Welcome everyone to the fourth week of our Corporate Finance course. We are now beginning to talk about risk. I said that in such a solemn voice if you will because unfortunately, risk is the area that is oftentimes misunderstood. And even if it is understood properly, that does not lead to immediate practical implications. Because in the study of risk, our assumptions have to be modified to some extent. And unfortunately, we have to employ some advanced mathematical instruments. Now, our emphasis in this course on this risky area will be somewhat practical. And by that I do not mean that I will completely ignore the theory behind that. And we actually will talk about that in the first half of this week. But our job will be to see by the end of the week what we can use in a real life. And what is more or less, remains open for discussion or for further studies. Because in this area, unfortunately, quite a few questions that are unanswered. Even by the most advanced specialists in the area. Now as oftentimes happens, when people realize that the risk of investments is affected by very many different factors. And you can hardly see the exact line of influence, and therefore, it's very difficult to come up with a certain logic behind that. And to actually follow through with steps that result in this or that risk exposure. When it comes to this point, when people say, well, what can I do with all of this? And as again oftentimes happens, the solution comes from a completely different area. The most renowned specialist in the area, who started all that was Harry Markowitz, who wrote his pioneering articles back in the early 50s. But then most of the study of risk as it exists right now is based on that. And remarks with its idea was very simple as for most really breakthrough ideas. You said that, what if we studied the returns on the assets as random variables? We cannot see the exact line of influence, line of impact, so why would we just take that as random? And then, we will be able to use the powerful apparatus of statistical mathematical approach and that of statistics. And that will allow us to come up with certain conclusions about that. And basically, this whole line of argument goes at the start from saying we treat returns as random and then we build up on that. So our journey this week will be as follows. We will say a few words about how this all started, how it develops and produces very practical results. And then, at the very end of this week we will come back to our first episode, if you will. And you will see well, what are these steps that follow from the very basic assumptions of tweeting returns as random, to what everyone can do when they study projects? Now, what exactly it means to treat these returns as random variables? It's not sufficient to say that they're random, you also have to make an assumption about the distribution of these random variables. And it's quite well known that normally people say, well returns they're random. Not only that, they are normally distributed. Well, not everyone agrees with the fact that normal distribution, if you will. Not everyone agrees with that, because normal distribution basically places an extremely low probability on an unlikely event. So that is far from being average, but the good news about the normal distribution that often times is represented as the falling chart. So if this is the pi, the probability of finding a certain return and this is the expected average return, now the curve looks somewhat like this. Well it's symmetric and this is the return actually and this is the pi. So the probability of finding it. So it's most likely that you find this average thing here, this is a great probability. And then, the probability of finding the return far from average somewhere here is very low. The area under this curve is 1, as for all distributions. And this is basically the approach and you can say well, why do we care about these distributions and develop this math? And here I would just lay the foundation of the process, how we will proceed from here. Well we say, if we use this approach, then we know how to calculate expected returns and also standard deviations. This is basically the width of this distribution. Not for just one asset but for combinations of assets. And combinations in this special area have a special name. They are called portfolios. And you can say, so what? Because we will see that calculating these returns and standard deviations for our portfolios is although straightforward but kind of cumbersome. And but the idea is that the ideology goes that if we know how to deal with these portfolios, we can identify certain special portfolios that will be first of all, few. And second of all, these portfolios play a really pivotal role in our approach to finding our expected return for any asset in this market. So, it is the need to proceed through special portfolios that sort of forces us to first study how people calculate expected returns in standard deviations for the portfolios of assets. Again, most of the mathematics that is used here is given in our hand outs, so I clearly will not produce most of the formulas on these flip charts. And that is for those who would like to go deeper in this. But for us, the key idea again is to the extent we agreed to treat returns as random. Then we have to proceed to these special portfolios that will allow us to come up with extremely simple and important formulas just a few episodes later. So here, I am wrapping up and starting from the next episode we will see how exactly people deal with calculating special returns and standard deviations. I should put for the first time this famous sigma here for the portfolios of securities or assets