Hello, welcome back. In previous lectures, we described return distribution on an asset using simply the average, the mean, and the standard deviation. This is because investment management is so much easier if we can approximate returns by the normal distribution. In other words, if the probability distribution of returns looks like the familiar, nice bell shape, then all we need to describe the probability distribution is, basically, the mean and the standard deviation. Now what if returns deviate from the normal distribution? In this lecture, we're going to look at how a distribution of returns may differ from the normal, and talk about some additional measures of risk that will allow us to describe risk better. In this graph, you see the cumulative return on a dollar invested in two different assets. The red line shows you the cumulative return that you would have had if you invested a dollar in the S&P 500 index in 1989 and kept it through 2012. The blue line shows the cumulative return on a short volatility strategy. Now, what is volatility strategy? If you've never heard of volatility strategy, it is basically an investment strategy that is based on selling protection against increases in volatility. Most people would like to protect themselves against spikes in volatility. We know volatility, for example, increases in financial crises. But on the other hand, if you have the stomach to weather large increases of volatility, basically the volatility strategy allows you to collect premium in stable periods by selling others protection against large increases of volatility. We're going to see these strategies later on, I just want to describe the distribution. Basically, it collects premiums in stable periods but stands to lose large, large losses when the volatility does spike up, for example, during crises periods, like the one we had in 2008 and 2009. What we see is that indeed the cumulative wealth on the dollar invested in the volatility strategy increases steadily. In other words, it increased steadily. There were some bumps along the way here and there with some occasional crisis periods but it increased pretty much steadily. Up until, of course, the financial crisis of 2008 and 2009, when the volatility went basically through the roof and this strategy suffered huge losses. It did recover by the end of 2012, and we see that the cumulative returns on S&P 500, and on this volatility strategy, were about head to head by the end of 2012. Now let's try to summarize the distribution of these returns for it's strategy. Let's first look at the returns on the S&P 500. The top graph here shows you the histogram of the returns on the S&P 500 index. Basically, the histogram shows the frequency of returns in the data. You can see that the distribution is centered somewhere greater than zero, but it's got that nice distribution. The second graph, on the other hand, below, plots the return distribution, probability distribution, and compares it to a normal distribution, represented in the red dash line. We see that except for some large negative returns, the probability distribution of the S&P 500 index is pretty well approximated by the normal distribution. It has that nice, familiar look of a bell shape curve. Now let's look at the distribution of returns to the volatility strategy. Again, the graph at the top is the histogram, which shows you the frequency of realized returns and the graph at the bottom, plots out the probability distribution and compares it to the normal distribution. Now we see a very different picture, we see a number of differences. First, we see that the return distribution of the volatility strategy is much more narrow and it shows a long left tail compared to the normal distribution. In other words, there is this much greater probability of large losses to the volatility strategy. Now let's look at the average volatility, etc, to summarize the returns. When we look at the average annual return, it was approximately the same, around 10% for both the volatility and the volatility strategy and the S&P 500 index. When you look at volatility as measured by the standard deviation, it was also very similar, around 15%. Now let's look at skewness, Which is a measure of the asymmetry of the distribution. The normal distribution has a skewness of zero, because it's symmetric. When the distribution is skewed to the left, that means extreme negative values will dominate the distribution and the skewness measure is going to be negative. This is when the standard deviation by itself is going to underestimate the risk. For example, the data shows that the volatility strategy shows large negative skewness. What does that mean? It's prone to large occasional losses. Finally, the last row in the table shows the kurtosis. The kurtosis measures the degree of fat tails, which is another important deviation from the normal distribution. When the tails of a distribution are fat, or more fat than the normal, this means that there is more probability mass for extreme events that would have been predicted by the normal distribution. Again, in this case, the standard deviation as a measure of risk will underestimate the likelihood of extreme events. Again, for example, we saw that the volatility of the volatility strategy and the return the S&P 500 index was similar. But we also saw that there's a large tail for the volatility strategy and that extremely amount of probability is much bigger. The volatility strategy has fat tails, as measured by the kurtosis. In this lecture, we looked at skewness and kurtosis as additional alternative measures of risk. This is especially important when the returns can deviate from the normal distribution.