Lesson 13 of 25 · 6 min
What a simulation inherits from its input distribution
Clip the random draws at two and a half sigma and every headline number survives; only the disaster disappears.
Track 4 lesson 10 set out the mechanics of a simulation step by step. The question left open is which outputs are properties of the machinery and which are copies of the input, and it can be settled by running the same machinery three times.
The setup, used for the next five lessons: collateral $1,470,000, a balance of $742,000, a maintenance threshold of 65 percent of market value, three years in monthly steps, a drift of 6.4 percent, a volatility of 23 percent, and 200,000 runs. The trigger value is $742,000 ÷ 0.65 = $1,141,538, which is a decline of 22.34 percent from where the collateral starts.
Three runs, identical in every parameter, differing only in the shape of the monthly draws. First, ordinary normal draws. Second, normal draws clipped at 2.5 standard deviations and rescaled so the volatility is unchanged. Third, fat-tailed draws rescaled to the same 23 percent.
The outputs, side by side as normal, clipped and fat-tailed:
- Median ending collateral: $1,646,163, $1,646,399, $1,647,286 — a spread of 0.07 percent.
- Fifth percentile: $855,545, $855,535, $860,911.
- Probability of touching the trigger value: 37.08 percent, 37.16 percent, 33.74 percent.
- Worst single month across all runs: −30.99 percent, −15.19 percent, −97.88 percent.
Everything from the median through the fifth percentile is the machinery and barely moves. The worst month is nothing but the input. A simulation cannot produce a month it was never allowed to draw, and clipping the draws hid the catastrophe without disturbing a single headline figure.
Key terms
- Innovation
- The random draw applied at each step of a simulated path. Its distribution is an assumption, not an output.
- Clipped draws
- Random numbers truncated at a chosen number of standard deviations. Used here to demonstrate that removing the extreme leaves the ordinary results untouched.
- Inherited property
- An output that reflects the input distribution rather than the model's structure. Deep percentiles and worst paths are inherited; medians are not.
Worked example
One machine, three inputs
Collateral $1,470,000, balance $742,000, threshold 65 percent, three years, 6.4 percent drift, 23 percent volatility, 200,000 runs each.
- Median, three inputs
- $1,646,163 / $1,646,399 / $1,647,286
- Fifth percentile
- $855,545 / $855,535 / $860,911
- Touch probability
- 37.08% / 37.16% / 33.74%
- First percentile
- $654,634 / $655,769 / $637,470
- One-tenth percentile
- $486,498 / $490,559 / $396,606
- Worst single month
- −30.99% / −15.19% / −97.88%
Indistinguishable.
The shapes finally separate.
Four of these six rows are the same three times over. The separation begins at the one-tenth percentile and is total at the worst month.
Check yourself
A simulation is rerun with its random draws truncated at 2.5 standard deviations. The median, the fifth percentile and the touch probability are all unchanged. What has actually changed?
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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.