So first thing we want to talk about is that diversification or thinking about the spread around the mean is important, because it reduces uncertainty. So let's consider a very contrived market. I have d different assets. All of them have the same expected return Mu, all of them have the same volatility Sigma, and the correlation between the assets is equal to zero. So each of the assets is basically identical. Now, let's think about two different portfolios. In one portfolio x, I invest everything in asset one. In the other portfolio, I equally distribute my initial dollar over all the assets. So in every asset, I invest one over d. Both of these are portfolios. The expected return of the first portfolio which is x is simply the expected return of asset one which is Mu. The expected return on the portfolio, y, which equally invest in all the assets is going to be the average of the returns of all the assets. Since each of them has the same return Mu, the average is also going to be Mu. So if we were just looking at the expected value, the two of these two portfolios cannot be differentiated. They both give me the same expected value, we should be as happy investing in x as we should be investing in y. But we are as we should be interested in reducing uncertainty, then perhaps these two portfolios are different. So if you look at the variance of the returns of the two portfolios, what do you get? Sigma x squared, which is the variance of portfolio x. We know that that just invest in asset one so it is nothing but volatility squared, Sigma squared. What happens to the variance of portfolio y? If you plug in the formula, every one of them has one over d invested in it. So it's one over d squared, Sigma squared. There are d terms, so ultimately you get that the variance associated with portfolio y it's Sigma squared over d. Think about d of the order of 100. Say I'm interested in the S&P 500 index. So in the one case I get volatility Sigma squared, in the other case my volatility has dropped down hundredfold. So just by diversifying between assets, identical assets, now I have been able to reduce my volatility a lot. The mean-variance portfolio selection problem in the end generalizes this idea. Here I'm taking a very simple problem. All the assets have the same return, all the assets have the same variability, and I know that's equal spreading is the best thing. Now what I want to do is, move this idea to a case where the mean returns are not the same, variances are not the same, the covariance may not be equal to zero. Think about this problem. How does one compute efficient portfolios? Meaning portfolios that have a good mean and variance properties is going to be the main focus of these sets of modules. So in 1954, Markowitz proposed a portfolio selection strategy. In his model, he suggested that the return and the return has been put in quotes, I want you to read this more as the benefit coming from a portfolio to be the expected return of that portfolio, and the risk associated with the portfolio to be the volatility of that portfolio, and what he suggested was that, these are the two quantities that are going to be interesting to investors. They would want to increase their return and decrease their risk. So what I'm plotting over here are the returns on some random portfolios. In the next module, I'm going to show you a spreadsheet which shows you how these random returns were generated. I had eight different assets, the details of which will be in the spreadsheet. I randomly generated positions on these eight assets, figured out what the return is going to be, figured out what the volatility is going to be, and I plotted the points. So all of these blue dots are actually various portfolios randomly generated and the efficient frontier, this line over here, is generated by the following procedure. You pick a particular value of volatility or risk and try to compute the largest return that you can get on a portfolio that has risk no larger than a particular bound. So let's say Sigma bar is the bound here. Figure out a portfolio, compute portfolio and I'll show you in this spreadsheet how these portfolios are computed, which has the largest return with risk not exceeding Sigma bar, and that point would be right here. Similarly, you take different values of these Sigmas, compute out what is the maximum return that you're going to get, and this blue line is generated by computing the maximum return for a given value of risk. That frontier is called the efficient frontier. Why is it a frontier? Because all portfolios, all feasible portfolios must lie below. This is all the part that is feasible. For any portfolio that you choose, its risk and its return values must be below the line. All of this space is unachievable. You cannot create a portfolio whose return and risk point lies in that region. Why is that? Just the way it's computed. I take the value of Sigma which is the risk, I compute the maximum possible return that I can get and that's how I get this point. This return up here is not achievable. So everything above the frontier is not achievable, everything below the frontier, below or equal to the frontier is achievable. But I would never want to be below the frontier. If I have a point over here, its risk is some quantity over here. Let's call it Sigma one. My frontier tells me that I can create another portfolio, a different portfolio from the one that generated that point whose return is going to be right here. It's going to be on the frontier. So I would never want this portfolio, I only want portfolios that lie on the frontier. Above the frontier unachievable, below the frontier inefficient, right at the frontier is placed where I want to be. So the question that we will answer in the next few modules is, how does one characterize this efficient frontier? How does one compute efficient or sometimes I'm going to call it optimal portfolios? Portfolios that lie on this efficient frontier. There are three different ways in which you can compute the mean-variance optimal frontier, that line that I showed you before which tells you the maximum return for a particular value of risk. One way is to minimize risk for a target return. You can set the target return that you want. You want to make sure that the expected value on the portfolio has to be greater than equal to r. Among all portfolios that satisfy that, you want to minimize the variance or minimize the volatility which is equivalent. If you expand this optimization problem, you can write it as sum of x_i is equal to 1. So this is a portfolio constraint and here you're saying that the expected return on that portfolio must be greater than equal to r, and this expression here just expands out whatever is written over there. For those of you who are mathematically sophisticated, this expression is nothing but the vector x transpose a matrix of covariance times x. What is this matrix? Is Sigma 1 squared, Sigma 1 2, Sigma 1 3 and so on. Sigma 2 1, Sigma 2 squared, and so on. So this is a variance covariance matrix, and this expression there is simply x transposed variance-covariance matrix times x. An equivalent way to get the entire frontier is going to be to maximize return for a given value of risk. Maximize Mu of x such that Sigma squared of x is below some target number Sigma bar squared. If you write this in terms of the x_i, it becomes sum of the x_i must be equal to one. Again the portfolio constraint, x transpose Sigma x or Sigma_ij x _i x_j summed must be less than equal to Sigma bar and you want to maximize the expected return. There's a yet a third way of trying to get the entire frontier, and that is to maximize a risk adjusted return. So maximize over portfolios x, Mu of x the expected return minus Tau which is called a risk aversion parameter, Sigma x squared. So the risk aversion parameter is always greater than zero. You don't like risks. Therefore, you want to subtract from the expected return a certain quantity that depends on the risk. If you again expand this expression, you get it sum of x_i is equal from x_i must be equal to one which is the portfolio constraint, and this is just Mu of x and that is Sigma x squared, and that's the Tau there. What do I mean by saying all these three will generate the same frontier? There are three parameters in all of this. All of these formulations are Sigma bar squared and Tau, and as you crank up these parameters for different values of these parameters, you will write out the same curve.