Say I have a limited sample, a month of daily returns, and I want to estimate the 99.5th percentile of the distribution of absolute daily returns.
Because the estimate will require extrapolation, I will need to make a distributional assumption. Is there a standard approach to take here if I want to include the higher kurtosis in my estimate?
What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at Morningstar has written a few papers on this topic, and you can find more by googling him and this topic. Here is a less technical, more introductory piece on the topic.
Also, you should consider other risk measures, most notably Expected Shortfall (ES), also known as Conditional-Value-at-Risk (CVaR), which examines not just what the 99.5% percentile is, but what the conditional expected return is beyond that point. I also recommend you read this MSCI piece on the topic.
