ECON 251: Financial Theory

Lecture 22

 - Risk Aversion and the Capital Asset Pricing Theorem


Until now we have ignored risk aversion. The Bernoulli brothers were the first to suggest a tractable way of representing risk aversion. They pointed out that an explanation of the St. Petersburg paradox might be that people care about expected utility instead of expected income, where utility is some concave function, such as the logarithm. One of the most famous and important models in financial economics is the Capital Asset Pricing Model, which can be derived from the hypothesis that every agent has a (different) quadratic utility. Much of the modern mutual fund industry is based on the implications of this model. The model describes what happens to prices and asset holdings in general equilibrium when the underlying risks can’t be hedged in the aggregate. It turns out that the tools we developed in the beginning of this course provide an answer to this question.

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Financial Theory

ECON 251 - Lecture 22 - Risk Aversion and the Capital Asset Pricing Theorem

Chapter 1. Risk Aversion [00:00:00]

Professor John Geanakoplos: Okay, but now I want to move to the next topic, which is the topic called the Capital Asset Pricing Model, and it’s in some points the high point of the class. It used to be the high point of finance. The theory hasn’t worked out as well as people thought in recent times, but it’s quite a great achievement and a lot of it was done here at Yale, so I want to explain it to you.

So you see we have a problem so far. If everybody’s trying to hedge that means everybody’s trying to get a completely riskless payoff. It’s impossible because, I mean, there’s real risk in the economy. And what do we mean by real risk? Well, something in one state is just going to be bad for the whole economy compared to another state.

Maybe we’ll run out of oil or something like that. It’s impossible that everybody can consume the same thing in every state, so it’s impossible that everybody can perfectly hedge, but everybody wants to perfectly hedge. So what has to happen? What gives? How does the theory have to change?

Well, the theory’s going to change in a simple way, which Shakespeare himself already knew and already told us about in The Merchant of Venice.

What’s going to happen is everybody is going to try and hedge as much as they can by diversifying, but because there’s some real risk in the economy, in some states things will be in the aggregate worse than they will be in other states. So what’s the consequence of that?

The consequence of that is if you’re going to buy an asset that pays something that’s riskless you’re going to pay the discounted expected return of the asset, but if you’re going to buy an asset that’s risky you’re going to need a higher rate of return so the price will be less than the expected discounted payoff.

So Shakespeare, remember, said exactly that. When the play begins with Antonio looking melancholy his interlocutor says, Salerio or somebody asks him whether he’s worried about his businesses. He says, “No, I’ve got a different ship–every ship’s on a different ocean so I’m diversified. I’m not that worried.” So Shakespeare knows about diversification and that’s what everybody should do, but then when it comes time to pick the caskets to get to marry the beautiful Portia, who by the way is not just beautiful but she’s rich so they’re looking for a prize, but they sign a contract that whoever picks the wrong casket not only doesn’t get Portia, he can never marry anyone in the future. So what’s the purpose of that contract? It’s to make it a very risky gamble. And so why did Shakespeare want to make it a risky gamble, so he could explain he understands risk and return.

So you remember the conversation where everybody says, “Well, I’m not going to pick this unless, you know, it’s only because she’s so rich and so beautiful that I’m willing to do this. The return is so high that it’s worth the risk to me.” So Shakespeare already understood that things that are risky are going to have to be priced less than their expected return–expected payoff–so the expected return, that is the payoff per dollar put into it looks higher to compensate you for the risk. So Shakespeare almost had the whole story.

What’s missing from Shakespeare? Well, what is the definition of risk is missing from Shakespeare, and it will turn out that it’s going to be a very surprising definition. So the purpose of the model I’m going to explain is, how do you measure risk, and how should that affect the price of things, and how does that affect all the analysis we’ve done so far? So that’s the topic of the next couple lectures.

Chapter 2. The Bernoulli Explanation of Risk [00:03:35]

So the first person to confront this problem and propose a solution, a mathematical solution, was the mathematician Bernoulli and his brother. So the Bernoullis were a famous mathematical family, and one of the brothers went off to St. Petersburg where he ended up dying shortly afterwards, but he noticed the following puzzle, and some of you have heard this, it’s called the St. Petersburg Paradox.

So suppose I offer you a bet. I say I’m going to flip a coin, and I keep flipping the coin until it comes up tails, and I count how many coin flips I’ve counted until you get tails, and if that’s N flips you get 2 to the N dollars. So if you flip it 1 time and it comes up tails right away, which is probability 1 half, you get 2 dollars. If I flip it 2 times and it gets heads and then tails, that’s with probability 1 quarter, you’ve flipped it twice you get 2 to the 2 or 4 dollars. If I flip it three times and I get heads, heads, tails the odds of that are 1 half times 1 half times 1 half, which is 1 eighth, but then you’ll get 2 to the 3 dollars. So 4 + 8 + 1 over 2 to the N times 2 to the N + ….

So Bernoulli, the one who died, told his brother Daniel about this and he said, “Well, I’ve offered this bet to a bunch of people and I asked them, how much would they be willing to pay for this risky asset?” I mean, what would you pay? Let’s hear some numbers? How many dollars would you pay if I offered you this bet? I’m just going to keep flipping a fair coin, count the number of flips until a tails and pay you 2 to the N dollars.

So this is the expectation which obviously equals infinity. So according to what we’ve said so far you should pay infinite amount of dollars for it, but Bernoulli couldn’t get anyone to offer him that much money. How much would you offer for this bet? I just want to hear some numbers.

Student: 1 dollar fifty.

Professor John Geanakoplos: 1 dollar fifty. Anybody else have any– it’s pretty conservative, I mean, that’s almost ridiculous, in fact. You’re guaranteed 2 dollars no matter what happens, right? So you’re paying 1 dollar fifty and you’re going to get 2 for sure, so that’s a pretty conservative number to say, anybody a little more venturesome than that? You can’t do worse than 2 dollars in this bet.

Student: 4 dollars.

Professor John Geanakoplos: What?

Student: 4 dollars.

Professor John Geanakoplos: 4 dollars. All right, so Bernoulli asked a bunch of people and the average of what they say happens to have been 4 dollars. That’s what they said on average. And so Bernoulli said, “Well, this is amazing. The expectation is infinite and they’re only willing to pay me a miserable 4 dollars for this.”

Now, the real reason might have been that they didn’t believe Bernoulli was actually going to pay the money, and they’d give up their money and they weren’t going to get anything back, but let’s ignore that temporarily and take it seriously. Bernoulli said, the solution must be that people don’t care about the dollar payoff. They care about the utility of the dollar payoff. So let’s put in a utility function.

So the utility of the dollar payoff would be [one half] U of 2 + 1 quarter U of 4 + 1 eighth U of 8 + 1 over 2 to the N U of 2 to the N + … And so then he said–well, of course why is that going to help? Well, because if the utility function, say, looks like this–so here’s X and here’s U of X–the more dollars you get, maybe it increases utility but by less and less, so you’re not really gaining much by getting these numbers way out here. They’re not adding really much to utility, so you only care about these small numbers. I mean, it’s good to get more, but not much better to have more. So he said, lo and behold, if I put in log natural as my utility function, which this looks like, that’s the graph of log natural, and I put this in.

Now, you see this is easy to solve, to compute, because log of 2 to the N is N times log of 2. So the log of 2s come out and it’s just the sum. It’s log of 2 times the sum of 1 over 2 to the N times N. So it’s N over 2 to the N. So this is equal to log of 2 times the sum, N equals 1 to the infinity, N over 2 to the N.

That’s what it turns out, that this thing is that. So it’s not just 1 over 2 to the N which would have added up to 1, but N over 2 to the N. So the point is because you have the log function here this actually equals the log of 4.

So anyway, I’ve worked out the arithmetic. It’s very simple. You all know how to sum 1 over 2 to the N. You probably don’t know how to sum N over 2 to the N. You never thought of doing it before, but the same trick gets you to be able to sum N over 2 to the N. It’s obviously more than 1 over 2 to the N, in fact, it’s equal to 2. So this sum is equal to 2 and 2 times log of 2 is log of 2 squared which is log of 4.

So by plugging in–instead of caring about expectation you care about the expected utility. You can explain why the average person was willing to pay 4 dollars because the expected log of this is equal to the log of 4. This is equal to the log of 4. So Bernoulli thought he’d brilliantly explained his paradox. So this is the other brother, Daniel.

The dead brother posed the problem, maybe solved it too for all I know, but the other brother who still lived came up with the solution, maybe with his brother, that people don’t look at expected payoffs, they look at expected utility of payoffs, and the utility should have this concave feature that more and more payoff adds on the margin less and less utility. So this function satisfies d squared U (X) / dX squared is less than 0. The second derivative is negative. So the marginal utility is declining as you get more and more. So that was the first advance on how to deal with risk.

Now, actually Bernoulli didn’t really solve the problem because just saying that you replace the payoff with expected utility of a concave function–this log wouldn’t have really solved the problem, because suppose that Bernoulli had offered instead a bet not of 2 to the N but of 2 to the 2 to the N, a much more generous bet? Then even with logs you would have gotten an infinite number. So basically he should have said that people care about a concave function of payoff where the function is bounded unlike log which is not bounded. But anyway, let’s leave that aside. It should be a concave function.

So to put it another way, a concave function has the property, if you look at it, let’s say it looks like that, that if you have this payoff XA and this payoff XB and you’ve got–so this is the utility now here, XA, and this is the utility U of XB, and this is the utility U of XA. If you have a 50/50 bet of either getting XA or getting XB you’re going to end up with this expected utility. Your utility is going to be 1 half of U XA + 1 half of U XB.

That’s what we had down here, 1 half of this utility plus half of that utility assuming nothing else can happen. But if you give the person half the amounts of money for sure then he gets this utility which is much bigger than that utility because this is a concave function. The extra you gain by winning the bet, compared to getting the average for sure, the extra you gain doesn’t drive the utility up very much because it’s flattening out.

Whereas losing the bet, even though you’re losing the same number of dollars because from here to here is the same as from here to here, the loss of the same number of dollars is more important to you than the gain of an equal amount of dollars, and that’s why you’d rather get the middle for sure than having a 50/50 chance of going on the extremes.

Chapter 3. Foundations of the Capital Asset Pricing Model [00:12:38]

So Bernoulli pointed the way to the modern theory of risk aversion, which is to just assume–risk aversion in modern economics means people care about expected utility, maybe discounted, expected discounted utility where the utility is concave. So whatever utility function we wrote in here, maybe it shouldn’t be log, it should be something else. How would people evaluate this? They’d evaluate it log of 4. In other words, they’d say take whatever that constant utility was, which was log of 4, that produces the same utility as the random gamble.

So this random gamble gives this expected utility which is equivalent to having that for sure. So here’s the 4. So 4 for sure gives a utility, log of 4, that puts you here which is the same as the expected utility of getting the random gamble. That’s the modern theory of risk aversion, and it explains why people would rather have things for sure, but it’s now quantifiable because if you can’t have something for sure then you know that it’s more dangerous, but with this concrete utility function you can find out exactly how much you’re willing to pay to transform this risky gamble into a safe gamble. You’d give up this much expectation in order to get the payoff for sure. So we’re going to turn a vague theory into something quantifiable and get a surprising conclusion.

So that’s step one. We now think about people maximizing utility. Well, of course we thought about that from the beginning. The very first class you had utility and diminishing marginal utility. So actually this risk aversion with diminishing marginal utility, fortunately for us, is exactly the same thing we’ve been thinking about all along anyway, diminishing marginal utility for consumption. So the very assumption of diminishing marginal utility that we made from the beginning is also explaining risk aversion. So it’s incredibly fortunate that we don’t actually have to change any of our mathematics and we’ve explained a new phenomenon.

Now, the most simple utility function is either the log one or the quadratic. So remember, U (X) = a + b X - c X squared. Adding a constant is never going to change anything, so I’m always going to write this as a X - 1 half alpha X squared. That’s going to be my utility function, my quadratic utility.

It could be like this or it could be like that. If you add a constant which doesn’t depend on X that’s not changing what anybody does so that’s irrelevant, and if I divide it by a constant like B that’s not going to change what everybody does so I might as well assume the quadratic utility is X - 1 half alpha X squared, quadratic utility. So that’s about as simple as we can get and we’re used to working with those kinds of utility functions.

Now, why is that such a good convenient thing for us to use? It’s because let’s suppose now that you’ve got this random payoff where with probability gamma1 you’re going to get X1, probability gamma2 you’re going to get X2, probability gammaS you’re going to get gammaS [correction: XS]. So what’s the expected utility, the analog of Bernoulli?

That means U is going to equal summation, s = 1 to S of (gammas Xs- 1 half alpha Xs squared). So that’s all we’re doing. We’re just saying that people don’t care about the payoffs. They have to evaluate getting X1, X2 or XS. They’re going to multiply the payoff by the expectation but not look at the payoff itself, look at the utility of the payoff. Now, quadratic is very simple and the reason why we’re going to get such a beautiful theory out of it is because this number you don’t have to keep track of all the X’s to express this utility. We’re going to be able to summarize it incredibly simply.

This is going to equal some function F of the expectation of X and the variance of X. So all we’re going to have to worry about is the expectation of X and the variance of X, and so many, many very complicated things we can think about very simply. So more generally if you put the log instead of the quadratic utility we couldn’t get this simplification and so the theory would have to be more complicated. So the beautiful theory, the Capital Asset Pricing Model, comes out of using this simple quadratic utility. So why does it get so simplified?

Well, if I just write it out, U is going to equal the summation of gammas Xs, so this is s = 1 to S (I’ve let my probabilities be gamma, I don’t know why I chose that) - 1 half alpha, summation s = 1 to S, of gammas Xs squared. Well, here we have the expectation of X already.

Now, what is this 1 F alpha gammas Xs squared? Well, if I wrote Xs - the expectation of X and I summed this squared that’s equal the variance of X–oops, times gammas. That by definition is the variance, but if I wrote this out what would I get? I’d get summation s = 1 to S, gammas Xs squared which is what I have over there and then I’d have–well, what would I have– minus 2 times summation gammas Xs expectation of X + summation s = 1 to S of gammas expectation of X squared.

But what’s this? The second term minus 2 times that, the expectation of X is a constant so I can take that out, minus 2 expectation of X, can take this out and notice that summation gammas Xs, that’s the expectation of X as well. So that’s just minus 2 times the expectation of X squared. And so this I can take the expectation of X squared out and the summation of the probabilities is 1. So therefore I just get equal to summation s = 1 to S, gammas Xs squared - (expectation of X) squared.

So therefore, this up here is equal to the expectation of X. So what have I got here? I’ve got this term. So I’ve got the variance of X equals this summation Xs Xs squared - (expectation of X) squared. So this term equals the variance of X + (expectation of X) squared. So therefore I’ve got this minus 1 half alpha (expectation of X) squared - 1 half alpha variance of X.

That’s what the algebra gives me, so why is that again? Because given quadratic utility up here, that thing–getting old–given the quadratic utility up there I can write it as this in this term. This term is obviously the expectation of X, but this term is just the variance of X plus the expectation of X squared. So when I subtract it I keep the expectation of X minus the (expectation of X) squared.

That’s the first term, and then I’ve got minus the variance of X from that term times the 1 half alpha. So you see that depends on the expectation of X in a positive way, assuming alpha’s a small number, and in a negative way on the variance of X. So, just as I said, somewhere, I said it was going to turn out like that, and it did right here. The utility is equal to the expectation of X and the variance of X in a positive way on the expectation of X and a negative way on the variance.

Chapter 4. Accounting for Risk in Prices and Asset Holdings in General Equilibrium [00:22:15]

So now we’re ready to start the analysis.

So far we’ve assumed people only care about the expectation and then we said, well, we know they don’t only care about the expectations. Hedge funds and everybody else, if they know what they’re doing and they’re trying to keep their investors happy they’re going to hedge.

We didn’t say why they’re going to hedge. We just asserted they like to hedge so their investors don’t get mad at them, but really what we had in mind is the investors have some utility function. They don’t like risk so the hedge fund is going to try and keep the payoffs steady. But there’s a tradeoff. You can’t eliminate all risk. So, how much is the hedge fund and the investor going to suffer if all risk isn’t eliminated? Now we have a way of quantifying it.

People care about the utility and not about just the expected payoff and so you add more risk to them–you replace a sure thing with a risky thing with the same expectation–they think it is worse, that much worse. And so we’ve said that of all the myriad of utility functions–we could use log, some exponential e to the minus aX, X to the r, there are lots of different utilities we could use–we’re going to deal with the quadratic because it has the simple property that in evaluating an entire risky proposition people care about the expectation, which is what they cared about before, but they’re punishing themselves for getting a bad variance. So because that’s such a simple thing to say we’re going to get a simple conclusion and a very surprising conclusion.

So let’s now analyze a problem and see what would happen. So the problem I’m going to choose to analyze is this one. I’m going to say that three things can happen in the economy. Anyway, those are the probabilities.

Now, there are many firms in the economy A, B and C, and let’s say the first firm, I don’t want to invent the numbers here so I might as well just write down the ones I picked. The first one’s going to be 50, 100 and 75. B, the other firm’s going to be 150, 180 and 365, and C is going to be 300, 220 and 60.

So those are the three things that can happen in the payoff of the three firms. Let’s say there are two agents, agent alpha owns A and also 133.5 units of X0. And beta owns B & C and–I may have reversed these two guys–66.5 units of X0. So here we go. So by alpha and beta, I mean, there are a million alpha agents and a million beta agents so everything could be scaled by a million because I want a big economy like we always have. So there we have it. We’ve got now a risky world. Things can happen. So what are the utilities?

Let’s say utility now of alpha is going to equal–sorry, I left out the main point. Utility of alpha is going to equal 1 half X0–I just made up these numbers, by the way, they’re not–so summation s = 1 to 3, gammas, that’s the gammas up there, times Xs - 1 over 400 Xs squared, the states. So there’s s1. So here are the states. This is state 1, s = 1, s = 2 and s = 3. So those are the three possible states just like we had before with this payoff and those are the payoff of all the assets.

Alpha owns firm A which is producing that output in the three states, and also owns 133.5 units of X0. So over here alpha’s owning 133.5 and beta’s owning 65.5 of consumption at time 0. So this is time 0. This is time 1, the end of the year. So by the end of the year something is going to happen. There’s a lot of uncertainty between now and then. Some of the firms are going to be paying off in some of the states and badly in other states and so on.

And so the utility function for alpha, so he cares about consumption at time 0 and also in each of the three states, but now he’s going to have these quadratic utilities. He’s going to say to himself, if I just hung onto my A in state 1 this would be–if I never traded, I just hung onto A the utility function would be this quadratic thing of 133.5. So it would be 133.5 - 1 over 400, 133.5 squared, plus he would end up with (50 - 1 over 400 times 50 squared) times 1 quarter + (100 - 1 over 400 100 squared) times 1 quarter + (75 - 1 over 400 75 squared) times 1 half. That would be his utility if he never traded. If he just stuck to A he’d eat his own endowment 133.5 at time 0.

Oh, that isn’t true. I wrote down the wrong utility at time 0. I said his time 0 utility is 1 half X0. So it’s 1 half 133.5. So we get 1 half 133.5, but in the future he’d get 50 in state 1, 100 in state 2, and 75 in state 3, and that’s the utility he’d end up with.

But that’s not very good for him because he’s running this gigantic risk. He’s got this risk at time, you know–state 1 is a disaster for him if he just sticks to that, so he doesn’t want to stick to that. So how should he evaluate the shares of firm A? How should he evaluate the shares of firm B which he could get if he gave up some of A, or how should he evaluate the shares of firm C? What should he do?

So beta has a similar utility. Beta’s utility, Ubeta is going to equal 3 quarters X0 + the summation, s = 1 to 3, (gammastimes Xs - 1 over 800 Xs squared). So I’ve made these guys–far from being impatient they seem to prefer consuming in the future until now. That was a poor choice of numbers. This number should be bigger than 1 and this should be bigger than 1, but anyway I put 1 half and 3 quarters. So there’s impatience built in except it goes the wrong way. That was just poor choice of numbers, but the rest of it expresses their risk aversion.

So alpha is looking at the expected payoff of what he gets to consume in the future, but he’s punishing himself by the variance. So you look at this formula you see it’s not just the expectation, but he loses something because of the variance.

And similarly, beta, he’s looking at consumption today, he’s adding to that the expectation of his consumption tomorrow for his utility, but he’s punishing himself for having variance in the future. So it’s exactly what we formalized, Shakespeare’s idea of people not liking to be exposed to variance, to uncertainty, which we’ve quantified by calling variance. Is everyone with me now? Yes? Good, I’m glad you have a question.

Student: I don’t understand how you got 1 over 400 and 1 over 800.

Professor John Geanakoplos: I just made up those numbers. That’s the utility of alpha and that’s the utility of beta. I could pick any people I wanted. I just picked those two people. Now, how do they differ? Which person is more afraid of risk than the other? Is alpha or beta more afraid of risk?

Student: Alpha.

Professor John Geanakoplos: Alpha is more afraid of risk, right? This 1 over 800 is smaller than 1 over 400, so beta doesn’t really care that much about risk, well cares, but is not going to punish himself too much by being exposed to risk. Alpha is not going to punish himself too much, but is going to punish himself somewhat more. So alpha is more risk averse.

Alpha is more afraid of risk, it seems. So I’ve taken two agents who are afraid of risk, one’s more afraid than the other, and I’ve put them in an economy where there are risky things that could happen. And so we now want to work out a more sophisticated version of pricing and of equilibrium than we had before.

So let me remind you that what we sort of have been supposing up until now is that the price–what would the price of A be if we didn’t think about risk aversion? So far what we would say–what would you say the price of A is, price of firm A? If we were naïve you might say it is 1 quarter–by the way, I hope I have those probabilities right–you’d say it is 1 quarter times 50 + 1 quarter times 100 + 1 half times 75. Is that what we would have said up until now? Even up until now we would have been more sophisticated than that.

Student: Discounted.

Professor John Geanakoplos: Discounted, times discounted. That’s what we sort of figured up until now. That’s the logical thing to do. Well, but we ignored risk aversion, and we ignored it at our peril because it’s obviously important.

I mean, Shakespeare, a literary person, he understood already 400 years ago that risk aversion was important, and there are facts that confirm what Shakespeare’s intuition is. The stock market historically has had a lot higher return than the bond market. Even with the last stock market crash, of course it came back a lot, averaged since 1926 the stock market’s made something like 9 percent a year compared to 2 and 1 half percent in the bond market. So there’s a huge disparity and after over such a long period of time it can’t just be it was luckier every year after year after year.

Somehow people must have realized the stock market is riskier, and so as Shakespeare said they wanted a higher return meaning they were paying a lower price, but how much lower? How can you figure out how much lower? So in this example, in other words, what is the price of A? So this is the wrong price of A, apparently, because it doesn’t recognize risk aversion. So that’s where we are. So any questions about what the question is?

We’re about to give an answer. So you see what the question is, that our old methodology for figuring out prices–that’s taking expectation and discounting–obviously can’t be right because it doesn’t recognize risk aversion. On the other hand, we always had a utility function in there from the beginning, even a quadratic one, so all we have to do is do what we did before and put in a quadratic utility and we’ll probably get the right answer. So that’s exactly what we’re going to do.

So Arrow, in 1951, this is the same guy who proved with Debreu the Pareto efficiency of equilibrium. He was my thesis advisor. He said we can do the same trick that Fisher did, only for some reason he never credited Fisher. I could never quite figure that out. He had some obscure Danish guy he credited.

But anyway, apply Fisher trick and assume firm dividends are part of endowments. Look for GE, the general equilibrium, trading outputs, trading goods, then go back to deduce value of firms.

Now, what goods are we trading here? That was a conceptual advance. We call them Arrow and then Debreu got involved too. Arrow-Debreu, so Debreu was the Yale Assistant Professor while Arrow was a Stanford Assistant Professor, so Arrow-Debreu State Contingent Commodities. So, just as Fisher said, an apple today and an apple next year, even though they’re identical apples, are different commodities with different prices because they come at different places in time. In fact, most people would prefer the apple today to the apple next year.

So Arrow said an apple in the top state is a different commodity from the same apple in the second state, so it should have a different price. So we’ve got just our conventional equilibrium according to Arrow where as long as you have these Arrow state contingent commodities that you can trade–trading today, you can imagine today buying an apple if state 1 occurs but not having to get the apple if state 2 or state 3 occurs, and that’ll have a price P1.

And today you could imagine buying the apple if state 2 occurs, a different price from the apple if state 1 occurs, and also an apple if state 3 occurs, which obviously is going to be more expensive, or it looks like it’ll be more expensive than the other apples because it’s 50 percent likely to happen, and those are the prices we have to look for. And that’s going to solve our problem because the prices of the Arrow securities are going to be different, maybe, from the probabilities and that’s what will reflect the fact that when everybody’s trying to hedge and not everybody can do it you’re going to have to change the tradeoffs.

So we’ve already seen this in our gambling thing, at least the prices. Remember with our bookies the bookies were effectively willing–remember there were two outcomes, the Yankees win or the Phillies win. You could get the bookie who thought the odds were 60/40, by paying 60 cents today the bookie was going to give you 1 dollar if the Yankees won, or paying 40 cents today the bookie will give you 1 dollar if the Phillies won.

So we’ve already had these Arrow contracts, these Arrow securities implicitly in our equilibrium. And those 60/40 odds those were the opinions of the bookie, maybe not the actual probabilities. We said the final betting odds depended on what the other bookies were willing to give. It didn’t have to correspond to reality. There might not be a reality even. So here there’s a reality, 25, 25, 50, but that doesn’t mean that the odds, the prices they’re going to quote in the market are going to turn out to be that. We have to solve for equilibrium and see what they are. So what’s going to happen?

Well, we can solve for equilibrium very easily because we’ve done this a million times before. And I’ve chosen linear quadratic utilities, the kind we did on the very first day of class, because those are easiest to solve for equilibrium. You don’t have to get involved in the budget set or anything complicated. You just set marginal utility equal to price.

So we know for alpha, sorry, we know that the marginal utility of alpha at time 0 divided by the price 0 is going to have to equal the marginal utility of alpha at each state s times the price Ps. So the equilibrium, the Arrow-Debreu equilibrium, is going to involve P0, P1, P2 and P3, the prices of the Arrow securities, the present value prices.

P0 is what you pay today to get the apple today. P1 is what you pay today to get the apple a year from now in state 1. So these are the present value (that’s what Fisher would say) state contingent prices. The state contingent is what Arrow added.

Now, you may ask whether there really are these Arrow securities floating in the economy, and we’re going to come back to that question, but you could imagine all these Arrow-Debreu state contingent prices and commodities, and those would be the prices we’d solve for equilibrium. So we get this over this, marginal utility of that. So what is this? And similarly for beta, marginal utilitybeta at 0 over the price of 0 equals marginal utilitybetas over the price of s. So what is this?

For alpha, his marginal utility of consumption is 1 half. We might as well assume one of the prices is 1. Let’s take this price to be 1. So beta, her marginal utility is 3 quarters and the price is 1. What’s his marginal utility in any state s? It is gammas times (1 - 1 over 200 times Xs). So I just differentiated this. I got 1 - 2 over 400 times Xs. And what’s her marginal utility? It is gammas in state s times (1 - 1 over 400 times Xs). So I know in equilibrium that’s going to imply that 1 half–well, now I have to screw around here, so how am I going to–so I’ve got this thing over here 1 half equals this thing over here. What?

Student: Over Ps.

Professor John Geanakoplos: Over Ps. Ah, glad that appeared. I was getting worried there. Thank you. Over Ps, that helps a lot. So that implies that something like X–so this is what alpha is going to do and this is what beta is going to do. So this implies Xalphas equals what? So if I multiply through by 200, and I bring Ps over gammas to the other side, and I do a bunch of stuff, I’m going to guess this is 200 - 100 Ps over gammas. How do you think that’s going to play in Peoria? Let’s see.

If I multiply through by Ps over gammas I get Ps over gammas times 1 half. Then I multiply everything through by 200. So I get 100 Ps over gammas, and then I get the 200 here, and the Xs goes to the other side, and the Ps over gammas goes to the other side. So it’s 200 - 100 Ps over gammas. And this one is going to be–Xbetas is going to equal–well, I have to do the same trick here except I’m going to be multiplying through by 400 and taking 3 quarters which is 300.

So it’ll be 300 minus–no, that was wrong, 400 - 300 Ps over gammas. Because if I multiply through by 400, put Ps over gammas on the other side I have 3 quarters Ps over gammas times 400, which is 300 Ps over gammas. This becomes a 400 and the Ps gammas went away so I have that.

So I know now if I could figure out what the prices are I know what everybody would demand in every state. So let me pause here. That was the first critical step. So what did I do? I said it’s a long story. A lot of years went into this.

I said people are risk averse. Shakespeare knew that. We want to quantify it so we say people have concave utility functions. That quantifies risk aversion. We want to make a simple concave utility function. We pick quadratic, but of course we don’t know what quadratic. Different people could have different quadratic utility functions. Then we do the Fisher trick and say that any equilibrium, as long as you can buy and sell every contingent commodity in the future, because all the Arrow securities are there, it can always be reduced to general equilibrium just like we did before.

And so now you have to feed the endowments into the agents’–I mean the payoffs and the dividends into the agents’ endowments. So we haven’t done that yet. And then we solve for supply equals demand. So all we have to do is we have to have Xalphas + Xbetas has to equal the endowment of alpha in s plus the endowment of beta in s.

All right, so we have to do that for every s. So this is 200 - 100 PsS over gammas equals–now we have to do it in state 1. So it equals whatever they are. So what is endowment of alpha of s plus endowment beta of s? We have to look at each state separately. And lo and behold I picked the numbers so that if you add these all together you get 500, and here you get 280 and 220 is also 500, and here you get 500 again.

So lo and behold there is no aggregate risk in the economy, although the individual stocks are risky the aggregate is totally un-risky. So no matter what s is, I could put in 500 here. It’s going to turn out that the total endowment of both people, because I’ve plugged the dividends into their personal endowments, added up the two people, it’s 500. So it means that Ps over gammas equals…

Student: <> you forgot.

Professor John Geanakoplos: What?

Student: You forgot the second term.

Professor John Geanakoplos: I’ve forgotten something for sure, what?

Student: Xbeta.

Professor John Geanakoplos: Oh, Xbeta. So that’s alpha. Thank you. Plus 400 - 300 Ps over gammas = 500. So if when I add this up I get 600 - 400 Ps over gammas = 500, so then I flip them to the other side and I get Ps over gammas = 1 quarter, because 500 from here is 100 and put the 400 on the other side and divide by it I get Ps over gammas equals 1 quarter for all s. So what did I find? So it’s the same. Ps over gammas is the same, same in all states. So what would the price of A be here? What’s the price of A in equilibrium? What’s the price of A?

I’m going to take the price of A should be P1 times 50 + P2 times 100 + P3 times 75, and what does that equal? Well, P1is just 1 quarter times gamma1, right? P1 over gamma1 is 1 quarter. P2 is 1 quarter times gamma2, and P3 is 1 quarter times gamma3, so I just got this multiplied by 1 quarter. So in fact all I did is I did what I had always done. I took the expected payoff and discounted it. The discount rate is 1 quarter. What’s the price of the riskless asset, pi of (1, 1, 1), is just going to be 1 quarter, because it’s 1 quarter times gamma 1 + 1 quarter times gamma 2 + 1 quarter times gamma 3, gamma 1 + gamma 2 + gamma 3 is 1 so it’s 1 quarter. So it implies the riskless interest rate is what? What’s the riskless interest rate?

Student: 300 percent.

Professor John Geanakoplos: 300 percent. So we’re discounting by 1 over 1 quarter because the interest rate is 300 percent. So basically nothing happened. We got all the prices exactly as we would compute them without, you know, just doing expectations we got the right discount rate. All we had to do was figure out the discount rate. So risk hasn’t played any role. And why didn’t it play any role?

Because although alpha started off owning A alone which exposed her–forgot who was her and who was him, let’s say her–exposed her to a lot of risk. She’s not going to sit there stupidly just holding A. She’s going to trade it for B and C for different shares. In fact she’s going to end up holding her consumption, this is her consumption, 200 - 100 Ps over gammas, this number doesn’t depend on s. She’s going to consume the same thing in every state, and how can she do that?

She can own equal shares of A, B and C. She’ll own a share of the whole economy. So in other words, by diversifying alpha and beta each get rid of all risk. So instead of calling it diversifying I could call it hedging, the same thing. She doesn’t just hold her A. She mixes B and C with it so that she gets a payoff of consumption that’s exactly the same in every state because Ps over gammas is independent of the state.

She’ll always consume the same thing. Everybody can hedge perfectly and there’s no problem because there’s no aggregate risk that anyone has to be stuck with, and therefore the price is just going to be the same as the probabilities discounted. And that’s the theory we’ve worked with so far.

So, so far you could say that everything we did was kosher it’s just that when we had these two different probabilities of things happening up or down we thought that the aggregate economy would have the same endowments here as it did there, and therefore the probabilities we used were the objective probabilities discounted. No reason to change them because nobody’s going to be forced not to hedge. Everybody’ll hedge.

So are there any questions about what I’ve said? I’m sure there should be a question because I can’t have said it as clearly as I ought to have. So would somebody like to say something? Yes?

Student: <> the old price that we found when we hadn’t done this, but that also change the new one?

Professor John Geanakoplos: This is the new price with the 1 quarter. This is the correct new price. So the theory so far hasn’t changed in any interesting way. We just found the discount rate. It just looks like expected utility, but you shouldn’t have expected it to change because the aggregate endowment was 500, the constant in every state.

There’s no reason why we can’t have everybody perfectly hedged and consuming a constant in every state, and in fact that’s what we did have, everybody–she consumed the same thing in every state. He consumed the same thing in every state. No reason why they both couldn’t hedge themselves perfectly and in equilibrium that’s exactly what they did. Any other… Yes?

Student: If the total endowments in every state hadn’t all added up to 500 would you create an expected endowment or would you just not do the problem?

Professor John Geanakoplos: So the next step is going to be–what I’m going to do now is I’m going to assume that the endowments don’t add up to a constant in every state. Then what’s going to happen?

So this is not at all obvious how to solve this and what to do, but it’s going to turn out to have a beautiful simple answer, shocking, not only be simple but also surprising. So before I do that I’m going to change the endowments so they’re not all a constant. Any questions about where we’re going? Yeah?

Student: Could you just repeat what you said about hedging <>?

Professor John Geanakoplos: Yes. Thank you for the question. So I went a little quickly. I said that what we proved by solving for the general equilibrium is that the price in every state was just going to be 1 quarter times the probability.

Chapter 5. Implications of Risk in Hedging [00:54:11]

That’s what we showed had to happen in equilibrium. Now, what’s the consequence of that? The consequences are twofold. Number one, the price of all the assets is the same expectation we naïvely would have taken before where we used the discount rate 1 quarter. That’s the first implication.

The second implication is that from the formula for consumption we noticed that she consumes the same amount in all three states because Ps over gammas is 1 quarter in all three states. Her consumption is going to be the same in all three states, and his consumption, which will be different from hers, but his will be the same in all three states as well. The two will add up to 500.

So then I took a little bit of a leap and I interpreted that conclusion that her consumption doesn’t depend on the sate. What’s the interpretation of that? She has obviously, somewhere behind the scenes, given up some of her A to get B and C and held them in a mixture so as to get the same consumption in every state. What must the mixture be? Obviously she holds the same proportion of A, B and C because those add up to 500, 500, 500. So she must have held the same proportion of A, B and C, a fraction of the market and got a riskless payoff.

So she diversified. She didn’t just stick with her A. She substituted a little bit of A, a little bit of B and a little bit of C, a different boat on every ocean, and now she runs no risk at all. So she diversified, but in the language we used last time I could call diversification hedging if I wanted to. She just, sort of, sold Arrow securities in the right proportions to turn her A into something that was completely riskless. So whether you call it diversifying or call it hedging she’s achieved the same end of totally balancing her consumption. He did the same thing and they both could do it because the aggregate consumption was a constant. Yes?

Student: What would an Arrow security actually look like?

Professor John Geanakoplos: In real life?

Student: Yeah.

Professor John Geanakoplos: The closest we’ve come to an Arrow security in real life is a CDS, and this is part of the reason why these economists, Larry Summers, my classmate, and Rubin who was the Secretary of the Treasury, and ran Citi Corp, and who was a Yale law school student and a Harvard undergraduate, and who I’ve sat on many committees with, they were seduced by the–so what’s a CDS?

A CDS pays 1 dollar if some bond defaults by 1 dollar. So that isn’t an Arrow security because an Arrow security is a much more detailed thing. An Arrow security says I’ll pay 1 dollar in state one.

An Arrow security says you get an apple in state 1, but state 1, remember, is not described by a single firm, state 1, the states of nature are total descriptions of everything that could happen in the economy. So an Arrow security really says if it stops snowing in Siberia, if Khomeini loses power in Iran, if there’s a favorable election outcome in Afghanistan, and if Obama wins reelection, and if the U.S. solves the energy problem then I’ll give you 1 dollar. So the Arrow security lists an incredible number of contingent things, every contingency possible and says in that case I’ll give you 1 dollar.

A CDS says if this thing happens I’ll give you 1 dollar whether or not Obama wins election, whether or not America discovers a new source of energy, whether or not Afghanistan turns around. Just so long as the bond defaults I’ll give you 1 dollar. So the CDS is an event contingent security. That’s the CDS, and an Arrow security is a much more finely specified thing. It’s a state contingent–you say everything that happens in the economy, so we’ll never get to Arrow securities, but CDS looks like we’re on the way to them.

And these guys blundered by thinking since CDSs are on the way to Arrow securities we should have as many CDSs and let people trade as much of them as we can, but we’re going to get to that in the last lecture about how all this theory, what’s wrong with all the theory.

So, any other questions before we–so let’s now make the change that he suggested up there. Let’s now change the economy just a little bit. Let’s eliminate C. So this just disappears. So obviously now the total endowment is very contingent. It’s 200, it’s 280 and it’s 440. Now what do we do? So beta owns B. So now what’s equilibrium going to be? What do you think is going to happen?

We want to quantify this. We want to give a beautiful simple theory that’s quantifiable, but what do you anticipate happening to P1, P2 and P3? So everybody’s going to say, alpha, she’s going to say, look, my A is risky. I don’t want to hold my risky thing. I want to start hedging and trading these Arrow securities so I get the same constant in every state. Of course beta who owns B, he’s going to do the same thing. So they’re both going to be trying to trade Arrow securities. What’s going to happen, do you think? Yes?

Student: Aren’t they both just going to be exposed to whatever the total risk of the economy is in <>?

Professor John Geanakoplos: Yeah, there’s no way that they can, exactly, there’s no way that each of them can be perfectly hedged. So no matter what they do, they’re going to be exposed to more risk in state, you know, state 3 is going to be a great state. State 1 is going to be a terrible state, so what do you think that means about the prices?

Everybody can’t be hedged, and so in fact what’ll happen is nobody will be hedged. Although, alpha will be, who hates risk more than beta, will be closer to hedged than beta will be. So beta will end up bearing more of the risk than alpha. And what do you think will happen to the prices of the Arrow securities relative to the probabilities? Yes?

Student: It won’t be constant.

Professor John Geanakoplos: There won’t be a constant ratio of 1 quarter as we had before, but can you be more specific?

Student: The price of the securities for state 1 will be greater relative to the probability than the price of the security in state 3.

Professor John Geanakoplos: Exactly. So that’s what’s going to turn out.

The world is short of commodities in state 1, there just aren’t many apples. That’s the disaster. That’s when we can’t solve the energy crisis. We’re totally screwed. Everybody wants to consume more in that state. Everyone’s going to try and hedge against that state. They’re all going to be trying to buy Arrow securities in that state, which means that because there aren’t as many to buy, there’s just not enough apples to go around, the price of Arrow securities in state 1 is going to be high relative to state 3. There’s plenty to go around there.

So she is going to sell some of her A and get some B to diversify, but B’s got so good in state 3 that all of a sudden she’s not going to be so worried about state 3 anymore, but state 1 she’s still going to be worried about, and there’s nothing to be done about that. So the price is going to have to be very expensive in state 1.

So all right, that’s all blah, blah, blah. Let’s solve for equilibrium and see what happens. We can solve immediately. Nothing’s changed. The utility functions are the same. None of this changed, so this board doesn’t change at all. That’s demand. Still depends on P0, P1, P2 and P3, but now we have to be a little bit more careful in state 1.

So demand in every state is 600 - 400 P1 over gamma1 equals endowment of alpha + endowment of beta. So in state 1, I’m going to now change this to a 1 although with my handwriting it looked like a 1 anyway, what’s the aggregate endowment in state 1? The aggregate endowment in state 1 is 200. This is a 1 now. That’s 200, so that means P1 over gamma1 = 1, right? Because 400 and 400 so it’s 1. So you’re not discounting the first state at all. You’re looking at the probability of it.

But what if I go to P2 over gamma2? Well, the demand is going to be the same. It’s the price that’s going to change to make up for the fact that the supply is much different, namely, namely what, 280. So now if I subtract I get 400, 280 minus that is 320 divided by 400 which looks like 4 fifths, maybe. 320 over 400 is 4 fifths, right? Because 320 divided by 400 is 4 fifths, so P2 over gamma2 is 4 fifths. So they’re not proportional anymore. And then P3 over gamma3 equals–now the outcome is 440, so if I subtract 440 from this I get 160 divided by 400, what’s that?

Student: 2 fifths.

Professor John Geanakoplos: 2 fifths, thank you. P3 over gamma3 = 2 fifths. So the prices turned out to be quite different. Now, the reason why they’re slightly higher on average, of course, than they were before is because there’s less consumption in the future. We’ve suddenly made our future much worse off. So people are more desperate to consume in the future, so that means the prices of future consumption are going to be higher.

So we have two effects here. These prices instead of being 1 quarter everywhere are higher, much higher than 1 quarter because the future looks so much worse. The interest rate is going to go down. It’s not going to be 300 percent anymore.

But more interesting is that the prices are no longer proportional to the probabilities. Just as he said over there the price in state 1 is going to be much higher relative to the probability, namely 100 percent of it, than the price in state 3 which is only 40 percent of it. So that’s the conclusion.

So now what do we do for our price? What’s the price of A? What’s the price of A? What do I plug in here? That, so that equals 1 quarter times 50 + 4 fifths times 1 quarter–1 fifth times 100 + P3 was 2 fifths times 1 half + 1 fifth times 75 which equals something, 20, 35 and 12 and 1 half, 47 and 1 half.

Student: Why would you <>?

Professor John Geanakoplos: Why did I what?

Student: Why would you <>?

Professor John Geanakoplos: So what is P1? P1 is equal to gamma1 and gamma1 is 1 quarter. So that’s how I got 1 quarter here. So that’s 1 times 1 quarter. P2 was 3 fifths. What was P2? Maybe I did it wrong anyway. P2 was 4 fifths times 1 quarter which is equal to 1 fifth, and P3 was 2 fifths times 1 half which is equal to 1 fifth.

So that’s how I got the prices. So all right, so you see that things changed, and we’ve captured the idea that people can’t hedge fully by making the price of the Arrow security in the state where the economy’s worse off, much smaller than it was before, I mean, much higher than it was relative to the probability than before. So we haven’t gotten close to the punch line, sorry. Yes?

Student: Can you repeat the part where you said stuff about the future looks so much worse they need to increase consumption?

Professor John Geanakoplos: Two things happened to the prices compared to before. One is that we no longer have the prices proportional to the probabilities, right? Their proportion is 1, 4 fifths, 2 fifths instead of the same constant 1 quarter everywhere, and that’s because of the relative scarcity. People are much more worried about the first state than the fourth state and that’s why, relative to the probability, the price is much higher than the third state. You agree with that, right?

Student: Yes.

Professor John Geanakoplos: But there’s a second effect which is that all these numbers, 1, 4 fifths, and 2 fifths they’re bigger than the 1 quarter, 1 quarter, 1 quarter we had before, but that’s obvious. That’s because we wiped out the future.

Half the endowment in the future disappeared, so naturally people are willing to pay more for the future because they’re poorer there. In the first day of class we said that the interest rate, or the third week, the interest rate according to Fisher would go down if you got poorer in the future. So that’s part of the reason that’s happened.

By the way, what is the riskless rate of interest? So P1 + P2 + P3 equals what now? It’s equal to 1 quarter + 1 fifth + 1 fifth, so 1 quarter + 1 fifth + 1 fifth. These are the prices, 1 quarter + 1 fifth + 1 fifth and that’s equal to 20, 10, 14 over 20, so that’s 7 over 10, so therefore the interest rate 1 + r = 10 over 7 so r = 3 sevenths which is like 40 percent.

So the interest rate went from 300 percent to 40 percent, but that’s because we lost all this future consumption. But that’s not what I’m concentrating on. Fisher would have already known that. What I’m concentrating on is the fact that the prices are no longer proportional to the probabilities. You’re discounting every probability, but adjusting the probability because people are much more worried about the first state than the third state.

Student: So people are much more worried about A?

Professor John Geanakoplos: Not A, they’re more worried about the first state. The firms are A and B. The states are 1, 2 and 3, so they’re much more worried about the first state where the payoff is 200, than they are about the third state where the payoff, total dividends in the economy are 440. Are you with me?

Student: Yeah.

Professor John Geanakoplos: Oh boy. That sounded so unconvincing.

Chapter 6. Diversification in Equilibrium and Conclusion [01:09:40]

I want to say the punch line. So I’ve got three more minutes to go. There are two punch lines. I haven’t gotten to the stunning conclusion.

So far I’ve said stuff which Arrow and Debreu had already figured out, but now I want to go to the thing that Tobin and Markowitz figured out, which is one more step we haven’t noticed yet.

Arrow has already figured out that because not everybody could hedge that means that the price of an Arrow security is not exactly equal to the probability. It’s relatively high if the economy’s poor like in state 1 and relatively lower if the economy’s rich like in state 3. That’s common sense. Now, what’s not common sense is the extraordinary conclusion I’m about to show you.

Let’s look at what the consumption is; the final consumption of these two people. So if we look at the final consumption of these two people, what’s her final consumption, so XA. In the three states it’s 200 - 100 times 1 which is 100. What is it in the second state? It’s 200 - 100 times–what was Ps over–times 4 fifths which is, help, 160, maybe. And the last step was 200 - 100 times 2 fifths. 2 fifths is 20 so this is 180. That’s hers.

And his consumption in the future–I’ll put a tilde, I haven’t talked about X0 yet–is 400 - 300 times 1 which equals 100, and here’s it’s 400 - 300 times 4 fifths which is equal to, help! 4 fifths of 300 is 160, and here it’s 400 - 300 times 2 fifths which is equal to 280. Is this right? 100, 160, who told me it was 160? Yes and what’s that? 2 fifths is 120. This is 280.

Student: The number <>, like 200 - 100 times 4 fifths is like 120.

Professor John Geanakoplos: What?

Student: The first <>.

Professor John Geanakoplos: Which mistake is there here?

Student: No, the second <>.

Student: The second 120 <>.

Professor John Geanakoplos: Is this the wrong one?

Student: The wrong one.

Student: 160 should be 120.

Professor John Geanakoplos: Here. 200 - 80 is 120. Thank you. So these are all right now?

Student: 180 should be <>.

Professor John Geanakoplos: 180 should be, okay, this is 40 so this should be 160. Thank you. That’s it, great.

So now what’s so shocking about those numbers? That I finally got them right? Thank you. What’s shocking is this consumption is just the sum of the aggregate endowment–what’s the aggregate endowment? Remember the aggregate endowment is just 200, 280 and 440. So let’s say you take 1 quarter of this.

Let’s take 1 quarter of that. That’s 50, 70–that’s 50, 70 and 110. So 1 quarter of this plus if you add to that 150 you’re going to get all these numbers. So this person, alpha, A, I claim, just holds 50 of the riskless bond, pays 50, 50 plus 50, 70 and 110. No. Is this the right–let’s just check the numbers. Sorry, only one more second.

I should have–so 100, 120 and 160, that’s the right number and that’s equal to 50 of the bond plus 1 quarter of this thing. So 50 + 50 is–1 quarter of this is 50, 70 and 110, right? So if you hold 50 of the bond plus 1 quarter of this you get 100. 50 of the bond plus 70 is 120. 50 of the bond plus 110 is 160.

And this guy is going to hold 3 quarters of the aggregate endowment plus minus 50 of the bond, so 3 quarters of the aggregate endowment, 3 quarters of this thing, 3 quarters of the aggregate endowment is 150 - 50 is 100. 3 quarters of this is 210 - 50 is 160. 3 quarters of that, is 330 - 50 is 280.

So what they’ve done in equilibrium is everybody, despite having a million stocks to choose from and thousands of states and all that stuff, what everybody does is hold the riskless bond, puts money in the bank and holds the whole stock market. So the first theorem we’re going to prove next time is called The Mutual Fund Theorem which is that everybody diversifies by holding the aggregate economy, all stocks in the same proportion, plus money in bank.

So that theorem of Shakespeare of diversifying, what did it amount to do? We have a very concrete thing. You hold 10 percent. This person’s holding 25 percent of every stock in the whole economy plus putting some money, 50 dollars in the bank. The other person is doing 3 quarters of every stock in the whole economy plus lending the money to the first person. So that’s the first of the two amazing results and I’ll start next time by explaining it.

[end of transcript]

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