In the next few modules, we're going to focus on implementation difficulties associated with mean-variance. In the theoretical modules on mean-variance, we showed that mean-variance portfolio selection has some very nice properties in markets with only risky assets, two mutual funds suffice to satisfy all investors, in markets with a risk-free asset, the sharp optimal portfolio, and the risk-free asset is sufficient to satisfy all investors. This led to the capital asset pricing model with many deep and theoretical implications. In these sets of modules, we're going to focus on what kind of difficulties arise when you want to use mean-variance in a practical setting, meaning estimating data, constructing portfolios from estimated data, and trying to understand what the performance of these portfolios are going to be. There are many aspects of the implementation details of mean-variance that one could focus on, we chose to focus on the three most important ones. First has to do with parameter estimation. The parameters that go into a mean-variance portfolio selection problem in practical situations is never known. The true mean vector and the true covariance matrix of the assets is unknown. All we have is historical data, and we will have to estimate these parameters using these historical returns. As a consequence, we end up making statistical errors. For the mean vector, the data is often sufficient, but when you start estimating the covariance matrix, the data is never sufficient. The reason is that this covariance matrix has order d squared independent parameters. In order to have sufficient data to estimate these d squared parameters, you have to collect returns over a very long period and over this long period the market parameters shift. So you are playing a game where you can never able to get enough data to estimate these parameters sufficiently. Moreover, the portfolios that you compute turn out to be very sensitive to estimation errors and we'll focus on this in one of the modules. We're going to show you why this happens, how you could correct for it and what are the current state of the art on taking estimates and constructing portfolios from them. We're also going to focus on how does one get negative exposures in the Excel module that goes with the mean-variance theoretical modules. We showed you that very often the optimal portfolio has short positions. Taking on short positions is very dangerous, particularly because it has an unlimited downside. You can lose a lot of money because the price could suddenly jump very high and you end up losing a lot of money on the short positions. It's for this reason that it's not very often allowed for wealth managers. One way to get negative exposure is to use a leveraged exchange traded fund or leveraged ETF. But if you use leveraged ETF, you have to be very careful. In one of the modules, we're going to focus on how do ETFs work, what are the difficulties associated with ETFs, how should you interpret the returns of ETFs. Finally, we're going talk about whether variance itself is a good measure for risk. Mean-variance portfolio selection focuses on variance as a risk measure or equivalently volatility as the risk measure, does it make sense to use this risk measure? What are the limitations of variance? What can you do to mitigate some of these limitations is going to be the focus of another module. In this module, we will mainly focus on the issues associated with parameter estimation. The starting point of this module is that the true parameters that we're after, which is the mean vector and the covariance matrix of the assets, is never known. We are going to use historical returns to compute estimates for mean return and the covariance matrix. The easiest way to do that is to estimate the mean return by the sample average of the returns over some period M. Once you have the sample average for the mean, you can compute the covariance matrix by just substituting, instead of the true mean, the estimated mean to get an estimate for what the variance is. What I've done on this plot that goes on this slide is I simulated the returns using the mean vector and the covariance matrix given in the spreadsheet that goes with these modules, I simulated 60 months of data and using those 60 months of data, I estimated the mean. Each of these green dots on this plot are an estimated value of the mean using one particular simulation of 60 months worth of data. I'm only plotting the estimated mean for Asset 1 and Asset 2. The point that I want you to focus on is that the estimated mean can often be very far away from the true mean. The true mean has been plotted on this plot with the red square, here's where the true mean is. This is a valid estimated mean generated from 60 months of data and as you can notice, it's very, very far away from what the true mean is going to be. What we do know is that if I estimate the mean and I construct a 95 percent confidence interval around it, so here's one particular value of the estimated mean, here is the 95 percent confidence interval around it and because we are talking about two assets, this integral becomes an ellipse, it's 95 percent confidence ellipse, then with probability 0.95, the true mean lies in the ellipse. So in this particular case, the true mean barely made itself into the 95 percent ellipse. So the question you should ask yourself is, does parameter error matter? In this slide, I want to tell you that parameter error is often very serious for mean-variance portfolio selection, and what I'm describing on this is the same experiment that I described on the last slide taken one step further. I estimated the mean and the covariance matrix using 60 months of data. So I take one sample from all those green dots that I showed you on that slide, I have a mean vector, I have a covariance matrix, so I can construct an efficient frontier using that data. I'm going to call that the estimated frontier. So the green line here on this slide, this one, is the estimated frontier. It's the frontier that has been computed using an estimate for the mean and estimate for the covariance matrix. The blue line is the true frontier. This is the frontier corresponding to the unknown true mean and the unknown true covariance matrix. The red line is labeled the realized frontier. What that means is, I take a frontier portfolio on the green estimated frontier, compute the true mean return on that portfolio and the true volatility of the portfolio and plot it and the line that I get from doing that is the red line. So this diamond here actually gets moved to this diamond when you replace the estimated mean with a true mean and the estimated covariance matrix with a true covariance matrix. As you can notice, there is a big gap between what the estimated return on that portfolio is going to be and what the true return on that portfolio is. The estimated return is around 6.4 percent and the true return, or the realized return if you were to use that portfolio in the market, will be close to 4.4 percent, a good two percent drop. Why does this happen? Is this generic? Or did it happen just for one of the samples? In the next slide, I'm going to show you that in fact the situation is much worse. In this slide, I'm plotting the estimated frontiers corresponding to five different simulation runs. I simulated 60 months of data, five different times, computed the estimated mean, the estimated covariance matrix and I plotted the corresponding estimated frontier. The green lines on this plot are five different estimated frontiers. As you can see, these frontiers are extremely unstable. Not only are the frontiers unstable, the difference between the frontiers, and the estimated frontiers, and they realized frontiers can also be very large. So we want to understand why this happens, why is there such a big gap between what happens in the estimated frontier and what is actually realized? Why is the estimated frontier so unstable and is there anything that we can do to remove this gap and remove this instability?