[MUSIC] In the mean variance framework of this modern portfolio theory, as it is officially called, what we are trying to do is to combine the different securities available to maximize this effect of diversification. Here I have randomly chosen a portfolio P, was 20, 20, 60 invested in the different securities. I could have chosen something completely different. So what I'm going to show you now is an example of portfolio construction, but we're going to do this completely randomly. So you're going to watch now a little animation where we are going to compute the expected return and risk of many different portfolios, but we're going to construct them in a completely random fashion. We are just going to impose that each of these portfolio is fully invested, meaning that the sum of all the weights add up to 100%. So let's have a look at what this implies in terms of risk and return, if we combine in a completely random way asset A, B, and C. Okay, so now we've added all these points on the graph, they all corresponds to portfolio. We can see that some structure emerged from this exercise. There is some kind of a limit to the diversification effect. For example, you can see that we can construct portfolios which have a similar level of return than stock B that have lower risks. They're located to the left of the B along the horizontal line. But there seems to be a limit and there seems to be some kind of an envelope around all these points. If we look at our candidate portfolio P, the one with 20 in A, 20 in B, 60% in C, this portfolio is apparently not the best we can actually construct with a level of return of 6.6%. In particular, if you look at Portfolio Q, this is another portfolio that was randomly generated by our little exercise before, and this portfolio also has an effect of diversification. But among all the portfolios that reach a level of 6.6%, they are all aligned on the red dotted line. This portfolio Q seems to be the best one. And if you look at the graph of all these randomly generated portfolio, for each level of expected return, there seems to be an optimal portfolio with a minimum level of risk. So there is a way of actually directly constructing this limit. And the way we followed here by using randomly generated portfolio is perfectly equivalent. But we could've used an optimization procedure to compute the position of all these portfolios that minimize the level of risk for a given level of expected return. And this approach of minimizing the level of risk for a given level of expected return. This is the minimum variance, or min variance framework, developed in the 50s by Harry Markowitz. So if we look at portfolios that are similar to portfolio Q, in the sense that they minimize the risk for a given level of return, we can actually draw the entire envelope. This is this thick black line that you now see on the graph. This envelope is called the efficient frontier. This is the collection of all portfolios, that for a given level of return, minimize the risk of the portfolio. They're attained by adequately choosing the weights in the available security. And you'll see with only three assets we're able to create a large variety of portfolios, and we are also able to perfectly use this effect of diversification. If we use it to it's full extinct we will reach this thick black line. Some remarks are in order regarding this efficient frontier. This black line that constitute the envelope of all these randomly chosen portfolios. They provide the portfolio on the black line the optimal level of diversification. But we can see that one portfolio is a little bit particular. The one on the extreme left of the graph constitute the minimum level of risk that we can obtain by combining asset A, B and C. It is at the summit of this hyperbola, completely to the left. We call this portfolio the Global Minimum Variance Portfolio. It is the minimum level of risk we can reach by combining the three assets. Another remark on this efficient frontier is that, for example, if you want to attain a level of risk at 15%, there seems to be two candidate portfolio. One with a low level of return around 1%, and one with a rather high level of return around 7%. If your objective is to minimize risk for a given level of return or maximize return for a given level of risk, it wouldn't be reasonable to consider the portfolio for level of risk of 15% that only attain the 1% expected return. So when we consider the efficient frontier, we actually only look at the upper half of the envelope. Starting at the minimum level of risk to a global minimum variance portfolio and considering only all the portfolio that have higher expected return. [MUSIC]