Beware Of Zeros, Averages, & Non-Llinearity

Beware Of Zeros,  Averages, & Non-Llinearity
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Article by Investment Masters Class

Over the two decades that I have been advising institutional clients on markets, I’ve experienced my fair share of extreme stock market volatility. This has included market crashes, stock price death spirals and short squeezes. During this time I’ve often witnessed whole analyst communities get blindsided by unexpected outcomes, rendering their forecasts and recommendations completely obsolete. I’ve also seen investors lose significant sums. Some of these outcomes may have been avoided by recognising a few simple mathematical concepts, namely Averages, Zeros and Non-Linearity.

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Understanding these three concepts, which as you'll see, can often be inter-related, can help you avoid the permanent loss of capital.


Averages are a form of simplification. They can summarize a lot information into one key output. You'll often see market commentators reference an average to support a recommendation ...  "Historically, the market has rallied/fallen x% when xyz happens."

The danger in relying on averages is that the range of historic outcomes may be very wide; the range may contain a zero, or the future outcome may end up being far outside those historic outcomes. It's also quite likely the outcome will be nowhere near the historic average.

The average annual stock market return over the last century is a case in point. While the average return for the S&P500 has been c10% pa, the typical yearly return is far from that. Buffett made that point in his 2004 letter ...

"In one respect, 2004 was a remarkable year for the stock market, a fact buried in the maze of numbers on page 2. If you examine the 35 years since the 1960s ended, you will find that an investor's return, including dividends, from owning the S&P has averaged 11.2% annually. But if you look for years with returns anywhere close to that 11.2% - say, between 8% and 14% - you will find only one before 2004. In other words, last year's 'normal' return is anything but."

Whenever I hear references to an 'average', I remind myself of the story of the 6-foot man who drowned crossing the river that was, on average, 5-feet deep.

“Over the  past 100 years, it is generally understood that the stock market’s annualized return is approximately 10%. That 100 year annualized return for the S&P 500 Index masks a great deal of volatility, or variability around the average. It calls to mind the parable of the 6-foot man who drowned in the 5-foot average depth river. The lesson: beware of averages!Chuck Akre

Another limitation of averages is that the future might look a lot different to the past. Billions of dollars were lost in the Global Financial Crisis as people relied on the notion that`On a national basis, US house prices never go down'. Never forget that absence of evidence is not evidence of absence.

“The mother of all harmful investment errors is mistaking the absence of evidence... for the evidence of absence. In other words, assuming that just because historically infrequent and potentially catastrophic events, known as Black Swans, haven’t happened… they won’t.”  Frank Martin

As an investor you must structure your portfolio to cope with the unexpected. This means stress-testing individual ideas and the assumptions that underpin them and taking the time to think about what a worst case scenario might do to each position and the portfolio as a whole.

"We should all be humble enough to realize that once every 20 or 30 or 40 years, values go to real extremes. Any investment program must take into account the impossibility of knowing when and to what extent such extremes might occur."  Paul Singer


Averages also mask historic outcomes that resulted in zeros. Regardless of the potential return, if there is a possibility of a complete loss, investors should steer clear.

“Makes sure that the probability of the unacceptable (i.e: the risk of ruin) is nil.Ray Dalio

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Zero means game over.

"Never forget that anything times zero is zero. No matter how many winners you’ve got, if you either leverage too much or do anything that gives you the chance of having a zero in there, it’ll all turn to pumpkins and mice.”  Warren Buffett

"In business and also investment, success is measured through the compounding of a series of returns.  Mathematically, the biggest risk to a compounded series of returns is large negative numbers or even a single negative number, if large enough.  Take however many spectacular annual outcomes and multiply them by just one zero and the answer is of course, zero."   Marathon Asset Management

“There is a difference between opportunities missed and capital lost, with which most investors, anecdotally, do not appear adequately familiar. You can miss a million opportunities in a lifetime and still become very rich. Every asset that has risen in price that one didn’t purchase was an opportunity lost.  Capital losses are not so forgiving. If you lose 100 percent of your capital – just once – you're broke.” Frank Martin

"One single loss can eradicate a century of profits." Nicholas Nassim Taleb

"No matter what price you pay for a stock, when it goes to zero you've lost 100% of your money." Peter Lynch

And, as we saw above, history may not be a good guide to the future. To manage that risk requires creative thinking, consideration of potential alternative outcomes and acknowledging worst case scenarios.

“In my book 'The Most Important Thing', I mentioned something I call “the failure of imagination.” I defined it as, “either being unable to conceive the full range of possible outcomes, or not understanding the consequences of the more extreme occurrences.”  Howard Marks

"In life, both financial and social, sometimes events swing to extremes that seem inconceivable to conventional minds."  Barton Biggs

As Warren Buffett noted above, a common cause of investment ruin is leverage. I've seen plenty of over-leveraged companies turn into zeros.

"The floor for any business is different. If a company is highly levered, the floor could be zero." Mohnish Pabrai

"More than anything else, it's debt that determines which companies will survive and which will go bankrupt in a crisis. Young companies with heavy debts are always at risk."  Peter Lynch

Another danger is complex businesses that are difficult to understand or lack transparency. I recently witnessed a well-capitalized insurance company turn to dust when a fraction of its long-tail reinsurance liabilities blew up the balance sheet.  Earlier this year Buffett said: "You can make big mistakes in insurance.”

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It's no wonder the Investment Masters generally steer clear of companies with lots of debt, lack transparency or that are difficult to understand.


Humans are wired to think in a linear fashion. Most things in life work that way, but we often fail to see the potential for non-linear outcomes.

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

“Our intuitions are not cut out for nonlinearities. Consider our life in a primitive environment where process and result are closely connected. You are thirsty; drinking brings you adequate satisfaction. Or even in a not-so-primitive environment, when you engage in building, say, a bridge or a stone house, more work will lead to more apparent results, so your mood is propped up by visible continuous feedback.” Nassim Nicholas Taleb

“Decades of research in cognitive psychology show that the human mind struggles to understand non-linear relationships. Our brain wants to make simple straight lines. In many situations, that kind of thinking serves us well: If you can store 50 books on a shelf, you can store 100 books if you add another shelf, and 150 books if you add yet another. Similarly, if the price of coffee is $2, you can buy five coffees with $10, 10 coffees with $20, and 15 coffees with $30.  But in business there are many highly non-linear relationships, and we need to recognize when they’re in play. This is true for generalists and specialists alike, because even experts who are aware of non-linearity in their fields can fail to take it into account and default instead to relying on their gut. But when people do that, they often end up making poor decisions.” Whitney Tilson

But in markets small changes can have large impacts on outcomes. Sometimes things don't work in a linear fashion.

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“With linearities, relationships between variables are clear, crisp, and constant, therefore platonically easy to grasp in a single sentence, such as, "A 10 percent increase in money in the bank corresponds to a 10 percent increase in interest income and a 5 percent increase in obsequiousness on the part of the personal banker." If you have more money in the bank, you get more interest. Non-linear relationships can vary; perhaps the best way to describe them is to say that they cannot be expressed verbally in a way that does justice to them.” Nassim Nicholas Taleb

Jamie Dimon touched on this topic in his recent annual letter ...

"I am a little perplexed when people are surprised by large market moves. Oftentimes, it takes only an unexpected supply/demand imbalance of a few percent and changing sentiment to dramatically move markets. We have seen that condition occur recently in oil, but I have also seen it multiple times in my career in cotton, corn, aluminium, soybeans, chicken, beef, copper, iron – you get the point.

Each industry or commodity has continually changing supply and demand, different investment horizons to add or subtract supply, varying marginal and fixed costs, and different inventory and supply lines. In all cases, extreme volatility can be created by slightly changing factors.

It is fundamentally the same for stocks, bonds, and interest rates and currencies. Changing expectations, whether around inflation, growth or recession (yes, there will be another recession – we just don’t know when), supply and demand, sentiment and other factors, can cause drastic volatility."

Over the years I've witnessed numerous events cause unexpected negative outcomes due to non-linearity. Here are a few ..

High Fixed Operating Cost Businesses - when businesses have high fixed operating costs and low profit margins, small changes in top line revenue can have a huge impact on a businesses profitability. When sales are booming this is a great benefit, but when things turn down, profit can disappear quickly.

Highly Leveraged Businesses - when businesses carrying a lot of debt turn down, profit can disappear quickly. While the value of the enterprise might decline by 50%, if the business is 50% geared it means the equity is wiped out. These can be very profitable short candidates..

“You have to remember that if you are shorting a leveraged company, with 90% of the capitalization in debt and 10% in equity, a 50% decline in the stock only wipes out 5% of the total capitalization. You have to look at the total capitalization.”  Jim Chanos

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Commodity Companies - Analysts often expect supply and demand curves to be linear. Often they are not. A good example was when China entered the iron ore market in the early 2000's. The chart below shows a stable iron ore price from c1985 until 2009 when a supply bottleneck saw prices spike dramatically. Analysts expected the iron ore to be supported by the high Chinese marginal cost of production post c2012. Huge increases in supply and large debt loads that needed to be serviced saw iron ore prices collapse as the supply and demand curve proved non-linear.

permanent loss of capital

Peter Brandt, a CTA since 1976, summed it up nicely in regards to commodities trading below costs ... "But, you might say, this kind of drop is impossible because producers must make money. Who says?? Markets in supply surplus tend to go the production price of the most efficient producers. Plus, who would have ever believed when Crude was at $148 in mid-2008 that prices would retreat to below $40 in just six months. So take your pet macro-economic/fundamental scenario and burn it with the trash!"

Non-linear outcomes can lead to extreme events, also know as 'tail risks'. They too, don't show up in past averages.

Averages mask non-linearity and lead to prediction errors.”  Bart de Langhe

Having an understanding and awareness of the possibility of non-linear outcomes can better prepare you and your portfolio for success in the markets.

"Don’t project along a straight line." Jim Tisch

"Nonlinear outcomes, those exponentially greater than the apparent precipitating causes, are a great threat to financial and economic stability. Having even a crude understanding of power laws, as they are known,particularly in the area of fat tails, is critically important for effective risk management, for appreciating the potential magnitude of rare market upheavals."  Frank Martin

I particularly like the anecdote about the six foot man to explain most of this. Consider that if the average depth of the river he wanted to cross was 5', and the shallowest depth was 0', the deepest part of the water could be well in excess of his 6' height. Which is why he drowned. The message is don't rely on averages. Or expect that all things will conveniently follow a straight line. Buffett's comment that the market's returns bore a resemblance to the average in only one year across the 35 years he was reviewing, is quite simply, astonishing. And let's face it, who wants to drown? Or suffer a permanent loss of capital?


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