This module is the first of three modules where we'll be walking through the main theoretical ideas behind mean-variance portfolio selection. We'll first define what a portfolio is, we'll talk about what the return and risk of a portfolio is going to be, we're going to define what an efficient frontier is, we're going to define how inefficient frontier changes, when there is a risk-free asset, and how all of that leads up to the capital asset pricing model. The overview of what we're going to be talking about here is that we're going to define assets, we're going to define portfolios, we're going to define how to measure random returns on assets and portfolios, and for the mean-variance optimization as the name suggests, we are going to quantify the random asset and portfolio returns by their mean and variance. We're going to define something called mean-variance optimal portfolios or mean-variance efficient portfolios, we're going to define something called the efficient frontier and the portfolios that lie on this efficient frontier, and how does one compute them. Along the way, we're also going to talk about Sharpe ratio and Sharpe optimal portfolios, we're going to define something called a market portfolio, and after defining the market portfolio, we'll get to something called the capital asset pricing model. These are the various modules that we're going to be walking through in this bigger topic of mean-variance optimization. So what's the goal here? What I really want to do is I've got a certain amount of money and I want to split it among various assets that are available for investment. I'm going to characterize an asset by its price. More often than not, I'll be interested in returns on these assets. I'm going to invest in them today, I'm going to sell them tomorrow, and whatever difference is the return that I make on it. I would like to maximize this return in some appropriate sense. I'm going to define the random gross return on a particular asset to be simply the price one time step later. A time step could be a quarter, could be a year, could be six months. So it's Pt plus one divided by Pt. The net return is simply Rt minus one. It's going to be Pt plus one, minus Pt divided by Pt. I want to point out that both Rt and rt are random quantities. These are random because the price at time t plus one is random. Price at time t which is right now is known, but the price at time t plus one is random, and therefore these returns are going to be random. I've got d different assets, and I want to split an amount W, that I'm going to assume is strictly positive. This is the capital that I have, and I want to split it over the d assets that I have. w_i will be the total dollar amount that I've invested in asset i. If w_i is greater than zero, I'm going to say that that's a long investment, if w_i is less than zero, it's going to be a short investment. For the purposes of modeling, we allow w to be both positive or negative. R_w will be the net rate of return or net return on the position w. By position I mean the valves in the various assets. So what is the definition? It's simply the total value of this one time step later, which is the gross returns on each of the assets times the amount of money that was put into those assets, minus the initial wealth which is sum of the ws, divided by the initial wealth which is capital W. If you do the math, this one can be subtracted from R_i and this is going to be the net return little r_it, w_i divided by the sum of the w_is, rearrange them a little bit and you end up getting that the net return on position w, is the net return on each of the assets r_it, which is a random quantity, times w_i divided by capital W. So what is important to the net return is not the absolute amount of wealth that is invested, but the relative amount or the fraction of the total wealth that is invested in a particular asset. So instead of worrying about positions, we just have to consider a portfolio vector. So x is a portfolio vector, it could be positive or negative. X_is represent the fraction invested in a particular asset. So sum of the x_is must be equal to one. One thing that I want to point out here is that I've been talking about time t. In reality, the return that you get, whether it's a gross return or the net return, changes overtime. The random properties change over time, the actual values that are realized change over time. But in this set of modules, we are going to be looking at a very myopic investment strategy. I'm sitting at a particular time t, I only want to invest at time t plus one. When we get into more advanced topic, we will talk about how to extend this idea to a multi-period optimization. People are interested in multi-period optimization because at the end of the day they want to save money for retirement, they want to save money for buying a house, and so on. Which is not a one period problem but a multi-period problem. A multi-period problem is not simply a concatenation of one period problems because the space over which you can optimize becomes much larger when you look at multi-period problem. But in this set of modules, we are going to be concerned only with one period optimization, one quarter, one year, and so on. So how does one deal with randomness? The return on the portfolio r_x is going to be defined as r_i, x_i, and notice over here I've dropped the r_i, the time in the r_i because I'm focusing mainly on myopic optimization. This is a random return. Why is it random? Because each of the net returns r_is are going to be random. How does one quantify these return? Should I simply look at the maximize the expected value? Is that the right thing to do? Should one be worried about the spread around the mean? So the expected value or the mean value tells you what happens when you repeatedly invest. Most of us don't have the ability to repeatedly invest. If you go bankrupt, an investment is over. You don't have the ability to return from bankruptcy, then you might have to worry about what happened to the spread around the mean. How does one quantify the spread around the mean is going to be a question that we'll have to deal with. The way we're going to do this in these set of modules is, we're going to talk about the mean and we're going to quantify the spread around the mean by the variance. So the values defining the asset returns are going to be the mean return, which is the expected value of the net return, the variance of the asset return, which is simply the variance of the random variables. Notice the Mu is and the Sigma is are assumed to be independent of time. Again, either the market itself is stationary, or we are only interested in myopic investment, therefore we don't have to worry about time. The covariance between the asset return is the covariance between a random return on a particular asset i and asset j. Correlation again it's between two assets and there's a relationship between covariance and correlation. Covariance is nothing but the correlation times the volatility of asset i and the volatility of asset t. All parameters are assumed to be constant over time. Now, I'm going to characterize the random return that I get on a portfolio by looking at the expected return of the portfolio, and the variance of the return on the portfolio. So the expected return on a portfolio x, Mu_x is going to be the expected return on the net return of that portfolio r_x. By using linearity of expectations, we end up getting that this is nothing but the expected return on each of the assets times the fraction invested in that asset, x_i. So it's just Mu_i times x_i, sum from i going from one through d. The variance of the return again, just by the expression, it's the sum of r_i times x_i, the variance of this random variable, and if you expand it out it becomes the covariance of r_i and r_j, times x_i, x_j. Here's just a simple example to try to walk you through it. So I have two assets with normally distributed returns with mean Mu and variance Sigma square. So r_1 is one asset, it has a mean one and it has a variance 0.1, r_2 is another asset, it has a mean two and has a variance 0.5. The correlation between these two assets is negative 0.25. If you translate this statement into those parameters, Mu_1 is one, Mu_2 is two, Sigma_1 squared is 0.1, Sigma_2 squared is 0.5. Sigma_12, which is the covariance between r_1 and r_2 is going to be the correlation, whose numbers are right here, times Sigma_1, Sigma_2, you plug in the answers you'd end up getting it's 0.0559. In the two-asset market, portfolios are very easy to define. Remember portfolios were the fractions invested. So if I invest fraction x in asset 1, and since the fraction invested in asset 1 and the fraction invested in asset 2 must add up to one. I should invest one minus x in asset 2. Later on, we will see how we can use this x, one minus x to try to do efficient portfolio selection between these two assets. Plugging it in into the formula, the expected return on this portfolio is going to be sum of Mu_i, x_i, one through d. Two assets, the first asset has x, its expected return is one, so it's one times x. The second asset is one minus x, its expected return is two. So the total expected return that you get on this portfolio is x plus two, times one minus x. What is the variance associated with the return of this portfolio? The formula is Sigma_ ij x_ij x_i x_j summed from i equals one through d, which can be equivalently written as, sum of i going from one through d Sigma_i squared x_i squared plus two times j greater than i Sigma_ij x_i x_j. Plugging it in, 0.1 which is the variance of the first asset, times the investment in the first asset squared, 0.5 which is the variance of the second asset, times the investment in the second asset squared, two times the covariance between the two assets which is just computed here, times the investment in the first asset and investment in the second asset. This exact thing can be generalized to multiple assets and that is what we'll do in later modules.
