[MUSIC] Welcome to this lecture on portfolio performance estimation using asset pricing models. Let me walk you through the main learning objectives of this session. So first of all, we're going to define what is alpha, what is risk-adjusted performance. Then we're going to talk about how we estimate the alpha, or the selectivity, of the manager using the capital asset pricing model. Next, we're going to turn to a more complicated model, the Fama-French 3-factor model, and ask the same question. Namely, how do you measure the alpha, or the selectivity, of the manager? And finally, we're going to ask, is estimated alpha really capturing the true managerial skills? So what is alpha in theory? In theory, alpha is simply the expected return generated by the hedge fund manager or the mutual fund manager. E[Rp] above E[Rb]. And what is E[Rb]? E[Rb] is the benchmark return in expectation, and typically the one would be provided by the capital asset pricing model. So that's the normal return, given a level of risk that the portfolio manager should on average outperform. And if he outperforms the alpha will then be positive. So when we measure alpha, we have to keep two things in mind. What does alpha represent? Alpha represents the selectivity skill of the manager. And the selectivity skill asks the following question, does the manager on average, generate a better, a higher return than the benchmark capital asset pricing model? And how would he do that? Well, he would do that by buying stocks that will outperform, and selling short the stocks that underperform. In order to estimate this alpha, we need to abstract from the timing ability. What is timing? Timing is the ability of a given manager to time the portfolio risk factor. Namely, entering into the S&P 500, if he thinks the market in the US will overperform. And to measure the selectivity, we assume no timing ability for this manager. And imperial studies have shown that indeed, most managers do not have any timing ability. So that's a pretty realistic assumption. So how do we do that empirically? Well empirically, on the x-axis of this graph, you have the return on the market portfolio in excess of the risk-free rate of return, that's RM-f,t. And on the y-axis you have Rp-f,t, which would be the returns in excess of the risk-free rate generated by managers. Each dot corresponds to one realization. We've run ordinary least squares regression, the beta of that regression is the slope, the systematic risk, followed by the manager. And the alpha, the intercept, would be precisely the selectivity that you are trying to capture. So in this case, we have a good manager who generated a positive, abnormal performance. So typically, we have many asset pricing models. And the problem is, which one would we choose? So we could choose the CAPM, as we did until now. Or we can choose a more sophisticated model, like the Fama-French 3-factor model and its extensions. For instance, some people would use a CAPM and add a liquidity risk factor, because liquidity risk is also important in the financial markets. Now, let me give you just a little bit of a theoretical background. Anytime you chose an asset pricing model, there's three facts that have to be noted. First of all, all the factors need to be systematic. What does it mean that they are systematic? It means that the impact of these factors cannot be diversified away,it's a shock to the return of the securities. All securities, for instance, in a given asset class. Secondly, the factor generates unexpected changes. What does that mean? It means, these are surprises that hit the returns. And finally, the factory is priced. It means, its risk premium. Namely, the difference between the expected return on the benchmark factor, and the the risk-free rate, has to be priced strictly positive in absolute terms. Now let's turn to the Fama-French 3-factor model. This equation looks maybe a little bit difficult, but in fact, it's not. The Fama-French 3-factor model is basically an extension of the capital asset pricing model, which rests on the market factor as the first factor. And here, we add two additional factors. First, the SMB factor, which is a factor where you go long the stocks that are of small market capitalization, and we show those stocks that have large capitalization. And then, an HML factor, which is the third factor, which is constructed by going long, high book-to-market stocks, and shorting low book-to-market stocks. So what does this equation tell us? It tells us that the expected return on the portfolio, in excess of the risk-free rate, is equal to its beta with respect to the market portfolio beta iM, multiplied by the risk premium for stock market risk, plus the beta of the security or the portfolio with respect to the SMB factor multiplied by the risk premium for the SMB factor, and finally the beta of the security with respect to the HML factor multiplied by the premium for how much HML risk the security, or the portfolio, is generating. So, many empirical tests have been conducted, and the question is, what is the right model? So typically, one thing that we know is that all these tests have rejected the CAPM and other theoretical model in the family of the capital asset pricing model. Typically nowadays, practitioners, and also academics, would use the Fama-French 3-factor model. And the reason why they would use it is surprising. Why is it surprising? Well, because the factors, per se, that means the book-to-market factor that has been added, and the Small Minus Big factors that had been added to the market factor, are totally add talk. But it turns out, that this model fits the data pretty well. So let me remind you of a crucial slide and a crucial point. The choice of the asset pricing model, the proper model, the accurate model, is very important. If the model is wrong, the selectivity alpha that you estimate is also wrong. And typically, what happens is that if you omit a risk factor, let's say that in fact there are three factors, the market, the Small Minus Big, and the HML factor. If you forget the HML factor, what's going to happen is you're going to artificially increase the alpha, and you're going to increase it by the premium for the HML factor. So basically, if you omit risk factors, you generate alpha which is not representing managerial skills. Now another point is important, you need to have enough data. Enough data means enough days, weeks, or months over which you estimate the performance. Because if the sample is too small, the confidence interval is too large, and the alpha is less precise. Now let me give you an insight into what this module will actually try to address. There's a key topic in the finance performance measurement literature, do active fund managers, do mutual fund managers, beat a model like the CAPM. And this goes back to the original paper by Jensen. And you have a graph here, where he actually estimated the average alphas of 115 mutual funds and showed that, on average, they generate an alpha of -1.1% net of the expenses. That means a negative average alpha, which is not recovering the costs faced by this mutual fund. That's if you believe this graph, if you believe many other empirical studies. It is very difficult for mutual funds for active managers to deliver true selectivity, true performance net of their expenses. [MUSIC]