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# ECON 251: Financial Theory

## Lecture 15

## - Uncertainty and the Rational Expectations Hypothesis: Applications to Predicting Stock Prices, Default Probabilities, and Hyperbolic Discounting

### Overview

According to the rational expectations hypothesis, traders know the probabilities of future events, and value uncertain future payoffs by discounting their expected value at the riskless rate of interest. Under this hypothesis the best predictor of a firm’s valuation in the future is its stock price today. In one famous test of this hypothesis, it was found that detailed weather forecasts could not be used to improve on contemporaneous orange prices as a predictor of future orange prices, but that orange prices could improve contemporaneous weather forecasts. Under the rational expectations hypothesis you can infer more about the odds of corporate or sovereign bonds defaulting by looking at their prices than by reading about the financial condition of their issuers.

On the other hand, when discount rates rather than payoffs are uncertain, today’s one year rate grossly overestimates the long run annualized rate. If today’s one year interest rate is 4%, and if the one year interest rate follows a geometric random walk, then the value today of one dollar in T years is described in the long run by the hyperbolic function 1/ √T, which is much larger than the exponential function 1/(1.04)^{T}, no matter what the constant K. Hyperbolic discounting is the term used to describe the tendency of animals and humans to value the distant future much more than would be implied by (exponentially) discounting at a constant rate such as 4%. Hyperbolic discounting can justify expenses taken today to improve the environment in 500 years that could not be justified under exponential discounting.

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html## Financial Theory## ECON 251 - Lecture 15 - Uncertainty and the Rational Expectations Hypothesis: Applications to Predicting Stock Prices, Default Probabilities, and Hyperbolic Discounting## Chapter 1. The Rational Expectations Hypothesis [00:00:00]
But the world is much more complicated than that. It’s a world of uncertainty, and in a world of uncertainty, economics comes into its own, I think, as a fascinating subject. So I spent a little time reviewing some mathematics for you last time that many of you already knew, so I’m going to take that for granted going forward and just start over, this time from an economic perspective instead of a mathematical perspective. So suppose today that we assume that you could buy a stock today whose price tomorrow could be 104 or 98 with 50:50 probabilities and we assume that everybody knew the probabilities. Know probabilities and maximize expected payoff next period, okay? Well, + payoff this period. If we assumed–and we’re going to drop this assumption, but I’m going to keep it for a little while–if we assumed that all people cared about was their expected payoff next period and of course they care about their payoff this period, what would the value of the stock have to be? Well, under the simple rule for how people act, you’d take 1 half times 104 + 1 half times 98, and that would give you–what would that give you? It’d give you 101, okay, because this is + 4 times 1 half is + 2; - 2 times 1 half which is - 1, so it’s + 1, so that’d give you 101. So you would say that the price of the stock today would have to be 101. Now we could slightly refine this utility function and say people maximize the discounted expected payoff next period + the payoff this period. And if the discount is 100 over 101, then we’re going to have to multiply this by 100 over 101 and we’ll get a price of 100. Okay, so that’s the basic first step. We can incorporate uncertainty by assuming people replace the uncertain outcomes with certain outcomes in their head, and then discount, just like we’ve seen before. Of course, before we had utility functions but I’m not going to do that quite yet. I’m just going to say, suppose that we just did that, right? That would give us a theory of how people manage uncertainty and react to uncertainty and how they set the prices. So it’s the expected–expectation theory of pricing. Now before we complicate the theory, I want to just take this literally as true and make some inferences from it. Well, the first inference you can make is that today’s price would then be the discounted expectation of tomorrow’s price. That’s just repeating the same thing, but what’s an implication of that? The implication of that is, if you didn’t know tomorrow’s price, know the expectation of tomorrow’s price, you could guess today’s price. I’m writing out this trivial thought because it’s such an important idea. Once you have a theory of how prices formed, you can always go backwards, and as the naïve, uninformed member of a society, you can learn, instead of learning about the stocks, you can learn all you need to learn, perhaps, by looking at the price. You may be interested in what the expected value of the stock is next period. To do that in a serious way, you’d have to study the firm, study the product, study the new inventions, the new technologies it was trying to adopt, get some idea of the quality of the manager. You’d have to do a million things to figure that out. But if you just look at the price today, maybe that’s going to give you a good idea of what the expected value of the firm is next period. And that’s another way–that also implies you can test the theory. Is it true that typically today’s price is a good forecast of the price tomorrow, the expected price tomorrow? Obviously you can’t just look at one instance, because you would just be looking–if things went up, you’d be just looking at the 104, and 100 wouldn’t be a good guess of 104. But if you did this the next day, and things were independent, on the second day, of the first day, you’d have a new price, 104 the next day, and you could see whether the price went up or down or not. And by doing this 1,000 times or 100,000 times, you’d get a good idea if, on average, today’s price was a pretty good predictor of tomorrow’s price. So here’s the–I did that experiment and here it is. Why is this so small? Okay, so from 1980–I didn’t do this. I got someone this morning to do this at my hedge fund. So what did he do? He said, suppose you had 1 dollar to put in each day starting in 1980, you could keep track of how many dollars–say you had 100 dollars, so it’s 100 dollars. You have 100 dollars to invest each day, starting January 1 Then I told the guy, or he decided himself, put 100 dollars in the second day, in the S&P 500 and see what happened to that the next day. Maybe it went up 3 dollars the second day. So your total after 2 days would be 4 dollars. Not your total return, although that’s how he’s written it. It’s just the first day, the stock when up by 1 dollar, the 100 dollars went up by 1 dollar. The second day it went up by 3 dollars, so altogether, it went up by 4 dollars. So the hypothesis is that today’s price is a good forecast of tomorrow’s price. So if you’re averaging the +’s and -’s over many days, so there are 250 days for 30 years here, that’ a lot of days you’re averaging. And this is the cumulative total of what would have happened to you, okay? Well, that 100 dollars, if you did that experiment 7,000 times, you know, 30 times 250. 30 times 250 is 7,500 times, you would have ended up with 350 or 400 dollars by the end. So does that contradict or confirm the hypothesis that we’ve just made, that today’s price is a good forecast of the expectation of tomorrow’s price? What would you say?
So you’re making–so this is percent. I’ve already divided by 100 to turn it into percentages. So you make a tiny return. On 100 dollars, you might go up on average the next day to 100 and 1 half dollars, but that’s making it–but this .0047 percent, so 1 dollar would be 1 percent, there would be a 1 here, but we’ve got a lot of decimal places there. So you’re dividing 250 by 7,500. Maybe I’ve got one decimal too many there. So this is a tiny number. So in fact today’s price is a pretty good estimate of tomorrow’s price. You have 100 dollars and maybe it will turn in on average to 100.0047 dollars tomorrow. So compared to knowing nothing, if you asked yourself what’s the average value of this stock tomorrow? No one’s telling you it’s normalized at 100. It could be 500, it could be 23, it could be 75. Who knows what the average of these stocks are? The S&P 500 are mixing stocks that are worth 3 with stocks that are worth 500 per share with stocks that are worth 75 per share. And it turns out that it’s such an accurate predictor that you only are off by a fraction of 1 percent, on average, each day. So compared to knowing nothing, you have a huge insight into what’s going on in the world and how valuable the stocks are going to be tomorrow. Tomorrow hasn’t happened yet. Already by looking at the prices today, you have a tremendous idea of what the prices are tomorrow. So that’s the first thing to notice, the theory’s kind of confirmed. The second thing to notice as well, it doesn’t seem perfectly confirmed, because this seems like a pretty positive thing. You know, it seems to be going up most of the time, and as he said, “Well, we haven’t done the discounting yet.” We should have done discounting, because tomorrow is not quite as important to you as today, so I shouldn’t have just been looking at return. I should have looked at return per day. So I should have discounted each day by whatever the interest rate is. Let’s say you think it’s 4 percent in a year, divided by 250, since there are 250 days in the year. So approximately, I should have gotten 250, I should have discounted by that. So when you do that, the number gets even much closer to 0, but it doesn’t come exactly equal to 0 and so we’re going to see that we need something else to make up the difference. But it’s such a tiny difference that needs explaining. So to summarize, we have this view that uncertainty is going to change everything that we think about the world and it will change a lot of things dramatically, but it’s not going to change the idea that today, the price today of things is a pretty good indicator of what their value is going to be tomorrow, if you replace value tomorrow, which is uncertain, by the expected value tomorrow. So you can still learn a tremendous amount about the world, just by looking at the market. ## Chapter 2. Dependence on Prices in a Certain World [00:12:18]That’s a very important lesson, so let’s go a little further though. Suppose that you thought, well, maybe people–maybe I want to ask a more complicated question. I want to say, suppose I only look at stocks that went up yesterday. I only look at stocks that went up yesterday. Now maybe there’s something about the market that, you know, momentum will keep carrying those stocks up tomorrow. So once the market gets rolling, maybe the market’s not such a good forecast. Maybe, as Shiller says, there are all these psychological forces at work, and once things get rolling and prices have gone up yesterday, the price will keep going up tomorrow. So today’s price won’t be such a good indicator of tomorrow’s price, because tomorrow’s price is probably going to be higher. So I’m going to repeat now exactly the same experiment, except instead of putting 100 dollars in all the S&P 500, I’ll only put the 100 dollars into the stocks that went up yesterday. Or I might even refine it by selling short the stocks that went down yesterday. Okay, so what does that do? Does that change the numbers? So if I blow this up, maybe it is blown up. Can’t do any better than that. Does that change the numbers? Well, no. In fact, it makes it worse. It’s closer to 0 now. So again, over all this time, it kind of went up to the same peak, but fell down even further, so this thing–so again, the stock prices today, even if I try to refine it and get more clever. I try to fool the market. I say, okay, the market does a good job on average. Today’s price is pretty good on average of predicting tomorrow’s prices. What about today’s prices of those stocks that went up yesterday, you know, the momentum thing? Maybe that’s not such a good–maybe on that subset the market’s not so good. Well, the market is pretty good on those too. So again, you have to do the discounting and you have to realize that there are a huge number of days here, so this tiny return is really nothing, averaged over all those numbers of days. Well, let’s see if we can come up with another strategy. I forgot what other strategy I tried out here. Oh, suppose you could say, “I want to choose only those stocks that went up 20 days ago, or 25 days ago or 14 days ago.” This number here, these bars here, represent for everything for the S&P 500, you try to say, “What’s the correlation?” That’s like the covariance but normalized so it’s between 0 and 1. What’s the correlation of a move yesterday and a move today? Does the fact that a stock went up yesterday mean that it’s going to go up today? Or does the fact that a stock went up three days ago mean that it’s going to go up today, between today and tomorrow? So if it went up two days ago–if it went up yesterday, from yesterday to today, does that suggest that it’s likely to go up from today to tomorrow? So what this says–of course, if you only did the experiment once, you’d always find that it did something. Okay, so you have to do many experiments, and then figure out, it’s a statistical thing, to sort of guess what the correlation is, estimate the correlation. Then you have to see whether it’s significant. So anything in this blue band means the numbers are insignificant. So these bars represent what the correlation is. So no matter how far back you go, you basically, knowing which way the stock went 27 days ago tells you almost nothing about which way the stock is going to go today. There’s almost no correlation. If it went up 27 days ago, it’s statistically, over the last 7,500 days, slightly more than half the time, it went up again today. But such a small fraction of the time did it go up again, the more times up than down, that it’s statistically insignificant. If you only do it five times, it’s going to have to be one way or the other, so if you do it an odd number of times, it can’t be exactly even so you just figure out what the statistical significance is. So none of them hardly, almost none of them, are statistically significant. So once again, it’s not only the case that today’s prices are good forecasts of tomorrow’s prices, but today’s prices, even if you add some information to it, seem to be–even if you try to refine your set and look at only buying stocks that 27 days ago went up, the prices of those stocks are still going to be a reasonably accurate forecast of tomorrow’s prices. So I did one more experiment, or Rashid did one more experiment for me, in case he hears this in a year. He did the same thing on a portfolio of stocks. So he looked at a 12 month rolling average. He looked at the stocks that had done particularly well in the past 12 months and he bought those, and then he looked at the stocks that had done particularly badly in the last 12 months and he shorted those, and here’s what his returns would have been, just taking the daily thing and just adding it. And you see, you get almost exactly back to 0. So this was the original compelling evidence, things like this in the 1970s and 1980s, led people–economists–to say that the prices of very many things seem to be very accurate guides to future prices, and they called it rational expectations. So the high water mark of this theory was in 1983, I think, the most amusing example, was Richard Roll, who taught at UCLA, and oranges. So Richard Roll did the following experiment. He said, it turns out that for concentrated orange juice, 97 percent of the oranges that are used for concentrated orange juice are grown on trees that are very close to Orlando, Florida, where the weather is pretty much the same. I mean, it’s a small area, so whatever the weather is, it’s that weather over the whole area. It’s amazing that so many of the oranges are grown in the same place. I’m talking about concentrated orange juice. California’s no competition for Florida. In fact, no competition for Orlando, Florida when it comes to concentrated orange juice, not oranges in general. So he said to himself, “How good is the market at predicting the price of orange juice, at predicting next period’s price of orange juice?” And he found, just like we did here, it’s quite good. But then he said, “Maybe there’s other information that the market doesn’t know about.” So he said, “What about the weather?” So the weather has a tremendous effect on orange juice prices, because if it’s–four hours of freezing temperatures starts to kill the trees, then you get less supply of oranges for the concentrated orange juice, and then the price goes up. So he said, since 1970 or so, the US Weather Bureau has spent 250 million dollars building all these weather forecasting units that make daily–in fact, they make 36 hour, 24 hour and 12 hour forecasts of what the weather’s going to be next period. So he said, “Really, if the market is so good and the market price today is really telling an uninformed investor what the price ought to be tomorrow, let me see now, by getting a record of the weather reports, could I improve on the market price? By putting together today’s market price and the weather report today, the weather prediction today, could I make a better forecast of the market price tomorrow?” And he found out, no. Statistically, the weather + prices does not improve price forecast. So how could you interpret that? How could that possibly be that knowing the weather reports doesn’t help you predict the price better the next day than today’s price? How could that be? What’s the obvious reason for that? Yeah?
So that was a pretty stunning conclusion, but he didn’t want to stop there. So what did he do next? What if you were–you know how in comp. lit. they always says things backwards, the reader is detective or the detective is reader, you know–anyway, when I took comp. lit., that was the gist of every course, was to do everything backwards. That’s how you knew you were clever in comp. lit. What would a comp. lit. person have done? Yes?
So he said, suppose that whatever the forecast is here, you now say since the price went up, we’re now going to forecast that the weather guys–the price going up means they’ve learned something here about the weather probably being bad. So the question is, did the weatherman learn the same thing? So he says, “Let’s test the hypothesis that when the price went up, these guys learned more about the weather than the weather predictors did, so that in fact the actual weather from this prediction is likely to go down.” And that’s just what he found. You can’t use weather to improve the price prediction of prices, but you can use prices to improve the weather prediction of the weather people. That was one of those stunning confirmations of the rational expectations hypothesis, so what could explain that, by the way? Is that just crazy or an accident, or is there some logical explanation for that? Yeah?
## Chapter 3. Implications of Uncertain Discount Rates and Hyperbolic Discounting [00:24:42]Okay, so that’s the efficient markets hypothesis, which seduced many in the economics profession, and there’s still a tremendous amount of truth to it, at least at the level–if you don’t know anything and you want to know something about the future, look at the prices today. That’s going to tell you a tremendous amount about the future. Now the question is whether it’s as precise as Roll seems to suggest, and we’re going to see that it’s not going to be. But anyway, for a while, these people, the rational expectations school, which is mostly in Chicago, they had the view, and Fama was one of the leaders, they had the view that this rational expectations pricing was the best-documented truth in all of the social sciences. That was what Fama said. So we’ll have to come back to see that that’s not always the case, but certainly looks good in these graphs. Okay, that’s the first idea. Now the next idea that we looked at was, what is the most important thing to be uncertain? Well, there’s output that you’re uncertain about, but the next most important thing is the discount, the interest rate. After all, that’s the most important variable in the whole economy according to Fisher. Who’s to say the discount is always exactly the same thing, so, uncertain discounts. So now we said–and you’ve done in the problem set–if the interest rate is 100 percent, it might go up to 200 percent, say, or it might go down to 50 percent, and now you want to ask, what’s the value of 1 dollar here? It’s a little subtler, because here, the expectation was all that mattered, the expectation of the payoffs. If I change this to 106, and I change this to 96, I haven’t changed the expectation, so the price is going to stay the same. So the variance has nothing to do with what the price is. But things can get subtler. Let’s suppose that what’s changing is the discount rate. Now the variance is going to have a big effect on what the values of things are. So if we think this is happening with 50:50 probability, the guy–so what do I mean by this model? Today you know that the value of something tomorrow is going to be 1 half of what it pays tomorrow, up + 1 half what it pays tomorrow down, discounted by 100 percent. Tomorrow, you’re not sure whether you’re going to be discounting it 200 percent or discounting it 50 percent. So then the value today is going to be 1 over (1 + 100 percent), times (1 half times 1 over (1 + 200 percent) of 1 + 1 half times 1 over (1 + 50 percent) times 1). Because over here, you know that this dollar’s only going to be worth 1 third to you. Over here, you know this dollar’s going to be worth 2 thirds. So that’s the 2 thirds here and the 1 third here, and there’s a 50:50 chance of each of these, and you’re going to discount it by 1 over 100 percent, so that’s what the value is to you today. So now you did a problem set where you had to do a bunch of these things. We’re going to call that D(2), because that’s what you would pay today to get 1 dollar for sure at time 2 in the future, and D(1) is going to be just 1 over (1 + 100 percent), which is 1 half, which is what you would pay today to get 1 dollar for sure at time 1. And I could compute D(3), and any other D that I wanted to. Okay, so we’re going to see that interest rate uncertainty is the most important uncertainty in the economy. The value of everything is going to change. If the interest rates go up, all the bonds are going to go down in value. All the mortgages are going to change in value, although sometimes they go in surprising directions. But everything’s going to change in value when the interest rate moves. That’s going to subject everybody to tremendous amounts of risk and we have to figure out, how are they going to cope with all that risk? Before we answer that question, we want to answer the simpler question, how are they going to value things? And here we just have the same tree that we had before. So that’s what we did last class, and I just wanted to finish that thought, which I didn’t get a chance to do. So for period 3, we could have done a 3 period thing and assumed that this went up to 400 percent and would have gone to 100 percent or down to 25 percent, and then still paid 1, 1, 1. Okay, so that’s payoff for 1 dollar for sure, but now we’ve got still more uncertainty in the interest rates. So you figured out in the problem set what the value of that’s going to be and you got D(3). So I could have done this for D(4) and any other T that I wanted to and in fact, that’s also what you did in the problem set, you did it for all the way up to D(30). Okay, so I want now to just say one thing about the environment, before–we’re going to come back and analyze this over and over again to see the risk the whole economy’s exposed to and how people cope with that risk with interest rates changing, but I want to make one observation. These numbers, D(1), D(2), D(3), D(4), they reflect people’s attitudes towards the future. What would you pay today to get 1 dollar at time 1? What would pay today to get 1 dollar at time 2? What would you pay today to get 1 dollar at time 3? So what is the shape of that function? Well, in the case of certainty, with a constant discount rate, that function would have to decline exponentially. So it would be an exponential decline. Why? Because this would be equal to 1 over (1 + r), and this would be equal to 1 over (1 + r) squared, and this would be equal to 1 over (1 + r) to the fourth, etc. So after 100 years or 500 years, you wouldn’t care, as long as r is .03 or something percent, .03, r is 3 percent. As long as it’s a number like 2 percent or 3 percent, after 500 years, you just don’t care at all about what’s going to happen. If the whole economy, the society, is discounting the future and trading it off like this, you don’t care at all about the future. So environmental improvements today, which don’t have an effect for 200 years, would be regarded as stupid ideas. And environmentalists have been trying desperately to make an argument that 200 years from now really matters. So of course, they argue about the interest rate, but really, all they’re doing is they’re arguing that the interest rate, instead of being 3 percent should be 1 percent or something like that, and that’s not really helping, because even 1 percent, if you keep doing that for 500 years, you’re going to get a pretty trivial number by the end. So let me ask you the following question. Suppose you could have 15 dollars today, or–so this is an experiment Thaler ran, who was a behavioral economist. So, next month, 1 year and 10 years. How much money would you want next month instead of the 15 dollars today? Somebody give a number, shout out a number. What seems equivalent to you? 20. It happens to be exactly what the average–do you know the Thaler experiment? That’s precisely the average. Thaler did a class like this, averaged all the numbers, he got 20, amazingly. What about for 1 year, what would you say? 50 to 100. And what about 10 years? 200. Okay, so I’ll tell you the Thaler numbers. I stupidly forgot them all. What a turkey. Okay, so the numbers of Thaler–let’s just go with those numbers, but what do you think about those numbers? So Thaler got–it’s amazing–Thaler got 20, 50 and 100 were Thaler’s numbers, so very close to what you’re telling me. 50 and 100. So what’s the matter with those numbers. Let’s go with Thaler’s. They weren’t that different from yours. What’s the problem with Thaler’s numbers?
In fact, the reciprocal of that number is the discount, is .75 to the 1. So this is .75 obviously, and then (15 over 50) to the 1 tenth–1 twelfth–is .9, and then (15 over 100) to the 1 twentieth is .98. So you’re discounting by 33 percent, something like that, by 10 percent and then by 2 percent. So your discount rate is falling rapidly. You do experiments with animals, you get the same conclusion. You ask the animals–you can make an animal work and then they’ll have to wait a certain time to get the food. Or if they work harder, they can get more food, but they have to wait a longer amount of time. So you try and do the experiment. I’m not sure these actually are believable, but anyway, they do these experiments and they figure out how much the animal is –you know, these are birds and mice and all kinds of things–trading off waiting for getting a bigger reward, and they get the similar kinds of numbers to what Thaler got by talking to psychological experiments with real people. How can you explain it? In the world of constant discounting, you couldn’t possibly explain it. Now of course, you could explain it by saying, “Everybody’s discount rate is going to get smaller and smaller over time.” Their annual discount rate is getting smaller and smaller over time. But that’s totally unbelievable, you know. It’s just, you know that 1 year from now, if you were asked the same kinds of questions, you’d give the same kind of answers. So your discount between today and next month is going to be the same next year as it is now. So it’s not the case that the 1 month discount happens to be high now because you’re in college, and then the day you get out of college, you’re going to be more mature and so you’re going to have a smaller discount rate. When you get to my age, you’re going to be even more mature and have a smaller discount rate. That doesn’t happen. The discount rate doesn’t go down like that. In fact, if anything, if you’re rational, it ought to go up. I’m closer to death than you are. If I don’t get the stuff now, who knows when I’ll ever get it? So the discount rate should be going up, not going down, and yet it seems like there’s so much evidence that it goes down. So this is a big puzzle in economics. So I just offer, again, I’m going to make a habit of offering theories. I’m not saying this is the right theory. I’m just simply pointing out that if you had this random discount, put uncertainty into the discount, put uncertainty into the interest rates. Uncertainty in the interest rates is the heart of finance. Every single person, every single serious finance person, thinks about–what do you call it?–uncertain variability in interest rates. So I take the simplest possible process, where the interest rates can go up or down by the same percentage. So for example, you could start at 4 percent, and then the variation or the standard deviation could be 16 percent, which means that 4 percent basically goes to 4 percent times 1.16 or to 4 percent divided by 1.16. That process actually, times e to the .16 or times e to the -.16, which is very close to times 1.16, that geometric random walk is the basic model of finance. And what you found in your homework is, you were supposed to find, that as you go out further and further, the effective discount rate does go down. And what I forgot to say, the punch line, Thaler’s numbers here confirm what all the behavioral economists suggest, which is that there’s hyperbolic discounting. So what they confirmed in these experiments is that if this is D(t), this should go down like t to some power, you know, t to the - some power, t to the -2 or t to the -1 half, or something like that. They don’t pin down what this number is, but t to the -a, so it goes down much slower than the exponent, which is some exponent like .9 to the t. That goes down much faster than that does. This is a polynomial in t. This is an exponential in t. So it’s going down much faster. So this is a classic–Thaler’s numbers are a classic polynomial. In fact, with exponent 1 half. Thaler’s numbers fit t to the -1 half, if you do the right starting point. So what did I show? I showed that any geometric random walk, no matter where you start, no matter what r(0) is, no matter what you start, no matter what standard deviation you go, if you figure out the sequence of numbers, D(1), D(2), D(3), D(4), not up to 30 years, which is where everybody else stopped, because bonds end at 30 years, but you do it for 100 or 200 or 500, D(t) is always eventually going to be equal to some constant times t to the -1 half, exactly consistent with Thaler’s numbers. So I don’t know if that’s the explanation for hyperbolic discounting, but I thought it was pretty interesting, and anyone could have done it if they just didn’t stop at 30 years, just kept going. And then there’s some mathematics, you could compute examples, but there’s some mathematics to prove that asymptotically, that’s the right formula. Okay, so in fact this paper, I wrote this with a co-author, Doyne Farmer, whose daughter is a sophomore here and whose son just graduated by the way. He’s in Santa Fe. So if you look at the picture here, you can see that these are the D(t)s when you exponentially discount. I’ve got it on a logarithmic scale, so if you exponentially discount, things go–the Ds drop off really fast. That’s the dotted line, really fast. But if you do this random thing, you get the thing that goes much slower, and it goes with a slope of -1 half. Since I plotted things on a log scale, that’s just what this means. Taking the log of this, you get a straight line with slope -1 half, and that’s just what we found, and we managed to prove that that always has to happen. So if you look 500 years in the future, you start with 4 percent and you assume a constant discount rate. After 500 years in the exponential, nobody could possibly care about 500 years from now, But 500 years from now is 1 percent as important as now [in the uncertain case] if you discount–if everyone knows the interest rate is 4 percent now, and it’s going to go up or down and keep going forever, so it’s quite shocking. Okay, so that’s it. We’re going to come back over and over again to this, and this is the yield curve that you get, the 0 yield curve like in the problem set, goes up and then starts coming back down. All right, does anyone want to say anything about discounting or how to compute this stuff? You know how I did this. Yes?
The reason is because if the interest rate is moving in a geometric random walk, so it’s doubling or getting multiplied by 1 half, the geometric average is, it stays where it was before, but that means since the arithmetic average is always bigger than the geometric average, the arithmetic average of 200 percent and 50 percent is actually bigger than 100 percent. So at the beginning, you’re sort of going to be doing this arithmetic average and things are going to be getting bigger for a while. But when you go out farther and farther, why doesn’t that matter? So what is the intuition? And by the way, this is a common thing in finance with–someone named Weitzman, who was at Yale and now is at Harvard, did suggest this idea in economics. He said, for the environment, you should always use the lowest possible interest rate, and why is that? Let’s do an example. Suppose the interest rate was going to go to 100 percent, so you’re going to multiply by 1 over 2 and then keep multiplying by 1 over 2 forever, the interest rate stayed the same. Or let’s say the interest rate was going to be less discounting, 2 over 3 and was going to stay there forever. Okay, and you get 1 here and 1 here. So I’m doing a very simple case where 50 percent of the time, it stays at 100 percent forever and 50 percent of the time, it goes to 50 percent. This is 100 percent and this is 50 percent as the interest rate. Stays at 100 percent forever or 50 percent forever. So you multiply by 2 thirds forever or by 1 half forever. So this could happen with probability 1 half and this could happen with probability 1 half. If you average this, multiply by all the 1 halves, and this multiplied by all the thirds, by all the 2 thirds, the 1 half is irrelevant, because this is such a tiny number compared to this one, right? Because every time, you’re multiplying by such a small number up here compared to this, this thing is just negligible compared to this. So really, the total here is entirely given by what happened down here. Okay, so it’s 2 thirds to the Nth power times 1, plus a totally negligible thing. Okay? So you’re going to have half of this value is going to be the value here. So the high interest rate, the 100 percent interest rate, didn’t matter. It’s only the low interest rates that matter. So why is that? Because in the random walk, when you follow a random walk, it goes like that, so if you end up with a really low interest rate at the end–so here we start with 4 percent. By the end, because it’s a random walk, you don’t know where the final interest rate’s going to be. It’s going to be some normally distributed thing like that. You don’t know what the final interest rate’s going to be, but the low interest rate’s here at the end. Here’s where the interest rate was the same as where you started, back to 4 percent. So I’m not saying that people typically go down here. That would be a ridiculous assumption. I’m saying on average, they’re at the same level they were today. But the paths where the interest rate ends up high probably were high the whole way along, so they kept getting discounted, so they don’t make any difference. The paths where the interest went low, the path was probably low the whole way along, and that’s why those are much more relevant paths than these. So when you take your average, to get it, it’s going to be, in this particular example, as if it was 2 thirds, 50 percent discounting forever, but of course, you’re only averaging over these low paths, so I have to put 1 half in front of it. That’s why it’s not t to the -1 half, it’s an a times t to the -1 half. Okay? So that’s a vague intuition, but it maybe helps a little bit figuring out why that happens. Okay, so I don’t know, this may have some significance for the environment. So I personally think that we should do something about the environment, even if it’s only going to be 500 years away. I don’t think we should just discount it to 0 because the interest rates are 4 percent and 4 percent to the 500 ## Chapter 4. Uncertainties of Default [00:46:53]Okay, so I’m going to march on now if there are no questions. What’s the next most important kind of uncertainty that you see in the market all the time? It’s the chance of default. Now we’re going to see very shortly that default and the possibility of default changes a lot of things. But you could still be a rational expectations guy and believe that default is just no big deal. It’s just that the payoff, which over here was 104 and 98, the default just makes the payoff lower. So what’s the typical thing that defaults? It’s a bond. So a typical thing that defaults is a bond. So suppose I had a 1-year bond from Argentina that could pay 100 or it could pay 0. This is an Argentine bond. So you’ll have to forgive me if you’re from Argentina. And then we have the American bond that can pay 100 or it can pay 100. Okay, so what do you see? These both promise 100. The American bond, if you look at the market today, is going to sell for a higher price than the Argentine bond. Why is that? Because people assume that the American bond is not going to default. So even if you put a 0 here, they assume that the probability for the American bond is 1 here and the probability of the Argentine bond is some number, .8 or .2 or something. The question is, what’s the number that they put here? So there’s uncertainty about defaulting, and if defaulting means paying 0–we’re going to think a second about what it really means–if it means paying 0, that’s no big deal. We just calculate–in the expected payoff we have to take into account, not the usual dividends and all that stuff + 100. We also have to take into account the possibility things default. So let’s look at some of those curves. Oh no. Oh no, say it ain’t so. Did I forget the curves? Hang on. So I’ve got another one on my…oh dear. I got another one of my–I think I’m on the internet in here by the way? No. Oh, I didn’t realize this would happen. When I lost the internet–I had opened the file, but it doesn’t–yes. Okay, this will only take a second. Yeah, wireless, connect. Connection successful, okay. So I can close this and this and now I can–sorry, it will only take me one more second. I beg your pardon for this. I had it. It disappeared when I walked over here. It’s going to take a second for me to get on the internet. So what could we do here? We could figure out what the price of the Argentine bond was. So suppose the price of the Argentine bond is 80, and the price of the American bond is 95. What do you think–what does the market think the chance of Argentina defaulting is? How would you figure that out? So let’s write d here and 1 - d here. You don’t know anything about Argentina. You know it’s a great country, they have wonderful everything, music, beautiful people, everything, you know. Okay, but their bonds happen to sell for a lower price than American bonds do. So assuming the American bonds can’t default, because we’re just going to print the money, and Argentina might default, because maybe they’ve tied their payments to the dollar, so they can’t just print the money, what do you think D is? How can you figure out what D is? <>
So you see that–oh no, this is JP Morgan. What are the chances that after 1 year, JP Morgan’s going to go out of business? The market thinks it’s surprisingly high, 1 percent and 1 half. I should have asked you what you thought. After 10 years, they think that JP Morgan, the leader, the great investment bank which is now a regular bank and the most successful thing, they think 10 percent, the market thinks it will be out of business within 10 years. So how did we know how to get that number? We can do another one. We can do Citibank. Citibank is a totally lousy American bank that ought to have gone out of business already but it’s being propped up by the government. So of course, people think the government’s going to keep it propped up, so over a year, it’s actually got a smaller probability or about the same probability as JP Morgan, because everybody knows, the government’s going to keep propping it up. But then, you know, eventually maybe the government’s going to stop worrying about Citibank and so after 10 years, Citibank, what used to be the biggest bank in the world, has got a 25 percent chance of going out of business, 25 percent it won’t even be here. Okay, so how did I know what those numbers were? How did the Ellington trader figure that out? Every morning they figure out the interest rates and they figure out the implied default probabilities. So what is the implied default probability of this Argentine bond? How would you figure that out? Well, according to our theory, what is the price of the Argentine bond? It’s 80. What should I write that equal to? What?
So the payoff of the Argentine bond is (1 - d) times 100 + d times 0. So that’s the expected payoff. That’s what you expect to happen here. But she went–so she not only did that, but she went one step further and she said, “How would you–you have to discount it.” So how does she know how much to discount it? Well, you could buy an American bond, just as well as an Argentine bond, so basically we know that the discount rate, the world discount, everybody has–the Argentines can buy the American bonds and so 100 dollars for sure is worth 95. So according to our hypothesis, you take the expected payoff and then multiply by the discount, 95 over 100. So that’s = just as she said, to (1 - d), times 95. That’s what she said and she was exactly right. So therefore you can figure out that 1 - d is 80 over 95 okay, and so d is 1 - (80 over 95), which is something like 15 percent, a little bit more. Maybe it’s 16 percent, something like that. So it looks like there’s a chance of 16 percent that Argentina is going to default. So that’s how they figured out what all these default probabilities are. Any questions about that? Let’s see if we could do a 2 period version, okay? So they’ve done 1 year. Now I’m not going to show you what Argentina is. Last year I got to show everybody what Argentina was. Unfortunately, my hedge fund’s emerging market trader went out of business last year in the crisis, lost a lot of money, so we closed it down. So I can’t show you what–it’s too complicated. I didn’t bother to get all the countries’ prices and the default curves. We don’t bother to compute them anymore, because we’re not trading them. But we still are trading all these potential corporate bonds. All right, suppose it was 2 years. Suppose we had a 2-year thing, so this is the US. Now I’m going to do a simplified version first and then we’re going to have to complicate it. Okay, so I guess I’m assuming that we’re doing–okay, so let’s do the case where we’re doing 0s. So here’s America and here’s Argentina and we’re just going to be trading 0s, okay? So it’s going to get a little more complicated with their dividends, but not so much complicated. So there’s a 2-year…those curves should be parallel, so here’s Argentina. Now let’s say that the American–so here we’ve got the 1 year bond, pays off in yellow. So 1, 1 here. And let’s assume–it doesn’t really matter, but let’s assume that that price is .9 and then the 0, the 2 year 0 in America, which pays off 1, 1, 1 here. So I’ll just write that in pink, 1, 1, 1, is worth 70–what did I do? .72. Okay, now let’s do the same thing in Argentina. Let’s say the 1-year bond, which pays off 1 here and 0 there, this is default, so it’s probability d. Let’s say the probability of d is always the same. The 1 year Argentine bond let’s say is worth .54 and the 2 year is .216, let’s say. So now what does the 2-year–? So now we have to look at these paths. What is the 2-year Argentine bond going to be worth? It’ll be worth 1 here, 0 here. But now if the 1 year Argentine bond defaults, it’s the same country, so if they’ve gone out of business and aren’t going to pay their 1 year, they’re not going to pay their 2 year either, so it’s going to be 0 here and here. So let’s assume that’s the payoff. So here we know the 1-year American bond is 90 cents, the 1-year 0. The 2 year American 0, 72 cents and the 1 year Argentine 0 is 50–what did I say?– 54, and the 2 year is 21.6 cents. So how are we going to figure out what these–and why assume the same default probability? I think I’ll make it more interesting and assume d So solving for d
Now what’s the chance of default in the second year, assuming you haven’t defaulted already in the first year? If you default in the first year, you’ve wiped out everything anyway, including the 2-year 0. So what should I write now to get d
So actually, it’s quite interesting. We know that the probability–I wonder whether this was cumulative default. Must be cumulative default. So we know that things are getting worse in Argentina. The first year, there’s a 40 percent chance of default, but even if you get through the first year, the next year there’s going to be a 50 percent chance of default. Okay, so things are getting worse and worse and worse in Argentina in this example. I’m not saying in real life, but in this example. But by doing this, for any bond of any corporation or any country, you can learn a lot about what the market thinks about that country. So the market doesn’t think very much of Citibank. It thinks Citibank in 10 years could have a 25 percent chance of going out of business. And for JP Morgan, it thinks a lot better of JP Morgan, but surprisingly not as much better as you would have expected. They could go out of business with 10 percent probability. There’s very little chance they’re going to go out of business in the next year or two, mainly because the government is there protecting them all. But in 10 years, you know, it could be very different. And so that’s shocking to most people, I think, a shockingly high probability of those things going out of business. You wouldn’t know yourself what those things were, except if you looked at the market. Now, I actually could have computed the prices this way, which is the way we used to compute them at Ellington, but there’s a more direct way of computing them. There’s something called a credit default swap. A credit default swap pays 1 dollar in case a bond defaults within some time period. It actually pays 1 dollar for every dollar of principal, 1 dollar in case a bond defaults within some time period. So I assume here that when you default, you get 0. You don’t always get 0. Sometimes the guy is willing to work out something and pay you part of what he owed you, because after all, Argentina, if they default, and the US is angry about it, it can put a lot of pressure on Argentina, refusing to trade with it, doing all sorts of other things. Not that much pressure, but some pressure, and so maybe Argentina, if it can’t pay, it’ll agree to pay less and say, “Let’s forget about the whole thing. You understand why we can’t pay. We’re just–bad things happen. It wasn’t our fault. It was unlucky, so don’t hold us to it. Take a little bit less and let us get on with our lives.” So instead of putting 0s down here, maybe you would put a recovery down there. So we’ll have to come back to that. So in case that there’s a recovery, the credit default swap pays only the gap between what it was promised and what it actually paid. So it pays 1 dollar–pays 100 percent of the loss for any bond that defaults. So it pays 100 percent of loss, in case a bond defaults within some time period. Now that’s if you buy 1 credit default swap. You could buy 50 credit default swaps on the same bond, so then you’d get 50 times the loss. So we’re going to come back–this is going to be one of the causes of the crisis, that these credit default swaps got written that were so big. So you wouldn’t have to actually do the computation I actually showed you. You could just look at what the price of the credit default swap is. Because here if the payoff is 0, that means the credit default swap is going to pay the whole 100, so its price is 16. That’s telling you that everybody thinks–it wouldn’t be 16–so what would the price of the credit default swap be over here, by the way? It wouldn’t be 16 as I just said. That was wrong. What would the credit default swap price be over here?
This is d. Over here is just d. So knowing d of course, that tells you the credit default swap. Knowing the price of the credit default swap, you could get d. So you could deduce d in two different ways, either from the American bond price or from the credit default swap. In either case, you have to know the American bond price in order to figure out what the discount rate is. So the credit default swap is sort of overkill. It’s another way, it used all the information plus more to get the same answer more quickly. Now what would the credit default swap be worth over here? It’s a little subtler. What’s the credit default swap worth here? So credit default swap on Argentine 2 year bond = what? What would it be worth? How much would you pay for the credit default swap in this case? Well, 2 year bond over 2-year horizon, okay, so it’s only a tiny bit subtler than before. The 2-year bond could default in any one of two cases. So it could default here or it could default here, so you’re going to get the American .9, that’s the discount. I don’t know if you can see it over there, so let’s write it over here. You could get the American–so over here, what’s the value of going down here? It’s 1 - d Well, we have to–the payment’s coming in the second period, which in America is discounted at the rate of .72. So that sum is going to give you the value of the credit default swap. So d All right, so I want to end this discussion of default with one observation, one theorem, which is that you can get all these numbers incredibly fast. How can you get these numbers incredibly fast? What’s a trick? If recovery is 0–I’m only going to talk 2 more minutes here. I realize I’ve come to the end of time–if recovery is 0, the chance of default–the defaults, you know, if you default the first period, you default on all the bonds. If you default the second period, you default on all the bonds. Then the trick that the young lady who asked the first question pointed out right away is that–I don’t know where I wrote it–is that the chance of default from the very first equation is going to be very simple to compute, because you’ve got the–oh, I lost her equation. Anyhow, okay, because from the first equation where we had the chance of default here, we just got 1 - d is this 80–okay, how did we get this? We had the price of the Argentine bond is 80, compared to the–okay, so the American bond price is 95, so we just took 80 over 95. That ratio was the chance of not defaulting in the first year. Okay, so she did this incredibly quickly. This was a faster way of doing it. The Argentine bond is worth 80 ninety-fifths of the American bond. They’re only paying in one state. That means the chance of Argentina paying divided by the chance of America paying, that’s the only state where you get any money, must be 80 over 95. So that’s a very fast way of figuring out what 1 – d And for the 2 period thing, it’s equally fast, okay? So you just do 1 - d [end of transcript] Back to Top |
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