[MUSIC] And the way we're going to start this session is by looking at three hypothetical assets and the reason why they're hypothetical you're going to understand a few minutes from now, I need it to behave in a certain way. and, and that is what were going to see in just a minute. So there you have ten years, three assets and lets think of those as annual returns. It doesn't matter whether these are Dollar returns or Euro returns, whether this is our total returns or,. Just the capital gains. Just think of these are the annual returns of the assets. And if you want to draw a parallel, to the data that we've seen in the previous session. You can think of those as being total returns and all of them measured in In terms of Dollars okay? And as we've done before. We can calculate the arithmetic mean return. Simply by adding up all those observations and divide it by ten. And we can calculate the volatility of those assets in a, in a little more convoluted way but you're going to find again the expression for that in one of the two technical notes that goes with the first session. By the way, that brings it to a point that this session is also going to be complemented by a technical note, and that technical note will help you in the calculation in the implementation of the measures that we're going to discuss. Here, as we did in session one, we're going to try to magnify the intuition. We're going to focus on understanding the concepts. And then the actual application of the concepts, then you're going to go into the technical notes, look at the formal expression, and then you're going to try to work out a problem set that will evaluate whether you understood the concepts that we discussed in both session one and session two together. Because again, these are sort of two sessions running into each other. So back of the, back to the data that you have. In front of you, we have three assets over ten years. And as you see, you know, those assets, as any others, fluctuate over time. They give different, mean returns. In other words the average return over those last ten years has been different. Has been quite a bit higher for asset three than for asset two. And quite a bit higher for asset two. Than for asset one. But they also have very different volatility. So acid one fluctuated quite a bit. There are no negative returns there in any of the three assets. It doesn't really matter. But there are no negative returns in any of the three assets. But acid one, as measured by the standard deviation of 10% fluctuated a lot more than asset three. And asset three as measured by the standard deviation of 5%, measure fluctuated quite a bit more than asset two, which has a volatility of 1.5%. And remember that's the way. That we use this measure of volatility to evaluate uncertainty and variability in relative terms so we can say that asset one fluctuated a lot more than asset three which fluctuated a lot more than asset two, and therefore when I look ahead I have much more uncertainty. About what returns I can get in the future when I invest in asset one than when I invest in asset three, and then when I invest in asset three compared to when I invest in asset two so we're not going to go back to the interpretation of those, mean returns and volatilities, but we can calculate those mean returns and volatilities and what that actually shows you is that the most volatile asset of the three was asset one and the least volatile of the assets. Was asset two. Now, remember that what we want to do with this is to combine them in a portfolio. And we're going to focus on two portfolios that combine these assets. And the first portfolio in which we're going to focus. Is a very specific combination of asset one and asset two. And I say a very specific combination because the propotions of our money invested in asset one and asset two have been very carefully calculated, and you'll see in a minute why. They have been very carefully calculated. But we're going to consider that whatever capital that we have, and it's important that it doesn't matter whether you have $100 or you have $100 million, we're going to be thinking in terms of proportions. In finance, and particularly in portfolio management, we always think in terms of the proportion of your capital. So it's not about how much money you have, but it's actually how you split whatever amount of money. You may have. And so in this particular case, we're going to think about building a portfolio with asset one and asset two in very specific proportions, which are putting 13% of our money in asset one and 87% of our money in asset two. Now how would we calculate the return of our 13% in asset one, 87% in asset two portfolio on an annual basis? Well, that is very simple, and the reason it is very simple is because the return of a portfolio in any given period is equal to the weighted average return. Now we weighed average return by definition means the returns of each asset in the portfolio multiplied by the proportion of each asset in the portfolio. So if I multiply asset one had a return of 25%, and I'm putting 13% of my money in asset one so 25% multiplied by 13%. And asset two delivered a return of 21.3%, and I'm putting 87% of my money in that asset. So now, 25%, returning asset one, multiplied by 13%, proportion of my money invested in asset one, plus the same for asset two, that is 21.3%. Return of asset one multiplied by 87%. my, the proportion of my capital in asset two, that gives me the return of the portfolio in year one. If I do that, over, and over, and over again. For the ten years for which we have information, then we're going to end up with ten annual returns for a combined portfolio that is invested 13% in asset one. And 87% in asset two. And here comes the really interesting thing. If you actually do those calculations year after year after year, magic. You're going to get a portfolio that gives you exactly the same return give or take a decimal by exactly the same return of 21.7%. Now let me first show you a picture. That picture the blue line is actually the return of asset one. In the ten years that we have information. The green line is the return of asset two in the ten years for which we have information. And you can guess what the black dashed line is. Is basically the return of the portfolio fixed at 21.7%. Now it sounds pretty interesting. Doesn't it? That you actually put together two things that are volatile. Two things that fluctuate over time and you end up with something that is not volatile. That has no viability over time. Well, this is the magic of diversification. Before we get there we have one more thing to explore. But one thing that you should keep in mind for now is the fact that you are putting together. Two assets that have fluctuated over time. And we end up with a portfolio that has not fluctuated over time. Now we're going to do another combination. We're going to go back to assets one, two, and three that we saw at the beginning. But now we're going to combine assets one and three. Alright? And were going to do it in this case, it doesn't really matter the proportions. And because it doesn't really matter, which is going away to pic 50-50 so were going to put 50% of our money in asset one, and 50% of our money in this case, in asset three. All right? So you see the returns of asset one. You see the returns of asset three, and now you know how you calculate the portfolio in each of the return of the portfolio, in each of those years is 50% times a return of asset one, plus 50% times a return of asset two. So for period one. That would be 25% multiplied by 50% plus 32.5% multiply by 50%. That's going to give you the return of the portfolio in year one. And if you keep doing that over and over again. Over the ten years for which we have information. Then we're going to end up with the return of a 50/50 portfolio in asset one and asset three. And when you look at those returns. Awful. It doesn't look at all like the combination between asset one and asset two. All those returns are all oer the place. So now we're going to look at a picture. The picture is basically the blue line is exactly the same asset one that we had before. Now the green line, instead of being the asset two we had before, in this case is asset three. And the black dash line. Is the return of the portfolio, and as you can see this is a completely different picture from the one before. Before we actually put asset one and asset two together both of which fluctuated over time, and we ended up with a portfolio that did not fluctuate over time. Now we're combining asset one and asset three. And we end up with a portfolio and again look at the black line which fluctuates just as much as those two portfolios. In fact the variability of the portfolio is somewhere in between the variability of asset one and the variability of asset three. Now here's the interesting question. What is it that determines, that in case one, we combine two assets and we end up a portfolio that has no volatility what so ever? And in case two, we combine two assets and end up with a portfolio that is very volatile over time. That, in fact it's volatility somewhere in between the volatility of the two assets in the portfolio. Well if your asking that question and that is exactly what your wondering. You're exactly on the right track because the answer to that question is given by this incredibly coefficient that we call the correlation coefficient. [MUSIC]