0:16

So when we talk about allocating our wealth into different financial assets,

Â there are two important notion that always come into play.

Â The expected return and the amount of risk

Â that we take when we invest our wealth in different financial assets.

Â In this video, we're going to formalize the notion of expected return and

Â risk, and we're going to do this through a very simple example.

Â So let's start by looking at these two trajectories.

Â These are prices depicted at a monthly frequency of two financial securities,

Â Microsoft and IBM.

Â 1:14

In green, you have IBM, in yellow, you have Microsoft.

Â So this is a classical representation of what

Â the prices of financial securities look like.

Â There is a lot of information embedded in this graph.

Â And another way of representing the risk and return associated

Â with these two investment is to compute the simple return every month.

Â So what we're going to do is compute for every month the relative change.

Â So if the price increases relative to the initial value,

Â this is going to be a positive return.

Â If the price decreases, it's going to be a negative return.

Â 2:09

So here we have histograms of the returns of the two securities, Microsoft and IBM.

Â You see that now instead of having a representation through time,

Â we have a representation of the different possible returns.

Â The histogram represents the frequency,

Â how often we observe the various level of return.

Â And you see on the x-axis, on the horizontal axis,

Â we measure the level of return going from -20% to +20%.

Â Here it's written in decimal form, so we see we have -0.2 up to 0.2.

Â And the height of each of these blue bar represent the frequency.

Â So there is a very large bar here a little bit to the right of 0 for

Â Microsoft, which represent the most frequent observation of returns.

Â And then you have, for example, for Microsoft on the right-hand side,

Â a few little bars around 0.2.

Â These represent vary large monthly return, 20%.

Â But they occurred relatively rarely,

Â only a few occurrence were observable during that ten year period.

Â The same information is represented on the other graph for IBM.

Â You see that the two histograms are different.

Â They display the same type of information,

Â but the two financial securities have different return distribution.

Â In particular, we're very interested in observing a measure of tendency.

Â What's the average return?

Â What is the return we observe more frequently?

Â What is the average direction of the financial security and

Â how dispersed the distribution is?

Â The standard measure of tendency is going to be the expected return.

Â Whereas, the standard measure of dispersion

Â is going to be what we call the standard deviation, okay?

Â So from the histogram,

Â we can see that Microsoft seems to have a little bit more dispersion than IBM.

Â So a proper representation of dispersion would indicate that the metric

Â we used to measure dispersion is larger for Microsoft than it is for IBM.

Â So I've actually computed the average and standard deviation for

Â these two distribution and we're going to look at the result.

Â The standard deviation represents a measure of dispersion,

Â as I was saying, and it is computed by looking at the distance

Â of each observation from the average.

Â 4:48

These two histogram represent information separately for Microsoft and IBM.

Â But there is one other thing that we could do and

Â represent graphically is to see how the two returns occur simultaneously.

Â Do we observe high return in Microsoft when we observe high return in IBM,

Â or is it different?

Â Do we observe high return from Microsoft and low return from IBM on the same month?

Â One way of representing that is through a scatter plot.

Â So each of these red crosses corresponds to

Â a return of Microsoft measured on the x-axis and

Â a return of IBM measured on the y-axis.

Â So for example, if we take one of those red crosses in the upper right corner,

Â they corresponds to simultaneous occurrence of positive returns for

Â Microsoft and IBM.

Â The lower right corner would depict positive return for

Â Microsoft and negative return for IBM.

Â And we can see that, first of all,

Â we can see that points are scattered all over the graphs, so there isn't a clear

Â indication of a simultaneous occurrence of positive return or negative returns.

Â Sometimes we observe positive and positive, sometimes negative and

Â negative, and sometimes positive and negative.

Â So we say that these two returns are not perfectly correlated.

Â If they were perfectly correlated, they would all appear on a single line, okay?

Â We would have serve only positive return for IBM and Microsoft simultaneously, and

Â negative return for IBM and Microsoft simultaneously.

Â So here we see that there is some dependence.

Â They seem to occur relatively at the same time, but not always at the same time.

Â So a measure that we would like to add to the expected return and

Â the dispersion, so the standard deviation, is a measure of co-movement.

Â How do these two stock returns move simultaneously?

Â And the standard measure of co-movement is called the correlation.

Â 6:58

The correlation is a metric that takes a value between -1 and 1.

Â At 1, the two securities would be perfectly correlated and

Â they would align on an increasing line,

Â starting from the lower left corner and ending at the upper right corner.

Â On the other extreme,

Â a correlation of -1 would indicate perfect negative correlation.

Â In this case, it would mean that all the crosses, all the dots here

Â would align on a decreasing line starting in the upper left corner and

Â ending in the lower right corner.

Â In this case, this would correspond with a correlation of -1

Â to a situation where whenever we observe a high positive return for

Â IBM, we see a high negative return for Microsoft, okay?

Â Well, we see here that the situation is somewhere in between, so

Â we expect the correlation to be not at 1 but not at -1, either.

Â And actually if you look carefully,

Â we probably expect some level of positive correlation.

Â So what is your guess of the actual level of correlation?

Â We're going to compute it in just a second, but

Â do you think it's positive, negative, closer to 1, maybe close to 0?

Â Take a guess.

Â [MUSIC]

Â