Why Stock Return Ratios Do Not Necessarily Follow a Pareto Distribution
Summary
The document asks whether daily stock price ratios should follow a Pareto distribution and why their cumulative distribution functions do not appear linear on log-log plots. It describes plotting ratios for several stocks, first with both sides around a ratio of one and then with separate plots for declines and rises. The question highlights that a two-sided return distribution may need to be examined differently from a one-tailed wealth distribution.
The included answer compares histograms of price ratios, simple returns, and log returns for a small sample of stocks over roughly two years. It reports that fitting these representations produces several candidate distributions, including Cauchy fits. This illustrates that the chosen return measure affects the observed distribution and that a log-log plot alone does not establish a power law. The excerpt does not provide the plots, fitted parameters, formal goodness-of-fit tests, or enough evidence to identify a generally suitable distribution; its sample is limited and the question remains open.
Key ideas
- Price ratios, simple returns, and log returns are distinct quantities with different distributions.
- A two-sided return distribution should be assessed by considering its gains and losses separately when appropriate.
- A non-linear log-log cumulative plot is evidence against a simple Pareto fit, but does not identify an alternative distribution.
- The cited example reports varied fitted distributions, including Cauchy distributions, for a small stock sample.
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# Why stock prices changes don't follow Pareto Distribution?
# Why stock prices changes don't follow Pareto Distribution?
I calculated the distribution of the stock price changes (diffs). The diffs are multiplicative, $d_t=p_{t} / p_{t-1}$.
As far as I know the distribution should look like Power law distribution (Pareto distribution). With CDF being a line on log-log plot.
But the actual CDF is not looking like a line on log-log plot. Why?
I wonder could it be caused that price diffs distribution has two tails instead of one? It has two types of rare events, rare huge daily price drops with $d < 0.7$ and rare huge daily price rises with $d > 1.4$.
As far as I know the linear test for Power Law is used for one-tailed distributions. Like wealth distribution. Could it be also used for two-tailed distribution?
#### Example
The daily prices for 4 stocks for couple of years, normalised to be equal to 1 for the first day.
The CDF of daily diffs. The x axis is log scale, so the changes would look symmetrical around x = 1.
Let's plot it on log-log scale, and it's not looking like a line at all, nether one tail nor another.
On the previous log-log chart, one tail got crushed. So what I did instead I calculated two different CDFs, one for $d < 1$ and another for $d > 1$ and plotted it on log-log scale, so both tails could be seen. And there's same problem it's not looking like a line. Why?
P.S.
If it's not Pareto, what kind of Distribution could it be?
## Answer by user55753 (score 3)
https://quant.stackexchange.com/a/68042
For ~2 years of daily data for 3 stocks (SPY, AMD, BYND) a histogram of the ratio ($r_t=\frac{P_t}{P_{t-1}}$), simple returns ($r_t=\frac{P_t-P_{t-1}}{P_{t-1}}$), and log returns ($r_t=\log(P_t)-\log(P_{t-1})$) gives the following below. Fitting the distributions results in a variety of distributions, and 2 were Cauchy.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.