We also need to consider the weights of a portfolio, which is a percentage of the composition of a particular holding in a portfolio in dollar value terms whatever unit of currency using. So if you have $10,000, $6,000 might be in an asset a, so the weight would be 60%. The degree of risk reduction depends upon the weights in the portfolio. So if you have two assets, one is very risky and one is not so risky, if you put a lot of your money in the risky asset, then the the risk of a portfolio intuitively would go up. And likewise if you put all of your money in the safe asset, the risk goes down. So we we now need to figure out what are the optimal weights in the portfolio. In the previous examples, I just told you what they were, 16, 40%, but now we're really trying to figure out what is the optimal weight. We know there's securities out there, we know how to calculate the expected returns based on history. We can calculate the risk based on their variance, but now that we have done this for a suite of securities, what's the right mix? That's the question, and to reiterate in this class, I'm only going to assume positive weights, and this is realistic if you think of something like long only portfolios or mutual funds. These are also things that only have positive weights. So the efficient frontier represents the best combination of assets in order to get the best expected returns while considering risk. And there are two ways that we can think about this problem, we can hold expected return fixed and try to find the lowest possible amount of risk and find that portfolio, or find a portfolio that gives us the highest return given some level of risk. So, for example, I want to make a 10%, what kind of portfolio mix do I get and what's the least amount of risk I need in order to get that 10%? But on the flip side, maybe I'm thinking about how much risk I'm willing to take, I'm willing to take a lot of risk. So given that I'm willing to take a lot of risk how much more expected return can I anticipate? One thing to note is that portfolios below the efficient frontier are not desirable, I'll show you that in our graph. So let's look at a graph, so in this slide, we have a graph of the efficient frontier. And remember that what that means is for a given risk, you're trying to maximize returns or forgiven expect return you're trying to minimize the risk. And either case the efficient frontier will look something like this. It's a bullet shape and each of these dots represents different mixes of a portfolio. So if you have three assets, underlying asset securities, asset a, b, and c, what's the right mix? And those are the different weights, and for each of these different weights, you have a different return and a different level of risk. Risk is on the x axis, R-I-S-K, and return to the y axis, and to be strict about it, it should be expected returns, little hard to write on the screen. So expected returns of the y axis, risk is measured along the x axis, you increase risk as you go to the right and you increase returns as you go upwards. One thing to note is that this curve is a bullet shape here, you can sort of see that here, but ask yourself why would you not pick a portfolio down here, for example? Why are these gray dots not really something you would choose? So if you think about that, this says for this amount of risk, this amount of risk, whatever that number is. I get this amount of return, but for the same risk level assuming that's a straight line, I get that amount of return. So I would pick this portfolio on the top of the frontier and not one of these dots below and that's why this bottom part of the bullet curve as been grayed out. These are not rational choices, right? You want to minimize your risk and maximize the return. So if you are picking between two dots, you want to pick the one that's up and to the left. Okay, so anything above there is more desirable to this, but this is the best you can do. If you're trying to minimize the risk, given some expect a return, or you're trying to maximize your returns given some level of risk. Next, I want to talk a little bit about the shape of the efficient frontier and how that's driven by the correlation. But before I do let's look at the formula again to calculate the volatility of a portfolio. And that is calculated by the weight squared of asset a times the variance of asset a plus the weight squared of asset b times the variance of asset b plus 2 times the correlation of the standard deviations of the returns of a and b times their respective weights. And that's what we have here, and it is this term here that helps to drive the shape of the efficient frontier, so let's look at that. In this slide, I show some examples of how the correlation between the stocks affects the shape of the frontier. And I use a simple example of two stocks, two stock portfolio and in the bottom right we see the case of the efficient frontier when the stocks are perfectly correlated. This slide has four graphs of the efficient frontier trying to illustrate the relationship between the correlation and the shape of the curve. And the bottom right-hand quadrant, I have a to stock portfolio and these two stocks are perfectly correlated. And what we have here is a straight line, at the other end of this spectrum to stock portfolio where the assets are perfectly negatively correlated. We have this wedge-shaped here, and that represents some sort of arbitrage opportunity. More generally though and more often you'll see something that looks like these bullet shapes here. And in these two examples, we have rho = -0.05 and rho = 0. And as you play with different values of rho, you'll see that the efficient frontier will go from the straight line. It'll start to bow out and keep bowing out until it finally reaches a point and turns into this wedge shape that you see in the top left corner. And that wraps up our discussion about the efficient frontier.
