Lesson 15 of 25 · 6 min

The same setup simulated normal and fat-tailed

Matching the mean and the variance makes the heavy-tailed run quieter in the middle and much worse at the edge.

The expectation most people carry into this comparison is that fat tails make every number worse. Run it properly, with the mean and the variance matched exactly, and that is not what happens.

Same facility as lesson 13: $1,470,000 of collateral, a $742,000 balance, a 65 percent threshold, a trigger value of $1,141,538, three years, 6.4 percent drift, 23 percent volatility, 200,000 runs. One run uses normal monthly draws; the other uses fat-tailed draws scaled so the realised volatility is 23 percent in both, confirmed at 23.00 and 23.01 percent.

The median ending collateral is $1,646,163 against $1,647,286. The fifth percentile is $855,545 against $860,911 — the fat-tailed run is marginally better. The probability of touching the trigger at any point is 37.08 percent against 33.74 percent, again better under fat tails. And then the one-tenth percentile is $486,498 against $396,606, and the worst path out of 200,000 ends at $308,418 against $19,034.

The mechanism is that a fixed variance is a budget. A distribution spending more of it on rare enormous moves has less left for ordinary ones, so typical months are quieter and a threshold 22.34 percent away is touched less often. What it buys is a small number of months the other cannot generate at all.

Fat tails did not make the margin call more likely. They made the outcome where the collateral no longer covers the balance possible. Those two events have different cures, and only one of them survives the sale.

Key terms

Variance budget: The idea that a fixed total variance must be allocated across move sizes. Putting more into the extremes necessarily leaves less for ordinary months.
Moment matching: Calibrating two distributions to the same mean and variance so any remaining difference in results is attributable to shape alone.
Deficiency: The state in which collateral is worth less than the balance it secures. Distinct from a breach, because selling the collateral does not resolve it.

Worked example

Identical mean, identical variance, different shape

200,000 runs each. Realised volatility 23.00 percent and 23.01 percent, confirmed from the simulated paths.

Median ending collateral: $1,646,163 / $1,647,286
Fifth percentile: $855,545 / $860,911
Touch probability: 37.08% / 33.74%
One-tenth percentile: $486,498 / $396,606
Worst of 200,000 paths: $308,418 / $19,034
Worst single month: −30.99% / −97.88%

The fat-tailed run was better on four of the six most commonly quoted figures and catastrophically worse on the two nobody quotes.

Check yourself

Two simulations of the same facility use the same drift and volatility. The fat-tailed one reports a lower probability of touching the trigger, 33.74 percent against 37.08 percent. Is the fat-tailed model less risky?

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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.