With 2020 now (finally) in the rearview mirror, it’s a good idea for any investor to debrief the past year and figure out what worked and what went wrong.
Last year in particular came with a pretty unique set of challenges, including a massive market correction back in Q1 that stocks managed to rebound from in an equally historic way.
Looking back at the COVID crash from February 2020, it’s tempting to write that market shock off as an impossibly rare event – the kind that happens once a millennium or so.
And, sure enough, there were plenty of market commentators who said just that at the time:
“Did you know that last week's 14% plunge in the S&P 500 was so rare, by statistical measures, that it shouldn't happen once but every 14,000 years?” asked Zacks back in March. Others shared similar stats.
Only they're not true.
In fact, the crash was actually much less of an anomaly than those numbers suggest.
It’s true that the stock market selloff we saw back in the first quarter was the fastest 30% stock market decline ever, followed by the fastest recovery.
Depending on how you measure it and how big your lookback period is, the worst week during the selloff was something like a 5-sigma event.
That is, it was 5 standard deviations worse than the average weekly return for the S&P 500.
Whenever you see someone make a claim about the rarity of that sort of move, they’re making some assumptions about the underlying probability distribution. Sometimes those are good assumptions. More often, they’re bad.
If we assume that stock market returns follow a normal distribution, then a 5-sigma selloff is indeed exceedingly rare – the probability of seeing one on a given week is 0.000000114.
Sure enough, that means we should only expect to see something like that once or twice in human history.
Only, there are some lousy assumptions baked in there. First, there’s normality. It’s well known that stock market returns don’t follow a normal distribution. It’s practically a meme.
Would another probability distribution give us a more accurate way to evaluate things?
There’s been plenty of research proposing alternative distribution models for financial data, typically addressing the fat tails that lead to rare events happening much more frequently than expected.
(We’re also implicitly assuming that returns are i.i.d. here, which is far from true, particularly during volatile moves when serial correlation spikes.)
A better option yet is not to assume any distribution at all, and just decide whether a 5-sigma move is the kind of event we’d expect to see once in a lifetime or much more frequently.
Chebyshev’s inequality gives us an upper bound on how frequently we can expect a 5-sigma move in our data, irrespective of the actual distribution. Turns out, it’s 4%.
That’s dramatically more frequent than the guarantee the normal distribution gives us. At 4% we’d expect to see that sort of move once or twice a year!
Luckily for investors, Chebyshev provides a worst-case scenario, not necessarily an accurate one.
Unluckily for investors, there are more assumptions we’re making with Chebyshev that pose problems – namely that our returns are identically distributed from a distribution with finite expected value and non-zero variance. If those aren’t satisfied, our 4% worst-case scenario may actually be overly optimistic!
And in the real world, it’s likely that investment returns don’t come from a single monolithic distribution. Instead, a much more realistic model ascribes a different distribution to various market regimes (a mixture model). In that scenario, the big selloff we saw early in 2020 isn’t a 5-sigma move at all in the context of other volatile bear market plunges.
So, where does all that leave us?
After the jarring selloff that caught many investors and strategies off guard last year, it’s tempting to write off last year’s COVID crash as a statistical aberration that can't happen again.
But the reality is that markets aren’t statistically well-behaved enough to make that claim.
As the old Mark Twain quote goes, “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”
Likewise, it wouldn't even take a move quite so drastic to derail investor confidence this year.
With volatility at much cooler levels and the S&P 500 hovering around all-time highs heading into 2021, don’t make the mistake of thinking that another shakeout is a statistical impossibility. Rare and impossible aren’t the same thing, especially in equally historically unusual environments.