Hello, good to see you again. Hope you enjoy learning how to find the portfolio expected return in the last lecture. The next question we're going to ask of course is, well if we're holding a portfolio of securities, how do you measure the portfolio risk? Well, let me give you the punchline from the start. While we learned that the portfolio expected return is simply the weighted average of the expected returns of the individual assets, this is not really the case for portfolio risk. It is a bit more involved, okay? So, bear with me while I try to illustrate the intuition for how we measure portfolio risk. Okay, so let's first recall how we measure risk of a single asset. Remember, risk means there is uncertainty about what the actual realized term will be. We may have an expectation of what that may be. In other words, we have a measure of the expected return. But we don't know whether that will indeed be the realized return, all right? Risk measures how widely the actual realized return may differ from that expectation. In other words, we measure risk using the dispersion around that expected return. We measure risk using the standard deviation, or what we call, the volatility. And remember how we measure volatility, right? Given a probability distribution, right, so let's say that we have probabilities. All right? What is the variance? I better denote that is sigma squared. Well it is the, all right? Probability weighted. Squared deviations from the mean, all right? So the outcome in that state minus the mean squared. All right, so that's our definition of the variance, okay? So let's continue with our example from the previous lecture of three stocks, Toyota, Walmart and Pfizer, all right? So last time we computed the expected returns, let's now find the volatility for each stock, all right? So let's start with Toyota, all right, and find this volatility as measured by its standard deviation. So what is the variance of the return on Toyota? Remember, it's the probability weighted deviations from, squared deviations, from the mean, right? So the probability of an expansion, 10% times the outcome in that state minus the mean squared plus the probability of normal times 40%. The outcome in that state minus the mean squared, the probability of a recession, the outcome in that state minus the mean. Finally, the probability of a depression. The outcome in that state minus the mean square, right? So if you do all that, you're going to find that the variance is 16.14, right? What is the standard deviation? Well the standard deviation of Toyota is the square root of that number. It's going to be 4.02%. Okay, now that we have reviewed measuring the risk of an individual asset, let's next look at how we can measure the risk of a combination of assets.