Calculate CVaR for a portfolio

Question

I would like to calculate the Conditional Value at Risk for a portfolio. To be honest, I'm trying for a few days to find an example to calculate for an entire portfolio, not just for one security and I really have a hard time understanding. All the examples are for a single security. I need to add that I'm at the beginning of learning econometrics/statistics.

Let's say that I have a portfolio composed from 3 investments. If I want to calculate CVaR using Monte Carlo prices from the 3 investments, here is what I'm thinking: 1. create a simulated portfolio of 3 investments and take into the account the nominal value of every security and the direction (long/short). 2. run the above portfolio through Monte Carlo for n times and generate a distribution of P/L by calculating the difference between start and end of NAV for every MC iteration. 3. let's say I want to calculate at 99% confidence ratio. Then I get the mean of 1% of the worst losses resulted from the Monte Carlo distribution.

I would like to emphasize that I don't want to use a normal distribution.

Are the steps above correct for finding the CVaR of a portfolio? Also, does respect the coherent risk measure? Thank you.

Answer 1

This sounds correct, however step 2 is a little vague, so I will try to restate the steps here for you.

• The assets in your portfolio must be priced with respect to a set of risk factors (e.g. interest rate curve).
• Each scenario consists of a value for each of your risk factors. Given the value of your risk factors you can price your portfolio.
• You want to generate a number of scenarios ($S_i$). Your choice on the number of scenarios chosen depends on the number of your assets, the way you choose to model your risk-factors, what your confidence interval is and time/computational constraints. The way you generate these scenarios is important. For example take a look at short-rate models.
• You price your portfolio for each $S_i$.
• Sort the portfolio values $V_i$ for each $S_i$.
• For CVaR, as you say, you can average the $V-i$'s that correspond to a loss higher than the $a^{th}$ percentile. For more information. This is the most helpful introductory paper I have found on the topic: paper by Rockafellar and Uryasev. It has details on how to calculate CVaR.

And, yes, CVaR is a coherent risk measure.