MITOCW | watch?v=z2oQe6B1Qa4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: Well, let me pick up where we left off last time and give you just a very quick overview of where we're at now, because we're on the brink of a very important set of results that I think will change your perspective permanently on risk and expected return. Last time, remember, we looked at this trade-off between expected return and volatility. And we made the argument that when you combined a bunch of different securities that are not all perfectly correlated, what you get is this bullet-shaped curve in terms of the possible trade-offs between that expected return and riskiness of various different portfolios. So every single dot on this bullet-shaped curve corresponds to a specific portfolio, or weighting, or vector of portfolio weights, omega. So now what I want to ask you to do for the next lecture or two is to exhibit a little bit of a split personality kind of a perspective. I'm going to ask you to look at the geometry of risk and expected return, but at the same time, in the back of your brain, I want you to keep in mind the analytics of that set of geometries. In other words, I want you to keep in mind how we got this bullet-shaped curve. The way we got it was from taking different weighted averages of the securities that we have access to as investments. So every one of these points on the bullet corresponds to a specific weighting. As you change those weightings, you change the risk and return characteristics of your portfolio. So the example that I gave after showing you this curve where I argued that the upper branch of this bullet is where any rational person would want to be. And by rational, I've defined that as somebody who prefers more expected return to less, and somebody who prefers less risk to more, other things equal. So if you've got those kind of preferences, then you want to be in the Northeast. You want to be as north, sorry, Northwest as possible. And you would never want to be down in this lower branch when you could be in the upper branch because you'd have a higher expected return for the same level of risk. So after we developed this basic idea, I gave you this numerical example where you've got three stocks in your universe. General Motors, IBM, and Motorola. And these are the parameters that we've estimated using historical data. Now there's going to be a question, and we've already raised that question, of how stable are these parameters. Are they really parameters, or do they change over time. And I told you, in reality of course, they change over time. But for now, let's play the game and assume that they are constant over time, and see
what we can do with those parameters. So with the means, the standard deviations, and most importantly, the covariance matrix-- So this is the matrix of variances and covariances-- With these data as inputs, we can now construct that bullet-shaped curve. The way we do it is of course, to recognize that the expected return of the portfolio is just a weighted average of the expected returns of the component securities, where the weights are our choice variables. That's what we are getting to pick, is how we allocate the 100% of our wealth to these three different securities. And the variance, of course, is going to be given by a somewhat more complicated expression where you have the individual security variances entering here from the diagonals. But you also have the off diagonal terms entering in that same equation for that variance of the portfolio. And when we put these two equations together, the mean and the variance, and we take the square root the variance to get the standard deviation, and we plot it on a graph, we get this. This is the curve, the bullet-shaped curve, that we generate just from three securities, and from their covariances. And where we left off last time is that I pointed out a couple of things that was interesting about this curve. One is that unlike the two asset example, where when you start with two assets, the curve, the bullet goes through the two assets. In this case, with three or more assets, it's going to turn out that the bullet is actually going to include these assets as special cases, but they won't be on the curve. In other words, what this curve suggests is that any rational person is going to want to be on this upper branch. What that means is that it never makes sense to put all your money in one single security. You see that? In other words, if we agree that any rational investor is going to want to be on that efficient frontier, that upper branch, why would you ever want to be off of that branch? You'd like to be Northwest of that, but you can't. You'd never want to be below that branch, or to the right of that branch because you could do better by being on that branch. So what this suggests is that we never are going to want to hold 100% of IBM, or 100% of General Motors, or 100% of Motorola. If we did, we'd be on those dots, and those dots would lie on that efficient frontier. But in fact, they don't. So right away, we have now departed from Warren Buffett's world of, I want to pick a few stocks and watch them very, very carefully. Yeah, Brian? AUDIENCE: Would you expand that to say that you'd never want to invest in less than three stocks at a
given time? ANDREW LO: That's not necessarily true. There are points on this line where-- and they may be pathological, so in other words, they may be very rare-- but there may be points on the line where you are holding two stocks, but not the third. So you've got to be careful about that. But those are exceptions. As a generic statement, you're absolutely right. The typical portfolio is going to have some of all three of them. And if you had four stocks, the typical portfolio would have some of all four. Yeah, [INAUDIBLE]. AUDIENCE: You answered my question, which is if you take one more stock, you'll always have your package [INAUDIBLE] n stocks include all the [INAUDIBLE], all the n stocks so, at the limit you should have an infinite number of stocks [INAUDIBLE] ANDREW LO: Well, let me put it another way that may be a little bit more intuitive. What this diagram suggests-- you guys are already groping towards-- is the insight that the more, the merrier. As you add more stocks, you cannot make this investor worse off. So in other words, I've now shown you an example with three stocks, we used to do two. Is it possible that by giving you an extra stock to invest in, I've made you worse off? Yeah? AUDIENCE: No ANDREW LO: Why AUDIENCE: Because you can just not invest in that stock. ANDREW LO: Exactly. I can never make you worse off in a world where you're free to choose, that is. Because you always have the option of getting rid of the stock that you don't like. You can always put 0 on it. So to your point, [INAUDIBLE], as I add more stocks, first of all my risk- reward trade-off curve will get better. What does it mean to get better? What does it mean for the risk-reward trade-off to be better? Yes? AUDIENCE: It means you get a higher return for the same level of risk. ANDREW LO: That's right. A higher return for the same level of risk, or a lower risk for the same level of
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