Text / Yao Bin
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Back in the 1990s, Michael Maubousson, one of Meglaison Asset Management’s “Big Three” (the others being Bill Miller and Robert Hagerstrong), worked in Santa Claus. Study at the Philippine Institute. At that time, he knew the importance of the tail events of the fat tail effect.
In our real world, the vast majority of events depend on the tails of distributions, physicist Philip Anderson, one of the founders of the Santa Fe Institute, wrote in his article “On Distribution Theory in Economics” (limit state), not the mean or average; it depends on exceptional times, not the average; it depends on catastrophic events, not steady ups and downs; it depends on a handful of rich people, not the “middle class.” Therefore, we must free ourselves from the idea of ”averaging”.
Mauboussin pointed out the origin of the “tail distribution” theory in “Devil Investments”, which is the deep thinking of the “normal distribution” by the complexity scientists in Santa Fe. Traditional financial theory treats price changes as independent distribution variables and assumes that returns follow a normal, or logarithmic, distribution. The advantage of this assumption is that investors can use probabilistic calculations to know the mean and variance of the distribution, and thus can predict the percentage of price change with statistical accuracy. A large part of this assumption is reasonable. However, traditional standard financial theory is not suitable for extreme situations. As Philip Anderson said, what determines the real world is the tail of the distribution. The defeats of LTCM and Victor Niederhof may explain why.
The normal distribution is the basis of the “random walk” theory, the pricing of capital assets, the value at risk, and the Black-Scholes option pricing model and other financial theories. For example, the purpose of a value-at-risk model is to quantify the losses that a portfolio may suffer under a given probability. Although value-at-risk models come in different forms, the most basic version assumes the standard deviation as a measure of risk. When the normal distribution is known, we can intuitively measure the standard deviation and the risk level represented by the standard deviation, but if the price fluctuation does not conform to the normal distribution, measuring the risk level with the standard deviation may be misleading. Financial economists assume that fluctuations in stock prices are random, like the movement of pollen in water under the shock of molecules.
The so-called fat-tailed distribution refers to the fact that the probability of large price fluctuation events is much greater than the statistical distribution of the traditional normal distribution. The tails are significantly larger than the normal distribution, hence the name “fat tails”. The extreme value swings of the fat-tailed distribution occur far more frequently than predicted by the standard normal distribution model, and this has a significant impact on portfolio performance, especially for leveraged portfolios. For example, in October 1998, the S&P 500 plunged by more than 20% and deviates from the mean by 20 standard deviations. To this end, Roger Lowenstein, author of Saving Wall Street, pointed out: “Economists later discovered that, based on historical data on market price fluctuations, even assuming that the market existed from beginning to end from the beginning of the universe, this collapse occurred on this day. The probability of such a crash is also extremely small. In fact, even if the lifespan of the universe is repeated 1 billion times, such a crash is theoretically ‘impossible’.”
Self-organizing criticality refers to the critical state in which complex systems undergo mutation. A fat-tailed distribution is a sign of a system in a state of “self-organizing criticality”, which is the result of interactions between individual mediators (in securities systems, investors) that do not require any leader to be organized. core. In critical conditions, small-scale fluctuations can lead to many types of events. Self-organizing criticality is characteristic of many complex systems such as earthquakes, power outages, and traffic jams. Markets generally function effectively when sufficiently diverse investors interact. Conversely, when this diversity is disrupted and investors all adopt similar patterns of behavior, markets tend to become extremely vulnerable. A large body of literature on the “herd effect” illustrates this phenomenon. The herd effect is when a large number of investors make the same choices based on observations of the behavior of others, ignoring their existing knowledge entirely.
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Most states of nature—including artificial stock markets—do not conform to the concept of a normal distribution. Many natural systems have two characteristics: small-scale components are increasing in number, and elements of the same shape are distributed in the system at different scales. For example, a tree with a large trunk and many small branches, smaller branches… The small branches are basically the same in shape as the relatively large branches. That is, these systems are fractal. Unlike the normal distribution, no mean value can adequately represent the actual characteristics of a fractal system.
Using the normal distribution to characterize the fractal system of financial markets can be very dangerous, but theorists and practitioners still use this tool with pleasure. The difference between a normal system and a fractal system ultimately boils down to probability and reward. Fractal systems have a small number of large observations that do not fit a normal distribution. The most classic example is the stock market crash in 1987. The odds of market yields falling by more than 20% in a single day are slim and practically nil. At the same time, the stock market value has reached a stifling $2 trillion.
The same can be said by comparing the normal coin tossing game with the St. Petersburg game. Suppose the coin lands heads up, and you get $2; if it lands tails, you get nothing. The expected value of the game is $1, which is also what you are willing to pay to take part in this fair game. Assuming a simulation of 1 million coin toss trials, 1 million toss per round, and plotting the payoff, we get a near-perfect normal distribution, as expected. Although the underlying process is random, the distribution of the results follows the power law. For example, in a game, the probability of getting $2 is 50%, and the probability of getting a return under $4 is 75%. At this probability, 30 rounds of experiments can achieve a maximum return of $1.1 billion, but we only have a chance of achieving such a stunning return in 1.1 billion experiments. Fractal systems are characterized by a large number of small-scale events and few large-scale events. In the St. Petersburg game, the payoff per game is also erratic, so any average cannot describe the long-term outcome of the game.
Benoit Mandelbrot, the founder of fractal geometry, pointed out that as long as the speed of price change is accelerated or slowed down, the price series will show significant fractal characteristics. Infrequent large price movements are scattered among many small price movements, while price movement curves are very similar in shape, differing only in proportion (e.g. daily return, weekly return or per monthly income). Mandelbrot called this time series of financial returns a “multifractal”.
In “Why the Stock Market Crash”, geogeologist Didier Solnet points out that the price distribution of the stock market consists of two distinct sample populations: the main body (the part that can be modeled by standard theory) and the tail (depending on a completely different distribution mechanism). Solnet’s analysis of market downtrends convincingly disproves the assumption that stock returns are independent of each other, a cornerstone of traditional financial theory. His research provides a new and profound perspective on the shortcomings of classical financial theory:
(a) Causal thinking. One of the fundamental characteristics of self-organizing criticality systems is that there may be a nonlinear relationship between the amplitude of the waveform and the resulting consequences. In some cases, small-scale inputs can lead to large-scale events. This shatters our self-imposed attempts to find reasons for all outcomes.
(b) Risk and Reward. Nonlinearity is an inherent feature of self-organizing criticality systems such as stock markets. Investors must always keep in mind that financial theory should reflect data from the real world. Statistically widely used measures are not suitable for the market.
(c) Portfolio Structure. Designing a portfolio by the standard deviation may underestimate risk. This is especially notable for portfolios that use debt leverage to increase returns. Most failures for hedge funds are the direct result of fat-tailed (extremely low probability) events.
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In the 1930s, the Harvard linguist George Zipf noticed the number and frequency of all words used in novels. If you use a logarithmic scale to represent this distribution of word frequency, you will get a straight line extending from the upper left corner to the lower right corner. He summed it up in his handed down book, The Principle of Least Effort: An Introduction to the Ecology of Human Behavior, and came up with “Zipf’s Law”. This law states that we will see a small number of frequently occurring words and a large number of infrequent words in an article, thus forming a power-law distribution.
Since Zipf published his findings, scientists have discovered power laws that exist in many fields, including physical and biological systems. For example, scientists have used power laws to explain the relationship between animal body weight and metabolic rate, earthquake frequency and amplitude, and avalanche frequency and magnitude. Power-law distributions also play a pivotal role in social systems, such as changes in income distribution, city size, Internet traffic, company size, and stock prices. In the field of financial investment, according to the power-law distribution, the cyclical stock price fluctuations that occur infrequently are far larger than those predicted by traditional financial theory. This fat tail phenomenon is important for determining portfolio structure and leverage. At the same time, power-law distributions also reveal some fundamental rules in self-organizing systems. Although the scientific community has not been able to fully explain the mechanisms that cause power-law phenomena in social systems, there is enough evidence to show that we can indeed make structural predictions about the future state of certain systems through power-law distributions.
Zipf described the laws he advocated with the following formula: order × scale = constant. The formula shows that the measure of the subjects studied is inversely proportional to persistence. According to Zipf, we can multiply a sequence of numbers by a constant of 1, 1/2, 1/3, 1/4, and get a sequence. Take Spain’s urban population size, for example. If the largest city, Madrid, has 3 million inhabitants, then the second largest city, Barcelona, should have half the population of Madrid, the third largest city, Valencia, one third, and so on. However, this is clearly not in line with reality. Although Zipf can describe some systems well, its applicability is very limited, and many systems with power-law characteristics are not suitable for this formulation.
On the surface, the size and frequency distributions of species, cities, and firms appear to have little in common. But in fact, they all obey a power-law distribution, and in the logarithmic coordinate system, the distribution law between scale and frequency appears as a straight line. The power-law distribution tells us that the logarithm of size and frequency is inversely proportional, that is, large-scale events are rare, but small-scale events are common. In nature, the number of ants is countless, but the number of elephants is very small. Likewise, in the business world there are a lot of small-scale enterprises, while the number of large-scale enterprises is much smaller.
Why are large beasts such as tigers relatively rare, while smaller animals such as termites are everywhere? Ecologists explain the problem: all animals have their own habitat – a niche, which is not only embodied in spatial location, but even in the broader sense of physical existence, all living things have their own A world. Not only does a species need to survive in its own home, but it also needs to share that home with other animals and plants and learn to coexist with other animals.
However, the concept of niche still cannot explain the distribution law of species. Charles Elton of Oxford University has noticed that the survival of large animals depends on the presence of small animals, and the survival of small animals is rarely dependent on large animals. From this, Elton reasoned, as species increased in size, their numbers should have decreased. He called this biological phenomenon the “pyramid of numbers.” Large beasts are rare because their food sources are relatively scarce compared to smaller animals. The power-law distribution of species is an inevitable product of their interaction under the constraints of natural laws.
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Investors need to pay attention to such a special distribution law. because
(a) Businesses, like biological species, have a niche of their own. If we explore the current state of these niches and how they are changing, perhaps we can get a glimpse of the growth potential of businesses.
(b) There is substantial evidence that large firms are less variable in size than small firms (although the distribution of median growth rates across the population is quite stable). In addition, the growth of large enterprises is generally more sluggish. As a result, when investors recalibrate expectations in these markets, it often results in valuations that are significantly below average market performance.
(c) Investors often derive future growth based on past growth rates, so that even if growth continues, failure to meet initial expectations will still result in shareholder returns that are not satisfactory to investors. Given this bias in expectations, these unnecessary annoyances may not arise if investors recognize the true pattern of corporate growth.
Businesses need to find niches that meet their conditions. In much of the literature on competitive strategy, especially game theory, many aspects of these strategies address how firms should find profitable niche markets, and why. On this issue, the core idea is that as technology develops, regulations change, and new players enter and old firms exit the industry, the environment and thus the niche market is also changing over time. changes over time.
Imagine a small workshop-style processing plant and an integrated steel company, or contrast an Internet-based retailer with a traditional “bricks and mortar” type of enterprise, and the world is self-evident. Every time a new market gap emerges, new businesses flock to it. With the changing environment, this ability is the basic driving force for the development of enterprises, and enterprises with this ability are not common in reality. Therefore, for a particular industry, there may not be an optimal enterprise size at all, and enterprises with different business models are not comparable at all, and in this case, comparing their values is meaningless.
Mauboussin revealed four “indestructible” facts by studying the distribution of enterprise size and the level of growth rate:
①The distribution of firm size conforms to Zipf’s law. The most important thing for investors to realize is that the more this distributional law is in times of major economic change, the more it will show its suitability. That said, even in the future, the ratio of large to small firms is unlikely to change.
②The change of the growth rate of the enterprise decreases with the increase of the scale. In a large sample of US public enterprises, the median growth rate is relatively stable, while the change in growth rate decays rapidly with the increase of enterprise size. At scale, such a result is plausible—large corporations account for a sizable share of GDP, so it would be nearly impossible to achieve such projected growth rates if extrapolating from historical growth rates into the future. In GDP, the Fortune 500 has a share of more than 20%. Those that make it to the Fortune 500 tend to experience enviably strong growth, but that undoubtedly leaves investors with unrealistic expectations for the future. These empirical conclusions are fully consistent with models of stochasticity such as Gilbert’s Law. Gilbert’s Law, also known as the Law of Proportional Effects, states that a firm’s growth rate is completely independent of firm size. With some modifications, we can obtain a Zipf distribution from a sample of firms via Gilbert’s rule. However, classical microeconomics fails to provide us with satisfactory models to explain these observations.
③ The growth rate of large enterprises is often stagnant. Research by the Corporate Strategy Board (CSB) shows that once a company reaches a certain level of sales, its growth rate stagnates. At the end of the decade, the stagnation level has remained between 20 billion and 30 billion. That is to say, before entering the top 50 in sales, most of the companies showed high growth, but once they entered the top 50, there would be symptoms of weak growth. And the high growth rate in the first year reflects the huge effect of mergers and acquisitions on the company’s entry into the Fortune 500.
④Most industries have the same life cycle. At the beginning, an industry tends to show a trend of rapid growth, accompanied by a large number of enterprises entering the industry. Then, some enterprises exit strategically, while the surviving enterprises gain high-speed growth. level of economic returns. After that, the growth acceleration will gradually decrease. After the industry enters the mature stage, the growth rate of enterprises will become stable, and the economic rate of return will be close to the equilibrium level formed by competition. Large enterprises are mostly in the mature stage.
It is undeniable that large enterprises will certainly face greater difficulties in innovation than small enterprises, for a variety of reasons. In “The Innovator’s Answers,” published by Clayton Christensen in collaboration with Michael Ray, managers are provided with a basic framework for enabling innovation. However, one thing is beyond doubt, it is impossible for any enterprise to achieve permanent high growth.
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