Power laws: As in nature, so in the stock market? 

My home country, New Zealand, has between 50 and 80 earthquakes every day. 

In the aftermath of the devastating Christchurch quakes in 2011, there were so many each day that I began only to take notice if the picture frames started falling from the walls. 

And even then, I wasn’t exactly running for the door. 

Those were just the quakes you could feel, too. 

The ground beneath us is constantly shifting and vibrating. We just don’t notice. 

Until, of course, a big earthquake hits. 

A place can go decades or centuries without a significant earthquake. 

Then suddenly, a big quake unleashes more power in seconds than every preceding tremor combined. 

When that happens — like in Christchurch — the impacts are A) massive and B) long-lasting. 

I know someone who went from regular student life one day to recovering dead bodies from concrete rubble the next. 

Today, the city is vastly different as a result of its rebuilding and rethinking the nature of risk. 

But I’m not writing about earthquakes today. Not really. 

I’m writing about mathematical laws.

The Gutenberg-Richter Law dictates that while small quakes are common, they are geologically irrelevant. 

The ‘long tail’ of the distribution — those rare, massive events — carries nearly all the cumulative energy of the system. 

A Magnitude 9 event, for example, isn't just slightly bigger than a Magnitude 5; it’s one million times more powerful.

It’s easy to forget this. Because we live most of our lives in ‘Mediocristan’.

It seems as though every aspect of life clusters around a predictable middle. 

Most days are pretty much the same. 

Our routines seldom deviate massively from what they condition us to expect. 

We trust that everything will ‘average out’ on a long enough timeline…

But in reality, big changes happen suddenly, from a tiny percentage of events. 

Before the latest Christchurch earthquakes, for example, there’d been about 80 years of relative seismic peace in the region. 

About 1.6 million to 2.6 million insignificant earthquakes went largely unnoticed. 

Until just two came seemingly from nowhere, and changed everything very quickly — and forever, considering the many people, properties and businesses lost. 

There is a set of mathematical laws that describes this phenomenon. 

Power laws: what are they?

Here’s the statistical definition: 

A power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent.

Translation: 

Things don't scale by adding; they scale by multiplying.

Instead of a buy one, get one situation, it’s buy one, get a million.

In a normal (linear) world, if you double your effort, you double your results. 

In a power law world — the real world — doubling your effort might give you 10X, 100X, or 1,000X the results. Small changes in the input don't just nudge the output — they kick it into another dimension.

Most of us are taught to view the world through the lens of the average.

But in a universe governed by power laws, the average is a mathematical ghost — dominance isn't distributed, it's concentrated in the ‘long tail’ of extreme events.

As you can see from the chart, while most things happen in the middle of the normal distribution bell curve, the ones that make the biggest difference happen right out on the edge: 

Normal distribution, or average, does work for some things. 

Human height, for example — out of 10,000 people, even the tallest person won’t be that much taller than the average. 

But there are many situations in which normal distribution doesn’t describe reality. 

Power laws in nature

Kleiber's Law (Biology)

One of the most famous power laws in nature, Kleiber's Law describes the relationship between an animal's metabolic rate and its body mass. 

An animal's metabolic rate scales to the 3/4 power of its mass. 

Which means that while a cat is roughly 100 times more massive than a mouse, its metabolic rate is only about 32 times higher (rather than 100 times higher).

This pattern holds true across a vast range of organisms, from microbes to blue whales.

Zipf's Law (Linguistics)

Zipf's Law reveals a consistent mathematical pattern in human communication. 

It states that the frequency of any word in a large body of text is inversely proportional to its rank in the frequency table. 

For example, the most frequent word (such as "the" in English) will appear approximately twice as often as the second most frequent word, and three times as often as the third most frequent word. 

This structure exists across nearly all human languages.

Richardson's Law (Political Science)

Richardson's Law relates the frequency and severity of violent conflicts like wars and terrorist attacks. 

It demonstrates that the number of deaths in a conflict follows a power-law distribution.

While minor skirmishes are very common, massive, high-fatality wars are rare but follow a predictable mathematical trend relative to their size. 

Despite their rarity, World Wars I and II alone accounted for approximately 60% of all conflict-related deaths between 1820 and 1950.

Power laws in money and markets

So power laws exist in nature, language, and human conflict. 

It’s no surprise, then, that they exist in finance, too. 

Here’s three of the most prominent financial power laws at play in the world:

The Pareto Principle of Wealth (The 80/20 Rule)

Perhaps the most famous financial power law, originally observed by Vilfredo Pareto, states that 80% of a nation's wealth is typically held by 20% of its population. 

In modern digital markets, this distribution often becomes even more extreme, trending toward a 99:1 ratio where a tiny elite captures almost all the gains. 

This is not a linear growth pattern but a ‘preferential attachment’ process where capital naturally gravitates toward existing nodes of wealth.

The ‘Long Tail’ of Stock Market Returns

In a mature stock market, the ‘average’ return is a mathematical ghost. 

The vast majority of stocks actually underperform the market index or fail entirely over time. 

Market dominance follows a power law where a tiny fraction of companies — the giants like Apple, Nvidia, Saudi Aramco, or Palantir — drive the overwhelming majority of the total market's cumulative value. 

In fact, Hendrik Bessembinder, a finance professor at Arizona State University, published research in 2018 that showed market wealth follows a brutal power law. 

He analyzed the lifetime returns of nearly 26,000 US common stocks from 1926 to 2016, and found that just 4.3% of companies (roughly 1,100 out of 26,000) were responsible for all of the net wealth creation (roughly $35 trillion) in the stock market above the returns of one-month Treasury bills.

The Scaling Law of Market Crashes

While traditional finance treats a Black Monday or a total currency collapse as a once-in-a-century statistical error, power law math shows they are structurally inevitable. 

Just as the Gutenberg-Richter Law predicts the frequency of earthquakes, financial power laws show that the more time passes, the more certain a Magnitude 9 market event becomes. 

The ‘87 Black Monday crash… the 2000 Dot Com collapse… 2008… these events are rare but far more extreme than normal — extreme enough that they inflict massive and long-lasting impacts on investors who don’t prepare or can’t process the reality of these Black Swans. 

The power law model of Bitcoin price

I first stumbled upon the idea of power laws in the course of researching Bitcoin. 

Trying to model or predict where Bitcoin’s price is headed is one of the community's most favourite pursuits. 

Some argue there’s a four-year cycle connected to Bitcoin’s halving events.

Others that the asset’s behaviour is better explained by a ‘stock-to-flow’ model. 

There are many other views, of course. 

But this is the power law perspective on Bitcoin:

The Bitcoin Power Law isn’t just a speculative price model.

Rather, it is a scaling law for a digital network — which is what Bitcoin is. 

Just as Kleiber’s Law dictates how a biological organism’s metabolism must scale with its mass, the Bitcoin Power Law holds that value scales predictably as a function of time and network growth. 

But this isn’t the only power law characteristic at play with Bitcoin. 

The data from 2025 confirms that Bitcoin's performance is similar to seismic events.

While the asset ended the year down roughly 6.28%, that figure hides a year of massive, concentrated energy releases.

October 10 was the most violent energy release in Bitcoin's history. On this single day, Bitcoin printed its first-ever $20,000 red daily candle, plunging from more than $122,000 to roughly $104,000 inside 24 hours. 

It’s proven similar for Bitcoin bull markets, too. 

The record for the largest single-day USD gain is November 11, 2024.

Bitcoin added roughly $8,400 to its price within 24 hours. 

This move alone was larger than the entire price of Bitcoin in early 2019.

Five years earlier, Bitcoin surged 42% in a single day, jumping from approximately $7,500 to $10,500.

The trigger? A single black swan political event — Chinese President Xi Jinping’s public endorsement of blockchain technology.

So not only does the Bitcoin Power Law hold that the asset’s value appreciates according to a mathematical model…

But its price action clearly mimics that average-versus-rare dynamic we see in events like earthquakes. 

As fortune would have it, Stanford PhD Fred Krueger posted this while I was editing this Benchmark:

The truly big risks and opportunities live in the long tail

We live in a world of power laws. 

Average is often an illusion. It’s the long tail events that drive the big, lasting, changes.

Whether it’s a Magnitude 9 earthquake or the 4% of stocks that create all market wealth, the real energy of any system is always concentrated at the extreme edge. 

Most of the time, it feels as though we’re living in Mediocristan.

But the mathematical truth of power laws proves this is not the case. 

This week's quote:

“The greatest shortcoming of the human race is our inability to understand the exponential function.”

— Albert Allen Bartlett

Invest in knowledge,

Thom

The Benchmark

Read more: What happens when gold is as common as copper?

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