Sequence of returns

The Sequence of Returns Score

You’ve heard of sequence of returns (SOR) right?

That fascinating phenomenon where in retirement you can experience good investment performance on average, but due to an unlucky sequence of those returns you go bust in retirement and end up living on cat food!

SOR can be ugly.

But we all know what it looks like right? Just steer clear of a big market crash when you retire.

 

But this leads me to so many questions – many, many questions on SOR. So sit tight as I try to answer them. 

Sequence of Returns

If you need a quick primer on sequence of returns then I’ve written a number of articles on this subject in the past. It’s one of my favorite topics, other than bashing investment managers. There are plenty of other resources including a paper from Wade Pfau in Advisor Perspectives that I will also refer to. 

All papers about SOR sadly shake their heads at the prospect that “a prolonged recessionary environment early in retirement could jeopardize the retirement prospects for a particular cohort of retirees” [Pfau 3/10/15] and some offer solutions such as spending less or diversifying and de-risking the investment strategy.

However I want more a definitive field-guide for how to spot a potential SOR event. Just because it may look and quack like a duck – is it really a duck?

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A Field-Guide to SOR

So… some obvious questions to ask are the following:

  • How bad does this recessionary event have to be to create a SOR problem?
  • Does it have to be a single one-year shock, or could you have a few years of middling-poor performance?
  • What if you have a big bounce-back after a big downturn. Will that be enough to mitigate the SOR impact?
  • Does it have to be a downturn in the year after you retire? What if it was two years after? Or three years, or four years after?
  • Given two sequences of returns is there an easy way to evaluate which one would produce the largest SOR risk? [This is actually my primary question, and I think the crux of this investigation]

I don’t want to see any more hand-waving arguments I want to see some genuine numerical rigor put around the answers to these questions.

And this quest has led me to the Sequence of Returns Score – “the SOR Score”.

The SOR Score

If you give me a sequence of investment returns I want to be able to quickly evaluate whether that particular sequence will adversely impact your retirement prospects simply from being a “bad” sequence of returns.

Let’s have a quick quiz to test our ability here.

 

The following table shows three ten-year sequences of returns; A, B and C. All these sequences have the same compounded return over the period – 5% a year. In other words the compound annual growth rates (CAGR) for A, B and C are all identical at 5% (that ain’t coincidence ya know – I’ve put some thought into this!)

Sequence of returns

I want you to put them in order of severity of SOR. Which would have the greatest adverse impact on your retirement pot of money, and which would be the most beneficial sequence of returns? And then as a follow-up question where would you place the constant sequence of 5% a year relative to A, B and C in terms of SOR impact?

It’s not easy is it? I think we can easily believe that B and C would have a worse SOR impact than A given there are more negative returns early on, but how would you position B and C? And would A have a more adverse SOR impact than the constant 5% a year sequence?

Answers later….

Why is This Important?

SOR can have a big impact on your retirement spending. Look at the following chart from Pfau’s paper. I’ll explain what it means. 

Sequence of returns

He plotted the historical safe withdrawal rates for a thirty year period in blue. (This is where the 4% rule of thumb is derived.) He then takes each 30 year period and calculates the CAGR and re-runs the safe-withdrawal calculations on thirty year periods with a constant return equal to the CAGR. This is plotted in green and it’s the safe withdrawal rate that would apply with constant returns.

The gap between the green and blue lines shows the size of the SOR impact on that cohort of retirees. When the blue line is above the green then SOR is beneficial (relative to the constant sequence of returns) – see the roaring twenties 1920-29 – and when the blue line dips below the green then SOR has a negative impact – see the Great Depression 1929-1939.

 

The gap roughly spans +/- 1.5% in safe withdrawal rates. That’s a big level of uncertainty and so SOR can have a very large impact. 

Back to the SOR Score

I think this is pretty interesting but what’s missing for me is a quick way to determine whether any particular sequence is good or bad for a retiree without having to resort to the long and detailed calculations run by Pfau. And then…. if I can find a quick recognition method  I might be able to get some insight into potential mitigation. 

So what I did was look at many sequences of returns and simply put a constraint on all the sequences that the CAGR was 5% a year. (5% is not important, it works for any CAGR). 

Instead of looking at the safe withdrawal rate (like Pfau) I simply looked at the final value of your pension pot at the end of the period after annual withdrawals. I then compared the ending account value with the account value resulting from the constant sequence of 5% every year. 

Let’s revisit the three example sequences from above.
Example

The three ending values are in the last column. 

  • Sequence A- You can immediately see that sequence ‘A’  was beneficial to the tune of +4% (relative to fixed 5% a year). Recall that +4% means that if the fixed 5% a year sequence results in a final amount of $1000 say, then sequence A resulted in $1040. This sequence of returns helps you. 
  • Sequence B – had you experienced sequence B you would have 9% less money at the end of the period that the constant sequence. That’s a big SOR ‘ouch!’. 
  • Sequence C – is neutral in the sense that it actually presents no SOR relative to constant returns of 5% a year. 

The result for A is perhaps not a surprise given the high returns in the early years, but it’s not clear (at least to me) that B should be so much worse than C. Also it’s not obvious that C has no sequence risk. 

"But how could we have determined these results more easily without having to do all those tiresome calculations? Gimmee a simple method!"
financial independence
Actuary on FIRE
Blog Founder and SOR Jockey
sequence of returns

Finally! The SOR Score

I looked at a whole load of sequences of returns with an identical CAGR of 5% and calculated

               [ 10x yr1 return + 9x yr2 return + 8x yr3 return + …. + 1x yr10 return] / 55

The way to think about this is that it’s like the arithmetic return over the period but giving a greater weighting to returns that are early in the retirement period. You can see that the first return after retirement gets a weighting of 10, the next year a weighting of 9 and so on. The total of 10+9+8+…+1 is 55. Hence the denominator of 55. This is for a ten year period. For example if you were to assess a 30 year period, then the first weight would be 30 and the denominator would be 465.

 

I know that this is a bit of an odd thing to calculate, but look what happens if we plot this SOR Score against the ending value. 

Sequence of returns

The SOR Score is directly correlated with the ending value. In other words the higher the SOR Score the more beneficial that particular sequence is, and the lower the SOR Score the more adverse that particular sequence of returns is.

 

Note that this example is for a 10 year period with sequences with a CAGR of 5%, but this isn’t important. The relationship above still holds irrespective of these variables.

Trade-Offs in the SOR Score

You can see from the scatter chart that the relationship is not exactly linear; for very beneficial sequences it over-states the impact, and for very adverse sequences it under-states the impact. However, the fit consistently has an R^2 of around 0.95 which is good enough for a rule of thumb.

What I like about the formula above is that it is easy to calculate and it’s very intuitive. You can quickly and easily calculate the SOR impact on any sequence of returns without running a load of tiresome projections.

 

Let’s revisit our three examples with the SOR Score. 

Examples

I’ve added the SOR Score in the final column. 

SOR Score

A SOR Score of 5% (the CAGR) should come from a sequence  that is pretty neutral and you can see that for C the SOR Score is close to 5%.

A SOR Score greater than 5% should be beneficial from a SOR perspective, and you can see that applies to A that has an ending value of +4%.

A SOR Score less than 5% should be adverse from a SOR perspective, and you can see that applies to B that has an ending value of -9%.

Just for fun let’s try another example. Take the following sequence.

Sequence of returns

Again this sequence has a total CAGR of 5% but starts with a strong return of 17%. However there are four sharp down-years and we should be suspicious that so much of the return is loaded in the final year. We should suspect that this sequence has a pretty adverse SOR impact, but it’s difficult to be sure since the initial year is not bad at all.

We can easily calculate the SOR Score:

10×17.0% + 9×1.0% – 8×9.0% + … + 1×90.6% / 55 = 0.8%

That’s a pretty low SOR Score since it’s quite a bit lower than 5%. Given this, we should expect a low end value relative to the constant sequence of 5% every year.

 

Here is the SOR Score and the End Value. 

Sequence of returns

The SOR Score is 0.8% that is low. It’s lower than example B above that had a SOR Score of 2.2%. And we see that the ending value for sequence D is -15%. That’s a bad sequence of returns and worse than example B that had an ending value of -9%.  

What Have We Achieved?

I wanted a way to evaluate a sequence of returns to see if it was a ‘good’ or a ‘bad’ sequence, and to evaluate how good or bad it was relative to the constant sequence.

I wanted that method to be simple and intuitive.

I also didn’t want to have to do all that projection stuff.

The SOR Score is simple to calculate and intuitive. It’s a weighted arithmetic return where a higher weighting is given to the most recent returns since you retired.

Stay tuned for a future episode where I look at the SOR Score in a historical context and what conclusions we can draw.

Author Bio: I started actuary on FIRE as I did not see any actuaries taking a prominent role in the personal finance area and wanted to remedy a shortage of actuary jokes and write for those that appreciate rigor with fancy charts. In my regular day job I advise corporate US on investment and retirement strategies. I’m a qualified actuary, investment adviser and have a PhD in mathematics and reserve the right to have the occasional math post. 

5 thoughts on “The Sequence of Returns Score”

  1. Very interesting analysis, and I think it does indeed serve to compare the SORs from one period to another or between hypothetical examples, as you demonstrate. The big question is whether or not this is actionable in any way.

    1. I knew someone would call me out for not being actionable! But you, my friend…?!
      You’re right – from this article it’s not at all apparent how this can be used in any meaningful way for a retiree or prospective retiree. I should have sign-posted some of the upcoming results I will have on that. But honestly, I haven’t completed the research yet. Too much Netflix to catch up on, etc…
      Thanks for reading, as always.

  2. My understanding of SOR is that the few years leading up to retirement are just as important as the few years after. However you might have more flexibility if it hits the fan 2 years prior to retiring rather then after.

    Any way to account for the preceding years?

    Very interesting stuff.
    Great Work!

    1. You’re right the behavior of the market prior to retirement has a big impact on the level of your spending in retirement. There is a strong connection between the level of the market (usually measured by the CAPE) and the subsequent safe withdrawal rate you can use. For more info see this post https://actuaryonfire.com/200-years-of-safe-withdrawal-rates-in-one-cool-chart/
      This post was quite an abstract analysis of sequence of returns in the spendown phase. In particular I wanted to figure out what is it that makes a particular sequence ‘bad’ and how to evaluate that.
      Thanks for dropping by!

  3. Why do you care about SOR? There is only really one reason because you are worried about running out of money before you run out of breath. The real thing you need to project is the likelihood of running out of money. I don’t know what Wade Pfau does except try to hawk annuities and reverse mortgages. I use a Monte Carlo calculator which projects the statistical likelihood of a portfolios success given a historical asset allocation, a period of retirement and a given WR and starting portfolio value. The calc I use is at portfolio visualizer and allows you to calculate “10,000 possible futures” and then orders them along a Gaussian curve from most likely to least likely above and below the mean likelihood. The calculator tells you % success for a given WR, length of retirement, starting portfolio size and asset mix. It allows you to stress test the SOR by front loading all the bad sequences into the beginning years, and allows you to adjust for inflation and withdrawal model.

    So you can model Ray Dalio’s All Weather portfolio using real assets (not asset classes) @ a 4% WR and a nominal SOR and historic inflation and realize a 99.44% projected statistical survival for you 4% WR. You can stress test 5 years of initial bad SOR (the first 5 years are the worst) and realize a 9,579/10,000 chance of success. You can parameterize inflation to 4% and see success drop to 8069/10.000 simulations, but even that portfolio tends to last 27 years. You can change to a Boggle Head 3 portfolio and compare to the Dalio All Weather and see a BH3 with historic inflation and regular SOR stress testing succeeds only 8223/10,000 simulations for the same WR and starting portfolio.

    The question was what can you do? What you can do is adjust the risk profile, and leverage on the portfolio by a different asset allocation that takes better account of the free money realized by non correlated asset diversity. SOR is greatly magnified in portfolios that are concentrated in one asset class. A BH3 has 80% stocks an extremely concentrated allocation. A Dalio has only 30% equities and so the negative effect of concentration is greatly reduced. When you buy an asset what you are buying is risk. What you are hoping for is return over time. Return is never guaranteed for your purchase, it is merely implied. What you actually own is the risk. In retirement the best plan IMHO is to effectively minimize your relative risk for a given portfolio size and WR by efficiently adjusting the asset allocation to one of least risk for a constant return. The best plan is to work until you can correctly allocate into such a relatively low risk portfolio with a high degree of probable success.

    What you can further do is use a 2 portfolio model. 1 large portfolio holds the market risk, and 1 small portfolio holds mostly bonds. In the “B” example simply use some money from the low risk portfolio to live on during the -13% downturn. This would neutralize the -13% loss’s effect because the main portfolio is closed to withdrawal. If you set -13% to 0 (because you withdrew no money) the SOR score becomes 4.344 instead of 2.2 nearly double and in the ball park of the 5% standard. If you got rid of the next -4% the SOR score increases to 4.925 nearly identical to 5%. This implies having some money to spend from a low risk bucket early in retirement eliminates a lot of SOR risk. SOR risk is asymmetrical. You don’t care if you die with too much money only if you die with not enough.

    You only need eliminate the SOR risk once, since the remaining sequence has a weighted average that gives a normal and sustainable return for the remainder of the portfolio’s life. This is not a typical “bucket strategy” where the bucket gets refilled so do not confuse it with that. It is a once and done strategy which changes the SOR of the portfolio to a different more favorable sequence You do NOT have to refill the small portfolio from the high risk portfolio. 1 year of low risk portfolio effectively immunizes against the SOR forcing you to run out of money. 2 years is even better. 3 years except in the worst SOR won’t make any difference to portfolio longevity because the risk is asymmetric. Once you get close to the constant return your portfolio will survive. The cost of this technique is 2 extra years of savings, which can be added as extra spending to your retirement after about half the length of retirement expires. I ran simulations on this technique against several recessions of the last century and it worked even in the 1973 recession. I don’t have Big Ern or AoF grade databases so my analysis is amateur, but using the data I could generate over the internet and using a 2 fund stock and bond low and high risk portfolios, this thing works.

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