!

1. Yes its matlab. Its the only way to handle all the data so quickly. There is a function to permits the elements of a vector.

5. Your data includes only the US market. A balanced portfolio since 1984 should have included foreign and emerging market returns, as well as some bonds.

1. Correct this post only has US stocks. The later post “Reprise! Sequence of Returns Risk, considers stocks and bonds.