Hello, welcome back. In the previous lectures I illustrated for you that portfolio risk is not simply the way the average of the individual asset will facilitate. What effects portfolio risk is how the assets in the portfolio comove together which we measured by the covariance measure or the correlation measure. And if they move in a way that is less than perfectly correlated, portfolio risk will always be less than the average of its components. This is the intuition behind why combining assets into a portfolio is able to diversify risk. We're finally ready to put it all together and write down an expression for portfolio risk that directly incorporates covariance measures and the variance measures, okay. So let's look at the risk of a two asset portfolio. All right, so suppose there are two assets. Asset one and asset two. All right, we combine them in a portfolio with weight w1 and w2. And let's also denote the variance of asset one as sigma 1 squared and sigma 2 squared, right, the individual variances. And finally, the covariance between the two, let's denote them as sigma 1 2 or the correlation between the two which is the correlation between r1, r2 right. So now we can write down the variance of a portfolio that consists of asset one and asset two, directly as the portfolio variance. It's going to be the variance of the first one times the weight of the first one. Plus the variance of the second one times the weight of the second one plus, right? So, so far, I've only taken care of the variances of the individual assets. Finally, 2 times the weights, times the covariance between the two, right. Well since covariance can also be written as the correlation coefficient times the volatilities, that just comes from the definition of the correlation. We can plug this whole thing into here and rewrite the portfolio variance again as this time with the correlation coefficient times the volatilities, right? So now we have the expression for portfolio variance that directly incorporates the variances and the covariances of the individual assets, okay. So now let's verify this using our example from before for the portfolio of Toyota and Pfizer Okay, so recall, for example, that the covariance between Toyota and Pfizer was minus 17.820. All right, and the correlation was minus 868, right? What are the weights? Well we're equally weighted. Right, 50%. Okay? So in this table, these are all numbers that we found before, right, the expected return to volatility, right? And now let's apply our formula. What is the portfolio variance? Well it's going to be the weight of the weight squared times the variance of Toyota plus the weight of Pfizer squared times the variance of Pfizer, right, times 2 times the weight. Times the covariance, right. Again, if you do this you're going to find that the variance is going to be 1.66, and if you take the square root of that you're going to find that it is 1.29, which is exactly the same number as we found before. Okay so, what's happening here? When we combine Toyota and Pfizer, right, the correlation is negative and it's quite strong so we get lots of diversification, right? The average standard deviation of the two stocks is 4.56, but the portfolio standard deviation is only 1.29. Where is that coming from? Well it's coming from the fact that Toyota and Pfizer have a very strong negative correlation, all right? When we combined Toyota and Walmart together the average standard deviation of the two stocks is 3.21, right, but the portfolio standard deviation is a little less, still a little less, 3.16, so we get, we still get some diversification. It's not a lot. Why? Because as you found before, these two stocks are very strongly positively correlated. They're not perfectly positively correlated but they are nearly perfectly positively correlated. So in general, right? Portfolio risk between two assets, right, is going to depend on the correlation coefficient between those two assets, right? So if two assets are perfectly correlated such that correlation coefficient is 1, there won't be any risk reduction or any diversification benefits. At the other extreme, if you have two assets that are perfectly negatively correlated, right. Then combining them would enable us to eliminate all risk. As long as they are less than perfectly correlated, right? There will be some risk reduction, some benefit to diversification. So even if it's 0 correlated, right, there will still be considerable risk reduction that will be possible. Okay, so in this lecture, you'll learn how to finally compute the portfolio variance for a two asset portfolio. You'll also learned, I hope, the intuition behind why portfolio risk is not simply the weighted average of the individual assets. All right, why? Well it's how the assets comove together is what ultimately affects portfolio risk.