All right. Welcome back. So we ended last session with a discussion of a risk-adjusted measure that we called the sharp ratio. We made the point that it's the ratio of the excess return that you would get over what you could get with no risk. In other words, the excess return over the risk-free rate per unit of volatility. We're going to now look at another measure of risk, which is not volatility. The argument here is that some people say that volatility really is not necessarily a bad thing because it's what is it. Remember, it's just deviation from the mean. If you deviate from the mean on the upside, well, that's actually not a bad thing. What this line of thought says is that, what is risk is really the possibility of losing money. So if I have money and I lost it, that's bad, that's risk. So it's a reasonable point of view. So what we're going to look at is a very popular measure of risk that is called The maximum drawdown. So what is The maximum drawdown it is the maximum loss that you could have experienced, if you had been unlucky enough to buy the asset or the strategy or whatever you're looking at. At its very peak, and you sold it at the very bottom. It is the worst return of the peak to trough that you could have experienced over the time that the return series are being analyzed, that makes sense? So one way of thinking about it is, this is the worst possible return you would have seen if you had been unlucky enough to buy it at its peak, and sell it at its trough, at the worst point. That is called the maximum drawdown. It's a very interesting worst-case measure. It is not the loss you would have experienced if you'd been in the strategy necessarily, it just is the worst case if you had timed it absolutely horribly. So now, let's figure out how we're going to take a return series and convert it to a drawdown. Let's walk through the steps. So I'm going to take the same. We've seen this return series already before, the small cap and the large cap, US large cap stocks. So the first step is you take the return series and convert it to what's called a wealth index. A wealth index is just a fancy way of saying, what would have happened if I had taken let's say a dollar or a $1,000, and invested it over time. Just buy and hold. Just kept it in that asset all the way through that period. So you can see that sometimes it goes up, sometimes it goes down. But $1,000 by the time you've done with this 40-year period is approximately $300,000 if you had invested in small cap. But you see here that there are periods where it was as high as $175,000 and then just a short while later, it was down to a much lower number. That's a drawdown. A drawdown is from the peak to the trough at any given point in time. So let's look at large-cap, similar thing. You see that you've gone up at some point, then you go down. You go back up, and go back down. That's the drawdown, the up to the down. How do we measure it? So I took the returns. The first thing that I did is converted into what's called a wealth index. The next step is to compute the peaks, the previous peaks. So at any point in time, you want to keep track of what is the highest value that I have experienced in this strategy at that point of time since inception? So that is marked with a green line here, and you can see that when the wealth rises, that is your peak. But then when you fall, when your wealth falls, because the market is fallen, then your peak stays the same. That is your previous peak. The drawdown is nothing more than the distance between the blue line and the green line at any given point in time. That's the drawdown. The distance between the blue line and the green line at any point in time is basically the drawdown, why? Because that is the money you had in some theoretical sense, at least, the unrealized gain even though you didn't actually sell the asset and take cash, you had that much money in your account. So the distance from that green line to the blue line is how much you feel you've lost. This is part of why the drawdown measure is so popular. Because it actually is very consistent with a lot of behavioral effects that we know we're subject to, behavioral biases that were subject to. Once we have that money in our statements, we feel like we have it, and when we lose it after that, we really really hate it. We don't like losing money that we thought we had. So that's why drawdowns are so popular I think. Let's look at the same diagram now for a large cap. Again, you can see that there are drawdowns, and let's look at the extent of the drawdown. So remember, the drawdown is the distance from the green line to the blue line. What we want to know is what is the maximum drawdown? In the case of the small cap stocks, we're in the small cap stocks here. The case of the small cap stocks, you can see it's horrible, it's gut wrenching, it's 63 percent. By the way, it is not completely a theoretical number that I lost 63 percent. It is theoretical if you somehow just held onto it because you still might make that money back. But if you're that one unlucky person who bought it on the peak and sold it at the trough, you would really have lost 63 percent of your wealth. Now, that's drawdown. You can plot drawdown in a couple of different ways. It's very useful to plot a time series of the drawdowns, and that's what we have here in the form of that red line. You can see that the lowest point that the red line goes is below 50 percent. That's in fact 63 percent. That's the drawdown in small cap. If you look at it in large cap, not as bad, but still certainly not fun. In fact, you see that the drawdown in large caps was much much less, or at least this data series during the 2008 crash. So that's drawdown. That's how you compute it. We're going to actually do all of these in the lab. You could be wondering, "Well, since this is a risk measure, why don't I use this as a way of doing a risk adjusted return?" In other words, "Why don't I look at the excess return divided by the drawdown?" You in fact could do that. There's a name for it, it's called the Calmar ratio. So if you thought of this by yourself, congratulations. But the basic idea in the Calmar ratio is you look at your trailing 36 months drawdown. That's how they typically do it. The maximum drawdown just over the last 36 months, and then you look at the return in the numerator. Usually, they actually don't look at the accessory term, but you certainly could. Now, I'm going to end by just pointing out some problems with drawdowns. As you can see, they're very sensitive to, they essentially dependent on two data points, and you don't want to use statistics that are dependent on essentially two data points. Though it's not the best, but it is a very very popular measure. You're going to encounter it at some point. So you might as well get used to it. I think the other problem that you should watch out for in terms of drawdowns calculation is, drawdown on a daily basis is very different from drawdown on a weekly basis. Again, because it depends on just one point. If you look at drawndown on a daily basis, you're going to see the worst worst-case. If you look at it on a weekly basis, the worst-case would have essentially disappeared because you are only looking at weekly data. If you read monthly data, it's even less. So it's very very sensitive to the granularity of the data. There are better measures of this kind of extreme behavior as we call it, these outliers, these worst-case. So if you're looking at a worst or if you're looking for a worst-case measure, you're better off looking at other more robust well studied extreme measures. There are things like VAR and CVAR, and that is in fact the topic of the next few lectures that Legionella is going to be doing, and he's going to walk you through these extreme measures, how they're computed, etc. So max drawdown is useful. You're going to encounter it as a practitioner. But really I would say, I like looking at VAR and CVAR much more. Thank you very much.