# Fitting (marginal/multivariate) distributions to financial return data

I have calculated the simple arithmetic return on a number of different financial securities and am fitting both a Student-T and Generalised Pareto Distribution.

My question is can I just use the simple returns as my input observations or is it better practice to convert the data into log-normal (i.e. take the exponential) or standardise the data (subtract mean and divide by standard deviation).

I know this may seem like a very simple question, but I am looking for the soundest statistical practice and why.

• Use log-returns. Pareto and t-dist probably won't fit to most stocks. Rather they're usually logistic, normal, and sometime Log-normal. Once and a while they are Cauchy.
– user55753
Sep 21, 2021 at 18:37

I would personally go for a normal returns, because you do not make any assumptions about the data or returns.

When we use log returns we assume that prices are distributed log normally (which, usually is very far from the truth).

Moreover if you will investigate different distribution you will not use the log returns features like time additivity or approximate raw-log equality.

And if you think about T-Student Distribution this is worth considering:

Mathematically there’s a problem: when you assume a student-t distribution (a standard choice) of log returns, then you are automatically assuming that the expected value of any such stock in one day is infinity! This is usually not what people expect about the market, especially considering that there does not exist an infinite amount of money (yet!). I guess it’s technically up for debate whether this is an okay assumption but let me stipulate that it’s not what people usually intend.

This happens even at small scale, so for daily returns, and it’s because the moment generating function is undefined for student-t distributions (the moment generating function’s value at 1 is the expected return, in terms of money, when you use log returns). We actually saw this problem occur at Riskmetrics, where of course we didn’t see “infinity” show up as a risk number but we saw, every now and then, ridiculously large numbers when we let people combine “log returns” with “student-t distributions.” A solution to this is to use percentage returns when you want to assume fat tails.

• Thanks @Robert I agree re: log returns. What about your opinion on standardising/scaling data. My inclination was to not standardise as I am not assuming a standard-normal, however from various other posts there seems to be a suggestion that non-standardised data will not generate a good multivariate distribution fit (I am modelling it in R) Oct 6, 2015 at 5:48
• Glad to help. You can try a safe approach. Try with non-standardized data and check the distributions (if they fit). If you will be unsatisfied with the results, try to standardize them. It should not be very hard and time consuming to implement the second option when you have the first and vice verca. Oct 6, 2015 at 8:05