Fat-tailed distributions are graphical representations of the probability of extreme events being higher than normal. In many domains fat tails are significant, as those extreme events have a higher impact and make the whole normal distribution irrelevant. That is the case when it comes to power laws. Therefore, understanding the properties of those extreme events become critical to business survival and success.
Understanding fat-tailed distributions
Typical bell curve graphs depict the probability distribution of data with the apex of the curve representing the mean, mode, or median. The width of the bell relative to the apex is determined by its standard deviation. This normally distributes the data and forms the shape of the bell curve with two “lean” tails of outlier data on either side.
Normal distributions can be analyzed to predict stock market volatility and make educated predictions around future stock prices. Bell curves can also be used by educators to compare test scores and also in the assessment of employee performance.
However, data are not always normally distributed. Some bell curves have fatter tails with a higher prevalence of data significantly different to the mean. Fat-tailed distributions are said to decay more slowly, allowing more room for outlier data to exist sometimes 4 or 5 standard deviations above the mean. As a result, extreme events are more likely to occur.
Lean tail curves, on the other hand, have distributions that decrease exponentially from the mean. This means that extreme events are highly unlikely, which helps to mitigate risk in a variety of situations.
Examples of fat-tailed distributions
Some of the more obvious fat-tailed distributions include:
- Wealth – mean annual income globally is approximately $2,000. Yet there is a high number of millionaires and even billionaires who are many, many standard deviations above this mean.
- Urban populations – the vast majority of cities worldwide have populations in the tens to hundreds of thousands, but the increasing prevalence of megacities such as Tokyo, Delhi, and Shanghai skews normally distributed data.
- Costs of natural disasters – climate change is increasing the severity of natural disasters, leading to higher insurance claims. For example, the costliest hurricane in the US was Hurricane Andrew in 1992 at $41.5 billion. Just 13 years later, Hurricane Katrina set a new record inflicting $91 billion worth of damage.
Implications for fat-tailed distributions in business
Normal distributions tend to understate asset prices, stock returns, and associated risk management strategies. This was highlighted during the 2008 Global Financial Crisis (GFC), where conventional financial wisdom was unable to predict fat tail risks brought about by unpredictable human behavior.
Insurance companies rely on normally distributed, historical data to generate profits. However, claims relating to flood and crop damage in particular are challenging historical assumptions of normal distribution. Health insurance claims are also rising as obesity rates soar in many developed western nations.
Companies that offer uncapped insurance contracts are at an increased risk of bankruptcy as climate change and more sedentary lifestyles challenge assumptions of lean-tail distribution.
- Book Publishing:
- Scenario: A publishing house is analyzing the sales of their books over a decade.
- Normal Distribution: A certain consistent number of books sell around 10,000 copies.
- Fat-tailed Distribution: Occasionally, a breakout bestseller, like a new fantasy series, might sell millions, skewing the average sales distribution. Most books don’t achieve this level of success, but those that do have a significant impact on the publisher’s revenue.
- Viral Videos:
- Scenario: A content creator uploads videos on a platform like YouTube.
- Normal Distribution: Most of their videos get a consistent 5,000 views.
- Fat-tailed Distribution: Occasionally, one video might go viral and achieve 5 million views. This outlier drastically affects the creator’s average view count and potential revenue.
- Startup Success:
- Scenario: An investor is examining the returns from their portfolio of tech startups.
- Normal Distribution: Many startups yield moderate returns or even fail.
- Fat-tailed Distribution: Occasionally, a startup might become the next unicorn, like Uber or Airbnb, and provide returns many times over the initial investment, overshadowing the performance of other investments.
- Natural Disasters:
- Scenario: An insurance company is assessing claims related to natural disasters over several years.
- Normal Distribution: Most years, claims remain within a certain predictable range.
- Fat-tailed Distribution: Some years, a catastrophic event like a super typhoon or a mega earthquake can lead to claims that are multiple times the average, affecting the insurance company’s profitability.
- Scenario: A hospital is analyzing patient admission rates.
- Normal Distribution: On most days, the hospital admits a consistent number of patients.
- Fat-tailed Distribution: Occasionally, events like a disease outbreak can lead to a sudden spike in admissions, requiring the hospital to mobilize extra resources.
- Stock Market:
- Scenario: An investor is examining stock returns.
- Normal Distribution: Most of the time, stock returns fluctuate within a certain expected range.
- Fat-tailed Distribution: Rare events, like a global financial crisis or a pandemic, can lead to extreme stock market crashes, causing severe losses for investors.
- Internet Traffic:
- Scenario: A website owner is analyzing daily website traffic.
- Normal Distribution: The website receives a consistent number of daily visitors.
- Fat-tailed Distribution: Occasionally, being featured on a popular site or getting shared by a celebrity can cause a surge in visitors, overwhelming the server.
- Fat-tailed distributions are found on bell curves with a greater prevalence of outlier data. These distributions suggest a higher probability of extreme events than would be typical in a normally distributed bell curve.
- Fat-tailed distributions decay more slowly than lean-tailed distributions, resulting in outlier data that is often 4 or 5 standard deviations above the mean.
- Fat-tailed distributions explain variation in the distribution of global incomes and urban population size. In the finance and insurance industries, external stressors are challenging historical assumptions of normal distribution and in turn, profit potential.
- Fat-Tailed Distributions: Graphical representations of the probability of extreme events being higher than normal, with a greater prevalence of outlier data.
- Normal Distributions: Typical bell curve graphs with lean tails and normally distributed data.
- Properties of Fat-Tailed Distributions: Decaying more slowly than lean-tailed distributions, resulting in a higher probability of extreme events occurring.
- Examples of Fat-Tailed Distributions: Wealth distribution with a high number of millionaires and billionaires, urban populations with increasing prevalence of megacities, and costs of natural disasters with rising severity.
- Finance: Normal distributions understate asset prices and risk management strategies, leading to challenges during financial crises.
- Insurance: Historical assumptions based on normal distributions face challenges due to climate change, flood and crop damage, and rising health insurance claims.
- Fat-tailed distributions have a higher probability of extreme events.
- They decay more slowly than lean-tailed distributions.
- They explain variations in global incomes and urban populations.
- In finance and insurance, challenges arise when dealing with extreme events.
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