Lesson 6 of 25 · 7 min

Power-law behaviour, and the scaling that identifies it

Counting past three thresholds gives two independent estimates of one exponent; their agreement is the evidence, not the plot.

The previous lesson defined a fat tail by a constant ratio. A power law is the distribution for which that constancy is exact, and it has one parameter.

P(X > x) = (x ÷ x₀)^(−α)

x: The size of the move being asked about, as a percentage decline.
x₀: The level at which the tail behaviour begins. Below it the distribution can be anything.
α: The tail exponent. Lower means fatter; the whole character of the tail is in this one number.

The exponent can be recovered by counting, with no curve fitting. Take the illustrative century from lesson 3: 25,200 trading days containing 312 declines worse than 3 percent, 43 worse than 6 percent, and 6 worse than 12 percent. Each step doubles the threshold, so each ratio of counts is 2 raised to the exponent. From the first pair, α = ln(312 ÷ 43) ÷ ln 2 = 1.98186 ÷ 0.69315 = 2.859. From the second pair, α = ln(43 ÷ 6) ÷ ln 2 = 1.96944 ÷ 0.69315 = 2.841.

Two independent stretches of the tail returned 2.859 and 2.841. That agreement is the finding. Plotting the counts against the thresholds on logarithmic axes and observing a straight line is the same fact drawn rather than computed, and the drawing tends to be more persuasive than it deserves to be.

Take α = 2.85 and extrapolate one doubling further. Declines worse than 24 percent in a day should number 6 ÷ 2^2.85 = 6 ÷ 7.21 = 0.83 per century. The normal model, refitted at 1.15 percent daily, gives 2.2 × 10⁻²¹ of a day. The honest caveat is that the 12 percent bucket holds six observations and the exponent is estimated largely from them, so 2.85 carries real uncertainty — which is the subject of lesson 18.

A haircut is a quantile of this distribution. Under a normal tail, buying more confidence costs almost nothing in extra haircut; under an exponent of 2.85 it costs a great deal, and the difference is the whole argument about how large a haircut needs to be.

Key terms

Power law: A tail whose probability falls as a fixed power of the size of the move, so that doubling the move always costs the same factor.
Tail exponent: The number α in that relationship. Around 3 for equity returns in most estimates; below 2 would mean the variance does not even exist.
Log-log scaling: Plotting counts against thresholds on logarithmic axes, where a power law appears as a straight line whose slope is the exponent.

Worked example

Recovering one exponent from three counts

Illustrative century of 25,200 trading days. Counts of declines worse than each threshold.

Worse than 3%: 312 days
Worse than 6%: 43 days
Worse than 12%: 6 days
Exponent from the first pair: 2.859
Exponent from the second pair: 2.841
Extrapolated count worse than 24%: 0.83 days per century

Two estimates from disjoint parts of the tail agreed to within 0.02. That is what identifies a power law; a straight line on a chart is only the picture of it.

Check yourself

An illustrative record has 312 days worse than −3 percent and 43 worse than −6 percent. What exponent does that imply, and what does it predict for days worse than −12 percent?

Saved in this browser only. No account, nothing sent anywhere.

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.