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49:49
Transcript
0:17
Today we're going to talk about
0:19
a very important topic
0:21
topic in economics, which is
0:22
expectations. We have barely mentioned
0:24
expectations when we talk about the
0:26
Phillips curve. We talked about
0:28
expectations when we
0:30
when we discussed the UEP and so on. But
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0:33
expectations is a much bigger issue in
0:35
economics. In fact, most decisions by
0:37
firms, by consumers,
0:39
governments involve considerations of
0:42
the future. And it plays an even bigger
0:44
role in finance, in which essentially
0:46
everything is about the future. The
0:48
price of an asset today is meaningless
0:51
in itself. You have to compare it with
0:53
what you expect to get out of that asset
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0:55
in the future. So, it's all about
0:56
expectations and so on.
0:58
So, that's what we we're going to do
0:59
today. We're going to talk about
1:01
uh expectations, the how to value things
1:05
that that you expect to receive in the
1:07
future,
1:08
uh and how to compare those things with
1:10
things that you have in the present.
1:13
Um
1:14
but before doing that, actually, let's
1:16
talk a little bit about the news. Who
1:18
knows who First Republic Bank is?
1:21
Remember that a few weeks ago I told you
1:24
that um Silicon Valley Bank,
1:27
you read it. I I I I just mentioned it
1:30
that
1:31
uh
1:32
uh
1:33
we'll discuss it that that you know, we
1:35
had the second largest bank by asset in
1:38
in in US history. It was Silicon Valley
1:41
Bank was the second largest
1:43
asset bank in terms of assets
1:46
uh to collapse in the US. The first one
1:48
was uh many years ago.
1:50
Uh and then we had this bank that had
1:52
more than $200 billion in assets that
1:55
essentially collapsed in a few days. It
1:57
was a run on deposits. They had problems
2:00
before,
2:01
but what really did as it always the
2:03
case with banks is they had a run on
2:05
deposits, funding.
2:07
Uh
2:08
well, it's no longer the second largest
2:11
collapse in US bank history. Now we have
2:14
over the weekend
2:15
the the new second largest bank to
2:17
collapse, which is First Republic Bank,
2:20
that was essentially
2:22
liquidated and sold to JP Morgan over
2:26
today morning, very very early in the
2:27
morning. Okay? So,
2:29
you have an account in First Republic
2:31
Bank, you sooner likely to have an
2:32
account in JP Morgan.
2:34
But again, what made it collapse was
2:37
something very similar to what made
2:39
Silicon Valley Bank collapse, which is
2:41
that they had invested on on a series of
2:44
things that were very vulnerable to to
2:46
the fast pace of hikes
2:48
uh
2:49
in interest rates in the US.
2:51
And when they had those losses,
2:53
depositors became became worried about
2:55
it and eventually they decided not to
2:57
wait, just run.
2:58
And see what happened. They
3:00
First Republic Bank lost about $100
3:02
billion in deposits just last week.
3:04
Okay? Um the last few days of last week.
3:07
So, so, so
3:10
so that
3:11
it was obvious that that
3:13
it was not going to survive and that's
3:15
the reason
3:16
something was arranged over the weekend
3:18
to avoid the panics associated with
3:20
collapses of a bank and so on. Okay? But
3:23
anyways, by the way, this is all about
3:24
expectations. This is you know, this is
3:27
if people had expected the deposit to
3:29
remain in the bank, then probably this
3:31
bank would not have collapsed. It's all
3:32
about people anticipating what other
3:34
people will do and so on and so forth.
3:38
Okay, but now let me get into the
3:40
specific of
3:41
of this lecture.
3:43
So, there you have this is the most
3:44
important index of of equity equity
3:47
index in the US, S&P 500. It's a very
3:50
inclusive index that captures all the
3:52
large most of the large companies in the
3:55
US,
3:56
all of the large I think companies in
3:58
the US. And uh that's an index. It's an
4:00
average weighted average by by this
4:04
uh capitalization value of each of the
4:07
shares. It's a weighted average of the
4:08
major the main shares in the US, equity
4:10
shares in the US.
4:12
And one thing you see is that it moves a
4:14
lot around, you know? Here, for example,
4:16
when when
4:18
we became aware that COVID was going to
4:20
be a serious issue,
4:21
the US equity market collapsed by 35% or
4:24
so. That's a very large collapse in a
4:26
very short period of time.
4:28
And then, as a result of lots of policy
4:31
support, actually we had a massive
4:33
rally. Uh so, up to the end of 2021, the
4:37
equity market had rallied by 114%.
4:40
So, a big rally.
4:41
Then we got inflation and the Fed began
4:44
to worry about
4:45
inflation, so they began to hike
4:47
interest rates. And when they hike
4:48
interest rates, that eventually led to a
4:50
very large decline
4:52
uh
4:53
in in asset prices of the order of 30%
4:55
or so, 25% or so, actually, from the
4:58
peak to the bottom.
5:00
And then, since the bottom, which is was
5:02
more or less October of last year, we
5:04
have seen a recovery of about 16% or so
5:07
of the equity market. Okay? And if you
5:09
look at the Nasdaq, which is another one
5:11
index that is very loaded towards
5:13
uh
5:14
technology companies, then you can see
5:16
swings that are even larger than that.
5:19
Now, why do these prices move so much?
5:22
Well,
5:23
a lot of it has to do with expectations.
5:27
You know, are things going to get worse
5:29
in the future? Will the Fed cause a
5:30
recession? Uh
5:33
how much higher will be the interest
5:35
rate? And things like that matter a
5:37
great deal.
5:38
Another thing that matters a great
5:41
deal is
5:42
how much people want to take risk at any
5:44
moment in time. And if you're very
5:46
scared about the environment, you're
5:47
unlikely to want to have something that
5:49
to invest on something that can move so
5:51
much,
5:52
and so risk is well known. So, it's
5:54
called risk off when when people don't
5:56
want to take risk, these asset prices
5:58
tend to collapse. Okay? Of the risky
6:00
asset. Equity is a very risky asset.
6:03
But that's not the only thing that moves
6:05
these assets around. It's not just the
6:06
risk that the companies underlying
6:08
company may go bankrupt or anything like
6:10
that.
6:11
Here you have, for example, the movement
6:13
of a
6:14
for it's an ETF, but it doesn't matter.
6:16
It's a portfolio of
6:18
bonds of US Treasury bonds of very long
6:20
duration. Maturities beyond
6:22
uh 20 years and so.
6:24
So, this is incredibly safe bonds, you
6:26
know? Because it's US Treasuries. So,
6:28
there's no risk of default or anything
6:29
like that.
6:30
Still,
6:32
the price swings can be pretty large. I
6:33
mean, in over this period, you know,
6:35
there you have seen an an an increase in
6:37
value of 45%,
6:39
then uh a decline in value of of of
6:42
about 20%. Another increase in 15% here.
6:45
There was a huge decline, 40%,
6:48
since since essentially
6:50
uh uh uh
6:51
What do you think happened here? Why is
6:53
this big decline in in in in bonds?
6:56
You're going to be able to answer that
6:57
very precisely later on, but but I can
6:59
tell you in advance that that was
7:01
essentially the result of monetary
7:02
policy tightening.
7:04
You know, increasing interest rate
7:07
caused the the bonds to decline. So,
7:09
even these instruments that are very
7:10
safe in the sense that you if you hold
7:12
it to maturity, you will get your money
7:13
back and all the promised coupons along
7:15
the path, well, still their price can
7:18
move a lot. And it's obvious that that
7:20
movement in price
7:22
is something you need to explain in
7:23
terms of expectations, what people
7:25
expect things to to happen. In this
7:27
case, it's not whether people expect to
7:29
get paid or not, because you will get
7:31
paid, but it's expect but in this
7:34
particular case, it's about expectations
7:35
about future interest rate. If you think
7:37
the interest rate will be very high,
7:39
then the price of bonds will tend to be
7:40
very low and so on. But it's all about
7:42
the future. Okay?
7:45
So, the a key concept
7:48
uh that we're going to discuss today and
7:50
then we're going going to use it to
7:52
price a specific asset
7:54
uh is the concept of expected present
7:57
discounted value. This this is a loaded
7:59
concept. There's lots of
8:01
terms in there and we need to understand
8:03
what each of these terms means.
8:06
So, the key issue
8:08
that we're going to discuss is how how
8:10
do we decide, for example, if you see
8:11
the price of an asset out there
8:13
that is 100, how do you decide whether
8:16
that price is fair or not, looks cheap
8:19
or not? Okay? Uh and and and and and
8:23
that question means you have to decide
8:25
whether that price that you're paying
8:26
today is consistent with the future cash
8:29
flows that you're going to get from this
8:31
asset. I mean, that's the reason you buy
8:32
an asset is because you'll get something
8:34
in return in the future. Okay? But how
8:37
do we compare that? How do we compare
8:39
the price today with those things that
8:41
will happen in the future?
8:46
So, answering that question, which is
8:48
what we're going to do in this lecture,
8:50
involves the following
8:52
concepts. First, expectations, big
8:55
thing.
8:57
That's a you know, this is expected
8:59
present discounted value. The E part is
9:01
for expectations. That comes there.
9:04
You
9:05
Expectations are really crucial because
9:07
these are things that happen in the
9:08
future. You need to expect. Even if if
9:11
it's a bond that promises you to pay,
9:13
you know, 50 cents per dollar every 6
9:16
month,
9:17
you still may have an expectation that,
9:19
you know, if it is a bond issued by
9:20
First Republic Bank, it may not pay. So,
9:23
so, so you need to have an expectations
9:25
about that.
9:26
Uh
9:28
so, crucial term is expectation.
9:30
Then you need some method
9:32
uh to compare payments received in the
9:34
future with payments made today. I mean,
9:36
if you buy an asset, you pay today,
9:39
but you're going to receive things
9:40
returns on for that asset in the future.
9:42
So, how do I compare that that that
9:45
Suppose I pay one today and I receive
9:47
one 1 year from now.
9:49
Does that seem like a good asset?
9:52
Probably not. I mean, you know,
9:54
probably not.
9:56
Uh
9:57
Um and that's what the word discounted
9:59
really means. You know, when you say
10:01
expected present discounted value
10:04
it says
10:05
somehow that things I receive in the
10:07
future are valued less than things I
10:09
have today.
10:10
Okay? So, if you're going to tell me
10:12
that you're going to pay me a dollar in
10:13
the future and I have to pay you a
10:15
dollar today, most likely I won't take
10:17
that deal.
10:18
So, I need In other words, I'm
10:20
discounting the future.
10:22
How do we discount the future? Well,
10:23
something that we're going to have to
10:24
figure out.
10:27
So,
10:28
let's let me first shut down this part,
10:30
the expectations, and then we'll
10:31
introduce it. So, assume for now that
10:33
you know the future.
10:34
Okay? And I'm going to derive all the
10:36
equations with assuming that you know
10:39
the future. So, there's no issue of
10:41
trying to figure out what the future is.
10:42
You know it. But still you have to
10:44
decide whether
10:45
what is the right value for for an
10:48
asset.
10:52
Okay, so
10:54
let's start with the case where you know
10:56
the future. Sorry.
10:58
And let's do the comparison uh
11:02
Let's try to understand how do we move
11:04
flows, how do we value flows at
11:05
different points in time.
11:07
This is the thing is think first about
11:10
comparing an asset that gives you a
11:11
dollar in the future,
11:13
how much do you think it's worth today?
11:16
Well, the easiest
11:17
way to get to that value is is to think
11:19
on the alternatives. As suppose I have a
11:21
dollar today, what can I do with it?
11:26
Well, in terms of investment.
11:28
Well, suppose that you have available
11:31
one-year bonds, treasury bonds, and that
11:33
the interest rate is I
11:34
t. That's the interest rate on an I
11:36
one-year bond.
11:38
So, if you want if you if you
11:40
have a dollar,
11:42
you have the option to invest it in that
11:44
asset, in that bond, which give will
11:46
give you 1 + I dollars
11:48
uh
11:49
next year.
11:51
Well,
11:52
that means that I can get $1 next year
11:57
by investing 1 over 1 + I dollars today.
12:01
No?
12:02
Because if if I invest 1 + 1
12:05
rather than $1, I invest 1 over 1 + I
12:08
today, then I multiply this by 1 + I
12:11
and I get my dollar in the future.
12:14
So, that tells me that say the interest
12:16
rate is 10%, then with $1 today I can
12:19
get 1.1 dollars in the future.
12:22
That means that investing 90% 90 cents
12:25
today, more or less,
12:27
I can get $1 in the future.
12:29
That tells me that a dollar in the
12:31
future is equivalent to 90 cents today.
12:34
That's the assumption. Okay?
12:36
So, that's the reason when I told you
12:38
the deal of me, look, I have an asset
12:40
that cost cost you a dollar, but gives
12:42
you a dollar in the future, well, that's
12:44
not a good deal if the interest rate is
12:45
positive.
12:47
If the interest rate is 10%, then then a
12:49
right a fair comparison is 90 cents with
12:51
$1, not $1 with $1.
12:54
Okay? So, that's the discounting of the
12:56
future. You can The most obvious way of
12:58
discounting the future
13:00
is to discount it by the interest rate.
13:02
Uh
13:04
which interest rate to pick? That's more
13:06
subtle. That depends on risk, depends on
13:08
many other things which we're going to
13:09
discuss to some extent here. But for
13:12
now, let's make it very simple. And in a
13:15
world in which you really know the
13:16
future, really the right interest rate
13:17
to use is the safe interest rate, the
13:19
interest rate of of treasury bonds and
13:21
things like that.
13:23
Okay? So, that's that's that.
13:25
What about a dollar that you receive
13:27
What about if you're thinking about what
13:29
is the value of a dollar two years from
13:31
now?
13:32
Well,
13:34
you know, if I get a dollar to I can do
13:35
the same logic. If I if I
13:38
I can use the same logic. If I get a
13:39
dollar today,
13:41
I can convert that into 1 + I t * 1 + I
13:45
t + 1 dollars. Okay?
13:48
So, say 10% and 10%, I get 1.1 next year
13:52
and then I get 1.1 * 1.1, 1.21 or
13:55
something like that. Okay?
13:57
That's my final
13:58
result.
13:59
So,
14:01
well, then how much is it worth to have
14:03
a dollar, an asset that gives you a
14:05
dollar two years from now?
14:09
Well, it's going to be that dollar
14:11
divided by the product of these interest
14:13
rates.
14:14
Okay? Why is that? Well, because with
14:16
this amount of
14:18
dollars today,
14:20
it's point 80 cents or something like
14:22
that, I can generate a dollar two years
14:24
from now.
14:25
That means a dollar
14:27
two years from now
14:29
is worth about 80 cents today.
14:32
Okay?
14:35
We're going to use a lot this type of
14:37
logic, so
14:38
and and I know that that it may not be
14:40
that intuitive the first time you see
14:42
it, but
14:43
ask questions.
14:47
You want me to repeat it?
14:54
Okay. The
14:55
The final goal is the following. We're
14:57
going to In the what comes next, we're
14:59
going to see if which happens again with
15:01
many decisions in life, but it perhaps
15:03
particularly for financial assets,
15:05
we're going to try to value something
15:07
that
15:08
whose payoff happens at different times
15:11
in the future. And the question is
15:13
how do I value an asset that pays me,
15:16
you know, $5 one year from now, $25
15:19
three years from now, uh
15:21
minus $10 10 years from now, plus $50
15:26
100 years from now?
15:27
What is the value of that? Of having an
15:29
asset like that?
15:31
And so, I needed some method
15:33
to bring it to today's value because
15:35
today I have a meaning of what a dollar
15:37
is, you know?
15:38
And and therefore I can compare it with
15:40
whatever price I mean
15:42
people are asking me for that asset.
15:44
So, what this is doing is is is that is
15:47
doing that. It's telling you how to
15:49
convert a dollar at different parts in
15:51
the future into a dollar today.
15:54
And by that logic,
15:56
the recipe is well, use the interest
15:59
rate because you could always go the
16:01
other way around. You could always with
16:02
a dollar you can ask a question, with a
16:03
dollar today, how many dollars can I get
16:06
two years from now, say?
16:08
That.
16:09
Well,
16:10
say X. Well, then I need 1 over X. Then
16:13
$1 there is worth 1 over X dollars
16:15
today. You know, that's that's the logic
16:18
because 1 over X * X is 1.
16:21
So,
16:23
that's too fast, probably.
16:28
So,
16:29
you know, with $1 today, oops,
16:35
I can generate, say,
16:38
$1.1
16:41
at uh uh
16:44
at t equal to
16:45
Okay?
16:46
Then I'm I'm The question I want to know
16:48
is how much is a dollar worth
16:51
How much is a dollar received at time t
16:53
equal to worth today?
16:56
That's the question I'm trying to
16:57
answer.
16:58
You know, because an asset will be
17:00
something that will pay you in the
17:01
future. So, I want to know how much is
17:03
$1 received in the future worth today.
17:08
And then the answer is
17:10
well,
17:11
then is I know the answer from this
17:13
logic because I know that with one
17:18
if I have 1 over 1.1 dollars today, I
17:21
can convert it
17:25
into one.
17:27
How do I know that?
17:28
Because
17:33
1 over 1.1
17:38
* 1.1
17:40
is equal to 1.
17:43
Okay? This if I invest these dollars
17:45
today,
17:47
I'm going to get this return on that.
17:50
And the product of these two things
17:51
gives me my dollar.
17:54
Okay?
17:55
So, if I tell you, do you prefer to have
17:56
a dollar two days from two years from
17:58
now or today?
17:59
You say, I prefer it obviously prefer it
18:02
today because I can get 1.1 dollars two
18:05
years from now.
18:07
But then then the more relevant question
18:09
is, no, no, but then you do you prefer
18:10
to have 90 cents today
18:12
versus a dollar in the future? And then
18:15
I'm I need to do my multiplication
18:16
because I have to multiply the 90 cents
18:18
by the 1.1 and see whether I get
18:21
something comparable to a dollar or not.
18:23
Okay?
18:24
But that's that's the logic behind that.
18:26
And And that's a So, the interest rate
18:29
is what we discount the future by.
18:33
And it's natural because if the interest
18:34
rate is very high If the interest rate
18:36
is zero, say,
18:37
then a dollar received two years from
18:39
now or a dollar received today is is the
18:41
same
18:42
because I can't If I invest a dollar
18:44
today and the interest rate is zero, I'm
18:45
going to get my dollar two years from
18:46
now.
18:47
If the dollar If the interest rate is
18:49
50%, it makes a big difference receiving
18:51
the dollar today versus receiving it two
18:52
years from now.
18:54
If you're in Argentina, the interest
18:56
rate I don't know what it is. It's
18:58
700%. It makes a huge difference whether
19:00
you receive it, you know, one year from
19:02
now than today.
19:05
And and and uh
19:08
So, that's that's the role of the
19:09
interest rate. The higher is the
19:10
interest rate,
19:12
the less
19:13
is a dollar received in the future worth
19:15
relative to a dollar received today.
19:17
Because you can get a much higher return
19:19
from the dollar you have today
19:21
if the interest rate is high. If the
19:22
interest rate is low,
19:24
you don't get that much. Okay?
19:26
Much difference. Okay, good.
19:28
So, this is a big principle. And and I I
19:30
mean
19:31
everything I'll say next builds on this
19:33
logic.
19:38
So, let me give you a general formula.
19:40
So, let's ask what is the value
19:43
of an asset
19:44
that gives
19:46
payouts of Z
19:49
t dollars this year,
19:51
Z t + 1 one year from now, ZT plus two,
19:55
two years from now, and so on and so
19:57
forth for N periods more. Okay?
20:01
Well,
20:03
I just need to do several of these
20:04
operations. I know that the dollar
20:06
received this year is is worth a dollar.
20:09
Okay? That's ZT.
20:10
A dollar received one year from now
20:13
is not
20:14
is not the same as a dollar received
20:16
today. It's the same as one over one
20:18
plus IT dollars received today.
20:22
So, that cash flow I'm going to receive
20:23
from this asset is worth this amount.
20:26
For a two something that I receive two
20:28
years from now, then it's not
20:30
it's not certain, it's much less than
20:32
receiving a dollar today. It's going to
20:34
be one over one plus IT one plus IT plus
20:38
one.
20:39
And that I have to multiply by the
20:40
number of dollars I will receive two
20:42
years from now. Okay? And I keep going.
20:46
So, that's that's the
20:48
the present value. Present discounted
20:51
value. Present because I'm bringing all
20:53
these future cash flows to the present.
20:56
That's what each of these terms is
20:57
doing. The one over that is bringing it
21:00
to the present.
21:01
Discounted because the interest rate is
21:03
discounting things. It's making them a
21:05
smaller.
21:06
And value because I'm trying to reduce
21:08
them to the current value. Okay?
21:12
That's the general formula. So, it's a
21:14
formula you need to understand.
21:15
It's just So, that that was an asset
21:18
that gives you Z dollars today,
21:21
ZT plus one, one year from now, so you
21:23
use this formula. ZT plus two, two years
21:26
from now, so you use this formula, and
21:29
then you keep going. Okay?
21:33
What if we don't know the future?
21:35
You know, I have to remove the expected
21:37
part.
21:39
Well,
21:40
if we don't know the future, then the
21:41
best we can do, in fact, we do fancier
21:43
things, but that's what we're going to
21:45
all that we'll do in this course.
21:47
Uh all that you can do is just replace
21:50
the known quantities we have here
21:52
for the expectations.
21:54
Okay? So, that's the closest. So, you
21:56
know, I know ZT, that's the cash flow I
21:58
get now,
22:00
but I don't know ZT plus one. So, I can
22:02
replace it by expectation.
22:04
I do know the interest rate on a
22:05
one-year bond from today to one year.
22:08
So, that's the reason I don't need an
22:09
expectation here.
22:10
But I don't know what the one-year rate
22:12
will be one year from now. So, that's
22:14
the reason I need an expectation there.
22:17
And so on.
22:18
And I don't know what the cash flow will
22:20
be two years from now. I have an
22:21
expectation about what the cash flow
22:22
will be, but I don't know it.
22:24
So, I have an expectation there. Okay?
22:26
So, so all that I've done here is say,
22:29
"Okay,
22:31
I acknowledge that this guy knew a
22:32
little bit too much. You know, he knew
22:33
exactly what the cash flows were going
22:35
to be in the future, and he knew what
22:37
the one-year rates were going to be in
22:38
the future."
22:40
This guy here knows less. He knows the
22:42
cash flow today. He knows the interest
22:44
rate today, but he doesn't know the cash
22:46
flows. Really, he has a hunch, but he
22:48
doesn't know the cash flows one year,
22:50
two years, three years, and so on for
22:51
the future, and he doesn't know the
22:53
one-year interest rate in the future.
22:56
So, all these expectations, here's
22:58
important the concept of time. This is
23:00
an expectation as of time T. At time T,
23:02
you have some information and you make
23:04
forecast about the future. Okay?
23:07
Use whatever you want, machine learning,
23:08
whatever, but you have information at
23:10
time T,
23:11
and then you have a forecast for the
23:13
future. At T plus one, you have you'll
23:14
have more information, so you make
23:16
another forecast, and so on and so
23:17
forth.
23:18
But in this we're valuing an asset at
23:20
time T, then all these expectations are
23:23
taken as of time T. That means given the
23:26
information you have available at time
23:28
T.
23:30
That's the reason these guys don't have
23:31
expectations in front of them because
23:33
you know this at time T.
23:36
Had we taken the value at T minus one,
23:38
we would have not known that, and then
23:40
we would have had to expectation because
23:41
it would have been expectation as of T
23:43
minus one.
23:46
Okay, so that's your big formula there.
23:48
So,
23:49
there are some examples that are sort of
23:51
well known and and and
23:53
and and
23:55
and and
23:56
So, let me let me show you. They have
23:58
nicer expressions. So, that's that's an
24:00
example
24:02
of the valuation of of this the same
24:04
asset,
24:05
but when the interest rate is constant,
24:08
then
24:09
then obviously I don't need all these
24:10
products in the denominator.
24:13
I have a constant interest rate, then I
24:16
just get powers of that interest rate.
24:18
That's one in which you have constant
24:20
payments. So, the interest rate may be
24:22
different,
24:23
but the payments are the same over time.
24:26
Okay?
24:27
So, that's that.
24:29
So, those are two
24:30
easy formulas. That's one in which you
24:32
have
24:33
both constant, the interest rate
24:36
and the payment.
24:37
Then you get a nice expression. That's
24:39
just uh
24:40
that. Okay? You'll recognize that.
24:44
If if you have a constant
24:46
uh
24:47
constant interest rate here, you see
24:50
that the value
24:52
is is declining is a is a geometric
24:54
series. You know? The value of a two
24:56
years from now is a square
24:58
of one over one plus some
25:00
it's a square of a a number less than
25:02
one.
25:03
You know? One over one plus I is some
25:05
number less than one. This is a square
25:07
of that, then the cube, and so on. So,
25:09
it's a geometric series that is
25:10
declining at the rate one plus I, one
25:12
over one plus I. Okay? Or declining at
25:14
the rate one plus I.
25:16
So, that's your geometric series.
25:18
Okay?
25:19
That's the value of that.
25:22
Constant rate and payment forever.
25:25
Suppose you have an asset that
25:27
it
25:28
that lives forever.
25:31
There are some bonds like that called
25:32
perpetuities. Uh uh
25:36
The US hasn't issued one, but the UK
25:38
has, and so on.
25:40
I
25:40
So, that's an asset, for example, that
25:42
pays you a fixed amount
25:44
forever. And if the interest rate is
25:46
constant, that's the trickier thing,
25:48
then the value of that asset you can see
25:50
that this this is going to zero.
25:53
So, the value of that asset
25:55
is that.
25:57
And actually a formula that you may see
25:59
that is very oftenly used as as a first
26:02
approximation is this one. This is
26:04
is is the same asset, but it's called
26:07
ex-dividend or ex-coupon. It's it's
26:09
after the coupon of this year has been
26:12
paid.
26:13
Okay? So, it's an asset that starts
26:15
paying at T plus one. It's ZT plus one,
26:17
ZT plus two, and so on.
26:19
Well, that
26:20
is the same as this minus the first
26:22
coupon, so is equal to that.
26:25
Okay?
26:29
That's an interesting thing, huh? Look,
26:31
what happened to this asset as the
26:32
interest rate goes to zero?
26:36
So, this is an asset that lasts for a
26:37
very long time.
26:39
And and and look, we got to a valuation
26:41
formula.
26:43
What hap- what is happening as the
26:45
interest rate goes to zero?
26:46
To the value.
26:49
Very large. It goes to infinity.
26:51
And a lot of what has happened in in
26:53
global financial markets
26:55
in the last few years has to do with
26:57
that.
26:58
Interest rates were very very very low.
27:01
And so, most assets that had long
27:02
duration had very high values.
27:05
Okay?
27:07
And it has a lot to that. Monetary
27:09
policy had a lot to do
27:10
whether it was the right monetary policy
27:12
or not,
27:13
that's something to be discussed. I
27:15
think on average it was the right
27:16
monetary policy, but one of the things
27:17
it did, it increased the value of many
27:20
assets. In fact, that's one of the
27:22
mechanisms through which monetary policy
27:24
works in practice. It's not something we
27:25
have discussed, but you can begin to see
27:27
here. Because if the value of all assets
27:29
go up a lot, people feel wealthier, and
27:31
that they will tend to consume more, and
27:32
so on. Well, this is one of the channels
27:34
monetary policy does. By the way, this
27:36
effect happens also to this asset that
27:38
has finite N. It's just that this goes
27:40
is
27:41
it's maximized when this asset lasts
27:43
forever. You know?
27:44
This this asset literally goes to
27:46
infinity
27:47
if the interest rate goes to zero.
27:51
Well, if an asset lasts for N periods,
27:54
it doesn't go to infinity. It goes to N
27:56
times Z.
27:58
You know? It's the sum.
27:59
If the interest rate is zero, you just
28:01
sum things.
28:03
See that?
28:04
If I if if an asset lasts for N periods,
28:07
and it gives me a payment of Z in every
28:09
single period,
28:11
then when the interest is zero, that
28:13
asset is worth N times Z.
28:15
Because I will receive Z coupons.
28:18
And I don't discount the future because
28:19
the interest rate is zero.
28:21
What happens is when the asset lasts
28:23
forever, then N times Z is a really
28:25
large number, you know? And that's
28:27
that's what this expression captures
28:29
here.
28:31
Okay.
28:34
So, let's talk about bonds now. We're
28:36
going to start pricing bonds.
28:39
Well, so bonds differ uh uh uh
28:42
along many dimensions, but one of them
28:45
is is very important for bonds is
28:46
maturity, the N that I had there
28:49
in the previous expression. Okay?
28:52
Uh so, so maturity means essentially how
28:55
long the bond lasts. Okay? When when
28:57
does it pay you back the principal? The
28:59
bonds typically pay coupons, and then
29:02
there's a final payment, which we call
29:03
face value of the bond or something like
29:05
that. And and when that final payment
29:08
takes place, that's the maturity of a
29:10
bond. Okay?
29:12
So, a bond that promises to make a
29:14
thousand-dollar final payment in six
29:15
months
29:17
has a maturity of six months.
29:20
A bond that promised to pay a hundred
29:22
dollars for twenty years and then one
29:24
thousand dollars final payment in twenty
29:26
years has a maturity of twenty years.
29:28
Maturity is different from duration. I
29:30
don't think I'm going to talk about
29:31
duration here, but but that's maturity.
29:33
Just when the when is the final payment
29:36
of of a
29:38
of a loan.
29:39
Of a of a bond. Okay?
29:42
Bonds of different maturities each have
29:44
a price
29:45
and an associated interest rate. We're
29:47
going to look at those things.
29:48
And the associated interest rate is
29:51
called the yield to maturity, or simply
29:53
the yield of a bond.
29:55
This is terminology, but we're going to
29:57
calculate these things later on.
30:00
The The relationship between maturity
30:02
and yield
30:03
is called the yield curve. Very
30:05
important concept. Big fuss about the
30:07
yield curve these days.
30:09
Talk a little bit more about that.
30:11
Or sometimes it's called the term
30:13
structure of interest rate.
30:15
Term, in the language of bonds, is
30:17
really maturity.
30:19
So, term structure of interest rate
30:21
really tells you what is the yield in a
30:23
1-year bond, 2-year bond, 3-year bond, 4
30:26
5 6 so on. You plot them, and that gives
30:29
you a curve.
30:30
Okay.
30:31
So,
30:33
uh for example,
30:34
look at the those These are two
30:36
different yield curves. This is November
30:38
2000,
30:40
and this is June 20
30:42
2001.
30:43
So, this tells you what the yield is
30:47
in on a 3-month bond, so a bond that
30:49
matures in three in 3 months, on a
30:51
6-month bonds and so forth, up to
30:53
30-year bonds. Okay.
30:56
What is the big difference between these
30:57
What do you think happened here in
30:58
between? Notice that these two curves
31:00
are more or less the same long-term
31:02
interest rate.
31:04
But they have very different This curve
31:06
This is a very steep curve, and this is
31:08
a very flat or even inverted curve.
31:12
What do you think may have happened
31:14
there?
31:16
Between November 2000 and June 2001.
31:20
People changed their expectations then.
31:22
Yeah, it's That's true. That's for sure
31:25
true about that. But look also that But
31:28
that that that this 3-month There is
31:30
very little uncertainty about 3 months.
31:32
It was a lot lower than that.
31:34
So, yes, people changed their
31:35
expectation, but why do you think they
31:37
changed their expectation?
31:42
Well, it's rising inflation. We have a
31:44
lot
31:46
Rising inflation from here to here.
31:47
These are These are nominal interest
31:48
rates.
31:50
Up to now I've been talking about
31:51
nominal interest rate.
31:56
What happens here
31:58
is there was a mini recession.
32:00
So, the Fed cut interest rate.
32:03
When you're in recessions, the curve
32:05
tend to look like this.
32:08
Because
32:09
the central bank is cutting interest
32:10
rates in the in the short run to deal
32:12
with the current recession.
32:14
What happens 30 years from now has
32:15
nothing to do with the business cycle
32:16
today, so that interest rate doesn't
32:18
need to move a lot. But the Fed is
32:20
bringing interest rate down a lot in the
32:22
front end. Okay. So, that's the typical
32:24
shape of a curve in a recession.
32:27
That's the typical shape of a of a curve
32:29
in the opposite situation where the
32:31
inflation is too high and so on. Because
32:32
what happens? The Fed is trying to The
32:34
Fed really controls the very front end
32:36
of the curve.
32:37
That's what the Fed really control. The
32:39
central bank in general, but the Fed.
32:40
They control the very front end of the
32:42
curve because they're setting the very
32:43
short-term interest rate.
32:44
So, this is a situation where they're
32:46
tightening the monetary policy very
32:47
tight.
32:48
Because they are a situation of uh
32:51
overheating in the economy. And in fact,
32:53
they got too carried away. That's the
32:54
reason they we ended up in a recession
32:55
here.
32:57
Okay.
33:00
How do you think it looks today?
33:06
That Do you think it looks more like
33:07
this or more like that?
33:09
Is inflation low or high today?
33:14
High. I mean, that's a problem, you
33:15
know? The Fed is trying to hike interest
33:17
rate. Now, recently, because of the the
33:19
mess in the banking sector, then the
33:21
expectations of interest rate began to
33:23
decline a little, but but but the
33:25
situation was was very important. Here
33:27
you are.
33:28
That's The green line is today.
33:30
Okay. So, it's very inverted.
33:34
Okay.
33:35
A year ago, it looked like that.
33:38
So, you see the the long end hasn't
33:39
changed much, but a year ago, there was
33:41
no sense that the inflation was getting
33:43
so much out of line.
33:46
It happened a little later than that.
33:47
There was some concern that interest
33:49
rate would would rise,
33:51
but but but now it's very clear the
33:53
economy is overheating. And this I
33:55
should have plotted you something for
33:57
for
33:58
a month ago. It would have been even
34:00
steeper. Okay.
34:03
Anyways, but that's because the Fed is
34:05
trying to slow down the economy. It's
34:06
hiking interest rates. That's the reason
34:08
the curve is very very inverted today.
34:12
So, let me let me calculate these rates.
34:14
How do we go about it? So, the first
34:16
thing we're going to do is we're going
34:17
to use the expected present discounted
34:19
value formula to calculate the price
34:22
of a bond.
34:24
And then we want to start
34:25
doing it for different bonds,
34:27
and we're going to construct uh the
34:29
yield curve.
34:30
So, suppose you have a bond that pays
34:33
$100,
34:35
nothing in between, $100 1 year from
34:37
now. So, this is a bond with maturity
34:39
1-year maturity.
34:41
I'm going to call that bond with 1-year
34:43
maturity
34:45
P1 the price of a bond with a 1-year
34:47
maturity at time T, P1T.
34:50
Well, that's easy to calculate. It's
34:51
expected present discounted value for If
34:53
you have the interest rate, whatever you
34:55
say, 1-year interest rate, then I know
34:57
that the price of the bond is 100
34:59
divided by 1 plus the interest rate, the
35:01
1-year interest rate today.
35:04
Okay. That's the price. That's expected
35:06
discounted value. So, I tell you what
35:07
I'm showing you is the relationship
35:08
between interest rates and prices.
35:11
Okay. Our price of a bond. The price of
35:13
that bond is just 100
35:16
uh divided by 1 plus the 1 plus the
35:19
1-year interest rate today. Okay.
35:23
So, important observation is that the
35:25
price of a 1-year bond varies inversely
35:28
with the current
35:29
1-year nominal interest rate. This is
35:31
all nominal, huh?
35:34
Why is it an inverse relationship?
35:36
Why is it the price of a 1-year bond is
35:38
inversely related to the 1-year interest
35:40
rate?
35:44
In other words, I'm asking
35:46
what do you think happens to the price
35:48
as a nominal as a nominal interest rate
35:49
rises?
35:50
And why do you think that's what happens
35:52
to the price?
35:55
Well, the first question doesn't have a
35:57
I mean, it's very easy, you know, the
35:58
answer to the first question. What
35:59
happens if I goes up? Well, it's obvious
36:01
that this price comes down.
36:03
But why?
36:08
And and and I'm And you use the concept
36:11
we have developed here. Remember we
36:12
spent
36:13
like 20 minutes in one slide. Well, you
36:15
start the slide for that answer.
36:22
Hint.
36:24
This $100 you're not receiving today,
36:26
you're receiving a year from now.
36:28
What happens with a dollar received a
36:30
year from now?
36:31
What is the value of a year
36:33
dollar received 1 year from now when the
36:35
interest rate is high?
36:39
Slow, because, you know,
36:41
you'd much rather have the dollar today,
36:42
invest it, and get this big return on
36:44
the on on the dollar.
36:46
That means, naturally, a bond that is
36:48
paying you $100 tomorrow is going to be
36:50
worth less
36:51
when the interest rate is very high.
36:53
It's going to be worth less today when
36:54
the interest rate is very high. You'd
36:55
rather have the money today, invest it
36:57
in the in in the interest rate, and get
36:58
the interest rate.
37:00
And and uh
37:02
No, I need to invest 1 over 1 plus I1T
37:05
dollars to get $100. That's another way
37:07
of saying it.
37:10
What about with the bond that pays $100
37:13
in 2 years?
37:15
Well, I need to discount that by this,
37:17
which is a You know, it's a product of
37:19
the two interest rate. And since I don't
37:21
know what the 1-year rate
37:23
will be 1 year from now, I have to use
37:25
expectation here rather than the actual
37:26
rate. But look at the notation. I'm
37:28
calling
37:30
P2T,
37:31
dollar P2T, the price of a 2-year bond,
37:34
a bond with maturity of 2 years,
37:36
as of time T.
37:39
Okay.
37:40
And this is a bond that has no coupons.
37:41
So, yes, pays you $100
37:44
at the end of the 2 years.
37:47
Now, note Note that this price
37:50
is inversely related to both
37:53
the 1-year rate today
37:56
and the expectation of the 1-year rate 1
37:59
year from now.
38:01
If either one of these goes up,
38:04
the bond is worth less today. You
38:05
discount more a dollar received
38:08
uh
38:09
um
38:10
2 years from now. I don't care which
38:12
one.
38:13
You know,
38:14
either of them that goes up is is bad
38:16
news for the for the price of a bond.
38:19
Okay.
38:24
Is this clear?
38:28
So,
38:29
there's an alternative So, this is the
38:31
way you price a bond bonds using just
38:33
expected discounted value
38:35
uh approach. Now, it turns out that in
38:38
practice, a lot of the asset pricing is
38:40
done by arbitrage. Meaning, you you
38:42
compare different assets, and that that
38:45
have similar risk, they should give you
38:47
more or less the same return. That's
38:48
what you do. So, let me let me do this
38:51
arbitrage thing. Suppose you're
38:53
considering investing $1 for 1 year. So,
38:56
that's your decision. I'm going to
38:57
invest one I need I have a dollar, which
38:59
I want to invest for 1 year.
39:02
But I But I I have two options to do
39:04
that. I can invest a dollar in a 1-year
39:07
bond.
39:08
I know exactly what I'm going to get,
39:10
you know, in that bond.
39:11
Or
39:13
I can invest in a 2-year bond
39:16
and sell it at the end of the first
39:17
year.
39:18
That's Those are two ways of, you know,
39:20
investing for 1 year.
39:23
Arbitrage has to be compared over the
39:24
same period of time and everything. It's
39:26
not the return of a bond that you hold
39:27
for 10 years versus one that you hold
39:30
for a 1 year. It has to be something a
39:31
similar investment. Suppose I need to
39:33
invest for 1 year.
39:36
Or you know, then then Okay, then if I
39:39
have these two bonds, the option is not
39:43
buy one or the other and then hold to
39:45
maturity because that would be comparing
39:46
an investment of 1 year with an
39:48
investment of 2 years.
39:49
I need to compare the strategies of
39:52
getting my return in 1 year.
39:54
In the 1-year bond, that's trivial
39:55
because I get my return at the end of at
39:57
the maturity of the bond. In the 2-year
39:59
bond, it means I need to sell it in
40:01
between after 1 year. Okay? So, those
40:03
are the two strategies I want to compare
40:06
and since I'm not take
40:08
considering riskier as a central
40:10
element,
40:11
those two strategies are going to have
40:13
to give me the same expected return.
40:15
Okay? That's arbitrage. That's what we
40:17
call arbitrage.
40:18
Okay?
40:19
Two
40:20
the two strategies have to give me the
40:23
same expected return.
40:26
So,
40:28
what do we get from this strategies?
40:30
Well, if I go through the 1-year bond, I
40:32
know I'm going to get my dollar times 1
40:34
plus I1T. That's what I get off a 1 year
40:37
out of investing a dollar in a 1-year
40:39
bond.
40:40
If I go through the 2-year bond
40:42
strategy, buy it and sell it at the end
40:44
of the year, then I'm going to get I I I
40:47
I
40:48
invest a dollar today,
40:49
no? I'm going to pay P2T.
40:52
That's what I paid today for a 2-year
40:54
bond. That's what I pay here for a
40:56
2-year bond and I expect to get the
40:59
price of a 1-year bond 1 year from now.
41:02
I mean, the 2-year bond will be a 1-year
41:04
bond
41:05
after a year has passed.
41:07
No?
41:08
It's a 2-year bond today, but
41:10
after 1 year, it's going to have only 1
41:11
year to mature.
41:13
So, that's the reason the price I need
41:15
to forecast is the is the price of a
41:18
1-year bond 1 year from now. That's what
41:20
this is here.
41:21
Okay? And that's my return on this
41:23
strategy because I'm going to pay this
41:25
today,
41:26
these dollars,
41:28
and I expect to get that 1 year from
41:30
now.
41:30
Okay?
41:32
So, arbitrage means I need to set these
41:35
two equal.
41:38
Okay?
41:42
So,
41:44
that means I have to get the same return
41:46
with the two strategies. That means I'm
41:48
investing the same, so I only need to
41:49
compare the the the the
41:52
the returns. This needs to be equal to
41:54
that.
41:57
That's what I have here.
42:00
Which tells you
42:01
that you're solving from here that the
42:03
price of a 2-year bond at time T
42:06
is equal to the expected price of a
42:08
1-year bond at T plus 1
42:11
discounted by 1 plus the 1-year interest
42:14
rate.
42:15
No?
42:16
This was like my cash flow.
42:18
My cash flow now is not the cash flow.
42:20
It's It's just a price. I'm going to get
42:22
a price for that asset. That's like the
42:23
Zs I had in my formula. Okay?
42:27
And for a 1-year strategy, I only need
42:29
to worry about the ZT plus 1.
42:31
There was no dividend at day zero.
42:34
Okay? And that's exactly that formula.
42:37
But notice that at T plus 1,
42:41
that will hold.
42:43
No? So, at T plus 1, I'm at T plus 1, I
42:45
don't need expectations. I know that P1T
42:47
plus 1 will be equal to 100 divided by 1
42:50
plus I1, the 1-year rate at T plus 1.
42:55
Therefore, the expected is something
42:57
like this, approximately. The expected
42:59
price is something like that.
43:01
Okay? I expect
43:04
I mean, this will be without the E will
43:06
be the price
43:07
of this 1-year bond at T plus 1. I don't
43:09
know exactly what the interest rate will
43:11
be next year, so I have the best I can
43:13
do is have an expectation. That's my
43:15
expectation, approximately.
43:17
Okay? But now I can stick this
43:19
expression in here.
43:22
No? I have this.
43:24
I'm going to go out and I can stick that
43:26
in there
43:28
and I get this expression. So, that's
43:30
the price for the 2-year bond.
43:32
Do you recognize this?
43:37
You saw it before.
43:46
You know?
43:47
That's the same expression that we got
43:49
when we used the expected present
43:50
discounted value formula.
43:52
Right?
43:53
We said, "Well, I'm going to get the
43:54
$100 100 years a 100 years a 2 years
43:57
from now. I know that discount factor
43:59
for that is 1 over 1 plus I1T times 1
44:02
plus I1T plus 1 expected."
44:05
Well, that's what I got.
44:08
That's from arbitrage.
44:09
Okay?
44:10
From an arbitrage logic. This is used a
44:12
lot in finance.
44:15
I I I'm going to say something
44:16
complicated, but but um
44:21
just ignore it if it's
44:26
uh
44:26
not really up for the for the for the
44:28
quiz or anything, but
44:30
you know, there's a big debate in the US
44:32
today about uh not big debate, a big
44:34
concern about
44:36
uh
44:37
the the
44:40
the US Treasury debt because there is a
44:42
debt ceiling, meaning there's a maximum
44:45
amount that the government can
44:46
of debt they can issue.
44:48
And and the and that ceiling has been
44:52
moved over time, but every time we get
44:54
close to a deadline when this needs to
44:56
be agreed again, there's a concern and
44:58
there's negotiations and so on.
45:00
And the and the
45:03
and the well, I mean, everyone at this
45:05
moment at least thinks that
45:07
as every as in every instance in the
45:09
past, they're going to reach some sort
45:11
of agreement the day before
45:14
of the deadline or not.
45:16
But if they don't and there is a mess,
45:18
this is huge for finance. It's huge for
45:20
finance because US Treasury bonds,
45:22
especially short-term bonds, are used
45:25
for pricing everything
45:26
through arbitrage and so on.
45:28
So, you get a mess there,
45:30
that's a mess in every single financial
45:32
market. You wouldn't know how to price
45:34
many financial assets, actually.
45:36
So, it would be a disaster. But uh
45:39
but the reason I describe I mention this
45:42
here is because
45:43
again, lots of prices are priced in
45:45
reference in as in finance are priced in
45:47
reference, especially derivatives,
45:49
options, and stuff like that.
45:50
Uh you price them relative to something
45:52
using this type of logic. So, if the
45:54
thing you use as a base as a reference
45:57
becomes highly unstable and uncertain
45:59
and risky, then obviously everything
46:01
becomes very complicated,
46:03
very risky, and and financial markets do
46:05
not like risk. That's for sure.
46:09
Anyway, ignore that. That's
46:11
irrelevant for your quiz, but that's the
46:13
reason this the whole discussion then
46:15
over the summer can get to be very very
46:17
tricky for finance.
46:21
So, the yield to maturity, remember I
46:22
mentioned this concept before, of an
46:24
N-year bond,
46:25
but it's also what we When you see
46:27
Whenever you hear the 3-year rate,
46:31
is that. It's the yield to maturity.
46:34
Uh which is different from Okay, let me
46:36
tell you
46:37
show you a formula that's easy to
46:38
explain then.
46:40
And it's defined, it's important, as the
46:42
constant
46:44
annual interest rate that makes the bond
46:45
price today equal to the present
46:48
discounted value or expected discounted
46:50
value.
46:51
So, notice notice the highlighted
46:53
is defined as the constant annual
46:56
interest rate
46:57
that makes the bond price today equal to
46:59
the present discounted value of future
47:01
payments of the bond.
47:03
Okay?
47:04
So,
47:06
for example, in our 2-year bond,
47:09
that's the price. Right? This is the
47:10
price of the asset.
47:12
We that we know the price. We already
47:14
got the price from the previous slides
47:16
of the bond,
47:18
which was based on the short-term
47:19
interest rate, 1-year interest rate, and
47:21
our forecast of the
47:23
short-term the 1-year interest rate 1
47:25
year from now.
47:26
I know that price. Take that as a
47:28
number.
47:29
So, then I the yield the yield to
47:31
maturity is calculated as that constant
47:34
interest rate
47:35
constant How do I see this constant?
47:37
Because, well, I'm using the same
47:38
interest rate for the first period and
47:40
the second period. I And now I'm calling
47:42
it I2T. It's a 2-year interest rate, but
47:44
it's constant. Constant doesn't mean
47:46
that it doesn't move over time.
47:48
It means I'm discounting all the cash
47:50
flows as a constant interest rate. This
47:51
It means I'm using
47:55
I'm using this equation.
47:57
Okay?
47:58
So, the yield to maturity is
48:00
find an interest rate
48:02
that allows me to use this constant
48:03
thing constant assume use this formula
48:08
and get back the same price
48:10
as I got by using the the
48:13
the expected discounted value or the
48:14
arbitrage or something like that. Okay?
48:17
So, that's that's the definition. Okay?
48:19
You have this price.
48:21
Now you you look for that interest rate
48:23
that allows you to match that price.
48:26
Okay?
48:27
And that's called the yield. That's the
48:29
thing I Remember I plotted this the some
48:32
curves?
48:33
Well, those those interest rates in
48:35
those curves were computed this way.
48:39
Now,
48:40
notice that we know what this price is.
48:43
This price is by the expected discounted
48:44
value or the arbitrage approach is equal
48:46
to 100 divided by this.
48:49
So, I know that these two things are
48:51
this is equal to that,
48:54
which means that this denominator is
48:56
equal to that,
48:58
and that implies for a small interest
49:00
rate that this 2-year interest rate is
49:03
approximately equal to the average of
49:06
the expected interest rate 1-year rates.
49:09
Okay?
49:11
So, this is called actually the
49:12
expectation hypothesis, by the way. Is
49:15
that the the 2-year rate
49:17
is approximately equal to the average of
49:19
the 1-year rate this year
49:21
plus the expected 1-year rate 1 year
49:24
from now.
49:26
Okay?
49:30
So, that's an important concept. And I'm
49:31
going to start from here again
49:34
in the next lecture.
49:48
Mhm.
— end of transcript —
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