# 8. Theory of Debt, Its Proper Role, Leverage Cycles

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

bit of yellow. PROFESSOR ROBERT SHILLER: Oh,

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.