Hello there, welcome back. I hope you're enjoying this first module of our final course in our specialization. Well, in this lecture, we're now going to turn our attention to how we measure the volatility of the excess return. In other words, how much extra volatility is undertaken for the sake of the excess return. After all, right, that is the measure of risk we need for an actively managed portfolio. So in this lecture, you're going to learn about tracking error and residual risk, and what they mean, and how to measure them. These are fundamental measures in comparing the risk of an actively managed portfolio to that of this benchmark. All right, let's start with tracking error. If the average excess return is the reward of an actively managed portfolio, then the standard deviation of that excess return is what is usually viewed as the corresponding measure of risk, right? So then specifically, tracking error is simply defined as the standard deviation of the arithmetic excess return, right? So it is equal to, the arithmetic excess return, let's say it is defined this way, right? It's the volatility of that. Right, and it's also referred to as what's called as active risk. What is then residual risk? Well, to understand the concept of residual risk, think back to the CAPM or other factor models. It's actually what these models state, right, is that the return on an actively managed portfolio can be written as. Right, so here is the (rp- rf) of an actively managed portfolio = alpha + beta (rb- rp) + epsilon, right? So, let me define terms, what is rp, well, that's the portfolio return that we're interested in, right? What's the alpha? Well we're going to talk more about it, but of course it's the intercept coefficients. And or as we'll see it's the risk adjusted Excess return. What is beta? Well, that's the portfolio beta. All right? Or that portfolio's sensitivity, To the benchmark, Right. Of course in the CAPM this is simply the market beta, right, or the benchmark portfolio is the market portfolio. What is rb? Well, that is the benchmark portfolio return. Right, again, in the CAPM it's the market portfolio. And rf, of course, is the risk free rate. Right, so what's the epsilon? Well it's the error term or the residual term. So what is residual risk? Well residual risk is the defined as the volatility of this, of the residual. Right, the standard deviation of the residual is what's called the residual risk. Another way to use this equation of course, to note that what we're doing here is that the volatility or the portfolio return can be decomposed, right, into two components. One that is contributed by the benchmark, right? Multiply by the beta or the systematic risk. And the second component which is called the residual risk, right? Okay, so the volatility over the arithmetic excess return over the benchmark, is simply a special case where the beta is equal to one, right? Then the standard deviation of the residual becomes actually equal to the standard deviation of the excess return, right. So when beta equals one, the residual risk and the tracking error are exactly equal to each other. When beta is not equal to one however, the estimated alpha is not the same as mean excess return on the portfolio over the benchmark, right? And the tracking error and the residual risk are no longer the same, right? It's very common, however, to simply assume that the beta is one, and use the tracking error as residual risk interchangeably. All right, so in this lecture, you learned how to measure tracking error and residual risks. These are essential to risk measures of an actively managed portfolio compared to the benchmark portfolio. Now while tracking error and residual risks are similar in concept, the only time they're actually equal to each other is when the alpha is equal to the arithmetic excess return or put differently when the beta is equal to one.