Welcome back to the Risk Institute. Today we're going to see how to design a low carbon index tracking error. In the last video, we have seen how to replicate a Benchmark. And now in this video, we're going to add a new constraint related with carbon intensity. So just to start, I want to show you again, what is the carbon intensity? How do we compute carbon intensity for one equity portfolio? And we have to start from the carbon intensity of the single stock? That is the total emissions of the company divided by revenues. So we have a normalized measure of emissions. And to compute the carbon intensity of the portfolio, we just take the weighted average of all the carbon intensity of the constituents of our portfolio. And the weights are the same weights used to build the portfolio. Now, just to recall you, in the last video we found this solution. So we started from a Benchmark that was the black line and minimizing the tracking error. We got the red portfolio. That is the portfolio with the lowest tracking error against our Benchmark. And you can see that the the performances actually are quite similar between two portfolios. Right now, we will do again the same exercise, but we're going to add a new constraint. So we want to minimize the tracking error between our Benchmark and our strategy. We keep the constraint related to the sum of all the weights equal to one, because we want to have a full invested portfolio. We want also that all the weights have to be greater than zero because we don't allow to short sell assets. And then the new constraint, is that our portfolio, the carbon intensity of our portfolio has to be lower than a percentage of the carbon intensity of our Benchmark. And the percentage is Alpha, and we can select them as we know, as we want. So, for example, if we choose Alpha equal to 80%, we know that we are going to decrease the carbon intensity of our portfolio of 20% compared to the carbon intensity of the Benchmark. So what we found, this is the solution. And we can see that the green line is the new portfolio. And we see that the green line is more distant from the Benchmark compared to the red line. That was the first strategy that we found without carbon intensity constraint. This happens because we are adding a new constraint and we are reducing the set of faesible solutions of our problem. So the lower the Alpha and then the higher the carbon intensity restriction. The higher will be the tracking error because we are reducing the visible set of solutions. And then we have to choose among lower solutions. And so we have a sort of trade off between the tracking error of our strategy and the carbon intensity of our strategy. I have additional methods to decarbonize my Benchmark. These methods are called positive and negative screening. To apply this method, I have to sort all my stocks according a greenness measure, and it can be the carbon intensity. And between my stocks, I'm going to divide them between the worst in class stocks, the stocks with the highest carbon intensity. A group of stocks that are neutral stocks and the best in class stocks, that are stocks with the lowest carbon intensity. To choose the dimension of these groups, I just change the cut off related to the strategy. When I'm going to talk about negative screening, I'm taking out from my asset universe, the worst in class stocks. So I will remain with the best in class stocks and the neutral stocks. And I'm going to apply the optimization process without any carbon intensity constraint on this group of stocks. On the other side, when I talk about positive screening, I'm going to select only the best in class stocks. And with these stocks, I'm going to run the optimization process without adding any additional carbon intensity constraint. Apply positive or negative screening, it's like to add an additional constraint, where we are setting the weights of some stocks equal to zero. And the stocks for which we set the weight equal to zero, are the stocks that are taken out from our stock universe. Just to wrap up, to set a decarbonization strategy, we have two options. We may impose a carbon intensity constraint, that it was the first case in the tracking error minimization problem. But we can also apply positive negative screening. Where we are going to select the universe of assets before to apply the optimization process. A very important finding of this lesson, is that when we are going to add a very strict carbon intensity constraint, we have to accept that to increase the tracking error of our strategy. Because for the reason that we are reducing the feasible set of possible solutions. And then we have to accept a solution that is quite distance from the original solutions without any carbon intensity constraint. Now we have seen all the theoretical stuff that we needed. And from the next video, we will start to work on the Excel file.
