Okay, so so far we've assumed that there are no cash flows in and out of the portfolio, right? But what if we have cash flows in and out of the portfolio? How do you measure returns then? Well, when we consider investments over a period during which cash was either added to or withdrawn from the portfolio. Measuring the rate of return becomes slightly more difficult, right? So let's do an example. So suppose you buy a share of stock at time 0, let's say at $50, right? Let's say this is time, right? So at 0, you buy your 1st share at $50, all right? And then you buy another share, let's say a year later, all right? Buy a 2nd share at $53, okay? So you can think of these as your cash outlays, right? Okay, now suppose that in addition, right, you collect $2 in dividends. It's equals 1, right? From the 1st share that you bought, right? And then at t equals 2, you get $4 in dividends, because now you have two shares, okay? And let's say that at the end of the 2nd year, all right, you also sell your shares at $ 54 per share, right, so you collect a total of $108, right? So how do you find your rate of return? So these are your, if you will, your cash inflows. All right, your cash outlays, your cash inflows, all right? Well, we can find the average return over the two years by using the same, right, notion as the discounted cash flow method. Right, what do we do? Well, we equate the present value of the cash inflows to the present value of the outflows, all right? So what does that equal to? Well, you bought one share at time 0, and then you bought another share at time 1, right? So discounted, that gives you the value at 0. So these are the present value of your cash outflows. What is the present value of your cash inflows? Well, you received $2 at the end of year 1, right? And now you've received $4 in dividends and you sold your two shares at the end of year 2, right? And if you solve for r, you're going to find that it is, 7.117%, right? This is of course what we call the, what we've seen before, the internal rate of return or the IRR of the investment, right? And we sometimes call this the dollar-weighted return. It is called the dollar-weighted return because the stock's performance in the 2nd year, right, when you have two shares is a greater influence on the average overall return. So what is your time-weighted or your geometric average return? Well, let's think about it, your 1st year return is what, well, you collected $53,. Well, your share is now worth at the end of the year 1, $53, you collected the dividends, right? You spent $50 on it, divided by 50, right, that gives us 10%, right? So that's the return over the 1st year. What is your return over the 2nd year? Well, your 2nd year, you got $54, you collected $2 in dividends, you had bought it at $53. All right, so that's 5.66%, all right, so what's your average return? Well, let me write over here., the time-weighted return, right? So it's going to be (1 + 10%)(1 + 5.66%), right? That's over 2 periods, so I'm going to take the square root of that -1 and that gives me 7.81%. Right, so that is your time-weighted return or your geometric average return, right? Notice that I computed the per period returns, compounded them and took the geometric average. Which is different than the dollar-weighted average return. All right, so in this lecture, we reviewed measuring returns, all right? The performance of an investment is measured by its return. However, there are several ways of calculating a rate of return. In this lecture, you learned the definition, the computation and the different purposes for a variety of return measures.