Today we're going to talk about
a very important topic
topic in economics, which is
expectations. We have barely mentioned
expectations when we talk about the
Phillips curve. We talked about
expectations when we
when we discussed the UEP and so on. But
expectations is a much bigger issue in
economics. In fact, most decisions by
firms, by consumers,
governments involve considerations of
the future. And it plays an even bigger
role in finance, in which essentially
everything is about the future. The
price of an asset today is meaningless
in itself. You have to compare it with
what you expect to get out of that asset
in the future. So, it's all about
expectations and so on.
So, that's what we we're going to do
today. We're going to talk about
uh expectations, the how to value things
that that you expect to receive in the
future,
uh and how to compare those things with
things that you have in the present.
Um
but before doing that, actually, let's
talk a little bit about the news. Who
knows who First Republic Bank is?
Remember that a few weeks ago I told you
that um Silicon Valley Bank,
you read it. I I I I just mentioned it
that
uh
we'll discuss it that that you know, we
had the second largest bank by asset in
in in US history. It was Silicon Valley
Bank was the second largest
asset bank in terms of assets
uh to collapse in the US. The first one
was uh many years ago.
Uh and then we had this bank that had
more than $200 billion in assets that
essentially collapsed in a few days. It
was a run on deposits. They had problems
before,
but what really did as it always the
case with banks is they had a run on
deposits, funding.
Uh
well, it's no longer the second largest
collapse in US bank history. Now we have
over the weekend
the the new second largest bank to
collapse, which is First Republic Bank,
that was essentially
liquidated and sold to JP Morgan over
today morning, very very early in the
morning. Okay? So,
you have an account in First Republic
Bank, you sooner likely to have an
account in JP Morgan.
But again, what made it collapse was
something very similar to what made
Silicon Valley Bank collapse, which is
that they had invested on on a series of
things that were very vulnerable to to
the fast pace of hikes
uh
in interest rates in the US.
And when they had those losses,
depositors became became worried about
it and eventually they decided not to
wait, just run.
And see what happened. They
First Republic Bank lost about $100
billion in deposits just last week.
Okay? Um the last few days of last week.
So, so, so
so that
it was obvious that that
it was not going to survive and that's
the reason
something was arranged over the weekend
to avoid the panics associated with
collapses of a bank and so on. Okay? But
anyways, by the way, this is all about
expectations. This is you know, this is
if people had expected the deposit to
remain in the bank, then probably this
bank would not have collapsed. It's all
about people anticipating what other
people will do and so on and so forth.
Okay, but now let me get into the
specific of
of this lecture.
So, there you have this is the most
important index of of equity equity
index in the US, S&P 500. It's a very
inclusive index that captures all the
large most of the large companies in the
US,
all of the large I think companies in
the US. And uh that's an index. It's an
average weighted average by by this
uh capitalization value of each of the
shares. It's a weighted average of the
major the main shares in the US, equity
shares in the US.
And one thing you see is that it moves a
lot around, you know? Here, for example,
when when
we became aware that COVID was going to
be a serious issue,
the US equity market collapsed by 35% or
so. That's a very large collapse in a
very short period of time.
And then, as a result of lots of policy
support, actually we had a massive
rally. Uh so, up to the end of 2021, the
equity market had rallied by 114%.
So, a big rally.
Then we got inflation and the Fed began
to worry about
inflation, so they began to hike
interest rates. And when they hike
interest rates, that eventually led to a
very large decline
uh
in in asset prices of the order of 30%
or so, 25% or so, actually, from the
peak to the bottom.
And then, since the bottom, which is was
more or less October of last year, we
have seen a recovery of about 16% or so
of the equity market. Okay? And if you
look at the Nasdaq, which is another one
index that is very loaded towards
uh
technology companies, then you can see
swings that are even larger than that.
Now, why do these prices move so much?
Well,
a lot of it has to do with expectations.
You know, are things going to get worse
in the future? Will the Fed cause a
recession? Uh
how much higher will be the interest
rate? And things like that matter a
great deal.
Another thing that matters a great
deal is
how much people want to take risk at any
moment in time. And if you're very
scared about the environment, you're
unlikely to want to have something that
to invest on something that can move so
much,
and so risk is well known. So, it's
called risk off when when people don't
want to take risk, these asset prices
tend to collapse. Okay? Of the risky
asset. Equity is a very risky asset.
But that's not the only thing that moves
these assets around. It's not just the
risk that the companies underlying
company may go bankrupt or anything like
that.
Here you have, for example, the movement
of a
for it's an ETF, but it doesn't matter.
It's a portfolio of
bonds of US Treasury bonds of very long
duration. Maturities beyond
uh 20 years and so.
So, this is incredibly safe bonds, you
know? Because it's US Treasuries. So,
there's no risk of default or anything
like that.
Still,
the price swings can be pretty large. I
mean, in over this period, you know,
there you have seen an an an increase in
value of 45%,
then uh a decline in value of of of
about 20%. Another increase in 15% here.
There was a huge decline, 40%,
since since essentially
uh uh uh
What do you think happened here? Why is
this big decline in in in in bonds?
You're going to be able to answer that
very precisely later on, but but I can
tell you in advance that that was
essentially the result of monetary
policy tightening.
You know, increasing interest rate
caused the the bonds to decline. So,
even these instruments that are very
safe in the sense that you if you hold
it to maturity, you will get your money
back and all the promised coupons along
the path, well, still their price can
move a lot. And it's obvious that that
movement in price
is something you need to explain in
terms of expectations, what people
expect things to to happen. In this
case, it's not whether people expect to
get paid or not, because you will get
paid, but it's expect but in this
particular case, it's about expectations
about future interest rate. If you think
the interest rate will be very high,
then the price of bonds will tend to be
very low and so on. But it's all about
the future. Okay?
So, the a key concept
uh that we're going to discuss today and
then we're going going to use it to
price a specific asset
uh is the concept of expected present
discounted value. This this is a loaded
concept. There's lots of
terms in there and we need to understand
what each of these terms means.
So, the key issue
that we're going to discuss is how how
do we decide, for example, if you see
the price of an asset out there
that is 100, how do you decide whether
that price is fair or not, looks cheap
or not? Okay? Uh and and and and and
that question means you have to decide
whether that price that you're paying
today is consistent with the future cash
flows that you're going to get from this
asset. I mean, that's the reason you buy
an asset is because you'll get something
in return in the future. Okay? But how
do we compare that? How do we compare
the price today with those things that
will happen in the future?
So, answering that question, which is
what we're going to do in this lecture,
involves the following
concepts. First, expectations, big
thing.
That's a you know, this is expected
present discounted value. The E part is
for expectations. That comes there.
You
Expectations are really crucial because
these are things that happen in the
future. You need to expect. Even if if
it's a bond that promises you to pay,
you know, 50 cents per dollar every 6
month,
you still may have an expectation that,
you know, if it is a bond issued by
First Republic Bank, it may not pay. So,
so, so you need to have an expectations
about that.
Uh
so, crucial term is expectation.
Then you need some method
uh to compare payments received in the
future with payments made today. I mean,
if you buy an asset, you pay today,
but you're going to receive things
returns on for that asset in the future.
So, how do I compare that that that
Suppose I pay one today and I receive
one 1 year from now.
Does that seem like a good asset?
Probably not. I mean, you know,
probably not.
Uh
Um and that's what the word discounted
really means. You know, when you say
expected present discounted value
it says
somehow that things I receive in the
future are valued less than things I
have today.
Okay? So, if you're going to tell me
that you're going to pay me a dollar in
the future and I have to pay you a
dollar today, most likely I won't take
that deal.
So, I need In other words, I'm
discounting the future.
How do we discount the future? Well,
something that we're going to have to
figure out.
So,
let's let me first shut down this part,
the expectations, and then we'll
introduce it. So, assume for now that
you know the future.
Okay? And I'm going to derive all the
equations with assuming that you know
the future. So, there's no issue of
trying to figure out what the future is.
You know it. But still you have to
decide whether
what is the right value for for an
asset.
Okay, so
let's start with the case where you know
the future. Sorry.
And let's do the comparison uh
Let's try to understand how do we move
flows, how do we value flows at
different points in time.
This is the thing is think first about
comparing an asset that gives you a
dollar in the future,
how much do you think it's worth today?
Well, the easiest
way to get to that value is is to think
on the alternatives. As suppose I have a
dollar today, what can I do with it?
Well, in terms of investment.
Well, suppose that you have available
one-year bonds, treasury bonds, and that
the interest rate is I
t. That's the interest rate on an I
one-year bond.
So, if you want if you if you
have a dollar,
you have the option to invest it in that
asset, in that bond, which give will
give you 1 + I dollars
uh
next year.
Well,
that means that I can get $1 next year
by investing 1 over 1 + I dollars today.
No?
Because if if I invest 1 + 1
rather than $1, I invest 1 over 1 + I
today, then I multiply this by 1 + I
and I get my dollar in the future.
So, that tells me that say the interest
rate is 10%, then with $1 today I can
get 1.1 dollars in the future.
That means that investing 90% 90 cents
today, more or less,
I can get $1 in the future.
That tells me that a dollar in the
future is equivalent to 90 cents today.
That's the assumption. Okay?
So, that's the reason when I told you
the deal of me, look, I have an asset
that cost cost you a dollar, but gives
you a dollar in the future, well, that's
not a good deal if the interest rate is
positive.
If the interest rate is 10%, then then a
right a fair comparison is 90 cents with
$1, not $1 with $1.
Okay? So, that's the discounting of the
future. You can The most obvious way of
discounting the future
is to discount it by the interest rate.
Uh
which interest rate to pick? That's more
subtle. That depends on risk, depends on
many other things which we're going to
discuss to some extent here. But for
now, let's make it very simple. And in a
world in which you really know the
future, really the right interest rate
to use is the safe interest rate, the
interest rate of of treasury bonds and
things like that.
Okay? So, that's that's that.
What about a dollar that you receive
What about if you're thinking about what
is the value of a dollar two years from
now?
Well,
you know, if I get a dollar to I can do
the same logic. If I if I
I can use the same logic. If I get a
dollar today,
I can convert that into 1 + I t * 1 + I
t + 1 dollars. Okay?
So, say 10% and 10%, I get 1.1 next year
and then I get 1.1 * 1.1, 1.21 or
something like that. Okay?
That's my final
result.
So,
well, then how much is it worth to have
a dollar, an asset that gives you a
dollar two years from now?
Well, it's going to be that dollar
divided by the product of these interest
rates.
Okay? Why is that? Well, because with
this amount of
dollars today,
it's point 80 cents or something like
that, I can generate a dollar two years
from now.
That means a dollar
two years from now
is worth about 80 cents today.
Okay?
We're going to use a lot this type of
logic, so
and and I know that that it may not be
that intuitive the first time you see
it, but
ask questions.
You want me to repeat it?
Okay. The
The final goal is the following. We're
going to In the what comes next, we're
going to see if which happens again with
many decisions in life, but it perhaps
particularly for financial assets,
we're going to try to value something
that
whose payoff happens at different times
in the future. And the question is
how do I value an asset that pays me,
you know, $5 one year from now, $25
three years from now, uh
minus $10 10 years from now, plus $50
100 years from now?
What is the value of that? Of having an
asset like that?
And so, I needed some method
to bring it to today's value because
today I have a meaning of what a dollar
is, you know?
And and therefore I can compare it with
whatever price I mean
people are asking me for that asset.
So, what this is doing is is is that is
doing that. It's telling you how to
convert a dollar at different parts in
the future into a dollar today.
And by that logic,
the recipe is well, use the interest
rate because you could always go the
other way around. You could always with
a dollar you can ask a question, with a
dollar today, how many dollars can I get
two years from now, say?
That.
Well,
say X. Well, then I need 1 over X. Then
$1 there is worth 1 over X dollars
today. You know, that's that's the logic
because 1 over X * X is 1.
So,
that's too fast, probably.
So,
you know, with $1 today, oops,
I can generate, say,
$1.1
at uh uh
at t equal to
Okay?
Then I'm I'm The question I want to know
is how much is a dollar worth
How much is a dollar received at time t
equal to worth today?
That's the question I'm trying to
answer.
You know, because an asset will be
something that will pay you in the
future. So, I want to know how much is
$1 received in the future worth today.
And then the answer is
well,
then is I know the answer from this
logic because I know that with one
if I have 1 over 1.1 dollars today, I
can convert it
into one.
How do I know that?
Because
1 over 1.1
* 1.1
is equal to 1.
Okay? This if I invest these dollars
today,
I'm going to get this return on that.
And the product of these two things
gives me my dollar.
Okay?
So, if I tell you, do you prefer to have
a dollar two days from two years from
now or today?
You say, I prefer it obviously prefer it
today because I can get 1.1 dollars two
years from now.
But then then the more relevant question
is, no, no, but then you do you prefer
to have 90 cents today
versus a dollar in the future? And then
I'm I need to do my multiplication
because I have to multiply the 90 cents
by the 1.1 and see whether I get
something comparable to a dollar or not.
Okay?
But that's that's the logic behind that.
And And that's a So, the interest rate
is what we discount the future by.
And it's natural because if the interest
rate is very high If the interest rate
is zero, say,
then a dollar received two years from
now or a dollar received today is is the
same
because I can't If I invest a dollar
today and the interest rate is zero, I'm
going to get my dollar two years from
now.
If the dollar If the interest rate is
50%, it makes a big difference receiving
the dollar today versus receiving it two
years from now.
If you're in Argentina, the interest
rate I don't know what it is. It's
700%. It makes a huge difference whether
you receive it, you know, one year from
now than today.
And and and uh
So, that's that's the role of the
interest rate. The higher is the
interest rate,
the less
is a dollar received in the future worth
relative to a dollar received today.
Because you can get a much higher return
from the dollar you have today
if the interest rate is high. If the
interest rate is low,
you don't get that much. Okay?
Much difference. Okay, good.
So, this is a big principle. And and I I
mean
everything I'll say next builds on this
logic.
So, let me give you a general formula.
So, let's ask what is the value
of an asset
that gives
payouts of Z
t dollars this year,
Z t + 1 one year from now, ZT plus two,
two years from now, and so on and so
forth for N periods more. Okay?
Well,
I just need to do several of these
operations. I know that the dollar
received this year is is worth a dollar.
Okay? That's ZT.
A dollar received one year from now
is not
is not the same as a dollar received
today. It's the same as one over one
plus IT dollars received today.
So, that cash flow I'm going to receive
from this asset is worth this amount.
For a two something that I receive two
years from now, then it's not
it's not certain, it's much less than
receiving a dollar today. It's going to
be one over one plus IT one plus IT plus
one.
And that I have to multiply by the
number of dollars I will receive two
years from now. Okay? And I keep going.
So, that's that's the
the present value. Present discounted
value. Present because I'm bringing all
these future cash flows to the present.
That's what each of these terms is
doing. The one over that is bringing it
to the present.
Discounted because the interest rate is
discounting things. It's making them a
smaller.
And value because I'm trying to reduce
them to the current value. Okay?
That's the general formula. So, it's a
formula you need to understand.
It's just So, that that was an asset
that gives you Z dollars today,
ZT plus one, one year from now, so you
use this formula. ZT plus two, two years
from now, so you use this formula, and
then you keep going. Okay?
What if we don't know the future?
You know, I have to remove the expected
part.
Well,
if we don't know the future, then the
best we can do, in fact, we do fancier
things, but that's what we're going to
all that we'll do in this course.
Uh all that you can do is just replace
the known quantities we have here
for the expectations.
Okay? So, that's the closest. So, you
know, I know ZT, that's the cash flow I
get now,
but I don't know ZT plus one. So, I can
replace it by expectation.
I do know the interest rate on a
one-year bond from today to one year.
So, that's the reason I don't need an
expectation here.
But I don't know what the one-year rate
will be one year from now. So, that's
the reason I need an expectation there.
And so on.
And I don't know what the cash flow will
be two years from now. I have an
expectation about what the cash flow
will be, but I don't know it.
So, I have an expectation there. Okay?
So, so all that I've done here is say,
"Okay,
I acknowledge that this guy knew a
little bit too much. You know, he knew
exactly what the cash flows were going
to be in the future, and he knew what
the one-year rates were going to be in
the future."
This guy here knows less. He knows the
cash flow today. He knows the interest
rate today, but he doesn't know the cash
flows. Really, he has a hunch, but he
doesn't know the cash flows one year,
two years, three years, and so on for
the future, and he doesn't know the
one-year interest rate in the future.
So, all these expectations, here's
important the concept of time. This is
an expectation as of time T. At time T,
you have some information and you make
forecast about the future. Okay?
Use whatever you want, machine learning,
whatever, but you have information at
time T,
and then you have a forecast for the
future. At T plus one, you have you'll
have more information, so you make
another forecast, and so on and so
forth.
But in this we're valuing an asset at
time T, then all these expectations are
taken as of time T. That means given the
information you have available at time
T.
That's the reason these guys don't have
expectations in front of them because
you know this at time T.
Had we taken the value at T minus one,
we would have not known that, and then
we would have had to expectation because
it would have been expectation as of T
minus one.
Okay, so that's your big formula there.
So,
there are some examples that are sort of
well known and and and
and and
So, let me let me show you. They have
nicer expressions. So, that's that's an
example
of the valuation of of this the same
asset,
but when the interest rate is constant,
then
then obviously I don't need all these
products in the denominator.
I have a constant interest rate, then I
just get powers of that interest rate.
That's one in which you have constant
payments. So, the interest rate may be
different,
but the payments are the same over time.
Okay?
So, that's that.
So, those are two
easy formulas. That's one in which you
have
both constant, the interest rate
and the payment.
Then you get a nice expression. That's
just uh
that. Okay? You'll recognize that.
If if you have a constant
uh
constant interest rate here, you see
that the value
is is declining is a is a geometric
series. You know? The value of a two
years from now is a square
of one over one plus some
it's a square of a a number less than
one.
You know? One over one plus I is some
number less than one. This is a square
of that, then the cube, and so on. So,
it's a geometric series that is
declining at the rate one plus I, one
over one plus I. Okay? Or declining at
the rate one plus I.
So, that's your geometric series.
Okay?
That's the value of that.
Constant rate and payment forever.
Suppose you have an asset that
it
that lives forever.
There are some bonds like that called
perpetuities. Uh uh
The US hasn't issued one, but the UK
has, and so on.
I
So, that's an asset, for example, that
pays you a fixed amount
forever. And if the interest rate is
constant, that's the trickier thing,
then the value of that asset you can see
that this this is going to zero.
So, the value of that asset
is that.
And actually a formula that you may see
that is very oftenly used as as a first
approximation is this one. This is
is is the same asset, but it's called
ex-dividend or ex-coupon. It's it's
after the coupon of this year has been
paid.
Okay? So, it's an asset that starts
paying at T plus one. It's ZT plus one,
ZT plus two, and so on.
Well, that
is the same as this minus the first
coupon, so is equal to that.
Okay?
That's an interesting thing, huh? Look,
what happened to this asset as the
interest rate goes to zero?
So, this is an asset that lasts for a
very long time.
And and and look, we got to a valuation
formula.
What hap- what is happening as the
interest rate goes to zero?
To the value.
Very large. It goes to infinity.
And a lot of what has happened in in
global financial markets
in the last few years has to do with
that.
Interest rates were very very very low.
And so, most assets that had long
duration had very high values.
Okay?
And it has a lot to that. Monetary
policy had a lot to do
whether it was the right monetary policy
or not,
that's something to be discussed. I
think on average it was the right
monetary policy, but one of the things
it did, it increased the value of many
assets. In fact, that's one of the
mechanisms through which monetary policy
works in practice. It's not something we
have discussed, but you can begin to see
here. Because if the value of all assets
go up a lot, people feel wealthier, and
that they will tend to consume more, and
so on. Well, this is one of the channels
monetary policy does. By the way, this
effect happens also to this asset that
has finite N. It's just that this goes
is
it's maximized when this asset lasts
forever. You know?
This this asset literally goes to
infinity
if the interest rate goes to zero.
Well, if an asset lasts for N periods,
it doesn't go to infinity. It goes to N
times Z.
You know? It's the sum.
If the interest rate is zero, you just
sum things.
See that?
If I if if an asset lasts for N periods,
and it gives me a payment of Z in every
single period,
then when the interest is zero, that
asset is worth N times Z.
Because I will receive Z coupons.
And I don't discount the future because
the interest rate is zero.
What happens is when the asset lasts
forever, then N times Z is a really
large number, you know? And that's
that's what this expression captures
here.
Okay.
So, let's talk about bonds now. We're
going to start pricing bonds.
Well, so bonds differ uh uh uh
along many dimensions, but one of them
is is very important for bonds is
maturity, the N that I had there
in the previous expression. Okay?
Uh so, so maturity means essentially how
long the bond lasts. Okay? When when
does it pay you back the principal? The
bonds typically pay coupons, and then
there's a final payment, which we call
face value of the bond or something like
that. And and when that final payment
takes place, that's the maturity of a
bond. Okay?
So, a bond that promises to make a
thousand-dollar final payment in six
months
has a maturity of six months.
A bond that promised to pay a hundred
dollars for twenty years and then one
thousand dollars final payment in twenty
years has a maturity of twenty years.
Maturity is different from duration. I
don't think I'm going to talk about
duration here, but but that's maturity.
Just when the when is the final payment
of of a
of a loan.
Of a of a bond. Okay?
Bonds of different maturities each have
a price
and an associated interest rate. We're
going to look at those things.
And the associated interest rate is
called the yield to maturity, or simply
the yield of a bond.
This is terminology, but we're going to
calculate these things later on.
The The relationship between maturity
and yield
is called the yield curve. Very
important concept. Big fuss about the
yield curve these days.
Talk a little bit more about that.
Or sometimes it's called the term
structure of interest rate.
Term, in the language of bonds, is
really maturity.
So, term structure of interest rate
really tells you what is the yield in a
1-year bond, 2-year bond, 3-year bond, 4
5 6 so on. You plot them, and that gives
you a curve.
Okay.
So,
uh for example,
look at the those These are two
different yield curves. This is November
2000,
and this is June 20
2001.
So, this tells you what the yield is
in on a 3-month bond, so a bond that
matures in three in 3 months, on a
6-month bonds and so forth, up to
30-year bonds. Okay.
What is the big difference between these
What do you think happened here in
between? Notice that these two curves
are more or less the same long-term
interest rate.
But they have very different This curve
This is a very steep curve, and this is
a very flat or even inverted curve.
What do you think may have happened
there?
Between November 2000 and June 2001.
People changed their expectations then.
Yeah, it's That's true. That's for sure
true about that. But look also that But
that that that this 3-month There is
very little uncertainty about 3 months.
It was a lot lower than that.
So, yes, people changed their
expectation, but why do you think they
changed their expectation?
Well, it's rising inflation. We have a
lot
Rising inflation from here to here.
These are These are nominal interest
rates.
Up to now I've been talking about
nominal interest rate.
What happens here
is there was a mini recession.
So, the Fed cut interest rate.
When you're in recessions, the curve
tend to look like this.
Because
the central bank is cutting interest
rates in the in the short run to deal
with the current recession.
What happens 30 years from now has
nothing to do with the business cycle
today, so that interest rate doesn't
need to move a lot. But the Fed is
bringing interest rate down a lot in the
front end. Okay. So, that's the typical
shape of a curve in a recession.
That's the typical shape of a of a curve
in the opposite situation where the
inflation is too high and so on. Because
what happens? The Fed is trying to The
Fed really controls the very front end
of the curve.
That's what the Fed really control. The
central bank in general, but the Fed.
They control the very front end of the
curve because they're setting the very
short-term interest rate.
So, this is a situation where they're
tightening the monetary policy very
tight.
Because they are a situation of uh
overheating in the economy. And in fact,
they got too carried away. That's the
reason they we ended up in a recession
here.
Okay.
How do you think it looks today?
That Do you think it looks more like
this or more like that?
Is inflation low or high today?
High. I mean, that's a problem, you
know? The Fed is trying to hike interest
rate. Now, recently, because of the the
mess in the banking sector, then the
expectations of interest rate began to
decline a little, but but but the
situation was was very important. Here
you are.
That's The green line is today.
Okay. So, it's very inverted.
Okay.
A year ago, it looked like that.
So, you see the the long end hasn't
changed much, but a year ago, there was
no sense that the inflation was getting
so much out of line.
It happened a little later than that.
There was some concern that interest
rate would would rise,
but but but now it's very clear the
economy is overheating. And this I
should have plotted you something for
for
a month ago. It would have been even
steeper. Okay.
Anyways, but that's because the Fed is
trying to slow down the economy. It's
hiking interest rates. That's the reason
the curve is very very inverted today.
So, let me let me calculate these rates.
How do we go about it? So, the first
thing we're going to do is we're going
to use the expected present discounted
value formula to calculate the price
of a bond.
And then we want to start
doing it for different bonds,
and we're going to construct uh the
yield curve.
So, suppose you have a bond that pays
$100,
nothing in between, $100 1 year from
now. So, this is a bond with maturity
1-year maturity.
I'm going to call that bond with 1-year
maturity
P1 the price of a bond with a 1-year
maturity at time T, P1T.
Well, that's easy to calculate. It's
expected present discounted value for If
you have the interest rate, whatever you
say, 1-year interest rate, then I know
that the price of the bond is 100
divided by 1 plus the interest rate, the
1-year interest rate today.
Okay. That's the price. That's expected
discounted value. So, I tell you what
I'm showing you is the relationship
between interest rates and prices.
Okay. Our price of a bond. The price of
that bond is just 100
uh divided by 1 plus the 1 plus the
1-year interest rate today. Okay.
So, important observation is that the
price of a 1-year bond varies inversely
with the current
1-year nominal interest rate. This is
all nominal, huh?
Why is it an inverse relationship?
Why is it the price of a 1-year bond is
inversely related to the 1-year interest
rate?
In other words, I'm asking
what do you think happens to the price
as a nominal as a nominal interest rate
rises?
And why do you think that's what happens
to the price?
Well, the first question doesn't have a
I mean, it's very easy, you know, the
answer to the first question. What
happens if I goes up? Well, it's obvious
that this price comes down.
But why?
And and and I'm And you use the concept
we have developed here. Remember we
spent
like 20 minutes in one slide. Well, you
start the slide for that answer.
Hint.
This $100 you're not receiving today,
you're receiving a year from now.
What happens with a dollar received a
year from now?
What is the value of a year
dollar received 1 year from now when the
interest rate is high?
Slow, because, you know,
you'd much rather have the dollar today,
invest it, and get this big return on
the on on the dollar.
That means, naturally, a bond that is
paying you $100 tomorrow is going to be
worth less
when the interest rate is very high.
It's going to be worth less today when
the interest rate is very high. You'd
rather have the money today, invest it
in the in in the interest rate, and get
the interest rate.
And and uh
No, I need to invest 1 over 1 plus I1T
dollars to get $100. That's another way
of saying it.
What about with the bond that pays $100
in 2 years?
Well, I need to discount that by this,
which is a You know, it's a product of
the two interest rate. And since I don't
know what the 1-year rate
will be 1 year from now, I have to use
expectation here rather than the actual
rate. But look at the notation. I'm
calling
P2T,
dollar P2T, the price of a 2-year bond,
a bond with maturity of 2 years,
as of time T.
Okay.
And this is a bond that has no coupons.
So, yes, pays you $100
at the end of the 2 years.
Now, note Note that this price
is inversely related to both
the 1-year rate today
and the expectation of the 1-year rate 1
year from now.
If either one of these goes up,
the bond is worth less today. You
discount more a dollar received
uh
um
2 years from now. I don't care which
one.
You know,
either of them that goes up is is bad
news for the for the price of a bond.
Okay.
Is this clear?
So,
there's an alternative So, this is the
way you price a bond bonds using just
expected discounted value
uh approach. Now, it turns out that in
practice, a lot of the asset pricing is
done by arbitrage. Meaning, you you
compare different assets, and that that
have similar risk, they should give you
more or less the same return. That's
what you do. So, let me let me do this
arbitrage thing. Suppose you're
considering investing $1 for 1 year. So,
that's your decision. I'm going to
invest one I need I have a dollar, which
I want to invest for 1 year.
But I But I I have two options to do
that. I can invest a dollar in a 1-year
bond.
I know exactly what I'm going to get,
you know, in that bond.
Or
I can invest in a 2-year bond
and sell it at the end of the first
year.
That's Those are two ways of, you know,
investing for 1 year.
Arbitrage has to be compared over the
same period of time and everything. It's
not the return of a bond that you hold
for 10 years versus one that you hold
for a 1 year. It has to be something a
similar investment. Suppose I need to
invest for 1 year.
Or you know, then then Okay, then if I
have these two bonds, the option is not
buy one or the other and then hold to
maturity because that would be comparing
an investment of 1 year with an
investment of 2 years.
I need to compare the strategies of
getting my return in 1 year.
In the 1-year bond, that's trivial
because I get my return at the end of at
the maturity of the bond. In the 2-year
bond, it means I need to sell it in
between after 1 year. Okay? So, those
are the two strategies I want to compare
and since I'm not take
considering riskier as a central
element,
those two strategies are going to have
to give me the same expected return.
Okay? That's arbitrage. That's what we
call arbitrage.
Okay?
Two
the two strategies have to give me the
same expected return.
So,
what do we get from this strategies?
Well, if I go through the 1-year bond, I
know I'm going to get my dollar times 1
plus I1T. That's what I get off a 1 year
out of investing a dollar in a 1-year
bond.
If I go through the 2-year bond
strategy, buy it and sell it at the end
of the year, then I'm going to get I I I
I
invest a dollar today,
no? I'm going to pay P2T.
That's what I paid today for a 2-year
bond. That's what I pay here for a
2-year bond and I expect to get the
price of a 1-year bond 1 year from now.
I mean, the 2-year bond will be a 1-year
bond
after a year has passed.
No?
It's a 2-year bond today, but
after 1 year, it's going to have only 1
year to mature.
So, that's the reason the price I need
to forecast is the is the price of a
1-year bond 1 year from now. That's what
this is here.
Okay? And that's my return on this
strategy because I'm going to pay this
today,
these dollars,
and I expect to get that 1 year from
now.
Okay?
So, arbitrage means I need to set these
two equal.
Okay?
So,
that means I have to get the same return
with the two strategies. That means I'm
investing the same, so I only need to
compare the the the the
the returns. This needs to be equal to
that.
That's what I have here.
Which tells you
that you're solving from here that the
price of a 2-year bond at time T
is equal to the expected price of a
1-year bond at T plus 1
discounted by 1 plus the 1-year interest
rate.
No?
This was like my cash flow.
My cash flow now is not the cash flow.
It's It's just a price. I'm going to get
a price for that asset. That's like the
Zs I had in my formula. Okay?
And for a 1-year strategy, I only need
to worry about the ZT plus 1.
There was no dividend at day zero.
Okay? And that's exactly that formula.
But notice that at T plus 1,
that will hold.
No? So, at T plus 1, I'm at T plus 1, I
don't need expectations. I know that P1T
plus 1 will be equal to 100 divided by 1
plus I1, the 1-year rate at T plus 1.
Therefore, the expected is something
like this, approximately. The expected
price is something like that.
Okay? I expect
I mean, this will be without the E will
be the price
of this 1-year bond at T plus 1. I don't
know exactly what the interest rate will
be next year, so I have the best I can
do is have an expectation. That's my
expectation, approximately.
Okay? But now I can stick this
expression in here.
No? I have this.
I'm going to go out and I can stick that
in there
and I get this expression. So, that's
the price for the 2-year bond.
Do you recognize this?
You saw it before.
You know?
That's the same expression that we got
when we used the expected present
discounted value formula.
Right?
We said, "Well, I'm going to get the
$100 100 years a 100 years a 2 years
from now. I know that discount factor
for that is 1 over 1 plus I1T times 1
plus I1T plus 1 expected."
Well, that's what I got.
That's from arbitrage.
Okay?
From an arbitrage logic. This is used a
lot in finance.
I I I'm going to say something
complicated, but but um
just ignore it if it's
uh
not really up for the for the for the
quiz or anything, but
you know, there's a big debate in the US
today about uh not big debate, a big
concern about
uh
the the
the US Treasury debt because there is a
debt ceiling, meaning there's a maximum
amount that the government can
of debt they can issue.
And and the and that ceiling has been
moved over time, but every time we get
close to a deadline when this needs to
be agreed again, there's a concern and
there's negotiations and so on.
And the and the
and the well, I mean, everyone at this
moment at least thinks that
as every as in every instance in the
past, they're going to reach some sort
of agreement the day before
of the deadline or not.
But if they don't and there is a mess,
this is huge for finance. It's huge for
finance because US Treasury bonds,
especially short-term bonds, are used
for pricing everything
through arbitrage and so on.
So, you get a mess there,
that's a mess in every single financial
market. You wouldn't know how to price
many financial assets, actually.
So, it would be a disaster. But uh
but the reason I describe I mention this
here is because
again, lots of prices are priced in
reference in as in finance are priced in
reference, especially derivatives,
options, and stuff like that.
Uh you price them relative to something
using this type of logic. So, if the
thing you use as a base as a reference
becomes highly unstable and uncertain
and risky, then obviously everything
becomes very complicated,
very risky, and and financial markets do
not like risk. That's for sure.
Anyway, ignore that. That's
irrelevant for your quiz, but that's the
reason this the whole discussion then
over the summer can get to be very very
tricky for finance.
So, the yield to maturity, remember I
mentioned this concept before, of an
N-year bond,
but it's also what we When you see
Whenever you hear the 3-year rate,
is that. It's the yield to maturity.
Uh which is different from Okay, let me
tell you
show you a formula that's easy to
explain then.
And it's defined, it's important, as the
constant
annual interest rate that makes the bond
price today equal to the present
discounted value or expected discounted
value.
So, notice notice the highlighted
is defined as the constant annual
interest rate
that makes the bond price today equal to
the present discounted value of future
payments of the bond.
Okay?
So,
for example, in our 2-year bond,
that's the price. Right? This is the
price of the asset.
We that we know the price. We already
got the price from the previous slides
of the bond,
which was based on the short-term
interest rate, 1-year interest rate, and
our forecast of the
short-term the 1-year interest rate 1
year from now.
I know that price. Take that as a
number.
So, then I the yield the yield to
maturity is calculated as that constant
interest rate
constant How do I see this constant?
Because, well, I'm using the same
interest rate for the first period and
the second period. I And now I'm calling
it I2T. It's a 2-year interest rate, but
it's constant. Constant doesn't mean
that it doesn't move over time.
It means I'm discounting all the cash
flows as a constant interest rate. This
It means I'm using
I'm using this equation.
Okay?
So, the yield to maturity is
find an interest rate
that allows me to use this constant
thing constant assume use this formula
and get back the same price
as I got by using the the
the expected discounted value or the
arbitrage or something like that. Okay?
So, that's that's the definition. Okay?
You have this price.
Now you you look for that interest rate
that allows you to match that price.
Okay?
And that's called the yield. That's the
thing I Remember I plotted this the some
curves?
Well, those those interest rates in
those curves were computed this way.
Now,
notice that we know what this price is.
This price is by the expected discounted
value or the arbitrage approach is equal
to 100 divided by this.
So, I know that these two things are
this is equal to that,
which means that this denominator is
equal to that,
and that implies for a small interest
rate that this 2-year interest rate is
approximately equal to the average of
the expected interest rate 1-year rates.
Okay?
So, this is called actually the
expectation hypothesis, by the way. Is
that the the 2-year rate
is approximately equal to the average of
the 1-year rate this year
plus the expected 1-year rate 1 year
from now.
Okay?
So, that's an important concept. And I'm
going to start from here again
in the next lecture.
Mhm.