Hi there, welcome back. In the previous lectures, we have been little by little laying the ground work for what we call the mean-variance analysis framework. Which forms the basis of modern portfolio theory. Mean-variance analysis or mean-variance investing is a paradigm that transformed the investment management world when it was first introduced. While it clearly has some limitations that we will also discuss, it still remains as one of the main ideas behind optimal portfolio choice. So, a little history. Modern portfolio theory was first developed by Harry Markowitz in the 50s. He was a young PhD student studying at the University of Chicago, and published his dissertation on portfolio selection in the Journal of Finance in 1952. Markowitz's contribution transformed the way finance practitioners thought about risk at the time. Basically, he came up with a theory that analyzes how investors should optimally choose portfolios. In other words, how we should allocate wealth in an optimal way, in assets that differ in their expected return and risk. Of course, almost 40 years later this idea earned Markowitz the Nobel Prize in Economics. The main assumption behind mean-variance framework is that asset returns can be characterized entirely by their risk and reward trade-offs. Where we define reward as their expected return and risk as measured by their volatility. Now this is why we're plotting assets and combinations of assets, portfolios in the expected return on volatility space in all those graphs, right? Because we wanted to illustrate the trade-off between the expected return and volatility. Mean-variance investing is all about diversification. The main takeaway is that the interaction of assets with each other allows for one's gain to make up for the another's loss. And therefore it creates the reward, the returns, while reducing the risk. So diversification reduces total risk, as we combine imperfectly correlated assets. Now, as I mentioned, the crucial assumption here is that asset returns can be entirely summarized by their means and variances. Now, in the next module we will look at investors who choose optimal portfolios if indeed they act this way. Right, if indeed this is all they care about. Mean and variances. Expected returns and volatility. But let me play the devil's advocate for a second here. What if, as investor, you also care about downside risk? That tail event. Or maybe some other higher moments of risk other than variance that we looked at in the previous module? Now the beauty of mean-variance investing is that when you combine imperfectly correlated assets, variances always decrease, right? So its diversification in this sense always help reduce risk. What if however, other measures or risks, let's say for example downside risk, do not decrease when you combine assets? For example, a portfolio can be more negatively skewed and have more downside risk than the downside risk of each individual assets that make it up. In fact, many investors care about more risk measures as simply variance, right? So mean-variance framework and diversification clearly will not be able to address such concerns. Furthermore, what if you are risk seeking, right? What if you are hoping to hit the jackpot by bidding on the next Apple, or Google, by investing everything into one thing? Right, clearly then diversification is not for you either. Diversification, by definition, eliminates that idiosyncratic risk and therefore that extremely high lottery like payoff that would come from a very concentrated portfolio. Now, having said all that however, it is probably not a bad assumption to make that most investors would probably shy away from such risks. And they would like to avoid the potential catastrophic losses that can come from failing to diversify a way appropriately. But again, right, all that depends on investors preferences, which we're going to take up in the next module. So in summary, mean-variance analysis framework is the basis of modern portfolio theory. It assumes that as it returns can be entirely summarized by their expected returns and risk as measured by their volatilities. And as long as assets are not perfectly correlated, diversification will help reduce total portfolio risk.