ECON 251: Financial Theory
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ECON 251 - Lecture 24 - Risk, Return, and Social Security
Chapter 1. Testing the Capital Asset Pricing Model [00:00:00]
Professor John Geanakoplos: Anyhow, so today I’m going to try and wrap up a few loose ends. So I’m going to try and talk about the high water mark of finance which is basically CAPM and Black-Scholes, both of which we’ve almost talked about, almost finished talking about, and I’m going to talk in the middle about Social Security which we never finished, my Social Security plan. So I hope to deal with all three of these things today.
So just to wrap up the CAPM, the CAPM had two main ideas. The first idea was diversification and the second main idea, as we said, was the tradeoff between risk and return.
Shakespeare had both of these ideas. He understood that you’re safer if you’ve got each of your boats on a different ocean, and he understood that if you’re going to take a high risk you’d better get a high return otherwise nobody is willing to take it. And so he made that very clear.
In fact I think the whole point of the play is about economics, The Merchant of Venice, and he made both of these principles very clear, but he couldn’t quantify them and he couldn’t mathematize them, and the Capital Asset Pricing Model gives a mathematical, quantifiable form to both and a quite shocking recommendation in both cases.
So just to draw those two pictures, for those of you who didn’t hear it last night, if you do Tobin’s famous picture, this is the Markowitz-Tobin Capital Market Line they called it. It’s very important to keep straight what’s on the diagram. There’s the standard deviation of any asset per dollar. So here I’m writing the price. So it’s per dollar, and here is the expectation of the asset per dollar. So pi of Y, I’m calling that the price. So this is per dollar.
And I’m not going to go over what we derived before, but we noticed that if you took all the risky assets, any risky asset’s going to have per dollar, if it’s priced, some standard deviation and some expectation. You can write down all the risky assets, and you can combine them, and you’ll get a possibility set that looks like this. So the diversification already shows up in this picture because if you combine this asset and this asset you’re going to get possibilities by putting half of your money in each, not on the straight line between them, but on some curved line that’s much better because you’re reducing your risk.
So Markowitz already had that curve and this picture in mind. Tobin added the riskless asset, which he put here, and then he said, well, if there’s a riskless asset everybody’s going to do the same thing and everyone’s going to choose the same point. It’s better for everybody, so this is the optimal risky portfolio and it has to be equal to the market, because if everybody’s smart enough to figure out what to do they’re all going to do the same thing, and whatever everybody holds that’s by definition the market. So Tobin showed that everybody should choose something on this capital market line and so everybody should split his money between the market and a riskless asset.
Put your money in the bank or own the entire market. That was his recommendation and that was the beginning of the whole idea of index investing. So that was the first idea, first, most important idea.
Now, how could you test this? Well, you could simply test every fund in history and compare so called experts and see if those guys could get you above this line. So by the way, the slope of the line, one more thing, is the slope equals Sharpe ratio. So everybody’s trying to get the highest Sharpe ratio they can, and if you have this portfolio you’ve made a mistake, you haven’t held the optimal one, then the slope of your performance will be a lower Sharpe ratio than that. The slope of this, which is the Sharpe ratio of that point, is worse than the slope of that, which is the Sharpe ratio of this point.
And the Sharpe ratio, remember, is (the expectation per dollar minus what you can get in the riskless asset per dollar) divided by the started deviation of Y per dollar, let’s say. So that’s just the slope of this line. So we can measure performance of a portfolio manger by the Sharpe ratio. Everyone should try and get the highest Sharpe ratio. The upshot of diversification is you should buy a little bit of every stock in the economy no matter how small, no matter how big, no matter how safe, no matter how risky, you should spread your money across everything. That’s the recommendation of Tobin.
And it relies, of course, on everybody else being rational, so everybody else is choosing the right thing. Then you can just piggyback and do what everybody else is doing. So that’s the first thing.
Very shocking, as I said, at first glance you’d think someone who loves risk would just invest in risky stocks and people who are very risk averse would just invest in safe stocks. Tobin says, no, no, no, no. If you’re risk averse you should just put most of your money in the bank and hold a little bit of every stock. So you’re here somewhere. If you’re very risk tolerant you shouldn’t just concentrate on the risky stocks. You should buy the same proportion of all stocks as everyone else, but you should borrow the money to do it and maybe way up here. And people who borrow money to invest that’s called leveraging.
So that’s idea number one, the crucial idea. There are a lot of ideas mixed in that, and as I said, you can test it by seeing if anyone can beat the market–is there anyone historically that has had a higher Sharpe ratio than the market.
In 1967 when all of this was being tested there was a famous book called A Random Walk Down Wall Street written by Malkiel, which I recommend you read. He gives the history of tests of whether people could beat the market. Of course many people beat the market for short periods of time. Then they totally collapse. And then some people beat the market for 20 years in a row and then retire, but there are so few of those people that it looks like just by random dumb luck you could get almost those people.
On the other hand there are some people like Warren Buffet who seem to beat the market for 40 or 50 years. It’s really hard to believe that that’s all just luck. And so that’s one test of the thing.
The second idea of the Capital Asset Pricing Model is risk and return. And so now if we assume–so all this graph, by the way, assumed quadratic utilities, quadratic or mean variance utilities, quadratic utility and common probabilities. So everybody knew what the probabilities were of the different states, so you know how to measure the covariance and the variance and all that because you know what the probabilities are to measure over.
So here, having assumed that, it turns out that you just apply the principle that we saw the first few days of class, that the price is going to equal the marginal utility, not the total utility.
Price is going to equal marginal utility and if the price equals the marginal utility the marginal utility is going to be–depends, according to this assumption of quadratic utilities as we showed, depends on expectation. So this is the price of Y, so let’s just call it pi of Y like I did before. It depends on the expectation of Y and it depends on the covariance of Y with the market.
We’ve assumed with quadratic utilities that people only care about the mean and the variance, so you’d have expected the marginal utility of any single asset to depend on its mean and its variance, but it’s the marginal utility. It’s what it adds to what you’ve already got. So it adds expectation because expectation’s additive, but when you add another stock to what you already have, the variance of the whole portfolio changes according to the covariance of the new thing with what you already had, and everybody already had the market.
So given that you know that the price, mathematically therefore, the price has to equal something times the expectation of Y plus something times the covariance of Y with the market. But that’s true for all Y. In particular it’s true for the riskless asset. So the riskless asset pays 1 for sure, has no covariance with anything like the market, and its price is 1 over (1 + r). So therefore A has to be 1 over (1 + r).
So we know that this has to be (1 + r). And then B could be whatever it is. The covariance, by the way, is going to be bad, so this is minus. And then B, you have to figure B out. So you don’t know what B is a priori. You don’t know what the interest rate is either. You had to see that in the market.
Once you know the price of one asset, and you know its correlation with the market, if you knew this price, and you know the expectation of Y, and you know the covariance with the market that will tell you B. So once you’ve given one price of a risky asset, like of the market as a whole, you’re going to be able to figure out the price of every other asset because from that one price you can deduce B, and once you have B you can price everything. So that’s the second idea, risk and return.
And you can see it graphically and more dramatically, because if I take that same equation, this is called the covariance pricing formula, you take that and divide by pi of Y you get 1 equals expectation of Y divided by the price of Y–dividing both sides by pi of Y– minus B times the covariance of (Y over the price of Y) with M. So that just tells me the expectation and the covariance of a dollar’s worth, just like this was a dollar’s worth.
This was per dollar and this is per dollar, so if you put the covariance of Y and the pi of Y down here, and M over here, and here you put expectation Y over pi of Y, this just says this is just a constant, and this is a constant, so it says this and this are linearly related.
These are all securities who have price 1. There has to be a linear relationship between their expectation per dollar and their covariance per dollar with the market. And as the covariance gets higher the return gets higher. So if you take a stock with a high covariance it’s going to have to have a high expected return. That’s exactly what Shakespeare said.
If it’s more risky people won’t hold it unless it’s got a higher return, which means you can buy it for a cheap price. So its price is less than its expectation. But the difference with Shakespeare is we’ve quantified risk. It’s not what you would have thought, variance, it’s covariance. And that’s it.
Those are the two big ideas and you could test–so this is called the Security Market Line, and you could test this by asking is every stock–so if you look at every stock last year, you look at its covariance, you put a point here, then you look this year at its expected return, of course some stocks are going to be wild, but you average over a bunch of stock and a bunch of years. Do you get things that lie in a straight line or close to a straight line?
And Sharpe, in 1967, found that amazingly when you did that you got–so I described last night exactly what his test was. You got something that looked incredibly close to a straight line like that. So this was the greatest triumph in the history of the social sciences, it was called, because you had huge amounts of data, huge capacity to test things, a shocking conclusion, a shockingly precise theory to test, and it fit the line so incredibly closely.
And it just swept, it made a huge impression in finance and in economics and lots of Nobel Prizes were given for this. Sharpe won a Nobel Prize. Markowitz won a Nobel Prize. Tobin won a Nobel Prize. Lintner died. He would have won a Nobel Prize too. So this was the first major triumph, the most important triumph of finance, the Capital Asset Pricing Model.
Now, as it happens, as the theory’s been tested over and over again every year since 1967 it’s performed more and more poorly. And so now when you do this you get points that look like this, and like that, and like this, and they don’t seem to lie on a straight line at all, and it’s pretty shocking how up until ‘67 or the early ’70s that it seemed to work out perfectly and now it doesn’t work very well.
Chapter 2. Evaluation of Fund Management Performance Using CAPM [00:14:08]
And this business they’ve also discovered, according to this you couldn’t beat the market. So how do you test whether something beats the market? You can pick a bunch of people and say, “Could they beat the market,” or you can actually just make up a strategy yourself like buy only new stocks. Just every year buy the stocks that have come onto the S&P 500 for the first time, or just buy the stocks that are the biggest 10 percent. And it turns out some of those strategies actually do beat the market.
So I don’t have time to go into the details, you know, that is they lie above or below here. So the theory has come into–so theory and doubt. And so what I’m going to talk about next week is, and it certainly doesn’t explain the current crisis, so I want next week to explain crisis. So next week I’m going to end by giving another theory which is going to explain the crisis and be different from this one, but of course it hasn’t been tested over 30 years or 50 years like CAPM.
CAPM looked great for the first 20 years, so whatever I say next week will probably turn out to be 90 percent wrong, but at the moment it looks pretty good. And I didn’t want to spend half the course on it in case in 20 years it seemed to be ridiculous. But anyway, I’m going to present it anyway to those of you who want to hear it next week, and it’ll be, I think, understandable and down to earth and it’ll invoke a very basic thing. Nowhere in this model did we take into account, you know, when we say people leverage that means they’re borrowing money and choosing to go up here.
We haven’t taken into account that they have to pay the money back, and who’s going to trust them to pay the money back? And forcing them to do something to guarantee they’re going to pay the money back is going to change the theory in a critical way, which I’ll get to next week. Now, one last thing about the theory, as I say, you test it by seeing–nobody should be above this line and nobody should be above this line, but of course you can find people above this line.
Swensen, to take a famous Yale example, is above this line. And you can find people above this line, Ellington. So Ellington, let’s say, above this line. All right, so hedge funds. Let’s call it an anonymous hedge fund since we’re on camera now, hedge fund, hedge fund E. So the question is, is that luck, they just did it a few years in both cases like 15 years apiece, or does it mean something?
All right, well, so you can measure the, you know, if you believe this theory and you just say, okay, let’s say the theory’s 99 percent true, it’s just that there are a few people who really do beat the market, then you can decide by looking at these graphs just how good is Swensen relative to the market? How good is this hedge fund relative to the market? And so you have a way of calibrating and measuring how well people do. So I don’t have time to get into that, but I would measure the quality of a hedge fund by its marginal Sharpe ratio. How much does having access to this hedge fund, if you added it to your portfolio, increase your portfolio hedge ratio [correction: Sharpe ratio]?
That’s, I think, the right measure of how good a hedge fund is. And the right measure of how good Swensen is, is, what’s his Sharpe ratio? And so there’s a different criterion. I keep emphasizing this. Swensen is managing an entire portfolio so you have to measure him on how’s he done for Yale, because Yale’s not going to combine him with something else. It’s just holding what he holds. So it’s his Sharpe ratio.
But Ellington, you’re going to combine Ellington with some other stuff. So Swensen might invest in Ellington, but ten million other things. And so you have to say how much is Ellington going to help Swensen’s Sharpe ratio? So instead of Swensen’s Sharpe ratio how much is Ellington going to help the market Sharpe ratio?
So that’s how you should measure a hedge fund, its marginal Sharpe ratio, and Swensen by his Sharpe ratio. So I’d say measure Swensen performance by Sharpe ratio, measure hedge fund performance by marginal Sharpe ratio, because you’d be crazy to put all your money in a hedge fund. The only people who do it are the partners of the hedge fund who are sort of forced to do it because that’s how they can persuade investors that they’re really paying attention to their fund, because all their money’s at risk. So that’s it.
That’s the basic bottom line. So are there any questions before I now move on from this? We talked about this last night and we talked about it for the last two lectures. All right, so it’s a high water mark as I say; incredibly precise prediction and incredibly close to the facts, at least in 1967 but not after that. Now, I want to move to an aside of Social Security before I come back to Black-Scholes which is the last high water mark of the standard theory.
Remember the upshot of what we’ve talked about so far. The upshot of this is that the market will get a higher return. Oh, by the way, I’d say one other thing. What is the standard critique of this theory? So of course everybody else has seen that the theory doesn’t work. So what is their explanation for why? What’s the old explanation for why the theory isn’t working so well now?
The explanation is that in the old days what was thought of as the market was just all American stocks. But now it’s pretty obvious, in fact it should have been obvious from the beginning. And so Swensen realized it early on that Markowitz’s advice is to diversify, diversify, diversify, not just into all American stocks, into everything all over the world and all kinds of assets that aren’t stocks at all. So what you measure as the market shouldn’t just be American stocks.
It should be the world portfolio, but how do you measure the world portfolio? That’s hard to do, and so people have tried to maintain that the reason the theory doesn’t work is because we’ve got the wrong M in all these equations. So that would be a nice way of fixing the theory. I doubt if that’s going to help in the end. So my problem about collateral is a different look at it.
But anyhow, so the market according to the theory will get a higher return than bonds, and of course that’s true. The market has a very high positive covariance with itself so a very low price up here. It’s high covariance with itself, so a low price. Compared to its expectation the price is low, so you’re paying much less than the expectation. So then when you see what actually happens it’s going to look much higher than that. It’s not just this. The price is lower therefore the final outcome’s going to be much bigger on average.
So to put it another way, if you had two possibilities which you knew were equally likely, 1 half and 1 half, and the market paid off something good here and something bad here, so everybody’s rich here and everybody’s poor here. The price of the Arrow security of this state is going to be much lower than the price of the Arrow security of this state.
So this state is going to have a price, let’s say .8, and this one might have a price .2 because here the bad state, that’s when everybody’s poor, so you’re going to, you know, this formula basically tells you that if you’re paying off here you’re negatively correlated with the market, you’re going to have to pay a very high price for it because you’re hedging.
By buying this security you’re balancing off what you have with the market. By buying the market, people don’t like that because it adds more risk to what you already have. And so people don’t like it so they’re not going pay very much, so the price will be much less than the actual probability of 1 half. So that’s the conclusion of what we’ve done. I want to now connect this to Social Security.
Chapter 3. Reassessing Assets within Social Security [00:22:30]
So you remember in our Social Security model we had land that went here, here, here, paid 1, 1, 1, 1, 1, and we had these people, overlapping generations-people like that. And we discovered a very important thing. The reason why Social Security seems to be broke is because we gave away so much money to the people in the first generation.
It’s not that Social Security wastes money like George Bush seems to imply whenever he talks about it. It’s not like they just throw it away. It’s that we gave a huge gift to the people in the ’40s, the ’50s and the ’60s and so everyone after that has to pay for that gift.
But there’s a little bit more to George Bush’s story than I’ve allowed so far, which is that he wants to put the money into stocks. And we know that stocks are going to get a higher return than bonds. That’s the whole point of this theory. You put money in stocks, the expected return on stocks is higher than the expected return on bonds. So if you want to make this mathematical we have to add the possibility of risk and stocks and it looks like it’s going to be hopelessly complicated, but we can do it very easily.
So I’m going to reproduce the same picture with uncertainty in stocks in it, and then we’re going to come back to Social Security and spend 15 minutes talking about what’s the right plan for Social Security. So how could I take this simple model, which already seemed complicated, it’s already a pretty complicated model, had no uncertainty, and now put uncertainty in it?
Well, I’m going to suppose that land–so this is the land dividends I’m now going to describe. And by the way, I had people with endowments here. They were (3, 1) and (3, 1). Remember, this is their endowments, but we didn’t say where those endowments came from. So I’m going to take the land dividends. I’m going to suppose the dividend of the land today is 1, but things might get better or worse in the future.
Maybe the dividend goes to 2 or down to 1 half, say. I’m taking some extreme case where it doubles or halves, and maybe after that it could double again or it could go back to 1 and here it could go to 1 or it could go to 1 quarter. So what dividends are doing is a random walk. If the economy starts getting more productive then it’s in a better position, and from that point things get even more productive or maybe go back to where they were. If things get worse they’re worse, but maybe things will start to get better again or continue to get worse. So this is what I imagine as my process of dividends.
So if you own the land you get all these dividends in the whole infinite future, all right? And now let’s say that there’s labor as well, so labor income. So I have to make an assumption about labor income. So I’m going to assume only young work. So here we kind of made the same assumption. You get a lot when you’re young and not much when you’re old.
I’m going to assume only the young work, and I’m going to assume that this productivity gains or losses, and by the way, you might think you never go backwards. Maybe it goes from 1 to 2 or 1 to 1.1. So it’s always growing, but random. But I’m just going to keep it here since I solved this example last night. I might as well stick to that one, but there’s no reason why–I could have made the thing always, no matter what, either stay the same or get better. That might have been a better example to do, more realistic because we don’t really have tremendous productivity declines, but anyway, all I mean to do is to put some randomness here.
So only young work and wages proportional to dividends, so let’s just assume this, so I’m assuming–this is what’s called neutral technical change. So presumably the reason why productivity is going up, the dividends are getting higher, is there are more discoveries improving the product of the land, but they’re probably also improving the product of labor, and so let’s say they have same effect on both. So I’m going to assume that labor is actually–put the 1 above.
Let’s say labor always gets 3 times what dividends are. So this is a 6 and this will be 3 halves, and this will be 12 and 3 again and 3 quarters, so that’s what labor is getting, so labor income. I shouldn’t have switched colors, labor income. So that’s my complicated world, way more complicated than it was before.
So if you own the land you get all the dividends in the future. People live two periods, so I’m going to assume that utility–so you’re going to live when you’re young and then when you’re old and up and when you’re old and down. So you’re going to live–let’s say a guy born here, he’s young here, or she’s young here.
She can work here, and then she’s going to be old here and here, but when she’s old it’s not clear how good her investment’s going to turn out. So this is going to just be our simple log. I’ll call this 1 half log Y + 1 quarter log Zup + 1 quarter log Zdown. So you don’t discount the future, let’s say, you just take log of consumption today plus probability 1 half of log consumption in the up state, 1 half log consumption in the down state. So the probabilities here are 1 half. These are the objective probabilities. So maybe I should have written the utility–maybe better it would have been to write the utility like that. That’s what people know, what the probabilities are. So that’s it. That’s my set up. So what should the economy do? What should people do?
You see, if you buy–if you’re some person like this woman here, she could decide to hold bonds, and she could get whatever the interest rate turns out to be, which we’re going to compute in a minute, or she could hold stock in which case she’s going to get–let’s start her off here, maybe. She could buy land; buy stock. Now we don’t know what the price is yet. We have to compute that.
But if she buys the stock she’ll get a dividend of 2 here and be able to sell the land for a high price, because from here on it’s obviously pretty productive, better than it was back here, or she might get unlucky and the price might, you know, the dividend might be really terrible, just 1 half, and not only that but the land she sells when she’s old is also going to have a really lousy price.
So if she holds land she’s going to get a high dividend and a high capital return, or else a low dividend and a low capital return. It’s really risky for her, or she could just buy a bond at whatever interest rate it turns out to be and get something safe. So what should she do, and will that be a good thing for the economy or can we make everybody better off by using Social Security?
So how can we solve this? So it looks like a really complicated model to solve because we’ve got a generation born here, another one born here, the same generation might be born up here under different circumstances, and then this generation here could be born under any of three different circumstances, so there’s a lot to solve for.
But what we know is that when we compute the Arrow price, starting from any point, this Arrow price is going to be bigger than 1 half and this one’s going to be less than 1 half because the economy is doing much worse down here so people know they’re all going to get screwed down here and consume a lot less. So to hedge that they’re all going to want to buy, they’re going to be desperate to buy consumption down here.
Of course they can’t all do that, so it’s got to be the price that’s more than 1 half that discourages them from doing that. And they don’t really need that much to buy up here because their dividends, they’re going to be so rich anyway. So the price of the Arrow security is going to be less than 1 half because there’s so much that they have to end up holding and that’ll encourage them to buy.
I mean, sorry, I said the logic backwards. There’s so much that they’re going to have to end up consuming because the economy is so productive. How can you get them to plan to consume that much up there? It’s by having a low Arrow security price. So even in advance they know that they’re going to end up consuming a lot and they’re happy to do that because they’re willing to buy Arrow securities because they’re so cheap. So how do we solve all this? So everybody following the problem? It’s quite a complicated problem, but it turns out to have a very simple answer. Yep?
Student: So the down state, the price of consumption is high but the price of land is low?
Professor John Geanakoplos: Right, so. Well, so if you start–starting from here the price of consumption–if you want to buy at this point, consumption at this point, you’re going to have to pay more than 1 half to get it, but you know that once you get here the price of land is going to have dropped. So yes, I’m agreeing with you.
Student: I understand the first part because the CAPM price…
Professor John Geanakoplos: Right, now why should the price of land be lower here? Remember, the price of land is always in terms of the consumption good that period.
Every period I might as well take an apple to have a price of 1. So you see, the land here is producing an entire apple, and on average going forward it’s going to be producing 1 apple. The geometric average of all these numbers is 1 starting at this point.
Once you’ve gone down to here production has deteriorated. The land is only producing 1 half an apple here, and in the future instead of producing 12 it’s going to produce 3 apples here and only 3 quarters of an apple. So it’s crummier land down here, so you’re going to pay less for the land. Any other questions?
But I’m going to tell you exactly what you’ll pay for the land. So we’re going to solve this. So need to solve for land price in each state, and interest rate at each state, and price of up Arrow security, and price of down Arrow security at each state. So how can we do all that?
Well, first of all you notice that the Arrow securities are always the key. Once you solve for those you can figure out everything else. So we just have to solve for these, the price of the up and down Arrow securities in every state. Oh, this was me in a previous class today. I didn’t erase it. I figured the next guy would do it, serves me right. So how can we solve this?
Well, we can guess, let’s do it up here, guess that these prices, Arrow prices, the same at every state. So what do I mean by that? I mean, here, how many apples would you pay to get an apple here? I’m guessing that that’s the same amount you would pay here to get an apple here. And what you’d pay in apples at this point to get an apple down here is going to be the same as this generation would pay to get an apple down here.
That’s a guess. We have to verify that that’s going to work. It seems like everything is sort of homogeneous, and so guess that price of land is proportional to dividends in that state. Well, the whole thing is sort of homogeneous and everything, so if the land here is 4 times as good as the land here, the land here is 4 times as good as the land here, because the dividends are always 4 times higher if you go up one thing. So this is 4 times that, that’s 4 times that. So why not guess that the price of land is 4 times here than it is here, 4 times higher here than it is here. Sounds possible, anyway, and of course these are going to turn out to be correct guesses.
So once you make these guesses it’s going to be very simple to find the equilibrium, and then we can verify that the guesses are correct. So what does this guy want to do here? So assume you only work when you’re young. So what’s your income? What’s anybody’s income like this young guy here? Young at time 1, so this is time 1 here, time 2, time 3, time 4, so at time 1 their income is 3 because that’s what their wage was. They work when they’re young and they’ve got three. That’s it.
They don’t own the land when they’re born. They’re going to buy the land but they don’t own it now, and so what do they want to do? They want to consume when they’re young, their utility, remember? So what’s their budget set? Their budget set is they can consume Y, that’s consuming here, or they can consume here or here. So they can consume Pup times Zup + Pdown times Zdown. That’s what they can do with those Arrow securities. So what are they trying to maximize?
They’re trying to maximize 1 half log Y + 1 quarter log Zup + 1 quarter log Zdown. That’s what they’re trying to do. So what do we have to do? We have to clear this market and this market given what the young are going to do in the next period, but let’s see what this guy’s going to do when he’s young.
Everybody is going to face a similar, basically a scaled up version of this problem. So this is what the guy is doing here. He’s got an income of 3 and there’s the price of up and down Arrow securities. That’s going to tell him what to consume here and what to plan to consume up here and down here.
Student: Do you change your utility function?
Professor John Geanakoplos: So utility function is the same one as this. I just multiplied through by 1 half, right? So I put 1 half, 1 quarter, 1 quarter to get that. So I’ve just multiplied it all by a constant. So now the coefficients add up to 1; 1 half, 1 quarter and 1 quarter.
That’s his budget constraint, but what’s the budget constraint going to be of a guy down here? It’s exactly the same. He’s going to maximize the same utility function and his budget constraint, since the prices of the Arrow securities are the same, the only difference is going to be that he’s only got 3 halves here. So they’re pretty similar problems.
So let’s see what the guy at time 1 is going to do. That’s going to imply that Y = (Cobb-Douglas) 1 half times (his income) 3 divided by the price of Y which is just 1, so 1 half of 3 over 1 which is 3 halves. And what’s he going do when he’s old? Zup is going to be–well, his Cobb-Douglas is 1 quarter, and then his income is still 3, and then the price is Pup of the Arrow security. In the down case he’s going to have 1 quarter times 3 (that’s his income) divided by Pdown.
So to clear this market what do we have to do? We have to clear the market by doing what? Zup, that’s this guy, so 1 quarter (so this is the old guy) times 3 divided by Pup. That’s what this guy wants to consume up here, plus what the young are going to do there, but we know what the young are going to do here. This young person is doing exactly what this guy did, except instead of having a starting wealth of 3 he has a starting wealth of 6. So instead of consuming young 3 halves he’s going to consume young 6 halves. So this is going to be plus 3.
He’s going to consume half of his wages just like this guy consumed half of his wages when he was young. So this new young guy, the young guys are always going to consume in the first period half of what their wages are. That’s what this equation tells us. If this is your budget set, this is your income, this is what your budget set is and this is what you’re maximizing.
This is a very special Cobb-Douglas case where you have endowments only in one good, so it’s clear that in that good you’ll always consume the Cobb-Douglas coefficient fraction of it, 1 half of that, so 3 halves. And this guy’s going to consume 1 half of 6 which is 3. So the old up here are going to consume that number. The young are going to consume 3, and what’s the total that’s available for them to consume? What’s available?
Professor John Geanakoplos: No, more.
Professor John Geanakoplos: 8; the young got 6 apples and the land paid 2 apples in dividends, so the total number of apples is 8. So that’s it.
And so we can solve that and that’s going to imply that multiplying–so that’s 5 and multiplying by 4 is 20, so PU, of course I’m going to screw this up now. PU is 3 over 20, because if you put 3 over 20 in here you get the 3s cancel and you get 20 over 4 which is 5, and 5 + 3 is 8, so it’s 3 over 20. Now meanwhile, what are we going to have in the up state? In the up state…
Student: In the down state.
Professor John Geanakoplos: Down state. That was the up state. In the down state the guy’s going to do 1 quarter 3 over PD. What is this other guy going to do? The young person here is going to consume what? So in the downstate what are we going to have?
The equation of the downstate is going to be–so this one 1 quarter 3 divided by Pdown plus what’s this guy going to consume when he’s young? Well, his income, his wages are 3 halves and he’s going to consume half of that so it’s 3 quarters, right, because when you’re young you always end up consuming half of your wages, so 3 quarters and that has to equal–what’s the total? 2? 2, so what does that leave us?
That leaves us 3 quarters, that’s 5 fourths, so let’s see. So 5 fourths equals 3 quarters over PD, so PD is going to equal 3 fifths, right, because the 4s drop out and you put PD up and the 5 down, it’s 3 fifths. So that’s it. So now we’ve got–we’ve figured out the prices. So this Arrow security was 1 half. I mean, the probabilities are 1 half, 1 half, but the prices, the Arrow security prices, are going to be less than 1 half and more than 1 half. I forgot what they were already, 3 twentieths, 3 over 20 and 3 over 5. Those are the Arrow prices.
By the way, you notice they don’t add up to 1. So what’s the interest rate? So if you want to get–a riskless bond pays 1 in each of these two states, so how much is that worth, 1 over (1 + r) equals (You have to buy both Arrow securities. We’ve done this before.) 3 twentieths + 3 fifths which equals 15 twentieths, which equals 3 fourths, so therefore 1 + r = 4 thirds. So r is 33 percent. So that’s what we did.
We’ve got the probabilities, 3 twentieths and 3 fifths. Just as we thought the price of the down Arrow security is going to be much bigger than the price of the up Arrow security even though objectively they have 50/50 probability, and why is that? Because everyone who’s rational is going to realize that they have to plan to consume much more up here than down here, but they’re not going to want to do that unless the price is much cheaper. So that’s how we got the price. So now what’s the price of land?
That’s the only last thing we have to figure out, so the price of land. So the price of land at time 1 equals what? Well, what if you buy the land? What do you get? You get 3 twentieths, then you get the dividend which is 2, but then you get to sell the land which is Plandup. And then plus if you get the down state you get a dividend–so the down state is worth 3 fifths to you, right? And then what do you get? What’s your dividend? 1 half + the price of the land you can sell, Planddown.
So that’s what the price of land is at 1. It depends, of course, on what you can sell the price of land for in the future, but we made a guess here that the price of land is just proportional to the price of the dividend. So whatever the price of land is here when the dividend is 1 it should be twice as high here and half as high here. So therefore this, instead of looking at it as the price of land went up it’s going to be twice the price that land was at the very beginning. So it’s just 2 times the price of land at the very beginning times that same price.
And this price of land at down, the dividend is half, so the land is sort of half as good from here on out as it was in the beginning, so let’s guess the price of land is half of what it was at the beginning. But now you see we’ve just got Pland in terms of other stuff and so we can solve for it. So the price of land is going to equal–well, now it’s complicated. So 3 tenths + 3 tenths times the price of land + 3 tenths + 3 tenths times the price of land, and so that means the price of land minus 6 tenths. So 4 tenths the price of land is going to equal 6 tenths. So the price of land is 3 halves at the beginning. So we figured out the price of land here. Price of land is 3 halves at the very beginning.
So we’ve solved for the whole equilibrium. It took a long time, but I’m at the end of this story now. So let’s just review what we did. And so I don’t think it’s that complicated a story.
The story is that you’ve got land which has risky dividends. So instead of having one straight line of what can happen in the future we’ve got this infinitely expanding tree. So it’s very hard to imagine solving it, but then we’ve built in somewhat special assumptions. We’ve got all this homogeneity and that allows us to solve it because it’s the same problem getting repeated over and over again, just like we did before. So the price of land turns out to be 3 halves.
So what is this guy going to do? He’s going to take his land, his income which is 3, eat half of it, which was 3 halves, that’s what we said he always does when he’s young, spend half his income on consumption, the rest of his income, 3 halves, he uses to buy all the land.
Now, with the land he gets a payoff of 2, but he gets to sell the land now for double the price he bought it, namely for 3. So 2 + 3 is 5 and that, by the way, the price of up was 3 twentieths so this is 5, right, because if I put in 3 twentieths up here that’s the Arrow price of up, then Zup is going to be 5. And so that’s exactly how we got it. With the land he gets the dividend of 2 plus he sells the land for 3. That’s 5, so he does consume 5. Zdown, incidentally, the price of down was 3 fifths. So the consumption of down–what’s consumption of down going to be?
Student: 5 fourths.
Professor John Geanakoplos: 5 fourths, thank you. That’ll be 5 fourths, so let’s see that that’s going to work out too. So the guy buys the land, he gets his dividend of 1 half, and then he sells the land for half of this price which is 3 quarters, and 1 half plus 3 quarters is 5 fourths, exactly what he’s supposed to have. So you see it works out that everybody’s doing exactly what they’re supposed to and all the markets are clearing.
Anyhow, the point is not so much the numbers, the point is the following; that sure enough, just as we said before, the price of Arrow securities are going to be expensive when things are going down and cheap when things are going up.
The upshot of that is the return on stocks, like land, is going to be very high. You put in 3 halves and what happens with the 3 halves? With probability 1 half, that’s what you put in, with probability 1 half you get 2 plus you get to sell the land for 3, which is 5. And with the other probability 1 half you get–the dividend is 1 half, plus you sell the land for 3 quarters. That’s what you’re eating, 5 and 5 fourths. That’s equal to 5 halves + 5 fourths + 5–I don’t know what this is. What is this? 1 half + 3 quarters is 5 fourths times 1 half…
Student: 5 eighths.
Professor John Geanakoplos: It’s 5 eighths divided by 3 halves. So this is a very high number. So what is this? 4 times that, that’s 25 over 8 divided by 3 over 2 which is…
Student: 25 over 12.
Professor John Geanakoplos: Is what?
Student: 25 over 12.
Professor John Geanakoplos: 25 over 12, is that what you said?
Student: 25 over 12.
Professor John Geanakoplos: 25 over 12, I’m just believing you now. So that is over 200%, right? So your return is 100%. Whatever you put in you’ve more than doubled your money in expectation, whereas the interest rate, we said, was 33 and 1 third percent. So of course you’re getting a tremendous return by putting your money into capital because that’s the whole point of the theory. It’s risky.
The whole economy is going to be rich or poor together, so the price of capital, on the margin it’s a very risky thing to do. It just adds to this inequality in your consumption. So the price is going to be low. You’re going to get a very high return.
That’s all I wanted to do. I just solved it out concretely so you can see very clearly that you get a high return in the good state and–you get a high return on capital, much higher than the riskless rate of interest.
Chapter 4. Reconciling Democratic and Republican Views on Social Security [00:53:04]
So there we are. We’re back at the beginning now where George Bush–we can see that somewhere in the middle here things are really disastrous. These people who are getting Social Security, which means they’re getting a fixed amount here and here are doing really badly, at least in expected return, compared to what they would do in the stock market. And so what should we do about that, and is it true that just because stocks make a higher return, we should privatize?
So what do you think now? Where are we in this argument? Has anything materially changed? So George Bush is saying, and many Republicans say, that we should let people, you know, get them invested in stocks. There’s a higher expected return. So what do we think about that?
What do you think about that? This is a real policy issue here. We’ve got this problem in America. Right now we’ve got this huge deficit in Social Security. The rate of return has gotten really poor for your generation. You’re looking forward to not making very much money, and now we know theoretically, confirming what we’ve seen in the past, stocks are making higher returns than bonds, and Social Security is paying people a fraction of the wages of the young.
So let me remind you of the arguments. So the issue has come up because Social Security is running out of money. So it came up because of this problem. There’s going to be a deficit. If we continue to pay people at the same rules that we’ve used up until now we’re not going to be able to afford to do it much longer, so we’re going to have to lower what we pay the people going forward, lower it in some clever way like they don’t get money until they’re older. Instead of 65 make them wait until 70, or do something like that to pay them less.
That’s one way of balancing things. And because we’re going to have to make changes like that, that’s why the whole issue of Social Security has come up, and now deeper thinkers are questioning the whole idea of Social Security. So let me just remind you that America was ahead of everyone in the world when we created Social Security under Roosevelt. Nobody had anything like it.
Then everybody in the rest of the world gradually copied us, and now everybody’s facing the same problem we are, which we know why you have to face it. You’re giving away stuff to the people at the beginning and so they’re facing it by changing Social Security in some crazy way or another, and we haven’t changed anything at all, but obviously we have to, and the question is how should we change it and what’s the right thing to do.
And just to remind you, again, of the issues, the Democrats and Republicans seem to be totally opposed to each other. The Democrats say, “Oh, we want to continue to give people who have done poorly in their lives a better deal on Social Security because we want to redistribute. We want to help the poor.” The Republicans, actually some of them agree with that, but some of them don’t agree with that, but anyway, even the ones who agree that’s not the most important thing they think of in Social Security.
Then the Democrats say, “Social Security is so important because we’re sharing risks across generations. If the next generation has poor wages so they end up down here with bad wages, you know, the economy takes a turn south with bad wages, then the old people should suffer along with them. And if the young do better with high wages because things went up then the old should also get better Social Security dividends,” and that’s what the current situation promises.
It’s also indexed according to inflation, all right, so once you retire then you’re protected against changes in inflation. And then another important thing of Democrats, you don’t have an opportunity to make a mistake. If you invest in stocks and stuff like that and the stock market suddenly collapses you’re not going to lose money in Social Security because Social Security depends on the wage of the next generation, and that moves much slower than the stock.
The Republicans, on the other hand, say, “This is just terrible. Nobody knows what their property rights are in Social Security. No one actually understands how much money they’re getting when they’re old. They just know now it’s going to be bad compared to what they’re paying in taxes, but you don’t even know what it is, so how can that be good? It’s not transparent, but not only that, whatever you think it is now it’s going to turn out to be worse because in ten years somebody’s going to say, ‘Oh, the system’s not balanced. We have to delay benefits or something to put it back into balance, and we’re going to take away what you thought you were going to get.’”
So Republicans say, “That’s horrible. Nobody knows really exactly what it is, and they get an idea of what it might be, but it could always be taken away. Really what they want to know is, ‘What’s the value of my Social Security account, and I want it to be mine. I can also then tell what the redistribution is. If I know what I’m paying in taxes and I know the value of the benefits I’m getting I can see that my taxes are much more than my benefits. And if I happen to be a nice guy and think it’s a good idea to have redistribution, fine. But if I’m a bad, you know, not a bad guy, if I just don’t believe in that much redistribution I should be aware of it and have the chance to vote against it.’”
Also, a key thing is these Republicans say, “People should get equity like returns, not this wages of the next generation, those grow slowly. Stocks might have a much higher return on average. People should get equity like return, so let’s put them in the stock market.” And then Republicans think choice is always great. Even if people make mistakes it’s their own mistake, so let them make mistakes.
So those are the opposite sides of the argument. They seem totally at odds with each other and impossible to reconcile. So I want to now tell you my plan. So everyone’s got the problem straight. We’re running ou
Chapter 5. Geanakoplos’s Personal Annuitized Average Wage Securities [00:59:32]
All right, so my plan, in five minutes because I want to end with Black-Scholes, my plan in five minutes is, okay, to make a few observations. First of all, yes it’s true that you get a higher return on stocks than you get on bonds, but you face a higher risk. So it’s not like you get it for free. It’s a bigger risk. So that’s number one.
So no, actually, point zero is you’ve still got the problem because you gave money to all those people in the ’40s, and the ’50s, and the ’60s, and the ’70s. That money is gone. We’ve essentially borrowed to give them money and somebody’s got to pay back that debt. And so there’s no way around that. There’s this huge 17 trillion dollar debt hanging out there that we have to pay back and you can’t get around that. So putting the money into equities, or wherever you put the money there’s still that tremendous debt.
Now the question is–so you can’t just make the problem go away by saying that equities have a higher return. Bonds also have a higher return than the return people are getting on Social Security because the Social Security return includes the tax that you’re paying to make up for the original generation. So that’s the first thing. So the next thing is maybe it’s not such a big difference, this equities and not equities, because notice in this model when the stock is paying higher, when the stock dividends are higher, 2 instead of 1, the price of stocks goes up.
That’s a great return from the stock market, but the wages were also higher here as well. So in fact I claim that in the long run, and it’s amazing that the Republicans haven’t noticed this, in the long run over 30 or 40 years it’s obvious that the stock market and wages are correlated. I mean, if the stock market collapses that’s disaster for America and you can be sure the wages are going to go down too. If our stock market’s booming and America’s incredibly successful it’s a sure thing that in 40 years if the stock market is just booming along wages are going to be higher as well.
So if you get the wages of the generation 30 years from now in the Democratic plan you’re actually getting equity like returns, but you’re getting an advantage which is that the wages move slowly. So although the stocks are bouncing up and down, so somebody who retires in the year 2007 if he held stock and sold it when he retired he could get a huge pension, where someone who retired in 2008 and sold the stock just when he retired he would be crushed because the stock market lost 50 percent. So that will never happen if you just pay according to wages because they’re much more stable. But wages in the long run are correlated with stocks in the long run. In the short run they just don’t fluctuate at much. So what is my plan?
So those are just some preliminary observations and I’ll give you the plan. Then as I said this is obviously controversial. I think I’ve identified the problem in a scientific way. That is, everyone would agree with me if they knew enough economics, but now my plan, even people who do know as much as me might disagree with me, but this is what I would–okay so here’s the plan.
As I said, other plans just say, “Well, things are bad. Let’s just keep the same system.” So there are two basic ideas. The Democratic idea is, “Let’s just keep adjusting at the margin. We’ll make up for it by giving people money later. We’ll do all kinds of things like that. Maybe we’ll raise some tax, reduce some benefits, balance the system.” The Republicans want to junk the whole thing and privatize and let people, you know, force them to save, but put it in the stock market in their own account.
Now, here’s my plan. I don’t have time to talk about Bush’s plan. So here’s my plan. Number one, the fact that we gave all this money away to people in the ’40s, and ’50s, and ’60s and ’70s, why should that be the responsibility of the workers today? I mean, our government decided it was a good idea. Roosevelt and Frances Perkins decided we had to rescue the old of the ’40s, and our Congresses kept that up for the next 20 or 30 years even though the old weren’t in such dire shape, but they kept it up.
Maybe that was a good idea. It’s probably a very good idea, but because we have that huge debt why should workers be the only people responsible for paying back that debt? I would start by imposing a legacy tax on everybody, not for Social Security, just for the debt we accumulated by giving it to those old generations. So I believe that would be something like 1 percent. I wrote 2 to 3 percent, but I think it’s closer to 1 percent, actually, on all income. If I did it on all income it would be like 1 percent and that fund would pay off for the Social Security legacy.
So then if we were starting the system afresh how should we do it? And some countries like Chile decided they’re going to start afresh. So how should we start afresh having gotten rid of this old debt that was hanging over our heads? Well, what would I do? I would have what I call Progressive Personal Accounts. So I like all the Republican ideas of knowing what you’ve got, making it yours, and making it transparent.
I think that’s all an extremely good idea, but why should it be that that comes from putting your money into stocks? I mean, stocks, this is an old fashioned security and it’s incredibly risky. From one year to the next it totally changes. I like the idea of indexing something to wages. So I want to create a new security that I called Personal Annuitized Average Wage Security.
So what this does, this is a security that just pays proportional to the wage in the whole country at the year you retire. So whenever you earn money on Social Security–I’m running out of time so I’m going to skip over this pretty quickly. So if you earn 1 dollar and you pay, let’s say, 12 percent in Social Securities taxes, that income, I’m going to say that’s your income, it’s your 12 percent.
You’re going to be forced to save it, and what you’re forced to do is to buy these securities. So it’s a new kind of security that pays proportional to the wage. So it might pay 1 percent of the average wage. So it would pay 12 cents here, 3 cents here, and 3 quarters of a cent there. That’s the wage in the economy and it would pay 1 percent of that. And how much of it would you get to buy, as much as you could afford to buy with your security, with your tax contribution.
Now, that almost replicates the current system. The only difference is that it doesn’t replicate the current system, there’s nothing progressive about it yet. So what I would do is I would make it a progressive tax. If you’re making a hugely high income I wouldn’t give you the whole 12 percent. I would take part of it.
And if you’re making a really low income I would have the government subsidize it a little bit. So instead of getting 12 percent of your low income you’d get a little more than 12 percent. If you had a high income your 12 percent tax the government would take 1 of those 12 away from you and you’d only have 11 percent that you got to put in your personal account.
Now, but it has a tremendous advantage, this system, which is that if you can price the PAAWS, if you know what the market price of PAAWS is then you’re balancing the system, because every time you hand somebody money you don’t make some crazy promise that when you get to be old I’m going to pay you such and such dollars like the current system is. You’re letting the guy, or the woman, let’s say, who’s working; she is buying her own Social Security benefits. So the benefits have an equal value to what she’s paying for. So you’ve balanced the system.
The legacy tax you got rid of by having the tax on everybody of 1 or 2 percent, and the balancing the system going forward is occurring because you’re forcing people to buy their Social Security benefits. So anyway that’s my, in a nutshell, my system.
And the last thing is how are you going to get PAAWS priced by the market? Well, I’m going to get the market to trade them, and I think that would be a tremendous boon to the economy if we had these securities paid. Why is that? So how would I do that?
I’d force everybody to invest only in PAAWS, which I claim are like stocks in the long run, but in the short run they’re less risky, so it’s a better investment vehicle, but they’d have to sell exactly 10 percent of their PAAWS into the market, and with that 10 percent of money they could hold stocks or whatever else they wanted to, and those 10 percent would all be pooled together and traded in the market. And the market then would have an idea of–a new instrument to price what they think future wages are going to be. So anybody giving a pension plan that’s indexed to future wages would know what the market price of those things are, so it would dramatically help pension plans as well. Anyway, that’s my idea for Social Security.
Chapter 6. The Black-Scholes Model [01:08:48]
So I have four minutes left and I want to end with one last thought, one last idea, which is Black-Scholes. Now, we’ve already done Black-Scholes in a few problem sets. So what is the idea of Black-Scholes? It’s just like the idea of the example we gave. By the way, if you do come next week and you want to question me about my Social Security plan and criticize it I’d be thrilled to be criticized because you can’t learn anything unless you’re criticized and I think I can defend it too.
But anyhow, this is an idea which obviously hasn’t caught on yet because Social Security reform stopped. We’ve got worse problems to worry about, but if we get through this crisis the very next thing on Obama’s agenda is going to be reforming Social Security. So I’m ready. So let me just end with Black-Scholes.
So in 1972 Black and Scholes wrote a famous paper. And what did they do? They started off by saying, this model we had, you see, of the stock market which follows a geometric random walk, it can go up by a certain percentage or down by a certain percentage. We’ve used that with interest rates too. They can go up or down by a certain percentage. Here the stocks can go up or down by a certain percentage.
We saw that you could solve models like that very easily. So Black and Scholes, Fischer Black who is a great economist, and Myron Scholes who won the Nobel Prize, but was not as mathematical as Fischer Black. Anyway, he started the hedge fund Long Term Capital. Myron Scholes was one of the people. They looked in 1972 at the returns each day on the S&P 500, and they binned them up like this. So here are the number of times the return was between .47 percent and some other number.
And then they compared that to a normally distributed random variable, and look how close to normally distributed it is. It’s practically like that. So this thing over here is exactly normally distributed, and this is the frequency graph of what actually happened that year, incredibly close to normally distributed. If you take the cumulative occurrence of each thing and compare it to the normal it’s incredibly close to the normal.
So they said, “Gosh, isn’t it great, this model of things going up or down by a certain percentage every year, maybe with a drift, that’s exactly the model the stock market seems to follow, and maybe interest rates follow that sort of model too,” and that’s why we’ve worked with all those models.
Well, it turns out if you make those assumptions and then you try to solve for option prices you can do it very quickly through some formula, which I’m not going to have time to present. It’ll only take one minute, but I’m going to skip that. So what happened then? So this was the second high point of, you know, there’s the CAPM model and then the Black-Scholes model. You’ve done problems with Black-Scholes now in the problem set to figure out the value of a call option. So I’m just pointing out that you could put it in a spread sheet like I’ve done, which will be on the web.
And you can do daily returns, and fix the standard deviation and stuff, and do backward induction incredibly quickly, and figure out the value of the call option, even get a closed form formula for it, and then explain why call option prices have the form that they do.
So it was a great triumph and it relied on things being normally distributed. I’m down to 30 seconds. So what happened? If you do the same thing recently–so you do exactly the same thing and bin everything up like Fischer Black did, not in 1972–shit, sorry. Not in 1972 but the last five years, say, they only did one year, but you do the last five. Actually I did this a couple years ago, so from seven years ago to two years ago.
If the pink thing is normally distributed and we do the same binning that Black and Scholes did, look everyday at what the return is and stick it in a bin and just do the frequency thing, what do you see? You see that there are a lot more times where in reality the move was very small, but there are a lot more times in reality where the move was very big.
So this is precisely what you call a fat tail, that you get smaller moves than the normally distributed ones but also bigger moves than the normally distributed ones. So you can have some big negative moves. And so we know we’ve had some huge negative shocks.
So I’m ending now with this thought that finance produced remarkable theories, remarkably precise predictions, and for decades at a time it seemed those were borne out in practice. But then looking back 50 years later or 30 years later on these discoveries we see that they don’t do quite as well as they seem to be doing at first glance. So something’s missing in the theory, and I think the subject is so exciting because it’s so connected to the world.
Everybody talks about finance nowadays. Anyone in the world is spending half their time thinking about finance and the theory is still up in the air because the old classics of the theory no longer hold. And so it’s an exciting time to be developing a new theory and maybe you’ll think about it in the future. So I’ll see you, some of you, next week.
Professor John Geanakoplos: You don’t have to clap. Thank you. So good luck in the exam, remember, it’s going to be simple problems exactly like the problem sets. I don’t expect you to do anything original on the exam. Just like the problem…
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