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# ECON 251: Financial Theory

## Lecture 23

## - The Mutual Fund Theorem and Covariance Pricing Theorems

### Overview

This lecture continues the analysis of the Capital Asset Pricing Model, building up to two key results. One, the Mutual Fund Theorem proved by Tobin, describes the optimal portfolios for agents in the economy. It turns out that every investor should try to maximize the Sharpe ratio of his portfolio, and this is achieved by a combination of money in the bank and money invested in the “market” basket of all existing assets. The market basket can be thought of as one giant index fund or mutual fund. This theorem precisely defines optimal diversification. It led to the extraordinary growth of mutual funds like Vanguard. The second key result of CAPM is called the covariance pricing theorem because it shows that the price of an asset should be its discounted expected payoff less a multiple of its covariance with the market. The riskiness of an asset is therefore measured by its covariance with the market, rather than by its variance. We conclude with the shocking answer to a puzzle posed during the first class, about the relative valuations of a large industrial firm and a risky pharmaceutical start-up.

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html## Financial Theory## ECON 251 - Lecture 23 - The Mutual Fund Theorem and Covariance Pricing Theorems## Chapter 1. The Mutual Fund Theorem [00:00:00]
So much of what economists know was already known to businessmen, because it’s common sense, and even to literary writers like Shakespeare, who was himself quite a successful businessman. So you might remember that Shakespeare already had the idea of diversification. In the Merchant of Venice, Antonio says “Each boat on a different ocean.” He said it a bit more poetically than that, “each sail on a different argosy” or something. I’ve forgotten what it was, but each boat on a different ocean. So he wasn’t very worried because he was diversified. He also had the idea of risk and return. Nothing ventured, nothing gained. Nothing ventured, so ventured means risk. Nothing gained. So another way of saying this diversification is “Don’t put all your eggs in one basket.” So what have economists done that wasn’t already known to every business person and every clever literary writer? Well, economists have quantified this and turned this into a usable, practical piece of advice. So the diversification theorem in CAPM– so I should have moved CAPM. I should have called this Shakespeare and this CAPM. CAPM becomes the mutual fund theorem. So the mutual fund theorem has very practical advice. It says if you’re investing in the stock market, don’t try to pick out individual stocks. Hold every stock in proportion to its value in the whole economy. If you want to be more venturesome, don’t pick riskier stocks. Just simply leave less money in the bank. So divide all your money between the index and the bank. It says, hold all your money in stocks in the same proportion everyone else is holding them. Hold the market, in other words. Hold an index like the S&P 500, of all the stocks, in proportion to how big they are. And if you want to get more venturesome–if you’re cautious, put some of your money in the index and some of it in the bank. If you want to be more venturesome, take some money out of the bank and put it into the stocks. But in the index. If you want to be even more venturesome than that, borrow the money to put the money into the stock market. That’s called leverage. But you should, according to this theory, not try to pick stocks. So if you’re interested in risk, you shouldn’t pick high technology startup companies. You should pick the same mixture of blue chip companies like General Electric and these startup companies that everybody else is choosing. So that’s what it means to diversify. Hold a little bit of everything in the same proportion as everyone else. So it’s much more precise, much more surprising than Shakespeare’s advice. ## Chapter 2. Covariance Pricing Theorem and Diversification [00:03:47]Then the second thing, risk and reward, it says that the price of a security, so we’ll call it pi We know that you have to adjust for risk somehow, which we haven’t taken into account before. But what’s so shocking is, you might have thought you’d subtract maybe something times the variance of the return of asset J, but that’s not going to be the case. It’s going to be, surprisingly, subtracting something times the covariance, with the market, of asset J. So you penalize–the price of a stock should go lower if it gets riskier. Its risk is not defined by what its variance is, but by what its covariance is with the market. So I want to develop these two ideas in this lecture, spend the whole lecture on just developing these two ideas, which are the modern mathematical version of what Shakespeare already knew 400 years ago, but made much more concrete, much more precise, with the mathematics wholly behind them. So what have we got here? We imagine a world, the situation I’m going to describe, is the one we had last time, where we know that there are different things that can happen. These are the states with different probabilities, so there’s states. s = 1 to big S and each of them have probabilities, with probabilities gamma So here’s A You can also compute the variance, so I’ll write that sometimes as A bar So those three numbers, the expectation, the variance and the covariance between any pairs, those are numbers that anybody could compute, knowing the payoffs of the assets and knowing the probabilities. And what the theory is going to say, the capital asset pricing model, that model is going to explain how the prices of all the assets depend on their expectation, their variances and their covariances, under the assumption that people have quadratic utilities, or more generally, assume people only care about the mean–that’s the expectation–and variance of final consumption. So we wrote that last time. We said if you had quadratic utility for consumption in every state, you could summarize these thousands–there could be millions of states and thousands of securities. You add them all together, what you hold, it only depends on the end about what the expectation and the variance of your final portfolio is. You like expectation and you hate variance. So under that hypothesis, we’re going to have a very concrete interpretation of what Shakespeare was talking about from the beginning, which is going to allow us to quantify what you should do, how people in the market are doing. We’ll be able to answer a series of questions. And then we’ll be able to test the theory too. So now the theory I’m about to explain was developed in 1950. It began in 1950 when Koopmans was a professor at the Cowles Foundation, the director of the Cowles Foundation, was in Chicago. I told you, the Cowles Foundation was started by Alfred Cowles, who’d been a Yale undergraduate, and his family owned the He was embarrassed to realize that he had predicted that everything was going to be fine, and he sent around questionnaires and read everybody else’s, his competitors’ newspapers, and realized they’d been predicting everything was going to be fine too, except for one guy, Jones, the Dow Jones guy. Except for that, everybody had been making the same rosy predictions. So he decided the whole subject of economic forecasting, in fact, economics altogether was a fraud, and he wanted to put money in encouraging the use of mathematics in economics, instead of the touchy-feely more qualitative analysis that was the norm at the time. So he went to see Irving Fisher, who was the great Yale economist, and said, “What should I do with all the money? Where should I put this institute, and who should I invite to the institute?” So Fisher and he set up the whole thing and it started in Colorado Springs, because Fisher believed in fresh air, and he wrote a lot of books about fresh air and how it staved off tuberculosis and things like that. And no one wanted to go to Colorado Springs, so they moved it to Chicago where the family was originally from, the Cowles family. The director then in the ’50s was someone named Koopmans, and Koopmans had this idea of trying to examine risk by making people with quadratic utilities. And he said, “Okay, student Markowitz, why don’t you try to develop a theory?” And Markowitz began the theory of the capital asset pricing model. Now the theory, just to get a little ahead of the story, Markowitz went quite a ways, won a Nobel prize many years later for what he did, but didn’t quite get to the end. And while he was working on it, Koopmans didn’t want to be director any more of Cowles, and he invited Tobin, the great Yale economist, to come and be director of Cowles. And Tobin got there and the people at the University of Chicago said to Tobin, “Well, we’re glad you’re coming. You’ll be right next to us, but don’t try and get us mixed up with all those mathematical people at the Cowles Foundation. They’re all crazy. We would like to keep that high tech mathematics out of our department. That’s just Cowles’ crazy idea.” So Tobin refused the job, came back to Yale, and Markowitz was so angry–not Markowitz. Koopmans, his advisor, was so angry that he went to Cowles and said, “They’re not treating us right at the University of Chicago. Let’s move the entire Cowles Foundation to Yale.” So in 1955, they moved it to Yale. But in these conversations Tobin had with Koopmans and the Cowles Foundation, he got to know what Markowitz was doing, and so he added the final crowning achievement to–the mutual fund theorem–to the Markowitz setup, and he subsequently won the Nobel Prize for that. In fact, the So anyway, that’s the theory that I have to explain now, that it’s a little bit less trivial than just saying, “Don’t put all your eggs in one basket. So, the idea is that you like expectation, you hate variance. So the first thing we’re going to do is notice that covariance–I’m sorry, standard deviation is defined as the square root of variance. So to say that you don’t like variance means you don’t like standard deviation. They’re the same thing, but you’re going to see, it makes the diagrams much prettier. So if we had a portfolio that you could choose among some combination of putting your money in the A’s, you’d get some expectation of some standard deviation for your total portfolio. And people like more expectation and like less standard deviation. So the indifference curves are going in that direction, unlike the usual picture. That was the first step. Now the first thing we noticed last time, this was three weeks ago or so, we made an extremely important observation. We said, what is the covariance of part of your money in X, the variance of part of your money in X and the rest of your money–let’s say you have 1 dollar–so t in X and (1 - t) in Y? For example, 1 half in X and 1 half in Y. Well that is by definition the same as the covariance of this portfolio with itself, and now we know this from this theorem, from this definition of covariance, that if you look at the covariance of J with K, so fixing K–this is just linear. This is a constant, multiplying the differences from J and its expectation. So this is a linear in each variable, so therefore this is going to equal the covariance of tX with tX + twice the covariance of t–so + the covariance of tX and (1 - t)Y, + the covariance of (1 - t)Y and tX + (1 - t) squared (so I’ll put that one over here) + the covariance of (1 - tY) with itself. So I’ve taken this with this, then this with that, then that with that and that with the thing on the end. But it’s linear again, so I can pull out all these things. So that’s the formula. Now what we noticed is that if 2 stocks, X and Y, are independent, their covariance is 0. So I’m assuming you remember that, so those terms just go away. And then it’s also linear, so this is just going to equal t squared, covariance of X with X, which is just the variance of X. So assume XY independent, then that is what allows me to cancel these two terms like that. So I get t squared times the covariance of X + (1 - t) squared times the variance of Y. If I take t = 1 half, I know that the variance of 1 half X + 1 half Y = 1 quarter times the variance of X + 1 quarter times the variance of Y. We derived this three weeks ago. I’m just repeating it. So the amazing thing is, if X and Y have the same expectation and the same variance, but are independent of each other, by splitting your money between two independent things, you’re obviously going to have the same expectation, but only half the variance. And that last step is, the variance of X = variance of Y. So amazingly–or it’s not so amazing, I guess, if you’ve thought about it–but if you haven’t thought about it, you’ve got 2 equal risks. Neither one is better than the other, so why bother with one rather than the other? They’re both the same. But if they’re independent, you should put half your money in each of them. You get the same expectation, but you’re going to get only half the variance. So if you have 2 dice, let’s say the first die will give you–you pay 100 dollars for a dice, you can get 100 + the number of dots showing. So if it gets a 1 that’s 101 dollars, you get a 6, it’s 106 dollars. So if you held 1 dice, 1 die, you’d get on average 103 and 1 half dollars. You could pay 100, get 1 die, which on average paid 103 and 1 half. Instead, you could put 50 dollars in the first die, get half of that, 50 dollars in the second die, get half of that. So on average, you’re still getting 103 and 1 half, but the variance you get is going to be half of what it was before. So in this picture, you start off with X and Y with the same standard deviation and the same expectation, but they’re independent and now you put half your money in each and you cut the variance in half, which means you divide–the standard deviation is the square root–so you divide it by the square root of 2. So you’ve improved. You’ve moved to the left. You’ve gotten less standard deviation without constricting your expectation at all. So that’s the most important thing to notice. I’m going to say one more thing, one more way. I’m going to go back to this formula, which is where the variance was, here. Now suppose–let’s look at that formula. Suppose I take 2 stocks, which I’m going to switch now, X and Y. That was most unfortunate. But suppose I take 2 stocks and I’m at the point where all my money is in Y. So I’m going to take the case t = 0. So if I took t = 0, I would just have the variance of Y. That’s actually quite good, the variance of Y. So I’d have all my money in Y. I’d have this standard deviation and this expectation. Now what happens if the two are independent, X and Y? Let’s drop the fact that they have the same variance now. Maybe X has way more variance than Y does. So look, X is much riskier than Y. Maybe X has a higher expectation than Y, but it’s so risky. People who are risk averse wouldn’t ever want to subject themselves to all this risk, say. So maybe they should just stick with Y, you would think. But no, the answer is, you shouldn’t, because what happens if you start to mix some of your money in Y–take a little out of Y and put a little bit into X? Your expectation, if you take a little money out of Y and put a little into X, your expectation’s going to go up, because it’s going to be the average of X and Y. So the expectation’s going up. But then you say, “Oh, but my variance is going up.” So I wouldn’t want to do it, but no, that’s wrong. The variance isn’t going up, because they’re independent. Even though X has a way higher variance than Y, mixing a little between the two lowers your variance. So how could that be? Seems quite shocking. Well, let’s take the derivative of this with respect to having all your money originally in Y. So this is the variance of putting t dollars in X and 1 - t in Y. So we simply take the derivative of that, which I’ve done here, I suppose. Why don’t I just write it out? Take the derivative of that, d/dt of this = 2t times the variance of X + 2 times (1 - t) times (okay, but this is a -, so I take the derivative) it’s 2 times this times - 1, so - 2T, 2 times (1 - t), variance of Y. Let’s evaluate that at t = 0. The first term is 0, the t is 0, so this doesn’t matter. The second term, t is 0, so this is - 2 times the variance of Y. It’s going down. The variance is actually going down here. So just as this says, the variance starts down. Eventually, it has to end up here, so as you move your money, all of it originally in Y and start moving some of it into X, your variance is going to go down. So those two things put together are the argument for diversification. If they’re independent and identical, then you should put half your money in each. But even if they’re not identical, you shouldn’t stick to Y. You should mix Y with some X. So there’s why diversification is such a good idea. So it’s not all diversification. It’s diversification of independent risks. There’s a bit of a mathematical nuance. Diversification of independent risks is bound to help you. Everybody should diversify a little bit, because you can reduce your risk. That’s the first important thought. Now I don’t want to do too much in the mathematics. So I prove in the notes, but I won’t do it here, suppose we didn’t assume X and Y were independent. Suppose X and Y were arbitrarily chosen. They have big covariance, whatever you want. Then what is going to happen when you combine your money between X and Y. So there’s a proof, which I’m not going to give, because I’m a little bit behind, so the proof is this. Suppose we have exactly the same situation, where you started with Y and you’ve got X, and you could think of moving you money in between X and Y. Well, it turns out, if they’re perfectly correlated, so really they’re the same thing, Y is just 80 percent of X, it’s the same stock, so moving back between them is not really changing–X is just a scalar improvement on Y, so it has more expectation and more standard deviation. If that were the case, you’d just move on that straight line that connects the two points. I must have a way of doing this. Yes, if they were perfectly correlated, you would just be moving along this line by putting your money in between. If they’re independent, you’re going to move along that line. But no matter how correlated they are, as long as they’re not perfectly correlated, it’s going to turn out the curve you get by moving between them is going to look like that. If they’re negatively correlated, then you can really improve yourself and get a curve that looks like that. Independent will be like this, positively correlated, but not perfectly will be like that, and perfectly positively correlated, no diversification gains, you get the straight line, but you’re always above the straight line. So that’s a kind of–anyway, it takes a little bit of algebra to prove that, you know, the Cauchy-Schwarz inequality, but I’m going to skip that. But anyway, that’s a mathematical fact. You see in the independent case, we proved that the line has to start going that way and eventually it has to end up here, so it should be no surprise that it looks like that in the independent case. And then it turns out that every other case is sort of in between that or at the other extreme. It always has this bowed out shape. So there’s always gains to diversification, something you’re getting out of diversification. The risk is somehow going down by mixing things up. ## Chapter 3. Deriving Elements of the Capital Asset Pricing Model [00:25:19]That’s it, that’s the mathematics and that’s the general principle. Now, what can you get out of this? I want to derive these two famous theorems, which we already saw in action in the example that we calculated, but it seemed like a miracle. So what did we find in the example we calculated last time in class? We found that–we only had two stocks, but if there had been hundreds of stocks and millions of states of nature, and each person had a different utility, different risk aversion, but all quadratic, and you calculated the competitive equilibrium, assuming there were Arrow securities, it would end up that every consumer held the same mix of all these stocks, + positive or negative amounts of the bond. So the mutual fund theorem turned out to be true, that the best thing everybody could do was hold the same amount, the same proportion of stocks and bonds, and maybe more or less of the riskless asset, maybe even negative of the riskless asset. So I want to show you why that’s true now. Then we’re going to have to talk about whether we really believe it or not. So let’s see what an implication is. Suppose now what I do is I write down–I have all these stocks, X, Y and Z. So you can imagine–you have a certain amount of money–so you can imagine putting your money–so let’s suppose that I’ve written down all the stocks. The prices are given. Let’s say prices of stocks are in equilibrium. So I’ve solved for the equilibrium like we did last time, and I write down X, Q, Y and Z. These are the expectation and standard deviation per dollar of the stocks, or per I dollars of the stock. Maybe the person has exactly I dollars, so let’s say it’s I dollars of each stock, what their expectation, return and standard deviation would be. If 1 dollar, you get some expectation, standard deviation, 10 dollars of the same stock, you’d be able to buy 10 shares, which means you would–10 times as many shares, which means you’d have 10 times the expectation and 10 times the standard deviation. So whatever the guy’s income is, they’ll fix the returns as per his income, because he’s going to spend all his income on these stocks. So what could he do? He could put all his money into X, he’d end up here, or all his money into Y, he’d end up here, or all his money into Z, and he’d end up here. But maybe he wants to divide part of his money in Y, part in Z and part in Y. He’ll be somewhere like here. Or maybe he wants to instead put part of his money in Z and part of it in X. Well, he could end up here. But why should he stop there? Maybe he wants to take some combination of the money that was in Z and X and combine it with some combination of things that was in Z and Y, combine this and this. That would get him out to here. So there are lots of different combinations he could hold. So graphically, it looks like that. Mathematically, what’s he trying to do? He’s trying to–so R now. I’ve replaced A with R. So R Now what does he want to do? He wants to choose his money, how much money he’s going to put in each asset, which is omega So the question is, how should he divide his money between the assets, and here’s the kind of stuff he could imagine doing. Looks very complicated, but it’s going to turn out to be very simple. So are there any questions about what we’re doing so far? I want to get a simple rule to tell you, practical advice about what to do. Markowitz posed this problem. This is the Markowitz problem, okay? It turns out, by this argument, you see if I keep combining things over and over again, I’m getting this sort of blob. But the blob is always pushing itself further and further out. It’s always convex, because given any two points, I can always do the thing to the left of it. So if I could do this and I could do this, and I look at the line connecting them, I can always be above that line by spreading my money between those two assets. Or given this combination of assets that produces this expectation and standard deviation, and given that one up there, I could combine the two and do better than the straight line connecting them, do to the left of it. So therefore, all of the combinations put together have to, geometrically, look like this in the end. So I’m going to do this algebraically in a second. They have to look like that if I did every possible variation. It couldn’t end up with a picture where after I do every possible variation–if this is the expectation, this is the standard deviation, I couldn’t get something that looked like that, because I would just combine this thing with this thing. Whatever portfolio produced this, whatever portfolio produced that, I’d put half my money into each of the stocks that this told me to do, half the money into each of the stocks this told me to do. I’d still end up investing 1 dollar and instead of getting the line, I’d get something that looked like that. So I couldn’t have the thing going in like that. It has to go out. But then of course, I can combine this and this and get that, and so I’m going to end up with this kind of shape. Now I’m going to prove that algebraically. But if I had that kind of shape, then what would the guy do? He’d choose something like that, where he’s tangent. That’s what Markowitz basically did, except he did it algebraically. So that’s what Markowitz did. That doesn’t seem to have gotten us very far. You notice that different people are going to make different choices. If somebody is incredibly worried–someone’s not very worried about the standard deviation and cares a lot about expectation, they’re going to have a flatter indifference curve. So instead of looking like that, it will be a flatter thing. And so if it’s flatter, the guy who cares–sorry, if it’s flatter, it means he cares about expectation and not standard deviation, because a little bit of expectation can compensate him for a lot more standard deviation, so his indifference curve is flatter. He’s going to choose further up. So he’s going to choose a point like way up there. So he’s going to get a higher expectation and higher standard deviation. Someone who’s more risk averse is going to choose where that point is. So that’s qualitatively what’s going on. So Markowitz played around with this, played around with this. Then Tobin appears on the scene. Tobin says, “All this gets so much easier if you have a riskless asset.” Let’s suppose that there’s a bond. You haven’t mentioned the bond yet. Let’s suppose we have a bond that pays something for sure, and we’re going to ignore inflation, Tobin said, which is a big problem. We should come back to that if we have time. We’re going to ignore inflation, so US Treasury’s going to pay a certain amount of money for sure. So it has an expectation like this, but no standard deviation. It’s 0. Now why is this so important? Let’s imagine combining this riskless bond with some other stock X, 1 dollar’s worth of X. 1 dollar’s worth of the riskless bond gets you 1.06 say. 1 dollar’s worth of X gets you much–you know, maybe your average is 1.12, but you’re also running a big risk. What happens if you put your money part way in between them? So suppose you put part of your money in X, and the rest of your money in the riskless bond. What’s your expectation going to be? Your expectation, let’s say X is 12 percent and Y is 6 percent, 1.06 and 1.12, the average is going to be 1.09 if you put half your money in both. But what is the standard deviation going to be? I claim it’s just going to be on this straight line. Why is that? Because the riskless thing has covariance 0. If X is riskless, its covariance is going to be 0 with Y. So these terms are going to disappear. Look at the covariance. If X–ifY is the riskless bond, it pays always its expectation. So these numbers are always 0. Therefore the covariance of that thing with anything, the covariance of Y with any X is going to be 0. And not only that, but the variance of Y itself is also 0. So all you get is, for the variance of the t dollars in X and 1 - t dollars in Y, you just get t squared times the variance of X. But then when you take the standard deviation, it’s just t times the standard deviation of X. So in other words, the standard deviation of the mixture of Y and X lies right on the line. So unlike–to put it another way, something that has no variance is perfectly correlated with everything else, it makes no difference. When the other thing goes up or down, it doesn’t tell you–you still know what Y’s going to be. So unlike everything else, where you have this bow shaped thing, the riskless asset just connects everything to a straight line. So if there were a riskless asset and you could do this with some X and here was a Z, you could also do that, or you could do that. Is this clear? Okay, so the punch line, which we’re now going to see algebraically, the punch line is–and by the way, if I combine, what does it mean to extend that line? I should have extended that line. That was a huge mistake. Suppose instead of–so here’s all the money in the riskless asset. Here’s half the money in the riskless asset and half in X. Here’s all my money in X. What would happen if I extended the line here? What would that correspond to? I’m now at a point on the extension of the line. How do I get that?
So knowing that, now what should I do? Is there something really simple? Well, what would you do here, faced with this choice? Which combination of stocks and bonds would you hold? I say it’s really simple. You just look for a line through the riskless point that’s tangent to this blurb, which happens to be here, and now you should choose your point anywhere along this line, independent of what your mean-variance utilities are. No matter how risk averse you are or how risk loving you are, you should choose somewhere along this line. Why is that? Well, I’ve drawn, with this blob, I’ve drawn all the possible combinations I could get by mixing different risky assets. What I left out of the picture is all the combinations I could get by adding the riskless asset. I’ve got all the risky asset combinations. That’s the blob. Now I want to do–adding the riskless asset means I can take part of my money in here and part in there. So in particular, I can put part of my money here and part of my money there, or I can go short. So whatever this combination was, that’s the best thing I can do, because now I can get everything along this line. Nothing else can do better. What other combination could there be? It has to be the riskless asset + a bunch of risky assets. But combining the risky assets just puts me in the blob. So it might put me here in the blob. And then once I combine that with the riskless asset, I’ll be on that line. This line is always below that line, so this is the best possible thing I can do. No combination of the riskless asset and any other combination of risky assets, that’ll always put me on some line from here, through some point in the blob, which is going to be below this tangent point. So this is the best thing I can do. And it’s independent of whether I was risk averse or a risk lover. If I’m risk averse and want to play it safe, my utility function–I forgot what those looked like. My indifference curves–I’ll be here. I’ll put most of my money in the bank and not very many in the risky stock. If I’m more of a risk lover, I’ll be maybe here, and I’ll choose to put a good fraction of my money in the stock and only a little in the bank. If I’m really reckless, I’ll borrow money and put it in the stock market and be way up there. But everybody will do the same thing, no matter what their preferences are. They’ll all be along the same line. So that’s the mutual fund theorem. Why is that the mutual fund theorem? Because it says everybody should invest in the same index of stocks and put more or less money in the bank. So every single person ought to be doing the same thing. This is the mutual fund that everybody should hold. Whatever combination of stocks that got me to this point is the mutual fund that everybody should hold, and combine that with putting money in the bank. ## Chapter 4. Mutual Fund Theorem in Math and Its Significance [00:40:25]Okay, so let’s just prove that algebraically, in a special case, which is going to illustrate the point, the idea of it anyway. I’m doing a very special case here, but just to illustrate it algebraically. So suppose that the Rs are, as we said, the returns, up here, per dollar of stock. The R So how should you do it? Well, you care about your expectation and you hate variance, so I’m doing just a special case to make it illustrate the idea. Suppose I write down your utility as if it were the expectation, - alpha times the variance. It could be a more complicated function. We said already that the quadratic utilities give you expectation - alpha expectation squared, - some other alpha times the variance. That’s the utility we derive from quadratic–when people have quadratic utilities over consumption, their induced utility of portfolios is this thing. So it’s not exactly the one I’ve written down, but the argument would be just the same and this is shorter and simpler. So just to illustrate the flavor of it, let’s say all you care about is your expectation - some constant, how risk averse you are, times the variance. Now let’s suppose, to keep it even simpler, that the assets are all independent. The theory doesn’t rely on that. I’m just saying, suppose it were so we can algebraically get it without having to work very hard. So you’re maximizing your expectations, so the bar above means the expectation. So what’s your expectation of omega dollars in here, omega Now what’s your variance, the thing that you don’t like? Your variance is going to be–is going to be the variance of these. They’re independent. So the variance is just going to be what I’ve written. So call the variance of R This is the variance you have to watch out for. This is the expectation you want. How should you pick omega So why is that? We just reduce this algebraically, solve it algebraically. So the constraint is a nuisance. To maximize something with the constraint’s a nuisance, so if you just notice that if you satisfy the constraint, omega R bar So with respect to omega But it depends on alpha in a very trivial way. So if I took the fraction, the relative amounts of money I put in i and j and divided this by omega So unlike this old Markowitz case, the Tobin case looks like that, and everybody chooses the same combination of risky assets and then this safe asset. So it may be that different people put different combinations of safe and risky, but everybody’s proportion of risky assets is proportional to everybody else’s. That’s what we just proved algebraically. So graphically, a sort of general proof, algebraically, a special case. And then if you were in a graduate class, you’d get an algebraic proof in general, which is actually much faster than either of these, but you have to know a little linear algebra. And so that’s the proof. It’s a very simple thing. It’s a remarkable thing, and here’s the famous Tobin diagram. Everybody should do that, put their money in the same combination of risky assets and the riskless asset. Are there any questions about this? It’s kind of shocking. I could swear that most people think, maybe even rightly, if you really want to go for it, put your money in the young Microsoft or something. It could be a tremendous success or it could go out of business. That’s what you should do, but no, this says not at all. If you really want to go for it, take the same index that everybody, the S&P 500 and just leverage it. Borrow a lot of money and go out that straight line, way out to there, to the right. Don’t put your money in Microsoft. Just leverage the hell out of the S&P 500. Okay, so after this theory was created–I’m going to tease out all the implications of it. After this theory was created, the mutual fund business took off. Vanguard and all of these places, what did they do? They said, “Look, economic science has taught us, the best you can do is not try to pick individual stocks. The best you can do is to hold the same index, the whole market. “Now of course, it’s very expensive for you to hold the whole market. Are you going to go to a broker and tell him ‘Buy me 3 shares of this and 7 shares of that and 9 shares of that, 14 shares of that’? It’ll take the guy forever to buy all that stuff and you have to pay a commission on every single thing you buy. It’s going to cost you so much money to do what economic science says. Come to us at Vanguard. We’ll buy the whole market for you, call it our index fund and charge you almost no commission at all, just a tiny little bit, because we’re doing it in such bulk, everyone’s–we’ve got lots of investors like you. They’re all buying the same thing. We have huge volume, so we’re going to get a really small brokerage fee, so you should put your money in with us.” So that’s what everybody started to do. So all these mutual funds and money managers, they all grew up, originally by guaranteeing to produce an index fund, the very same thing economic science recommended, at almost no cost. If you read Swensen’s book, his first, most important piece of advice: if you are a standard investor who doesn’t know how to manipulate all the complicated instruments on Wall Street and has transactions costs and things like that, put all your money into an index. That’s what he recommends doing. And he says everyone else who tells you otherwise is just trying to steal your money, rip you off. So that’s the main financial advice that economists give now. It’s Shakespeare’s advice. Don’t put all your eggs in one basket, but it’s very specific. Put it in an index where you buy everything in the same proportion. Now of course, if you’re buying everything in the same proportion and everybody’s doing that, what does this have to be? I forgot to say that. If everybody’s buying every stock in the same proportion, that proportion must be the market, because that’s what everybody’s holding. So this doesn’t turn out to be any old portfolio. It’s the market portfolio. So the index, that’s a huge conclusion here, the index that they should offer is not just some magical index that’s better than everything else, because if everyone’s doing the right thing, everyone’s choosing the same index–that’s what we proved–if everyone’s choosing the same index, it must be the market index, because that’s all there is to choose. Supply has to equal demand. So therefore the advice is to put all your money into the market index, not just an index, the market index + maybe put money in a bank. Or even borrow from a bank to put it into the market index. That’s the first surprising theorem that Tobin proved and it had dramatic–so it created the indexed investments. It created Vanguard. So these people who run these companies, I forgot the guy’s name who runs Vanguard. He’s always giving these speeches. So that’s how he describes how he got his start in his business. All he says is, “I take all the recommendations of economic science,” he says, “and I allow people to carry them out. That’s why I’m doing great good for society. Everybody else, all these hedge funds and stuff, they’re ruining society. I am just doing what Tobin told us to do.” That’s his basic speech. ## Chapter 5. The Sharpe Ratio and Independent Risks [00:52:36]So Markowitz was a little bit more precise even than that. Markowitz’ formula and Tobin’s special case, he not only told you what to do. He told you, “do what everyone else is doing.” He told you what that should be. So let’s suppose that I’ve got 2 stocks, i and j, with the same return, same expected return, but one has standard deviation 3 times higher than the other. So in my picture here, that standard picture, we’ve got the standard deviation, we’ve got the expectation. We’ve got 1 stock here and we’ve got another stock here with 3 times the standard deviation. So you might have thought, “Well, these 2 stocks, same expected return, what’s the difference? Put all my money in one or the other, doesn’t matter.” A little bit further thought would say, “Well, this one is much better than that one. I’ve got smaller standard deviation, so I should put all my money into this one.” But then on further thought, given that these two are independent, what should you do? How should you allocate your money between this stock and that stock? What does the formula tell you to do? It’s very clear. Well, the ratio of money in one stock to the other stock looks like this ratio. But I’ve told you that the expectations of the 2 stocks are the same. So those things are the same. I’ve told you that one has 3 times the standard deviation than the other, so therefore, what should you do? Yeah.
So the first concrete advice is, the first concrete thing we’ve gotten out of this is, buy the index. Now the index typically in America, they’re easy to buy, are indexes of stocks. But you know, the second piece of advice is, and this is also Swensen’s second most important point–I’m quoting Swensen because we’re here at Yale, not because he’s the only one to recognize these things, but he was a student of Tobin–so the second important point is that you should not just buy the index, because the indexes are the wrong index. The index should be the index of everything. It should include American stocks, Chinese stocks, French stocks, Greek stocks, Israeli stocks. You should have all stocks in the index, but you can’t really very easily buy all those other things. You can only very easily buy index of American things. It’s getting easier and easier now to buy the global index. But anyway, you should buy the global index. That’s what this says. So every time you find a new independent risk you should look for it. So you should buy the global index, seek out independent risks. So what did Swensen do? One of his most important things was, Yale is always looking globally to find independent risks. So let’s go back to the–so you should always find independent risks, and you suffer if you don’t. So you see, by putting 90 percent here and 10 percent here, Markowitz is telling you, you’re going to do better. What else do I have to say? So let’s go back to our dice example. Suppose that you had 1 dice. Now by the way, I want to introduce one more term, which I’ll call the Sharpe ratio. So the slope of this line–so it’s the Tobin diagram. It should be called the Tobin ratio, but this other guy at Stanford snuck in and managed to give his name to it. He ended up getting a Nobel prize too. So here this line, the slope of this line, this is the riskless asset–the slope of this line, what is the slope of this line? So this is standard deviation, this is expectation. The slope, what’s the slope? Well, it’s the expectation of whatever you’re holding, X, - the riskless asset divided by the standard deviation of X. That’s the Sharpe ratio of X. So you notice that every point you could conceivably choose, which is somewhere in this blob, then connect it to the riskless asset–so every point on this line obviously has the same Sharpe ratio. If you leveraged your position and went to here, you wouldn’t change your Sharpe ratio. It has the same slope. If you put half your money in the bank and half in the riskless asset, you’d be here, same Sharpe ratio. Anything else you could do has a lower Sharpe ratio. So another thing to say is that an investor managing all his money should maximize the Sharpe ratio. You could in fact tell if somebody had screwed up. Instead of doing the global index, they just did the American index, pick something here, you could look at the slope of where they ended up and it would be a worse slope and a worse Sharpe ratio. You could even judge people, how they’re doing, by looking at their Sharpe ratio. So let’s take the case of the 2 dice. Suppose you had 1 die. So if you had 1 die, remember, if you invest 100 dollars, you could get 101, 102, through 106. Let’s say the interest rate is 3 percent. So you could get 101, 102, 103, 104, 105, 106, with 1 sixth probability of each. So the average of that is obviously 103.5, right, because 1 and 6 is 7, and divide by 2, is 103.5. So the expected payoff for putting your money into 1 dice is the average return, is 1.035. The riskless return is going to be 1.03, so it’s .035 - .03, .005. And now what’s the standard deviation? What’s the variance of this? Well, you could take 101 - 103.5 squared, + 103 - 103.5 squared, etc. and take the square root of it, and you’d get .017. So you notice the chance of losing money, by the way, is 50 percent. 50 percent of the time, you’re going to lose money, and your Sharpe ratio is .005 over .017. So that’s about 3–no, it’s about .3. Now that happens to be the Sharpe ratio of the stock market, just about. Shockingly low, isn’t it, if you look at the history of the stock market–so it depends what year you start and what year you end, if you do it right after a crash or before a crash, but just to pick vague numbers it’s 9 percent and let’s say the interest rate is 4 percent, something like that. It could be 5 percent or 3 percent. Depends again the period. Maybe it’s 2 percent, but somewhere like that. And the standard deviation of the stock market is 16 percent or 12 percent, again depending on what period. But always around 12 to 18 percent, let’s say 16 percent. That’s the one I like to use, because it’s percent a day. So that’s = to 5 percent over 16 percent, which is about 1 third. It’s about .3. That’s exactly what this dice has, about .3. So that’s all you can get out of the stock market. Not a very good Sharpe ratio. So you could measure Swensen’s Sharpe ratio. You could measure Ellington’s Sharpe ratio and see what they are. You can see who you think is a better manager. So I won’t tell you those numbers just yet. But now let’s suppose, what else could I do? So let’s suppose you put your money in two dice. That’s the whole point of this. Suppose you put half your money in the first dice and half your money in the second die. Now what are the chances–what’s going to happen? Well, again, your expected return hasn’t changed. It’s still .035, but the Sharpe ratio has drastically changed. In fact, you could see that. What are the chances of ending up with 101 dollars now, if you put half your money in each, the worst case? It’s not 1 in 6. You have to get the worst case in both dice. So to get 101 dollars, it’s 1 in 36. It’s vastly less likely that you end up with this extreme bad outcome once you put half the money in each die. So if you compute the standard deviation, it’s going to be a lot smaller number. It’s going to be .012. So the Sharpe ratio’s gone from .3 to .4, just by finding the second die. So what you should be doing is looking, like Swensen does, like everybody does, for those independent risks that the rest of the market is too stupid to see. Even after they got the idea of the index, for 20 years, they didn’t realize the index should mean everything, not just in America. So you’d have a huge opportunity to do better than the rest of the managers. While everyone else is screwing around on this line, you can be screwing around on that line. So you could run the same risk this guy did. So Swensen says, you could leverage–not that he does, but if you’re Harvard and you’re on this thing instead of this thing, you could leverage and go up here and say, “Look, I’ve got the same standard deviation everybody else does, and look at my returns. They’re so much higher than everybody else’s.” So that’s what Harvard is trying to do. They’re trying to globally diversify, cut down their standard deviation and then leverage, and compared to everybody else, get a higher return with the same standard deviation. So that’s the first implication of the theory. Now the second implication is the best part. Let’s hope I get to it. I’m going to skip over leverage. The second story is coming right up. ## Chapter 6. Price Dependence on Covariance, Not Variance [01:04:19]How are these things priced? I have to get to the second part of Shakespeare. What can we say about the pricing of individual stocks? It looks like an individual stock is going to be worth more if it contributes no variance, less variance. But that turns out to be completely wrong. It’s shocking that that’s wrong. So why is this? The crucial thing is, remember back to our first principles. What is price = to? What is the key ingredient of price? What is it always = to in basic economics?
Everybody’s holding a market, maybe with some riskless asset. So they’re holding the market, plus maybe they’re long or short the bonds, but that’s not creating any covariance. So if I’m holding the market and I hold a little bit more of a stock, so if I could pay a little bit more and buy t units of stock X, what good would it do me? Well, the expectation, the change in expectation, with t dollars of X would be, as I increase t a little bit, the derivative of my expectation is just the expectation of X. So by holding the new stock, I get expectation of–I get a little more. I get t times the expectation of X, I hope. By coming up with t extra dollars, I could increase my expectation by t times the expectation of X obviously. But what would happen to my variance? If I look at the variance of this, what happens to my variance? Which is = (for variance I could write) is the covariance of this with itself, tX + M. So everybody’s supposed to hold M + some riskless thing, negative or positive in the bond. But the negative or positive in the bond is riskless asset. That contributes nothing to the covariance. I might as well just look at the covariance of a little bit of money in X with the market. And what’s that covariance going to be? Well, it’s covariance of t, tX and, tX, covariance of tX and M (these are definitely not 0 anymore) + covariance of–so it’s 2, 2t, covariance of tX and M. And then it’s the, + covariance of M with itself. So it’s the covariance of tX with itself, tX with tX, twice the covariance of tX with M and then this tX with M, 2 covariance tX with M, and the covariance of M with itself. Now I want to differentiate that thing with respect to t. If I put just a little bit of money, that’s being on the margin, what’s the marginal effect on my variance? I hate variance, but see the mistake people have made in the past is they said, “People hate variance. Therefore a stock that has a lot of variance is bad.” Absolutely wrong reasoning. That was Adam Smith’s puzzle about water and diamonds. You know, why is water less expensive than diamonds when water’s so much more important? It’s not how important the whole thing is; it’s how important one tiny extra drop of it is, the marginal utility. So if you add a tiny bit of X to what you’re already holding, which might involve a bunch of X already, so you add a tiny bit more of X to what you’re already holding, what’s going to be the change in your variance? So it’s the derivative of this with respect to t, when t = 0. So this, by the way, = t squared, variance of X + 2 covariance–2t times the covariance of X and M + the variance of M. I’m differentiating this with respect to t. Right? Because I’ve just rewritten the same thing. That’s just t squared, the variance of X. The t comes out, 2t covariance of X with M, and here’s the variance of M. But if I differentiate with respect to t, I get 2t variance of X (remember, it’s the derivative of t when t is 0), 2t variance of X + 2 covariance of X and M + 0, because the variance of M doesn’t depend on t. So at t = 0, this is also 0. So you see the change in my variance, when I add a little more of X, is twice the covariance of X and M. So what is it that X adds? It adds some expectation and it also adds some variance, but not according to the variance of X, according to the covariance of X with M. When I add some more–the price of water is how much extra utility I get, given what I’ve already got. I’ve already got a huge amount of water. That’s why an extra amount of water’s not doing much for me. That’s why the price is low. So X itself might be very dangerous, but if it’s independent of all the other stuff, M, that you’re holding, in fact, you’re only adding a drop of it, it won’t change your variance hardly at all. So this is 0. It’s the covariance. So that’s the crucial idea, that the marginal contribution of every stock depends on its expectation and its covariance, not its variance. And the expectation is good and the covariance is bad. And all of this is linear, because it’s differentiable, so it just says that–this is the change in covariance. So remember the guy, the marginal utility of expectation. How am I doing in time? 1 more minute. Marginal utility of expectation times the expectation of X, that’s the change, when you add 1 dollar’s worth–a tiny bit of X, your expectation goes up by the expectation of X. So this is the marginal utility of expectation, which is the good thing, - the marginal utility of variance. That’s the bad thing. And what’s the change? The thing that affects–the contribution to variance is the covariance of X with M. So the punch line is, everybody has a linear tradeoff between expectation and covariance. So for any fixed person, it’s just a constant times the expectation of X - a constant times–this doesn’t depend on the portfolio. This is the marginal utility at his consumption. Marginal utility of expectation at his consumption. You face him with any possible new thing that he could buy, just like a consumer in the first day of class. He could buy apples, he could buy oranges, he could buy pears. The marginal utility of each one of those at the point he’s already chosen to consume. This is a number that’s fixed, this is a number that’s fixed. Whatever X I put in here, that’s what I get. So there’s a linear tradeoff between expectation and covariance. Expectation good, covariance bad. So the final picture is that in equilibrium, what you should have is every stock–this is called the security market line. This is the expectation of X. This is the covariance of X with M. So what we’ve found is that it should look like this. There has to be a linear relationship. In order to want the stock–if it’s got a higher covariance, that means it’s adding bad stuff. You wouldn’t want it unless it had a higher expectation. So here’s the riskless asset right here. So this is a different diagram than the Tobin diagram. It’s expectation and covariance. Every stock is along this line. Should be priced along this line. So let me end with a puzzle, the one we started with on the first day of class. If I have two companies, General Motors and AIDS–an anti AIDS company. Some scientist at Yale discovers a cure for AIDS. He calls it AIDS. He discovers a cure for AIDS. If the thing works, he’ll make a fortune. If the thing doesn’t work, of course, it’s going to go bankrupt and pay anyone no money. Let’s say we calculate the expected profits of General Electric. General Motors is too junky now. General Electric. Let’s say the expected profits of General Electric = the expected profits of the anti-AIDS company. Which will sell for a higher price?
All right, you’re anonymous. I didn’t see who it was, so exactly. That’s what you would have thought before this class began. Everybody hates variance. The AIDS company is so risky, you could get a fortune or 0. You couldn’t get riskier than that. General Electric is not going to go bankrupt very easily and it’s not going to suddenly multiply its value by 100 times either. It’s much more solid. So which would you pay more for? Everyone at the beginning of the class, I think, would have said like he did, whoever he is, would have said “General Electric, you’d pay more for.” But the answer’s not General Electric. Why is that? Yes?
It’s General Electric is the one with the higher risk, because it’s correlated with the market. It’s correlated with the market. Obviously if people are rich, they’re going to buy more refrigerators and engines and stuff like that. If business isn’t doing very well, they’re not going to buy that stuff. It’s very correlated with the market. Anti-AIDS, if it cures AIDS, people are going to buy it. If they’ve got AIDS, they’re going to buy it no matter what, otherwise they won’t, so it’s uncorrelated with the market. So therefore the price of the AIDS company is going to be its expected payoff, discounted. The price of General Motors [correction: General Electric] is going to be much less, because it’s going to be punished for having a correlation with the market. Therefore the return on General Electric is going to be much higher, because the same expected payoff and a lower price, your return’s going to be higher in General Electric. That’s the shocking thing that Shakespeare couldn’t have noticed. So yes, we’re just doing what Shakespeare said in the beginning, but in a way he couldn’t possibly have done without any mathematics. Okay, now the problem set, you’ll have to see if you can do the problem set. It’s due on Tuesday. [end of transcript] Back to Top |
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