耶鲁大学金融市场英文文本Financial-MarketsLecture-04-Transcript.doc

Financial Markets: Lecture 4 Transcript Professor Robert Shiller: Todays lecture is about portfolio diversification and about supporting financial institutions, notably mutual funds. Its actually kind of a crusade of mine-I believe that the world needs more portfolio diversification. That might sound to you a little bit odd, but I think its absolutely true that the same kind of cause that Emmett Thompson goes through, which is to help the poor people of the world, can be advanced through portfolio diversification-I seriously mean that. There are a lot of human hardships that can be solved by diversifying portfolios. What Im going to talk about today applies not just to comfortable wealthy people, but it applies to everyone. Its really about risk. When theres a bad outcome for anyone, thats the outcome of some random draw. When people get into real trouble in their lives, its because of a sequence of bad events that push them into unfortunate positions and, very often, financial risk management is part of the thing that prevents that from happening. The first-let me go-I want to start this lecture with some mathematics. Its a continuation of the second lecture, where I talked about the principle of dispersal of risk. I want now to carry that forward into something a little bit more focused on the portfolio problem. Im going to start this lecture with a discussion of how one constructs a portfolio and what are the mathematics of it. That will lead us into the capital asset pricing model, which is the cornerstone of a lot of thinking in finance. Im going to go through this rather quickly because there are other courses at Yale that will cover this more thoroughly, notably, John Geanakoploss Econ 251. I think we can get the basic points here. Lets start with the basic idea. I want to just say it in the simplest possible terms. What is it that-First of all, a portfolio, lets define that. A portfolio is the collection of assets that you have-financial assets, tangible assets-its your wealth. The first and fundamental principle is: you care only about the total portfolio. You dont want to be someone like the fisherman who boasts about one big fish that he caught because its not-were talking about livelihoods. Its all the fish that you caught, so theres nothing to be proud of if you had one big success. Thats the first very basic principle. Do you agree with me on that? So, when we say portfolio management, we mean managing everything that gives you economic benefit. Now, underlying our theory is the idea that we measure the outcome of your investment in your portfolio by the mean of the return on the portfolio and the variance of the return on the portfolio. The return, of course, in any given time period is the percentage increase in the portfolio; or, it could be a negative number, it could be a decrease. The principle is that you want the expected value of the return to be as high as possible given its variance and you want the variance of the return on the portfolio to be as low as possible given the return, because high expected return is a good thing. You could say, I think my portfolio has an expected return of 12%-that would be better than if it had an expected return of 10%. But, on the other hand, you dont want high variance because thats risk; so, both of those matter. In fact, different people might make different choices about how much risk theyre willing to bear to get a higher expected return. But ultimately, everyone agrees I-thats the premise here, that for the-if youre comparing two portfolios with the same variance, then you want the one with the higher expected return. If youre comparing two portfolios with the same expected return, then you want the one with the lower variance. All right is that clear and-okay. So lets talk about-why dont I just give it in a very intuitive term. Suppose we had a lot of different stocks that we could put into a portfolio, and suppose theyre all independent of each other-that means theres no correlation. We talked about that in Lecture 2. Theres no correlation between them and that means that the variance-and I want to talk about equally-weighted portfolio. So, were going to have n independent assets; they could be stocks. Each one has a standard deviation of return, call that . Lets suppose that all of them are the same-they all have the same standard deviation. Were going to call r the expected return of these assets. Then, we have something called the square root rule, which says that the standard deviation of the portfolio equals the standard deviation of one of the assets, divided by the square root of n. Can you read this in the back? Am I making that big enough? Just barely, okay. This is a special case, though, because Ive assumed that the assets are independent of each other, which isnt usually the case. Its like an insurance where people imagine theyre insuring peoples lives and they think that their deaths are all independent. Im transferring this to the portfolio management problem and you can see its the same idea. Ive made a very special case that this is the case of an equally-weighted portfolio. Its a very important point, if you see the very simple math that Im showing up here. The return on the portfolio is r, but the standard deviation of the portfolio is /(n). So, the optimal thing to do if you live in a world like this is to get n as large possible and you can reduce the standard deviation of the portfolio very much and theres no cost in terms of expected return. In this simple world, youd want to make n 100 or 1,000 or whatever you could. Suppose you could find 10,000 independent assets, then you could drive the uncertainty about the portfolio practically to 0. Because the square root of 10,000 is 100, whatever the standard deviation of the portfolio is, you would divide it by 100 and it would become really small. If you can find assets that all have-that are all independent of each other, you can reduce the variance of the portfolio very far. Thats the basic principle of portfolio diversification. Thats what portfolio managers are supposed to be doing all the time. Now, I want to be more general than this and talk about the real case. In the real world we dont have the problem that assets are independent. The different stocks tend to move up and down together. We dont have the ideal world that I just described, but to some extent we do, so we want to think about diversifying in this world. Now, I want to talk about forming a portfolio where the assets are not independent of each other, but are correlated with each other. What Im going to do now-lets start out with the case where-now its going to get a little bit more complicated if we drop the independence assumption. Im going to drop more than the independence assumption, Im going to assume that the assets dont have the same expected return and they dont have the same expected variance. Im going to-lets do the two-asset case. Theres n = 2, but not independent or not necessarily independent. Asset 1 has expected return r1. This is different-I was assuming a minute ago that theyre all the same-it has standard-this is the expectation of the return of Asset 1 and r2 is the expectation of the return-Im sorry, 1 is the standard deviation of the return on Asset 1. We have the same for Asset 2; it has an expected return of r2, it has a standard deviation of return of 2. Those are the inputs into our analysis. One more thing, I said theyre not independent, so we have to talk about the covariance between the returns. So, were going to have the covariance between r1 and r2, which you can also call 12 and those are the inputs to our analysis. What we want to do now is compute the mean and variance of the portfolio-or the mean and standard deviation, since standard deviation is the square root of the variance-for different combinations of the portfolios. Im going to generalize from our simple story even more by saying that, lets not assume that we have equally-weighted. Were going to put x1 dollars-lets say we have $1 to invest, we can scale it up and down, it doesnt matter. Lets say its $1 and were going to put x1 in asset 1 and that leaves behind 1-x1 in asset 2, because we have $1 total. Were not going to restrict x1 to be a positive number because, as you know or you should know, you can hold negative quantities of assets, thats called shorting them. You can call your broker and say, Id like to short stock number one and what the broker will do is borrow the shares on your behalf and sell them and then you own negative shares. So, were not going to-x1 can be anything and x-this is x2 = 1-x1, so x1 + x2 = 1. Now, we just want to compute what is the mean and variance of the portfolio and thats simple arithmetic, based on what we talked about before. Im going to erase this. The portfolio mean variance will depend on x1 in the way that if you put-if you made x1 = 1, it would be asset 1 and if you made x1 = 0, then it would be the same as asset 2 returns. But, in between, if some other number, itll be some blend of the-mean and variance of-the portfolio will be some blend of the mean and variance of the two assets. The portfolio expected return is going to be given by the summation i = 1 to n, of xi*ri,. In this case, since n = 2 thats x1r1 + x2r2, or thats x1r1 + (1 - x1)r2; thats the expected return on the portfolio. The variance of the portfolio -this is the portfolio variance-is = x1 1 + x2 2 + 2x1 x2 12; thats just the formula for the variance of the portfolio as a function of-Now, since they have to sum to 1, I can write this as x1 1 + (1 - x1) 2 + 2x1 (1 - x1) 12 and so that together traces out-I can choose any value of x1 I want, it can be number from minus infinity to plus infinity. That shows me then for any value of x1, I can compute what r is and what is and I can then describe the opportunities I have from investing that depend on these. Now, one thing to do is to solve the equation for r and x1 and I can then recast the variance in terms of r ; that gives us the variance of the portfolio as a function of the expected return of the portfolio. Let me just solve this for-lets solve x1 for r. Ive got-this should be x-did I make a mistake there-so it says that r1 - r2 = x1 r1 - r2, so x1 = (r - r2)/(r1 - r2) and I can substitute this into this equation and I get the portfolio variance as a function of the portfolio expected return r. Thats all the basic math that we need. If I do that, then I get whats called the frontier for the portfolio. I have an example on the screen here, but it shows other things. Let me just-rather than-maybe Im showing too many things at once. Let me just draw it. Ill leave that up for now but were moving to that. What were doing here is the-with two assets, if I plot the expected annual return r on this axis and I plot the variance of the portfolio on this axis, what we have-Im sorry, the standard deviation of the portfolio return. It tends to look-it looks something like this-its a hyperbola; theres a minimum variance portfolio where this is as small as possible and there are many other possible portfolios that lie along this curve. The curve includes points on it, which would represent the initial assets. For example, we might have-this is asset 1-and we might have something here-this could be asset2. Depending on where the assets expected returns are and the assets standard deviations, we can see that we might be able to do better than-have a lower variance than either asset. The equally-weighted case that I gave a minute ago was one where the two assets had-were at the same-had the same expected return and the same variance; but this is quite a bit more general. So thats the expected return and efficient portfolio frontier problem. I wanted to show an example with real data that I computed and thats whats up on the screen. The pink line takes two assets, one is stocks and the other is bonds, actually government bonds. I computed the efficient portfolio frontier for various-its the efficient portfolio frontier using the formula I just gave you. The pink line here is the efficient portfolio frontier when we have only stocks and bonds to invest. You can see the different points-Ive calculated this using data from 1983 until 2006-and I computed all of the inputs to those equations that we just saw. I computed the average return on stocks over that time period and I computed the average return on bonds over that time period. These are long-term government bonds and I-now these are-since theyre long-term, they have some uncertainty and variability to them. I computed the 1, 2, r1, and r2 for those and I plugged it into that formula, which we just showed. Thats the curve that I got out. It shows the standard deviation of the return on the portfolio as a function of the expected return on the portfolio. I can achieve any combination-I can achieve any point on that by choosing an allocation of my portfolio. This point right here is, on the pink line, is a portfolio 100% bonds. Over this time period, that portfolio had an expected return of something like a little over 9% and it had a standard deviation of a little over 9%. This is a portfolio, which is 100% stocks, and that portfolio had a much higher average return or expected return-13%-but it also had a much higher standard deviation of return-it was about 16%. So, you can see that those are the two raw portfolios. That could be investor only in bonds or an investor only in stocks, but I also show on here what some other returns are that are available. The minimum variance portfolio is down here. Thats got the lowest possible standard deviation of expected return and thats 25% stocks and 75% bonds with this sample period. I can try other portfolios; this one right here-Im pointing to a point on the pink line-that point right there, 50%stocks, 50% bonds. You can seeYou can also go up here, you can go beyond 100% stocks, you can have 150% stocks in your portfolio. That means youd have a leveraged portfolio, you would be borrowing. If you had $1 to invest you can borrow $.50 and invest in a $1.50 worth of stocks. That would put you out here; you would have very much more return, but youd have more risk. Borrowing to buy stocks is going to be risky. You could also pick a point down here, which is more than 100% bonds-how would you do that? Well, you could short in the stock market, you could short $.50 worth of stocks and buy $1.50 worth of bonds and that would put you down here. Any one of those things is possible its just the simple math that I just showed you. Do you have any idea what you would like to do, assuming this? Well if youre an investor, you dont like variance. So, you probably dont want to pick any point down here, because youre not getting anything by picking a point down there because you could have a better point by just moving it up here. Youd have a higher expected return with no more variance. Its getting kind of complicated, isnt it? We started out with just a simple idea: that you dont want to put all your eggs in one basket and if you had a lot of independent stocks you would want to just weight them equally. But now, you see there are a lot of possibilities and the outcome of your portfolio choice can be anything along this line. Im not going to tell you what you want to do except to say, you would never pick a point below the minimum variance portfolio, right? Because, if you did, then you would always be dominated. You could always find a portfolio that had a higher expected return for the same standard deviation. But beyond that, if you were confined to just stocks and bonds, it would be a matter of taste where along this frontier you would be. You would call it an efficient portfolio frontier. It would be anywhere from here to here, depending on how much youre afraid of risk and how much you want expected return. Now, we can also move to three assets and, in fact, to any number of assets. The same formula extends to more assets. In fact, I have it-suppose we have three assets and we want to compute the efficient portfolio frontier, the mean and variance of the portfolio. What I have up there on the diagram are calculations I made for the efficient portfolio frontier with three assets. So, now we have n = 3 and in the chart are stocks, bonds, and oil. Oil is a very important asset, so we want to compute what that-so now we have lots of inputs. Lets put the inputs-r1, r2, and r3 are the expected returns on the three assets. Then, we have the standard deviations of the returns of the three assets and we have the covariance between the returns on the three assets. There are three of them- 12, 13, and 23. Thats what we have to know to compute the efficient portfolio frontier with three assets. To make this picture, I did that. I computed the returns on the stocks, bonds, and oil for every year from 1983 and I computed the average returns, which I take as the expected returns, I took the standard deviations, and I took the covariance. These are all formulas, I just plugged it into formulas that we did in the second lecture. What is the portfolio expected return? The portfolio expected return-we have to choose three things now: x1, x2, and x3. x1 is the amount that I put into the first asset, x2 is the amount that I put into the second asset, and x3 is the amount I put into the third asset. Im going to constrain them to sum to one. The return on the portfolio is x1 r1 + x2 r2 + x3 r3. The variance of the portfolio, , is x1 12 + x2 2 + x3 3 - then we have to the count of all of the covariance terms - + 2x1x2 12 + 2x1x3 13 + 2x2x3 23. Is that clear enough? It seems like a logical