Welcome back. Today we are going to talk about Scientific Diversification and more precisely where I'm going to talk about global minimum variance, portfolios. Now, before we do so just let us remember that the most attractive before you can think of is the maximum sharpe ratio portfolio. It's really attractive because by construction it is the portfolio that gives you the highest possible performance per unit of risk. And given that investors have access to fairly limited risk budgets, having technology that allows you to maximize the performance for a given risk budget is something that is extremely useful and powerful. Now, remember that we also said that the bad news is that the expected return parameters that are critically needed for the maximum sharpe ratio portfolio are extremely difficult to obtain with any degree of accuracy. And as a result of that, large amount of estimation error, practical implementation of the maximum sharpe ratio portfolio tend to fail. So what do we do then if we are still trying to focus on scientifically diversified portfolio, yet recognizing that we do not have access to very meaningful or very reliable expected return estimate. Well, there's actually a very interesting remarkable portfolio on the efficient frontier another remarkable portfolio, which is the global minimum variance portfolio. Now, as we can see that portfolio as the name suggests, solve the problem of minimizing the risk of the portfolio. That's why you see the risk portfolio is the one that sits at the extreme left corner of the graph on the efficient frontier parabola. Now, this particular portfolio is extremely appealing from a practical standpoint because there's no need to use any expected return estimate when implemented the minimum variance portfolio. The minimum variance portfolio is actually the only point on the efficient frontier that does not require any expected return estimate. And the reason why that's the case is pretty simple. It's because we are just trying to minimize risk without any expected return target. So this is the good news. We look at the expression for portfolio variance, which is essentially the sum of WI, WJ, Sigma IJ, where Sigma IJ is our portfolio covariance or the covariance between the the assets I and J, which can also be written as Sigma I times Sigma J times OIJ. Where OIJ is the correlation between stock I in stock J. And the goal of the minimum variance portfolio is to minimize this quantity, which is the variance of the portfolio. So it's pretty much a straightforward. Now, there is a problem though with the minimum variance portfolio. And there are different ways to think about the problem, one way to think about the problem is to recognize that when we estimate, implement a minimum variance portfolio. We think we are not using expected return estimate and that's true we don't need expected return estimate. But now we can rephrase the problem in a slightly different way, we can think about what kind of assumptions about expected returns would we need to make for that global minimum variance portfolio to coincide with the portfolio, which is the most attractive portfolio, namely the maximum sharpe ratio portfolio. Now, when you think about it what happens is you need to assume that all stocks have the same expected returns. And if you do so, if all stocks have the same expected returns, then the optimizer when trying to optimize the risk return trade off, only focuses on minimizing risk. Because when it comes to performance when it comes to return all those positions seems to be born equal. So in other words the global minimum variance portfolio does not require any expected return estimate. But yet at the same time, it implicitly assumes that all expected returns are equal. At least you need that assumption for that minimum variance portfolio to get close to the most attractive portfolio that you care about, that you should care about which is the maximum sharpe ratio portfolio. Now, in reality, of course different stocks of different portfolio constituents have different expected returns. And what happens with the minimum variance portfolio is you turned to overweight those portfolio constituents that have low volatility. In other words, the minimum variance portfolio is pretty straightforward. You want a portfolio with low volatility while we are going to heavily load that portfolio with low volatility constituents. So in other words the problem that goes with that is the minimum variance portfolio tends to be a low risk portfolio, which is what we expected. But it tends to be a fairly concentrated portfolio. So in other words by trying to focus on minimizing risk, we are kind of giving up on portfolio diversification. And that's a problem because that low risk portfolio will indeed deliver what it is meant to deliver, which is low risk. But at the same times it might as well deliver a low performance because of this high concentration in very few portfolio constituents. We have reasons to believe academic studies have shown that the global minimum variance portfolio is actually not consistently better than the equally weighted portfolio when it comes to looking at the out-of-sample sharpe ratio or the reward per unit of risk. How can we fix this? How can we improve the minimum variance portfolio so that it is also a fairly well diversified portfolio. And in that sense hoping to generate a higher reward per unit of risk compared to a naive portfolio such as equally weighted portfolio. Well, there are a few competing methods that actually can be used and they have been shown to actually work in terms of improving minimum variance portfolio. Well, the first method is to minimize portfolio variance but to do it subject to a minimum effective number of constituents target. So remember that the effective number of constituents is a quantitative measure of how many stocks you're actually holding. And what we saw last time is the S&P 500 index for example in its cap weighted version is actually equivalent to a hundred stocks index. So the effective number of constituents is no greater than a hundred. So what you could do is you can minimize portfolio variance but impose a minimum number of constituents may be equal to whatever 200 or 250 or 300. If you do so then you're forcing a minimum degree of naive diversification within this scientific diversification process and it has been shown to work nicely. In other words the minimum variance portfolio with minimum ENC or effective number of constituents tend to be better than the equally weighted portfolio in terms of risk reward trade-off. There's another methodology which is essentially achieving the same objective but does it in a slightly different way, which is with imposing that all stocks are all components have the same volatility. So we have rewrite portfolio variance in a simplified way by assuming that all volatility Sigma I are all equal to a constant Sigma term. That portfolio ends up in the end minimizing the sum of WI, WJ, OIJ. And for this reason this portfolio sometimes known as the max decorrelation portfolio. It's a pretty small portfolio in the sense that we are forcing the optimizer to leverage on the magic power of diversification, which is utilizing the correlation in a smart way. So as to start with potentially risky assets and but that weakly correlated, and use those weak correlation to turn these baskets of weakly correlated assets potentially high risky into a low risk portfolio. Wrapping up, well, the most attractive portfolio is the maximum sharpe ratio portfolio, but unfortunately, it's not practical due to the presence of large estimation errors in expected return estimate. The good news is there's another remarkable portfolio on the efficient frontier the minimum variance portfolio, the estimation of which does not require any expected return estimates. The problem though is minimum variance portfolio stand to be heavily concentrated in low vol components. But fortunately, we have a number of competing techniques that can be used to ensure a proper or minimum degree of naive diversification or to ensure that is minimum variance portfolio stays well balanced. [MUSIC]