(B) Since worldwide equity markets peaked on October 2007, most market indexes and funds have lost between 40% and 60% of their values. Depending on the index and the fund, the months of October and November 2008 have destroyed between 5 and 10 years of wealth. So is it wrong to believe as we often hear that the average annual gain on a well-diversified stock portfolio is 7% per year and so the value of your portfolio should double every 10 years giving you an easy way to assess how much you should invest to plan your retirement? Can you predict or find somebody who can predict for you when to be invested in cash instead of stocks in order to avoid a painful 40% loss in your portfolio like this year?

Until 1697, the Western world believed that all Swans were White until a Dutch explorer, named Willem de Vlamingh, documented the first observation of a Black Swan in Australia.

In his book “The Black Swan”, Mr. Nassim Nicholas Taleb, an ex-Wall Street quant and trader now Professor, leverages the metaphor of a Black Swan to describe “The Impact of the Highly Improbable.”

The Definition of a Black Swan

Mr. Taleb defines a Black Swan as a random event with three attributes:

“First, it is an outlier, as it lies outside the realm of regular expectations because nothing in the past can convincingly point to its possibility. Second, it carries an extreme impact. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.”

A Black Swan is a change of low predictability but of a large impact. What makes the impact of a Black Swan extraordinary is that it built over time a cumulative effect.

In his book, Mr. Taleb demonstrates that financial forecasts and models fail because they are based on what we know when they should be based on what we do not know because what we do not know is in the end what is going significantly to shape the outcome of the forecast or the model. Not what we know.

The Collapse of the US Financial System this Year is a Gray Swan

The collapse this year of the US financial system is not by definition a Black Swan since it already occurred during the Great Depression. Mr. Taleb considered market crashes as Gray Swans since we can somewhat take them into account but cannot predict accurately their occurrence and estimate quantitatively their impacts.

Applied to portfolio management, the analysis of Gray Swans led Mr. Taleb to two fundamental implications:

• First, that measures of uncertainty or risk cannot be based on standard deviations from an average. In other words, market returns do not follow a normal or Gaussian probability distribution represented by a Bell Curve where around an average return, the probability of a significant market decline decreases at an exponential rate.
• Second, while unpredictable significant market return deviations such as the -40% market return so far this year are rare, they cannot be dismissed as outliers because their long-term cumulative effects on a portfolio will determine the value of the portfolio.

Market Returns can be Explained by Fractal Geometry

Market returns do not follow a normal distribution or a Bell Curve with an average 7% yearly return. Pioneered in early 1960 by French American Mathematician Benoit Mandelbrot, a retired Professor from Yale University and an IBM Scientist and Fellow, Fractal Geometry introduced a better framework to analyze stock market returns, a framework that Mr. Taleb supports passionately.

Fractal Geometry establishes the repetition of geometric patterns at different scales. Stones look like rocks and rocks like mountains. Small branches look like bigger ones and bigger ones look like trees. The fundamental assumption of Fractal Geometry is that there is no qualitative change when the scale of an object changes. A property called “scale invariance”.

Applied to statistics and to stock prices, Fractal Geometry provides a representation of statistical measures that are “somewhat” preserved across scales. In a nutshell, Fractal Geometry establishes the repetition of market returns that can be 7%, -7%, 20%, -20%, 40% or -40% while a normal distribution of market returns fails to take into account the occurrence of large deviations.

The function that exhibits the property of a fractal is a power law , f(x) = ax^k + O(x^k), and the distribution of stock prices follows a power law distribution P(X>x) ~ O (x^-α).

When the market moves are mild, the distribution of the stock prices is a Gaussian distribution but when the moves are wild (such as this year), the distribution is a Cauchy distribution. A French mathematician Paul Levy demonstrated that both the Gaussian and the Cauchy distribution can be expressed by the same power law distribution, a theorem that Mr. Mandelbrot, a Ph.D. student from Mr. Levy, called L-stable for a Levy stable distribution.

In Mandelbrot Fractal Financial Engineering (M2FE) model, if α>2, the distribution becomes Gaussian. In that case, the distribution of the returns is “normal” and every single data point converges to an average return. And, the more you observe those market moves, the more you notice that the return on your portfolio will converge toward that average let’s say of 7% annually. But if α<2, the curve becomes surprisingly “abnormal”! A Gray Swan! But do not be a fool, it is normal (a fact that Mr. Taleb calls the round-trip fallacy). The distribution of the returns has in that case “fat tails” or “outliers” or as proofed by M2FE model: scale invariance.

Ce qu’il fallait demontrer (CQFD)!

Long-Term Portfolio Performance is Dependent of a Few Market Moves or Gray Swans

The damaging and complete collapse of the US financial system this year is a negative Gray Swan that has never been seen since the Great Depression. It will have a significant impact on everyone long-term portfolio performance for years to come.

The long-term performance of a portfolio is not smooth and steady over time but largely determined by a few significant market day moves generated by Gray Swans. The occurrence of any Gray Swans such as market bubbles (positive Gray Swans) or market crashes (negative Gray Swans) contributes to a large creation or destruction of wealth in a portfolio. That is when huge fortune can be made or lost.

To illustrate that point, following is a complete empirical dataset published in January 2008 by Professor Javier Estrada from the IESE Business School in Barcelona:

“Based on 15 international equity markets including Australia, Canada, France, Germany, Hong Kong, Italy, Japan, New Zealand, Singapore, Spain, Switzerland, Taiwan, Thailand, UK and the US and over 160,000 daily returns, Gray Swans have a massive impact on long-term performance.

From 1990 to 2006, on average across all 15 markets, missing the best 10, 20, and 100 days resulted in a reduction of 43.3%, 62.3%, and 95.2% in terminal wealth relative to passive investment. On average across all 15 markets, 10, 20 and 100 days represent only 0.23%, 0.47% and 2.34% of the total number of days considered. Avoiding the worst 10, 20, and 100 days, in turn, resulted in an increase in terminal wealth of 87.9%, 204.4%, and 6,268.5%, again relative to passive investment.

Across all 15 markets, and relative to a passive investment, missing the best 10 days reduced mean annual compound returns by over three percentage points to 1.9%; missing the best 20 days resulted in negative mean annual compound returns in 5 markets, and on average across all markets; and missing the best 100 days (2.34% of the days considered in the average market) resulted in negative mean annual compound returns in all markets. Avoiding the worst 10 days, in turn, increased mean annual compound returns by almost four percentage points to 9.4%; avoiding the worst 20 days resulted in more than doubling mean annual compound returns to 12.3%; and avoiding the worst 100 days resulted in mean annual compound returns of 30.5%, over five times higher than those of a passive investment.”

Note that how in Mr. Estrada’s analyzed portfolio, a very small number of days account for the bulk of stock market returns and that Gray Swans destroy more wealth than positive Gray Swans create it. So the timing of those small number of days when Gray Swans are going to work is everything: good market timing and you beat the average market index; bad market timing and you lost more than the average market index. That is what professional investors attempt to do.

Mandelbrot Fractal Financial Engineering (M2FE) takes very well into account the impacts of those Gray Swans.

M2FE proofs that if 1 < α < ∞ in the distribution of a fractal power law, a small number of market moves generate the final return of the portfolio. That result is established in the Multifractal Model of Asset Returns (MMAR).

Again, CQFD!

How to Take into Account Gray Swans in your Portfolio

So conventional investment practice proposes investing over the long term in diversified asset classes to avoid the challenges of market timing. That strategy is supposed to provide two advantages. First, ensure to be exposed to the positive Gray Swans that boost the performance of the portfolio even if that will likely result as well in the exposition to negative Gray Swans that will destroy some of the wealth of the portfolio. Second, hoping that the destruction of wealth by a Gray Swan in one asset class will be compensated by the creation of wealth by another Gray Swan in another asset class (see our article “Investing for Dummies” for establishing a pseudo structure of a portfolio).

However, this second argument did not hold this year since all asset classes including stocks, from every sector on every exchange in the world, commodities and hedge, private and venture funds have all significant negative returns.

A non-conventional investment practice is to pursue Mr. Taleb’s investment strategy: to be hyper-conservative when exposed to negative Gray Swans, the ones that crash the stock markets and to be hyper-aggressive when exposed to the positive Gray Swans, the ones that lift the markets.

For Mr. Taleb’s being hyper-conservative implies investing 85 to 90% of his portfolio in Treasury bills to avoid any market risks while being hyper-aggressive implies investing the remaining 10 to 15% of his portfolio in venture capital where the downside is limited (the worst case is that all the money is lost) but where the upside can be very significant (start-up acquisition or IPO).

Preventing the Next Big Negative Gray Swan

And, more importantly, always assume in your portfolio that there is no guarantee that even over the long term, the stock market will go higher.

And if you do not believe me, imagine what will happen to stock prices, when the cost of Global Warming will be factored into the stock market. Could that be the next big negative Gray Swan? Bigger than the 2008 Worldwide Financial Collapse?

We probably should do much more about Global Warming! What do you think?

Note 1: Mr. Taleb calls Gray Swans, Mandelbrot Gray Swans probably because they can be explained by Mandelbrot’s Fractal Geometry.

Note 2: Why the most challenging mathematicians that I have to study at school were either Russian or French?

Note 3: Due to popular demand, we will attempt to explain in more details M2FE soon in future articles.

Note 4: M2FE is an acronym that we like to use to refer to Mr. Mandelbrot’s outstanding achievements (soon Nobel Prize?) to Finance Theory.

Note 5: The picture above is from the coasts of Corse in France and a nice visual proof that Mother Nature‘s Geometry is Fractal.

References

• Estrada, Javier (2008), “Black Swans and Market Timing. How not to Generate Alpha Returns.” IESE Business School.
• Mandelbrot, Benoit and Hudson, Richard (2004). The (Mis) Behavior of Markets. A Fractal View of Risk, Ruin and Reward. Profile Books.
• Mandelbrot Benoit (1997), Fractal and Scaling in Finance, Springer.
• Mandelbrot Benoit (1982), The Fractal Geometry of Nature, W.H. Freeman.
• Taleb, Nassim (2007), The Black Swan. The Impact of the Highly Improbable. Random House.

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