Hello, welcome back. Okay, so in the last lecture we saw that the difference between two geometric means is not the geometric mean excess return. So then how do we correctly compute the geometric mean excess return? Okay, so to see this let's start by letting the zero denote the beginning value of our active portfolio, all right? And let's let vp1 to be its value after one month or one period, right? So this is our active portfolio. Now similarly, let's let V0 be the beginning value of the benchmark portfolio and we can let VB1 be its value after one period. All right, so this is our benchmark portfolio. All right, now accordingly we can compute the portfolio return and the benchmark return. The portfolio return RP, right, is as we've seen before is given by the ratio of the ending value divided by the beginning value, right? What about the benchmark return? Let's call that RB, right? Well it's given by the ratio of the ending value of the benchmark portfolio divided by its initial value, okay? Now remember from last lecture that we defined the account balance excess return in terms of the ending values of the active and the benchmark portfolio, right? In particular, we use the ratio of the ending values. Right, so now we can do the same here. Let's let rx be the countdown excess return so now we can write 1 + rx, right, to be the, Equal to the ratio of the ending value of the portfolio return, and the ending value of the benchmark return. Now, if I divide both the numerator and the denominator by b0, now what do you have? Well, you see that the numerator is of course this expression, right? So I can replace that by 1 + rp. The portfolio return, right? And the denominator is the 1 + rb, right? So now we have an expression for the excess return. The compound express return is simply (1 + the portfolio return) divided by (1 + the benchmark return)- 1, right? So in fact with a little algebra, we can even show that this excess return rx, all right, can be written as the portfolio return minus the benchmark return, divided by 1+rb. Right, so this, our x here, is called the compound excess return for a period, right, or the geometric, Excess return, right? So, notice that we define it similar to what we did before by computing the ratio of the end of period portfolio values. But that just happened to equal to 1 plus the portfolio return divided by 1 plus the benchmark return minus 1, all right? Now notice that the account balance access return is related to the arithmetic access return right which is this, right? But is not equal to the arithmetic access return. So let's now go back to the example from last time, right? And we can compute the compound excess return for each period as we just saw. So what is the mean excess return over this period? Well, the first return is 1 + 19.2%, right? That's the portfolio return divided by the benchmark return minus 2.0%, right? Times the second period 1 + -2.6% divided by 1 + 9.7% times the third period, 1 + (-15.6%) divided by 1 + -3.1%, all right? So that gives us the three months periods, right? So what is the geometric mean excess return? Take the one-third root of that -1 which gives us -2.02%, okay? So this is exactly what we did in the previous slide, right? To compute the compound excess return for each period. Now, in fact, the relationship between the compound excess return and the geometric return can be written as, All right, so this is the geometric excess return. Take the 1 plus the geometric portfolio return, all right? Divided by, the geometric benchmark return- 1. All right, so if I did that and we go back and point out the numbers to you, what am I comparing? I'm comparing this. Right, so we're going to be comparing this and this. So what is that number? Well it's 1 plus minus 0.67% divided by 1 + 1.37%- 1. And of course, that gives us example the same number, -2.01% or 2%, okay? All right, so that gives us the geometric mean excess return. Okay, so, so far I told you that arithmetic mean return can be misleading when evaluating past performance. Are there any situations where we should the arithmetic average? Yes, and these tend to be the situations where we are modeling expected values of future returns. All right, so as we saw previously the arithmetic average is the unbiased estimate of the expected return of future returns. All right, the expectation of future returns. So the arithmetic average or the arithmetic means provide estimates that are independent of the. And that's why they are unbiased. So, in the last two lectures, you learned that the arithmetic excess return can be misleading when evaluating performance. More accurately, we need a measure of the excess return that compares the ending values of the portfolio relative to what you would have earned with the benchmark portfolio. Right, this is why we prefer the account balances return to the geometric mean difference, or there's arithmetic excess return. Neither of these latter measures give us an accurate indication of the excess wealth over multiple periods. More generally, however, you'll learn that the term excess return can mean many different things, all right? Best practice is always disclose the specific calculation you use when reporting measures of excess returns.