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# ECON 251: Financial Theory

## Lecture 9

## - Yield Curve Arbitrage

### Overview

Where can you find the market rates of interest (or equivalently the zero coupon bond prices) for every maturity? This lecture shows how to infer them from the prices of Treasury bonds of every maturity, first using the method of replication, and again using the principle of duality. Treasury bond prices, or at least Treasury bond yields, are published every day in major newspapers. From the zero coupon bond prices one can immediately infer the forward interest rates. Under certain conditions these forward rates can tell us a lot about how traders think the prices of Treasury bonds will evolve in the future.

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html## Financial Theory## ECON 251 - Lecture 9 - Yield Curve Arbitrage## Chapter 1. Defining Yield [00:00:00]
Now, yield is an extremely common expression in finance, and it turns out not to be that well defined, often, or that useful. But the word is so important and has been used so often that it still hangs around, even when probably we should use different concepts. So remember the yield was an attempt to look at an investment, and without paying any attention to the market or anything outside the investment, just looking at the investment itself, try to assess, give a number, quantifying how attractive the investment was. So we said you could apply that to a bond–it has cash flows. You could apply it to a hedge fund that’s taking in money and paying out money, and the formula we came up with said that if the cash flows are given by C(1), C(2), the net cash flows, C(T) over the course of the period, and its price is some P(0), maybe it’s a negative cash flow, so C(0). So some of these cash flows might be negative and some of them might be positive, then we should just look at the number Y, such that discounting all these things at rate Y gives you 0. The Y that did that was what we defined as the yield of the investment. So we saw that that had some advantages. For example, in a hedge fund, if you just look at the rate of return it makes on its money every year, that doesn’t take into account that in some years, it’s got a lot more money. So if those were the years that lost money, and the years when it hardly had any money were the years it made money, just taking the average, the multiplicative average, the geometric average of all those yearly rates of returns, would give a misleading figure. Well, the yield also gives a somewhat misleading figure, and I don’t want to spend too much time on why it might be misleading, but I’ll give you just an example. Suppose that the cash flows happen to be 1, -4, and 3. Now what’s the yield to maturity? Well, there are two of them. You could have Y = 0, because 1 over (1 + 0), + - 4 over (1 + 0), + 3 over (1 + 0), is just 1 - 4 + 3. That equals 0, so the yield to maturity of 0 percent, the yield of 0 percent, makes this have present value 0. But also I could try Y = 200 percent, and then I’d have Y–I’d put a + 2 and a + 2 squared here, and I’d have + 1 - 4 thirds + 1 over 3 squared is 3 over 9, so it’s + 1 third. So it would be 1 minus 4 thirds, plus 1 third, which also equals 0. So is the yield to maturity, the internal rate of return 0 percent or 200 percent? It’s ambiguous. So yield to maturity can’t be the right way of doing things. To go back to the hedge fund example, you know, the hedge fund was taking in money, paying out money, taking in more money, paying out money, and we calculated the yield to maturity. Well suppose that there was some period, you know, here, at which point everyone had taken all their money out, so the hedge fund wasn’t actually doing anything for a bunch of years, maybe for a long time, and then it started up and took money in and paid money out and stuff. Well, because the gap in time was very long with nothing happening, if you take a positive Y, the stuff that happens in the second incarnation of the fund is hardly going to be making any difference, because by that time, it will all be discounted a lot. So the yield will depend too sensitively on stuff early rather than stuff late. And so again, you get into troubles yielding just yield to maturity, so that can’t be the right thing to do, even though people have done it for years. So the word, however, lives on, and there’s no getting rid of the word because it’s used in common vocabulary. Now what would Irving Fisher say you should do, if you had to summarize how good an investment was? What’s his lesson? What do you do? An investment where there’s no doubt about what the cash flows are going to be, what would he say you should do, to evaluate the attractiveness of it? What’s our lesson, our main lesson from Irving Fisher? What would he say? Yes.
You know what the market is doing, you know what the interest rates are. Use the market interest rates and figure out what the present value of all the cash flows is. So we’re going to now do that a bunch of times, okay, for the rest of the class, and see what that means. So we have to begin, the two thirds of the class is going to be spent on the question, how do you know what the market rates of interest are? So how do you know what the market rates of interest are? How could you find out what the market rate of interest is? What would you do to find it out? Yeah.
You would find, for different maturities–it used to be up to 30 years–for different maturities, you would find the yield on the various bonds, okay? So why would you find the yield? Well, the yield of various government bonds, of gov. bonds. Why do they quote the yield? Well, that’s just because, you know, a hundred years ago, people started using the idea of yield and so the vocabulary has been kept, even though it’s not the best way of describing what’s going on. So for instance, let’s just look at some of the yield curves you might have seen over the last 9 years, almost 10 years, since December 2000. You would see that in December 2000, the yield on the 1-year bonds, you know, the short yields–this isn’t a log scale, so this is 3, 6, 12, okay. So the shortest bonds usually have lower yields than the highest bonds, but sometimes, like in December 2000, the yields are almost all the same. It’s called the flat yield curve. Other times, like now, we’re in this light blue one here, right now the short bonds have very small yields and the long bonds have much higher yields, so the last one is the 30-year bond. So you get the yield on every single bond. ## Chapter 2. Assessing Market Interest Rate from Treasury Bonds [00:09:07]Now what do you notice about this picture, by the way? They can be very different at different time periods, so in December 2000, the interest rates were really high. The yields were 6 percent. I’m talking yields so far. We haven’t talked about interest rates. We have to figure out what the interest rates are, but anyway, they’re obviously going to be connected. So the yields were very high in December 2000, and they got much lower in December 2008, and they’ve stayed very low. So why are they so low now? What got them to be so low now? Yeah?
So you notice that the yield curve now, December 2008, was this blue one. So the Fed, in the crisis of 2008, you know, was terrified, and it dropped the interest rate almost to 0, virtually 0, and it’s kept it there, because from December 2008 till now, we’re at October 2009, September 30
Okay, so you’ve got a bunch of yields. Now what are the yields? Well, let’s just do an example here. So I’m going to make up an example, which is very easy to compute. So let’s try this one. Okay, so I’m reading here at the top, let’s just say that you’ve got a bunch of different bonds. A 1-year bond, a 2-year bond, a 3-year bond, a 4-year bond and a 5-year bond. Now each of the bonds was issued–I’m assuming here that they were all issued on the same day. So they’re issued with different coupons. Let’s say they were all issued today. We’ll come back at the end. Obviously the Treasury doesn’t issue new bonds every single day, so how does this change when you arrive on a day when they haven’t issued things? But let’s just keep it simple and suppose that today, the Treasury has issued 5 different bonds over these 5 different years. Now the Treasury has to set what do they do? They decide how much of these bonds they’re going to sell and they decide what coupon they’re going to set. They set the coupon. So let’s say the coupon they set was 1 dollar for the 1-year bond, 2 dollars for the 2-year bond, 3 dollars for 3 three-year bond, 4 for the 4-year bond, 5 for the 5-year bond. It’s easy to remember, that’s why I chose those numbers. And the face value, let’s say, is always 100. So why did they set those coupons? Well, because given how much they want to sell, they’re picking the coupon, hoping that the price turns out to be close to the face value. So let’s say, when they actually market these, and supply equals demand in equilibrium, the prices turn out to be 100.1, 100.2, 100.3, 100.4, and 100.5. So 100.5 is the price the market’s paying for the 5 year bond and if the coupon is 5, and they pay coupons once a year–they may pay twice a year, but let’s say they pay once a year for simplicity, you’re going to get 5, 5, 5, 5, and 105 the last year. If you bought the four-year coupon bond, you get 4, 4, 4, 104 the last year, right? So those are the bonds. Now the newspaper’s not telling you any of that, so you’re sort of losing that information, so you don’t actually know that from reading the newspaper. So then, what do you know? You know that–you know the yields on all these things, okay? So here, this tells you the yield on each of these bonds. Going back to where I was. Okay, knowing the yield, you could figure out what the price of each of the bonds is, or in fact, the way that they calculated the yield that the newspapers reported. How did the newspapers get the yield? The newspapers said, well, for the 4 year bond, we’re going to say that 100.4 = 4 divided by (1 + the 4 year coupon bond yield), + 4 over (1 + Y (4)) squared, plus 4 over (1 + Y (4)) cubed, plus 104 over (1 + Y (4)) to the fourth. In this case, because these numbers are all positive, it’s monotonic in Y (4), so there’s a unique Y (4) which you can use to solve this equation, price = the discounted value at that yield of the payments. So that’s how the newspapers, the reporter, that’s how he got all the yields to show you that graph. He looked at the market, or could call the bank or something like that, had a computer screen, talked to his friends on Wall Street. He knew the price of all the bonds, he knew the coupons of all the bonds, and then he produced the yield for all the bonds. So the yield, as I say, that’s the word that everybody uses, but really, the information that you want to deal with is the price, and what did the coupon actually pay? Okay, so that’s what you know. Everybody knows this every day, the information I have given. Every morning, maybe every few hours, people will update it. They’ll look at what are the coupon bonds paying and what are the prices? The thing that’s changing from hour to hour are the prices, but we’re taking a snapshot at the beginning of the day and looking at the prices. So now we’ve got prices of bonds, which I’m going to call capital Pis, Pi (1), Pi (2), Pi (3), Pi (4) and Pi (5). But now, what does Fisher say you should do? What’s the most important thing to do? The most important thing to do is find the interest rates. So what has this got to do with interest rates? Well, if you modernize Fisher a little bit, the most important thing–he didn’t put it this way, but this is really what he must have meant–the most important thing to do is find the prices of the zeros. So [little] pi (1) is today’s money price, today’s money price for 1 dollar at time 1. pi (2) is today’s money price for 1 dollar at time 2. pi (3), today’s money price at time 3, and pi (5) is today’s money price at time 5. Okay, now why do you want to find these things? Because once you know these things, you’d be able to value any investment. That original investment that we talked about, which maybe disappeared, this one. It disappeared. Anyway, once you have all the pis, if anybody tells you, if a hedge fund tells you, “This is the revenue I’m going to produce for you in the next five years,” if a company says, “This is our business plan. We’re going to build a factory today that’s going come out a certain amount of money and we’re going to get profits in the next five years, blah, blah, blah.” If a new bond comes on the market and you don’t know how to price it, and somebody offers a price for it, how do you figure out what it’s worth? All you do is you take the cash flows, the Cs that I not very cleverly erased, you take the cash flows and multiply them by the pis. So the correct price P is just whatever the cash flows you’re predicting times these pis that Fisher says is what you should really be finding. So nothing could be simpler. Now why is this the right price? Because if you can go in the market and buy 1 dollar at time 1 for pi of 1, and 1 dollar at time 2 for pi of 2, etc., you can buy all the cash flows from this investment project by spending this amount of money. So if the guy is offering you the investment opportunity at a higher price, it’s crazy to do it. You could have bought those cash flows yourself by paying this price. If he offers it to you at a lower price than that, then definitely you should do it, because it’s a bargain, because if you had to buy it yourself, it would be more expensive. In fact, you can make an arbitrage profit. If he’s offering it to you at a lower price, you can buy it and how do you buy it? By selling these very promises, C(1), C(2), C(T) on the market, and if people believe you that you’ll pay, you can sell it for this price. So you buy his project for a lower price. You sell it for a higher price. You make the difference, and when it comes time to keep your promises, the project is giving you the cash to keep your promises, so you lock in a profit for sure. So if you knew the pis, you would know for sure how to value any project where you knew for sure the cash flows, knowing the pis would tell you how to value it, and would tell you whether it was attractive or not attractive. You just look at how high the present value is. Okay, any questions about that? So we just need to figure out how to deduce what the little pis are from the data that we’re going to be given, and that we are given every day by the market. Okay, so I said literally pi (1) is the price you would pay today to buy 1 dollar tomorrow. Now how could you go about buying 1 dollar tomorrow, given that the only things you can trade on the market are these Treasury bonds, these government bonds? And I’ve told you what they pay off and I’ve told you what their prices are. So how would you go and buy 1 dollar tomorrow and how much would it cost you, 1 dollar next year? How would you do that? You can trade, buy or sell any of these Treasury bonds. So in the background, I keep saying that we’re going to have to worry about people defaulting. We’re not quite doing it yet. So buying a Treasury bond, you need the cash to buy it. Selling a Treasury bond means you promise to deliver what the Treasury bond promises, and your promise is as good as the government’s. So if you sell it to somebody, they’ll pay you the same government price for it. So obviously in the background, you’re going to have to do something to convince the guy you’re making the promise to that you’re going to keep your promise. So we’re going to worry about that later. So for now, when I say that the market for those Treasury bonds clears at those levels, I mean that anybody who wants to can buy Treasury bonds at those levels, or can sell them, even if he doesn’t have them, at those prices, by making the promise of what the Treasury bond does. Because we’re assuming that the government and you, everybody is just as reliable, everybody is going to keep their promise. So whether it’s the Treasury making the promise, or you making the promise, same thing. Okay, so how do you buy 1 dollar next year? What would you do in the market with the Treasury bonds to get 1 dollar next year? Yeah?
Okay, so this number, by the way, is some number which, actually, I of course worked out here. Happens to be .991, but we’ll come back to that. So it happens to be point 991. So now we know pi (1). Well, how would you buy 1 dollar in year 2? So there’s a way of directly buying 1 dollar in year 2, once you know how the Treasuries trade. So what I’m doing is I’m explaining the idea of replication, pricing. It’s giving me pricing and it’s going to lead to arbitrage. They’re all basically similar ideas here. So what I want to do is directly buy 1 dollar in year 2. So I could probably go to a bank and they would actually make that trade for me. I could just call up the bank and say, “I want 1 dollar in year 2,” and they’d tell me, pi (2), how much I have to pay for it. But how are they going to figure out what it’s worth? They’re going to see how–they’re going to go out and have to buy the dollar for me. So they’re going to go out and go to the Treasury market. And what are they going to do in the Treasury market to come up with my dollar in year 2? They’re going to replicate my purchase of 1 dollar with a more complicated portfolio that they can actually trade, and that’s how they’re going to figure out how to price my request for 1 dollar in year 2. So what is this bank going to do in the Treasury market? What does it have to do to get 1 dollar for sure in year two, and nothing else? Okay?
You bought the dollar in year 2. You also bought a little bit in year 1. You can now sell off the extra stuff you’ve gotten in year one to reduce your cost of buying that. So what should you do in year one? That’s exactly what you were thinking. You just didn’t quite say it right. So what should you do in year 1?
Okay, so does everybody see what’s going on here? To buy 1 dollar at time 2, you don’t get the whole 2-year Treasury, you buy 1/102 of the 2 year Treasury, so it costs you that amount of money. But that gives you a little bit of extra at time 1. How much extra does it give you? Well, you’ve got 2 dollars extra for every 2-year Treasury, but you didn’t buy a whole 2-year Treasury. You bought that fraction of it. So it gives you this much extra which you now get to sell off, so you’re going to sell it off for this price, pi (1). And of course, we can plug in for pi (1), by putting 101 down here and putting 100.1 up here. Okay, so that was pretty clear, right? So now any questions about that? So that’s going to be some number, which I calculated again, which happens to be 962, .962. So notice, of course, it’s getting cheaper to buy–how much does it cost to buy 1 dollar in year one? It’s that. To buy 1 dollar in year 2, is less. Now what about pi (3)? How would we get pi (3)? We’ll stop at pi (3). How would we get pi (3)? Then we’re going to find a very fast way of computing all these numbers. What’s pi (3)? How would you get that?
So he said, you buy the three-year bond. So by buying the three-year bond, you’re getting–if you bought the whole three-year bond, you’d get 3, 3, 103. You only want 1 at the end, so you have to divide by 103. Now we get 3 over 103, 3 over 103, 1, and so he’s saying we’ve got two extra payments. Let’s sell them off. And so he sold them off like that. That’s correct. So he sold them off, so this one he sold off at pi (2) and this one he sold off at pi (1), so he’s making use of the fact that we’ve already found out this price and this price. But actually, that’s slightly–okay, but we’re talking about not what you would do talking to the banker. We’re talking about what the banker would do, and he’s got to trade in the Treasury market. So how’s this guy going to do this? The banker and the Treasury market now, this Pi over 2 dollars, he’s going to have to hold this complicated portfolio–what’s he going to do to–? He’s going to have to combine the 1 and the 2-year to do this thing and then the 1-year to undo that thing. So it’s actually going to be–so in terms of trading, if you just had to trade Treasuries, what would you do? So this is the correct formula. That’s correct and we can figure out what that is, okay. And so the correct formula is 91.68. That’s .917. Okay, but you see what you’ve done is, these are the kind of fictitious things that Irving Fisher has told us to do. What you’re really doing in the market is trading the Treasuries. So here, you’ve traded a Treasury. You’ve bought 1 over 103 units of a 3-year Treasury. Now what else should you do? You’ve got to trade Treasuries. How can you sell off this amount of money in year 2? You have to sell some Treasuries to do that, so what would you sell?
So what have we done here? We’ve had to sell off this amount of money. So how can you sell off this amount of money? Well, by selling 1 over 102 2 year Treasuries, you’re selling off 1 dollar, but you don’t want to sell off 1 dollar, you want to sell off something smaller than 1 dollar, so it’s that amount. So you’re selling that amount of 2 year Treasuries. ## Chapter 3. Zero Coupon Bonds and the Principle of Duality [00:35:46]But now what do you have to do? Now, you see, you’ve bought some 1-year dollars by getting the 3-year coupon bond. But by selling the 2-year coupon bond, you’ve made some promises in year one, so you’ve got to net out all those things and do the right thing on the one-year coupon bond, right? So that looks a little complicated, but you can obviously do it by algebra. So everybody following? You’re not following what the right thing to do is, but let’s just say in words what we’ve done. In words, what we’ve done is we’ve said, there are things you can actually trade on the market. Those are the Treasuries. Those are our benchmark securities. Let’s call them benchmarks. Now what we’re interested in is some other maybe fictitious securities or new securities. The price of the zeros, those are the basic building blocks that will help us evaluate the present value of any investment. So the reason why we know these prices is because we can replicate them by trading only the benchmarks, only the Treasuries. So to get the 1-year zero, we just buy us the correct fraction of 1 year Treasuries. To get the 2-year zero, we have to buy the correct fraction of 2 year Treasuries and sell the correct fraction of 1 year Treasuries, and that gets us that thing. So we’ve replicated the 2-year zero by a portfolio consisting of being long the 2 year Treasury and short the 1-year Treasury. Right? To get the 3-year zero coupon, we have to buy the 3-year Treasury, sell the 2-year Treasury and do something complicated that we haven’t quite figured out yet with the 1-year Treasury, and that will duplicate the 3 year zero. And then we’ll just add up the cost of that portfolio that replicates this, and that must be the price of that thing, okay? So that’s what we’re doing. Any questions about that? Are you following this? Yes?
And now the trick–this is the trick we’re going to see over and over and over again–the subtlety in finance is that they don’t just tell you what the little pis are. You have to figure that out yourself, okay? And so how are you going to figure out the little pis? Well, you know the Treasuries. You can trade the Treasuries, and you know what those prices are. You can see it on the market. So by combining the Treasuries in a very clever way, you can end up getting the prices of all the zero coupon bonds, the things that pay just 1 dollar at the end. Why are they called zero coupon bonds? Because it’s like–you just get principal at the end, of 1 dollar, without any coupons in the middle. So the little pis are called the zero coupon prices, because the payments are just 1 dollar–you know, pi (3) is the price of 1 dollar at time 3. It’s as if there was a bond that paid no coupons and paid 1 dollar of principal at time 3. So the little pis are the prices of zero coupon bonds of various maturities, and those aren’t really traded directly in the market. What’s traded directly in the market, where pieces of paper change hands, are the Treasury bonds. But everybody, every day is calculating these zero coupon prices, because that’s what they need to do to evaluate every single project that they might conceivably do that day, and decide whether it’s a good project or a bad project. Is it worth the price or not worth the price? And it’s done by the principle of replication, just as we said. So this formula is going to be slightly complicated. I don’t know whether it’s worth writing down. So we’ve got, buying the 3-year Treasury, the right amount of that. Then we have to sell a certain amount of the 2-year Treasury, because we accumulated extra coupons. But now we’re also going to be able to sell a certain amount of the 1-year Treasuries, and so how much is that going to be? It’s going to be some formula, okay. So it’s going to keep track of everything we did and get a formula here. So I’m actually not going to bother, I think–I was going to write down the formula, but it’ll take 3 minutes to work it out–because there’s a much faster way of getting all these numbers. But is everybody with me here? You all understand how I could get this number if I wanted to do the work to get it? I’d figure out I had to sell–I’d sell some of the 1 year and buy some of the 2 year–I’d do something complicated here, okay? Sorry, I would do something with the 1 year Treasury here to compensate for the fact that the 3-year thing I bought is paying me coupons here. The 2-year thing I sold is reducing some of those coupons, and so it’s only the net coupon that I can sell, and I’m going to sell that by selling the 1 year Treasury, okay? So that’s how I would get the number there, and I added the cost of doing all these things together, and I get .917. So you’re silent, but are you following it? Who can I–okay. So it’s too complicated to just figure this stuff out all the time. So instead, there’s a very fast algorithm that you can do almost instantly, and that’s why it’s such a triviality to calculate these numbers ever day. So it’s called the principle of duality. You go backwards, and you say to yourself, “What I want is pi (1), pi (2), pi (3), pi (4), and pi (5), and I’ve started to figure out what the replicating–” so these are the prices of zeros. Prices of zero coupon bonds. That’s what I want–want prices of zero coupon bonds. I have the prices of the Treasuries and the way I’m figuring out the prices of the zero coupon bonds is by replication. Now if somebody stupidly, as happened 50 and 60 years ago, fairly routinely, if somebody was willing to give me 1 dollar in year 3 and only ask 90 cents for it, then I would be able to lock in a profit. How could I lock in a profit? Because I would just–he’s willing to give this to me for a low price of 90 cents instead of 91 cents. So what can I do? Let’s say he’s willing–he’d pay me, let’s say more likely–let’s say he’d pay me 93 cents. Say some guy came to me, I’m the banker, and he says, “I’ll pay you 93 cents today to get 1 dollar in year 3,” in other words, for a 3 year zero. Well, I’d say, “That’s wonderful.” I’ll sell them this promise in year 3, of 1 dollar for 93 cents. Then with that 93 cents, I’ll only use 91.7 of those cents and I’ll go out and buy the 3-year Treasury. I’ll sell some of the 2 year Treasury and I’ll sell a little bit more of the 1 year Treasury. And that portfolio which I’ve done by doing that will pay me exactly 1 dollar in year 3, enabling me to keep my promise to him, but it will only have cost me 91.7 cents. So I’ll have made a 1.3-cent profit for sure, with no chance–it’s a pure arbitrage. I made a profit of 1.3 cents with no chance of losing any money, okay, because I’ve done all the transactions today, and the government’s going to keep its promises. I don’t have to worry about the government giving me the money, and so I’ll be able to turn the money over to that guy in year 3. Meanwhile, he’s given me his 93 cents. So if you want to do an arbitrage and make your profit, you have to figure out what the replicating portfolio is, and the replicating portfolio also tells you the price. But it takes a long time to figure out what all these arbitrage-replicating portfolios are. And maybe nobody’s coming to you and offering a stupid deal like that. So you don’t need to figure out–so the principle of duality is, you don’t need to figure out the replicating portfolio to figure out what the pi (1), pi (2), pi (3), pi (4), and pi (5) are. I can find those numbers now just by clicking a button in Excel, trivially, without bothering to find the replicating portfolios. Then if some, you know, bad trader comes to me and offers me 93 cents for the 3 year zero coupon, then I’ll figure out the replicating portfolio and take advantage of that offer to make a pure profit for sure. So what I want to show you know is how to get pi (1) through pi (5) without having to go through this complicated calculation. And it just reasons backwards, okay. So please interrupt if you’re not following this logic. So you reason like this: we don’t know what pi (1) through pi (5) are, but if you did know them, you’d be able to price the very bonds that the market is trading. So you’d know that 100.1 had to equal 101 times pi of (1). And you’d know that 100.2, the 2 year zero coupon [correction: Treasury] bond, whose price is 100.2, would have to be 2 times pi of 1 + 102 times pi of 2, right? Because pi of 1, remember, is the price you pay today for 1 dollar 1 year from now. 101 dollars, 1 year from now, costs 101–if you knew pi (1), this would be the price. If you knew pi (1) and pi (2), you could figure out the price of the zero coupon bond–I mean, of the 2 year Treasury bond, because 2 dollars at time 1 cost 2 pi (1) and 102 dollars at time 2 cost 102 pi (2). And then the 3 year is 100.3 = 3 times pi (1) + 3 times pi (2) plus 103 times pi of 3, etc. Then we can go down to the last–well, I’ll just write them all. It doesn’t take a second. Okay, and the last one is, 100.5 = 5 times pi (1) plus 5 pi (2) plus 5 pi (3) plus 5 pi (4) plus 105 pi (5), okay? So you don’t know the little pis, but you do know these prices, because the market tells you, and you know the payoffs of all the bonds, because that’s just written on them, literally, so you can just read what the payoffs are. You know the government’s going to keep its promise. So rather than doing this complicated stuff, trying to figure out the pis, assume you had the pis. And then if you had the pis, it would tell you what the prices of everything were. So if you guessed the wrong pis, you’d get the wrong prices. But basically, you’re solving 5 equations and 5 unknowns, and that’s what Excel is so good at. It’s going to start with a wild guess of the pis, and then it’s going to move around the pis until you match all these prices. And since it’s 5 equations and 5 unknowns, and they’re all linearly independent, it’ll be a unique set of pis that it will calculate. But that 1 set of pis has to be the replicating portfolio prices, because there’s only 1 set of pis that are going to work and solve these equations, namely the ones you got by the replicating argument. So we can figure out the pis by solving 5 equations and 5 unknowns, so that’s what I do. So if you guess the pis, 1, 1, 1, 1, 1, any questions about what I’m doing? If you write 1, 1, 1, 1, 1, you’re going to get prices, you know. For the first one, it will just be 101, and for the 2 year, it will be 2 times 1 + 102 times 1, is 104. For the third one, with all the pis 1, which is obviously not the right thing, it would be 3 + 3 + 3. That’s 109, so those are bad prices. We’re trying to match what the market says the prices are. So all I do is, I subtract the market prices we’re trying to match from the actual prices. I look at what the error is, which we’re trying make 0, then I square the error. And then presumably this was adding the error, sum C16 to G16. I added the error, and so I now want to do my Excel thing. Hopefully–I haven’t done this, but let’s–it’s got to work. Okay, so minimize, good, that. Such that by changing cells–which are the cells I want to change. I want to change the pis. Those are the ones that are wrong, so I do that. So I’m minimizing the squared error by changing the pis, B18, okay, and I solve. Okay, and I’ve done it. And there is the answer. So you notice that I got the same prices that I told you about before by replication, .991, .962, and .917, etc. So we got the pis. So that’s step 1. All right, so that’s what every single financial firm in the entire world does every single morning, and sometimes every single hour. So are there any questions about what we just did? You have to do this in the problem set. Is there anything puzzling you about this? ## Chapter 4. Forward Interest Rate [00:50:31]Okay, now I’m going to start deducing all kinds of surprising things from this. I hope that you’ll be surprised, but I want to make sure you’ve got the concept of what we’ve done now. Anybody puzzled by it? Okay, so somehow, Fisher’s pi (1), pi (2), pi (3), pi (4), pi (5) are going to be deducible from what’s going on in the markets, every day. All right, so let’s ask one more thing that’s deducible. Suppose I go to a bank and say, “I promise to give you 1 dollar in year 2. How many dollars will you give me in year 3?” What do you think the bank’s going to tell me? Every bank will give me the same answer, if that yield curve–given this morning, and that was this morning’s yield curve, if I ask every bank in the world, “I’ll give you 1 dollar in year 2, you tell me how many dollars you’ll be willing to give me–” So what am I doing? I’m saying, “I promise today to hand over to you 1 dollar in year 2, and you know I’m going to keep my promise. And in exchange, I want a promise from you to give me a certain number of dollars in year 3.” How many dollars is the bank going to offer to give me in year 3? Every bank will give the same answer, and what will that be? So the thing I’m asking, is, I’m asking for what’s called the forward interest rate. So we’ve got these things, which are obviously very important numbers. Those are the most important things. Fisher would say those are the prices you use to get everything. But now I want to say something, I’m going to ask another important thing, almost as important. I want the forward rate. So 1 + i
So this is the number of dollars at t + 1 agreed today. So we agreed today that you’re going to pay this many dollars at time t + 1 in exchange for 1 dollar at time t. So this is like the interest rate that you might pay at time t. You give up a dollar at time t, how much do you get at time t plus 1? The interest rate. But we’re not there yet. We’re agreeing to it today. So today we’re agreeing to this interest rate. So what is the interest rate we’d agree to today, so we’ve locked in the rate? When it comes to time t, one guy’s going to hand over the dollar, and when it comes to time t + 1, the other guy is going to give back this many dollars. So what is the rate we would lock in today? It’s called the period t interest rate forward, because we’re locking it in today for a forward period of time, but it’s really just the normal time t interest rate for one year, but we’re locking it in today. So what would we lock in today? How do we compute that? We already know what that number is. What is it? Yes?
But today we already know what the ratio is of time t dollars to time t + 1 dollars. We know that pi (2)–in fact, we know what it is. pi (2) happens to be .962. That’s a bigger number than pi (3). From today’s point of view, 1 dollar at time 2 is worth more than 1 dollar at time 3. We already know how much more 1 dollar at time 2 is worth than 1 dollar at time 3. It’s the ratio pi (2) over pi (3). So that ratio, as he says, pi (2) over pi (3), has to be exactly the exchange rate that the people are agreeing to today. That’s what pi (2) and pi (3) mean. If you express it as an interest rate, it’s 1 + the forward interest rate. That ratio is 1 + i
Does that mean that pi (7) is less than pi (6)? Maybe, but it couldn’t really be that way. Okay, so let me translate his question. He’s saying, look, yield curves very often are flat. Mostly they go up, very often they’re flat. Sometimes they even start to go down. He said, “That worries me that maybe pi (7) is less than pi (6).” But that could never happen. That would be crazy, because that would mean that there would be a negative interest rate in the future, and with money, that can never happen. You can store the money. No one’s ever going to ask for a negative interest rate. He could just keep the dollar and keep it in his pocket. Why is that? Remember, pi (6) = 1 over (1 + the 6 year yield) to the 6 Okay, so we could get the i’s. The i’s will typically be going up. Suppose the yield curve is going up, by the way. Will the i’s be going up faster or slower than the yields? Yeah, if this is the yield curve and I calculate the forwards, do you think the forwards will be going up faster or less fast than the yields? All right, well let’s do it in the example. Let’s just go back to our example that we were doing. Yield curve spreadsheet, okay. So maybe I did it here. Hopefully I solved it all. Okay, so here we got the actual pis. You see the pis are always declining. And if we now look at the yield curve, you can figure out the yield. How can you figure out the yield? Because you solve that formula we gave at the very beginning. You take the price of the–I guess I’ve erased it now. Okay, you know what the price is. To figure out the yield on the 3 year, we just plug in 1 + Y of 3, 1 + Y of 3 squared, 1 over 1 + Y of 3 squared, 1 over 1 + Y of 3 cubed, and that gives us the yield. Remember, that’s how the newspaper reporter figured out the yields. So I figured out the yields in the spreadsheet down here, and these are the yields. So this is what would appear in the newspaper. The 1-year yield is a little under 1 percent. The 2-year yield is a little under 2 percent. The 3-year is a little under 3 percent. The 4-year is a little under 4 percent, and the 5-year is a little under 5 percent. Those are the yields. So what if we did the forwards? The forwards, remember, are just the ratios of these pis. What are the forwards? They’re down here. So what do they do? Sorry, I don’t know how I did that, but here are the forwards, over here. So the forwards have gone up much faster than the yields. They went from .008 percent, which is the same as that one, to 2.9, which is bigger than 2.8, to 5, which is way more than 3.8, to 7.2, which is way more than that, to 9.6, which is way more than that. So the forwards went up much faster than the yields. So why is that obviously going to be the case whenever the yield curve is upward sloping? So if we go back to our picture here. We go back to our picture. If the yield curve were completely flat, what do you think the forward yields would be? This is just common sense to see if you have any idea what’s going on. If you think about batting averages and how somebody’s average changes each time he bats–if the yield curve is flat, like in 2006, the forwards are going to basically be flat. But if the yield curve is upward sloping, then the forward yields are going to go up much faster. So why is that? Okay, well, remember, the yields, you know–when you do a 5 year coupon bond, you’re discounting all the cash flows, the previous 4 cash flows, using the same yield to discount them all. So if you go from year 4 to year 5, and you have to raise the yield a lot, it means, you know, like if you’re a batter, and your average goes up. It means the last thing you did was better than the average of what you’ve done before, so it’s going to be even higher. If your average was .300, and then you played a series against the Red Sox in which you did very well and your average went up to .320, in that series against the Red Sox, you obviously did even better than .320, because you have to average what you did then with what, your previous .300 to get .320. So if the average sort of is going up, remember the yield is the same thing over the whole history. If, when you take a longer history, the average has gone up, it must mean that the most recent thing went up really much more, okay? So the 4-year yield is sort of averaging the payoffs of the first 4 years. The 5-year yield’s averaging it over 5 years. So if that yield, the 5-year yield, has gone up, what happened in the 5 Okay, so we know now to summarize, everybody can look at these pictures every single day. From these pictures, they can deduce the pis. That’s the crucial variable in the whole economy, the pis. But a second crucial variable is the forward yields, the 1 + i The forwards are so important because, suppose you believed that–so the forwards, let’s go back to what we got. Let’s just look at the numbers here. Here are the forwards, remember, down here. Okay, suppose I said to you, “You tell me.” You don’t know anything about the economy, maybe, but you can read the newspapers and do mathematics like we’ve just been doing. What do you guess the interest rate’s going to be in year 2? So this is the 0 year forward, the 1 year forward, the 2 year forward, the 3 year forward and the 4 year forward. If I say in year 2, “What do you think the interest rate–guess what the interest rate’s going to be,” what would you guess? Yeah.
So my classmate, Larry Summers, introduced TIPS, Treasury Inflation Protected Securities, and he, you know, announced this was a fantastic idea and was going to change radically the whole markets. And then nobody traded them, and they offered astronomical interest rates, real interest rates, to get anyone to buy them. And so they were nicknamed–it’s really bad on camera–but their nickname became totally illiquid pieces of… and so the market has not used the TIPS to do most of its pricing. It uses the Treasury bonds, but yes, Fisher would say, if the TIPS were a reliable market, you would use the TIPS to get the real interest rate, and that’s really what you should care about, is the real interest rate, not the nominal interest rate. But we’re using the Treasuries here to get the nominal interest rate. However, you’ve now just dodged my question. My question was, if I asked you to predict, on the basis of the yield curve in this example, and what we’ve been able to deduce, what would you predict the interest rate was going to turn out to be 2 years from now, the 1-year interest rate? What would you predict? Yeah?
Because suppose that you knew for sure the interest rate was going to be 4 percent in year 2. How could the market get to a 5 percent forward? That means that some guy is promising today, “You give me 1 dollar in year 2 and I’ll give you 1 dollar 5 in year 3.” And we’re agreeing to that deal. But that’s a ludicrous thing for him to do, because when he got to year 2, he could simply–if he knew for sure what was going to happen in the future, and that the rate was going to be 4 percent, he would just–I said it backwards. What an idiot. Suppose he knew for sure the rate was going to be 6 percent, he would be–oh yeah, if he knew for sure the rate was only going to be 4 percent, he’d be a fool for promising to give the guy 5 percent today, because in the future, when he got the dollar, what would he do with it? Put it in the bank and get 1 dollar 4 next year? That wouldn’t cover his promise. He’d be screwed, okay. So if he knows for sure that the rate 2 years from now is going to be 4 percent, the forward rate would also have to be 4 percent. So to say it backwards, if you assume everybody knows for sure what’s going to happen in the future, then the forward rates would be exactly equal to what everyone is expecting to happen in the future. To say it slightly differently, if you happen to be the one ignoramus in the world who didn’t know what was going to happen in the future, but you knew that everybody else who was trading in the market did know what was going to happen in the future, and you saw a forward rate of 5 percent, then you could deduce, even though you were an ignoramus, that actually 2 years from now, the interest rate was going to be 5 percent. Okay, so just what you said, but just said a little bit more precisely. We’re assuming here perfect certainty about what’s going–we’re assuming the traders who trade today all are completely convinced of what’s going to happen in the future. ## Chapter 5. Calculating Prices in the Future and Conclusion [01:10:05]Okay, so let’s go back to our picture now. The picture of the zero yield curve. So what do you think this blue curve means? What are the traders convinced is going to happen in the future? The second half of the course is going to be dealing with uncertainty. We’re now assuming, like Irving Fisher, that everybody trading today has no doubt about what’s going to happen in the future. So what do you think these traders think about these prices, about the interest rates? We’re now in the world today, this is September 30
So they think rates are going to go way up in the future, okay, much higher than that, because you’re starting so low, staying low for a while. So the rates have to go up, the forward rates, really sharply to pull the average that high. So people are convinced that rates are going to go up in the future. That’s what that tells you. And why would they be convinced of that? Well, for the two reasons that you gave at the beginning. One of you said, “The market is going to get more productive.” Irving Fisher has already told us that when the market gets more productive, you know, you’re more optimistic about what’s going to happen later, the real interest rate goes up. And if inflation’s constant, and the real interest rate goes up, the nominal interest rate goes up. The other possibility is that the real interest rate stays the same, but there’s inflation in the future. The real interest rate + the inflation is the nominal interest rate. That’s another explanation for why people might expect the nominal interest rate to go up, okay? So you know that the market is predicting rates going up, and the two obvious explanations according to Fisher is that either inflation is going to go up or the real rate is going to go up. And why might the real rate go up? Well, there are a bunch of reasons, but most likely because productivity is going up. Okay, so I’ve got one more surprising conclusion to end this. So if you were certain about the future and you took the 5-year coupon bond, could you tell me what the price of the 5-year coupon bond was going to be next year? How would you figure that out? Assume that everybody is convinced that the 5 year coupon bond–that they know the future for sure. So therefore, from the zero curve–I erased my graph–this is the last question. I’ll let you go as soon as you answer this. You need to answer this to do the problem set. Wrong graph, shit. Sorry. Okay, it’ll only take 1 second to answer this. Okay, I’m telling you now that this is what everybody’s looking at in the morning, okay, these numbers. They’re getting all the forward rates and stuff like that. They’re making all the deductions that we made. Now if you suppose that those people are convinced, they don’t have any doubt about what’s going to happen in the future. Because they don’t have any doubt about what’s going to happen in the future, you can infer from these prices what they think about the future. So the question is, can you infer what the price of the 5-year Treasury, which is now 100.5, that 5-year bond is 100.5, next year it will only be a 4-year bond. Do we know what its price is going to be next year? Yes.
If there’s a world of certainty, you’re going to know what all the interest rates are in the future. And if you know what all the interest rates are in the future, obviously Fisher tells you, you can figure out the price is. However, to get the exact formula would take a few minutes and I unfortunately am a few minutes behind, so you’re going to have to figure out what the right formula is, but it’s what he said and what you were getting to. [end of transcript] Back to Top |
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