Welcome back. So I have illustrated to you, I hope convincingly, that when we combine different securities into a portfolio, portfolio risk is not simply the weighted average of the individual variances or volatilities. Well, what you might already be saying, of course, is that the reason is, we need to also consider how these assets co-move together, right? This is the key. When we combine different securities into our portfolio, the variance of that portfolio's return will be affected by how these securities co-move together, right? And so today, accept that they are less than perfectly correlated, we may expect them to cancel each other and to reduce the risk of the portfolio, right? So in order to quantify the extent to which assets co-move together, we need measures of co-movement. What are those measures? Well, covariance and correlation. You might remember from statistics, right? These are two measures that allow us to evaluate how securities move or do not move together. And so let me start with the covariance measure, right? So what is covariance? Well, covariance is a measure of the pair-wise co-movement between two securities. One of my colleagues, that I like very much, uses the analogy of butterflies moving together, right? Are they moving in the same direction or in opposite direction or not related at all, right? Another colleague of mine likes to think of two people dancing. Well hopefully you know when two people are dancing they're dancing together and they're moving in the same direction. I guess that's not really a requirement of dancing, but anyway, I'm digressing here. Anyway, what is covariance? So, mathematically it is the probability weighted average of the product of the deviations from the mean. So, let me illustrate that, so suppose we have two assets, right? And r1 and r2, right? And they have expected return r and expected return r2, and there are different possible outcomes with probability j, let's say there are outcomes. So what is the covariance between r1 and r2? We denote that sometimes as sigma 12, right? What is that? Well, it's the probability weighed average of the product of the deviations. All right, deviation of the 1 times the deviation of the r, right? So you can already see that if both of them are positive, right, they are moving in the same direction, if these are, if one is positive or one is negative, they are moving in opposite direction, right? So now let's use that expression, that definition, to find the covariance between Toyota and Pfizer, right? Again this is the familiar table now that you've been saying, right, let's now find it, the covariance between the returns of Toyota and Pfizer. Let me see if I have room here, yes I do. All right, so what is the covariance between Pfizer and Toyota? Now let me denote that as Toyota and Pfizer, right, like that. So what is that going to be? Well, remember, it's the priority-weighted product of the deviations from the mean, so a 10% probability in a expansion, this is the deviation of the Toyota and that's the deviation of the Pfizer. Plus the probability of normal times, times the deviations from the mean for each one of them. Plus. [COUGH] The probability of a recession, oops sorry. Minus the mean. Finally, the probability of the operation and the product of the deviations from the mean. All right, so if you did all that. I know that's a bunch of numbers, you would find the covariance to be -17.820. So what does that number tell us? Well, the magnitude doesn't really tell us much. All we can learn from that number is that the covariance is negative, right? Which tells us the stocks of Toyota and Pfizer move in opposite directions. We can observe that from the table, from the distribution of returns as well, right? See that when Toyota does well, Pfizer does okay, but when Toyota sanks, Pfizer does as really well, right? That's the nature of the negative covariance. Now, one problem with the covariance measure is, as I mentioned, is that the magnitude doesn't really tell us anything, right? The sign tells us that they co-variant negatively, but it doesn't really tell us about the strength of that relationship, right? Instead, we use the correlation measure for that purpose. So what is correlation? Well we typically denote correlation coefficient as using the Greek letter Ro. It's simply the scaled covariance, right, is defined as the covariance between, let's say, 1 and 2. The covariance between two assets scaled by the product of the standard deviations of each asset. Now notice that since the denominator, right, will always have a positive sign. The correlation coefficient will of course carry the same sign as the covariance. So, if the covariance is negative, the correlation coefficient will be negative. If the covariance is positive, the correlation coefficient will be positive. Moreover, notice that the correlation coefficient can only take values between -1 and +1. So if it's perfectly positively correlated, the correlation will be +1. If it's perfectly negatively correlated, the correlation coefficient will -1. So going back to our example, let's now find the correlation coefficient between Toyota and Pfizer, right? So, what is the correlation coefficient between Toyota and Pfizer? Well, it's going to be the covariance between the two stocks divided by the volatility of Toyota, and the volatility of Pfizer, right? I'll remind you what these numbers are. The volatility we found to be 4.02%. The volatility of Pfizer, we found to be 5.11%. And the core variance between the two we found to be -17.820, right? So now I can find the correlation which is 4 point, sorry, -17.820/(4.02)(5.11). Which gives us 0.868, right? So the correlation coefficient between the Toyota and Pfizer stocks is -0.868. Now look what this number is telling us. The stock return on Toyota and Pfizer are fairly strongly negatively correlated. They're not perfectly negatively correlated, right, they do not move one-to-one perfectly, but the negative correlation between them is fairly strong.