[MUSIC] Welcome to the module three of this course, which has a focus on portfolio diversification. And today we are going to talk about the benefits of portfolio diversification and why portfolio diversification is so exciting in the first place. Well let's talk about, let's call it the magic of diversification. Okay, where does it come from and why is there any magic in diversification? We sometime call diversification the only freelance that we have in finance. Well first of all let's take a look at the decomposition of expected return. Let me call it Up in terms of mutation. Expected return on the portfolio, and I would like to decompose the expected return on the portfolio in terms of the expected return on the individual stocks or individual components of the portfolio, whatever they are. And I call them Ui. And this is very simple equation that comes from the linearity of the expected return operator is telling you that the expected return on the portfolio is nothing but the weighted average expected return on individual stocks. Well, there's no magic there, that's a little disappointing, right? There's no magic because if you start with high expected return stocks you're going to end up with a high expected return portfolio. If you start with low expected return stock, you're going to end up with low expected return portfolio. There's no magic effect coming from diversification in terms of performance. Where the magic of diversification actually shows up is when we turn to the risk dimension. The risk is in most simple, setting measured in terms of the variance of the portfolio. Now, if you look at the expression for portfolio variance, it's even by the sum for all i, and j, of Wi, Wj, sigma ij. Or Wi, Wj, or again, percentage weights, percentage allocation in the portfolio to these asset i and j. And sigma ij is the co-variance between asset i and asset j. And what is magic in this case is that it is actually very easy to see that these variance of your portfolio is actually lower, quickly lower than the average variance of individual components in your portfolio. And that's where the magic of diversification comes along. In other words, you can perfectly well start with high risk components, very high volatility stocks or asset classes, and eventually cook up a portfolio by utilizing the magic of diversification in such a way that you end up with a low risk portfolio starting with a high volatility components. And that's really magic, that's where the magic comes along. Now, of course, this is only going to happen if the correlation of the co-variance between these components is not too high. Obviously the limiting case where all these assets are perfectly correlated, these assets are a perfect substitute, one with respect to another. And then there's no magic, magic diversification effects to expect. But in general for a assets that are weekly or imperfectly correlated, well then, this is where the magic of diversification comes up and shows along. This has very profound implications in terms of portfolio decision. Let me take this opportunity to introduce the Sharpe ratio here for portfolio p which is defined as the excess return of the portfolio Up. Expected return on the portfolio in excess of the risk rate or f divided by the volatility of the portfolio sigma p. And one of the goals in the scientific approach to portfolio diversification is to maximize the Sharpe ratio. Now, maximizing the Sharpe ratio kind of intuitively will come along very nicely if you think about the situation where the numerator is pretty high. So you're looking at assets that have high expected returns, presumably because these are high risk assets and that's why the high expected returns, regardless of how we measure risk. And then you divide that by portfolio volatility. And if you can somehow make portfolio volatility small by an efficient use of portfolio diversification, well, then you end up with a portfolio that has a high expected return and a low risk. Well, we'll like it because the, Reward per unit of risk will be high in this particular case. Well, that's exactly represented in this graph which looks at the standard efficient frontier. We are not going to go back to the details of how the efficient frontier is being constructed. But simply here we are going to recognize that if you look at the risk-free rates on the y axis, and if you look at this straight line that go from the risk-free rate, and that is tangent to the efficient frontier. Well, that's exactly what we want to do. We want to increase the slope of that straight line, the slope, that straight line being nothing but the Sharp ratio of the portfolio we are defining. And we want to increase it as much as possible. And the tangency point that we see on this graph is nothing but the maximum Sharpe portfolio, the one that has the highest reward per unit of risk. And that exactly what diversification will bring you. Now, there's an extremely interesting result that is not that well-known and that I think is worth mentioning at this stage. To think about the power of diversification, and also to think about why the maximum Sharp-ratio portfolio is such an attractive portfolio. To see this, let's just go back to single-factor model. And let me just use, for the sake of simplicity, not because necessarily that's the one we want to use in practical applications. But just let me use Is single factor model like the market model, in this case, right? There's a single factor maybe the factor is is the return on the market. Now, what happens is of course, I can use the factor model, the individual stock level, and I can also use it at the portfolio level. And the beta of the portfolio with respect to the market, which is my preferred factor in this case, is simply given by the weighted average of the beta as of individual stocks with respect to the market. And the residual component epsilon p is nothing but the weighted average of the residual component of the specific stocks. Now, there is a very important statement that we're going to make. If the fact the model that I'm using is the true asset pricing model, then what we can claim is that there is something extremely specific about the maximum Sharpe ratio portfolio. It is such that all unrewarded risk has been taken out, it has been diversified away, and hence the power of diversification. In other words, if I apply this decomposition for a particular portfolio which is the maximum Sharpe ratio portfolio, well, in this case there is no unrewarded risk. All the risk contain in the portfolio is rewarded risk. Everything unrewarded has been diversified away. And that's very intuitive because that's kind of intuitively explains how we can manage to maximize Sharpe ratio, well, simply by eliminating unrewarded risk and maximizing therefore the return per unit of risk [MUSIC]
