Hello there, welcome back. Thank you for joining me again. When we talked about risk earlier, I mentioned that there are several ways of measuring risk. Well, one shortcoming of the standard deviation measure is that it treats the upside deviations from the mean equally as it does deviations from the mean. Now if you're wanting to maximize your portfolio value there's good reason to think that you might be more concerned about the downside variation. Which we might call as bad variation relative to upside variation which might be considered as good variation. Furthermore, you might even have a threshold in mind, a target return that you would find values below as completely undesirable, right? So, in this lecture you're going to learn about the concepts of semivariance and measures of downside risk. These concepts are particularly useful if the return distribution is asymmetric. And this can happen with certain asset classes or portfolio strategies with, for example, option-like features or if the investor, for example, has a particular return target. All right, so what is semivariance? Well, semivariance is really very similar to what you know as variance. Except that, it started squaring all deviations from the means. Semivariance is computed by squaring only the negative deviations from the mean. And the square root of semivariance is called semideviation. So, how do we use a semivariance? Well we can generalize it by introducing a return target, or which we can call the minimal acceptable return, right? And measure the deviations from below this target right? This is what we call the target semivariance. In other words, target semivariance is the expected square deviations below that target, right? And what we call target semideviation is the square root of target semivariance. Target semivariance is usually what is referred to as downside risk. In fact, this notion of target semivariance is really just a special case of a more general methodology for dealing with downside variation called the lower partial moments or LPM methodology. So what the LPM methodology does is that it uses a downside return target or a minimum acceptable return. And then defines any bad variation below that threshold as bad variation. So what are some typical targets? Well, typical targets might be the mean or zero or it could be even the risk-free rate. So the formula for computing the lower partial moments of any degree p, for a given target is given by, let me denote the LPM p for any degree of p. All right, so for variance it's going to be two obviously, right? Is given by. I'm going to define each term in a minute d(i) right? And then the deviation from that target to the p power, right? So now, let me sort of go through this expression with you and explain what it is. All right, so we have an observations, right. What is d(i)? Well, it's an indicator function really, right, that takes a value of one. All right, if The observation, the return observation falls short of the targets. So that's your threshold, so any shortfall from that is a bad variation and it takes a value of 0, right? If the observation is greater than the target because we don't count that. What is wi? Well it's the weight applied to each return. Now of course we're equally weighted, for up, then this would be simply 1 over N, right? So P is for any moment when you think of that, if P is equal to 2, then you have the semi, the target semi variance, okay? All right, so now let's put this into use in an example Right? So what I have here is a table of quarterly returns for the Russell Top 50 Index for the US. And it illustrates how we calculate the downside deviation for a sample of returns given a target of zero, right? So here is my, here is my tow here, right? That's the target, right? Here are the quarterly returns, right? So the first two columns present the dates and the returns. The third column presents the indicator function, right? That takes a value of either one or zero depending on whether the quarterly return is above or below the target. In this case it's simply zero, right? And in the fourth column is the deviation from the target. All right, so using the formula that I just presented in the spreadsheet, I computed the targets in my variance, right? So where the p is now obviously it's equal to two, right? And now I first computed the target's semi-variance, right? And then computed the target semideviation. Now, you can work with this spreadsheet, you can find it on the website and play around with the spreadsheet and make sure that you get the same results. Okay. Upside risk, well just like downside risk, the upside risk is also known as upside semideviation. And it's the converse of downside semideviation. All right, the only difference now is that we measure the deviation above the target level, right? Since upside deviation is presumably good, we can use the statistic to, for example, identify those distributions that are more desirable than others. All right, okay, so, in this lecture, we reviewed volatility of returns as a measure of risk. You also learned how to measure downside risk using lower partial moment measures. In particular, the downside risk measure of semivariance is a a special case of the LPM methodology with p power equal to 2, and the target equal to the mean. Downside deviation statistics are especially useful in dealing with distributions that are not symmetric and in comparing distributions that have different means.