Hi everyone, welcome back. This course is about performance evaluation. And of course the foremost measure of investment performance is returns. So in this lecture, we're going to review how we measure returns. If you've taken their previous three courses, you should have pretty good idea of how we compute returns, compute average returns, etc. But it never hurts to review a little bit and get our minds refreshed and get everybody else up to speed. So, how do we measure returns? Well, in fact, there is a number of ways of calculating the return. So in order to evaluate performance correctly and compare apples to apples, it is important to understand how each type of return is defined. In this lecture, we're going to review what we mean by a holding-period return, a cumulative return, continuously compounded return, annualized return, etc. And in addition, you will also learn how to calculate what we call a dollar-weighted return versus a time-weighted return. And how they differ from each other. By the end of this course, you will know how to calculate investment returns in a number of ways and understand the differences between them. All right, so the rate of return over a holding period is simply the change in value between two points in time, all right? The time interval can be measured in days or months, or it could be quarters or years, that doesn't really matter, right? Whichever the unit of time is, what we need is the end of period portfolio values to compute the return. So, for example, if returns are measured monthly, we need portfolio values at the end of each month to compute the monthly rate of return, okay? So if, let's say, over a given holding period there are no cash flows in and out of the portfolio, and turns out that this is important, as we'll see how we need to adjust for that case, if there are cash flows in and out of the portfolio, right? So let's, for the moment, assume that there are none, there are no cash flows in and out of the portfolio. Let's let, v0, Denote the beginning value. And, V1 denotes the end of period value, right? Then of course your gross return over this period is, as you might have guessed, right? Is simply v1 over v0, right? And your holding period rate of return, so your net return is simply, let's call that r, right? Is v1 over v0- 1, all right? Or put differently, (v1- v0)/v0 which is the same thing, right? So for example, if you started out with $5,000, right? And at the end of the month your portfolio value is $5,048. What is your monthly return? Well, it will be 5,048 divided 5,000- 1, which gives us 0.96% or 96 basis points. All right, so of course you could compute a holding period return over a longer period, for example a year, similarly, right? But what if, you calculate the returns for each month for 12 months and you would like to compute your one year return from the 12 monthly returns, right? In other words, what is your 12th month cumulative return, all right? So now, I have t=0, t= 12 and then let's say that we have all the monthly returns, all right? So now let's v12 be the value at the end of 12 months, all right? V0 be the value at the beginning of the period. So remember your gross return over the entire year would be equal to what? v12 divided by v0, all right? But we can also write this as, think about it, v1 divided by v0 x v2 divided v1 x v3 divided by v2 and so and so forth, right? V12 divided by v11, right? What are these? These are the individual gross returns for every month. So I can write this as, (1 + r1), or gross return, times (1 + r2), your gross return over period 2, times (1 + r3) and so on and so forth, (1 + r12), all right? So now you have an expression for the 12th month cumulative return, or the yearly return, all right? What is your annual return in this case? Well, it is, or your net return, Zero- 1, right? Or put differently, (1+r1)(1+r2) all the way to (1+r12) all multiplied -1, all right? And this is of course what we called it, compounded return, right? So you may remember this concept from previous courses. In general, what is your t-period cumulative return, right? So, let me write over here, in general, right? T-period cumulative return, what is that? Well, it is simply the product, going from t = 1 to period T, of the individual period returns,- 1, okay? All right, okay, so now let's talk about average returns, right? Because we need average returns to summarize returns over time. Now remember, there are two methods of averaging, right? The geometric average and the arithmetic average. And they give different results for different purposes, right? So we just saw how we combine shorter period holding returns, the monthly returns, into a longer period holding return by accumulating returns, right? Now suppose we have the cumulative 12-month return, right? And we want to find the equivalent monthly return, right? By equivalent, I mean, what is the constant return per period that would produce the same cumulative return, right? In other words, what we're looking for is the geometric average, all right? That satisfies. This expression. All right,? I'm looking for the constant return per month, rg, such that when compounded, it gives me the same cumulative return over that period, all right? And solving for this geometric average of course, gives us what? Well, the product of the returns, the per period returns, And I'm going to take the 1 over T, root of it,- 1, okay? Now when we compute geometric average return, each return, each per period return, has an equal weight in computing the geometric average, right? So this is why the geometric average is sometimes referred as the time-weighted average, all right? In contrast, how do you compute the arithmetic average? Well, the ordinary average is simply the ordinary average of the collection returns is called arithmetic average, right? For example, so if I wanted to find the arithmetic average, it's simply adding up the per periods returns. All right, and then dividing them by T. or I can write it as 1/T summation of t plus 1/T, all right? That's simply the average return, okay? All right, so often we're going to want to express returns on an annual basis, right? So we need to also know how to annualize average returns, okay? So suppose we have the monthly geometric average return. What do we do to annualize it? Well, to find the annualized geometric return, we would have to, If we had the supposed monthly geometric average, we would compound it, By the number of periods, all right? What if we wanted to analyze the monthly arithmetic average return? Well, We simply multiply by the number of periods. So we would, We would multiply by the arithmetic average, by the number of periods. Suppose we want to come up with a measure of future expected returns, right? Which average return, using historical data, should we use when we need to come up with assumptions for the future expected return? Okay, well, let's think about that, all right? So, we have initial amount of, let say, v0, right? And it's going to grow over t periods. These are, let say, our monthly period returns, All right? Which is going to give us the end of the period value, all right? Well, you might say, if returns are identically and independently distributed, then we could take the expectation of each of these per-period returns and we would get something like, The following, right? It would be, The expectation of each of these returns would be the same. So we would get something like this. Hm, this look suspiciously similar to the geometric average return, doesn't it? So should you use the geometric average as a good estimate of the future expected return? Nope, unfortunately, that is not the case, all right? Geometric average is not what you should use for finding the future expected return. Why? Well, it's the arithmetic average return, which is a better statistic for the estimate of the future expected return, all right? The arithmetic average return is the unbiased, Measure of future expected return. Unbiased, all right? So if you're going to use past data to come out with the future expected return, you should use the arithmetic average return not the geometric average return.