### A New Perspective On Market Probabilities

Most people think that markets are normally distributed. That is, they believe market outcomes are shaped like a bell curve; which is dangerously incorrect.

In fact, markets follow Pareto distributions (or more broadly, Power Law distributions), more commonly known as the 80/20 rule.

Naturally, this reality changes how you should invest or trade in the markets – the question is how?

### How Does The 80/20 Rule Apply To Investing? 1

Most people understand on some basic intellectual level that few outcomes in financial markets (and life) are certain – that is, luck is a factor.

In turn, the existence of luck means that market outcomes are rarely binary. How does this affect our ways of thinking about markets? More importantly, what does the 80/20 Rule have to do with it?

In order to answer these questions, we have to begin thinking in *maybes*.

Our human minds seem predisposed to jumping to conclusions when we are faced with a paucity of information. Reddit’s *certainty* of how their GME short squeeze could only turn out in their favor is a good example of this.

Retail traders saw GME’s price chart go vertical and somehow that was enough to convince them the stock could *only* go higher. Of course, more than a few of them learned the *very* hard way that certainty is not a luxury anyone in financial markets can afford.

Perhaps it would be wiser not to think in terms of “yes”, or “no”, but in terms of “maybe”, and “maybe” to what degree?

Going back to the GME example, a retail trader, when faced with the question “Can GME only go higher?”, instead of immediately thinking “YES!” and rushing to execute the trade, would pause and ponder different possible scenarios – the *maybes.*

Maybe GME won’t go higher, or maybe it will. Maybe the short squeeze ends a few days after the trade is executed. Maybe it doesn’t and the stock price really does go to the moon.

It doesn’t take much effort to work out a few basic scenarios of what GME *may or may not* do. If this abstract level of uncertainty is tricky enough, the bad news is that it only gets worse after this point.

The general (and most widely accepted) idea at this juncture is to apply what is taught in school, which is to run through an expected value calculation.

Unfortunately, as we will see, this calculation isn’t fit for purpose when it comes to financial markets. This is due to the nature of uncertainty in markets, which precludes any sort of precision.

Expected value is calculated:

E.V. = P(X_{1})×X_{1 }+ P(X_{2})×X_{2 }+ P(X_{3})×X_{3 }+ … + P(X_{n})×X_{n}

Where P(X) is the probability of the scenario number (1,2,3,…,n) occurring, and X is the payoff the scenario gives.

As such, our hypothetical retail trader thinking about taking a punt on GME has to assign probabilities to each scenario occurring, as well as the payoffs he thinks are associated with each scenario.

Considering that our trader is already biased by all the GME short squeeze talk flying around, he will naturally overweight the most bullish scenario, say at 70%, and the associated payoff at $150.

The trader then considers the bearish scenario of GME’s share price falling like a stone, and weights this at 30%, since 70% + 30% = 100%, with the associated payoff -$150.

The expected value works out to be $60, which gives our trader even more confidence in going long GME!

Obviously something isn’t working correctly. How could someone come out of a thought exercise meant to highlight prevailing uncertainty with even *more *certainty ?

*To be continued…*

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### How Does The 80/20 Rule Apply To Investing? 2

At this point, it should be apparent that the entire expected value thought process is extremely subjective, and vulnerable to cognitive bias. Every trader who runs through this exercise will come up with not just different numbers, but also different scenarios.

For example, a more experienced trader would also consider the scenario of GME shares trading sideways. A more fastidious one might even include “middle” scenarios where GME moves higher/lower, but not as much as in the most bullish/bearish scenario.

Moreover, every trader will ascribe different payoff amounts to each scenario, even though all are trading the same stock, reading the same narratives, and looking at the same prices.

One trader might think a bullish scenario in GME means a $100 move higher, while another might think that it means a $50 move higher.

As such, applying the methods taught in probability/statistics textbooks to financial markets is difficult, to say the least. Unlike the simple games used in textbooks to teach probability, markets do not have discrete outcomes like getting tails on a coin flip or rolling a 6 with a dice.

Instead, we have stocks that can end the trading day anywhere between $0 and some astronomically high value, since there isn’t a cap on the upside.

It gets even more confusing and risky in other markets, like the oil futures market. WTI contracts on the CME have neither an upside cap nor a *downside *cap – oil futures can trade for less than $0 a barrel!

Complicating matters is the fact that trading scenarios have *another* dimension to them – time.

In textbooks, scenarios are always clear and simple – they end after the dice/coin settles on one of its faces. Roll, wobble a little, settle, and done.

Position management in trading/investing is not so simple due to the open-ended nature of trades. Depending on the trading strategy, the timing of the exit may not be clear. And the longer the position stays on, the more market realities change. This means that the expected value thought exercise that the trader had previously engaged in is now invalidated.

Not only will the possible payoffs for each scenario change as information and narratives develop, but the *probabilities* themselves will change too!

Consequently, the discrete and elegant expected value formula cannot accommodate the dynamic and continuous nature of markets.

Any trader who ignores this reality will find that ascribing probabilities and payoffs to different scenarios is an exercise in guesswork, and not reliable precision.

How then should we think about probabilities in the context of markets?

*To be continued…*

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### How Does The 80/20 Rule Apply To Investing? 3

The first, and most important point to understand about markets (and in many ways, life) is that outcomes * aren’t* normally distributed.

“Normally” in this case meaning the famous bell curve. This might come as a shock to a lot of people, who might not even realize that other distributions exist – such is the extent of the normal distribution’s dominance in modern day thinking.

Instead, market outcomes are distributed according to the Pareto distribution*, also known by its far more famous and catchy name, the 80/20 Rule. The 80/20 Rule was the brainchild of Viflredo Pareto, and differs from the normal distribution due to its fat tails.

Fat tails, in a probability distribution, simply signify a higher probability of extreme events occurring (relative to a less fat tail).

Extreme, in this case, meaning *life changing.*

Covid turning into a global pandemic was (and at the time of writing, still is) an extreme event. The many people who lost jobs, especially in Tourism and Aviation, had their lives severely disrupted almost overnight.

On a less gloomy note, rare life changing events can happen to the upside as well. Entrepreneurial success stories are a good example of this, as well as people who manage to, through some combination of luck and skill, profit from rare events that catch everyone else off guard.

A good instance of this is the people who made money shorting US housing and/or investment banks in the lead up to 2008.

Both Covid and 2008 also demonstrate how we live in a Paretian world. The Covid induced stock market plunge in early 2020 was about a 5 sigma (standard deviation) move, assuming it followed a normal distribution. 5 sigma moves are “supposed” to occur once every 14,000* years*.

Later on *in* 2020, amid murmurings that Covid vaccines were close to being a reality, stock markets reacted wildly. As traders/investors rushed to reposition portfolios to take into account the possibility of a faster than expected return to normalcy, momentum stocks were sold heavily.

So heavily that their move lower constituted a 15 sigma event, *only ~8 months after a 5 sigma event*.

That’s a 1 in 1.090e+048 (1 with *48* zeros behind) years event happening a *mere* 8 months after a 1 in 14,000 year event!

Furthermore, during the height of the credit crunch in 2008, the moves in markets were so extreme that David Viniar, then CFO of Goldman, commented that:

We were seeing things that were 25-standard deviation moves, several days in a row.

25 sigma moves, in a normally distributed world, are “supposed” to occur once every 1.309e+135 years. That’s once in, 1 with *135* zeros behind it, years. And not just for one day, but for “several days in a row”!!

As such, headlines or arguments that claim some rare event will not happen again, or was inconceivable because of it being an X Sigma event, are meaningless. This is because mean and standard deviation, crucial to a normal distribution, are meaningless in Pareto realities.

Furthermore, the normal distribution is reliant on independent outcomes. This means that, in a bell curve world, events have nothing to do with one another – getting heads on the first coin flip does not affect the outcome of the second flip.

On the other hand, Pareto distributions reflect the financial market reality of *dependent* outcomes. People buy when they see others buying, and they sell when they see others selling; both emotions and paradigms matter.

Clearly the financial markets (and world, and life) are not normally distributed!

*To be concluded…*

**of the more general Power Law family of probability distributions*

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### How Does The 80/20 Rule Apply To Investing? 4

What does the Rule imply, and how does this affect the way we think of and trade the markets?

A common way of thinking about the 80/20 rule, especially in corporate/personal improvement settings, is as a means to increase productivity.

In truth, the 80/20 rule is more an expression of the lumpy nature of life. Change tends to happen in big, and very noticeable ways after a prolonged period of very little change.

Ernest Hemingway described this perfectly in his novel The Sun Also Rises:

“How did you go bankrupt?” Bill asked.

“Two ways,” Mike said. “Gradually and then suddenly.”

From a broader standpoint, it means that the effects of a singular period of change dwarf the effects of the prior period of calm. This is also clearly expressed in the mainstream view of the 80/20 rule. If 20% of your time accounts for 80% of work done:

20% of time = 80% of work done

80% of time = 20% of work done

The *effects *of the 20% of time spent dwarfs those of the 80%, just like the *effects* of being bankrupt dwarf those of being not bankrupt, regardless of how long one spent being not bankrupt.

This is true for trading investing as well. As Kenneth Grant, in his book Trading Risk, describes:

For nearly every account in our sample, the

of all transactions ranked by profitability accounted fortop 10%for the account. In many cases, the 100% threshold was crossed at 5% or lower. Moreover, this pattern100% or more of the P/Lrepeated itself consistently across trading styles,asset classes, instrument classes, and market conditions. (emphasis ours)

Obviously, market realities are a lot more extreme than normal life, pushing the 80/20 Rule into something more like 100/10 (it’s possible to be greater than 100 because other trades in the portfolio caused losses).

If you pause and think about it for a second, you would realize the *sheer size* of the profits on those 10% of trades (5% in many cases).

In the best case scenario, the other 90% of trades net to zero, and the 10% constitute all the profits of the account.

In the worst case, every single one of the 90% would be losses (but small on an individual basis, as the trader didn’t blow up), which means that the 10% of trades had to make up for the losses *and *push the trader’s PnL into the black.

What Kenneth Grant did not mention was that the Pareto effect occurs in the opposite direction too. Because of survivorship bias, only profitable traders were studied, but what about those who blew their trading accounts up?

The 100/10 observation will apply here as well.

Just as a small handful of trades can make a trader profitable (even hugely profitable) for a year, a small handful of trades can also make a trader blow up.

Even if a trader doesn’t lose more than 100% (because of leverage), any big loss will see them lose their job, which makes recovering from the loss impossible.

Put simply, 5 – 10 percent of trades make or break a trader’s year. Those same 5 – 10 percent of trades can also make or break a trader’s *career!*

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### How To Exploit The 80/20 Rule In Trading & Investing 1

We live in a world where Paretian effects dominate, and financial markets are no exception. Understanding this reality can bring about a paradigm shift for people exposed to this idea for the first time.

Such a shift can, in turn, result in one adopting a more profitable approach to trading.

In terms of trading/investing, having a Paretian perspective means that traders must act to both guard against, *and* take advantage of, Paretian effects.

Beginning with Kenneth Grant’s 100/10 observation, we know that 90% of trades will either *not* be particularly profitable, or be straight up losses. That means, to be prepared for the worst outcome, we should expect to lose money on most of these trades.

Consequently, losses on these trades ** must** be limited.

Since we do not know beforehand which trades will fall into the 90% bucket and which will fall into the 10% one, losses on *all** *trades must be limited.

To see how this works out mathematically, we can use the expected value calculation first mentioned at the beginning of this series of articles. While expected value does *not* work well when used to analyze scenarios for individual trades, it works very well on the portfolio level, allowing us to see just how powerful Pareto effects are.

E.V. = P(X_{1})×X_{1 }+ P(X_{2})×X_{2 }

If a trader expects to lose $100, 90% of the time, while winning $1000, 10% of the time; the expected value works out to $10.

This illustrates the degree to which Pareto distributions affect outcomes: if the trader made 10 trades and 1 was a big winner, the profits on that one trade *can* make up for all the total losses on the other 9.

Of course, there is a limit to this, as can be seen from the table, if the trader expects to lose $200, 90% of the time, the expected value drops to -$80.

Which means that in order to fully take advantage of the Paretian nature of markets, the trader must be conscientious with risk management, keeping losses per trade small.

The second table shows Pareto effects working *against* the trader. In this scenario, the trader takes many small wins, but a few large losses, which is enough to push their PnL into the red.

If the trader expects to win $100, 90% of the time, while losing $1000, 10% of the time, the expected value works out to be a net loss at -$10. Assuming this happens over a total of 10 trades, 1 trade with a big loss is enough to wipe out all previous profits.

Obviously, this underscores the need for some form of risk management to limit large losses.

*To be continued…*

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