[MUSIC] All right, now we get to what I think is the coolest and most amazing part of finance. And when I first saw this I was like, this is really weird, this is magical. We're going to look at some magical stuff, and hopefully you'll find it just as exciting as I did when I first saw it. We'll see. What we are going to be doing in this and the next lecture is really the core of all of portfolio construction. So if you sort of really understand what's going on here and the next lecture, everything in portfolio construction will just seem, not just easy and obvious, but magical. All right, so let's start by just laying out what we're trying to do here. It's very simple. I'm going to take two assets, A and B, and I'm going to look at it in this sort of risk/return space, some people call it mean variance framework. Mean variance basically means on my x axis, I'm going to have the variance or the, I'm using standard deviations here, so you've got risk here, and you've got return here. And that's really how we're going to look at the portfolio. So I've got two assets that I can plot on this risk/return space, A and B. A has a variance of 10%, sorry, a standard deviation of 10%, and has a return of 4%. B has 14% volatility, and a 6% return. The question that I'm asking now is what happens when I build a portfolio consisting of A and B? So I'm going to combine A and B and I'm going to build a portfolio, and I'm going to see how does this portfolio behave. Let's start with something very simple, very obvious. What would be the risk and return of a portfolio that consisted of 100% in A? So I've got $100, I put all of it in A, what would be the risk and return of that portfolio? Well, we've got 100% of it in A, so it's going to behave just like A, which means it's going to give you a 4% return with a 10% volatility, okay? That's obvious. The next question I'm going to ask is also as obvious. What if I put 100% of that portfolio in B? Sorry, 100% of my assets in B, right? The portfolio consists of nothing but B. Okay, that's also very obvious. It's just B and nothing else, therefore it's going to behave exactly like B, which means it'll give you 14% volatility and 6% return. Okay, now I'm going to ask a question, which is, what if I split the difference? If I put half my money in A and half my money in B, what is that portfolio going to behave like? So you might make a reasonable guess, you might say, well, I've got half my money in A and half my money in B, and so I'm going to get halfway between the two. So I'm going to get a 5% return, that is 50% of A and 50% of B. Think of it like a weighted average of A and B, because my weights are 50/50. And my risk is going to be a weighted average of A and B. So my risk is going to be halfway between the two, which is 12%. And my return is going to be half way between the two as well, which is going to be 5%. You would be half right, and you would be half probably wrong. Let me explain why. The part that you have right is the return. The return on a portfolio is nothing more than the weighted average of all the components of your portfolio. That's easy, okay? So the return, no problem. There's no magic, there's no weird stuff going on here, it's perfectly okay. What is weird is on the risk side. So in fact, the answer to what is the volatility of this combination of 50/50 in A and B can not be answered with just the information I've given you, why? Because it depends on the correlation between A and B. If A and B are perfectly correlated they basically behave like the same asset, they go up together, they go down together. Then in fact you would in fact see that the volatility is kind of halfway in between. But if they are not correlated, one goes up while the other goes down, one zigs while the other zags. In that case you see that the combination will actually be less volatile than you might think. It is not going to be just half of it, it's going to be somewhere less than that. And the more decorrelated they are, the less the volatility of that portfolio. This is the basic sort of mystery or the magic of portfolio construction. Let's take an example. So let's look at the very, very simple case here. And let's come up with the expression for the return. We've already talked about that, it's just nothing more than the weighted average of the two assets, A and B. The expression for the volatility is a lot more complicated. So you'll see it is sigmas, also it's easier to write in term of variance. So it's the variance of A times the weight of A squared plus the variance of B times the weight of B squared. And then this term, which is twice wA times wB, weight of A times weight of B, times the volatility of A times the volatility of B times the correlation between A and B, okay? Now, let's plug in some numbers. So let's assume that the correlation between these is, let's say, 0.4. So now what you would see is the volatility of this 50/50 combination between A and B is actually 10.10%. The volatility is lower than the volatility you would have expected. What was the volatility you would have expected if they had been perfectly correlated? You would say, well, it's halfway in between, so let's split the difference, so to speak, so it's the weighted average of the two volatilities, it's 12%, but no, it's not. It's actually much less. It is less because these two are not perfectly correlated, and the actual volatility is 10.10%, okay? This is really the magic of diversification at work. So let's look at different weights, right? So if you look at this curve that I've put on the screen, what you've got at the left end of the curve is the portfolio that consists of 100% in A. And what you've got at the right top end of the curve is the portfolio that consists of 100% in B. But as you start moving from B and putting money in A, you see that the portfolio that is resulting from that has a return that is between the two. But the volatility actually drops far below what you would expect from just the weighted average of the two, all right? To the point where there is a point, there is a combination of the two weights that have a volatility less than either one. You see that there's a point that's called a nose of this curve. So this curve plots, this kind of, it's a convex hole, but the point is there's a nose to that, there's an edge there. And you see that the nose is actually less than the volatility of A and less than the volatility of B. This should be really weird to you. You've got two assets. This has some volatility. This has some volatility. I put them together in some ratio, and suddenly I've got something that has a lower volatility than either one of them. This is the magic of diversification. The magic of diversification is simply that you can take two assets, you can mix them in a certain ratio, and end up with a portfolio that has a lower volatility than either one of them. It was astonishing to me the first time I saw it, and I just assumed that I was misunderstanding what was going on. But you can actually see that it is a function of the correlation. It is purely the correlation that's doing it. In this example I've actually plotted a whole series of curves. So if the two assets are perfectly correlated you get that straight line. In other words, if the correlation between those two is one, then in fact you do just get a straight line between the two. But, if the correlation starts to drop off, let's say you go from, let's say these two assets are correlated, the correlation coefficient is, let's say, 0.9, well, you see that you get a little bit of a gradual curve. It's still less than what you would get from just having a perfectly correlated set of assets. And as you go more and more, as the correlations drop, you will see that curve becomes more and more pronounced. To the point where if these two have very low correlation, you will almost certainly be able to construct a portfolio as that has a volatility that is less than the volatility, significantly less than the volatility of either one of them. And this is exactly what the science of portfolio construction is for the most part. It is exploiting this one feature that says that, hey, if you put two things together that are decorrelated, you will get some sort of magical bang for the buck. This is what someone people call the only free lunch in all of finance. And the only free lunch in all of finance is basically this, that if you take two decorrelated assets and you put them together, you can construct a portfolio that has a lower volatility. In order words, this the power of diversification. And we're going to take this power in the next lecture and we're going to see how we can use this to construct portfolios that have exactly the sort of characteristics that we want. And in fact, as a preview of what we're going to see, we're going to see that there are going to be some portfolios, as a result of this, that you would never ever want to hold. Because there's always going to be another portfolio that is going to be a better portfolio to hold. And that's what we'll cover in the next section. [MUSIC]
