# Estimating distribution of rate of return

Let $$f[t]$$ be the price of a stock at time $$t$$. We can calculate the rolling rate of return of the stock in a window of length $$n$$ by computing: $$r[t] = \frac{f[t] - f[t-n]}{f[t-n]}$$ $$r[t]$$ is serially correlated, since neighboring values overlap by $$n$$ samples. I want to estimate a distribution for $$r[t]$$ empirically that is unaffected by autocorrelation. One way to do this is to thin the series (i.e., sample every $$n$$th value). This effectively means the windows used to create the $$r[t]$$'s that get sampled don't overlap.

1. How do I decide whether to select samples $$1, n+1, 2n+1, \ldots$$ or $$2, n+2, 2n+2, \ldots$$ or $$3, n+3, 2n+3, \ldots$$, and so on?
2. Is there a way to make use of all the samples (e.g., by creating distributions from each of the sets of samples above and then combining them) that is not sensitive to the autocorrelation?

I have done that. The distribution if there are no dividends, mergers or bankruptcy and if liquidity costs are ignored is $$\Pr(r|r^*;\gamma)=\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{r^*}{\gamma}\right)\right]^{-1}\frac{\gamma}{\gamma^2+(r-r^*)^2}.$$

It has no expected value. You can find a reduced form discussion here https://youtu.be/R3fcVUBgIZw.

If you attempt to tackle it directly as a ratio distribution, you end up needing data we never collected.

You can find a general solution here.

Autocorrelation is irrelevant for this discussion. Autocorrelation is an artifact of certain time series methods but not others. For example, Bayesian methods are not impacted by autocorrelation, so it is all but ignored. It is only important in Frequentist statistics because it interferes with inferences.

Your formula is not a time series. If it were, then a convolution would first have to happen in the numerator for the difference. However, as the right side of the numerator is also the denominator, it is no different than subtracting one. It is a shift variable and the difference of the values in the numerator can be ignored.

Your largest problem will not be autocorrelation, which can be ignored, but structural breaks due to interest rate, capital structure, dividend, tax and market changes. I strongly recommend a Bayesian method as you are almost only restricted to Theil's regression or quantile regression if you find that too computationally expensive.

There is no expectation so you cannot use squares minimizing routines, so also, no variance.

• Interesting answer. Do I understand correctly that you apply the Cauchy-distribution (in the formula with tan above). Which assumes independence (I assume, it does) of the numerator and denominator and additionally a Gaussian distribution for each of them.
– Ric
Apr 21 at 8:08
• @Ric I do not assume it. It follows from auction theory. If it were something like an accounting ratio, or if the trades were close in time in a thin market, then independence does not make sense. Apr 22 at 2:17
• @Ric that ignores dividends, liquidity costs, mergers and bankruptcy. Apr 22 at 2:18
• Thanks, DaveHarris and @Ric, for your answers. I'm reading your responses carefully and may need a couple of days to process. Will get back to you soon. Apr 22 at 4:34