[MUSIC] Welcome back, so last time, we talked about returns. We're now going to switch, and we're going to look at something that is as important, in fact, more important than returns, in a sense, and that is volatility. So let's start again, with the same return series that we saw before, we have the blue series and the orange series. They both have the same mean, but clearly the blue series is much less volatile than the orange series. And what that really means is that the volatile series, that is the orange series, are deviating away from that mean more often and more significantly than the blue series. That's really what we mean by more volatile, right? The average, the mean of these two series, are exactly the same. And the only difference really in a sense, is that the orange ones are moving away from the mean more often. So what we're going to do now, is just going to formalize that basic insight, right? So, how are we going to do this? So the first step is we take the return series, and we do what's called demeaning or just subtracting the mean. Okay, so now what you've got is you've got to return series, you've subtracted the mean from all of them. So all you have is the deviations from the mean. Then what we want to find out is how much do they deviate? What's a good measure of the deviation from the mean? Well, it's very simple, you square the demeaned return, so that gives you a measure of the deviation from the mean. Whether it's a negative number, a positive number, when you square it, they both end up being a positive number. And then just take the average of that, that number is called the variance. And of course, if you're familiar with that concept or that definition from statistics, this should have been very obvious and a complete waste of your time. But the point is, the variance of a set of returns is nothing more than the average of the square of the deviations from its mean. Okay, so that is your measure of volatility. Clearly, the orange series has a much higher variance than the blue series. Now, one adjustment that I'm going to make here is that the variance is really the average of the squares of the returns and so it makes it hard to compare with the returns themselves. So what we do typically is instead of using the variance directly, we will take the square root of the variance, because that's small comparable with the returns themselves. So that's a measure volatility, what is the measure volatility? It is the square root of the variance. And again, if you have familiar with the concept of standard deviation from your statistics class, this is all going be very obvious. But it's important for you to understand sort of the intuition behind it. It's nothing more than the square root of the variance, which itself is nothing more than the average of the deviation of the square of the deviation from the mean. All right, so now we've got a measure of volatility. Higher the volatility, the more the deviations from the mean. The one thing we have to do to be able to compare volatilities is, in fact, the same thing that we did for returns, and that is we need to annualize it. So if you've got returns that were calculated with daily data, you're going to see a lot more variation. Because you're going to notice that variation more, because you're going to have more data points. When you look at it over a month, of course, you won't see all those intermediate data points. So you can't compare the volatility that you obtain from daily data with the volatility that you obtain from, say, monthly data. But that's easy to adjust, the way you annualize volatility is you multiply the volatility that you get from the daily data by the number of days per year. And remember for us the number of days per year is 252 not 365. So if I have a volatility of let's say 0.1% that I computed on daily data. To annualize all I do is take that 0.1% and I multiplied that is a 0.001, and you multiplied by the square root of 252, and that gets you 15.8 %. So, very simple, you take the daily data or whatever granularity. Let's say you take daily data, you multiply it by the square root of 52, you get annualized volatility. Let's say you're working with monthly data. Then you take the volatility that you compute on monthly data multiplied by the square root of 12, you've got the annualized volatility. Okay, so now that we know how to compute returns and we have know how to compute risk, we can now look at some sort of method to compare returns that have different risk. Take a look at this graph. So this is the returns from 1926 to 2018 of two different portfolios. The blue line are small caps, US small caps, and the red line is US large caps. Clearly, US small caps are far more volatile. And if you look at the numbers, you will find that the volatility of the US small cap over these years, the annualized volatility is 36%. But they also give you a higher return, they give you 17% return. Large cap gave you only a 9.5% return over that same period annualized, but they did so with much lower volatility. So the question we have now is how do we compare these two? So one way of thinking about it is to simply compute the ratio, right? To say, how much return did I get per unit of risk? And you could do that, that's called a return on risk ratio. But I'm going to suggest an adjustment that you have to make to get an even better picture. If you just did the very simple computation that I said, that is the ratio of return to risk. It looks like small caps gave you about 0.47, large caps gave you about 0.50, it might suggest that even though large caps gave you lower returns. They actually gave you a better return per unit volatility, right? So you might think, okay, well, maybe small caps weren't that great after all. The adjustments that I'm going to suggest is very intuitive one and the idea is that you shouldn't just look at return. You should look at the excess return per unit of volatility over, excess return over what? Excess return over what I could have got with no volatility, right? And what you can get with no volatility is what we call the risk free rate. The risk free rate is basically the return that you would get with virtually zero risk, right? Put your money in, let's say, US treasury bonds or US key bills. Okay, so we know what that rate is, and for this math I'm just going to keep it very simple, I'm going to assume it's, let's say, 3%, right? So what I can actually do now is I can look at the ratio, not of the return to volatility, but I can look at the ratio of the excess return over the risk free rate per volatility. That ratio is called the Sharpe Ratio. And this is a number that you will see again and again and again no matter what you do. The Sharpe Ratio is probably one of the most Significant ratios that you will look at. And basically, what this is telling us here is that once you adjust for the risk free rate, it looks like small caps actually gave you a slightly better risk-adjusted return than large caps. Bookmark that fact, because we're going to come back to that later on in the course. But for now, I just want you to understand how we took the returns, adjusted it for risk, and came up with a measure of risk adjusted return that we call the Sharpe Ratio. That's what we have here for this section. What we're going to do in the next section is we're going to look at another measure of risk, which is much more popular in the practitioner literature. And that is called the drawdow, that's what we have for next time. Thanks. [MUSIC]