All right, let's proceed. And again, we will, here, produce the a more simplistic case of just two returns. This will be security one. We'll return, expected on R one, and then Sigma one, which is the standard deviation of this distribution. And then, security two will E of R2 and then Sigma two. Now, like I said, the key story is that we can describe each asset with just two parameters, this and that. This is the feature of a normal distribution. If there do exist other distributions, sometimes, that may be even closer to what has been observed on the market. But they have more parameters. And that does not make the calculation of the corresponding parameters for portfolios unrealistic. But this is much more cumbersome, and oftentimes it's used only by professionals in asset management, and there exist specific computer programs for that. For us on this ideology path, this is not important to the extent we stick with normal distributions. Now, the general formulas are very simple. First of all, returns are linear. So we can say that, the return for the portfolio, and P portfolio will be W one times one plus W two, and Ws are just weights, times two, whatever. And then, we can put the formula for returns is clear because this is linear. And there are symbols, so we can say that expected for the portfolio is just W one, E of R one, Plus, I will keep using different colors, W two, E of R two. But when it comes to the standard deviation of this portfolio, the formula becomes a little bit worse. So, Sigma portfolio squared is equal to, first of all W one squared. Sigma one squared, plus W two squared, Sigma two squared, and unfortunately, plus the term that I will use the red color for, that is two, W one W two. Sigma one, Sigma two, but that's not it. And then also there is a coefficient of Rho one, two, which is the correlation coefficient. So, and that all is oftentimes used as covariance, but well, is this formula real bad? Well, not quite because it looks a little bit intimidating, but it's a quadratic form. And you can see that if we went ahead and use not two portfolios, but n portfolios, then this formula will be just n members. This would be a little bit worse, because it would be like it would have, not just n, but n times n and minus one over two members. Because this is like the n by n matrix. This is a symmetric matrix. So, these guys are the diagonal members, and all these correlations are this triangle of non-diagonal pieces in this formula. So, all that gives you an idea that you can easily calculate all these things. By the way, the sum of all these weights is one. Because this is the overall sum of the weights in the portfolio. However, the funny thing is that sometimes these weights they don't have to be positive. If you borrow then the weight becomes negative. And for example, if you have one dollar over your own and you borrowed one more dollar and then you put together these two dollars and invested them in the portfolio, then you will have the overall weight of, W one will be two, W two will be minus one. Although, some of them will still be one. Although when squared, that will contribute to the risk of your portfolio much more significantly. Now, this whole approach just gives you an idea. Again, we could proceed with lots of problems and lots of generalizations of this formula. I wouldn't do that. I would just say that there is a way by which people calculate this. And the important thing is this member. So, that is the correlation member. And this whole idea of correlation, that means the dependency of the return of one asset to the other asset. That is the key story here, because in general, the behavior of returns on the assets, it is not independent. It in some special cases, is independent. But in general, they are correlated, and they are not correlated perfectly. Because you have pretty good idea of correlation just without any formulas. Now, if something moves in concert, that's perfect relation. If it's the other way, it's perfect negative correlation. And if one goes like this and the other is basically smooth, this is a zero correlation. But that is important for us to proceed in the next episode. But for now, I would just like to say a few words about these portfolios in reality. Again, most all the information can be found there handouts or in famous text books like Brealey and Myers that produce these charts of the behavior of US securities for over a hundred years. But what is important is that, if we take the most interesting portfolios for the people, let's say the portfolio of US Treasury securities, that are oftentimes used as a proxy for being riskless and not without success, and then the portfolio of US Stocks what we can say is that, I will reproduce a very simple chart here. So, this is sort of US securities, and this is from 1900 to 2011, this is taken from Brealey and Myers. So, what we can say is that if this is a certain portfolio, and let's say this is it's Sigma calculated as average this period of time. So, what we can see is that there is a portfolio of Treasury bills, this is a portfolio of Treasury bonds, and then the portfolio of, I will put it in red, of stocks. So, and we can see that, over all this period of time, the Sigma of treasury bills was about 2.8%, the Sigma of Treasury bonds was about 8.9%, and the Sigma of stocks was about 20%. So, one thing that we can see, even from this most simplistic table, is that the good news is that risk does grow with- Because, clearly, stocks are riskier than bonds. So, if we saw that the Sigma for them would be, let's say, the same or less, then you could say the approach of random variables is invalid. Unfortunately it's invalid, and therefore here we can see some qualitative connection, and we can say that clearly, at least indeed this Sigma is greater for the riskier securities. So, this is good news for the overall application of this random approach. So basically, we can say that we can also expect some risk premium in the form of return. And, again, I would not reproduce the returns for all these averages and all these portfolios, but it can be said, because it does depend upon the time period and here, unfortunately, if you take the longer term, here it's over a century, that may not produce the best proxy for that, because you know in some periods, you can see more growth in some periods you can see more correction, if you will. But the generally accepted idea right now, is that the risk of the market portfolio is about the risk free rate, plus the premium of five to eight percentage points over a long period of time. Now, and again, like I said in Brealey and Myers there is a lot more information on that. For us, the key idea from here will be that investors do require a premium for holding the assets that are risky, which is a very natural idea and then very initial assumption, it's more than an assumption, this is what is observed in all markets. So from now on, we will see how we can tackle this idea of a risk premium. So, how much more in terms of return the investors would require for holding portfolios that are not risk free?
