Welcome back to a Edhec Risk Institute. Today we are going to see how to replicate an equity benchmark. We will see what does it mean replicate a benchmark, and we will see how to build that replication strategy. In the last video, we have seen how to assess the performance of an equity portfolio, looking at their returns and their volatility. Today, we're going to use the same technology, more or less and we will use it to build this kind of replication portfolio. We can start looking at this chapter, and we can see that we have our benchmark the red lines, and other replication strategy. Later we will see how to build it. Now we want to assess how much this replication strategy is following the benchmark, so how much the two lines are close? For example, we can see that around 2014, the lines are almost perfectly in line, so the replication is almost perfect. But in 2017, they have much more divergence. We want to find a measure that tells us our strategy is following very well the benchmark or your strategy is not following very well the benchmark. This measure is the tracking error. If you look at the formula, it's very similar to the standard deviation that we have seen in the previous video. The only difference is that we are not looking the dispersion of the returns around its mean, but we are interested in the dispersion of our returns compared to the benchmark. If the returns of my strategy are equal to the returns of the benchmark, it means that the returns are exactly equal and my strategy is perfectly correlated with the benchmark, and then my replication is perfect. We are going to use this measure of tracking error to design the replication strategy, and just for your information, in the previous example, the tracking error of this strategy was about 13 percent every year, so it means that on average, the difference between my strategy and the benchmark is around 13 percent. Now we can set a problem. We pick an index that is the Standard and Poor's 500 energy index, so all the constituents that are in the Standard and Poor's 500 that are in the energy sector. Taking the constituents of this index, we want to try to replicate the index. Here you can see the black line is the index and all the other lines, the dashed lines are all the constituents. We want to find the a portfolio using this dashed lines that replicates very well, the black line. The problem that we have to set is this one. We want to minimize the tracking error for a portfolio for which we have to select the weights, under the following constraints we want to invest all our money in this portfolio, so the weights have to be equal to one. We don't want to short sell assets, so the weights have to be higher or equal to zero. If we run this problem, we end up with this picture. We find that the red portfolio is the portfolio with the lower tracking error. The other lines are other possible portfolios, and the black line is our index. Then an additional interesting measure is the information ratio that we use for the replication strategies. This measure gives us how much we outperformed or under performed the benchmark for one unit of tracking error. If you look at them, it's very similar to the sharp ratio. The tracking error can be seen like standard deviation. The difference of the returns can be seen like the excess returns. We can see that our strategy has an information ratio of 0.36, and two percent tracking error. Just to wrap up, we need the tracking error measure to set the replication strategy. The tracking error is a measure that tells us how much distant are returns of our strategy and of the benchmark. When we minimize it, we find a portfolio that is following very well the benchmark. An additional measure is the information ratio that tell us if our strategy out performed or under performed the benchmark for a given unit of tracking error. In the next video, we will see how to add an additional constraint, that is the carbon constraint. We will end up building a carbon efficient replication strategy, and then we will move on Excel, and we will see how to put all this knowledge in practice. See you in the next video.