The Pitfalls of 1-3 Standard Deviation Risk Metrics
Introduction
"How often do 1 in a million events happen in the stock market?"
It's amazing what questions come to my mind at 1am. I came across a piece on LTCM, explaining that the moves seen in August 1998 were a "1 in 80 trillion event" based on their risk models. So were they unlucky? Or do extreme events happen far more often than we think?
Just to clarify, roughly speaking a 1 in a million year event in the stock market means mathematically a 6 Sigma Event (see note below). If you're not familiar with Normal Distributions and Standard Deviations (SD), I'd recommend reading this first. If you are familiar, let's begin.
Note:
6 Sigma = 1 in 506,797,346
Divide 506,797,346 by 252 (because there's 252 trading days in a year) = 2,011,101. So it's actually a 1 in 2 million year proxy, so we're actually being more conservative
S&P 500 Case Study:
Let's examine the most famous stock index - the S&P 500. Quite simple methodology:
Download daily values of the S&P 500 from StockCharts.com, going back to 1970 (so call it half a century of data)
Work out Daily Returns, 1 Standard Deviation etc (The Mean Return is 0.03%, 1SD came to 1.1%)
Create a table showing what we expected to see (assuming a normal distribution) and what we actually see.
So a 6 Sigma event (aka move greater than 6 Standard Deviations), is any day where the S&P 500 returned approx. +/- 6.6% (see no.2 above, 1.1% * 6).
Now you could argue that looking at positives is pointless, because they're positive and thus not a risk. However, for those reading this who are market makers or running long/short funds (particularly with leverage), such moves could blow you up (hence the Hurt Locker picture). So I feel it best to look at both the positive and negative moves combined, to truly gauge risk.
For those who just want "the stat" to deploy at their next meeting, here it is:
On average, how often do you get a 1 in a million event in the stock market? Every 2 Years.
Quite different to a normal distribution right? let's have a look at the full results (have done some rounding to make it clearer), and talk through the key points:
S&P 500 Daily Returns between 1970-2023
1. Most of the time, nothing happens
See the first row (> 1SD)? It says that 1 in 5 days, you'll see a move outside this. Or put another way, 4/5 days a week the stock market hardly moves. The image of the immortal day trader using 2 phones at once to place multiple trades a day, is quite frankly nonsense. There's simply nothing much happening most of the time (Unless you're trading Stocks in Play like SMB Capital)
2. The Extremes happen way more often
Now on the flip side, we expected to see a move greater than 6 Standard Deviations (aka 6 Sigma Event) every 1 million years. Though in reality, they're every 2 years on average. Let's take a closer look at 10 such moves (daily returns +/- 6.6%).
What you can see, is they're not actually happening every 2 years. Instead, they're clustered. So it would be wrong to assumes that between every FIFA World Cup you get 2 of these. In reality, markets are very quiet most of the time (see previous section), then all of a sudden BOOM (again, Hurt Locker picture). So whilst the 2 year average is mathematically correct (okay 2.4 I did some rounding), it is rather misleading.
3. This is not stock market exclusive - extreme moves happen in other asset classes too
I had a quick look at the CRB Index (Commodities proxy), and the U.S 10 Year Treasury (Fixed Income proxy). Turns out 1 in a million day events, also happen far more frequently in these asset classes too. A recent example is during my day job, where I have heard various fund managers talk about 6 Sigma events being witnessed in March 2023 (Silicon Valley Bank, the U.S 2 Year Yield dropping like a stone etc). So with that in mind, how can investors better deal with such moves?
Recommendations to Investors
I try and make my research accessible to all, though given my prior experience in Market Risk might use some foreign concepts to you (unless you work in the industry). So feel free to drop a comment or message me if there's anything that doesn't make sense.
1. Look beyond 3 Standard Deviations
In all my roles (Buy Side, Sell Side, CB Capital Management's Stocks and Shares ISA), I have never encountered any firm/platform/practitioner looking beyond 3 Standard Deviations in their risk metrics. And as I've pointed out, the extreme's happen far more often than we expect.
Based off experience, the go to risk metric in different domains is as follows:
Fund Factsheet's = 3 Year Annualised Standard Deviation (1SD)
Banks = 1 Day 95% VaR (call it a 2SD proxy for simplicity, I know it's 1.65SD)
Investment Management = 1 Month 99% VaR (call it a 3SD proxy for simplicity)
I know Expected Shortfall is coming into Banks which is definitely better, though such move would certainly be welcome in Investment Management.
2. Model Volatility Clustering better
As demonstrated above, what tends to happen is one extreme move followed by more in quick succession (autocorrelation in quant terms). With that in mind, why don't we try and put a number on these things? For example, a useful insight would be something like (this is made up by the way)
"After a greater than 4SD move, there is a 70% chance of another within the next 2 weeks"
Such insights would be really valuable to all parties. For example, if it becomes clear that the moment the first shell hits (e.g. a move > 4SD), multiple shell's will hit your portfolio in quick succession. Then you should be freeing up capital ASAP to prevent further losses, and allowing you to trade opportunistically. Rather than just being shelled into oblivion. That said, I do appreciate the challenges with liquidity, having overseen large funds and trading desks during extreme periods (such as March 2020).
3. Calm Down
This one is far more implementable than the other 2, which require some modelling (and not the sort at Milan Fashion Week). It's very easy to panic when losing money, especially if your retirement savings or prestigious Stocks and Shares ISA are at risk (you have no idea how seriously I take mine).
Though based on the above analysis, you can see that extreme moves happen every few years. So when they do, don't panic. The world will go back to normal again (albeit with larger government debt, a conversation for another time).
Conclusion
To summarise, extremes events happen far more often than most models predict, in many asset classes. These extreme moves are clustered, so rather than facing a 1 headed dog every few years, you'll face an 8 headed dog at fairly random intervals. So build better models and don't panic, or just slay the 8 headed dog.
Thanks as always for reading, hope you found it useful
Chris