econ-252-08: Financial Markets (2008)
Lecture 10 - Debt Markets: Term Structure [February 15, 2008]
Chapter 1. Introduction [00:00:00]
Professor Robert Shiller: Before I begin, I just wanted to say that I was very — I found last period's lecture with David Swensen very interesting. Also, I liked being serenaded by New Blue, which is a first experience for me. Swensen — He gave, I thought, a very interesting talk. I've heard him talk before but it's always interesting. He really emphasized diversification. But, I don't know what you were thinking, we got a 28% portfolio return last year. There's something else going on besides diversification; diversification means we get the average return. And I was glad that you asked questions at the end. Some of your questions seemed to draw out other things that he doesn't plan to talk about, like what he really did to make money. One thing was that he said that they shorted the Int0ernet stocks in the late 1990s. That's a brilliant market timing device that — I think enlightened people thought that those prices were getting high — and also made some play on credit spreads and now on real estate, he said.
How does he do it? Again, it's my theory that there's no — you can't entirely be taught, but partly though I think that he does it, as do other good portfolio managers, by just keeping a very broad base of knowledge and listening to people and collecting information and watching the big trends and thinking about them. One of, I think, Swensen's best talents is he's a good listener and he incorporates basic facts and acts on them. You might have — it would be incorrect, I think, in listening to what he said and conclude that he just says diversify because he's obviously done something very different from that.
By the way, I have another speaker who told me that he would like to talk to our class and I hope we can work it out. His name is Carl Icahn, who is one of the biggest Wall Street — powerful and enlightened people on Wall Street — who can also maybe tell us something about how he makes money or how he makes it a better world. The problem is that he has a — these people have very tight schedules and he's involved in various takeovers and things right now. He said he could do it if we could arrange to meet after 3:00 p.m., so I'm going to — on a Monday or a Wednesday, perhaps at 3:00 p.m. or later. He said — his assistant said it might even be 7:00 p.m., so that's the way he works. I don't think he gets up at 9:00 a.m., apparently. So, I take that that's all right with you — that you can come to a special section of this class. We would have to see if we can arrange that. Again, I don't guarantee that it will happen because the reality is, someone who is involved in as many things as he is — it's going to be something. We do have Stephen Schwarzman coming February 22nd and we have Andrew Redleaf coming March 5th, so we have a really strong set of outside speakers this year. Again, the mid-term exam is Monday; sorry that I misstated that last week. You've already seen last year's mid-term and it's going to look a lot like that.
Chapter 2. The Discount and Investment Rates [00:04:25]
We're talking today about interest rates and bonds. Interest rates are an old, old thing. They go back to ancient times, but I'm going to talk about some of our modern institutions and I'm going to talk first about discount bonds. This is a little bit more of a technical lecture, but I find it just as interesting myself. We're talking about discount bonds, and then coupon-carrying bonds, and then talk about the term structure of interest rates and why we have interest rates. I think that's my Blackberry beeping. I'm trying not to live in too electronic a world. That's the way it is these days. It's giving its last gasp. Then finally, talk about inflation index bonds. The first thing is a discount bond. It's the most simple — or often called a bill — a discount bond does not pay interest; it's sold at a discount. You have the principle, which is the amount owed, which is, let's say it's $100, that will be paid to you at some date in the future, specified in the contract. And nothing, you would just get $100. Effectively, you get interest from this because it's sold at a discount, so the price that you pay is equal to one hundred minus the discount. You wouldn't buy this bond at par because — at par, meaning at $100 — because then you wouldn't get any interest on it. So you buy it at a discount and the return you get is of course a hundred minus — the return you get is the discount.
The U.S. Government is a big issuer of discount bonds and they're called Treasury bills. I have them up on the screen here. This is from a U.S. Government website called treasurydirect.gov and you can get on and access it at any time. It is showing — since the U.S. Government issues these bonds, it's showing its data on these bonds. The bonds are auctioned off on regular dates and you cannot participate in the auction; I assume you can't. You'd have to be an authorized participant, but if you become an institution you can get authorized to trade in the auction. These are the dates; the latest auction was February 14th. I guess that was yesterday, right? The term means it was a sixty-day Treasury bill; that means it pays not $100, it would start at $10,000. Treasury bills — maybe you have to realize there's a distinction between savings bonds and Treasury bills. Treasury bills are for serious investors and so they don't come in small denominations. The U.S. Government also issues small denomination debt to individuals to help them called savings bonds, but we aren't talking about those. At the February 14th auction, Treasury bills were — sixty-day treasury bills were sold and the auction price is given. Well, here's the price of the bill. The issue date was today, February 15th, and they mature in what should be exactly sixty days — on April 15th. If you want to buy one you would pay this price.
This number, CUSIP number, is a number that identifies any security. It's like you have a social security number or other ID number, which positively — there might be another person with the same name as you, God forbid, but you at least have your own number, which is guaranteed to be unique to you. So every security has an identical CUSIP, which identifies it.
I want to start by explaining the numbers here. How did they get — how do these numbers interrelate? Well, the discount rate is sort of the — you notice there are two different interest rates here and you might be confused by them. The discount rate that is shown is the number that you plug into a formula to get the price. What really matters to you as an investor is how much you have to pay today to get $100 in sixty days. So, how do we get the price from the discount rate? Well, there's a formula that's been used by bond traders for hundreds of years and it's a traditional expression, which goes from the discount rate to the price. In this case, price is $99.58 and what does that equal? It equals 2.51 — that's the discount rate that you see up there — well, (2.510 x 60)/360. As you know, the maturity is 60-days and, as you know, there are about 360 days in a year. By tradition, they divide by 360, not 365, and so you understand that dealers of Treasury bills tend to "discounts."
It's like a language, a language of finance. So what everybody knows is that the discount is converted into a price according to this formula. You might ask, well is this an approximate formula? No, it's an exact formula. Then you say, well why didn't they divide by 365, because I know there are 365 days in a year? The answer is — this goes way back — it's an old tradition and they used to have to do this by hand. They had to divide by hand and so they didn't like the number 365, so they thought, let's just round it to 360. As long as everybody knows that's what we're doing, what difference does it make, right? That's what you're supposed to know. If your dealer quotes you a discount rate of 2.51%, you know how to convert that into the price and that's all that matters — all that really matters is the price you have to pay. They also have something up here called the investment rate. Yes?
Student: I'm sorry, I don't understand since the left hand side of your equation does not equal the right hand side, isn't that discount, not price?
Professor Robert Shiller: I'm sorry, glad you — that's the difference between 1 and 99.58. Yes, thank you — between 100 and 99.58. Are you okay now? I'm sorry I made a — now the thing is, to convert that into — what is this other — there's another interest rate up here called the investment rate. Well, that's supposed to be your percentage return on an annualized basis. Remember, this thing only runs for sixty days and you could compute your sixty-day return, but people like to compare annual returns — once again, a tradition we have. Now, there's nothing special about annual — that's the time it take the Earth to go around the sun, but it doesn't have any relevance to finance — but we are just accustomed to using that. What we're going to do, however, is take account of another problem. That is, you might think that this quantity here is my return but you have to actually divide it by the price, which is less than a hundred, to get a return. You're not putting in $100 to this investment; you're putting in $99.58 for the investment and you're getting out the difference between $100 and $99.58. So, that is not — your return is actually higher than 2.51% on an annualized basis.
What we do — I'll show you how they got their number. 2.563%, which is the number you see under investment rate, is equal to 1/(.9958 – 1), corrected for the number of days in a year. How do we do that correction? When I first tried this, I thought, fine I'll multiply by 365/60, but I forgot this is a leap year. So, times 366/60. So, that's how you go from the discount rate to the investment rate. Do you understand why we had to do this? The investment rate is telling you how much money you're really making on an annualized basis, so it's very simple. I put in .9958 for every dollar I got out, so my appreciation of my money is 1/.9958. Subtract that by $1, which is what I put in, and that's how much money I made as a fraction of a dollar. All I have to do then is correct that for the number of days in a year, so that's what I did. That's how we go from the discount rate to the investment rate.
Now you might ask, well if we did 360 here, why don't we just do 360 over here? Well, it's one of the mysteries of Wall Street. They like to — when they're computing the investment yield, they want to be completely honest and not use any rules of thumb. So you have to understand that it's 360 a year because that's just a convention and everybody knows that it's just the way we quote prices. Rather than say $99.58, we just quote the 2.51 and everybody knows how to convert. When you ask for the investment rate, you want to know the truth — how much am I really getting — so they don't monkey around here. Also, they don't have to actually — in the old days, they didn't actually have to compute the investment rate. If you were talking to your broker — if you go really old days, you'd send a boy running over to your broker. They didn't have telephones; they used to have runners. The boy would go say, we're offering to pay a discount rate of 2.51 — that was the formula — and then anyone could do these calculations; they knew what it meant. This calculation wasn't done very frequently; that's just if you had one of the satisfactions of knowing how much money you were making.
Chapter 3. The Bid-Ask Spread and Murdoch's Wall Street Journal [00:19:12]
I think we pretty much explained — there's another thing I want to say about Treasury bills. I used to have up here a page — I used to take — instead of Treasury Direct, which is a website, I used to get a clipping from The Wall Street Journal that showed much the same information that you see on this chart. But, it showed additional information that you won't find on Treasury Direct; it showed bid and ask for dealers. Now, you have to understand that the Treasury has these auctions only on these dates. On February 14th, they only sold sixty-day ones, nothing else. On February 11th they sold only thirteen-week ones and nothing else. What if you want to buy a Treasury bill on some other day? Well, you can go to Treasury Direct as an individual and buy a Treasury bill, but it may not be the best place to buy. What you normally do is you go to a dealer and a dealer is someone, a professional, who participated in the auction and bought these up to accumulate an inventory to then sell off to you, the customer. So normally, you don't deal with Treasury Direct, you deal with a dealer.
The Wall Street Journal calls around to the dealer, which has an inventory of all these maturities and asks them for their bid and ask. And a dealer — what a dealer does is it maintains an inventory of, in this case, Treasury bills and then it stands ready to buy and sell. So, the bid price is the — they don't do it in terms of price, they do it in terms of discount — so the bid discount rate — there would be two numbers for each. One is the bid, which is what they'll pay you for a Treasury bill if you want to sell to them. The ask is what you have to pay them through this formula. The formula is — I could write it this way: price equals one hundred minus this. Normally, when you have a dealer, the dealer has a — when we talk in terms of price the dealer has an ask, which is higher than the bid. If you go to an antique dealer — maybe you're familiar or maybe you've done this if you've bought furniture for some apartment or something. The dealer will charge a higher price for you to buy the furniture than will give to you to purchase the furniture from you; that's how the dealer makes a difference. The difference between bid and ask is the profit for the dealer and the bid-ask spread is the profit for the dealer.
You can see these — you can see the bid and ask in The Wall Street Journal. It's interesting to look at it because you note that the bid-ask spread, which is the difference between the bid and the ask for the dealer, is narrower in the more liquid securities and some of the securities that are small and unimportant have a wide bid-ask spread. That means that it's harder to make a market for it, so the dealer wants to charge more. This doesn't show this here. Unfortunately, The Wall Street Journal stopped carrying this and I thought — it's an interesting story about what goes on. The Wall Street Journal is the most influential financial publication in this country but it has been going through financial difficulties with the Internet age.
In the last five years, the stock in Dow Jones Company has fallen by half. What's happening? Well, there are many factors, but one important factor is, the rise of the Internet has competed with — people used to buy The Wall Street Journal to get data like this. You see how I completely bypassed The Wall Street Journal and I went to Treasury Direct? There are a million websites that give away financial data so The Wall Street Journal was not doing as well. Do you know what happened finally? I'll give this as an aside; they reduced the size of the newspaper in 2005. It used to be a big, broad sheet and they made it smaller in 2005 and they started — they keep cutting out data. There's less and less data in The Wall Street Journal. It used to be this big, thick thing and you would go there everyday to look up — and everything was in there. But now, the Web is competing with it, so they're scaling down the size of the paper and trying to survive.
So, they created Wall Street Journal Online, which is WSJ.com, and they were trying to make money off of that, but people weren't willing to pay for it. You know how it is on the Web? You can get so much for free, why should I pay for WSJ.com? That wasn't working well, so last year — about one year ago — they announced, wallstreetjournal.com is absolutely free and we're going to give away all the data, basically all the data that used to be in the newspaper, free to anyone. I thought that was great. Now we can all get The Wall Street Journal without paying for it. There's some problem with that; it's not economic, so let's not be too jubilant.
What ended up happening is, you know the news, Rupert Murdoch bought The Wall Street Journal last fall and he announced that Wall Street Journal — WSJ.com — will no longer be free. So, this is the reality of the world. I thought it was interesting to look at Rupert Murdoch — this is finance. This may sound like an aside, but this is all finance. Rupert Murdoch, you may have heard of him, is a huge newspaper baron who buys up newspapers all over the world. It's interesting, he's been at it — he's continuing what his father did. To become the biggest publishing monolith in the world, it takes maybe a hundred years, so his father in Australia started buying up newspapers. His father was born in 1886 and now Rupert Murdoch is continuing. Rupert Murdoch is in his 70s and he's still buying newspapers. He kind of makes them survive and NewsCorp doubled its price — that's his company — doubled in the last five years.
You can see, this is how the unseen rule of finance — The Wall Street Journal is a venerable newspaper and source of information about finance, but it's not making money. The world is changing and The Wall Street Journal is flagging. So NewsCorp, whose price doubled when The Wall Street Journal's fell in half, eats up and The Wall Street Journal is gobbled up by the bigger company. You can see why stock prices matter. The Newscorp's rising stock price was a sign that Rupert Murdoch had some idea how to make money and The Wall Street Journal was not doing as well, so it got gobbled up. The question is, what will happen to The Wall Street Journal now? Well, they have to make money; that's the real world. So Rupert Murdoch tends to bring papers around.
I'm just going to go a little bit further on — Rupert Murdoch bought The Times of London in 1981 — that was twenty-seven years ago. The Times of London is one of the most venerable newspapers in the world and it was losing money fast; it might have disappeared if Murdoch hadn't taken over. But, Murdoch turned it around and it's still around. They turned it into a tabloid. It was the most dignified newspaper in the world and he kind of decided that, in order to survive, they had to get a little bit more down to earth. So, they reduced the size, so it looks like one of those — like New York Daily News newspapers now. It's still a great newspaper. He added celebrity gossip too, which The London Times would never do in the past. But you go to do it, right? That's the thing about the real world and finance is very much about the real world.
Murdoch claims he will not alter the editorial content of The Wall Street Journal. The thing that we have seen him doing is charging more, but I guess we've got to allow that. So we may feel annoyed that we now have to pay to get our Wall Street Journal Online. In the long run, we want The Wall Street Journal and so I guess that — we just have to accept that. If you want to find — by the way, if you want to read The Wall Street Journal online at Yale, you can do it no problem. You go to ABI Inform, which is one of the things that Yale subscribes to, on your laptop and it puts you right into the text of The Wall Street Journal. But it doesn't put you, as far as I can tell, into Wall Street Journal Online. We have other — lots of data sources — that Yale subscribes to and there are lots of free data sources, but Wall Street Journal Online is not free.
Chapter 4. Defining Bonds and the Pricing Formula [00:29:17]
Anyway, I want to talk — you understand now about discount bonds? It's pretty simple, right? I just wanted to get the pricing formula. The critical thing about a discount bond is it pays no interest. Treasury bills in the United States are limited to one year out; well, that's the name of it. We call an instrument of the U.S. Treasury with a maturity less than or equal to one year — we call that a bill — and they used to be the only discount bonds issued by the U.S. Government. Now, they also have longer maturity, called Treasury strips, but let me move to the other. So we have — U.S. Government issues bills and that's less than or equal to one year and they pay no interest. They also have government-issued notes and that's from one to ten years and bonds — this is just jargon — these are ten or more, well actually more than ten.
I'll show you from Treasury Direct — I have notes — this should be greater than ten. These are the recent auctions of the Treasury of notes. Now, these are different from bills, as I emphasized, because they pay interest. They carry what's called a coupon. And then we have the bonds. You see there aren't as many of these issued. We issue a lot of–there are a lot of auctions of Treasury bills and there are comparatively fewer of bonds. So, let's talk about a bond. A bond differs from a bill in that it carries what's called a coupon, which I will denote by the letter C, and it has a principal, which it pays out at the end — of one hundred. Of course, there would be larger denominations than $100, but by tradition, we speak of them as if they were bonds that you paid $100 to get. Now, another Wall Street tradition is that bonds pay a coupon; they pay C/2 every six months. The coupon is expressed as an annual amount; you get half of it every six months. The reason they call them coupons is that, in the old days, you used to actually — when you bought a bond, there would be a piece of paper and the piece of paper would have attached to it a lot of little coupons that you would clip. If it was a twenty-year bond, there would be forty coupons, one for each six-month period, and each one would have a date on it. What you used to do is, every six months you'd pull out your bonds and you'd clip the coupons with a pair of scissors; you take them to a bank and they would give you cash for your coupons. So, we still call them coupons, but now we don't do it — we do things electronically. You don't have to clip coupons anymore.
If you want to see bonds as they used to look, with their coupons, there are a number of them at the International Center for Finance down the street here with their coupons still attached. So they've got a lot of — Will Goetzmann and Geert Rouwenhorst are collectors of old bonds and they've got lots of bonds with their coupons still attached. You know what that means when they're framed there on the wall with their coupons still attached? It means the company went bankrupt and never paid; otherwise, the coupons would have been clipped. It's actually — our International Center for Finance is sort of a museum for defaulted bonds. The ones that are beautiful for framing are the ones that failed, so you can see all the coupons. Some of them have some of the coupons clipped and then they stop and you know that it was bad news when they stopped clipping them. We still call them coupons.
If you have a bond with an interest rate of 4.375% — that's not an easy one to divide by two — but you would get half of that every six months until maturity. We have to ask then, how do we get the price and the yield from this? What we do is we take the interest rate, which I'll call r, and plug it into a formula, which I didn't actually do the arithmetic to check — to check their number. The price is just the present value of the coupons at the interest rate r, so the price of a bond is the present discounted value of coupons and principal, at rate r. Now, you have to understand that when you buy a bond, if you buy it at issue, you get the first coupon in six months, the second coupon in one year, the third coupon in eighteen months, and the last coupon you get at the maturity date. So that means — what is the stream of payments? You get C/2 in six months, C/2 again in a year, C/2 again in eighteen months, and that continues until the last date — the maturity date — when you get 100 + C/2.
The price is just the present value of that stream, discounted at the interest rate. The formula can also be written and expanded out — P = (c/2) x [(1/(r/2) — which is the consol formula, if this were applied to an infinite stream — - 1/(1+(r/2))2T x 1/(r/2)]. Let me make sure I've got this right — +100 (1/(1+(r/2))2T. This should be obvious — if I did that right — this should be obvious from what you learned about in present value formulas. If you had a perpetuity, which paid C/2 forever, you already know from the perpetuity formula that the value of that would be C/2 divided by r/2 if r/2 is the discount rate. It's not forever because it terminates after 2T periods of six-month interval each. You want to subtract off the value of a perpetuity that starts after 2T, six-month intervals, so this is the present value of the perpetuity that starts after 2T, six-month intervals. Then you want to add the present value of the principal and that's the formula. That's another conventional formula that goes from interest rate to price of the security.
Chapter 5. Derivation of the Term Structure of Interest Rates [00:39:38]
Now, I want to talk about the term structure of interest rates and that's my next plot here. That's the term structure as of now on the chart. We've identified the prices and yields of bonds of various maturities. How do the prices and yields look at various points in time? So, I've got here a term structure; well, the term structure is the plot of yield-to-maturity against time-to-maturity. This is January of this year, before the Fed cut interest rates, and this is the term structure. I've got the Federal Funds Rate — it's the shortest interest rate, an overnight rate — it was at 4% at that time and then there was a sharp drop. The three-month Treasury bill rate is shown there — it was much lower — it was under 3%. Then the — so the term structure was downward-sloping until about two and a half years and then it was upward sloping.
The interesting question from the standpoint of economic theory is, why did it have that funny shape? I want to compare it with other examples of the term structure recently. This is the term structure just a short while ago — of December 2006 — very different. The Federal Funds Rate was 5.5% and then the whole term structure — all the way — almost all the way. Well, there's this funny glitch here between three-month and one-year, but then it just continued to decline. That's a strongly downward-sloping term structure. This is another example, not so long ago — this is December 2003. Now, the term structure was pretty much upward-sloping everywhere. At that time, the Fed had cut the Federal Funds Rate to 1%, we had very low short rates. And the three-month Treasury bill rate was about the same, at 1%. Going further out, the term structure just kept rising.
It's one of the questions of economics — what determines the term structure? You have to understand that it's determined in the market. The Fed has these auctions but it auctions them off at what price the public will pay, so the Fed doesn't determine the term structure. Neither do the dealers determine the term structure; the dealers have to buy and sell at prices that are in the market. It has to stay relevant to the market, so nobody really knows where these interest rates come from because no one person sets them. If you're a dealer, you've got to keep your bid-ask at market; otherwise, you'll get only one side or the other, right? You'll be selling too cheaply or you'll be buying too dearly. You've got to do it so you're right in the middle, so that the market is in the middle. Nobody really sees the reason for this; it's all a question of theory, so we have to think a little bit about theory.
The term structure is one of the most interesting things in economics because it shows the price of time at various maturities. In 2003, time was almost free out three months. It didn't — if you needed more time to get some business done you'd have to borrow, but you'd only have to pay 1% a year. That would be like a quarter of 1% to postpone your payment by another year; so time was really cheap. But if you wanted to postpone it over seven years or so, time got a lot more expensive. Why is this and why is the price of time changing? One thing to do is to go back and ask, what really are the reasons for interest? I wanted to talk about the theory of interest as presented by Professor Irving Fisher at Yale, who is famous for having exposited that. I imagine he did it at this very blackboard because, as I said, he had his office in this building and he didn't die until 1940s, so he must have been — this building was — this room goes back to the '30s. He had a diagram, which illustrated what is the ultimate cause of interest, and it helps us to think about this diagram whenever we try to understand the term structure.
What his diagram — this is the famous Irving Fisher diagram — depicted, there's only one period; there's today on this axis and tomorrow–well, I shouldn't say tomorrow, that suggests one day — next period on this axis. Let's say it's one year and I'm going to actually plot on this axis a person's income this year. I should say income today on this axis and on this, income next period. So for each person, there is a point representing my income today and my income next period. Let's assume that I know what I'm earning next year; there is actually uncertainty about it. If I draw this down this — this point is this year's income down here and this point here is next year's income; I'll use Y for income. This is Y today and this is Y next period.
A person has a budget constraint if the person can borrow and lend at the interest rate. The budget constraint is a straight line through this point with a slope of (1+r). I'll try to draw this line. That should be a straight line. It doesn't look that straight because I was running out of room up here. You see, that's a straight line. This point here then is the present value of your income; it's (Y today + Y next period) / (1 + r). What's this point up here? This point up here is the terminal value of my income, so the upper point is (Y today *(1 + r) + Y next year. In an ideal world, where you can borrow and lend freely, an individual could choose consumption along any point on this line. I could consume all of my income — if I borrow against my future income, I could — there's a problem here of starvation. If I consume it all this year, I will be starving.
I don't see how I can earn income next year but, in principle, that's what Irving Fisher said. Well, I could just not consume anything this year and I could wait until next year and consume the terminal value. I would take my income this year, invest it at the interest rate, and it would turn into Y today times (1 + R) and then I get income next year; so I could consume all that. I can consume at any point along that line as well. I could consume at today's income, or I could consume here, or I consume down here. If I consume less than my income today, I'm saving and my consumption would be lower this year. Am I doing that right? No, if I'm saving this year my consumption would be less than my income, so my consumption might be here and I would then have more next year. Each individual reaches a decision — each individual has an income point and decides how much to consume based on the budget constraint.
Now, the interest rate in Irving Fisher's world has to equate supply and demand in the market for debt. Each person who wants to borrow has to be met by somebody else who wants to lend, so the interest rate that we have in society is the compromise. It's the interest rate that clears the market for loans and that interest rate determines the market interest rate. Nobody can see the interest rate — why the interest rate's at the level it is in the market — because nobody can see all these individuals; but that's why the interest rate gets determined and is in equilibrium. Why the interest rate clears — it's a mysterious phenomenon, because it's a market phenomenon and each person only sees his or her own contribution to the market and not the whole market.
Irving Fisher also drew on the curve a production possibility frontier for society, which he made downward-curved. This is production possibility frontier and that is a curve — you've seen these before in economics — that says how the production of our society can produce different combinations of income this year and next year. If there were no credit markets, everyone would have to be on the production possibility frontier; there would be no other choice. But if we have credit markets, then people can individually choose to be off the production possibility frontier and at a higher level of consumption than otherwise would. You might have some people up here who are saving and other people down here who are dissaving and the production would be operating at the middle. So, that is Irving Fisher's diagram in a nutshell, but I think it's — so what does it say about the term structure? It says that at different horizons everything on this diagram is different. The production possibility frontier at different horizons is in different places. The budget constraint is going to have a different slope to it and preferences will be different over this horizon between consumption then and consumption now. So you — it doesn't — this theory doesn't say a whole lot about what the term structure will look like but it suggests that it's determined by the interplay of lots of economic factors.
Chapter 6. Lord John Hicks's Forward Rates: Derivation and Calculations [00:52:34]
I want to talk about a couple of other basic concepts in economics of interest rates. One of them is the forward rate and the other one is inflation indexed interest rates. Forward rates — I wrote a survey article years ago about the term structure of interest rates and I wanted to find out who was the originator of the term "forward rate." I asked my graduate student research assistant to research the whole literature and find out where did the word forward rate come from. My graduate student came back and said, it seems to have been Sir John Hicks in London in his 1931 — 1939 book, A Value in Capital. This is another aside. but I think it's motivational, and so I said, are you sure that J.R. Hicks invented the term "forward rate"? He said, how can I be sure? I mean, I've looked through everything; I can't find any earlier reference. And he said, I think it's J.R. Hicks. Then, I was talking it over with another graduate student and he said, well if you want to find out why don't you ask J.R. Hicks? And I said, wait a minute, is that guy still alive? He said, I think he is.
I found out he was living in London — Lord Hicks; I guess he got knighted for his contribution. I wrote him a letter and I said, basically, did you invent forward rates? Then, I didn't get any response for like three months and then to my surprise I got a hand-written letter, in kind of a trembling handwriting; I still have it. I should — I can put it up on the screen actually. He said, my apologies — he was very polite and diplomatic. He said, my apologies for not answering your letter but my health is poor and — but it was a long letter. He said, I'm trying to remember where I got the idea and, he said, I think it probably came from some of our coffee hours at The London School of Economics in the 1920s. He said, we were discussing that, and then he said, I thought it was in a book that my wife and I translated from the Swedish in the 1930s but, he said, I've looked and it's not there. So he said, I guess maybe it is my idea — I was the first person to write it up. Then he died shortly thereafter, so I got to him just in time.
I thought I would describe forward rates in terms of the coffee hour at The London School of Economics in the '20s. This is J.R. Hicks talking — so the year is 1925 — and we're talking about investing in discount bonds and we're going to put £100. Now suppose — the idea that Hicks invented is that they're, implicit in the term structure, there are future interest rates already quoted. I showed you the term structure. I showed you a one-year Treasury bill rate for right now — that's not right now, but you can see I have a one-year and a two-year Treasury bill rate. The idea of a forward rate is that, implicit in that term structure is also a quote for the one-year rate, one year hence, because if you look at the two-year rate, can't you infer back what interest rates are going to be in one year? Because the two-year rate is — you've got the one-year rate and the two rate, so between the two, what's left? The difference between those two somehow reflects what interest rates will be between one and two.
What Hicks said — this is the coffee hour conversation at The London School of Economics — he said, it's right now 1925 but if you want to invest or borrow in 1926 I can do it for you. I can lock in the interest rate right now. So let's say, okay I expect — suppose I expect to get £100 in 1926. Here we are, sitting in 1925, and I want to, today in 1925, lock in the investment return that I will get when I invest that hundred pounds. So, what we want to do — this is what Hicks discovered — I can do the following transaction to lock in, in 1925, the interest rate between 1926 and 1927. Here we are sitting in 1925; I'm going to buy in '25 this amount of two-period bonds — two year-bonds — (1 + r2)²/(1 + r1). These will mature and be worth £100 in '26 and I'm going to short how much? I'm going to short one one-period bond.
This is the number of bonds I have to buy and so let's analyze this. That's all I have to do and I have locked in the interest rate. r1 is the yield on the one-period bond and r2 is the yield on the two-period bond. The price of the two-period bond is 1/(1 + r2)² and the price of the one-period bond is 1/(1 + r1). If I buy this amount of two-period bonds, how much does it cost me? It costs me 1/(1 + r1). If I short the one-period bond it cancels out, so I've made no net purchase in 1925. I bought a number of two-period bonds such that the value of my purchase exactly equals the proceeds that I get from shorting the one period bond, so I've made a zero wealth transaction. It hasn't affected my portfolio — my cash position — at all.
What happens then in — that's in 1925, so there's no cash flow at all. In '26, what happens? Well, in '26, the — I've shorted the one-period bond and so I have to pay out one dollar, but that's exactly what I wanted to do. Remember, I said I'm doing this because I expect to have a hundred pounds — I said one dollar — I'm going to have to pay out a hundred pounds because this one-period bond, worth one hundred pounds principal, is coming due so I have to pay it out. That's what I wanted to do because I said I'd have a hundred pounds to invest. Nothing happens to the two-period bond because it just continues to mature. So 1926, I pay one hundred pounds. What happens in 1927? Now what happens is, I now get the maturity of this bond. What I get — I have purchased this number of bonds, so I'm going to get a hundred pounds times that amount there, the number of bonds — (1 + r2)²/(1 + r1).
You can see that by doing this transaction I have locked in a return between 1926 and 1927. I did it in 1925, but I've got it set up so that I will pay a hundred pounds in 1926 and I'll get this in 1927. He calls the forward rate equal to (1 + r2)²/(1 + r1). This is Hicks's discovery. You might say, this should have been obvious to someone, but it had never been written up well before. What Hicks said is that in these term structures, you don't just have today's interest rates — this is a map–actually, I've just showed the one-period, he had one-period forward rate — but you could do it over any combination and you can get forward rates of any maturity at any future date. That's Hicks's insight and it comes back then to Hicks's book, Value and Capital, in 1939.
He said that we shouldn't think that the — the simplest story of the term structure of interest rates, which he expounded there, is that forward rates equal expected future interest rates. That means, in the simplest — it's called the expectations theory of the term structure. It says that the forward rate, which you can compute from today's newspaper or from today's website — you can compute the forward rates for all future dates. What he says is those forward rates are what people think interest rates will be in the future and that's called the expectations theory of the term structure. Forward rates equal expected future spot rates. If you look at the December 2003, what's going on? It's very clear and that is the Fed had just cut interest rates to 1% — it was unusual — it was much talked about. People didn't expect that to hold and so people thought, well it's going to still be down at 1% maybe in three months. So, you can see the term structure doesn't go up between overnight and three months. But they are expecting that the Fed is eventually — and they were right of course in this case — the Fed is eventually going to raise rates. The upward-sloping term structure means that the forward rates are at higher levels. So, that's the expectations theory.
Hicks also talked about another theory in his 1939 book, which is — there's the expectations theory, but there's also a liquidity premium theory. This is the theory that there's risk in longer-term bonds, so there's a tendency for upward-sloping term structure even if expectations are flat. So, according to the expectations theory augmented with liquidity preference — This strongly upward-sloping term structure in 2003 would reflect two things. One is that people expected interest rates to go up and, secondly, that interest rates–longer-term bonds — are riskier, so there's also another effect pushing interest rates up.
Chapter 7. Inflation and Interest Rates [01:06:09]
The last thing I want to talk about is just — I'm running out of time here and I want to mention this before your mid-term exam, which is next Monday and covers everything through this lecture — something about inflation and interest rates. Inflation — we'll call that π — the inflation rate is π. So right now, it's something like 2% or it's a little bit higher last year, but maybe it'll be 2% going forward. The real interest rate, computed from nominal rates, is equal to one plus the nominal rate divided by one plus the inflation rate, so it corrects for inflation. If you have an interest rate — this is approximately equal to — one plus the real interest rate is equal to one plus the nominal interest rate, divided by one plus the rate of inflation. So we would say the real interest rate equals the nominal rate — approximately equals — the nominal rate minus the inflation rate.
If we have — right now, if we go back to our current term structure–well, this isn't complete. If you look at the current term structure, it's interesting. Look how — this is as of earlier this year; the Federal Funds Rate was at around 4% and it has this huge drop in the term structure and then it starts the upward-sloping. Why is that? That's because on the date that I got this term structure everybody knew the Fed was cutting rates and they got it exactly right. The Fed was cutting rates to 3%; they knew it was coming but not today. Right now, we have a Federal Funds Rate of about 3% and an inflation rate of around 3%, so right now the real interest rate is close to 0.
Finally, I just want to say that we have also a kind of bond called an indexed bond, which is a bond whose coupons are indexed to inflation. With an indexed bond you don't have to do this calculation to get the real rate. The yield-to-maturity on an indexed bond is already in real terms because the coupons are indexed to inflation. In my final slide is a picture of the first indexed bond; this is issued by the State of Massachusetts — I actually own this bond. I bought it for $1,000 on a website because I was interested in indexed bonds. I could have brought it to class and showed it to you, but this was issued during the Revolutionary War to help finance the war.
Now, what happened was the U.S. Government — or the Massachusetts Government — had created high inflation during the war and nobody wanted to lend money to it so they create here a price index. It says the price index contained this amount of beef, this amount of sheep wool, this amount of sole leather, and this amount of corn; that was the first consumer price index ever used for financial contracts. We now have something else called the Consumer Price Index. This bond was paid off, the U.S.–Massachusetts — didn't fail to pay on it in 1784. But anyway, indexed bonds are about 15% of the U.S. national — I'm sorry, about 8% of the U.S. national debt — about 15% of the U.K. national debt — and they are very important but still a minority of all of our fixed incomes. That concludes this lecture and I won't be here on Monday, but we'll have our lecture in this classroom on Monday.
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