All right, so last time we
began, or maybe two times ago, we began a discussion of
various vocabulary and facts you have to know about the markets
if you want to think about finance.
Today we're going to deal
mostly with the most important one, the most basic one,
the yield curve. And last time,
we introduced this word, "yield."
Now, yield is an extremely
common expression in finance, and it turns out not to be that
well defined, often, or that useful.
But the word is so important
and has been used so often that it still hangs around,
even when probably we should use different concepts.
So remember the yield was an
attempt to look at an investment,
and without paying any attention to the market or
anything outside the investment, just looking at the investment
itself, try to assess,
give a number, quantifying how attractive the
investment was. So we said you could apply that
to a bond--it has cash flows. You could apply it to a hedge
fund that's taking in money and paying out money,
and the formula we came up with said that if the cash flows are
given by C(1), C(2), the net cash flows,
C(T) over the course of the period,
and its price is some P(0), maybe it's a negative cash
flow, so C(0).
So some of these cash flows
might be negative and some of them might be positive,
then we should just look at the number Y,
such that discounting all these things at rate Y gives you 0.
The Y that did that was what we
defined as the yield of the investment.
So we saw that that had some
advantages. For example,
in a hedge fund, if you just look at the rate of
return it makes on its money every year,
that doesn't take into account that in some years,
it's got a lot more money. So if those were the years that
lost money, and the years when it hardly
had any money were the years it made money,
just taking the average, the multiplicative average,
the geometric average of all those yearly rates of returns,
would give a misleading figure. Well, the yield also gives a
somewhat misleading figure, and I don't want to spend too
much time on why it might be misleading,
but I'll give you just an example.
Suppose that the cash flows
happen to be 1, -4, and 3.
Now what's the yield to
maturity? Well, there are two of them.
You could have Y = 0,
because 1 over (1 0), - 4 over (1 0),
3 over (1 0), is just 1 - 4 3.
That equals 0,
so the yield to maturity of 0 percent, the yield of 0 percent,
makes this have present value 0.
But also I could try Y = 200
percent, and then I'd have Y--I'd put a
2 and a 2 squared here, and I'd have 1 - 4 thirds 1
over 3 squared is 3 over 9, so it's 1 third.
So it would be 1 minus 4
thirds, plus 1 third, which also equals 0.
So is the yield to maturity,
the internal rate of return 0 percent or 200 percent?
It's ambiguous.
So yield to maturity can't be
the right way of doing things. To go back to the hedge fund
example, you know, the hedge fund was
taking in money, paying out money,
taking in more money, paying out money,
and we calculated the yield to maturity.
Well suppose that there was
some period, you know, here,
at which point everyone had taken all their money out,
so the hedge fund wasn't actually doing anything for a
bunch of years, maybe for a long time,
and then it started up and took money in and paid money out and
stuff. Well, because the gap in time
was very long with nothing happening,
if you take a positive Y, the stuff that happens in the
second incarnation of the fund is hardly going to be making any
difference, because by that time,
it will all be discounted a lot.
So the yield will depend too
sensitively on stuff early rather than stuff late.
And so again,
you get into troubles yielding just yield to maturity,
so that can't be the right thing to do,
even though people have done it for years.
So the word,
however, lives on, and there's no getting rid of
the word because it's used in common vocabulary.
Now what would Irving Fisher
say you should do, if you had to summarize how
good an investment was? What's his lesson?
What do you do?
An investment where there's no
doubt about what the cash flows are going to be,
what would he say you should do, to evaluate the
attractiveness of it? What's our lesson,
our main lesson from Irving Fisher?
What would he say?
Yes.
Student:
> like to check?
Prof: Well,
let's say they're cash flows, so it's money,
money that you're going to get coming in and out,
yeah, he'd say, deflate by inflation and turn
them into real flows and then do what?
So just continue your answer.
So turn them into actual
potatoes every time, apples each time.
Deflate by inflation and then
do what with the numbers? This is a simple question.
You're thinking too hard.
Yes?
Student:
Compare the present value. Prof: Okay,
he'd say, "Just look at the present value of all these
things." So of course,
to do that, you'd have to know, what is the market rate of
interest with which to compute the present value?
So Fisher would say,
"It's ridiculous to evaluate how good an investment
opportunity is just by looking at the cash flows.
You're throwing away too much
information." You know what the market is
doing, you know what the interest rates are.
Use the market interest rates
and figure out what the present value of all the cash flows is.
So we're going to now do that a
bunch of times, okay, for the rest of the
class, and see what that means. So we have to begin,
the two thirds of the class is going to be spent on the
question, how do you know what the market rates of interest
are? So how do you know what the
market rates of interest are? How could you find out what the
market rate of interest is? What would you do to find it
out? Yeah.
Student:
You could go to a bank and see what they were estimating it to
be. Prof: And if you looked
in the newspaper, say, could you find it in the
newspapers? What would you find in the
newspapers? Yes?
Student:
You'd want to find a riskless investment, say like a T-bill.
Prof: Okay,
and so yes, you try and look at riskless investments like
government bonds, where there can't be any
default--at least, that's what they always used to
say-- can't be any default on an
American promise. America's government never
broke a promise and they can always print the money,
so presumably they don't have to break a promise--
so they're just promising money which they can print,
so why should they ever break their promise--
so what would you find if you opened a newspaper?
You would find,
for different maturities-- it used to be up to 30
years--for different maturities, you would find the yield on the
various bonds, okay?
So why would you find the yield?
Well, the yield of various
government bonds, of gov.
bonds.
Why do they quote the yield?
Well, that's just because,
you know, a hundred years ago,
people started using the idea of yield and so the vocabulary
has been kept, even though it's not the best
way of describing what's going on.
So for instance,
let's just look at some of the yield curves you might have seen
over the last 9 years, almost 10 years,
since December 2000. You would see that in December
2000, the yield on the 1-year bonds, you know,
the short yields--this isn't a log scale, so this is 3,6,
12, okay. So the shortest bonds usually
have lower yields than the highest bonds,
but sometimes, like in December 2000,
the yields are almost all the same.
It's called the flat yield
curve. Other times,
like now, we're in this light blue one here,
right now the short bonds have very small yields and the long
bonds have much higher yields, so the last one is the 30-year
bond. So you get the yield on every
single bond. Now what do you notice about
this picture, by the way?
They can be very different at
different time periods, so in December 2000,
the interest rates were really high.
The yields were 6 percent.
I'm talking yields so far.
We haven't talked about
interest rates. We have to figure out what the
interest rates are, but anyway, they're obviously
going to be connected. So the yields were very high in
December 2000, and they got much lower in
December 2008, and they've stayed very low.
So why are they so low now?
What got them to be so low now?
Yeah?
Student:
The Fed flooded the economy with money.
Prof: The Fed flooded
the economy with money. It wanted to drive the interest
rates down to 0. So we're going to see very soon
why the Fed might have wanted to do that, but these money rates
don't move totally on their own. They have to do,
and we said that Irving Fisher--
we haven't described Irving Fisher's theory of money and
nominal interest rates-- but somehow,
the Fed is controlling the nominal interest rates and it's
changed the yield curve. So you notice that the yield
curve now, December 2008, was this blue one.
So the Fed, in the crisis of
2008, you know, was terrified,
and it dropped the interest rate almost to 0,
virtually 0, and it's kept it there,
because from December 2008 till now,
we're at October 2009, September 30^(th),
2009, a long time has passed from this dark blue to the light
blue line, and the rate has been kept
fixed there. But in the intervening time,
the long rates have started to go way up.
Now why might that be the case?
What does that suggest to
anybody? Does anybody know?
Yeah?
Student:
> Prof: That could be one
reason, and could there be another reason?
Yeah?
Student:
Future expected inflation. Prof: Okay,
so those are the two reasons. So they somehow know that,
and some of you have no idea how they could possibly be
thinking that. And so I'm going to explain,
what information is there in the different yield curves.
Okay, so the point is that
every morning, every single financial analyst
wakes up and sees these yield curves,
you know, consults the market and sees where things are
trading, and can produce a yield curve
like that. Okay, so you've got a bunch of
yields. Now what are the yields?
Well, let's just do an example
here. So I'm going to make up an
example, which is very easy to compute.
So let's try this one.
Okay, so I'm reading here at
the top, let's just say that you've got a bunch of different
bonds. A 1-year bond,
a 2-year bond, a 3-year bond,
a 4-year bond and a 5-year bond.
Now each of the bonds was
issued--I'm assuming here that they were all issued on the same
day. So they're issued with
different coupons. Let's say they were all issued
today. We'll come back at the end.
Obviously the Treasury doesn't
issue new bonds every single day, so how does this change
when you arrive on a day when they haven't issued things?
But let's just keep it simple
and suppose that today, the Treasury has issued 5
different bonds over these 5 different years.
Now the Treasury has to set
what do they do? They decide how much of these
bonds they're going to sell and they decide what coupon they're
going to set. They set the coupon.
So let's say the coupon they
set was 1 dollar for the 1-year bond,
2 dollars for the 2-year bond, 3 dollars for 3 three-year
bond, 4 for the 4-year bond,
5 for the 5-year bond. It's easy to remember,
that's why I chose those numbers.
And the face value,
let's say, is always 100. So why did they set those
coupons? Well, because given how much
they want to sell, they're picking the coupon,
hoping that the price turns out to be close to the face value.
So let's say,
when they actually market these,
and supply equals demand in equilibrium,
the prices turn out to be 100.1,100.2,
100.3,100.4, and 100.5. So 100.5 is the price the
market's paying for the 5 year bond and if the coupon is 5,
and they pay coupons once a year--they may pay twice a year,
but let's say they pay once a year for simplicity,
you're going to get 5,5, 5,5, and 105 the last year.
If you bought the four-year
coupon bond, you get 4,4, 4,104 the last year,
right? So those are the bonds.
Now the newspaper's not telling
you any of that, so you're sort of losing that
information, so you don't actually know that
from reading the newspaper. So then, what do you know?
You know that--you know the
yields on all these things, okay?
So here, this tells you the
yield on each of these bonds. Going back to where I was.
Okay, knowing the yield,
you could figure out what the price of each of the bonds is,
or in fact, the way that they calculated the yield that the
newspapers reported. How did the newspapers get the
yield? The newspapers said,
well, for the 4 year bond, we're going to say that 100.4 =
4 divided by (1 the 4 year coupon bond yield),
4 over (1 Y (4)) squared, plus 4 over (1 Y (4)) cubed,
plus 104 over (1 Y (4)) to the fourth.
In this case,
because these numbers are all positive,
it's monotonic in Y (4), so there's a unique Y (4) which
you can use to solve this equation,
price = the discounted value at that yield of the payments.
So that's how the newspapers,
the reporter, that's how he got all the
yields to show you that graph. He looked at the market,
or could call the bank or something like that,
had a computer screen, talked to his friends on Wall
Street. He knew the price of all the
bonds, he knew the coupons of all the bonds,
and then he produced the yield for all the bonds.
So the yield,
as I say, that's the word that everybody uses,
but really, the information that you want to deal with is
the price, and what did the coupon
actually pay? Okay, so that's what you know.
Everybody knows this every day,
the information I have given. Every morning,
maybe every few hours, people will update it.
They'll look at what are the
coupon bonds paying and what are the prices?
The thing that's changing from
hour to hour are the prices, but we're taking a snapshot at
the beginning of the day and looking at the prices.
So now we've got prices of
bonds, which I'm going to call capital Pis, Pi (1),
Pi (2), Pi (3), Pi (4) and Pi (5).
But now, what does Fisher say
you should do? What's the most important thing
to do? The most important thing to do
is find the interest rates. So what has this got to do with
interest rates? Well, if you modernize Fisher a
little bit, the most important thing--he
didn't put it this way, but this is really what he must
have meant-- the most important thing to do
is find the prices of the zeros. So [little]
pi (1) is today's money price, today's money price for 1
dollar at time 1. pi (2) is today's money price
for 1 dollar at time 2. pi (3), today's money price at
time 3, and pi (5) is today's money price at time 5.
Okay, now why do you want to
find these things? Because once you know these
things, you'd be able to value any investment.
That original investment that
we talked about, which maybe disappeared,
this one. It disappeared.
Anyway, once you have all the
pis, if anybody tells you, if a hedge fund tells you,
"This is the revenue I'm going to produce for you in the
next five years," if a company says,
"This is our business plan. We're going to build a factory
today that's going come out a certain amount of money and
we're going to get profits in the next five years,
blah, blah, blah." If a new bond comes on the
market and you don't know how to price it, and somebody offers a
price for it, how do you figure out what it's
worth? All you do is you take the cash
flows, the Cs that I not very cleverly erased,
you take the cash flows and multiply them by the pis.
So the correct price P is just
whatever the cash flows you're predicting times these pis that
Fisher says is what you should really be finding.
So nothing could be simpler.
Now why is this the right price?
Because if you can go in the
market and buy 1 dollar at time 1 for pi of 1,
and 1 dollar at time 2 for pi of 2,
etc., you can buy all the cash flows from this investment
project by spending this amount of money.
So if the guy is offering you
the investment opportunity at a higher price,
it's crazy to do it. You could have bought those
cash flows yourself by paying this price.
If he offers it to you at a
lower price than that, then definitely you should do
it, because it's a bargain, because if you had to buy it
yourself, it would be more expensive.
In fact, you can make an
arbitrage profit. If he's offering it to you at a
lower price, you can buy it and how do you buy it?
By selling these very promises,
C(1), C(2), C(T) on the market, and if people believe you that
you'll pay, you can sell it for this price.
So you buy his project for a
lower price. You sell it for a higher price.
You make the difference,
and when it comes time to keep your promises,
the project is giving you the cash to keep your promises,
so you lock in a profit for sure.
So if you knew the pis,
you would know for sure how to value any project where you knew
for sure the cash flows, knowing the pis would tell you
how to value it, and would tell you whether it
was attractive or not attractive.
You just look at how high the
present value is. Okay, any questions about that?
So we just need to figure out
how to deduce what the little pis are from the data that we're
going to be given, and that we are given every day
by the market. Okay, so I said literally pi
(1) is the price you would pay today to buy 1 dollar tomorrow.
Now how could you go about
buying 1 dollar tomorrow, given that the only things you
can trade on the market are these Treasury bonds,
these government bonds? And I've told you what they pay
off and I've told you what their prices are.
So how would you go and buy 1
dollar tomorrow and how much would it cost you,
1 dollar next year? How would you do that?
You can trade,
buy or sell any of these Treasury bonds.
So in the background,
I keep saying that we're going to have to worry about people
defaulting. We're not quite doing it yet.
So buying a Treasury bond,
you need the cash to buy it. Selling a Treasury bond means
you promise to deliver what the Treasury bond promises,
and your promise is as good as the government's.
So if you sell it to somebody,
they'll pay you the same government price for it.
So obviously in the background,
you're going to have to do something to convince the guy
you're making the promise to that you're going to keep your
promise. So we're going to worry about
that later. So for now, when I say that the
market for those Treasury bonds clears at those levels,
I mean that anybody who wants to can buy Treasury bonds at
those levels, or can sell them,
even if he doesn't have them, at those prices,
by making the promise of what the Treasury bond does.
Because we're assuming that the
government and you, everybody is just as reliable,
everybody is going to keep their promise.
So whether it's the Treasury
making the promise, or you making the promise,
same thing. Okay, so how do you buy 1
dollar next year? What would you do in the market
with the Treasury bonds to get 1 dollar next year?
Yeah?
Student:
Can we just plug in 1 for our P in the bond prices and then
figure out which bond you want to purchase?
Prof: Well,
you're supposed to be telling me.
What do you want to do?
I want to know exactly what to
do. You can look at these numbers.
By the way, are you all
following these? Maybe this is mysterious.
There's a 1-year Treasury bond
that pays 101 dollars next year. The 2-year Treasury bond pays 2
dollars next year, then 102.
The 3-year Treasury bond pays 3
dollars at the end of the first year, 3 at the end of the second
year, 103 at the end of the third year, etc.
And the 1 year Treasury bond
happens to be selling at this price, the 2-year happens to be
selling for that price. So what can I do with all these
bonds to buy 1 dollar next year? All right, go ahead,
you've started. Student:
Buy a 2-year bond. Prof: A 2-year bond?
It's next year,
one year from now. Student:
You could just divide the price of the 1-year bond by
> Prof: You're a step
ahead of me. I'm saying, what do you
do--never mind how much does it cost?
What do you do today to get 1
dollar next year? What transaction can you make
today, what purchase can you make today to get yourself 1
dollar next year? Yes?
Student:
Just buy a 1-year bond. Prof: Buy a 1-year bond.
Well, that will give me 101
dollars next year. I want 1 dollar next year.
Student:
Take that fraction >
Prof: Okay,
well, that's exactly the point. I take that fraction.
That's what I wanted you to
tell me. So little pi (1) is going to be
(1 over 101) times the price of the one-year bond,
because the 1-year bond is paying 101 dollars.
So you take 1/101 of it,
you'll get 1 dollar. And whatever the price of the
1-year bond is--actually, we know what that is.
It's 100.1, that's how much it
costs. To buy 1 of them cost 100.1.
To get 1 over 101 of them costs
pi (1). Okay, so this number,
by the way, is some number which, actually,
I of course worked out here. Happens to be .991,
but we'll come back to that. So it happens to be point 991.
So now we know pi (1).
Well, how would you buy 1
dollar in year 2? So there's a way of directly
buying 1 dollar in year 2, once you know how the
Treasuries trade. So what I'm doing is I'm
explaining the idea of replication, pricing.
It's giving me pricing and it's
going to lead to arbitrage. They're all basically similar
ideas here. So what I want to do is
directly buy 1 dollar in year 2. So I could probably go to a
bank and they would actually make that trade for me.
I could just call up the bank
and say, "I want 1 dollar in year
2," and they'd tell me, pi (2), how much I have to pay
for it. But how are they going to
figure out what it's worth? They're going to see
how--they're going to go out and have to buy the dollar for me.
So they're going to go out and
go to the Treasury market. And what are they going to do
in the Treasury market to come up with my dollar in year 2?
They're going to replicate my
purchase of 1 dollar with a more complicated portfolio that they
can actually trade, and that's how they're going to
figure out how to price my request for 1 dollar in year 2.
So what is this bank going to
do in the Treasury market? What does it have to do to get
1 dollar for sure in year two, and nothing else?
Okay?
Student:
Do they buy a 2-year bond and sell a 1-year bond?
Prof: Okay,
so what are they going to do? How much of the two-year bonds
should they buy? Student:
What they're going to sell is a...
Prof: The 2 year bond.
You're talking about the 2-year
bond, so how much of the 2-year bond are they going to buy?
Student: 1/102.
Prof: 1/102, right.
That's very good.
Why is that?
Because in year 2,
we're talking about year 2 now, year 2, the 2-year bond pays
102. You get 1 over 102 of those,
you've got 1 dollar in year 2, so that's Pi of 2.
Which happens to be 100.2.
That's how much that costs.
Okay, but is that all you need
to do? Is that it?
Are you paying the right amount
or are you paying too much, or what are you doing?
You've got to do more than just
that. Why is that?
Yeah.
Student:
When you subtract 2 times the >
Prof: Close,
but not quite. Okay, so do your reasoning.
You told me what else to do,
which you slightly mis-said. So why do you have to do
anything at all? Tell me the reason why you want
to do something else. You're on the right track,
you just slipped up a little bit.
So why not just stop here?
Student:
Because you're also getting 2 dollars in year 1.
Prof: Exactly,
that's exactly right. By buying the 2-year Treasury,
you got 102 in period-- by buying this fraction of the
2 year, you got just what you want in
period 2, but you also purchased the
coupon in period 1, which you don't need.
So that's giving you more than
you needed to buy. You've bought extra,
so you're going to actually be able--
the cost of getting the dollar at the end of year 2 is a little
bit less than what we've written so far,
because you bought more. So far, this is buying too much.
You bought the dollar in year 2.
You also bought a little bit in
year 1. You can now sell off the extra
stuff you've gotten in year one to reduce your cost of buying
that. So what should you do in year
one? That's exactly what you were
thinking. You just didn't quite say it
right. So what should you do in year 1?
Student:
> Prof: I sell that.
Okay, I sell.
Okay, so I can get to sell 2 of
little pi of 1, right?
Because I know how much it
costs me to buy 1 dollar at time 1 now.
It's that number,
so I'm getting-- so is that correct, what I've written here?
That's what you said.
That's not quite right.
Yeah.
Student:
If you didn't actually buy 2 >
Prof: Okay,
so this is what he meant to say.
So that's fine.
Okay, so that's what you do,
exactly. So everybody's following?
You agree with me now, right?
But, you know,
we could plug in for this too, by the way.
So pi of 1, we know what that
is. Okay, so does everybody see
what's going on here? To buy 1 dollar at time 2,
you don't get the whole 2-year Treasury, you buy 1/102 of the 2
year Treasury, so it costs you that amount of
money. But that gives you a little bit
of extra at time 1. How much extra does it give you?
Well, you've got 2 dollars
extra for every 2-year Treasury, but you didn't buy a whole
2-year Treasury. You bought that fraction of it.
So it gives you this much extra
which you now get to sell off, so you're going to sell it off
for this price, pi (1).
And of course,
we can plug in for pi (1), by putting 101 down here and
putting 100.1 up here. Okay, so that was pretty clear,
right? So now any questions about that?
So that's going to be some
number, which I calculated again, which happens to be 962,
.962. So notice, of course,
it's getting cheaper to buy--how much does it cost to
buy 1 dollar in year one? It's that.
To buy 1 dollar in year 2,
is less. Now what about pi (3)?
How would we get pi (3)?
We'll stop at pi (3).
How would we get pi (3)?
Then we're going to find a very
fast way of computing all these numbers.
What's pi (3)?
How would you get that?
Student:
Buy the 3-year bond, divided by
> Prof: So the 3-year bond
costs 100.3 but we don't need all of it.
We need 1 over 103 units times
that, okay. So that's our main cost.
But then what else?
Student:
We need to get - 3 times >
Prof: - 3 over 103,
times little pi of 2. Right, because we got this
extra stuff that we didn't need. Student:
- 3 over 103 of little pi of 1, >
Prof: Okay,
so he's saying - 3 times 103 of pi of 1, okay,
because we didn't need that. So is that the right answer?
It's not the right answer.
It's close.
What did he overlook?
So he said, you buy the
three-year bond. So by buying the three-year
bond, you're getting--if you bought the whole three-year
bond, you'd get 3,3, 103.
You only want 1 at the end,
so you have to divide by 103. Now we get 3 over 103,3 over
103,1, and so he's saying we've got two extra payments.
Let's sell them off.
And so he sold them off like
that. That's correct.
So he sold them off,
so this one he sold off at pi (2) and this one he sold off at
pi (1), so he's making use of the fact
that we've already found out this price and this price.
But actually,
that's slightly--okay, but we're talking about not
what you would do talking to the banker.
We're talking about what the
banker would do, and he's got to trade in the
Treasury market. So how's this guy going to do
this? The banker and the Treasury
market now, this Pi over 2 dollars, he's going to have to
hold this complicated portfolio--what's he going to do
to--? He's going to have to combine
the 1 and the 2-year to do this thing and then the 1-year to
undo that thing. So it's actually going to
be--so in terms of trading, if you just had to trade
Treasuries, what would you do? So this is the correct formula.
That's correct and we can
figure out what that is, okay.
And so the correct formula is
91.68. That's .917.
Okay, but you see what you've
done is, these are the kind of fictitious things that Irving
Fisher has told us to do. What you're really doing in the
market is trading the Treasuries.
So here, you've traded a
Treasury. You've bought 1 over 103 units
of a 3-year Treasury. Now what else should you do?
You've got to trade Treasuries.
How can you sell off this
amount of money in year 2? You have to sell some
Treasuries to do that, so what would you sell?
Student:
The 2-year coupon. Prof: The 2 year coupon
bond, and so how much of that would you sell?
Well, this is the amount of
money you have to get. The 2-year coupon bond delivers
how much money? It delivers 102,
so if you did 1 over 102, you would get one,
so you have to divide this by 102.
Okay, and that you multiply by
the price, which is 100.2. Okay, so there's that term,
right? So what have we done here?
We've had to sell off this
amount of money. So how can you sell off this
amount of money? Well, by selling 1 over 102 2
year Treasuries, you're selling off 1 dollar,
but you don't want to sell off 1 dollar,
you want to sell off something smaller than 1 dollar,
so it's that amount. So you're selling that amount
of 2 year Treasuries. But now what do you have to do?
Now, you see,
you've bought some 1-year dollars by getting the 3-year
coupon bond. But by selling the 2-year
coupon bond, you've made some promises in
year one, so you've got to net out all
those things and do the right thing on the one-year coupon
bond, right?
So that looks a little
complicated, but you can obviously do it by algebra.
So everybody following?
You're not following what the
right thing to do is, but let's just say in words
what we've done. In words, what we've done is
we've said, there are things you can actually trade on the
market. Those are the Treasuries.
Those are our benchmark
securities. Let's call them benchmarks.
Now what we're interested in is
some other maybe fictitious securities or new securities.
The price of the zeros,
those are the basic building blocks that will help us
evaluate the present value of any investment.
So the reason why we know these
prices is because we can replicate them by trading only
the benchmarks, only the Treasuries.
So to get the 1-year zero,
we just buy us the correct fraction of 1 year Treasuries.
To get the 2-year zero,
we have to buy the correct fraction of 2 year Treasuries
and sell the correct fraction of 1 year Treasuries,
and that gets us that thing. So we've replicated the 2-year
zero by a portfolio consisting of being long the 2 year
Treasury and short the 1-year Treasury.
Right?
To get the 3-year zero coupon,
we have to buy the 3-year Treasury,
sell the 2-year Treasury and do something complicated that we
haven't quite figured out yet with the 1-year Treasury,
and that will duplicate the 3 year zero.
And then we'll just add up the
cost of that portfolio that replicates this,
and that must be the price of that thing, okay?
So that's what we're doing.
Any questions about that?
Are you following this?
Yes?
Student:
Just to clarify, so pi of 1 is today's price for
1 dollar at time 2 or...? Prof: Time 1.
Today is 0, so pi of 1 is what
you pay at time 0 to get 1 dollar at time 1.
pi of 2 is what you pay today
at time 0 to get 1 dollar at time 2, okay?
So knowing those little pis,
you can evaluate the price of anything, just by multiplying
the little pis by the cash flows in the future.
And now the trick--this is the
trick we're going to see over and over and over again--
the subtlety in finance is that they don't just tell you what
the little pis are. You have to figure that out
yourself, okay? And so how are you going to
figure out the little pis? Well, you know the Treasuries.
You can trade the Treasuries,
and you know what those prices are.
You can see it on the market.
So by combining the Treasuries
in a very clever way, you can end up getting the
prices of all the zero coupon bonds,
the things that pay just 1 dollar at the end.
Why are they called zero coupon
bonds? Because it's like--you just get
principal at the end, of 1 dollar,
without any coupons in the middle.
So the little pis are called
the zero coupon prices, because the payments are just 1
dollar--you know, pi (3) is the price of 1 dollar
at time 3. It's as if there was a bond
that paid no coupons and paid 1 dollar of principal at time 3.
So the little pis are the
prices of zero coupon bonds of various maturities,
and those aren't really traded directly in the market.
What's traded directly in the
market, where pieces of paper change hands,
are the Treasury bonds. But everybody,
every day is calculating these zero coupon prices,
because that's what they need to do to evaluate every single
project that they might conceivably do that day,
and decide whether it's a good project or a bad project.
Is it worth the price or not
worth the price? And it's done by the principle
of replication, just as we said.
So this formula is going to be
slightly complicated. I don't know whether it's worth
writing down. So we've got,
buying the 3-year Treasury, the right amount of that.
Then we have to sell a certain
amount of the 2-year Treasury, because we accumulated extra
coupons. But now we're also going to be
able to sell a certain amount of the 1-year Treasuries,
and so how much is that going to be?
It's going to be some formula,
okay. So it's going to keep track of
everything we did and get a formula here.
So I'm actually not going to
bother, I think--I was going to write
down the formula, but it'll take 3 minutes to
work it out-- because there's a much faster
way of getting all these numbers.
But is everybody with me here?
You all understand how I could
get this number if I wanted to do the work to get it?
I'd figure out I had to
sell--I'd sell some of the 1 year and buy some of the 2
year--I'd do something complicated here,
okay? Sorry, I would do something
with the 1 year Treasury here to compensate for the fact that the
3-year thing I bought is paying me coupons here.
The 2-year thing I sold is
reducing some of those coupons, and so it's only the net coupon
that I can sell, and I'm going to sell that by
selling the 1 year Treasury, okay?
So that's how I would get the
number there, and I added the cost of doing
all these things together, and I get .917.
So you're silent,
but are you following it? Who can I--okay.
So it's too complicated to just
figure this stuff out all the time.
So instead, there's a very fast
algorithm that you can do almost instantly, and that's why it's
such a triviality to calculate these numbers ever day.
So it's called the principle of
duality. You go backwards,
and you say to yourself, "What I want is pi (1),
pi (2), pi (3), pi (4), and pi (5),
and I've started to figure out what the replicating--
" so these are the prices of zeros.
Prices of zero coupon bonds.
That's what I want--want prices
of zero coupon bonds. I have the prices of the
Treasuries and the way I'm figuring out the prices of the
zero coupon bonds is by replication.
Now if somebody stupidly,
as happened 50 and 60 years ago,
fairly routinely, if somebody was willing to give
me 1 dollar in year 3 and only ask 90 cents for it,
then I would be able to lock in a profit.
How could I lock in a profit?
Because I would just--he's
willing to give this to me for a low price of 90 cents instead of
91 cents. So what can I do?
Let's say he's willing--he'd
pay me, let's say more likely--let's say he'd pay me 93
cents. Say some guy came to me,
I'm the banker, and he says,
"I'll pay you 93 cents today to get 1 dollar in year
3," in other words, for a 3 year zero.
Well, I'd say,
"That's wonderful." I'll sell them this promise in
year 3, of 1 dollar for 93 cents.
Then with that 93 cents,
I'll only use 91.7 of those cents and I'll go out and buy
the 3-year Treasury. I'll sell some of the 2 year
Treasury and I'll sell a little bit more of the 1 year Treasury.
And that portfolio which I've
done by doing that will pay me exactly 1 dollar in year 3,
enabling me to keep my promise to him,
but it will only have cost me 91.7 cents.
So I'll have made a 1.3-cent
profit for sure, with no chance--it's a pure
arbitrage. I made a profit of 1.3 cents
with no chance of losing any money,
okay, because I've done all the transactions today,
and the government's going to keep its promises.
I don't have to worry about the
government giving me the money, and so I'll be able to turn the
money over to that guy in year 3.
Meanwhile, he's given me his 93
cents. So if you want to do an
arbitrage and make your profit, you have to figure out what the
replicating portfolio is, and the replicating portfolio
also tells you the price. But it takes a long time to
figure out what all these arbitrage-replicating portfolios
are. And maybe nobody's coming to
you and offering a stupid deal like that.
So you don't need to figure
out--so the principle of duality is,
you don't need to figure out the replicating portfolio to
figure out what the pi (1), pi (2), pi (3),
pi (4), and pi (5) are. I can find those numbers now
just by clicking a button in Excel, trivially,
without bothering to find the replicating portfolios.
Then if some,
you know, bad trader comes to me and offers me 93 cents for
the 3 year zero coupon, then I'll figure out the
replicating portfolio and take advantage of that offer to make
a pure profit for sure. So what I want to show you know
is how to get pi (1) through pi (5) without having to go through
this complicated calculation. And it just reasons backwards,
okay. So please interrupt if you're
not following this logic. So you reason like this:
we don't know what pi (1) through pi (5) are,
but if you did know them, you'd be able to price the very
bonds that the market is trading.
So you'd know that 100.1 had to
equal 101 times pi of (1). And you'd know that 100.2,
the 2 year zero coupon [correction: Treasury]
bond, whose price is 100.2,
would have to be 2 times pi of 1 102 times pi of 2,
right? Because pi of 1,
remember, is the price you pay today for 1 dollar 1 year from
now. 101 dollars,
1 year from now, costs 101--if you knew pi (1),
this would be the price. If you knew pi (1) and pi (2),
you could figure out the price of the zero coupon bond--
I mean, of the 2 year Treasury bond,
because 2 dollars at time 1 cost 2 pi (1) and 102 dollars at
time 2 cost 102 pi (2). And then the 3 year is 100.3 =
3 times pi (1) 3 times pi (2) plus 103 times pi of 3,
etc. Then we can go down to the
last--well, I'll just write them all.
It doesn't take a second.
Okay, and the last one is,
100.5 = 5 times pi (1) plus 5 pi (2) plus 5 pi (3) plus 5 pi
(4) plus 105 pi (5), okay?
So you don't know the little
pis, but you do know these prices,
because the market tells you, and you know the payoffs of all
the bonds, because that's just written on
them, literally, so you can just read
what the payoffs are. You know the government's going
to keep its promise. So rather than doing this
complicated stuff, trying to figure out the pis,
assume you had the pis. And then if you had the pis,
it would tell you what the prices of everything were.
So if you guessed the wrong
pis, you'd get the wrong prices. But basically,
you're solving 5 equations and 5 unknowns, and that's what
Excel is so good at. It's going to start with a wild
guess of the pis, and then it's going to move
around the pis until you match all these prices.
And since it's 5 equations and
5 unknowns, and they're all linearly independent,
it'll be a unique set of pis that it will calculate.
But that 1 set of pis has to be
the replicating portfolio prices,
because there's only 1 set of pis that are going to work and
solve these equations, namely the ones you got by the
replicating argument. So we can figure out the pis by
solving 5 equations and 5 unknowns, so that's what I do.
So if you guess the pis,
1,1, 1,1, 1, any questions about what I'm
doing? If you write 1,1,
1,1, 1, you're going to get prices, you know.
For the first one,
it will just be 101, and for the 2 year,
it will be 2 times 1 102 times 1, is 104.
For the third one,
with all the pis 1, which is obviously not the
right thing, it would be 3 3 3. That's 109, so those are bad
prices. We're trying to match what the
market says the prices are. So all I do is,
I subtract the market prices we're trying to match from the
actual prices. I look at what the error is,
which we're trying make 0, then I square the error.
And then presumably this was
adding the error, sum C16 to G16.
I added the error,
and so I now want to do my Excel thing.
Hopefully--I haven't done this,
but let's--it's got to work. Okay, so minimize, good, that.
Such that by changing
cells--which are the cells I want to change.
I want to change the pis.
Those are the ones that are
wrong, so I do that. So I'm minimizing the squared
error by changing the pis, B18, okay, and I solve.
Okay, and I've done it.
And there is the answer.
So you notice that I got the
same prices that I told you about before by replication,
.991, .962, and .917, etc.
So we got the pis.
So that's step 1.
All right, so that's what every
single financial firm in the entire world does every single
morning, and sometimes every single hour.
So are there any questions
about what we just did? You have to do this in the
problem set. Is there anything puzzling you
about this? Okay, now I'm going to start
deducing all kinds of surprising things from this.
I hope that you'll be
surprised, but I want to make sure you've got the concept of
what we've done now. Anybody puzzled by it?
Okay, so somehow,
Fisher's pi (1), pi (2), pi (3),
pi (4), pi (5) are going to be deducible from what's going on
in the markets, every day.
All right, so let's ask one
more thing that's deducible. Suppose I go to a bank and say,
"I promise to give you 1 dollar in year 2.
How many dollars will you give
me in year 3?" What do you think the bank's
going to tell me? Every bank will give me the
same answer, if that yield curve--given this
morning, and that was this morning's
yield curve, if I ask every bank in the
world, "I'll give you 1 dollar in year 2,
you tell me how many dollars you'll be willing to give me--
" So what am I doing? I'm saying, "I promise
today to hand over to you 1 dollar in year 2,
and you know I'm going to keep my promise.
And in exchange,
I want a promise from you to give me a certain number of
dollars in year 3." How many dollars is the bank
going to offer to give me in year 3?
Every bank will give the same
answer, and what will that be? So the thing I'm asking,
is, I'm asking for what's called the forward interest
rate. So we've got these things,
which are obviously very important numbers.
Those are the most important
things. Fisher would say those are the
prices you use to get everything.
But now I want to say
something, I'm going to ask another important thing,
almost as important. I want the forward rate.
So 1 i_1--
Student: We can't see that.
Prof: You can't see that.
I'm glad you pointed that--can
you see this? Student: Yes.
Prof: Okay.
The cameraman told me this
board was great. But anyway, so how about,
I'll write 1 i_1^(f), forward, is--by the way,
am I calling that 1 i_0 or 1
i_1? Sorry, just want to get my
notation straight. Okay, so let's call 1
i_0 is the--1 i_t is the number of
dollars at t 1 in exchange for 1 dollar at t.
So this is the number of
dollars at t 1 agreed today. So we agreed today that you're
going to pay this many dollars at time t 1 in exchange for 1
dollar at time t. So this is like the interest
rate that you might pay at time t.
You give up a dollar at time t,
how much do you get at time t plus 1?
The interest rate.
But we're not there yet.
We're agreeing to it today.
So today we're agreeing to this
interest rate. So what is the interest rate
we'd agree to today, so we've locked in the rate?
When it comes to time t,
one guy's going to hand over the dollar,
and when it comes to time t 1, the other guy is going to give
back this many dollars. So what is the rate we would
lock in today? It's called the period t
interest rate forward, because we're locking it in
today for a forward period of time,
but it's really just the normal time t interest rate for one
year, but we're locking it in today.
So what would we lock in today?
How do we compute that?
We already know what that
number is. What is it?
Yes?
Student:
It's the ratio of pi (2) over pi (3).
Prof: Well,
I put t here, so something like that.
Student: pi t over t 1.
Prof: Okay, exactly.
That's pi (t) over pi (t 1).
That's it exactly.
So why is that?
Student:
It's a tradeoff between the dollar t, t 1.
Prof: Right,
so all we're doing is this forward rate.
We're exchanging time t dollars
for time t 1 dollars at this ratio.
But we're committing to do it
today. But today we already know what
the ratio is of time t dollars to time t 1 dollars.
We know that pi (2)--in fact,
we know what it is. pi (2) happens to be .962.
That's a bigger number than pi
(3). From today's point of view,
1 dollar at time 2 is worth more than 1 dollar at time 3.
We already know how much more 1
dollar at time 2 is worth than 1 dollar at time 3.
It's the ratio pi (2) over pi
(3). So that ratio,
as he says, pi (2) over pi (3), has to be exactly the exchange
rate that the people are agreeing to today.
That's what pi (2) and pi (3)
mean. If you express it as an
interest rate, it's 1 the forward interest
rate. That ratio is 1
i^(f)_t. Okay, any questions about that?
Yes?
Student:
What happens if your yield curve is downward sloping?
Prof: If the yield curve
is downward sloping, yes.
Student:
Do you agree to give them 1 dollar in year 2,
they give you less than 1 dollar in year 3?
Prof: Okay,
so you've made a little mistake in your premise.
Good question,
but let me phrase your question a little differently,
so you see the answer to it. The yield curve was almost flat
in year 2000. So in the year 2000,
the yield curve was almost flat.
In fact, there are moments
where the yield curve seems to go down.
So if the yield curve goes
down, what does that mean? Does that mean--between year 6
and year 7, the yield curve went down a little bit.
Does that mean that pi (7) is
less than pi (6)? Maybe, but it couldn't really
be that way. Okay, so let me translate his
question. He's saying,
look, yield curves very often are flat.
Mostly they go up,
very often they're flat. Sometimes they even start to go
down. He said, "That worries me
that maybe pi (7) is less than pi (6)."
But that could never happen.
That would be crazy,
because that would mean that there would be a negative
interest rate in the future, and with money,
that can never happen. You can store the money.
No one's ever going to ask for
a negative interest rate. He could just keep the dollar
and keep it in his pocket. Why is that?
Remember, pi (6) = 1 over (1
the 6 year yield) to the 6^(th) power.
And pi (7) = 1 over (1 the 7
year yield) to the 7^(th) power. So it could be that Y(7) is
less than Y(6) as it is there, and yet pi (7) is still small.
Could be that Y(7) is less than
Y(6), as it is over there,
but because you're taking this to the 7^(th) power and this to
the 6^(th) power, you still have pi (7) less than
pi (6). So just because the yield curve
is downward sloping, doesn't mean that the pis are
going down. The pis could never go down.
The pis are always going to go
up. So excellent question.
Any other questions?
Okay, so we could get the i's.
The i's will typically be going
up. Suppose the yield curve is
going up, by the way. Will the i's be going up faster
or slower than the yields? Yeah, if this is the yield
curve and I calculate the forwards, do you think the
forwards will be going up faster or less fast than the yields?
All right, well let's do it in
the example. Let's just go back to our
example that we were doing. Yield curve spreadsheet, okay.
So maybe I did it here.
Hopefully I solved it all.
Okay, so here we got the actual
pis. You see the pis are always
declining. And if we now look at the yield
curve, you can figure out the yield.
How can you figure out the
yield? Because you solve that formula
we gave at the very beginning. You take the price of the--I
guess I've erased it now. Okay, you know what the price
is. To figure out the yield on the
3 year, we just plug in 1 Y of 3,1 Y of
3 squared, 1 over 1 Y of 3 squared,
1 over 1 Y of 3 cubed, and that gives us the yield.
Remember, that's how the
newspaper reporter figured out the yields.
So I figured out the yields in
the spreadsheet down here, and these are the yields.
So this is what would appear in
the newspaper. The 1-year yield is a little
under 1 percent. The 2-year yield is a little
under 2 percent. The 3-year is a little under 3
percent. The 4-year is a little under 4
percent, and the 5-year is a little under 5 percent.
Those are the yields.
So what if we did the forwards?
The forwards,
remember, are just the ratios of these pis.
What are the forwards?
They're down here.
So what do they do?
Sorry, I don't know how I did
that, but here are the forwards, over here.
So the forwards have gone up
much faster than the yields. They went from .008 percent,
which is the same as that one, to 2.9, which is bigger than
2.8, to 5, which is way more than 3.8,
to 7.2, which is way more than that,
to 9.6, which is way more than that.
So the forwards went up much
faster than the yields. So why is that obviously going
to be the case whenever the yield curve is upward sloping?
So if we go back to our picture
here. We go back to our picture.
If the yield curve were
completely flat, what do you think the forward
yields would be? This is just common sense to
see if you have any idea what's going on.
If you think about batting
averages and how somebody's average changes each time he
bats-- if the yield curve is flat,
like in 2006, the forwards are going to
basically be flat. But if the yield curve is
upward sloping, then the forward yields are
going to go up much faster. So why is that?
Okay, well, remember,
the yields, you know-- when you do a 5 year coupon
bond, you're discounting all the cash flows,
the previous 4 cash flows, using the same yield to
discount them all. So if you go from year 4 to
year 5, and you have to raise the yield a lot,
it means, you know, like if you're a batter,
and your average goes up. It means the last thing you did
was better than the average of what you've done before,
so it's going to be even higher.
If your average was .300,
and then you played a series against the Red Sox in which you
did very well and your average went up to .320,
in that series against the Red Sox,
you obviously did even better than .320,
because you have to average what you did then with what,
your previous .300 to get .320. So if the average sort of is
going up, remember the yield is the same thing over the whole
history. If, when you take a longer
history, the average has gone up, it must mean that the most
recent thing went up really much more, okay?
So the 4-year yield is sort of
averaging the payoffs of the first 4 years.
The 5-year yield's averaging it
over 5 years. So if that yield,
the 5-year yield, has gone up,
what happened in the 5^(th) year must have gone up a lot to
bring the long run average up, okay?
So that's why the yield curve's
going to go up much faster [correction: the forward rates,
compared to the yield curve, are going to go up faster].
Okay, so we know now to
summarize, everybody can look at these pictures every single day.
From these pictures,
they can deduce the pis. That's the crucial variable in
the whole economy, the pis.
But a second crucial variable
is the forward yields, the 1 i^(f)s,
which you can just get by the pis.
Now why are the 1 i^(f)s so
important? We know that the pis are
critical, because they evaluate every project by multiplying the
cash flows by the pis. Why are the forwards so
important? The forwards are so important
because, suppose you believed that--so the forwards,
let's go back to what we got. Let's just look at the numbers
here. Here are the forwards,
remember, down here. Okay, suppose I said to you,
"You tell me." You don't know anything about
the economy, maybe, but you can read the newspapers
and do mathematics like we've just been doing.
What do you guess the interest
rate's going to be in year 2? So this is the 0 year forward,
the 1 year forward, the 2 year forward,
the 3 year forward and the 4 year forward.
If I say in year 2,
"What do you think the interest rate--
guess what the interest rate's going to be,"
what would you guess? Yeah.
Student:
I have a question. Could you make this all real
interest rates by doing this for TIPS?
Prof: Yes.
Right, I could do this real
interest rate by doing it for TIPS.
Fisher would say do that.
Trouble is, that TIPS are not
traded--they're becoming more and more freely traded in the
market. They used to be traded
very--people didn't want to trade them.
So my classmate,
Larry Summers, introduced TIPS,
Treasury Inflation Protected Securities,
and he, you know, announced this was a fantastic
idea and was going to change radically the whole markets.
And then nobody traded them,
and they offered astronomical interest rates,
real interest rates, to get anyone to buy them.
And so they were
nicknamed--it's really bad on camera--but their nickname
became totally illiquid pieces of...
and so the market has not used
the TIPS to do most of its pricing.
It uses the Treasury bonds,
but yes, Fisher would say,
if the TIPS were a reliable market,
you would use the TIPS to get the real interest rate,
and that's really what you should care about,
is the real interest rate, not the nominal interest rate.
But we're using the Treasuries
here to get the nominal interest rate.
However, you've now just dodged
my question. My question was,
if I asked you to predict, on the basis of the yield curve
in this example, and what we've been able to
deduce, what would you predict the
interest rate was going to turn out to be 2 years from now,
the 1-year interest rate? What would you predict?
Yeah?
Student:
The market predicts >
Prof: Which forward rate?
Student:
Sorry, which years were you >?
Prof: Year 2.
In year 2, what do you think
the interest rate will be between year 2 and year 3?
Student:
Year 2 >
Prof: It's today,
and we're asking, what do you predict the market
rate of interest will be in year 2,
between year 2 and year 3? Student:
The forward rate, it would be 1
i_2^(f). Prof: Okay,
and I_2^(f) happens to be right here,
5 percent. Okay, so that's the forward.
Student:
> Prof: So 5 percent you'd
predict. Student:
> position, but not the best
prediction. Prof: Okay,
so let me refine that a little bit.
If the world were one of total
certainty, so everybody trading today had
a perfect forecast of what was going to happen in the future,
then of course, the forward rates in the market
today would have to be exactly equal to the forward interest
rate. Because suppose that you knew
for sure the interest rate was going to be 4 percent in year 2.
How could the market get to a 5
percent forward? That means that some guy is
promising today, "You give me 1 dollar in
year 2 and I'll give you 1 dollar 5 in year 3."
And we're agreeing to that deal.
But that's a ludicrous thing
for him to do, because when he got to year 2,
he could simply-- if he knew for sure what was
going to happen in the future, and that the rate was going to
be 4 percent, he would just--I said it
backwards. What an idiot.
Suppose he knew for sure the
rate was going to be 6 percent, he would be--oh yeah,
if he knew for sure the rate was only going to be 4 percent,
he'd be a fool for promising to give the guy 5 percent today,
because in the future, when he got the dollar,
what would he do with it? Put it in the bank and get 1
dollar 4 next year? That wouldn't cover his promise.
He'd be screwed, okay.
So if he knows for sure that
the rate 2 years from now is going to be 4 percent,
the forward rate would also have to be 4 percent.
So to say it backwards,
if you assume everybody knows for sure what's going to happen
in the future, then the forward rates would be
exactly equal to what everyone is expecting to happen in the
future. To say it slightly differently,
if you happen to be the one ignoramus in the world who
didn't know what was going to happen in the future,
but you knew that everybody else who was trading in the
market did know what was going to happen in the future,
and you saw a forward rate of 5 percent,
then you could deduce, even though you were an
ignoramus, that actually 2 years from now,
the interest rate was going to be 5 percent.
Okay, so just what you said,
but just said a little bit more precisely.
We're assuming here perfect
certainty about what's going-- we're assuming the traders who
trade today all are completely convinced of what's going to
happen in the future. Okay, so let's go back to our
picture now. The picture of the zero yield
curve. So what do you think this blue
curve means? What are the traders convinced
is going to happen in the future?
The second half of the course
is going to be dealing with uncertainty.
We're now assuming,
like Irving Fisher, that everybody trading today
has no doubt about what's going to happen in the future.
So what do you think these
traders think about these prices, about the interest
rates? We're now in the world today,
this is September 30^(th), it's today.
That's this curve.
What does this curve mean?
Making the assumption that
traders today are convinced about what's going to happen
next, you know, in the future 10
years and 30 years, what do we know that they are
convinced of? Yes?
Student:
That interest rates are going to keep going up.
Prof: That interest
rates are going to go up and go up a lot.
They're not just going to go up
to 4 percent, because that's the 30-year
yield. They're going to go up--the
forward rates are going up much faster than that,
so we could have computed what they think.
So they think rates are going
to go way up in the future, okay, much higher than that,
because you're starting so low, staying low for a while.
So the rates have to go up,
the forward rates, really sharply to pull the
average that high. So people are convinced that
rates are going to go up in the future.
That's what that tells you.
And why would they be convinced
of that? Well, for the two reasons that
you gave at the beginning. One of you said,
"The market is going to get more productive."
Irving Fisher has already told
us that when the market gets more productive,
you know, you're more optimistic about what's going to
happen later, the real interest rate goes up.
And if inflation's constant,
and the real interest rate goes up, the nominal interest rate
goes up. The other possibility is that
the real interest rate stays the same, but there's inflation in
the future. The real interest rate the
inflation is the nominal interest rate.
That's another explanation for
why people might expect the nominal interest rate to go up,
okay? So you know that the market is
predicting rates going up, and the two obvious
explanations according to Fisher is that either inflation is
going to go up or the real rate is going to go up.
And why might the real rate go
up? Well, there are a bunch of
reasons, but most likely because productivity is going up.
Okay, so I've got one more
surprising conclusion to end this.
So if you were certain about
the future and you took the 5-year coupon bond,
could you tell me what the price of the 5-year coupon bond
was going to be next year? How would you figure that out?
Assume that everybody is
convinced that the 5 year coupon bond--that they know the future
for sure. So therefore,
from the zero curve--I erased my graph--this is the last
question. I'll let you go as soon as you
answer this. You need to answer this to do
the problem set. Wrong graph, shit.
Sorry.
Okay, it'll only take 1 second
to answer this. Okay, I'm telling you now that
this is what everybody's looking at in the morning,
okay, these numbers. They're getting all the forward
rates and stuff like that. They're making all the
deductions that we made. Now if you suppose that those
people are convinced, they don't have any doubt about
what's going to happen in the future.
Because they don't have any
doubt about what's going to happen in the future,
you can infer from these prices what they think about the
future. So the question is,
can you infer what the price of the 5-year Treasury,
which is now 100.5, that 5-year bond is 100.5,
next year it will only be a 4-year bond.
Do we know what its price is
going to be next year? Yes.
Student:
Yes, you can stick the cash flows in
> year 2,3, 4 and 5,
and multiply them by the >
Prof: By what pis?
Student:
The big pi, the price of the >
Prof: Those pis.
Those pis or something slightly
different from those pis? It's going to be a year later,
remember. Student:
Oh, by the equivalent of those in year 2.
Prof: How would you get
that? Yeah?
Student:
Discount by forward rates, so it's like
> 4 and 5, forward rate in 3,4,
> year forward,
add a discount >
Each >
would have one less forward in
it. Prof: Okay,
so what you both said is absolutely correct.
Unfortunately,
we're 2 minutes--we're ending now.
So let me just end by saying,
in the problem set, that's exactly the question.
What is the 5-year coupon going
to be priced next year? If there's a world of
certainty, you're going to know what all the interest rates are
in the future. And if you know what all the
interest rates are in the future, obviously Fisher tells
you, you can figure out the price is.
However, to get the exact
formula would take a few minutes and I unfortunately am a few
minutes behind, so you're going to have to
figure out what the right formula is,
but it's what he said and what you were getting to.
