All right, welcome back. So in this section, we're going to go right into the art of modern portfolio theory. Where we left off if you remember, was this very interesting point where we have two assets, and we were allocating our capital across these two assets, and we were finding that we were able to build portfolios that had lower volatility than either one of the two assets. That's great. What I'm going to do now is I'm going to completely change the picture just by adding one new asset. So we have exactly what we had before. We had A and B, and there is a curve between A and B that represents all the possible portfolios that you can make from A and B. Now, I'm adding C and therefore you will get another curve, which is the curve between B and C, consisting of all the portfolios that you could construct using B and C. So the point I'm trying to make is, every point on that curve corresponds to a portfolio. So for example, the point that I have marked as Y is just one point on that curve between B and C, and every point on that curve between B and C represents a possible portfolio that represents weights between B and C. Similarly, there is another portfolio which I've marked as X, and that lies on the curve between A and B, and so, X is nothing more than a portfolio that is allocating its capital across A and B in some weight. So that's not all that surprising. All we've done is add a new asset. So you think, okay, all I got is a new curve, but there's actually something very interesting going on here because, if you think about X now as an asset, after all it's a portfolio which means it's an asset, I can put money in X, and I can put money in Y. As a result really just by introducing the second curve, I can now build a portfolio that consists of some weight in X and some weight in Y. What that means is actually, I can define a new curve that represents all the portfolios that I can build by allocating capital to X and to Y. So now, I have this third curve. Well, you could keep going with this. You can actually draw curves from any point on the first curve to any point on the second curve, and then draw curves from any point on that curve, to any point on any of the other curves. What does this really mean? Is just by introducing this third asset. I have expanded dramatically the space of portfolios that I can build. If I have just three assets, I in fact no longer have just a curve that represents all possible portfolios, I have an entire region of portfolios. As we had before, every single portfolio that I can build out of all of these assets is a point within that region. Conversely, every point on that region, I can trace back to some combination of weights, to these underlying assets. So just by introducing a third asset, we've dramatically expanded the space of possible portfolios and we've actually got no longer, just a curve. We've got an entire region. So every point within that region represents a portfolio. So take a portfolio P. P is just a point somewhere in that region. The question that I am going to ask you is, would you invest in the portfolio P? The answer is, I would say, I really hope not. You should not invest in that portfolio P. Why? The reason is because, just look above P, there is another portfolio in that region called Q, and Q is a portfolio that has a higher return and the same level of volatility as p. So why on earth would anybody invest in P when they can invest in Q? Q is what we call a portfolio that dominates P. It dominates P in the sense that there is no reason why you would hold, there is no possible explanation for why you would hold P when you have the choice to hold Q. Why? Same volatility, high-return, you'd be crazy not to take it. In fact, there's another portfolio that also is better to be, better than P. That is if you go left instead of going straight up, if you go straight left, you see, there's another portfolio that I marked as R. What is R? R gives you the same return as P, but it gives you a lower volatility. So again, the question is, why on earth would you hold P, when you can hold R. Taking this to the logical extreme, the answer that you really are looking for in terms of what portfolio should I hold is, you would never hold anything in the interior of that region. You would never hold any portfolio in the interior of that region. Why? Because there's always at least two portfolios that are better. One is the one straight above you, going straight up, which is going to give you a higher return for the same level of volatility, or you can just go left. When you go to the extreme left, you're going to hit the edge of that region, and at that point you've found your portfolio that has the same level of return, but you cannot decrease the volatility anymore. What this is really saying is, out of all that entire region of portfolios that you could possibly hold, it turns out. You actually don't want to hold most of them. The ones that you care about holding, the only ones that are of interest to a rational investor are the ones sitting on that edge, sitting on the frontier, and that space is what we call that edge, that line is what we call the efficient frontier. The efficient frontier, are the only portfolios that a mean-variance investor should be interested in. A mean-variants investor, is someone who's looking at a portfolio's solution problem like this, which is "Hey all I care about is returns and volatility." If you care about returns and volatility, everything that you care about is captured by this diagram. Therefore, you would never look at any portfolio that is not on the efficient frontier. That's really the core insight of modern portfolio theory. So everything we do from this point onwards in this course, is really just playing with this very basic idea, that hey what do I need to do to get portfolios on the efficient frontier? It turns out that in practice, this is a lot harder to do than you might think, and there are significant problems that you face while trying to identify portfolios on the efficient frontier. Will identify the problems, will also identify what solutions we have to work around those problems. Before we go, I'm going to give you a little bit of a heads-up on something even more amazing. That it is going to turn out that, we actually don't even care about all of those portfolios on the efficient frontier. It's going to turn out that actually, we just care about one portfolio on the efficient frontier, but to do that you're going to have to wait for Lee and L to explain that to you. In the meanwhile, we're going to put a pause on this, and I'm going to walk you through the process of how to actually draw this efficient frontier using code and using a computer to actually estimate that efficient frontier. So you can actually look at it. Then when I'm done with that, we'll come back and Lee and L is going to explain to you why there's one more twist to the story. It's not even the entire efficient frontier we care about for the most part, we care about one point on the efficient frontier. You're going to have to wait for Lee and L to tell you how that works. Thank you very much.