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!)
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?
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.
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.
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.
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.
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.
I’ve added the SOR Score in the final column.
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.
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.
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.