Lesson 4 of 20 · 6 min

Fat tails, with a concrete example

The excess is not in the middle of the distribution; it is concentrated exactly where it hurts.

Fat tails is a phrase that gets repeated more often than it gets quantified. Quantifying it makes the consequence clear.

Take a borrower with $600,000 of collateral, a $300,000 balance and a 70 percent threshold. From Track 1, the trigger value is $428,571, which is a 28.6 percent decline. Now ask how likely that is over a year.

Under a normal distribution with a 7 percent expected return and 20 percent volatility, a 28.6 percent decline is about 1.8 standard deviations below the mean, which sits at roughly a 3.8 percent chance — about one year in 26. That figure is what a bell curve implies, and it sounds tolerable.

Now consult history rather than the curve. Declines of 28 percent or more in broad US equities have occurred in the early 1930s, 1937, 1973 to 1974, 2000 to 2002, 2007 to 2009, and briefly in 2020 — six or so episodes in about a century, which is closer to one year in sixteen than one in twenty-six. And that is for a diversified index. A concentrated or high-beta portfolio reaches the same decline on much smaller market moves, as Track 1 lesson 11 showed.

So the model understated the frequency by roughly sixty percent for an index, and by more for a typical individual portfolio. The lesson is not that the number should be one in sixteen — history is a small sample too. It is that the tail estimate is the least reliable part of any distribution and the part every leverage decision depends on.

Key terms

Tail event: An outcome far from the centre of the distribution. Rare individually, and the driver of most large losses.
Frequency understatement: The gap between how often a model says an extreme outcome occurs and how often it has occurred. For large equity declines the gap is a factor of two or more.
Small-sample problem: The difficulty that a century of history contains only a handful of tail events, so even the empirical frequency is imprecise.

Worked example

One trigger level, two estimates of how often it is reached (illustrative)

Collateral $600,000, balance $300,000, threshold 70%. Trigger value $428,571, a 28.6% decline.

Normal model, 7% mean and 20% volatility: 1.8 sigma below the mean
Implied probability in a year: About 3.8%
Episodes of ≥28% decline, broad US equities, last century: About six
Implied historical frequency: Roughly one year in 16
Understatement factor: About 1.6×
For a portfolio with beta 1.5: The same decline needs only a 19% market fall

The model said one in 28. History says closer to one in 16. For a high-beta portfolio the bar is lower still.

Check yourself

A normal model says a 28.6% annual decline has a 3.8% chance, roughly one year in 26. History suggests closer to one in 16. Why does this matter more to a leveraged holder than an unleveraged one?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.