Consider the following types of financial time series for a single publicly-listed stock:
- Price data
- Log returns
- Cumulative returns
Each is computed from the item listed before it: log returns are based on differences of prices, and cumulative returns are cumulative products of log returns.
- Which of the random variables listed above possess a probability distribution function (PDF),
- which have a cumulative distribution function (CDF), and
- which have both a PDF and CDF?
- for what sort of financial applications is the CDF preferred over the PDF, and vice versa?
I ask because the following post says all random variables have a CDF, but not all of them have a PDF. So I wanted to see how this applies to commonly used financial data, which are prices and returns. Graphical depictions of the above datas' CDF and PDFs displayed side-by-side would help in the explanation.
I'm particularly curious about cumulative returns. Since they're cumulative, it automatically makes me think it corresponds and is represented best by a CDF, so in a way I'm wondering if cumulative returns are more useful than they're made out to be, despite being non-stationary.