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I know that the pdf of a power law distribution is $$ p(x) = \frac{\alpha-1}{x_{\text{min}}} \left(\frac{x}{x_{\text{min}}} \right)^{-\alpha}$$

But what does it intuitively mean if, for example, stock prices follow a power law distribution? Does this mean that losses can be very high but infrequent?

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This is an heavy tailed distribution, since the cdf is $$ F(x) = 1 - \left( \dfrac{x}{x_\min} \right)^{1-\alpha} $$ So the probability to exceed $x$, $(x/x_\min)^{1-\alpha}$ can be made arbitrarily close to $1$ by the proper choice of $\alpha$. For instance, if one wants the probability to exceed $10^u x_\min$ to be at least $0.9$, one should pick $\alpha$ to be at most $$ 1-\log_{10}(0.9)/u $$ a curve represented below, with the first axis being scaled by $u$, not by $10^u x_\min$... R curve rendering of the above function

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It's not a peer-reviewed source, but I like this note by CMU stats professor Cosma Shalizi. He's also an author on this article, about estimating such things from data.

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  • $\begingroup$ That is why I asked my question. I read that article already. Without equations, what does it mean for something to follow a power law distribution? $\endgroup$ – Thomas James Feb 4 '12 at 17:16
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    $\begingroup$ Welcome to the site, Thomas! You might consider editing your question to give some indication of what piqued your interest initially. Generally, the more information, the better. For example, stating that you'd read Prof. Shalizi's note and it made you wonder about X not only preempts answers that would suggest precisely that, but also shows more clearly your train of thought, which tends to elicit better answers. :) (For example, have you read M. Mitzenmacher's review article in Internet Mathematics?) $\endgroup$ – cardinal Feb 4 '12 at 17:32
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The paper Power Laws in Economics and Finance may help gaining intuition about power laws. Xavier Gabaix states that power law (PL) is the form taken by a large number of surprising empirical regularities in economics and finance. His review surveys well-documented empirical PLs regarding income and wealth, the size of cities and firms, stock market returns, trading volume, international trade, and executive pay.

Intuition for the Pareto distribution

Pareto (wikipedia) originally described the allocation of wealth among individuals: a large portion of the wealth of any society is owned by a small percentage of people. His idea expressed more simply as the Pareto principle or the "80-20 rule" says that 20% of the population controls 80% of the wealth.

The right tail of income and wealth distributions often resemble Pareto

If income distribution is Pareto, then one can derive simple expressions for the share of top 1%, or top 10%. Then the top qth percentile's share of total income can be derived as:

$$\left(\frac{q}{100}\right)^{\frac{\alpha-1}{\alpha}},$$

where $\alpha \geq 1$ is the shape parameter. This expression implies that a lower $\alpha$ corresponds to a thicker tail of the Pareto distribution and thus to a greater share of total income being captured by individuals at higher percentiles of the distribution. For example, with $\alpha=2$, the top 1% share is 10%, and with $\alpha=3$, it is 4%.

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One interesting property of the power-law distribution comes from looking at it on a log-scale. If we have $X \sim \text{Power}(x_\min, \alpha)$ then the logarithmic transformation $Y = \ln(x/x_\min) \sim \text{Exp}(\alpha-1)$. That is, the values of $X$ have an exponential distribution on the logarithmic scale.

Now, one important property of the exponential distribution is that it has a constant hazard rate. Writing out the hazard rate for $Y$ via first-principles (as a conditional density in its limit form), and adjusting it to frame it in terms of $X$ we obtain:

$$\begin{equation} \begin{aligned} \alpha -1 = \lambda_Y(y) &= \lim_{\epsilon \downarrow 0} \frac{1}{\epsilon} \cdot \mathbb{P}(y \leqslant Y \leqslant y + \epsilon| Y \geqslant y) \\[6pt] &= \lim_{\epsilon \downarrow 0} \frac{1}{\epsilon} \cdot \mathbb{P}(\ln(x) \leqslant \ln(X) \leqslant \ln(x) + \epsilon| X \geqslant x) \\[6pt] &= \lim_{\epsilon \downarrow 0} \frac{\mathbb{P}(x \leqslant X \leqslant x e^\epsilon | X \geqslant x)}{\epsilon} \\[6pt] &= \lim_{\delta \downarrow 1} \frac{ \mathbb{P}(x \leqslant X \leqslant \delta x | X \geqslant x)}{\ln \delta}. \\[6pt] \end{aligned} \end{equation}$$

We can see from this hazard characterisation that $\mathbb{P}(x \leqslant X \leqslant \delta x | X \geqslant x) \approx (\alpha-1) \ln \delta $ for any small values of $\ln \delta$. Notice that this probability does not depend on the conditioning value $x$, which is the result of the constant-hazard property. Hence, for any conditioning values $x, x' > x_\min$, and any small value $\ln \delta$, we have:

$$\mathbb{P}(x \leqslant X \leqslant \delta x | X \geqslant x) \approx \mathbb{P}(x' \leqslant X \leqslant \delta x' | X \geqslant x').$$

Hence, we see that the power-law can be characterised by the fact that this conditional probability is approximately the same regardless of the conditioning point. In the context of stock prices, if these follow a power-law then we can say that, the probability that the stock will "rise" by some proportion is not dependent on its present value$^\dagger$.


$^\dagger$ We use "rise" loosely here, since we are talking about a single random variable, and we have not modelled a time-series of stock prices. Within out present context we refer to the probability of a "rise" in the stock price in the sense of a conditional probability that the price is within some interval above a lower bound, conditional on this lower bound.

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