Hello, everyone, welcome back. In the last few lectures, we talked about the different ways of measuring returns to evaluate portfolio performance. Of course, the next piece in evaluating performance is measuring risk, right? After all, we would like to know if the return provides an appropriate compensation for the risk involved in the investment. So in this lecture, we're going to revisit how we measure risk. If you've taken the previous courses, this should be a refresher for you. If not, it should help you get up to speed. All right. So what creates risk? There are several different sources of risk or risk factors. Right? What we call risk factors. Some of them are related to economy-wide uncertainties or fluctuations in the overall economy. Others only apply to some asset classes more than others. So, for example, here's a list of the different types of risks that can be identified when investing. You have business cycle risk, credit risk, interest rate risk, exchange rate risk, political risk and so on. Now, this list is by no means a completely exhaustive list for every asset class. And not every asset class is exposed to each type of risk. But you can see that there's a wide range of risks that might be encountered. All of these sources of risk or risk factors that cause asset prices to change as the factors change. We will not have a full discussion how to measure each and every type of these risks. But you can think about the wide variety of risk factors that assets can be exposed to. Okay. With that brief introduction to sources of risk, how do we measure risk? As you remember from our previous courses, we define risk as the variability or the dispersion in returns. So the most common, therefore, the most common measure of risk is the volatility of returns. Right? Which is measured as the standard deviation of the return distribution. So, given a sample of N returns. N observations. How do we estimate the standard deviation, sigma, from a sample of N returns? Well it is simply 1 over N-1, by going from 1 to N. All under the square root. Where our i is basically the return for observation i. Our bar is the mean return And is the number of observations we have. Why is this N minus one? Remember? This is the sample standard deviation. N-1 in the denominator is used to correct the bias since r bar is an estimate itself. If you replace N by N minus one with N. In other words, with one over N then you would get the population. Standard deviation. So remember, what is the standard deviation. Well, it's the square root of the average squared deviations from the mean. Now, if we also assume that returns are, for all practical purposes, normally distributed. Then the standard deviation has various useful statistical properties. For example, for a normal distribution, we would expect 68% of all return observations to fall within the range of plus or minus one standard deviation around the mean. We would expect about 95% of all the observations to fall within plus or minus two standard deviations from the mean. And finally, we would expect 99.7% of all observations to fall within plus or minus three standard deviations from the mean. Right? This is important because, this allows us to make probability statements about how often good or bad events are expected to occur. So for this reason standard deviations are fundamentally used for measure of risk for distribution of asset returns. Okay, so let me give you an example. So here I have three years of quarterly returns for the Russell 3000 Index from March 2003 to December 2005. I calculated the mean, which is 12.82%, which, by the way, is pretty high by historical standards. And the sample standard deviation was 16.13%. Now, under the normal distribution assumption, you would expect 68% of the observations that we have to fall within the plus or minus one standard deviation from this mean. So, in the second column, I computed the deviation from the mean for you. And in the last column, I noted whether the value falls, indeed, within this plus or minus one standard deviation. So understand it under normal distribution assumption, we would expect about 8 out of these 12 observations to fall within this range. What do we see? Well, 1, 2, 3, 4, 5, 6, 7, 8, 9 quarters out of this 12 quarter period. We're indeed within the range. Okay. So to compare returns over different periods, we usually convert them to a standardized measure such as annualized returns. So it's useful to know also how to convert the standard deviation for one holding period to a annualized measure. So say you have a monthly volatility measure. How do we annualize this volatility? Well, we multiply this volatility measure by the square root of the number of holding periods in a year. So if you have a monthly volatility we're going to multiply it by the square root of 12 to get the annualized volatility. So for example in the previous example, the standard deviation of the Russell 3000 quarterly returns was 16.13%. This is the quarterly volatility. All right. In order to annualize this volatility, we need to Multiply it by the square root of four, right, which is going to give us 32.26%. We have the daily volatility, then we would compute the annualized volatility by multiplying it by the square root of the number of days, weeks etc. Same idea. All right, so in this lecture we revisited how we measure risk. Volatility is the most common measure of risk that we use. And we measure volatility by the standard deviation of returns. We also have the normal distribution. Then we can make probabilistic assessments of how likely certain outcomes are, depending on how widely distributed, how far away from the mean, they are.
