Hello, everyone. Welcome back. I hope you are enjoying our discussion of the risk-adjusted return measures so far. All right, so in this lecture, we're going to turn our attention to what is known as the Jensen's alpha. Or sometimes simply called the alpha. Alpha is what every active manager is after. The term alpha is used in many different contexts. And may sometimes be used to mean different things. For example you might hear a portfolio manager say that she is trying to generate passive alphas. All right? So she basically means that she wants the portfolio to out perform. So basically in it's simplest form alpha is used to mean out performance. In general, alpha represents the excess return of the portfolio over the return of a benchmark. Then if it's higher then it's positive, if it's lower then it's negative. It is a measure of the risk-adjusted excess return after accounting for the risk. Relative to some benchmark. So increasing the alpha means increasing the excess return. Without increasing its exposure risk to some systematic risk factor. So when the reference instrument is, for example, the portfolios. The portfolio manager's benchmark. Then the alpha may be referred as the benchmark alpha. When the reference portfolio is the market portfolio. We call it the CAPM alpha. Or if the reference model for example is the multi-factor model. We may refer to the alpha as the multi-factor alpha. So in our examples for this lecture. We will often be referring to the CAPM alpha. Okay, so remember what the CAPM predicts? It says that the expected excess return. Any asset or portfolio, is given by its beta times the expected risk premium on the market portfolio. So using this framework, we can decompose to realize the exposed excess return. Or an asset or a portfolio into an alpha and a beta. So let me call. Defined excess return on an asset at time t. All right, alpha plus beta times, This is i. This is market. This is return, plus some epsilon. All right? Now then we can use statistical analysis methods to estimate the alpha and the beta using this model. Which is sometimes called an index model. So this is a single factor index model. Now think about what each term represents. Just like in CAPM, beta is the market risk or the systematic risk. The portfolio's tendency to follow the market. What is the alpha? Well, the alpha here. Or in the CAPM world, of course, we know that the alpha should equal to zero. So a statistical significant positive alpha will suggest that the portfolio is providing an excess return. Above and beyond what is adjusted for the risk, the market risk. On the other hand, remember, if you have a portfolio manager with a positive alpha. It does not automatically suggest skill on the part of the manager. It could also mean that the manager was just lucky. We'll think about that later. And finally, the epsilon is the idiosyncratic risk. All right, now let me relate this to what you might remember from previous lectures. The CAPM's main prediction represented by this security market line, Remember this security market line? All right, and the x axis you have the beta. On the y axis, you have the expected returns. And CAPM says, all securities lie on the security market line. Beta equal 1 is the market return. Beta is equal to 0 is the risk-free rate. So basically, the alpha here are deviations from the security market plan. The alpha as a performance measure is called Jensen's Alpha. Because it was Michael Jensen who first used it to evaluate the performance of a bunch of mutual funds in the 1960s. All right, okay, so. Let me give you an example. All right? Here I have five years of monthly return data for the Alcoa stock in the U.S.. Along with the return on the S&P 500 market index. Okay? Now, we can use, as I mentioned before, regression analysis. To estimate the alpha and the beta for the Alcoa stock. Essentially what we're estimating is the index model that just I showed you, the market model. To obtain estimates of the alpha and the beta. And the graph here presents a scatterplot of the relationship between the market return and the Alcoa return. So the x axis here is the S&P 500 return, the return on the S&P 500 index. And the y axis is the Alcoa stock return. And the scatter plot shows the relationship between the two. And what we want to do is use statistical tools to obtain the estimates of this alpha. This alpha and the beta measures. Now in this next graph. What I did is I added the line that shows this relationship. And here you have the equation for that line. Now of course the slope corresponds to the beta estimate. And what is the intercept? The intercept is the estimate of the alpha. Now I have one more statistic here. This R squared variable. What is that? It is a statistic that shows us how well this model fits. Or put differently, how much of the variation in our core return is being explained by the variation in the S&P 500 index. So this is a visual representation of the kind of analysis that we're after. We could also obtain these estimates. By estimating the regression. Which you can do in Excel or using some other statistical software tools. So what I have here is the output from the regression analysis just using Excel. The top panel here reports a number of statistics. Along with the R square measure that I just mentioned. It describes how well the model fits. Now I won't dive into all the output. I suggest that you refer to a statistics textbook to review some of these concepts. But what we're most interested in, however. Are the coefficient estimates. Which are reported down here. Again the intercept is the alpha. And the slope coefficient is of course, the beta. Now, you see that the intercept is 0.0008. So this is the estimate of the alpha. And the slope coefficient estimate is the beta. And that's equal to 1.90576. Which is the same numbers that I showed in the previous slide. Now, remember if CAPM holds, the alpha should be on average zero. Because all the securities should lie on the security market line. So, can we reliably say that this alpha that we just estimated, is statistically different that zero? Or say, if you were estimating the alpha for a portfolio manager. How can you conclude that your alpha estimate is significantly different than 0. Significantly positive. So that the portfolio manager should be compensated. Well, to assist the statistical significance. We look at the t statistic measures. Now, as a rule of thumb, if the t stat is less than 1.96. You can conclude that the estimate is not statistically different than 0. So for example in this case. The t stat for the beta estimate is statistically significant. It is 9.44851. So the beta estimate is statistically significant, different than 0. But the alpha estimate is not. Since the t stat associated is only 0.10. Which is of course less than the 1.96 variable. So in the previous example we looked at the CAPM alpha. We used the single factor index model. Now it's fairly easy to generalize this to a multi-factor model such as the Fama-French three factor model. Right? You would simply be estimating a multivariate regression. Instead of a regression with one single independent variable. So in this case, what would the model look like? Well, it will look something like this. Your portfolio takes this return on your portfolio = alpha + the market beta times their excess return on the market, plus now you're going to have other factors. The size beta times the size factor plus the value beta times the value factor plus the epsilon. And again you would be estimating this model to compute the three factor alpha. Now okay, remember, What the alpha represents. It is the excess return that is being generated above and beyond what is accounted for risk. So when we're estimating the Fama-French three factor model. We're accounting for presumably all the different risk factors. So, the alpha is really the maximum amount that you should be willing to compensate a portfolio manager. For their active management of the portfolio. Think about that. So, for example, if a firm has an alpha of 0.0015. Let's say monthly. Let's say that this is pre expense right? This means that we should be willing to pay up to 0.15% per month. Or 1.8% per year in expenses. Does that make sense? Alternatively, another way to think about this is if you were to compensate your portfolio manager this much. Basically, your after-expense alpha would be 0. And if you compensated anything more than that. Of course, you're just giving away free money. Now by the way, most studies of actively managed funds do not show statistically significant pre-expense alphas. In fact most fund alphas are statistically negative. Which means that most funds are performing, or most actively managed funds are performing. Underperforming their benchmarks. All right, okay. So in this lecture, you learned how alpha is used as a risk adjusting return measure. How to estimate it and how to interpret. Outperformance of a benchmark or of the market, basically corresponds to a statistically significant positive alpha.