Expected shortfall is an extension of value at risk (VAR). For a discussion on VAR, refer to the article where VAR is determined using Monte Carlo simulation. Expected shortfall is also known as conditional VAR.
Suppose we have determined VAR for our portfolio. Let's say we have a VAR for daily returns at 95% confidence level. 95% of the time, the maximum loss will be VAR. 5% of the time, losses will exceed VAR. This begs the question, how much can be lost in these extreme events? Expected shortfall attempts to provide somewhat of an answer.
Given the 5% where we exceed VAR, expected shortfall is the expected value of the losses exceeding VAR. In other words, given losses greater than VAR, what is the expected value of this 5% of outcomes?
We will use our simulation from the VAR article to determine expected shortfall. The model for determining VAR from our portfolio of five investments is shown below.
After running the simulation, the VAR is determined to be 41,161. Now let's look at the 5% of outcomes that exceed VAR. To determine the expected shortfall, we need to find the expected value of the 5% of outcomes where losses exceed VAR.
Using the raw data from the simulation we can filter out any outcome greater than or equal to VAR. Part of the simulation data sheet is shown below.
Now let's filter out any ending portfolio value greater than the 5th percentile of outcomes. The 5th percentile ending value is 9,958,839. Set up the formula shown in cell J2 below where any value less than 9,958,839 is shown and any value greater than or equal to this value is left blank.
Copy cell J2 down to the last data row.
Now we calculate the average of the filtered data points in cell K2 as shown below.
To calculate expected shortfall, we subtract the expected value (average) of the bottom 5% of portfolio ending values from the beginning portfolio value of 10,000,000.
Expected Shortfall = 10,000,000 - 9,906,068 = 93,932
To interpret expected shortfall, given that our losses have exceeded the VAR of 41,161, our expected losses will be 93,932.
It's important to note that since we are dealing with a small number of the total iterations (5% in this case), we need to run enough iterations to sample a sufficient number of extreme data points. This is also where selection of the return probability distribution is extremely important. The distribution may over or under estimate the magnitude of losses that are possible.
Expected shortfall expands on VAR to give an idea of extreme losses that VAR fails to show, but it is also imperfect at showing absolute risk. It is merely an average of the extreme losses. Nevertheless, it can be a useful next step beyond VAR.
Simulation was performed using Simulation Master.