0:25

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 risk.

Â 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

And green you have IBM, and 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 and

Â another way of representing the risk and return associated

Â with these two investments is to compute a simple return every month.

Â And 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, this is 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 levels of return.

Â And you see on the X axis, on the horizontal axis,

Â we measure the level of return, going from minus 20% plus 20%.

Â Here, it's written in decimal form, so we see we have minus 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 zero for Microsoft.

Â 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, these represent very large monthly return,

Â 20% but they occur relatively rarely.

Â Only a few occurrence where 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 but

Â the two financial securities have a different return distribution In

Â 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?

Â And how disperse 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.

Â 4:06

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 we use

Â 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 distributions, and we're going to look at the result.

Â 4:48

These two histograms represent information separately for Microsoft and IBM.

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

Â 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 returns from Microsoft and

Â low returns 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 a return

Â of Microsoft measured on the X axis and 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 first of all, we can see that points are scattered all over the graph.

Â So there isn't a clear indication of

Â simultaneous occurrence of positive returns or negative returns.

Â Sometimes, we observe positive and positive.

Â Sometimes negative and negative.

Â 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.

Â We would observe 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 There seem to occur relatively at the same time.

Â But not always at the same time, so

Â 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 this two stock returns move simultaneously?

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

Â 6:58

The correlation is a metric that takes a value between minus one and one.

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

Â they would align on an increasing line starting from the lower left corner and

Â ending at the upper right corner.

Â On the other extreme,

Â a correlation of minus one would indicate perfect negative correlation.

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

Â all the dots here would align on a decreasing line

Â starting in the upper left corner and ending in the lower right corner.

Â In this case this would correspond with the correlation of minus 1 to

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

Â IBM we see a high negative return for Microsoft.

Â We will see here that the situation is somewhere in between.

Â So we expect the correlation to be not at one but not at minus one either.

Â And actually,

Â if you look carefully, 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.

Â But do you think it's positive, negative, closer to one, maybe close to zero?

Â Take a guess.

Â [MUSIC]

Â