# Extreme Return

What makes a stock return "extreme"? The financial literature is littered with different definitions for abnormally large price shocks. Unfortunately, most are based on measures of standard deviation (e.g. any return of more than 3 standard deviations from the mean). Besides being computationally intensive (since they require a rolling measure of volatility to scale one that needs to be updated every day), these relative methods are deeply flawed because they wrongly assume returns follow a normal distribution. It is well established that stock market returns do not coalesce into a neat bell shape when plotted on a graph. For example, extreme returns (greater than 3 standard deviations, which work out to about +/- 2.92% in percentage terms) on the S&P 500 have happened 229 times in the last 100 years according to Six Figure Investing. Under a Gaussian distribution, it shouldn't happen more than 44 times. Clearly, the underlying pricing process is far from normal – or even lognormal!

This skewness problem is actually worse on individual stock returns, which don't benefit from the tempering effects that value-weighted-averaging confers on market prices. This fact is especially relevant for us because we will be focusing on individual returns, not market ones.

Our goal here is to capture large price movements in a way that doesn't fluctuate from stock to stock or for different moments in time. In our opinion, a more practical solution is to use an absolute cutoff. Our chosen criteria will be pre-determined, static over time, and common to all stocks. The fact that most blue-chip stocks will never trip the threshold isn't a weakness in the model, it's by design (i.e. with a standard deviation approach, minute changes in price can be misinterpreted as extreme, giving the incorrect impression that all stocks have an equal chance of experiencing a price crash, which isn't the case).

### Why We Aren't Satisfied With The Prevailing Methodology

We argue in favor of absolute thresholds to identify extreme/significant price movements, even if they are more primitive in a mathematical sense. Normally, you would have clustering issues with such event-driven definitions. For example, in instances where the aggregate market is especially bullish or bearing, volatility would normally surge, triggering a rash of hits on the rule being used to classify returns. However, we should point out that measures of standard deviations also suffer from such volatility clustering, albeit to a lesser extent. Besides, clustering over time isn't a problem for us, since we focus on firm-specific returns that aren't really correlated with the overall market sentiment due to their outlier nature. Stocks don't lose half their value in a day as a result of market pressures – rather, that particular company was surely hit with some sort of unexpectedly bad news!

Although extremeness in the literature is most often determined by relative measure of standards deviations, some researchers have bucked the trend and used absolute measures. Unsurprisingly though, those types of absolute cut-offs also vary by the few researchers that use them:

• Brown and Van Harlow (1988) use 20% to 65% (over a 1 to 6 month span)

• Brewer and Sweeney (1991) use 7.5%, 10% and 15%

• Howe (1986) uses 50% within 1 week

• Benou and Richie (2003) use 20% within 1 month

• A 2021 report from JP Morgan uses 70% from peak, without a time limit (only losses are examined)

Note: For simplicity, our measures are also not adjusted for the market return (hence, the cutoff is applied on raw returns, not market-adjusted ones). Granted, it is an unsophisticated approach, though one chosen for its simplicity.

Even though we advocate using such fixed limits, we still find the cut-offs mentioned above too easy to trip. They result in too many returns being classified as extreme. For instance, an increase or decrease of 7.5% in a day is pretty standard for technology stock with a small market cap. A single stock could have a hundred or so of those returns in a year. Of course, the S&P 500 as a whole won't likely see double digit declines during the year, but remember, we are focusing solely on individual stocks here. Our approach instead uses much higher cut-offs, given us a more limited (and manageable) set of observations.

We also want our cut-offs to work at different timeframes. Scaling the threshold requires a formula to translate the x (i.e. number of trading days) into our desired y (i.e. price crash cutoff in percentage). To determine this time-to-price function, we start with arbitrarily defining two points:

• 1 day extreme return of 55%

• 1 year extreme return of 95%

To fill in the gaps in between the two bookends, we use a log-based equation to minimize the degree of error. Ultimately, we pick a trendline of: y = 0.073ln(x) + 0.55. The two parameters used, 0.073 and 0.55, are admittedly subjective. However, they do a good job minimize the number of false positives. At a month, we are looking at a decline of 77%, which is certainly a seriously rare event for most stocks. Any such change in price on the downside would most likely be caused by a catastrophic loss in investor confidence. That type of extreme return in a relatively short timespan would warrant further investigation. And that's what we want: easily isolating extreme events from countless reams of historical price data!

### Remember: It Takes A Gain Of 25% To Counteract A Decrease Of 20%

One remaining issue to address is whether to correct returns for continuous compounding. A loss of 55% in a day isn't the counterpart to an increase of 55%. Those two returns are not two sides of the same coin. By virtue of the mechanics of calculating returns – where the base is always the previous price – a stock may gain more than 100% in a day, but by definition cannot sustain an equivalent loss (because prices are bound at zero). In order to balance out that asymmetry, we adjust all upper limits to match the continuous compounded returns, for example:

• 1 day: -55% is equivalent to a +122% return

• 1 year: -95% is equivalent to a +1900% return

These adjustments are meant to identify a similar number of extreme returns on the downside as the upside. Due to the tough threshold needed to meet the high bar on the upside (+1900% represents a 19 fold increase in the share price, a pretty unlikely scenario), most academics do not use continuously compounded returns (e.g. Howe's price change of 50% in a given week works both ways; it is not -50% on the downside and +100% on the upside). However, we would argue that a gain of 95% at the one year horizon is a fairly easy target to reach. Most acquired companies, especially those with price run-ups and multiple bids, experience positive returns of that magnitude (though we admit that the average merger premium is closer to 40% than 95%). Hence, we feel that standardizing absolute returns by taking its log is the best way to ensure parity and tackle the inherent biased nature of stock return calculations, where losses are capped at -1.

The methodology we propose is unlikely to gain much traction in academic circles. Some researchers might argue the cut-off is subjective (it is!), or the calculations are mathematically naïve (fair point!), or even the preference for absolute measures is based on faulty reasoning (true, relative measures have been much more popular in the literature, where 3 or 3.09 or 5 or 7 standard deviations are the norm). Others may be put off by the very high bar needed to clear the threshold for identification. Regardless, we are certainly open to criticism, and welcome the opportunity, to have our assumptions and conclusions questioned.