[MUSIC] So what we want to do when we discuss performance attribution is distinguish between asset allocation contribution and stock selection contribution. So this is the return decomposition, the active return, the return in excess of the one generated by the benchmark fund. So let's look at a numerical example. In this example, I have the benchmark composition and the fund composition. I've chosen a very simple example, where there are only three sectors, and we can think of this fund as being completely invested in stocks. Okay, so there are three categories of stocks, three industrial sectors that the fund is active in, which I've labeled technology, financials, and biotech. The benchmark has 30% invested in technology, 30% in financials, and 40% in biotech. Whereas the actively managed fund has an overweight in technology, 35 versus 30, underweight in financials, 25 instead of 30, and identical allocation in biotech, okay? So to be able to distinguish the performance contribution of stock selection and asset allocation, we must use some information. The portfolio weights is one aspect. The other aspect, of course, is the performance of the various sectors in the benchmark and in the fund. And this is the information we have in this next slide. So here you see that the technology stock in the benchmark had a performance of 8%, financials, 5%, biotech, 7%. Whereas in the actively managed fund, where, remember, the choice of securities within the technology sector can be very different from the technology sector in the benchmark. Here, in my example, the fund has a 1% excess performance in technology, 1% excess performance in financial, and an equivalent performance in biotech, okay? So let's first compute the overall excess return generated by this actively managed fund. We see that the fund does better in technology and financial and just as well in biotech. So the excess return is going to be positive. So let's compute it explicitly. So what we call the total active return is the difference between the fund performance and the benchmark performance. Each of these performances are computed as weighted average of the performance of each of the individual sectors. So for the fund, the second bullet point here, we have 35% in the first sector, we generate a 9% return, 30% at 6% performance, and the remaining 40% invested in the third sector with the performance of 7%. The total return is 7.45%, whereas the performance of the benchmark is 6.7%. The difference between the two is a little bit less than 1%, 75 basis points to be precise. Now we're going to decompose the 75 basis points between asset allocation effect and stock selection effect. The way we measure the asset allocation effect is by looking at the differences in weights. Did we over allocate in technology, under allocate in technology? But to see whether this distinction between the allocation and the fund and the benchmark is the source of performance, we're going to multiply this weight difference by the benchmark performance, to see if just over weighting one particular sector yields an excess return. In particular, in my example, we see here that if we over weight the technology sector, even if we select exactly the same stocks than the benchmark, the fund is going to over perform because the technology sector has a slightly higher return than the other sectors. So let's look at this asset allocation effect. And by computing the weight differences in technology, financials, and biotech, we obtain a total contribution of 15 basis points. We can measure something very similar when looking at the stock selection effect by looking at the benchmark weight multiplied by the performance differences. This time, what we're doing is looking at the asset allocation in the benchmark and looking at the excess performance generated by the correct selection of securities within each sector. And to measure that, we just look at the differences in performance between the fund and the benchmark, in each of the sector. For example in the technology sector, we have an excess return of 1%, in the financials, an excess return of 1%, and in the biotech, no excess return. We weight all these excess returns by the benchmark weight to obtain the overall stock selection effect. In this case, we obtain 60 basis point, a little bit less than a percent, almost half a percent, actually. Finally, the performance attribution must also take into account the interaction terms. So what are these interaction terms? We're going to look at the weight difference and the performance difference. Again, we could overweight a particular sector, which performs better than the others. This is the case for technology. But we could poorly select the securities within that sector, creating an overall underperformance in terms of interaction. So we're going to look at the product of weight differences and performance differences. And here in my example, there is a positive interaction term in technology. We overweight a sector in which we have a very large performance larger than the benchmarks, so this creates a positive interaction. But there is a negative interaction terms for financials. Why? Because we have an over-performance, we select the securities in financial in a good way in the activity managed fund, but we underweight solely that sector. So this creates an overall negative interaction effect. The sum of the technology and financial effect, given that the biotech effect is 0, the sum of the two cancel out, and we have 0 basis point coming from the interaction. So to summarize, in this example, we have a contribution coming from the asset allocation decision of 20%, stock selection of 80%. Out of the total over performance, 80% of the performance comes from the ability of the fund manager to select the correct stock within a sector. The performance attribution approach that we have just presented is called the return decomposition. It's not the only way that we can address that problem, there are other approaches. I'm listing here two of these alternative approaches. One is called the risk factor analysis, where instead of looking at the overall performance in terms of return, we decompose it into exposure to different risk factors. This requires a preliminary statistical analysis. Another approach is to look at a style analysis, where we decompose the return contribution into the exposure to different management styles, such as momentum, contrarian strategies, and so on. These two approaches are complementary to the methodology we have discussed here, but the return the composition approach is the first one that was developed to analyze the performance attribution. Thank you. [MUSIC]