# Probability using empirical distribution function of stock returns

Is it good idea to create empirical distribution of stock returns and make function of its density (using density() in R)? Then I can integrate it and have probability of value less than assumed. Another aproach is to model distribution using normal distribution but as we know stock returns are not normal distributed. What is better solution to get probability?

• What are you going to use the density for? – Dave Harris Jul 31 '18 at 2:09

I don't think it's a good idea to blindly use the density function; looks like it uses gaussian kernels by default, for Kernel Density Estimation.

This means its kurtosis is underestimated compared to stock returns. Which means you underestimate extreme results (such as, say, -5% or +5% in one day, for S&P500). A popular book about financial implications of this is The Black Swan.

What I would do instead, in general:

1. Optimize a distribution's parameters to fit your data; for R (which I don't use) it looks like you need fitdistr, or fitdist from the fitdistplus package.
2. Check the various distributions' fit to your data (both as a numerical score, and by visually inspecting them (look using a linear and a log Y-axis, as well as a Q-Q plot maybe).
3. Choose the one you think is best, in light of the data.

People argue a lot about which distribution is best for stock returns, but they generally accept that the Normal is not acceptable.

Here are some (hopefully) better options:

The more scholarly way of choosing a distribution is to also consider the theory behind their relationships, and to know why one might be a better fit than the others. While it's the safer option, it also takes lots of learning. I'm not an expert.