Welcome back. Today, we are going to introduce a new methodology for handling the curse of dimensionality in this parameter estimation in the context of portfolio construction. This methodology is known as a shrinkage approach to estimating the covariance matrix. It's based on the great paper with a great title. The title of the paper is 'Honey, I Shrunk the Covariance Matrix'. I love that paper. I love that title, from a French guy called Olivier Ledoit. So let me try and explain what the methodology is all about, because it's very smart and very useful in practice. Remember that we are facing a trade-off between sample risk and model risk. So for example, the sample-based estimates for covariance parameters has lot of sample risk, too many parameters to estimate, but there is no model risk, I'm just looking at data and trying to estimate it. On the other hand, I have other methodologies like the constant correlation methodology or the factor base methodology that suffer from a lower degree of sample risk, because they've allowed me to reduce the number of parameters to estimate. But that came at the cost of me introducing some kind of structure, from some kind of assumption, and therefore there is some fair amount of model risk. The idea behind the shrinkage method is that, you're not going to have to choose either to go for higher sample risk or higher model risk, you are going to take two methodologies and you're going to mix them. Here is what it looks like. It's actually very simple in various [inaudible] in the way it has been designed. So statistical shrinkage estimators are mathematically designed to deliver the optimal trade-off between model risk can sample risk. Typically, they take on the following form, your estimate for the covariance metrics parameter of the covariance matrix itself is given by, delta time a first estimator, let's call it F, for the factor model-based estimator for the covariance matrix, plus one minus delta times your estimate for the covariance matrix that's not based on a factor model that kind of purely based on the data. Let me call that one S. So you've got to covariance metrics estimate, one called F, and the other one called S. F has some model risk based on the choice of the model that I'm using, the factor model that I'm using, but less sample risk and S has lot of sample risk, but no model risk at all. What I'm going to do is, I'm going to diversify with these risks by looking at a mixture delta percent of the first one and one minus delta percent of the second one. I'm going to choose the delta optimally that's explaining that paper on that I was referring to a few minutes ago. If you do that carefully, then you find the optimal trade-off you're kind of diversifying away some of the sample risk and model risk by looking at the combination of those two estimators. Turns out that it actually works fairly well, and academic research has shown that out-of-sample performance of these portfolios based on this sample on this shrinkage estimate for the covariance matrix, was actually better than the out-of-sample performance of the optimized portfolio based on either one of the two ingredients, the F covariance matrix or the S covariance matrix. So the mixture is better than each one of the two components. Therefore it's better than the average of the two component. That's a pretty powerful result. Now, we can actually reinterpret this methodology in terms of weight constraints. There's a very interesting paper by Jagannathan and Ma, published in the Journal of Finance in 2003 actually. That paper actually shows that imposing constraints on weights is actually strictly equivalent from a mathematical standpoint equivalent to performing statistical shrinkage. The intuition behind the result is very simple, even though the math are a bit complicated. Think about the intuition, when you're doing shrinkage, you're actually shrinking the dispersion of the input parameter value. So the sample permanent value has parameters that estimates that are too high, let's say 0.9, and some of them that are too low, say -0.7. By mixing this guy, which has lot of dispersion, with another one which has much less dispersion because it's more structure, it's based on the factor model. Well, then you're reducing the dispersion of the inputs. Now, what the paper by Jagannathan and Ma shows, is that reducing the distribution of the inputs is equivalent to reducing the distribution of the outputs. The outputs speeding meaning portfolio weights. So you don't get with these methodologies because they are robust, you don't get things that are too crazy in terms of maximum weights that are too high and minimum weights that are too low. So the bottom line is, it was well known by practitioners for a long time that, if you do portfolio optimization without any constraints, you get crazy portfolios with massive sharp positions and a lot of leverage. So practitioners have been imposing minimum weights, maximum weights, so as to come up with more reasonable portfolios. Well, it turns out that if you're improving the covariance matrix in the first place, if you're improving the quality of the inputs, you will need to worry a little less about the crazy outputs, because you would have been able to generate much more reasonable outputs on the basis of the more reasonable inputs. So conclusion, statistical shrinkage is one of the most advanced state of the art methodology for estimating the covariance matrix parameters. It actually works well. It's based on taking an average of two competing covariance metrics parameter estimates, and that is done in such a way so as to find the best trade-off between sample risk and model risk. In the end of the day, performing statistical shrinkage is somewhat equivalent to introducing minimum and maximum weight constraints. So this is all intuitive and meaningful. Of course, in practical implementations, we always recommend that you should still have some reasonable minimum and maximum weight constraints in case your efforts at improving the covariance matrix parameters was not as successful as what you hoped. Introducing the weight constraints will allow you to get reasonable and meaningful portfolios.