We’re talking about the theory
of debt and interest rates. So, I want to talk about a
number of technical topics first. We’re going to start with
a model, an Irving Fisher model of interest. And then
I’m going to talk about present values, and discount
bonds, compound interest, conventional bonds, the term
structure of interest rates, and forward rates. These are all technical
things. And then, I want to get back
and think about what really goes on in debt markets. There’s two assignments
for this lecture. One is several chapters out of
the Fabozzi et al. manuscript. And then, there’s a chapter from
my forthcoming book that I’m currently writing, but
that is the most meager chapter that I’ve
given you yet. The book is not done, so I think
the real reference for this is the Fabozzi et al. manuscript, at this point. And then, Oliver will give a TA
section that will clarify, I think, some of the points. So anyway, what we’re talking
about today is interest rates. The percent that you earn on a
loan, or that you pay on a loan, depending on what
side of it you are. And interest rates go back
thousands of years. It’s an old idea. Typically, it’s a few percent
a year, right? The first question we want to
try to think about is what explains that. Why is it a few percent
a year? And why not something completely
different? And why is it even a
positive number? Do you ever think of negative
interest rates? Well, these are basic
questions. So, I wanted to start with the
history of thought and an economist from the 19th
century, Eugen von Boehm-Bawerk, who wrote a book
on the theory of interest in the late nineteen century. Actually it was 1884. And it’s a long, very verbose
account of what causes interest rates. But basically, he came up
with three explanations. Why is the interest rate
something like 5%, or 3%, or 7%, or something
in that range? And he said, there’s really
three causes. One of them is technical
progress. That, as the economy gets
more and more scientific information about how to do
things, things get more productive. So, maybe the 3% percent, or the
5%, whatever it is, is the rate of technical progress. That’s how fast technology
is improving. But that’s not the only
cause that von Boehm-Bawerk talked about. Another one was advantages
to roundaboutness. That must be some translation
from his German. But the idea is that more
roundabout production is more productive. This isn’t technical progress. If someone can ask you to make
something directly right now, you’ve got to use the simplest
and the most direct way to do it, if you’re going to
do it right now. But if you have time,
you can do it in a more roundabout way. You can make tools first and do
something else that makes you a more efficient
producer of this. And so, maybe the interest
rate is a measure of the advantages to roundaboutness. And the third cause that von
Boehm-Bawerk gave is time preference. That people just prefer the
present over the future. They’re impatient. This is behavioral economics,
I suppose. This is psychology. That, you know, you’ve got a
box of candy sitting there. And you’re looking at it and
you’re saying, well, I should really enjoy that next year. Well, maybe it would
spoil by next year. Next month. But somehow you don’t. You have an impulse
to consume now. So, maybe the rate of interest
is the rate of time preference. Why is the interest rate 5%? It’s because people are 5%
happier to get something now than to get it in the future. So, he left that train
of thought for us. This was not Mathematical
Economics. It was Literary Economics. But the next person I want to
mention in the history of thought is Irving Fisher, who
was a professor at Yale University. But he wrote a book in 1930
called The Theory of Interest. And it is the all-time
classic, I think, on this topic. So, Irving Fisher is talked
about in your textbook by Fabozzi et al. He’s talked about in
many textbooks. He graduated from Yale. I mention this, since you’re
Yale undergrads. He was a Yale undergrad. He graduated, maybe
it was in 1885. And he got the first Economics
Ph.D. at Yale University in the 1890’s. And he just stayed here in
New Haven all his life. And if you were living in New
Haven in the early twentieth century, you’d know him because
he was a jogger. Nobody else jogged. He would exercise and run around
campus, so everyone would see him. In the 1910’s or ’20’s,
nobody did that. But he did. He was a health nut,
among other things. I could talk a lot about him,
he’s a fascinating guy. I’ll tell you one more
story about him. He would invite students to
his house for dinner. And he would explain to them
before dinner that he believed that proper eating required that
you chew every bite 100 times before you swallow it. So, he would tell his
students to do that. And it slowed down
conversation at dinner a great deal. But that’s not what
he’s known for. What he’s known for is, among
other things, his theory of interest. So, this is what’s
talked about in your textbook and I wanted to start out with
Irving Fisher because — by the way, I don’t know when
this room was built. Does anyone know? Because he died in ’47. He probably lectured from this
same blackboard, right? So, I don’t know, this same
slate, it could be, right? It could go back to that time. So, I’m going to put back on the
board what he had on this board, I’m assuming in some
time in the 1930’s. What your author, your textbook
author, Fabozzi, emphasizes for a theory of
interest is something that came from Fisher that’s
very simple. And he says, the interest
rate — this is Fabozzi’s distillation
of Irving Fisher — the interest rate is the
intersection of a supply and demand curve for savings. So, I’m going to put saving,
s, on this axis. And on this axis, I’m going
to put the interest rate, call that r. I don’t know why we commonly use
r for interest. It’s not the first letter, it’s in
the middle of the word. And the idea is that there’s a
supply of saving at any time. That people then wish to put in
the bank or someplace else to earn interest. And the theory
is that the higher the interest rate, the more people
will save. So, we have an upward-sloping supply curve. Now this S means supply,
whereas this S down here means saving. And then there’s a demand for
investment capital, right? The bank lends out your savings
to businesses, and the businesses want to know what
the interest rate is. The lower the interest rate,
the more they’ll demand. So, we have a demand
curve for saving. And then the intersection of the
two is the interest rate. Well, it gives the interest
rate on this axis and the amount of saving on
the other axis. That’s a very simple story. And that’s what Fabozzi et al. covers in your text. But I wanted to go back to
another diagram that Fabozzi et al. did not include in your
textbook, but it also comes from the 1930 book, The Theory
of Interest. That is a diagram that shows a two-period story. And the thing I liked about
this two-period diagram is that it brings out the von
Boehm-Bawerk causes of interest rate in a very
succinct way. So, this is the second Irving
Fisher diagram. I’m going to do a little story
telling about this. Remember the book,
Robinson Crusoe. It was written by Jonathan
Swift in the 1700’s. It was the story of a man named
Robinson Crusoe, who was marooned on an island all by
himself and had to live on his own with no help. This is a famous story called
a Robinson Cruise economy. There’s only one person in the
economy, so, of course, there’s no trade. But we’ll move to a little
bit of trade. I’m just telling you a story of
the rate of — there’ll be a rate of interest on Robinson
Crusoe’s island. I’m going to show here
consumption today. And on this axis, consumption
next year. All right. I don’t remember the novel. Did anyone here read it? Somebody must have read
Robinson Crusoe. But I’m not going to be
true to the story. The story I’m going to
tell is that Robinson Crusoe has some food. That’s all that the whole
economy just does. Let’s say it’s grain. I don’t know how he got
that on the island. But he’s got grain. And he’s deciding how much to
eat this year, and how much to plant for next year. So, the total amount of grain
he has is right here. So, that is his endowment
of grain. That’s the maximum
he could eat. But if he eats it all, there
won’t be any grain to plant for next year, OK? So, he better not eat it all,
or he’ll starve next year. Now, in a simple linear
production — with technology that’s linear,
he can choose to set aside a certain amount of grain, which
is the difference between what he has and what he’s
consuming. And then that will produce
grain next period. So, I’m going to draw
a straight line. That’s supposed to be
a straight line. These are all supposed to
be straight lines here. And that is his choice set
under linear technology. I’m drawing it with no
decreasing returns. The idea is that for every
bushel of grain that he plants, he gets two bushels
next year, or whatever it is, OK? And so, if he were to consume
nothing this period, he would have — if I drew this thing with the
right slope of minus 2 — he would have twice as much. This is the maximum he could
have next period, OK? And so, he could consume
anywhere along this line. And this would be the simplest
Robinson Crusoe economy. So, what does he do? Remember from elementary micro
theory, he has indifference curves between consumption
today and consumption tomorrow. Remember these? These are like contours
of his utility. And we typically draw
them like this. So, what does he do? He maximizes his utility and
chooses a point with the highest indifference curve
touching the production possibility frontier. This is the PPF, the production possibility frontier. And that determines the amount
that he consumes and the amount that he saves. And he consumes this
amount here. And the difference between his
endowment and his consumption is his saving. And then next period, he
consumes this amount. All right, that’s simple
micro theory. That’s familiar to you? So, in this case the interest
rate, the slope of this line — this slope is equal to minus
1 plus r where r is the interest rate. So, in this case, I’ve told
a very simple story. It has only one von Boehm-Bawerk
cause, its roundaboutness. But maybe there’s technical
progress, too, I don’t know. It has maybe a couple
of his causes. If, as time goes by, Robinson
Crusoe figures out better how to grow grain, there could be a technical progress component. But preferences don’t matter in
this story, the preferences I represented by his
indifference curves. And since I’ve got a linear
production possibility frontier, impatience
doesn’t matter. The interest rate in this case
is decided by the technology, the slope of the curve. So, we don’t have all of von
Boehm-Bawerk causes yet. Next step. That was the simplest
Irving Fisher story. The next step is, let’s suppose,
however, that there are diminishing returns to
investment in grain. That means, for example, maybe
when he grows a little bit of it he’s very good at it and
he produces a big crop. But as he tries to grow more
grain, he gets less productive. Maybe he has to do it on the
worst land or he’s running out of water, or something
is not going right. Then we would change the
production possibility frontier, so that it
concaves down. Something like that. You see what I’m saying? Diminishing returns
to investment. As you keep trying to add more
and more grain to your production, as you save more
and more, you get less and less return. So now, we have a new production
possibility frontier that is more
complicated. So now, what happens? Forget this straight line, which
I drew first, and now consider a new production
possibility frontier that’s curved downward. Well, what does Robinson
Crusoe do? Well, Robinson Crusoe picks the
highest indifference curve that touches this production
possibility frontier. So, that means he finds an
indifference curve that’s tangent to it. And he chooses that point. OK? So, this is what Robinson
Crusoe would do. Now, the interest rate is the
slope of the tangency between the indifference curve and the production possibility frontier. It’s the same for both. And this was the insight that
von Boehm-Bawerk maybe had a little trouble getting. There’s two different things
determining the interest rate. One of them is the production
possibility frontier, and the other one is the indifference
curves. Now, we have all of von
Boehm-Bawerk causes. We’ve got roundaboutness, we
have technical progress, and we have impatience. Well, the impatience would be
reflected by the slope of the indifference curves. So, let me put it this way. Suppose Robinson Crusoe really
wanted to consume a lot today. He was very impatient. That means that his indifference
curves — did you give me colored chalk? STUDENT: There’s a little
we have a little yellow. All right. Suppose Robinson Crusoe
is very impatient. He wants to consume. Now, he doesn’t care
about the future. Then, his indifference
curves might look — I’ll just draw a tangency — his indifference might
look different. They might look like this. So, he would have a tangency
further to the right, consuming more today and
less in the future. Now, the slope here
is different than the slope here, right? Because I haven’t changed the
production possibility frontier but I’ve moved to
a different point on the production possibility
frontier. So, you can see that if Robinson
Crusoe becomes more impatient, his interest
rate goes up. Now you understand that the
interest rate in the Robinson Crusoe economy is not just
about Robinson Crusoe. Even though there’s only one
person in this economy, it’s about all of Eugen von
Boehm-Bawerk’s causes. The technology is represented by
the technical progress and the roundaboutness, and the
preferences are represented by the indifference curves. And you can see that the actual
rate of interest in his economy is determined
by the tangency. Now on the other hand, suppose
Robinson Crusoe were very patient and really wants
to live for the future. Then the highest indifference
curve that touches the production possibility
frontier might get up here, right? Now that’s another Robinson
Crusoe with a different personality who’s
more patient. Then the tangency would be up
here and the interest rate would be much lower, because the
interest rate would be the slope of the line that goes
through that tangency point, tangent to both the indifference
curve and the production possibility
frontier. So. this is just a one-person
economy. Is this clear? So, I’ve drawn a lot of
lines, maybe I should start all over again. We’ve now gotten all of von
Boehm-Bawerk’s causes of interest. And we’ve got
an interest rate. We’ve tied it to production — technology, represented by the
production possibility frontier, and taste,
represented by the indifference curve. But now I want to add a
person to the economy. So, let me start
all over again. There’s two Robinson Crusoes
in this island. And let’s start out
with autonomy. They haven’t discovered
each other yet. They’re on opposite sides
of the island. They have the same technology. They have the same production
possibility frontier, but they live on opposite sides of the
island, and they don’t trade with each other. So, let me start out again. This is the same diagram. We have consumption today, and
consumption next year, again. And we have a production
possibility frontier. That’s the same curve
that I drew before. And the technology is the
same for both of them. And let’s suppose they have
the same endowment. But let’s suppose that Crusoe A
is very patient, and Crusoe B is very impatient. So, Crusoe A, his indifference
curves form a tangency down here. So, this is A. And Crusoe
B’s indifference curves are up here. This is B. And so, they
are planning to plant. That means that Crusoe A will
be saving very little. I mean, will be consuming
a lot. A is the impatient one,
the way I’ve drawn it. Consuming a lot now and not
saving much for the future, but is maximizing his utility. That’s why we have the highest
indifference curve shown here with this tangent. And Crusoe B is the
very patient one. And is consuming very little
this year and plans to consume a lot next year. So, let’s say they’re
about to plant, according to these tastes. And then they find each other. Now, they realize, there’s
two of us on this island. Now, we’re getting a real
economy with two people. So, what should they do? Well, the obvious thing is that
there are gains to trade. And the kind of trade would
be in the loan market. This Crusoe B is suffering a lot
of diminishing returns to production. So, he really shouldn’t be
planting so much grain, because he’s not getting
much return for it. Whereas this other guy on the
other side of the island has very high productivity. He can produce a lot for
a little bit of grain. So, he should tell Crusoe A, you
should plant some of this grain for me. You are more productive,
because you’re not doing as much. Well, in short, what will
happen is, they’ll do it through a loan. I will loan you so much grain. There’s no money. A wants to consume a lot. So, B will say, instead of
planting so much, we’ll strike a loan to allow you to consume
along your tastes. And what will happen in the
economy is, we’ll find an interest rate for the economy
that looks something — I’m going to draw a tangency. Like, that’s supposed to
be a straight line. And on this tangent line, we
have Crusoe B has maximized his utilitiy subject to that
tangent line constraint. And Crusoe A maximized his
utility subject to the same constraint. And it has to be such a way that
the borrowing market as shown over here clears. And when we have that kind of
equilibrium, you can see that both A and B have achieved
higher utility than they did when they didn’t trade. So, this is the function
of a lending market. So, A who wants to — did
I say that right? A, who wants to consume
a lot this period — the production point is here. And B lends this amount of
consumption to A, so that A can consume a lot. He can consume this much. And B, since he’s lent it to
A, consumes only this much this period. But you see they’re
both better off. They’ve both achieved
a higher utility. And what is the interest
rate in the economy? The interest rate is the
slope of this line. Well, the slope of this
line is minus 1 plus the interest rate. So, that is the Fisher
theory of interest. And now, it’s much
more complicated. You can see how all of Eugene
von Boehm-Bawerk’s causes play a role. But the interest rate is not
something you could have read off from any one person’s
utility. It’s not just impatience. We’re both complicated people. We both have a whole set
of indifference curves. And it’s not necessarily easy
to define whether or how impatient am I. It interacts
with the production possibility frontier in a
complicated way to produce a market interest rate. So, this is the model for the
interest of the economy that Irving Fisher developed. And so, I wanted to just
take that as a given. Now, when you put it this way,
it all looks indisputable that the loan market is a
good thing, right? I can’t think of any criticism
of the two Robinson Crusoes going together and
making a loan. There’s nothing bad about
this loan, right? They’re just both consuming
more as a result. But I want to come back to
criticisms of lending at the end of this lecture, because
I want to try to make this course into something that talks
about the purpose of finance and the real
purpose of finance. And this story is not the whole
story about real people and how they interact with
the lending market. But before I do that, though,
I want to do some arithmetic of finance. Let me move on to what I said
I would talk about, mainly different kinds of bonds
and present values. The Irving Fisher story was very
simple, and it had only two periods. So, that’s too simple
for our purposes. So, what I want to talk about
now is different kinds of loan instruments. And the first and the simplest
is the discount bond. When you make a loan to someone,
you could do it between a company or between
a government and someone. A discount bond pays a fixed
amount at a future date, and it sells at a discount today. It pays no interest. I mean it
doesn’t have annual interest or anything like that. It merely specifies this bond
is worth so many dollars or euros as of a future date. And why would you buy it? Because you pay less
than that amount. So, let’s say that it’s worth
$100 in T periods. T years. I’ll say T years. And I made that a capital T.
So, what is a discount bond worth today? Now, we have an issue of
compounding, which I want to come to in a minute, but let’s
assume, first of all, that we’re using annual compounding, and T is in years. Then, the price of the discount
bond today, the price today, is equal to $100
all over 1 plus r to the T-th power. Where T is the number of
years to maturity. T years to maturity. And that’s the formula. In other words, 1 plus r to the
T-th power is equal to 100 over P. So 100 over P is the
ratio of my final value to my initial investment value if I
invest in a discount bond. And I want to convert that to
an annual interest rate. So, this is the formula that
allows me to do that. So, r is also called
yield to maturity. And the maturity is T,
the time when the discount bond matures. So, it says if it’s paying an
interest rate, r, once per year for T years — We can infer an interest rate
on it even though the bond itself has a price, not
an interest rate. I mean, we can calculate
the interest rate by using this formula. Now, let me come back
to compounding. This is elementary, but let me
just talk about putting money in the bank here. So, compounding. If you have annual compounding,
and you have an interest rate of r, and you
put your money in the bank with annual compounding, and the
interest rate is r, that means you don’t earn interest
on interest until after a year. If you put in $1 today, 1/2 a
year later you’ll have 1 plus r over 2 dollars, right? With an interest rate r. 3/4 of a year later, you’ll
have 1 plus 3/4r dollars. And then a full year later
you’ll have 1 plus r. But now, after one year, you
start earning interest on the 1 plus r. So, a 1/2 year after that, you
would have 1 plus r times 1 plus r over 2. And then two years later,
you’d have 1 plus r squared, and so on. That’s annual compounding. But the bank could offer you
a different formula. They could offer you every-six-months compounding — twice-a-year compounding. Then, here’s the difference,
after 1/2 a year, you’d have 1 plus r over 2, as before. But now, after 3/4 of a year,
you would have, instead, 1 plus r over 2 times 1 plus
r over 4, and so on. Now, what Fabozzi et al. likes to do is compounding
every six months. This is what might make the
textbook a little confusing. Because we naturally think
of annual compounding. A year seems like a
natural interval. But in finance, six months
is more natural. Because, by convention,
a lot of bonds pay coupons every six months. So, Fabozzi et al. uses the letter z to
mean r over 2. And his time intervals
are six months long. So, that means that the formula
that Fabozzi et al. gives for a discount bond
assumes a different compounding interval. The Fabozzi et al. assumption. He writes P equals 100 all over
1 plus z to the lower case t, where the lower
case t is 2T. And so, that’s the
Fabozzi et al. formula for the price
of a discount bond. Of course, it only applies at
every six months interval. He’s not showing what it is at
six and a half months, or something like that. So, is that clear about
compounding and about discount bonds? Now, a fundamental concept
in finance is present discounted value. If you have a payment coming
in the future — So, I have a payment in T years
or 2T six months, or we could say semesters. Then, the present value,
depending on how I compound — Well, let’s talk about
annual compounding. The present discounted value
of a payment in T years is just the amount, which is $ x
divided by 1 plus r to the T. Or if you’re compounding every
six months, it would be x all over 1 plus z to the
lower case t. Lower case t equals 2T. So, whenever we ask a question
about present values we’ll have to make clear what
compounding interval we’re talking about. By the way there’s also — I shouldn’t say by the way,
it’s fundamental. There’s also continuous
compounding. I talked about compounding
annually, or twice a year. I can do it four times a year. If I do it four times a year,
that means I pay 1/4 of the interest after three months,
and then I start earning interest on interest after
three months, and so on. What if you compound
really often? You could do daily
compounding. That would mean you would start
earning interest on interest 365 times a year. The limit is continuous
compounding. And the formula for continuous
compounding is e to the rT, where e is the natural
number 2.718. r is the continuously compounded
interest rate. So, your balance equals
the initial amount — what did I say? $1 times e to the rT. That’s continuous compounding. The unfortunate thing is that
present values allow us to compute present values in
different ways, depending on what kind of compounding
I’m assuming. But if I have a sequence of
payments coming in, the present discounted value
of the sequence — and suppose they come
in once a year — then it’d be natural to use
annual compounding. And then the present discounted
value, PDV, is the summation of the payments. And what am I calling
them here? x sub i all over 1 plus r. Oh no, I say x sub t over one
plus r to the t, from t equals 1 to infinity. And that’s the present
discounted value for annual compounding of annual
payments. And suppose the payments are
coming in every six months, as they do with corporate bonds,
then it might be natural to do compounding every six months. So then, we do PDV is equal to
the summation, t equals 1 to infinity, x sub t, all over
1 plus r over 2 to the t. If I wanted to do continuous
compounding — suppose I have payments that are coming
in continually — then the present bond’s
discounted value would be the integral from 0 to infinity of
x sub t e to the minus rt dt. And that would be a continuously
compounded present value for a continuous
stream of payments. So, if someone is offering me
payments over time, then the payments have to be summed
somehow into a present value. In finance, it often happens
that people are promising to pay you something at various
future intervals over time, and you have to recognize that
payments in the future are worth less than payments
today. Just as a discount bond is worth
$100 in five years, but it’s not worth $100 today, it’s
worth 100 all over 1 plus r to the T, appropriately
compounded. And that’s true generally. Anything in the future
is worth less. So, present discounted value is
one of the most fundamental concepts in finance. That whenever someone is
offering me a payment stream in the future, you discount
it to the present using these formulas. So, for example, if you are
lending to your friend to buy a house, and the person is
promising to pay you over the years, then you’ve got to figure
out, well, what is that payment worth right now? And you would take the
present value of it. There’s a few present value
formulas that are essential. And I’m going to just briefly
mention them. The present value of a consol,
or perpetuity. A perpetuity or a consol is an
instrument that pays the same payment every period forever. It was named after the British
consols that were issued in the 18th century. They were British government
debt that had no expiration date. And the British government
promised to pay you forever an amount. OK, what is the present value? Now, we’ll call the
amount that the consol pays its coupon. And let’s say the coupon
were GBP 1 per year. If it was paying GBP 1 per year,
and we’re using annual compounding, then the present
discounted value is equal to GBP 1 over the interest rate. That’s very simple, because
this bond will always pay you GBP 1. And so, what is the interest
rate on it? Your GBP 1 is equal r over the
present discounted value. So, the prices of the bond of
the consol should be the payment divided by the
interest rate. Another formula is the formula
for an annuity. An annuity is a different
kind of payment stream. It’s a consol for a while
and then it stops. An annuity pays a fixed
payment each period until maturity. So, it pays, let’s say, GBP x. Let’s not say GBP 1. If it pays GBP x every year,
then the present discounted value — it pays GBP x from t equals
1 to T. And then it stops. T is the last payment. Then the formula is x over r,
times 1 minus 1 all over 1 plus r to the T-th power. So, that’s the annuity
formula. And that’s very important,
because a lot of financial instruments are annuities. The most important example
being a home mortgage, a traditional home mortgage. You might take out a 30-year
mortgage when you buy a house. And the mortgage will generally
say in the United States — it’s not so common in
other countries — but in the United States it will say
you pay a fixed amount — well, usually it’s monthly,
but let’s say annually for now — a fixed amount every year as
your mortgage payment. And then you pay that
continually until 30 years have elapsed, and then
you’re done. No more payments. The final thing I want to talk
about is a corporate bond, or a conventional bond, which is
a combination of an annuity and a discount bond. And so, a conventional corporate
bond, or government bond, pays a coupon
every six months. So, a conventional bond pays
coupon c, an amount c, in dollars, pounds, or whatever
currency, every six months. And principal plus
c at the end. So, that means that it’s
really an annuity and a discount bond together, right? And so, the present discounted
value for the conventional bond would be the sum of the
present discounted value using the annuity formula for x equals
c plus the present discounted value of the
principal, which is given by the — well, it would be this one,
where you have r over 2, because it’s every six months. And then the final — I think it’s the final concept
I want to get at before talking a little bit about
other matters. I want to talk about forward
rates, and the term structure of interest rates. Now, at every point in time
there are interest rates of various maturities quoted. And we want to define the
forward rates implicit in those maturity formulas. And this is covered carefully
in your textbook of Fabozzi et al. I’m just going to do a very
simple exposition of it. The concept of a forward rate,
forward interest rate, is due to Sir John Hicks, in his 1939
book, Value and Capital. About 20 years ago, I was
writing a chapter for the Handbook of Monetary Economics
about interest rates. And I was trying to confirm who
invented the concept of forward interest rates. So, I’ll build a little
story around this. I thought it was Sir John Hicks,
reading his 1939 book, and I couldn’t find any
earlier reference. So, I asked my research
assistant, can you confirm for me that the concept of
a forward interest rate is due to Hicks. And my graduate student looked
around and tried to find earlier references to
it and he could not. Then, one day the graduate came
to me and said — this is like 20 years ago. The graduate said, why
don’t you ask Hicks? And I said, wait a minute. This book was written in 1939. Is that man still alive? And he said, I think he is. So, I wrote to the
United Kingdom. I found his address. I forget, Cambridge or Oxford. And I said, did you invent
the concept of forward interest rates? And then six months went
by and I got no answer. Then I got a paper letter — they didn’t have email
in those days — from Sir John Hicks. And it was written with
trembling handwriting. And he said, my apologies for
taking so long to answer. My health isn’t good. But he said, to answer your
question, he said, maybe I did invent the concept of forward
interest rates. But he said, well
maybe it wasn’t. Maybe it was from coffee hour
at the London School of Economics, where he was visiting
in the 1920’s. So, here we go. Sir John Hicks is reminiscing
to me about what happened in coffee time in the 1920’s. I’m just trying to convey what
he told me they were thinking. At any point of time, you open
the newspaper and you see interest rates quoted for
various maturities. That’s called the term structure
of interest rates. For example, you will
find Treasury — well, you’ll find one-year rate
quotes, there’ll be a yield on one-year bonds. There’ll be a yield
on two years. There’ll be a yield quoted
on three-year bonds. Right? Let me tell you right now, for
most of the world today, if you want to borrow money for one
year, it’s really cheap. In Europe, or the U.K.,
U.S., over much of the world it’s like 1%. In the U.S., it’s
less than 1%. It depends on who you are, what
your borrowing rate will be, depending on your
credit history. But if you have excellent
credit, one-year interest rates are really low. But if you want to borrow for 10
years, it’s more like 3.5%. It’s higher, right? And if you want to borrow for
30 years, they might charge you 4% or 5%. This is the term structure of
interest rates, and it’s quoted every day in
the newspaper. Well, I should say,
I’m thinking 1925. In 1925, you’d go to the
newspaper to see it. Now you go to the internet
to see it. So, newspapers don’t
carry this anymore. But I’m still in the mode of
thinking of Sir John Hicks. We’re in 1925. So, you open up the newspaper in
1925 and you get the yield to maturity, or the interest
rate on various maturities. All for today. Everything that’s quoted in
today’s paper is an interest rate between now, today, and so
many years in the future. The one-year interest rate
quoted is the rate between now and one year from now, right? And the two-year interest rate
quoted is the rate from now to two years from now, and so on. So, Hicks and his coffee hour
people were saying, well, it seems kind of one-dimensional,
because all the rates that are quoted are rates between now
and some future date. But what about between
two future dates. And then they thought about
this at coffee hours, and someone said, well, it’s kind of
unnecessary to quote them, because they’re all implicit in
the term structure today. And this is where the
concept of forward interest rate comes. And it’s explained
in Fabozzi et al. But I thought, I’m going to just
try to explain it in the simplest term. Once we get the concept,
it’s easy. And I’m going to assume annual compounding to simplify things. But Fabozzi, he being a good
financier, does it in six-month compounding. So, now the year is 1925, and
we’re in coffee hour. Suppose I expect to have GBP
100 to invest in ’26. It’s ’25 now, this is a whole
year from 1926, OK? And I want to lock in the
interest rate now. Is there any way to do that? I mean, I could try to go to
some banker and say, can you promise me that you’ll give
me an interest rate in 1926 for one year. The banker might do it, you
know, but I don’t need to go to a banker to do that. Once I have all of these bonds
available, and if I can both go long and short them,
then I can lock it in. So, here’s what I want to do. This is what they were
discussing at coffee hour. Buy, in 1925, two-year
bonds in an amount — you’ve got to buy the
amount right — you’ve got to buy 1 plus r sub
2, which is the two-year yield, squared, all over
1 plus r sub 1 bonds. Discount bonds. They’ll mature in two years. And then I have to short, in
1925, one-period bond that matures at GBP 100 So, suppose
I do that, what happens after one period? Well, after one period,
I owe GBP 100, right? Because I just shorted
a one-period bond. So, I pay GBP 100, that’s
like investing GBP 100. At the end, I get this amount
times GBP 100, right? Because this is the number
of bonds that I bought. So, what is the return
that I get? The return that I get
is the ratio — well, we’ll call that the
forward rate between ’26 and ’27 as quoted in 1925. And so, that forward rate, we’ll
say 1 plus the forward rate, f, is equal to 1 plus r
sub 2, squared, all over 1 plus r sub 1. It’s just the amount
that I get. What I’ll get if I bought this
number of bonds, I get GBP 100 times this number in two periods
in 1927, but I put out GBP 100 in 1926. So, the ratio of the amount that
I got at the end, in ’27, to the amount that I put in
in ’26 is given by this. So, that’s 1 plus the interest
rate I got on the bond. So, you can compute
forward rates. I’m just showing it for a
one-year ahead forward rate for a one-year loan. But you can compute it for any
periods further in the future over any maturity. And this is the formula given,
it’s on page 227 of Fabozzi et al. I’m not going to show you
the general formula. The expectations theory of
the term structure is a theory that — I’ll write it down. Expectations theory says that
the forward rate equals the expected spot rate. So, do you see what
I’m saying? Here in 1925, I open The London
Times, and right there I have printed the whole
term structure today. And I can then compute, using
forward rate formulas, the implied interest rates for
every year in the future. Even out to 2010, they could
have computed — if they had bonds that
were that long — I think they had a few. Let’s see, 1925. Some bonds go out 100 years. If you wanted to do the one-year
rate in 1925, for the year 2011, you’d have to
find a pair of bonds. One of them maturing in
2011, and another one maturing in 2012. And if you did that, you
could get an interest rate for this year. So, that was kind of the
realization that Hicks got, that the whole future
is laid out here in this morning’s paper. All the interest rates for — maybe not out to 2011 — but out to a long time
in the future. And so, what determines
those interest rates. So, Hicks, in his book, wrote,
the simplest theory is the theory that these forward rates
are just predictions of interest rates on those
future dates. So, we could go back and see,
what were they predicting? They weren’t thinking so
clearly, definitively, about 2011, but they must’ve been,
because they were trading these bonds. And so, you could test whether
the expectations theory, whether people are forming
rational expectations by looking at those forecasts and
seeing, were they right? Now there’s a huge literature
on this, but Hicks said that — I’ll just stop with this — Hicks said that the expectations
theory doesn’t quite work, because there’s
a risk premium. That the forward rates tend
to be above the optimally forecasted future
spot rates — spot meaning, you know, as
quoted on that date — because of risk. And people are uncertain about
the future, so they demand a higher forward rate then
they expect to see happening in the spot rate. So, I will stop talking
technical things. I wanted to say something. I have so much more to
say, but I’ll have to limit due to time. What I’ve laid out here is a
theory of interest rates. And I’ve done some interest
rate calculations. And I’ve pointed out the
remarkable institutions we have that have interest rates
for all intervals, out maybe 100 years. And so, it’s all kind of like
the whole future is planned in these markets. It seems impressive,
doesn’t it? And when I told you the Robinson
Crusoe story, didn’t that sound good? Like when the two Robinson
Crusoes discover each other, aren’t they obviously doing the
right thing to make a loan from one to the other? And I like that. I think basically everything
I’ve said here is basically right. But I wanted to say that one of
the themes of this course is about human behavior and
behavioral economics. And I wanted to talk a little
bit about borrowing and lending, and how it actually
plays out in the real world. And how our attitudes are
changing, our regulatory attitudes are changing. So, let me just step back. You know, I think this
literature, involving Irving Fisher and von Boehm-Bawerk,
and many others who’ve contributed to the understanding
of interest rates, is very powerful
and important. And it supersedes anything that
had been written in the last thousands of years. They had interest rates for
thousands of years, but that simple diagram, that
Fisher-diagram, came just a short time ago. It’s hardly long ago at all. But I wanted to step back and
think about what people said about interest rates going
way back in time. And so, I was going to
quote the Bible. There’s a Latin word. Do you know this word? [WRITES USURA] Do you know what that
means in Latin? Well actually our English word
“use” comes from it. I don’t know how to pronounce
it. ”Usura” in Latin means use. And it means also interest.
Because, what is interest? You’re giving someone the
use of the money. You’re not giving
them the money. They’re getting the
use of the money. And they had other words for
interest. But this ancient word had a negative
connotation. It kind of meant something
immoral. And so, we have a word
called usury. You know this word. This is English now. It goes back more
than 2000 years. I actually have it
here in Latin. I’m just curious about these
things, but I can’t pronounce it right. It must have been written
in Greek, or Aramaic, or something originally, but it
uses the word, “usury,” usura. But the quotation, it says in
Exodus, “If thou lend money to any of my people that is poor by
thee, thou shalt not be to him as an usurer. Neither shalt thou lay
upon him usury.” Now what does that mean? Because usura could mean both
interest and excessive interest. So, it’s not clear
what the Bible is saying about lending. It sounds like it’s telling
you you can’t lend — you can lend someone money, but
don’t take any interest. That’s what it seems to be
saying but it’s ambiguous. I was going to quote
the Koran. I don’t speak Arabic. I think there’s a similar
ambiguity in Arabic. And I’m quoting an English
translation of the Koran: “Oh you who believe. Be careful of Allah, and give
up the interest that is outstanding.” Or usura. That has been interpreted by
modern Islamic scholars as that charging interest
is ungodly. And it was interpreted by
Christian scholars. They go back and they try to
figure out what was meant, and they couldn’t figure it
out, either, then. Times change over
the centuries. But for thousands of years the
Catholic Church — or maybe not thousands. I don’t know the whole
history of this. It depends on which century
you’re talking about. But for many centuries, the
Catholic Church interpreted this, as do many Muslims today, that interest is immoral. And therefore, the only people
who were allowed to loan were Jews, because they weren’t
subject to the — even though it’s actually
the Book of Exodus — but they weren’t subject to
the same interpretation. So, it was considered immoral. I wonder, why is that? Why is it immoral? Because we just saw
the logic of it. Now, the Robinson
Crusoe story. I had two different men on the
other side of the island. And I had one of them wanting
consumption today and of one of them wanting consumption
later. Your first question
is, maybe they were wrong to be different. Maybe they should both be
doing the same thing. Why is one of them different
than the other? The guy who’s going to consume
a lot today, maybe I should have a word with this guy. You know, don’t do it,
you’re going to be really hungry next year. Why are you doing this? So, instead of forming a loan
between the two, we should advise them. And maybe they don’t
need a loan. So, this comes back to what are
we doing with our loans. And are we giving people
good advice? Or do we have a tendency in the financial world to be usurious? Are we going after and
victimizing people by lending them money? I think that there
is a problem. And these thousands of years
of history of concern about usury have to do with real
problems that develop. So, just in preparing for this
lecture, on an impulse, I got on to Google. And I searched on
vacation loans. I found 1.6 million websites
that were encouraging you to take out a loan to
go on a vacation. Now, is that socially
conscious? I was wondering about that. Is it ever right to borrow money
to go on a vacation? I mean, I’ve thought about it. And then I remembered, Franco
Modigliani, who was one of the authors of your textbook,
and he was my teacher. I still remember these moments
from classroom. And he was teaching us
about these subjects. He was thinking about examples
of investments — and he said, you know what? One of the best investments I
can think of is a honeymoon. When you get married, you
go on a vacation. Now, why are you doing that? Is it for fun? Probably not. In fact, I have a suspicion
that most honeymoons are not fun. I think it’s just people are
too uptight and tense. What have we just done? And I bet that’s right. So, why do you do it? Well, you do it as an
investment, right? You want this photograph
album. You want the memories. You’re kind of bonding. I think he’s absolutely right. You should go on a honeymoon. So, I did another search. I searched on honeymoon loans. And I got 1.7 million hits. It beat vacation loans. So, there are many lenders
ready to lend. And you should do it. If you’re just getting married
and you don’t have any money, go to the usurious guy and ask
for the honeymoon loan. So, I’m not sure whether
it’s bad. This is a question. I think that there
are abusers. And I wanted to just close
with Elizabeth Warren. I first met her just
a few years ago. Well, actually, I remember
her book. She wrote some important
books. She’s a Harvard law professor
who wrote books. One of her books was published
by Yale, called The Fragile Middle Class. And it’s about people who
go into bankruptcy. And she points out that in the
U.S., even back in the old days when the economy was
good, we had a million personal bankruptcies a year. This is because of borrowing. You don’t know how many
bankruptcies there are, because people are ashamed when
they declare bankruptcy. And they try to cover
it up from as many people as possible. There are as many personal
bankruptcies in a normal year as there are divorces. But you don’t hear about them. You hear about all kinds
of divorces. People are ashamed
of divorces, too. But they can’t cover them up
because everybody knows. But they can pretty well cover
up a bankruptcy, and so they don’t talk about it. So, what Elizabeth Warren is
saying, she thinks that the lending industry is victimizing
people. It’s advertising for vacation
loans and the like, and then they don’t tell people about the
bad things that will come. So, she wrote an article, and
this is interesting, it was in Harvard Magazine. And that’s a magazine that I
suspect none of you read. Anyone read Harvard Magazine? It’s the Harvard alumni
magazine. It goes out to all graduates
of Harvard. So, you don’t read it, and you
probably will never read it. You will be a reader of the Yale
Alumni Magazine, which will start arriving in your
doorstep after you graduate. And it will also include your
obituary in the next century when that comes. But the Harvard alumni magazine
published this wonderful article about
Elizabeth, describing all of the abuses that happen in
lending in the United States. I don’t know how I ended
up reading it. I think it was just such a
nicely written piece that it just became one of their
success stories. Most people don’t read that
magazine, but I read it, and a lot of people have read it. And she was so successful in
convincing the public — this is just two years — or 2008,
three years ago — she was so successful that she
got a Consumer Financial Protection Bureau inserted
into the Dodd-Frank bill. And we now have a regulator, a
new regulator, that’s supposed to stomp on these usurious
practices. It’s kind of an inspirational
story, but the downside of it is, she got too carried away
criticizing the lending industry in that nice article. And it makes them sound worse
than they really are, and so Obama could not appoint her to
head the Consumer Financial Protection Bureau, because it
would be too politically controversial. So, she is now the person trying
to find someone to head her bureau. But I think that this is just
another step, and it’s happening in Europe and
in other places. The financial crisis has
made us more aware of bad financial practices. And so, usury is again
on our minds. Usury is abusive lending that’s
taken without concern for the person who’s
borrowing. And I think what it means
to me is that — we’ll come back to talk
about regulation in another lecture — but that the original Irving
Fisher story and von Boehm-Bawerk story about
interest was right. And even vacation loans,
especially honeymoon loans, are right. But we need government
regulation to prevent abuses. And we do still have abuses
in the lending process. So, I’ll stop with that and
I’ll see you on Monday.