Risk in a Fractal World
In a fractal world, pure chance has an asymmetrical impact. The expected value rule creates a bias towards high-risk bets with lottery-like payoffs. The resulting thrust towards fragility makes the system highly vulnerable as demonstrated in the 2008 Global Financial Crisis.
An Investment Puzzle
Consider the following two problems:
Problem 1
You have a share (of company A) with a price today of $100. Tomorrow’s price can be up or down by 10% with equal probabilities. What would be the expected price after 100 days?
Problem 2
You have 100 shares (of company A) the price of each today is $100. Tomorrow, each can go up or down by 10% with equal probabilities. What would be the average share price tomorrow?
Standard statistics treat these two problems equivalently: the expected value of the price of 1 share after 100 days is essentially equivalent to that of 100 copies of that share after 1 day. The expected value approach treats time and space symmetrically. But there is a difference.
Arithmetic vs. Geometric Mean
To compute the price of a share over a certain period, you need to take into account how the price today is related to the price yesterday. Although a 10% change is equally likely each day, its impact on the price level depends on the price level of the previous day. This information is crucial for evaluating the progression of the price over time.
Suppose the stock price rises on day 1 by 10% and then goes down on day 2 by 10%. How much would be the price? On day 1, it will be $110, then on day 2, it will be 90% of that and thus $99. Suppose instead that, on day 1, the price goes down by 10% and then rises by 10% on day 2. How much would be the price? Again, it will be $99 ($90 on day 1 then $99 on day 2).
Of course, the price might go up on both days or go down on both. What is the most likely scenario? Up-then-down and down-then-up are the most likely scenarios. The number of up-days follows a binomial distribution, and thus its expected value will be the probability of up-days times the total number of days. (The number of down-days is simply the total number of days minus up-days.) For two days, this means the expected number of each of the ups and downs is 1.
Hence, given the price on day 1, the expected price on day 2 should be:
($100)(1.1)(0.9) = $99.
This formula is called “the geometric mean.” The formula takes into account the information obtained from the previous day.
If we ignore the relationship of the price on day 2 to the price on day 1, we will obtain the “arithmetic mean”: After 2 days, the average price is:
($100)[ (1.1 + 0.9) / 2] = $100.
The arithmetic mean is the same after 100 or 1000 days (we simply multiply (1.1 + 0.9) by 50/100 or 500/1000 etc.).
In contrast, the geometric mean of the price after 100 days will be:
($100)[ (1.1)(0.9) ]⁵⁰ = $60.5
(In 100 days, there will be 50 days up and 50 down, so each will be repeated 50 times.) The general formula for the geometric average price is:
Where Pₙ is the price on day n, and P₀ is the price on day 0. The price can go up by r+ with probability ρ or can go down by r- with probability (1 — ρ). Plugging the numbers, P₀ = $100, n = 100, r+ = r- = 0.1, ρ = 0.5, we get P₁₀₀ = $60.5.
Although there are equal chances every day for a loss or gain, after 100 days, the price will most likely go down by 39.5%. Clearly, there is something strange going on here.
Volatility Drain
Mark Spitznagel called the 39.5% loss “Volatility Tax.” It is also called Volatility Drain or Drag. It reflects an inherent cost of volatility over time, much like entropy. It arises from the nature of compounded returns. If the price of a stock goes down by 10% from $100 to $90, it needs to go up by 11.11% to restore its original price.
The asymmetry of losses and gains explains why the first million dollars is much more difficult than the second, and the second more difficult than the third, and so on.
The rich can get richer faster than the less lucky can catch up. This is known as the “Pareto Principle,” also known as the “80/20 rule.” The principle is named after the Italian polymath Vilfredo Pareto (1848–1923). He made an extensive analysis of data on income distribution in several countries and found the same pattern: Income is not normally distributed. Roughly, 80% of total income is held by 20% of the population. Pareto saw the same pattern nearly everywhere, leading him to conclude that it is a social law (Mandelbrot 2004, p. 154).
The kind of distribution that Pareto discovered is now commonly called Pareto distribution. It is a classical example of “heavy-tailed distributions,” whereby a few extremely large values on the tail (usually the positive tail) dominate the distribution. As we shall see, the underlying process is quite common in natural and social domains.
A Fractal World
The French mathematician Benoit Mandelbrot (1924–2010) was another polymath with diverse interests, especially in the exotic applications of novel mathematical ideas. He pursued the path explored by Pareto with much more surprising results.
Mandelbrot integrated different strands of science and mathematics into a coherent theory he called Fractal Geometry. He then applied his theory to phenomena ranging from markets to forests to clouds to galaxies. In one application after another, his fractal geometry proved robust and insightful.
So, what is the key idea of Fractal Geometry? The following parable might help.
A farmer wanted to increase milk production, and asked his neighbor, a theoretical physicist, for advice. To analyze the problem, the parable goes, the physicist declared: let us assume the cow is a sphere!
Simplification generally is useful: While the Earth is spherical, locally it can be assumed flat just as Euclid would prefer. In standard geometry and calculus, if we zoom in enough onto a graph or a surface, it becomes increasingly smooth until it becomes locally flat or a straight line. This approach works well in many areas, but not all.
In a fractal world, a rough shape is irreducible to a smooth one, even as a first approximation, even locally. “Nature fails to be locally linear,” writes Mandelbrot (1989, p. 4). Aggregation does not smooth out fractal roughness. In the introduction to his The Fractal Geometry of Nature, Mandelbrot writes (1983, p. 1):
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
The Name of the Game is Recursion
There is no simple way to measure the edge of a rough shape. But it can be usually reproduced using a simple algorithm. A line of code, when applied recursively, can generate surprisingly complex structures (Peitgen et al., 2004, chapter 1).
Recursion implies that fractal processes are multiplicative rather than additive. For a multiplicative process, the geometric mean is the proper average, while the arithmetic mean is suitable for an additive one.
Multiplicative processes are more common in nature than additive ones, as Nair, Wierman, and Zwart (2022, pp. 128–132) point out. Moreover, they are fundamentally tied to the emergence of heavy-tailed distributions. As Ole Peters and Murray Gell-Mann (2016, p. 26) indicate, multiplicative processes lead to emergent properties of complex systems that additive processes fail to capture.
The recursive nature of the algorithm underlying a fractal ensures the following features:
- A complex structure can arise from a simple set of instructions; complexity emerges from simplicity.
- The basic structure is conserved at all scales of the fractal. An arbitrary small part of it would more or less reflect the whole. This “scale invariance” is a key feature of fractals.
- The distribution of the hierarchy of a fractal will be largely skewed.
To elaborate on the last point, consider the Sierpinski Triangle which can be generated by many algorithms.
The number of small triangles is much larger than the number of larger ones. There is only one big triangle that encompasses the entire figure. Then there are three large triangles (excluding the empty one in the middle), each with smaller three triangles, and so on.
The distribution of Sierpinski triangles differs significantly from that generated from a normal distribution. In a normal distribution, very small and very large triangles have equal and low frequencies, while the medium-sized represent the bulk in the middle. Instead, the distribution of the Sierpinski triangles is highly skewed.
Extreme values in fractals are the “North Star” toward which the entire distribution is oriented
The extreme values in a fractal distribution are not necessarily outliers or exceptions. Instead, they may very well represent the “North Star” toward which the entire distribution is oriented. In the Sierpinski triangle, the largest triangle is the one encompassing all the others. Hence, the heavy tail of the Sierpinski distribution is an inherent feature of the structure rather than an anomaly.
Fractal Volatility
Let us now return to our Volatility Drain problem. Recall that if we have a stock with a price today of $100, and it has equal chances to go up or down on any given day by 10%, then, in 100 days, the price most likely will go down to $60.5. In other words, pure chance translates into a predominant loss.
We conduct a simulation (10,000 iterations) of the Volatility Drain using Python. Below we show the resulting distributions of (1) the daily (percentage) returns of a hypothetical stock, and (2) the benchmark returns where we compare the final price on day 100 to the initial price of $100.
As can be seen, the daily returns follow the bell shape common to the normal distribution, while the benchmark returns’ distribution is highly skewed.
One way to verify heavy-tailed distributions is by plotting the logarithm of the data vs. the logarithm of the complementary cumulative distribution function, or the “log-log plot.” The linear curve of the benchmark returns indicates a heavy-tailed distribution. The concave curve of the daily returns on the other hand suggests the distribution does not exhibit a power law. (See: Nair et al., 2022, p. 184.)
As a reference, below is the log-log plot for normal (0, 1) and Pareto (α = 2) distributions. The similarity of the daily returns to the normal distribution on one hand, and the benchmark returns to the Pareto distribution, on the other, is striking.
Finally, one common method to quantify the heaviness of the tail of a distribution is the Hill estimator (Nair et al., 2022, pp. chapter 9). A larger value of the estimator (usually above 4) reflects thinner or lighter tails. The Hill estimator (with k = 10%) for daily returns is 10.58. For comparison, for a normal distribution (0, 1), it is 13.68.
On the other hand, the Hill estimator for the benchmark returns is 2.25, while for a Pareto distribution (α = 2), it is 2.12. Again, the close values of the respective distributions are remarkable.
Mean vs. Median
One important feature of the normal distribution is that the mean and median are not only finite, but they coincide and have the same value. For heavy-tailed distributions, in contrast, the mean can be undefined or infinite, unlike the median as we shall see.
For empirical samples whereby the mean and median can be readily computed, heavy-tailed distributions will exhibit a significant gap between the two. More importantly, the sample mean may not generally converge as the sample size is increased. Instead, the sample mean may fluctuate without reaching a stable value. The indeterminacy of sample mean may have devastating consequences, as we shall see.
The graph above shows the evolution of the mean vs. the median of the empirical distributions of benchmark returns. As can be seen, the empirical mean does not converge to a stable value. Akin to fractal roughness, it seems there is no reliable way to measure the empirical mean of a heavy-tailed distribution.
Like fractal roughness, the empirical mean of heavy-tailed distributions defies reliable measurement
Surprisingly, the empirical median quickly converges to the exact value of the formula stated earlier for Pₙ. (This is especially true if the number of days n was even. For an odd number, it may not converge.)
To Bet or Not To Bet …
Here is a simple example of how the two rules, the mean and the median, might lead to starkly different outcomes.
Consider a lottery with a prize of $10 million. The lottery ticket costs $1 and has a probability of 1 in 10 million to win the award. Should a rational person buy this ticket?
The expected value of the lottery is calculated by multiplying the probability of each possible outcome of the lottery times that outcome. We have only two possible outcomes: (1) To win $10 million, or (2) to win nothing. Hence, the expected value of the lottery L is:
Since the expected value of the lottery is $1 and the cost is also $1, then a rational decision-maker following the mean value will be neutral on whether to buy the ticket or not. But we will get a different result if we follow the median rule.
According to Morris DeGroot in his Probability and Statistics (2011, p. 241), a median m of a continuous or discrete distribution of a random variable X needs to satisfy two conditions:
- Pr(X ≤ m) ≥ Pr(X > m)
- Pr(X ≥ m) ≥ Pr(X < m)
With this definition, notes DeGroot, every distribution has at least one median. This is in contrast to the mean which may not exist for some distributions, especially those with heavy tails (Nair et al., 2022, pp. 12-13).
The reader can verify that, of the two possible outcomes of the lottery, x₁ = 10⁷, and x₂ = 0, only zero satisfies the above two conditions for the median. Since the cost of the lottery ticket is $1, then a rational person following the median rule shall not purchase the ticket.
Lotteries are a good example of a highly skewed discrete distribution. Other examples include the Zipf and Sierpinski triangle distributions. A particularly good example is the St. Petersburg Gamble, where the payoffs are infinitely positively skewed. For this gamble the mean is undefined, note Hayden and Platt (2009, p. 4). The median would be a more reliable estimator of the central tendency of the distribution.
Burn, Baby, Burn!
When risk is fractal, there is a good chance that a vicious circle will develop where a heavier tail leads to a higher sample mean. In the competitive pursuit of higher profits, traders willingly create instruments with increasingly highly skewed payoff distributions.
The table below shows how, in 10,000 simulations (with a controlled seed number), the Volatility Drain may lead to highly skewed payoffs. As the magnitude of the gain or loss, r, rises, the empirical mean of benchmark returns increases, in stark contrast to the median.
As volatility increases, the empirical mean increases but the median return deteriorates. The empirical distribution becomes even more skewed with heavier tails as indicated by the Hill estimator. Heavier tails, in turn, involve more extreme returns and thus a more unstable empirical mean. This opens the door for an unsustainable vicious circle that must end in a crash to correct the market misbehavior.
This shows how the assumption of normal distribution (and light-tailed distributions in general) in a competitive environment, can lead to what Hyman Minsky (1986) called the “thrust towards fragility.” This thrust, however, is not inevitable. The Efficient Market Hypothesis, with its religious endorsement of the normal distribution, played and continues to play a critical role in the frequent bubbles and crashes of financial markets.
In his The Misbehavior of Markets, Benoit Mandelbrot (2004) explains his endeavors over more than 40 years to refute the foundational assumptions of the Efficient Market Hypothesis. The Global Financial Crisis proved that he was closer to truth than his opponents.
Toxic Assets, Nuclear Waste, and Financial Alchemy
The credit bubble ahead of the Global Financial Crisis witnessed all kinds of “financial alchemy,” as Nobel laureate Joseph Stiglitz (2010) and former governor of the Bank of England, Mervyn King (2016), among others, would describe it.
One clear example is “tranching,” whereby a pool of loans of various credit qualities are structured in a manner to produce tranches with different credit ratings: AAA, AA, BB, B-, and, at the bottom, the “equity” tranche. In simple terms, the risk of each tranche is shifted to the one below, until all risks are concentrated at the bottom (equity) tranche.
As Charles Morris (2008, p. 41) and Frank Partnoy (2009, p. 121) report, these junior tranches were known on Wall Street, long before the crisis, as “nuclear waste” or “toxic waste.” They pay high yields but with a considerable risk.
Toxic assets exemplify the disastrous consequences of fierce competition armed with flawed models. These models often create severely skewed distributions with rare extreme profits but substantial median losses.
Without these “nuclear waste” or “toxic assets,” the whole securitization structure would have not been possible. This “alchemy,” as Stiglitz (2010) explains, played a key role in precipitating the Global Financial Crisis.
Conclusion
“Chaos theory and fractal geometry have corrected an outmoded conception of the world,” note Peitgen, Jürgens, and Saupe (2004, p. vi). Economic theory, judged by the Global Financial Crisis, seems to be still struggling with some outmoded conceptions.
Former governor of the Bank of England, Mervyn King (2016), argues that the Global Financial Crisis was “a failure of a system and the ideas that it represented.” At the heart of that system are flawed models adopting normally distributed risks and returns.
The normality assumption denies the recursion structure of risk and the inherent positive feedback loop that determines the behavior of the market. This denial leads to excessive leverage and speculation, resulting in a higher frequency of extreme events and greater financial instability.
By acknowledging the fractal nature of risk, we would be cautious against positive feedback mechanisms that destabilize the market. We would create an environment conducive to productive investments rather than debt-fueled, lottery-like, speculative bets.
Reference
- DeGroot, M., & Schervish, M. (2011). Probability and statistics (4th ed.). Addison-Wesley.
- Hayden, B., and M. Platt (2009). The mean, the median, and the St. Petersburg paradox. Judgment and Decision Making, 4, 256–272.
- King, M. (2016). The end of alchemy: Money, banking, and the future of the global economy. W.W. Norton.
- Mandelbrot, B. (1983). The fractal geometry of nature. W.H. Freeman & Co.
- Mandelbrot, B. (1989). Fractal geometry: What is it, and what does it do? Proceedings of the Royal Society of London: A, 423, 3–16.
- Mandelbrot, B., & Hudson, R. (2004). The misbehavior of markets: A fractal view of risk, ruin, and reward. Basic Books.
- Minsky, H. (1986). Stabilizing an unstable economy. Yale University Press.
- Morris, C. (2008) The two trillion dollar meltdown. PublicAffairs.
- Peters, O., & Gell-Mann, M. (2016). Evaluating gambles using dynamics. Chaos, 26, 1–9.
- Nair, J., Wierman, A., & Zwart, B. (2022). The fundamentals of heavy tails: Properties, emergence, and estimation. Cambridge University Press.
- Partnoy, F. (2009). Infectious greed: How deceit and risk corrupted the financial markets. (2nd ed.). PublicAffairs.
- Peitgen, H-O., Jürgens, H., & Saupe, D. (2004). Chaos and fractals: New frontiers of science. (2nd ed.). Springer.
- Poundstone, W. (2013). The recursive universe: Cosmic complexity and the limits of scientific knowledge (2nd ed.). Dover Publications.
- Smith, P. (2013). An introduction to Gödel’s theorems (2nd ed.). Cambridge University Press.
- Stiglitz, J. (2010). Free fall. W.W. Norton.
