In the previous lectures, I talked about how we measured risk given a return distribution, in other words, given a possible set of outcomes. We'll looked at the most common measure of risk, the dispersion of returns or the standard deviation of returns. This is what we call volatility. Now, we also looked at some other descriptive measures, such as skewness, kurtosis, to characterize a given return distribution. Now, it's not that volatility only applies to normal distributions, right? I don't want you to think that, that's not true. Rather volatility does not capture well the probability of crisis for non-normal distributions. So for a normal distribution, right, a nice normal distribution. Which is symmetric with lean tails, right, where the skewness is zero and kurtosis is three, right, that is it's pretty symmetric and it's got lean tails. An event that is two standard deviations away from the mean is pretty uncommon. An event that is five standard deviations away from the mean almost never happens, right? Now, that may not necessarily be true for some investment strategy returns if they are not normally distributed, and we looked at this with the flow data strategy the last time. For example, for some hedge fund investment strategies, two standard deviation returns maybe common, and five standard deviation events can actually happen pretty regularly, right? So while volatility is a perfectly fine risk measure, where the return distribution is symmetric, and there is no extreme crash risk, it is not an appropriate measure of risk for strategies with an extreme crash risk. So what do we do? Well, another risk measure that we can use is value-at-risk, VaR, which attempts to capture the tail risk, right? Basically what VaR measures is that measures the maximum loss that we can lose with the certain confidence. For example, the VaR maybe the most you can lose with a 95% or a 99% confidence. So we can say for example, that a 95% for a hedge fund, that a hedge fund has a 95% VaR of $10 million, let say. If the probability of the loss Being less than 10% is 95%, right? So any lose bigger than that is only happens with 5% probability, right? So how do you measure this? Well the simplest way to measure the VaR is to sort past returns and then find the return that has the 5% worse days and the 95% better days, right? And that 5% will be your VaR. This is the VaR since, if history repeats itself, you will lose less than this amount with 95% certainty. Now, one problem with VaR, on the other hand, is that it does not depend on how much you lose, if you do lose more than the VaR, right? So the magnitude of these extreme tail losses can be captured by another risk measure which we call the expected shortfall, right? What is the expected shortfall, well it is the expected loss that you would lose if you lose more than the VaR, right? So it's the expectation of the loss given the lost is for example > 10 million in our example. So, in this lecture, you learn two additional risk measures: value-at-risk and expected short fall. These risk measures are especially useful when volatility is not an appropriate risk measures, in other words, when there is an extremely bad risk, such as a crash, in the left tail.