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visit MIT OpenCourseWare at ocw.mit.edu Let me begin by first
asking whether there are any questions from last
class, which was a week ago. Hope you had a good break. Any questions? OK. Before we begin today's
topic-- question? AUDIENCE: No. ANDREW LO: No. Before we begin today's
topic on arbitrage and the pricing of multiple
fixed income securities, I want to just
take a few moments to talk a bit about what's
going on in financial markets, also to welcome the prospective
students that we have sitting here in class today. So for once, over the
weekend, unprecedented things didn't occur. And so I'm glad to report
that we're still here. Financial markets
are still around. And as you know, the government
has proposed some measures to deal with this
current financial crisis. And at this point,
it's still unclear as to what they're proposing. But we can actually
see from the data what the market's reaction is. Last time, remember, we
looked at the yield curve and literally, it was just
a week ago that the yield curve looked like this. Now remember that
we focused on what happened at the very short
end of the yield curve, which is three month
Treasury bills. And last week when we
looked at this graph, the yield was about three
basis points for a three month Treasury bill. And we pointed out that
that was telling us something about the market. In particular, it was telling
us that the market is panicking. Yeah? AUDIENCE: So aren't we sort
of in this whole situation because looking at the market
grossly mispriced things? ANDREW LO: Well, I
wouldn't say that it was looking at the market
grossly mispriced things. AUDIENCE: The market
obviously did not officially price this risk, which
looking at the market a year and a half ago, we would
have been, oh, here's this much risk, this
criteria [INAUDIBLE]. It turned out to be
right, totally right. ANDREW LO: Well, there are
a number of things that are priced into a security. It's not just a risk, but
it's also a reflection of supply and demand, right? So in other words, what's
going on here-- the question that we want to answer
from looking at the price is what do we know about what's
going on in the marketplace based on that price. What is it telling us? The cost of borrowing over
a three month period-- when it goes down to
three basis points, that's telling you that
the price of that security, the price of Treasury bills is
extraordinarily high relative to historical standards. Now let's take a
look at what happens more recently, in
particular, today, if we go to any of
these websites-- so in particular, let's go
back to the Bloomberg site where we originally
looked at this. First of all, this is
now the yield curve. And it's hard to
compare because I've got a different
slide for last weeks. This is just yesterday's,
the orange, not last week's. But the one thing you'll note is
that at the very short end, now instead of three basis points,
the three month Treasury is yielding 41 basis points. What does that tell
you about the price? Now, so Treasury securities,
short-term Treasury securities, have declined in price
over the last week. And that's one sign that perhaps
markets are not as panicked as they were last week. There isn't this mad rush to
get into Treasury securities in the short term. All right, so short
term means have gone up. Yeah. AUDIENCE: So would
you say that this would be a lot about the
psychology of [INAUDIBLE]. ANDREW LO: Yeah. AUDIENCE: So what I was
wondering is when the fact-- it seemed like, as you said,
there's a flight in liquidity last week. Now why was there
such a huge movement, or what led to the price
dropping so much when, I guess, it could be reasonably
expected that the price would come back up and so
people would just wait it out and take a shot. ANDREW LO: Which price are
you talking about dropping? AUDIENCE: I should probably
say the yield dropping. ANDREW LO: The yield dropping
and the price going up. Well, so there are a
number of factors at play, but the current
perspective that most of us have in financial
markets about last week-- and this is just perspective. Remember, last week
is not that far away. What happened last
week, by most accounts, is that there was a
very significant rush to the exits by investors. By rushing to the exits, I mean
getting out of risky securities and into safer securities. And at this point,
it doesn't seem like there's much
of a safe haven other than US
Treasury securities, and in particular,
short-term securities because you know you can get
the money out over a relatively short period of time. So that's what the yield
curve told us last week, that three basis points means
that basically people didn't care about the yield. All they wanted to do was
to get into US treasuries at almost any price. OK. This week it's different. In particular, not only has
the short-term yield gone up, so now instead of three
basis points, we're up to 41. But look at the long
end of the yield curve. Before, the long end
of the yield curve-- let me just go back and remind
you what that looked like. At the 30 year, a week ago,
the 30 year yield was 4%. Let's take a look
at what it is today. It's now, the 30
year yield, 4.37. That's another big movement. Why is that? Why would the yield for
the long-term bond go up? What is the market
thinking today? AUDIENCE: They might be more
worried about inflation. The government has
promised $700 billion. ANDREW LO: OK, so
inflation has now been incorporated, just
over the last seven days. So question-- is
this price correct or was last week's
price correct? Getting to your point,
I mean, what do we do? Is the short end of the
yield curve appropriate today at 41 basis points, or was it
really appropriate at three basis points? There's no answer
to that question because there is
no right answer. These prices are a reflection
of the current expectations of all the market participants. Right or wrong,
it really reflects the combined either
wisdom or fear or greed of the marketplace. And so our approach is to try
to understand what that is. We want to explicate
the information that happens to be in
prices, but you have to understand that these are the
same imperfect kind of prices that we came up with
on day one when you bid for that little package. And it turned out
that you got lucky and got an iPod
for whatever, $45. But it could've
gone the other way. And in fact, , in the second
class it did go the other way. So we won't talk about that. The prices reflect all aspects
of the economy, the rational as well as the irrational. And so last week, was
it irrational for people to pull their money out from
all sorts of investments and put them into
Treasury bills? Well this goes to the heart
of why the Treasury acted so quickly, and why Chairman
Bernanke has said that he wants to get a quick resolution. Something very significant
happened last week. And I don't know how many of
you really got wind of it. Certainly, the Treasury
knew what was going on and the Fed did, but it
wasn't really highlighted in the newspapers
in the way that I would have thought
it should have been, given the importance. Anybody know what
I'm talking about? Yeah. AUDIENCE: Stopping
the short sell-- ANDREW LO: Well, that
was one piece of news. The SEC mandated that
for a period of time, to be determined,
we are not allowed to short sell financial
stocks because they wanted to stop the kind
of run that there has been on these securities. I'm going to come
back and talk about it in the end of this lecture
because we're going to talk about short sales. But that's not what
I was referring to. That's certainly a concern, but
that's not the major concern that I think the market
was responding to. Yeah. AUDIENCE: Was it the $8 billion
of redemption in money funds? ANDREW LO: That's right. Where was that coming from? What was going on with that? Why did that happen? AUDIENCE: Because people
are losing confidence in the short-term
debt securities to put in money market funds. ANDREW LO: And
was there a reason for that loss of confidence? I mean, money funds, what does
that have to do with mortgages and Lehman and Goldman? What's the connection? AUDIENCE: Well, I
just think more of it was a psychological reaction
than it was an actual-- ANDREW LO: Absolutely, it
was a psychological reaction, but did something trigger that? Is that psychological reaction
completely unreasonable? If your grandmother asked
you what she should do now with her money market
fund, should you tell her, don't worry about
it, just stay the course and see what happens? Something happened last
week that is related exactly to that issue. So you're on to something. What it that? Megan. AUDIENCE: Well, one of the major
money funds broke the buck. ANDREW LO: Broke
the buck, exactly. Which money fund? Do you remember? AUDIENCE: Lehman, no, no. ANDREW LO: The Reserve Fund. AUDIENCE: It's the Reserve. ANDREW LO: The Reserve
Fund is one of the first, if not the first,
money market funds. And what is a money market fund? Do we know what that is? You all know what that
money market fund as? You all probably have money
in a money market fund. Whether you know it or not. Money market fund is
a fund that contains relatively short-term
and supposedly riskless securities, like
CDs, treasuries, and other kinds of
very, very safe assets. And what does it mean
to break the buck? AUDIENCE: That for every
dollar that people invested in money market funds had
invested what they would be able to redeem at that time. ANDREW LO: Right. AUDIENCE: --less
than already spent. ANDREW LO: Right. Breaking the buck means
that when you put in $1, money funds are supposed to be
so safe that at the very least when you withdraw the money,
you're going to get $1 back. Breaking the buck means
that if you withdraw, there is a possibility that what
you withdraw is less than $1. Now that's scary because
think about a bank-- when you put your money into
a checking account for a bank, you expect to get
that money out, maybe not with a lot of interest,
maybe even with no interest if things don't go well, but you
expect to get what you put in. You expect to get the
principal back, right? Well, money market funds
are very much the same way. People use them as if they
were checking accounts. In fact, there are
money market funds where you can write
checks on them, right? And so breaking the buck
has been a major concern, not just among the money
funds, but among regulators. Because if it turns out
that retail investors, ordinary consumers are
scared about what's going on with their
money market accounts, they will do in mass
what happened last week, which is pull out
huge sums of money from these money market funds. And as I mentioned
earlier, no business can sustain a massive withdrawal
of all capital all at once. It's just not possible
for a business to be able to be
conducted in that manner. If that happens, we
will see mass failures of financial institutions that
will make the last four weeks look like the good old days. And that's what the
Fed is concerned about. That's what the Treasury
is concerned about. And so the hope is that the
measures that they put in place will calm the fears
of the public. That's the first
order of business. It's calming the
psychological kinds of effects that these headlines
have produced. And so the hope
is that once they put these measures
in place, that will take care of the concerns. What they've done is to propose
to guarantee money market funds the same way that the
FDIC guarantees your banking accounts. And the way that the SIPC
guarantees your brokerage accounts. And there are other measures
that have been proposed. We won't have time to
talk about them here, but over the next
couple of weeks, the finance group
at the Sloan School will be putting together some
kind of a panel discussion that will focus exactly
on these issues. So we'll let you know
when that happens. And we'll take on these issues
head on in that session. OK, but it looks like
for the moment, at least from the yield
curves that we saw, that things are
actually quieting down. We'll see on a day by day basis. So this is obviously last week. This week, we have yields
going up a little bit, so that suggests
that there isn't the same kind of pressure. But every day is another
day and with another set of revelations. So by looking at these
pieces of information, we can actually glean what
the market is thinking. Is it right? Of course not. All forecasts are
by construction incorrect to some
degree, but it's a window on exactly what's
going on in the marketplace and what people are thinking. Yeah. AUDIENCE: Sorry, what's the
connection between this money market issue and
the yield curve? Could you make the
connection again? I missed it. ANDREW LO: Yeah, sure. So the money market
concern is that what people thought were
safe, apparently is not as safe as people thought. And the Reserve Fund breaking
the buck-- by the way, breaking the buck
in that case meant that if you put in a
dollar, when you withdrew, as of last week, you
would have withdrawn $0.97 so you'd lost $0.03
to the dollar, which may not seem like much, but if you went
to your Bank of America ATM and you did a withdrawal. And for every dollar you put
in, you'd get $0.97 back, you'd be pretty
ticked off, right? So it's something that
is of great concern to retail investors. Anyway, so what happened
last week was that-- actually the estimate, I think,
is $90 billion. $90 billion of money came out of
these funds in a week and were put into either
cash in the mattress or were put into
Treasury securities like the three month T-bills. That's what pushed the
prices of those T-bills up and therefore
depressed the yields. And now we're back to a
somewhat more reasonable level. I say reasonable because having
this kind of a short-term yield of 40 basis points by historical
standards is still pretty low. So there are still many
nervous investors out there that are trying to figure
out what's going on and are waiting for the Treasury
to come up with something. This is another reason
why Chairman Bernanke said we have to act quickly
because markets are not going to stand and wait
for the Treasury or the Fed to do something. Markets will react. And if we wait too long, the
fear of this breaking the buck could actually return. And then once you
have a mass panic, it's very, very
hard to stop that. Anybody who's ever seen one of
these old animal kingdom type of movies about a stampede-- if you've got water
buffalo stampeding, it's pretty hard to try to just
say, oh, calm down, stop it. Slow down. You can't easily do
that once it begins. So you've got to stop
it before it actually gets to that critical point. And that's exactly what the
government is trying to do. Yes. AUDIENCE: When you say that--
if you interpret these, like the prices went down so now
there's less demand so it means people are more relaxed-- ANDREW LO: Yeah. AUDIENCE: What if you say, like,
if the interest rate went up, it means they are considering
the Treasury bills to be more risky or riskier? ANDREW LO: Well, remember
that Treasury bills don't have any default risk,
at least as far as we know. We have to be careful
about stating. All these unprecedented
things have happened. The reason that Treasury bills
don't have any default risk is because what the
Treasury security is an IOU from the government
that says I owe you a certain number of US dollars. And because the Treasury
owns the printing press, they can always print out more
dollars to give it to you, as long as you're
willing to take it. And at least from this graph,
it seems like a lot of people are willing to take it. They really want
Treasury bills right now. And they're happy with that. Maybe they're not happy
with it, but that's the smallest of all the evils
that they can think of in terms of putting their money. AUDIENCE: So decreasing
the yield in Treasury bonds do not mean an increase
in the default risk? ANDREW LO: We hope not. I mean, I guess it
could be possible that people are betting
that the United States is going to default in 30 years. But my sense is that
what's more likely, given that these are default
free in the sense that their nominal bonds-- so these bonds are
going to be paid off in the little
certificates called US dollars that the
printing presses can always come up with. There's no risk that they
can't print up more dollars. The risk is that $1
30 years from now isn't going to be
worth as much as we thought it was
going to be because of inflationary expectations. So as of today, that 30 year
yield is not 4%, it's 4.37% and the 0.37 one could attribute
to an inflationary expectation by the marketplace. AUDIENCE: Do you think that the
short-term change in the yield has anything to do with
the recent devaluation of the dollar against
foreign currencies? ANDREW LO: It could be that
because of that devaluation, dollars are cheaper
and people are putting more money into US securities. That's also a possibility. But another way of putting
that is that foreign investors are now finding treasuries more
attractive for whatever reason. So yes, that's also part of
that supply and demand story. OK, last question. Yeah. AUDIENCE: Historically,
what is a reasonable yield? You mentioned that before,
like [INAUDIBLE] or before all this time? ANDREW LO: Well, I'm glad you
asked that question because we have a graph. These are the historical yields
of the three month, six month, one year, two year, five
year, 10 year, and 30 year from 1962 to I think it's 2004. And it depends on what
flavor you're looking at. And it's kind of hard to
read this graph because of the colors. But if you look at
the dark blue line, they actually all move
together pretty much. But at one point, that
short-term yield was 4%. 4% for a three
month Treasury bill. Now these are all
annualized remember, so 4% doesn't mean
4% over three months. It means 4% on an
annualized basis, which is why 41 basis points-- that's an annualized yield-- for a three month loan
just seems ridiculously small by historical standards. But when people are scared
about not getting paid, that kind of fear, that
psychological pressure can be overwhelming. And do you believe
in these prices? Does it make sense? Well, that I'm
hoping to get you not to ask the question in
that way, but rather to ask the question, given
market prices and what I know about it, what can I
interpret from what's going on and how does that
affect me in terms of the financial decisions
that I want to make. So if you are thinking about
pricing other securities based upon these kinds of
numbers, you need to ask yourself
whether you believe the numbers make
sense or are they just completely out of whack. And the only way to do
that is to understand what the basis is for these numbers. So that's where we're
going next in trying to understand how to measure
the various different characteristics of these
numbers to get a sense of what's reasonable and what's not. Question. AUDIENCE: I had a question. I was wondering, those sort
of funds, Treasury bills, [INAUDIBLE], is that really
individual people buying the Treasury bills or is it
more hedge funds and that sort of thing that are systematically
hedging the risk against other securities they have that are-- ANDREW LO: Well, obviously,
it's very difficult to tell because we don't see
who's making the purchases and sales, but we can tell
from certain mutual funds and other money
market flows that it seems like most of the
flows last week came not from hedge funds, but rather
from retail investors that were taking their money out
of these money market accounts and then putting them into
certain mutual funds that buy only Treasury securities,
as well as Treasury securities directly. All of you can actually purchase
Treasury securities directly. There is a website
called TreasuryDirect.gov and you can give
them your credit card and register as a
user and actually participate in Treasury auctions
and buy Treasury securities. So it was a
combination of those. But it really seemed like
it was the retail sector, not institutions, not
sophisticated hedge funds that we're trying to do
some kind of complex arbitrage. It was just investors
saying, I'm really scared, I want to put my money
into something that's real and that will be there. And so short-term treasuries
seemed like an answer. And as we saw from last week,
gold was the other answer. It's not an answer
that I would propose for the typical investor because
gold prices are quite volatile. And so you have
to be very careful when you make an
investment in that security or in that particular asset. But it is something that
reflects the state of panic of the marketplace. And by the way, I'm telling
you something that you probably already know in the sense that-- my guess is that deep
down inside, all of you are feeling stressed out, right? I mean, you're
probably stressed out about things like what does
this mean for the job market, for career prospects, and so on. I would urge you all
to take a deep breath and not get panicked about
that because, as I said, this is the kind of
dislocation that, while very traumatic for market
participants today and for those on the
losing end, there are as many
opportunities created as there are taken away. And so my guess is
that in a year's time, the job market is going to look
extraordinarily attractive, particularly for
those individuals that are trained in the science
and art of financial analysis. So you're all going to be
very well equipped for that, even though you may
not feel that way right now because of what's
going on in the marketplace. So I wouldn't panic certainly. And by the way, you can
see the opportunities that are already being created. Warren Buffett just
spent $5 billion purchasing a stake
in Goldman Sachs. And we're going to talk about
that in a couple of lectures when we do common
stock because I want to use it as an example of
getting a good deal in markets. I mean, first of all, Warren
Buffett given the success he's enjoyed as an investor, you
know that when he plunks down $5 billion in
cash, he's probably doing it for a good
reason, not out of charity. And by the way, Goldman
raised another $5 billion from additional rights issues. That's $10 billion
of capital that they raised relatively quickly. Also Nomura is in
the negotiations to purchase certain
assets of Lehman Brothers. Lehman has a terrific
franchise and has some very significant operations
in Asia, as well as in the US. And it's a very smart
move on Nomura's part to take advantage of that. So these are the
kind of opportunities I'm talking about. And when Nomura
buys Lehman, they're going to have to
hire people in order to run the operations
because you can bet that the whole
dislocation ended up shaking loose a number of
very talented professionals from those organizations. So they got to hire. It's going to mean the
next two or three months, there may be some
difficulties in getting the attention of
these organizations because they're in the midst
of trying to figure out exactly what kind of organization
they are going to have when all the dust settles. But there are plenty
of opportunities that are being created
today, including, by the way, the opportunity for
the US government to take advantage of all
of these distressed assets. So one of the things that you
should be careful about when you read that there's
a $700 billion bailout, that is somewhat
misleading in the sense that, first of all, we don't
really know at this point exactly what the $700
billion will be for, how it will be used,
or how it's dispersed, or is it really $700 billion. A lot of it depends upon
how the money is spent and also what happens
to housing markets. There is a scenario
that I can imagine where the actual amount expended
is either zero or negative. In other words, the government
actually makes money from the current state of
the markets because they can buy assets very cheaply,
hold onto them forever until they pay off, and
then gain the kind of profit that the original financial
engineers were expecting but could not take advantage
of because of this liquidity crunch. So we'll talk about that
over the next few lectures because we're going to
develop some techniques to be able to illustrate how these
kind of arbitrage strategies work. Yeah. AUDIENCE: Thinking about
Nomura, is it more certain that it's going to happen
or are there many plans that are trying to
buy the [INAUDIBLE]. ANDREW LO: Well, certainly
there are a number of banks. I wouldn't say many,
simply because there aren't that many banks that
are well capitalized enough to be able to take on a
large unit of a business as big as Lehman Brothers. So Nomura is one of a handful
of banks that are engaged, but they seem to be the
front runner at this point. AUDIENCE: Generally,
what I've heard kind of from the Asian
analyst in Lehman that I had spoken to is that
it's quite likely that that might be not be true
as well because we find a lot of [INAUDIBLE]. ANDREW LO: Oh, absolutely. There are a number of
issues that will come up in any kind of a deal. And so you don't
know whether or not something is going to go through
until it actually goes through. The same thing could be
said for what happened with Merrill Lynch, with AIG. All of these deals are
sort of put together at the last minute. And either they get
consummated or there's some hitch at the end
that makes it difficult. So yeah, I mean,
with a grain of salt, you should take all
of these news reports. And literally until
the deal is signed, you will not know whether
or not something's on. But the point I'm illustrating
is that these assets are not completely worthless. What's happened is a very
significant liquidity crunch and panic. When that happens, the
pricing of all these assets becomes questionable. OK. We're going to actually
see an example of that. So if you wouldn't mind, let me
put that off for a few minutes. And then if you have further
questions about this, we can come back to it. Let me start-- so
this is lecture six. And what I want
to do is to start where we ended last time, which
was a discussion of coupon bonds and how you price
coupon bonds simply as a package or a portfolio
of pure discount bonds. Underlying this approach
to pricing coupon bonds is a very important principle. That's a principle
that was given to you in the very
first day of class where we talked about the
six fundamental principles of financial markets. And it's the principle
of the law of one price. So what I want to do
today is to focus on that and talk about the
law of one price and what it means for things
like arbitrage, leverage, short selling, and
relative pricing. Those are the key concepts
we're going to cover today. So let me talk about
the law of one price and remind you all what it is. It's a very simple idea. It's so simple that you
might think it's obvious, but it's got some very,
very dramatic implications. The law of one price says that
two identical cash flows must have the same market price. OK. Let me repeat that. Two identical cash flows
must have the same price. Now remember that when
we think of an asset. We think of an asset as just
the sequence of cash flows. That's what an asset is. So all I'm saying is that when
you have two identical assets, they have to have
the same price. That's not a very
controversial statement. And this principle is one
of the most important ideas in all of modern
finance because it leads to the pricing of
all sorts of securities, including all the
derivatives that have ever been priced on Wall Street. They use this idea of
the law of one price, OK. Yeah. AUDIENCE: But wouldn't
you have to qualify that by saying it's at equilibrium? ANDREW LO: No, no. I don't have to
qualify that at all. First, because this
is a free country and I don't have to do
anything I don't want to do. But more importantly,
it's because I don't want to restrict
it to an equilibrium. By equilibrium, you mean when
supply equals demand, right? I don't care about
supply and demand. Supply may very well
not equal demand. That's OK with me. This principle of the law of one
price, that two identical cash flows have to have
the same market price, the only assumption that
I need in order for that law to be true is that people
prefer more money to less money. And it's not even people. I just need one person in
the economy that prefers more money to less money. And I'm happy to volunteer
for that position, OK? Why is that? It's because if that law is
violated, if you can show me two identical cash flows that
sell for different market prices, first of all, tell
nobody but me about it. What I'm going to do is I'm
going to buy the cheaper asset, I'm going to sell the
more expensive asset. So as of today, I
make money, right, because I bought the cheap,
I sold the more expensive. That difference is
positive for me. And then I argue that
from that point on, I have no further risk and, in
fact, no further obligations. I can just forget about the deal
and take my money and spend it. Why? Because I've bought and
sold identical cash flows. So from that point
on in the future, all the cash flows cancel out. So I'm done. That's called an arbitrage, or
more technically, a free lunch. I've been able to create
money out of nothing. It doesn't assume
supply equals demand. It doesn't assume any kind
of mathematical formula for any kind of instrument. All it assumes is
that more people prefer more money to less. AUDIENCE: So that means so
that day when you auctioned the iPhone, then
the only difference between an open package
and the concealed packaged was that in one case
you basically did not give enough information. ANDREW LO: Right. AUDIENCE: And it wasn't
reaching a fair value and so-- ANDREW LO: Well, wait,
wait, wait, wait a minute. Well, when you say fair
value, that's a loaded term. What do you mean by fair value? It was a fair value,
given all the information that the market had. AUDIENCE: But it's not because
[INAUDIBLE] of that object. When the person buys
objects that's packaged, once they open it,
they'll know that the cash flow of that object is the same
as if it were not packaged. And they're going to
read the [INAUDIBLE], as we just mentioned. ANDREW LO: That's
right, but don't you think there's a difference
between the package wrapped and the package unwrapped? AUDIENCE: Yeah, one of them is-- I mean, it's just
like a factory which will produce the same stuff
but is valued at the market at a lower price than
what it would be-- ANDREW LO: Right, but if
one factory didn't tell you how it produced it and
another factory did. You think that they would
sell for the same price? AUDIENCE: The factories
wouldn't, but their future cash flow into revenue
could be the same. ANDREW LO: Only if it turns
out that, as a matter of fact, it is identical. But you don't know
that ahead of time. You can only price something
with the information you have, OK. So when I say two cash
flows are identical, I'm saying that we
acknowledge that the cash flows are, in fact, identical. And we know that
they're identical. If I put two packages upfront
in the room, one is an iPod and the other one is wrapped
so it looks like an iPod. It's got the same shape,
the same dimensions, but it's wrapped. Would you say that
they're identical? You can't know that. If you knew that, then you
would price it accordingly. By the way, that gives
you a very important piece of information. Suppose that I did
that experiment, I had the iPod that was clearly
unwrapped and it was an iPod. And then I had
another package that was the identical dimension
but it was wrapped, OK. And I auctioned them off and
it became clear initially that the two were priced
at about the same. Well, if you saw
that, you would then know that it was
quite likely that what was in the wrapped package
was the same as what was in the unwrapped
package, right? But the only reason
you would know that is because somebody
in the audience apparently bid the same price. Now why would they do that? Either they're knuckleheads
or they know something that you don't know. And now that you
looked at the price, you actually learn something
about what's in there. So what matters for pricing is
what the entire market knows, not just what you know, but
what the entire market knows. Now going back to
this arbitrage, let's not worry about
informational asymmetries. So let's assume that we all know
exactly what there is to know, which is that these two
securities have the same cash flow. If they have the same
cash flow, they've got to have the same price. And that's exactly what we saw
last time with this example. Same cash flows
and therefore, they have to have the same price. If they don't have the
same price, then instead of feeling upset and despondent
that somehow finance theory is in question, the most exciting
thing for a finance professor is to see that this
theory breaks down, because then we actually can
go transact in the marketplace and make money. So if the law of
one price fails, instead of calling me up
and complaining about it, you should call
me up and tell me what it is so I can take
advantage of it, all right? It's a great thing. And here's the example. This price of a
coupon bond is got to be equal to the prices of
the discount bonds multiplied by the number of bonds
you need in order to yield an identical cash flow. OK. Now if you have an
identical cash flow, the left-hand side has to
equal to the right-hand side. If it doesn't, if the
price of the left-hand side is greater than the prices
of the right-hand side, then you should feel very happy
because what you're going to do is to make money. Now how much money
are you going to make? Well, let's see. What you're going to
do is you're going to buy the right-hand side. You're going to buy
those bonds and you're gonna sell the left-hand
side bond, so what you make is that difference, right? Because you're buying,
you're spending a certain amount of money. That's the right-hand
amount of money. And the left-hand side is
what you're going to get. And so you're
actually going to be able to make that difference. But here's the key. When you make that difference,
it's not a risky investment. There is no risk. By the way, how much money
did you have to spend? How much of your
own personal wealth did you have to
commit to this trade? AUDIENCE: Just the risk. ANDREW LO: Just the risk? What risk? AUDIENCE: The difference
between them the moment they can buy and sell. ANDREW LO: Did you
have to spend that? AUDIENCE: You can guarantee it. ANDREW LO: That's
money that you get, not that you have to spend. Is there any money that
comes out of your pocket to do this trade? No, because what you
are buying is actually financed by what you sold. And then on top of that, you
have a little extra leftover, OK? So this is like one of these
infomercials at two o'clock in the morning, real
estate, how to make a million dollars in real estate
with no money down, right? You've put no money down. You haven't spent any money
because what you've done is you've sold the
bond and you've gotten this amount of cash. With that cash, you
can buy these bonds and how do you know you
have money left over? By assumption, I'm
assuming that this price is greater than that price. So now you have money left over. You've put no money down. You have cash in the pocket. You have no risk. You have no obligations because
all the future cash flows that you owe are financed
completely by the bonds that you bought. One for one, dollar
for dollar, it matches. So literally, as of today, you
walk away from this transaction a richer individual. Yeah. AUDIENCE: Professor,
isn't, like, by the time that you recognize a an
arbitrage opportunity, it corrects itself? ANDREW LO: Well, let
me get back to that. That's an interesting point. The question is, can
these things really exist, because once they
do, it's almost too late because they're gone. AUDIENCE: And then follow-up,
is it just my imagination or do arbitrage desks exist
in financial firms today? ANDREW LO: Arbitrage
desks absolutely exist in financial firms. And what they do is look
for this stuff all day long. That's what they do. And so is it true that by the
time you identify, it's gone? Well, it's true if you're
a finance academic, as well as a retail investor. But you guys, you're going
to go out in the job market and you get to be hired by
some of these arbitrage desks. So you're going
to be doing this. In fact, you're going
to be doing something quite a bit more complicated. I'll get to that in one minute. Before I do that,
I want to make sure that everybody is with
me about this arbitrage. Yes. AUDIENCE: Is there
any assumption that there is no
transaction cost? ANDREW LO: Very good point. Yes, in this context,
there's an assumption that you can buy and sell freely
with no transactions cost. Obviously, in the real world,
there are transactions cost. You've got to stick
those in and figure out whether you can still make
money with transactions cost. Now one form of transactions
cost is not a numerical cost, but it's a friction. What do you have to
be able to do in order to do this transaction? What financial operation
do you need in order to be able to get
this deal done? Yeah. AUDIENCE: Well, you have
to be able to short sell. ANDREW LO: Short sell, right. Remember we talked about
short selling last time. That's selling
something you don't own. So you have to borrow
the bond from somebody and then sell it and
get those proceeds. What if it were the case that
somebody imposed a constraint that you can't short sell? Who would ever do that? Well, we've seen what happens
in the marketplace that can generate that
kind of a constraint. What happens if you
can't short sell? Yeah. AUDIENCE: You need
to upfront the cost. ANDREW LO: What's that? AUDIENCE: You need to put up the
cost of the original security. ANDREW LO: Well, in fact, yes,
you need to put up the cost, but it really defeats the
purpose because if you're going to buy the bond
and then sell the bond, you've not really
done anything, right? This arbitrage argument
relies on the fact that you can sell
something you don't own by borrowing and shorting
it and getting the proceeds, and then giving it back to the
person you borrowed whenever they want, OK? If I don't allow
you to short sell, this argument
doesn't work anymore. And what that means is that
this pricing relationship, this left-hand side has to
equal the right-hand side, that relationship
goes out the window. For the next several weeks
and possibly several months, finance theory is
going to be on vacation because the government
has suspended short sales for certain securities. And so the kind of
force of markets that drive prices towards not
an equilibrium, but a pricing relationship that does
not depend on equilibrium, but depends on the
law of one price, that goes out the window
if you can't short sell. Yeah, Megan. AUDIENCE: I guess
I'm just wondering why the Treasury, just looking
for insight on this, why the Treasury wouldn't just
make the short interest on those short sales so high
that it was almost prohibitive, but people who really
wanted to short sell would, these instruments
to hedge portfolios, could do that if they
wanted to pay that. ANDREW LO: You know, that's
a fantastic alternative. I think that would
have been far better. You're absolutely right. Increase the cost
of short selling by raising borrowing costs. By the way, when you short
sell, as we said last time, it's not free. People aren't going to lend
you the security for free. They're going to
charge you for it. So what if instead of
forbidding it altogether, why not just triple the
cost or quadruple the cost? Therefore, only people who
really, really need to do it will do it. The problem with that is
more of a political one. The political problem is
that we want these evildoers, these short sellers
that are driving the prices of
financial securities to stop their bad activities. And so what we're going
to do is to mandate by law that they can't do that. Now, that may be
a reasonable thing from a political perspective,
but it's not a reasonable thing from an economic perspective,
because what it does is it disrupts relationships
like this, pricing relationships like this. Yeah. AUDIENCE: Can you actually
change the borrowing costs for a few types of
security [INAUDIBLE] by selling the short [INAUDIBLE]
and shorting [INAUDIBLE] the interest rate [INAUDIBLE]. ANDREW LO: Well, if
you're the government and you're in a
time of crisis, it seems like you can
do anything you want. So I mean practically,
yes, you can say, for all financial stocks
that are on this list, we will now charge an
extra high rate of interest for borrowing those stocks. And, in fact, there exists a
mechanism even before this rule was put in place that for
certain stocks that are quote "hard-to-borrow"-- that's a
technical term that Wall Street firms, brokerage firms use. Securities that
are hard-to-borrow means that they're
very actively traded and it's very hard to
find a counterparty that's willing to let you
borrow it from them. The hard-to-borrow
stocks typically are lent out at a premium. So they are charged
higher prices for those that are
very, very popular. And by simply
eliminating short sells, you basically make
those costs infinite. That's exactly what's going on. So would it have been better to
make them finite, but bigger? That would have been
better than what they did, which is making it infinite,
but from the purely financial markets perspective,
it would be best if there were no restrictions
and no transactions cost at all. Obviously, transactions
costs are inevitable. So when you price
these relationships, the equality is going to
be up to transactions cost. There might be a
little bit of a wedge between the left-hand side
and the right-hand side, but the difference will
have to be small enough that apart from
transactions costs, they are not
significantly different. Yeah. AUDIENCE: Could it
be that they also stopped the shorting because
of some illegal transactions or inside information? ANDREW LO: Well, if it
were illegal transactions, the way they should have
done it was to prosecute the illegal parties, as
opposed to stopping short sales for everybody, right? That would be a far
more effective way of dealing with
illegal activities. That's not what they were
concerned about primarily. The SEC has an enforcement
division whose sole job it is to check on all the
kind of transactions that may have occurred to
see whether or not there's any kind of
illegal activity going on. It wasn't the illegal activity
that prompted the short sales restriction. It was really the concern
that financial firms were being pounded
by short sellers that were betting on them failing. And the more pressure that
they impose on the stock price going down, down,
down, the more likely it is that people would
lose confidence and then all of a sudden stop
doing business with it. In many ways, that's what
happened with Bear Stearns. At least from the
historical record, it seems like what
happened was that there was a rumor that Bear Stearns
was not going to be solvent, even though they didn't
have any particular pressure to pay certain debts
at that point in time. And somehow that rumor
grew into general fear and the short sellers got in
and started shorting the stock. The stock price went down,
people looked and said, oh my god, the
stock is going down, I'd better take my
business elsewhere. And everybody
started doing that, and once everybody
started doing that, the firm began having
great difficulties in maintaining a business. Yeah. AUDIENCE: Even the days that
short selling is not allowed, then the investors who
own the more expensive-- ANDREW LO: That's right. AUDIENCE: --asset are going
to sell them because they don't think if they
don't sell this today, I'm buying the right-hand-- ANDREW LO: That's
right, that's right. AUDIENCE: So again,
you do the [INAUDIBLE]. ANDREW LO: That's
a very good point. Let me repeat it. The point is that even if you're
not allowed to short sell, then the folks who own
the left-hand side, who own the coupon bond,
they can say, hey, my coupon bond is worth 110, but
I can get the exact same payoff by buying a bunch of
pure discount bonds. And it's only cost
me 100 to buy them. I'm going to sell my
coupon bought at 110 and I'm going to buy $100 worth
of these this discount bonds and I just made $10. Now that can happen as
long as two things occur. One, the person who owns
the bond knows about this. And two, they actually
want to take the trouble to do the trade or they're
able to do the trade. The problem is that
most of the folks that bought the bond on
the left-hand side are pension funds that are
not in the business of doing this kind of arbitrage
transactions. So they're trying to do that. All they want to do is they've
got a bunch of pension plan participants. They need to pay
out their benefits. They've got contributions. They just want to match
assets to liabilities. They're not in the business of
doing this kind of high speed transaction. So what that means is
that you're still right, that if they realize this
relationship is there, and they are in a
position to do the trades, they will sometimes. And then that will
force prices closer, but it's not going to be the
same kind of numerical identity that has to hold when you've
got greedy people like myself trying to do the
trades and being able to do the trades
at a moment's notice. AUDIENCE: I'm especially
more interested about if these
players are big enough in the financial markets,
then it will correct and it could get out of that. ANDREW LO: Exactly, right. There are all sorts of
additional complexities of being able to do the
trade that as a pension plan, you're not even allowed to
do, never mind whether you want to do them or not. So these are the
kind of restrictions that would present this gap. But let's for now assume
that there's no gap. We're going to assume
that there's no frictions. We're going to assume that
there's no short sales prohibitions. And when that happens,
this pricing relationship has to hold. Now I promised you-- I want to show you
something more complicated that all of you may
do when you get out of here, and that's this. Instead of looking at
just one bond, what if we looked at a whole
bunch of coupon bonds? Now, let's take a look
using this pricing that's based on the law of
one price and basic greed. We've got n bonds
here, one through n, and each has its own
coupon, whatever that is. I'm just going to use
notation to write down that each bond has its own
particular coupon, right? So you've got a 3% 10 year bond. You've got a 4% 30 year bond. You may have 5 and
1/2% five year bond. So some of these
coupons may be zero because capital T
I'm going to assume is the most extreme
30 year period, right? So a five year bond
would have coupons for the first five
years and then for all of the coupon
payments after that, I'm going to have zero, zero,
zero, zero, zero, right? So this should hearken back to
your high school algebra days, where you have
multiple equations with multiple unknowns. Now what's unknown here
in these equations? What are the unknowns? Well, you observe the prices,
right, from the marketplace, so the left-hand
side is not unknown. What about the right-hand side? What are the unknowns? What do you observe,
what are the knowns? How about that? Let's start with that. Yeah. AUDIENCE: The coupons. ANDREW LO: The coupons, right. If I tell you that I've got
a 30 year 4 and 1/2% bond, you know what the coupon
is going to be, right? AUDIENCE: Because we
don't know the yield. ANDREW LO: What's that? AUDIENCE: The yield. ANDREW LO: The yield
is what we don't know. All right. That we either get from
the marketplace or we can try to solve it
from these prices. But how many different
yields do we need in order to price each bond? These are T year bonds. How many yields? AUDIENCE: T. ANDREW LO: Yeah, we need T
yields, one for each year. We have T yields
in these equations and they're the same across
the different bonds, right? Because we're using
the same pure discount bonds to replicate each
of these coupon bonds. I've got T unknowns. How many equations do I have? AUDIENCE: n. ANDREW LO: n bonds, right? Now, on any given day, you
might have 200 to 300 bonds that are trading. But you've only got 30 unknowns. So let's think back to your
high school algebra or college linear algebra days. If you've got 200
equations and 30 unknowns, how many solutions do you have? Well, let's get simpler. Suppose you had two
equations and two unknowns. AUDIENCE: Just one solution. ANDREW LO: You
have one solution, assuming certain
kind of conditions that hold like invertability,
which we're going to talk about in a minute. Two equations, two unknowns. You have one solution. How about one equation
and two unknowns. How many solutions? Infinite. OK, that's right, because
you've got that extra degree of freedom, right? There's lots of
different solutions. What about three equations
and two unknowns. Now how many
solutions do you have? Yeah. AUDIENCE: Because it's
linear, you only have one. ANDREW LO: It's linear. Are you guaranteed to have one? AUDIENCE: You might have zero. ANDREW LO: You might have zero. What are the conditions under
which you would have one? AUDIENCE: If the
solution of two equations applied for the third one. ANDREW LO: Right. There is a possibility
you have one solution if the solution of
the two equations actually applies to the third. Do you remember under
what condition-- AUDIENCE: Exhausted. ANDREW LO: --that, what? AUDIENCE: Exhausted. ANDREW LO: Exhausted? That's not quite the term
that mathematicians use. There's something called
linear dependence. That's a very
complicated word that describes the fact that
if you have these two equations and two unknowns,
and you have one solution, you can just take an average
of those two, a combination of those two, and you'll
get the third one. You can actually replicate the
third equation from those two. What happens if you don't? What happens if you cannot take
combinations of the first two equations to get the third? What does that mean? AUDIENCE: One of the
numbers is wrong. ANDREW LO: One of
the numbers is wrong. What do you mean by that? AUDIENCE: It just couldn't
be-- it can't be a solution. ANDREW LO: It can't
be a solution. OK, now this is getting
really interesting because on the one
hand, you're probably getting confused about what any
of this has to do with money. In a minute, I'm
going to tell you it has everything to do with
money, as you might expect. If we've got two equations
and two unknowns, and we have one
solution, that basically says that there are two yields,
a one year yield and a two year yield that is able
to price both bonds. And that better be
the case because we're going to use these two yields
to replicate the bonds. And so this kind of a
relationship has to hold. Now if we add a third bond,
if we add a third bond, and those same two yields that
worked for the first two bonds, they don't work for the third
bond, something's wrong, right? It means that this
third bond, its price does not satisfy the
relationship between those two yields. That's evidence of a mispricing. Something is wrong. But in this case,
what's wrong is that there's something
mispriced between those two bonds and the third. So when you run into a
situation like that-- and we're going to give you an
example in the problem set-- when you come across
that, what that means is you should be extremely
excited, as opposed to depressed in math class, you know,
gee, there's no solution. In finance, what
no solution means is that there is a transaction. There exists a linear
combination of the first two bonds and the third that A,
costs you no money down; B, will generate cash
flow today; C, will require no future payments
of any sort, so it's riskless. It is a free lunch. It is an arbitrage. Now that's with three
equations and two unknowns. Anybody can do that, right? That's easy. What if it were 200
equations and 30 unknowns? Now, not so easy. Now you actually have to know
something about linear algebra. Now the whole notion of what
an invertible matrix is, what the eigenvalues are, all
the kind of infrastructure that you can build
for understanding this becomes relevant
as a quant trading on a proprietary trading desk. In the 1970s, a number
of MIT graduates were hired by Salomon Brothers. They knew very little about
fixed income securities. They knew the basics, which
is what was taught here, but they didn't know much about
market realities in practice. And so when they went to solve
their three equations and two unknowns, they observed
inconsistencies. And they didn't do three
equations and two unknowns, they did 200 equations
and 30 unknowns. And in the 1970s, that was not
easy to do because we didn't have PCs, we didn't have Excel. We didn't have a lot of the
tools that we have today. And what they did was they
took these simultaneous linear equations. This is high school algebra. Even back then it was
high school algebra, OK? And they just cranked through
and looked for mispricing, looked for no solutions. And they found a lot of
cases with no solutions. In one of the years
during the 1970s or 80s, one of these MIT grads was paid
an annual bonus of $22 million for doing this, for solving
simultaneous linear equations. So this is an extraor-- and if that was
what he got paid, you imagine what he generated
for Salomon Brothers. It was a lot more
than $22 million. This activity is known as
fixed income arbitrage. There are many other
versions of it, but this is the plain
vanilla version. And by the way, the
plain vanilla version, it still works in the
sense that occasionally, if you're quick enough and
you have the right tools, you can identify mispricings and
take advantage of them quickly. It doesn't last long, so
you're absolutely right. It doesn't last long. But the person who
gets paid is the person who can do this
the fastest and is able to understand
the interrelationships among the securities. And you can understand
now why retail investors have no chance of doing this
kind of thing on their own. This is something you definitely
don't want to try at home, OK? I know you guys have MATLAB and
you can do matrix inversions, but there are lots
of other frictions, transactions costs, and
imperfections that you have to build into this analysis. But once you do, all sorts
of interesting things start popping out. All right, question, yeah. AUDIENCE: Can you
give a couple reason why this does still exist? Like, why hasn't the arms race
between everyone narrowed it down to basic efficiency? ANDREW LO: Well, it's
actually much narrower now than it used to be. So in fact, the market
has gotten much, much more competitive. But it's not down
to zero precisely because there are frictions
and other aspects. For example, some
of these bonds, they have weird features,
like they're callable. Or in some cases, they
may have certain types of other requirements and market
institutional imperfections that require you to build
in those constraints when you do the analysis. And so it's really about
who has the better model. And frankly there's
an arms race that's been going on for the
last three decades as to who has the
fastest computers. The first supercomputer
that was ever installed on Wall
Street, a Cray-2, was installed at
Salomon Brothers doing simultaneous linear
equations, among other things. And so this is where technology
has played a really big role in the developments
of market prices. Another question? OK, so this very, very simple
idea, and again, it is simple, has all sorts of important
ramifications for the pricing of bonds and other securities. What that tells us is
that market prices have all sorts of information that
are incorporated into it. And one of the things
that we want to understand is how to interpret
that information. In particular, the
first thing I want to do is to understand the
risks, all right, because we talked about the
fact that some of these market prices may have missed
the risks that were implicit in some of the trades. And so the question is
why and how do we measure the risk of a bond portfolio. So what I want to
do is to now look at the market price
of a bond in terms of a function that has inputs
and the price is the output. And I want to ask the
question, what kind of fluctuation in the input
will yield fluctuation in the output? You know, we now know how
to price these things. We now know how the
relationships must work from a present value approach. And now what I want to
do is to ask whether we can measure the sensitivity. So one way to do
it is to just graph the price of a bond as
a function of its yield. And we know that there's an
inverse relationship, right? Just like we saw with treasuries
this week versus last week, right? Last week it was a big run
on treasuries, lots of people wanted to get into them. Price goes up, yield goes down. And this week, less pressure,
so we have yield going up and the prices coming down. That provides us with one
kind of measure of risk. So the kind of measure
I'm talking about is the slope of this line. The instantaneous
slope that tells us for a bit of a tweak
in interest rates or yield, what does that
do to the market price? Because obviously when
you're an investor, you're focusing on
the price, right? Yield is a convenient
way of summarizing the properties of a
bond, but ultimately what you care about in
your portfolio is price. And so the question is for a
move in the interest rate, what does that do to the price? Well, it turns out
that there is one way to get at that that is somewhat
more intuitive than just looking at this
kind of fluctuation. It's called duration and it's
named after a person Macaulay who first proposed it
as a way of measuring how risky a bond is. What Macaulay noted was that the
longer the maturity of a bond, the more sensitive is the
bond price to the yield. So for example, a 30 year bond,
when you move the interest rate by one basis point, will have
a much, much larger price fluctuation than a
three month bond, right? Why is that? Anybody give me some intuition
for why that should be? Why that makes sense? Why should a longer
maturity bond be more sensitive
to changes in yield? Yeah, Ken. AUDIENCE: Because
you're hanging on, because the cash
is tied up longer. ANDREW LO: Yeah, and? AUDIENCE: And because the risk-- because the opportunity
cost essentially of having that money tied
up for that long means you can't spend it on all these
other things over those 30 years. ANDREW LO: That's right. You're investing for a
longer period of time so, in fact, the
opportunity costs, as measured by the
foregone interest, is going to be much larger. In other words, the
discount rate that you use, this one plus r, you've
raised that to the 30th power, not to the 1/4 power,
and so something that's raised to the 30th power
in the denominator, it has a much larger impact when
you perturb that denominator by a little bit, because you're
actually increasing that power. You're increasing that
quantity by that 30th power. Well, so if that's
the case, if it's the case that the longer the
maturity, the more at risk you are per basis point of
interest rate fluctuation, then why don't we just
measure the average duration of the bond? By average duration, I
mean how long does the bond last when you weighed it
by the coupon payments that it pays you? So for a pure discount bond,
the duration is simple. It's just the
maturity date, right? If you've got a 30 year strip
that pays you $1 in 30 years, the duration of that
bond is just 30 years. But what if you had a
coupon bond that paid you coupons all along the way? Well, then, you should take
a weighted average of the 30 years and give some
weight to the early years too, because the early
years are years where you're going to receive cash. And therefore, interest
rate fluctuations are going to affect the
value of those cash flows. So the weighted average
term to maturity is simply equal to the sum of
the date, 1, 2, 3, 4, 5, 6, 7, multiplied by a weight
factor that sums to one. And let's just use,
as a weight factor, the proportion of present value
of cash flow on that date. So if you take the
weighted average where you take the weights as
the present value of the coupon divided by the present
value of the bond, those weights
certainly sum to one. And then you're weighting
that by the date, the year that you get paid,
that number will give you what's called Macaulay duration. So it turns out that the reason
that Macaulay duration is interesting is that when you
take a look at bond prices and you ask the question, for
a certain percentage change, a certain basis point change
in the interest rate, what does that do to the bond
price as a percentage of its current price? So this is the sensitivity. It turns out that you can
show that that's actually the negative of the
Macaulay duration divided by one plus the bond yield. So this is a long way of stating
that the duration gives you a measure of how
sensitive the bond price is to changes in yield. The longer the duration,
the more sensitive the bond is to changes in yield. And duration now just
means a weighted average of all of the payout dates
that a bond will have. If a bond pays a lot
of its cash up front and very little
in later periods, is the duration high or low? AUDIENCE: Low. ANDREW LO: Low. And so if a duration
is low what that says is that there's not
a lot of sensitivity to changes in yield on
the bond price itself. If on the other hand,
all of your payment is out in the future, way
out in the future, that's going to make it very
interest rate sensitive. And the negative
number here indicates that there's an inverse
relationship between price and yield. So when bond investors look at
a particular portfolio of bonds, and this is key, they look
at a portfolio of bonds because that's what they're
going to be investing in. When you put your money
in a money market fund or in a bond fund,
medium term, long term, you're not putting
it in one bond, you're putting it in
a whole set of bonds. Your natural question
is, how sensitive is that portfolio to
changes in interest rates? And the answer is it's
related to the duration. And so if I told you
that this portfolio has a duration of five
and 1/2 years, that will give you some
intuition for how sensitive or how risky that portfolio is. So duration is a measure
that I've defined for a bond, but you can define it
for a portfolio of bonds, simply by taking the cash
flows at every period and then computing a weighted
average where the weights are the present value of those
cash flows as a function of the entire portfolio. OK, so here's an example where
I've got a four year Treasury note with a face value of $100,
7% coupon selling at a $103.50, which yields 6%. That's the yield,
that's the why that makes this 103.5 equal to
the present value of all of those coupon payments
plus the return of principal. And so here what I've done
is to calculate for you the cash flows of the bond,
the present value of those cash flows, and then the
respective product of t times the present
value of cash flows. So you can actually compute for
yourself that duration number. And you can get a sense
of exactly what that is. So the duration is
about 7.13 years. So duration is a
measurement that is in units of years
or half year units. Sorry, 7.13 half year units. And the modified duration is
going to be given by 6.92. Price risk at a
yield of 3% therefore is going to be given by just
this expression right here. What that says is that
if the yield moves up by 1/10 of a percent
or 10 basis points, the bond price is going to
decrease by 68 basis points. That will give you a sense
of how exposed you are. If you have very long
duration portfolios, that means that you're going
to be in for a wild ride as interest rates
swing around a lot. And if you're willing to take
on that risk, that's great, but you're going to be at
least compensated for that. Macaulay duration, you can
compute it in this way. For intra-year coupons,
that's a very straightforward calculation. So you want to make sure
you know how to do that. You simply just use
the usual discounting and dividing the yield by the
appropriate payment periods. And now the last concept
that I want to cover today is convexity. Convexity is another
measure of risk. It's the second derivative. What it measures is how the
sensitivity itself changes. And it turns out that
convexity, as a measure, gives you sort of a higher order
approximation both to the price of the bond, as
well as how the bond is going to move with
respect to interest rates. So let me just show you
what the derivation is. It's pretty straightforward,
but if you have any questions, I'm happy to discuss it. You take the price of a bond and
you take the second derivative with respect to the yield. What you're going
to get out of it is an expression
that looks like this. Now that expression
in and of itself has relatively little intuition. But let me just write
down the percentage change in the first
derivative as v sub m, and then I'm going to show you
an interesting relationship. For those of you who remember
your high school calculus or college calculus
class, you'll remember there's something
called a Taylor approximation, right? Taylor series. This is a method of
approximating nonlinear relationships using
polynomials, powers of the variables in question. If you took the bond price
as a function of the yield, it's a nonlinear function,
of course, because you've got that discounting going on. Then you can ask
the question, how can I approximate the bond
price as a function of the yield if I'm willing to take a couple
of terms of the approximation? And when you do
that, you get what may look like an
awful expression, but actually it's quite
beautiful in its own right. What this says is that the price
of the bond at a new interest rate, at a new yield, if
the yield changes, then price of the bond at
the new interest rate is going to be equal to
the price of the bond at the old interest
rate, multiplied by a factor of
something or other. And that factor is
going to be given by one minus a term that's a
linear function of the yield, plus another term that
is a quadratic function of the change in the yield. So this is what
I mean when I say this is a second order
approximation to the pricing relationship. And this basically
gives us a way to figure out when yields
move, what does that do to my portfolio. Now you might be
thinking, gee, this is an awful long way to go
to try to figure that out. Can't we just use
an Excel spreadsheet and then move the interest rate
and then recalculate and see what that does? Today, you can, but in
the 1970s, you couldn't. You didn't have that ability. So much of this framework
was developed in the 1970s because people wanted to have
easy ways of not only pricing bonds, but figuring out what
the risk of their positions were as bond traders. And to do that quickly
is very difficult. You remember the
story I told you about when I got a
mortgage 20 years ago and the bank loan officer
just could not figure out what my mortgage payments were. She had to go look for a book
and try to thumb through what those calculations are. Well, imagine if you're a
bond trader trading literally every minute of
the day, and there was bond trading going on in
the 1970s, believe it or not. You have to figure out
these pricing relationships. And it was not so easy. So a number of
mathematicians came up with these kind of
relationships for bond prices. In fact, it's funny. Bond pricing is actually
quite a bit more of a mathematical art
than equity pricing simply because
with a bond, there aren't that many moving parts. And so they aren't subject
to mathematical analysis like this, whereas
with stock prices, since there are so many
factors impinging on it, the actual tools that
we use are quite a bit less complex, at least from
a historical perspective. So the purpose of
this is really just to approximate the risk
of a bond portfolio. You want to know when
you change yields what that does to the portfolio. And you also want to
know if the yield changes in volatility, what does
that do to the portfolio. So this first term
here, this term tells you about shifts
in the interest rate. What this term does is
tell you about fluctuations or volatility of interest rates. And it turns out
that volatility has an impact on bond
portfolios, as does changes in the level of interest rates. Just looking at
this, can anybody tell me off the top of their
head what direction this goes? In other words, if you are
holding a bond and interest rates rise, we know that
bond prices will fall. But what if the volatility
of interest rates rise, as they have over
the last few weeks? Yields have bounced
around a lot more lately than they
have in the past. What does that do to a bond? Does it make it more
valuable and less valuable? Let's put it that way. How do you know? Less, more? We have some volatility here. Take a look at that expression. v sub m is going to be
the second derivative, and then you've got a
term there that's going to be the change in
the yield squared. If the change in the
yield squared goes up, other things equal, and
other things are never equal, but economists like to
say other things equal, it actually makes
bonds more valuable. In this respect, owning a
bond actually is sort of like owning an option. Now you don't know
about options yet. We're going to come
to in a few lectures. But it turns out that having an
option when volatility goes up can be very valuable. And similarly for
bonds, bonds have option-like characteristics. And we see that here
with this approximation. That second term will actually
yield some actual value when volatility goes up. Now I do a numerical
example here that I'd like you to
look through on your own where I compute for you both
the first and second terms of that approximation. And you can actually see how
good the approximation is. So I'd like you to take
a look at that, make sure you understand
it, and next time, be happy to answer
questions about this. What we're going
to cover on Monday is risky debt, which of course
is directly related to what's going on in markets today. And so I'd like you to read
up on that in the textbook. And when we come
back on Monday, we're going to talk about debt
ratings and possibly what went wrong with all of
these subprime securities.
