Lesson 3 of 25 · 7 min
How not-normal returns are, stated in counts of days
At three percent the error is a factor of nine; at seven percent it is hundreds of millions.
Track 4 lesson 3 established that returns are not normal and stated the discrepancy in sigma counts. Counts of days are harder to wave away, because a count is a thing you could in principle stand there and tally.
Take an illustrative century of a broad equity market: 25,200 trading days, with a daily standard deviation fitted to the ordinary days of 1.0 percent, which annualises to about 15.9 percent. In this illustrative record there are 312 days worse than −3 percent, 73 worse than −5 percent, and 28 worse than −7 percent. These counts are invented for the lesson, but their relative sizes are the point and they are not generous.
The normal model, given the same 1.0 percent standard deviation, predicts 34.0 days worse than −3 percent, 0.0072 days worse than −5 percent, and 0.000000032 days worse than −7 percent. Set those beside the record: the model is short by a factor of 9.2, then by a factor of 10,100, then by a factor of 868 million. The error is not constant; it explodes as you move out.
A natural response is that the standard deviation was fitted too narrowly. Widen it to 1.15 percent, a 15 percent increase, and the counts become 114.5, 0.173 and 0.0000145 days. The three-percent error falls from 9.2 to 2.7, which is real improvement. The seven-percent error falls from 868 million to 1.93 million, which is not. Rescaling repairs the shoulders and leaves the tail untouched, because the problem is the shape and not the width.
A haircut sized to survive the worst day a normal model admits is sized against the 34-day count, not the 312-day one. The two differ by a factor of nine at a level that occurs several times a decade.
Key terms
- Exceedance count
- The number of days on which a return was worse than a stated level. A direct observation, and the least arguable form of tail evidence.
- Scale versus shape
- Whether a model fails because its width is wrong or because its form is wrong. Widening a normal distribution fixes scale errors and cannot fix shape errors.
- Fitted standard deviation
- A volatility estimated from a sample. Fitting it to ordinary days and fitting it to the whole record give different numbers, and neither repairs the tail.
Worked example
Predicted days against counted days, over an illustrative century
25,200 trading days. Illustrative record beside a normal model with a 1.0 percent daily standard deviation.
- Worse than −3%
- 312 counted, 34.0 predicted
- Worse than −5%
- 73 counted, 0.0072 predicted
- Worse than −7%
- 28 counted, 0.000000032 predicted
- Refit at 1.15% daily
- 114.5 / 0.173 / 0.0000145
- Errors after refitting
- 2.7× / 421× / 1.93 million×
- What the refit achieved
- The shoulders, not the tail
Short by 9.2×.
Short by 10,100×.
Short by 868 million×.
Fifteen percent more width cut the three-percent error by two thirds and left the seven-percent error at nearly two million times.
Check yourself
A normal model with a 1.0 percent daily standard deviation predicts 0.0072 days worse than −5 percent per century, and the illustrative record contains 73. Someone proposes fixing this by raising the volatility input. What happens?
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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.