[MUSIC] Hello and welcome to this video. Where we're going to talk about the distribution of returns. So when we talk about allocating our wealth into different financial assets, there are two important notions that always come into play. The expected return, and the amount of risk that we take when we invest our wealth in different financial assets. In this video, we're going to formalize the notion of expected return and risk. And we're going to do this through a very simple example. So let's start by looking at these two trajectories. These are prices depicted at a monthly frequency of two financial securities, Microsoft and IBM. I've changed little bit the level of the initial price that they start at the same level. These trajectories represent 10 years of data, so we have 12 observation per year, 1 every month. And this is for a period of ten years. And green you have IBM, and yellow you have Microsoft. So this is a classical representation of what the prices of financial securities look like. There is a lot of information embedded in this graph and another way of representing the risk and return associated with these two investments is to compute a simple return every month. And so, what we're going to do is compute for every month the relative change, so if the price increases relative to the initial value, this is going to be a positive return. If the price decreases, this is going to be a negative return. And once we compute the monthly return for this period of ten years for each security, we're going to represent this in a slightly different way, also in a graph but a little bit different. Let's have a look. So here we have histograms of the returns of the two securities, Microsoft and IBM. You see that now, instead of having a representation through time, we have a representation of the different possible returns. The histogram represents the frequency, how often we observe the various levels of return. And you see on the X axis, on the horizontal axis, we measure the level of return, going from minus 20% plus 20%. Here, it's written in decimal form, so we see we have minus 0.2 up to 0.2. And the height of each of these blue bar represent the frequency. So there is a very large bar here a little bit to the right of zero for Microsoft. Which represent the most frequent observation of returns. And then, you have, for example, for Microsoft on the right-hand side, a few little bars around 0.2, these represent very large monthly return, 20% but they occur relatively rarely. Only a few occurrence where observable during that ten year period. The same information is represented on the other graph for IBM. You see that the two histograms are different, they display the same type of information but the two financial securities have a different return distribution In particular we're very interested in observing a measure of tendency. What's the average return? What is the return we observe more frequently? What is the average direction of the financial security? And how disperse the distribution is? The standard measure of tendency is going to be the expected return. Whereas the standard measure of dispersion is going to be what we call the standard deviation. So from the histogram, we can see that Microsoft seems to have a little bit more dispersion than IBM, so a proper representation of dispersion would indicate that the metric we use to measure dispersion is larger for Microsoft than it is for IBM. So I've actually computed the average and standard deviation for these two distributions, and we're going to look at the result. The standard deviation represents a measure of dispersion as I was saying. And it is computed by looking at the distance of each observed from the average. These two histograms represent information separately for Microsoft and IBM. But there is one other thing that we could do and represent graphically, is to see how the two returns occur simultaneously. Do we observe high return in Microsoft when we observe high return in IBM? Or is it different? Do we observe high returns from Microsoft and low returns from IBM on the same month? One way of representing that is through a scatter plot. So each of these red crosses corresponds to a return of Microsoft measured on the X axis and a return of IBM measured on the Y axis. So for example, if we take one of those red crosses in the upper right corner, they corresponds to simultaneous occurrence of positive returns for Microsoft and IBM. The lower right corner would depict positive return for Microsoft and negative return for IBM. And first of all, we can see that points are scattered all over the graph. So there isn't a clear indication of simultaneous occurrence of positive returns or negative returns. Sometimes, we observe positive and positive. Sometimes negative and negative. And sometimes positive and negative. So we say that these two returns are not perfectly correlated. If they were perfectly correlated they would all appear on a single line. We would observe only positive return for IBM and Microsoft simultaneously and negative return for IBM and Microsoft simultaneously. So here we see that there is some dependence. There seem to occur relatively at the same time. But not always at the same time, so a measure that we would like to add to the expected return and the dispersion, so the standard deviation is a measure of co-movement, how do this two stock returns move simultaneously? And the standard measure of co-movements is called the correlation. The correlation is a metric that takes a value between minus one and one. At one, the two securities would be perfectly correlated and they would align on an increasing line starting from the lower left corner and ending at the upper right corner. On the other extreme, a correlation of minus one would indicate perfect negative correlation. In this case it would mean that all the crosses, all the dots here would align on a decreasing line starting in the upper left corner and ending in the lower right corner. In this case this would correspond with the correlation of minus 1 to a situation where whenever we observe a high positive return for IBM we see a high negative return for Microsoft. We will see here that the situation is somewhere in between. So we expect the correlation to be not at one but not at minus one either. And actually, if you look carefully, we probably expect some level of positive correlation. So what is your guess of the actual level of correlation? We're going to compute it in just a second. But do you think it's positive, negative, closer to one, maybe close to zero? Take a guess. [MUSIC]
