Hi, everyone, welcome back. In the last lecture, we reviewed how we calculate the return on an investment. Now, in order to evaluate performance, we often compare the return on our investment to some reference point. And this reference point could be a cash return, for example, similar to a risk free return. In which case, we would be measuring the additional return we obtained from investing in a risky portfolio versus a risk free asset. More often however, the reference point will be the return on another portfolio, right? What we call a benchmark portfolio that may be similarly constructed as our own portfolio. And in this case, the comparison is similar to measuring the additional return we obtain from actively managing our portfolio under consideration relative to some index or benchmark. So in this lecture, you're going to focus on measuring excess returns relative to a benchmark portfolio. How should we calculate the excess return? Well, comparing the portfolio return to a benchmark return generally gives us what we called the excess return, all right? In other words, the excess return is the additional return that our portfolio creates in excess of the benchmark portfolio. Now it sounds simple, right? But there are different ways of calculating excess returns, under different situations. And they do not all give us the same result. All right, so here we go. So, for a single holding periods, right. Say, a one month or one year, we can simply subtract the benchmark return from the portfolio return. Right. And call the difference the excess return. Simple enough. Now, what if we want to summarize performance over multiple periods? This is where the problems arise. Think about what we were interested in. What is the benchmark? Well the benchmark really, right? Simply represents the investor's opportunity cost, right? So the investor is really interested in the portfolio's value at the end of the period. And how that wealth compares with the wealth that would have been earned if the benchmark had been chosen instead, all right? So how should we measure that excess return properly? All right, so let's do that by looking at an example. Right, so suppose we have data on the past three months performance of an actively managed portfolio. Let's call this our portfolio here. All right. And the benchmark. So before we move on, let's see if you can find the geometric average, the arithmetic average, and the three months cumulative return for both our portfolio of interest and the benchmark portfolio. Okay, I hope you got the right answers. So now, let's compare the performance of our portfolio with that of the benchmark. Well, first of all, if you take a quick look, it seems certain that our actively managed portfolio. All right. Has underperformed the benchmark portfolio over this three month period. All right. But the question is, by how much? Which measure should we look at? One way to measure the average access return is to look at the differences in average returns. We can look at the difference in the Arithmetic average return. And call the difference the arithmetic mean excess return. Or we can look at the difference between the geometric average, which we call the geometric mean difference. So what is the arithmetic mean excess return in this case? Well, the portfolio return, the arithmetic average in our portfolio is 0.33%. Right? The benchmark return, the arithmetic average on the benchmark was 1.53%. Right, 0.33, all right, let me change that color so that you can see more properly. All right. 0.33- 1.53 gives us the -1.20. That's this number here. All right? So that's the arithmetic mean excess return. What is the geometric mean difference? Well, the geometric average is 0.67- 1.37% which gives us -2.04% Right? And it's this number here. Right? And this is the geometric mean difference. And what is this? This is the arithmetic mean Excess return, right? But neither of these measures gives us really the right measure, right? Because neither of these measures measure the difference in investor's end of period wealth, right? Why? Well, think about what we're trying to do. Right? So let's suppose That we started out with $100. Right? The cumulative return on our portfolio is what? Well, it is -2.01%. All right. So the ending portfolio value on our portfolio is going to be what? Well 100 times 1- 0.0201. Right. That's going to give us $97.99. All right, so we started out with $100, our ending value is $97.99. What about the benchmark? Well, the cumulative return on the benchmark portfolio is 4.17% right? So if we had invested the $100 in the benchmark portfolio, the ending portfolio value would have been, 1 + 0.0417, right? That would give us $104.17. All right. What is the difference or the excess wells? Well the difference is $104.17 minus $97.99. The benchmark portfolio had a $6.18 in excess value. Or the excess return can be calculated from the cumulative return on the portfolio minus the cumulative return on the benchmark which gives us the 6.18%. Now, clearly neither the arithmetic average, nor the geometric average difference gave us this result, right? What's going on here? Why? So to see this, again think about what we're trying to do. We're trying to evaluate how our active portfolio value at the end of the period, compares with the value that would have been earned with the benchmark portfolio. Right. So in order to make this comparison, it's the ratio of these values that we should use. And this would give us what? The ending value of our portfolio is 97.99, the ending value for the benchmark portfolio is 104.17. So what is the monthly excess return over the three month period? So what is the per month excess return? So I'm going to take the 1 over 3rd root of that minus 1, well, that gives me -2.02%. This is the monthly excess return. Right? So -2.02%, monthly excess return over the benchmark. Right? So now given this value, we can compute the cumulative excess return over three months. All right. What is that? 1- 0.0202 compounded for three months- 1, which gives us -5.94%. So this is the cumulative excess return, all right? Based on the end of period portfolio values, right? So think about what this cumulative excess return means, right? This is a loss of $5.94, right? On $100 right, per $100 of ending value of the benchmark portfolio, right? So the benchmark portfolio ended up at 104.17 x (-0.0594) which gives us exactly the $6.18 that we were looking for before. And this is exactly equal to the difference in our portfolio ending value and the benchmark ending value. Right. And that's what we wanted to find. All right, okay so in this lecture you learned how to compare the return on two portfolios in measuring excess returns. It's probably not surprising that the arithmetic excess return is not the appropriate measure since it does not involve any compounding. All right? Perhaps more surprising is the fact that the geometric excess return is not either. All right? What's important here is that the difference between two geometric mean returns is not itself a geometric mean excess return. Right? What we need to do is to compare the ending portfolio values for our portfolio and the benchmark in order to come up with a true measure of the excess return.