I'm trying to adapt tools from portfolio theory for another use, and I have a question about how I might do so.
Suppose that instead of having normally distributed returns, the return $R_i$ is either, say, 3 or 0. So, $E(R_i)=3P(R_i=3)$.
On the wiki page for Modern Portfolio, it says
Note that the theory uses standard deviation of return as a proxy for risk, which is valid if asset returns are jointly normally distributed or otherwise elliptically distributed.
I'm mainly interested in computing expected returns and risk a la wiki. I'd like to use those as metrics in comparing a relatively small number of possible portfolios. But obviously in the case I'm interested in, the returns aren't normally distributed. What are the consequences of still using standard deviation of returns as a "proxy for risk"? Is there an alternative measure that makes more sense?
How bad would it be to pretend that the returns are normally distributed, centered at the mean of the bernoulli return, with the same variance?
The stakes aren't particularly high and currently the only tool for what I'm trying to do is human judgment and experience (the application isn't in finance).