[MUSIC] Welcome to module 4 of this online course. And module 4 has a focus on regimes and why regimes can be important in both for the decisions. And also, how machine learning techniques can be used to improve our process for identification of regimes in trying to keep track of those important patterns in portfolio construction decisions. First of all, in terms of motivation, the key component to keep in mind is that risk and return parameters of different assets and asset classes aren't constant over time. We have reasons to believe that they actually are time varying across the business cycles. And I don't think we need a lot of conviction to try and argue that capturing these time variations is of critical importance in portfolio decisions. Just let me give a very simple example, you'll be tempted to decrease risk taking when the riskiness or volatility of the risky asset tends to increase in your portfolio. So if you're trying to keep, for example, the riskiness within your portfolio constant at a level that's consistent with your appetite for risk taking, well, you do that by mixing a risky component with a safe component. If the risk of the risky component goes up, you would like to decrease the allocation to the risky component. So as to keep the overall level of risk constant in your portfolio. There's a line by Professor Angelo, one of our colleagues, who say, well, very clearly, when you think about volatility management as having who's controlling your car. If you want to maintain a constant speed in that what you get with cruise control, you don't have to be an engineer to use it and that's very useful. But you have to kind of adjust how much gas will be injected in the matter in the engine as a function of what the road is doing. Well, that's exactly what we do here. We react to changes in market conditions, if you will, so that's very important. Now, how do we keep track? How can we capture changes in risk parameters? Well, there are a lot of different statistical methodologies, if you stick to the standard kind of statistical toolkit. Well, there's a lot of methodologies that you can use for capturing time variation in risk parameters. So we are going to say a few words about these different methodologies. And as a conclusion, we are going to argue that eventually machine learning techniques will help us kind of improve on those techniques, and bring new insights into the identification of those regimes. Make it more robust, hopefully. Okay, first of all, let's start with the most straightforward estimate for volatility, right? So we're looking at returns, Rt, which is the return on a given asset, the Date T, we take the average. Let me call that R bar, the average over the sample period. And then we look at variance, which is as usual, the average squared distance with respect to the mean, right? That's how we define the variance. Now there's an improvement that one can be tempted to come up with. Which is, well, instead of assigning equal weights, when you look at the average either in terms of the mean value or in terms of the variance, we take the average of those observations. What you could say is that well the impact or the relevance of those observations tend to fade away over time, right,? And the most recent observations tell you about the world as it is now as what as it was not long ago. While the oldest observation tend to talk about the world as it was then. So what you may be tempted to do is having kind of a decreasing weight applied to different observations. Well, that's precisely what you can do, for example, with this exponentially weighted moving average model. Then the weights assigned to the returns or the squared returns regardless of the context, they tend to decline exponentially as we move back in time. Okay, and that very simple equation here gives you the alpha T, which is the weight assigned to Date T. And that's just a function of these Lambda, this weighting scheme, where Lambda is the exponential weighting scheme. And the lower the Lambda, the higher the weight assigned to most recent observation. Of course, if Lambda is equal to 1, you get equality, you get back to equality weighting. And as you get Lambda smaller and smaller, then you get increasing weight to the most recent observations. Okay, so that's kind of an improvement that allows you to capture time variation in the volatility estimates. What's next? Well, in an improvement over this methodology, improvement, we call an Arch Model, A-R-C-H model, what we could do is we could also assign some weight to long term variance. Let me call it VL. In other words, what you're going to do is, what you're going to say is that you're going to say that your best estimate for the variance today is equal to some long-term estimate for variance. And mixed with kind of what you learned from recent observations. So there's a gamma weight assigned to the long-term variance, and these Alpha T terms that we talked about, the weights assigned to your sample based observations. And what you want to impose of course is the sum of all these parameters, gamma plus sum of all these Alpha T, you want them to sum up to 1 eventually. And in the case of Arch one model is very simple, you get only two parameters, which is long-term value for volatility of variance and last observation. Now you can come up with a further improvement that we call a Garch Model. And let me here, just for simplicity, take a look at a very simple, the simplest example of a Garch model, that we call the Garch (1,1) model. So what we do is we do different things. So we take the model that we had before, the Arch model that we had before. So there's gamma times VL, where VL is the long-term variance, plus then Alpha times the last return, which is Rt squared, the last rate of squared. Plus then we're going to assign some weight to your previous variance estimate, which is, in this case, Sigma squared T minus 1. Now, the reason why we do that is because we want to capture some kind of volatility clustering that we see in the data. Which sounds like when volatility is high, it tends to stay high, when volatility is low, it tends to stay low. So there's some stickiness in the way that volatility over there, it evolves over time. And we do this again, imposing the constraint that the sum of these weighting, these weights, is equal to 1. Well, that's what we call a Garch model. And Garch models are routinely used in, that's very much part of the investment applications, and Garch models are very much part of the statistical toolkit that you need when you're doing financial analysis of different markets, financial econometrics. Now, what you could do as opposed to using a Garch model, which is you trying to track the changes over time of variance while recognizing that it doesn't tend to vary a lot. In the sense, that there are periods of time when it tends to be high, and periods of time when it tends to be low. Well, as opposed to capturing the time variations of these parameters, what you're going to be doing now is you're going to try and think about the state dependencies of these parameters. In other words, you're going to have a regime based model. So we are now introducing the concept of regime, and we are going to say, well, let me recognize that Sigma T, which is volatility at time T is not exactly a function of time, or precisely, it's a function of time to it being a function of some state of the world. So let me call St the state of the world. And so for example, St could be state of the world with high volatility or state of the world with low volatility. So in this case, you can think about the world being into regime, separated it out in two regimes, the high volatility regime and the low volatility regime. And now what we're trying to do here is we're trying to first estimate in which regimes we are and also estimate the probabilities of either staying in the same regime or switching from regime one to regime two, or from regime two to regime one. So clearly, we have, at our disposal, standard statistical techniques such as likelihood maximization, that can be used to try and estimate the parameters of these regime switching model, as we call it. And so by parameters, I mean the level of, in this example, the low volatility level, the high volatility level, and also the transition probabilities. Now, In contrast to the use of these parametric techniques based on some assumptions like Gaussian assumption. What we would like to do in the subsequent lectures and this is my colleague, actually, my colleague John, who's going to take over. And he's going to introduce you to machine learning techniques and see how these nonparametric procedures can be used for a more robust, hopefully, identification of regimes, and also for a more robust prediction of regime changes. So John will cover a number of different methods for identifying regimes and also discuss the implications for investment decisions. [MUSIC]
