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visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: The other item
that I auctioned off-- that's a good question. Well, first of all,
before I tell you that, you'll note that I put
those two packages up here in the front of the room
for this first section. One was bigger than the other. And you guys got the
smaller one, right? And you obviously
know what that was. The bigger one turns
out to have been a book. In fact, it's a
book that I recently published on hedge funds. Now it turns out that the
bidding of that second item went for $60. The bidding of the item
that I sold to this auction went for $45. And the fact that it's a
smaller package I think makes some sense of why it
went for a smaller amount. That shows you the
power of information, or lack of information, in
terms of determining values. Yeah, question? AUDIENCE: Yeah, I
was just going to ask if there was some
insider information that would up the value
of that second box? ANDREW LO: Well,
I don't think so. And one of the reasons that
I had two different packages and I showed this entire section
the two different packages is to try to eliminate a bit
of that private information. I don't believe there was any. If I used the same
two items, then I think there would
have been a concern that there might have been
some information spill over. So I don't think that was it. In fact, I think what it was
the size of the packages, believe it or not. So anybody that tells you that
size doesn't matter I think is not being realistic. Was there another
question back here? AUDIENCE: I had
the same question. ANDREW LO: OK, yeah? AUDIENCE: How much did
the book retail at? ANDREW LO: The book
retails for $45. Now of course, I
autographed the book, so-- [LAUGHTER] --so that probably
reduced the resale value. We played-- I took the $60. No tears. Fair is fair. So this class, the individual
who bid $45 for that iPod did very, very well indeed. And the student who
bid $60 for the book didn't do quite as well. But that's the point
of this example. It's that when you don't
know what you're bidding for, when there's literally
no information, you're obviously not going
to get it exactly right. But you have to admit that in
the grand scheme of things, it's actually not that
far off, and not that far off in a very particular way. I want to, in particular,
talk about the iPod. Because we are
looking at something like a 70% discount
off of the retail value of that for this class. Now I want to comment on
that in the context of what's going on today. I want to spend a few minutes,
before we start the lecture, talking about current events. Because finance of course
involves current events. And right now we are going
through an extraordinarily interesting set of times. I think that's a Chinese curse-- may you live in
interesting times. Well, we're going through some
really extraordinary times. Because what happened
over the weekend is something that has
probably not happened since the Great Depression. What happened over the weekend
is that the federal government took over two of the largest
government-sponsored financial entities in the world,
Fannie Mae and Freddie Mac. These are two
government-sponsored entities. They're private
organizations but they have the backing
of the government in some ambiguous sense. And it was ambiguous, up until
this weekend, what that meant. What happened was that these
large financial institutions were responsible for
purchasing the mortgages from various different regional
banks and lending institutions. And why would they do that? What was their function? Anybody know? Yeah? AUDIENCE: To provide a
secondary market for mortgages so that the banks that
are the initial lenders can generate more mortgages
and make credit more available. ANDREW LO: Exactly. Their goal was to really
make credit more available in the housing market, and
the student loan market, and the auto loan market--
these consumer finance markets. The purpose of these
two organizations was to buy up the various
different obligations from banks and other
financial institutions so that the banks and
financial institutions could then go out and lend more
money to more individuals that wanted these kinds of loans. So the purpose of
these two organizations was really to support and
grow these various markets. And it turns out that
over the last 20 years, these markets have grown
to epic proportions, tremendous proportions. And now, as the housing
market has turned down, the original lenders that
made these obligations are facing some
severe pressures. And that's carried
the ripple effects all the way through
the financial system, to the point where these
two organizations literally could not stand on their own. And if there had been a default
of these two organizations, there would be some
major repercussions. We're going to talk about
that in about four lectures. The material that we're going
to develop over the next three lectures are going to allow
us to analyze the situation and really try to
understand what's going on in these markets. But I just wanted to
alert you to the fact that something really
big has happened. You won't understand
fully what it is. You might read the
newspapers, and I would encourage you to do
so, but you won't fully appreciate the importance of
it for another three or four lectures. But you're going
to see this event unfold before your very
eyes this semester. And that will provide
the motivation for developing the tools to
understand what's going on. Yeah? AUDIENCE: Can I ask you
now, or can we ask you now, do you think that
was the only option, or do you think
that was the right-- ANDREW LO: Well, that's
a very tough question. In terms of the possibilities
for the Secretary of the Treasury Paulson
and the government, there aren't too many
left in the sense that these two organizations
are in a real bind in that they have issued lots of securities
based upon their activities. And in order to maintain the
quality of those securities, in order to make good on those
promises that it has made to other counter-parties-- very important
counter-parties I might add, including the governments
of India, China, and a number of sovereign wealth
funds that we depend on to loan us the money that has kept us
in the style to which we have become accustomed
in this country-- in order for us to maintain
those kinds of relationships, we have to make good
on our promises. And it's getting harder
and harder for these two organizations to do so. Because while they've
got all of these promises that they've made,
the money that's supposed to be coming in order
to allow them to make good on these promises is dwindling. And as they've tried to raise
more money in the open market, it's become harder and
harder, because people realize that they're in trouble, and
it's very difficult for them to raise that money. In fact, this is related exactly
to the point about the iPod auction that we
engaged in last time. Think about how little I
received for that $150 item. I received basically
$0.30 on the dollar. And why? It's because none of you knew
what was in that package. So now instead of an iPod
that's in a wrapped package, imagine if that wrapped
package contained paper issued by Fannie Mae or Freddie Mac. And now I'm going to ask you,
can you bid on this paper? Now I don't know whether Freddie
Mac or Fannie Mae will still be around in two years, but I'd
like you to bid on this paper anyway. And by the way, what you
get for this piece of paper, I don't really know,
so it's wrapped up. And you don't know either. You don't know. I don't know. You know I don't know. And I know you know
I know I don't know. So what's that piece of
paper going to go for? Well probably less than
$0.30 on the dollar, right? So what we saw happen in
this classroom last time is exactly what's happened
in spades to these two large organizations. At some point, there has
to be a day of reckoning. And the Treasury
decided rather than waiting for that
day of reckoning-- because if that day of
reckoning came before they were able to do something, it's very
hard to put the pieces back together. So in my opinion, I think
they had very little choice but to do something right now,
before it got out of hand. And we're going to
talk about this again. In fact, we're going to run a
trading game where all of you are going to get a chance to
engage in this kind of trading. It's not exactly the kind
that involves credit. But it'll be a
trading game, where you'll get to see what it's
like to be under pressure and to have to make
decisions and value securities in real time. And you'll have a much
deeper appreciation for the kind of issues
that we're up against. Yeah? AUDIENCE: Sorry, what did
the government exactly do to agree to pay for all their-- ANDREW LO: Yes,
so the government said, OK, you guys, Fannie
Mae and Freddie Mac, you can't make good
on your obligations because you've
got some problems. So we, the Treasury, will
stand by all of your claims. AUDIENCE: And the
government paid that? ANDREW LO: Absolutely. You know how? How can the government pay that? What? AUDIENCE: How, issuing money? AUDIENCE: [INAUDIBLE] ANDREW LO: Exactly. The government-- they
own the printing presses. So they just-- you want
to get paid dollars? I'll pay you dollars. Here, how many dollars
would you like? I'm going to run it. Here you go! AUDIENCE: So we're probably
getting hit with inflation? ANDREW LO: That's the concern. So the concern is that
if you do this too much and you do it without anything
backing up the pieces of paper you now print and
give to your debtors, then there'll be a problem. Because pretty
soon they're going to realize that
the piece of paper isn't really worth as much
as they thought it was. So inflation could be a problem. It could be a problem. We'll have to see. But right now the paper, the
dollars that are being printed, are backed by the full faith
and credit of the US Government. And for the time being,
that still means something. Yeah? AUDIENCE: Are all
the levels of debt being guaranteed with interest? How much of the Fannie
and Freddie is debt? ANDREW LO: So the
question is, is all of the debt being
guaranteed or is it just how much Freddie and Fannie issued? Well certainly it's
what they've issued. So all of the paper
that they've issued will be backed by
the government. In addition, there
is some new paper that they have to issue
in order to finance additional mortgages. And the government
will back that too. Now this is not for free. It's not just printing presses. That sounds a little too easy. Somebody has to pay for it. The money's got to
come from somewhere. Or I'll put it another way. That money that is spent now
backing these instruments cannot be used for other things. So if you believe in
balancing your budget as opposed to spending on credit,
which frankly is what got us into the problems to
begin with, then there will be additional reckoning. People have said that
the American taxpayer will pay for this. Well, that's true. But the question is
which set of taxpayers? Is it us? Is that your children? Is it your children's children? There are a lot of
intergenerational questions about where the burden lies. But the fact of the matter
is yes, we are on the hook, that the American taxpayers
are on the hook for tens of billions of dollars. Now in the grand scheme of
things, tens of billions of dollars is nothing. The federal budget is much,
much larger than that, so this is not a big deal. The concern is if this
spreads and ends up costing us more because
of ripple effects and knock on effects that
we don't yet know about, that could be a problem. And that, frankly, is
one of the reasons why the Fed and the Treasury have
decided to step in now to try to contain this problem. How good they will be at
doing that we don't know. Read the newspapers over the
next few weeks and let's see. Yep? AUDIENCE: Do you think
the common stock basically has to be worth zero
at this point now? ANDREW LO: Well, it's
not worth exactly zero, but it's pretty darn close. And in fact, it's
come down quite a lot, I think, over the
last couple of years. I don't know what the
peak of the price was. I haven't looked at it. But I think that
the decline has been on the order of 90%
to 95% decline of what the peak was over the last
couple of years, which is-- AUDIENCE: This
morning it's down 83%. ANDREW LO: Yeah, 83% since when? AUDIENCE: Since yesterday. ANDREW LO: Yeah, that
was just a day ago. But I'm talking
about since the peak. I think it's come
down dramatically. So effectively, the value
of the shareholders' equity is pretty close to zero. But that's what happens
in a bankruptcy. And by the way, we're going
to talk about bankruptcy too. That's going to be a very
important part of what we do when we analyze
fixed income instruments. So we're going to
get to that shortly. Yeah? AUDIENCE: Has
there been anything about potentially going
private again down the road? Is that-- ANDREW LO: There
have been discussions about them going private, but
private in a much smaller way. So the discussions
that have occurred at the Treasury--
and we don't know which way things
are going to end up because it's a political process
as well as an economic one. But there's a feeling that these
two institutions got too big, too quickly and
that there wasn't enough oversight to allow
them to be able to fail. In other words,
they grew so big, so quickly that
after a point they were in fact too big to fail. And when you're too
big to fail, all sorts of abuses and improper
risk management practices can go on, which we will
talk about in probably eight or nine lectures. So there's a
variety of proposals that have been put forward. And we'll talk about those
also over the next six or seven weeks. AUDIENCE: Who did the
government actually bought the shares from? Are they federal agencies? Are these private companies? Who-- ANDREW LO: So Fannie
Mae and Freddie Mac are called GSEs,
Government-Sponsored Entities, which means that they're
not a federal agency. But they are backed by
the federal government in some manner that has been,
up until now, relatively vague. So they're meant to
be private companies. They had a CEO. Both of them had CEOs, who
were fired over the weekend. They had separate
boards of directors. And they pay pretty attractive
salaries, private sector salaries. But they were related to the
government in the sense that their objectives were to try to
support and expand the housing market financing, which was
a government-dictated mandate to this-- AUDIENCE: Their main goal
is not to earn money? It's to bring a secondary
market for mortgages? ANDREW LO: Their main goal was
to create additional financing to allow the mortgage market to
grow and to allow people to buy homes at reasonable rates. Their secondary goal was to
make a profit on top of that. So that's why they're not
just a government entity, but they have a kind of a quasi
private sector type of mandate. And by the way, they
did a great job. They did a great
job in the sense that the housing market
has grown tremendously. Now you could argue that
that was irresponsible and that created
all these problems that we're facing today. But you talk to somebody who
has a subprime mortgage that otherwise couldn't have afforded
a house that now has made every single payment, and really
enjoys living in their home, and has a lifestyle that they
couldn't have otherwise had. You talk to them and you tell
them that it was a mistake. So it's not at all clear that
this is a total disaster. It's unfortunate. And it's a catastrophe
from the point of view of the current holders
of various pieces of paper that are related to
these subprime mortgages. But there's a whole
group of people out there that have benefited from
these subprime mortgages. And so you've got to balance
the costs and the benefits. And we're not doing
that right now. Right now we're just
panicking because we're all scared like hell
that something really is going to break. But there's a positive side
as well as a negative side to every story. And so we're going to have to
talk about that in a little bit more detail. But we can't do that unless
we have the framework to think about it. So that is going to lead
me to today's lecture. Yeah? AUDIENCE: Just to follow
up on another question, who were the shareholders in
Freddie and Fannie Mae system, and did the investment
go to zero overnight? ANDREW LO: So there are
two groups of individuals that are involved. One are the shareholders of
the companies themselves. And these are investors. A number of them are pension
funds, institutional investors. So therefore, you and
I, in our 401(k) plans-- we might actually
have an investment in one of these organizations
through the plan sponsor that invests in various
different entities. So they're a stock that was
traded on the exchanges, just like any other stock. That's one group. And those are going to
lose most of their capital. They're going to lose
their investment, just like when a
stock goes belly up, the investors of Enron, too bad. They lose everything. They're gone. They knew it. They took the risks and the
risks don't always work out, so that's that. The other parties
that are involved are the folks that do
business with these agencies, the counter-parties that bought
the paper that they issued, these IOUs and other complex
instruments that Freddie Mae and Freddie Mac issued. Those individuals,
we're hoping, will be relatively happy about
the outcome of the government backing the paper. Because now the paper that
they once held and thought was in trouble will have the
full faith and credit of the US government behind them. And that is a major concern. Because these counter-parties
are very, very substantial investors. If they decide that
this paper is no good and they start getting
rid of it wholesale, that will create some really
significant market dislocation and mass panic. And that's what the
Treasury stepped in to stave off this weekend. And hopefully they
will have succeeded. In fact, if you
look at the data, it looks like they did succeed. But I'm not going to talk about
that for another three or four lectures, because I want to
show you exactly how to think about these instruments. And then we're going
to look at the data and see whether or
not they've actually done something significant. And you'll be able to tell. By looking at market
prices, by looking at the outcome of auctions
like we did last time, we're going to be able to
tell exactly what happened. All right, well, that took a
little longer than I thought, but I think it is useful
for motivation for what we're going to do in the class. What I want to do
today is to start on lectures two and three,
present value relations. And I have to tell
you, this is one of the most interesting
lectures of the entire course, not because of the
underlying material, but rather because of the
novelty of the perspective that I want to give you
about these kinds of issues. This is the very first step
in changing the way you look at financial transactions. We're going to
start by discussing cashflows and assets. I'm going to define some terms. And I want to change the
way you think of an asset. And then we're going to
talk about the present value operator. And then we're
going to apply that to the time value of money. I want to make that
concrete so that all of you will start thinking
differently now about money today versus money next year. And then-- probably not
today, but next time-- we're going to cover two very
special kinds of cashflows, the annuity and the perpetuity. These are two
mathematical abstractions that provide some really
interesting insights into this whole notion
of present value. And then I'm going to
talk about a couple of technical issues,
compounding and inflation. And that will lead us right
to thinking about fixed income securities, and
how to value them, and how the market values
them, and whether or not the two are the
same or different. For the readings, I'd like you
to start on Brealey and Myers, Allen, chapters two and three. You should have already read
chapter two from last time. I'd like you to reread that
and also focus on chapter 3 for the next couple of lectures. So first order of business,
let's define some terms. I think you all know what
a cashflow is, right? It's just money
that's coming to you or going away from
you, a flow of cash. And you all know what
cash is, I presume. The next question I want to
take on is, what is an asset? We talk about valuing assets. But we have to start
first by agreeing on a definition for
what an asset is. And I've just put some examples
up here of what assets are. Business entity is an asset. If you guys decide
to do a startup, that startup has value. It's an asset. Property, plant and equipment,
patents, R&D, stocks, bonds, options, and even something as
difficult to value as knowledge and reputation are assets. Does anybody know of an example
of a really valuable asset that is not physical at all? You can't touch it or feel it. Yeah? AUDIENCE: The algorithms
that Google is using? ANDREW LO: The algorithms
that Google is using. That's right. Now how do you know
that those are assets? AUDIENCE: Because
they generate revenue. ANDREW LO: They
generate revenue, right. And has anybody turned them
into anything concrete, like any type of legal
structure to make them assets? AUDIENCE: Patents. ANDREW LO: A patent, exactly. A patent-- and Google
has many patents-- those have value. Not only are they
considered assets, but you can actually trade them. You can buy a patent, sell
a patent, license a patent. Yeah? AUDIENCE: Are they patents
or are they trade secrets? ANDREW LO: Well, they're both. So Google has patents. But there are also a
number of trade secrets that it uses to protect
its intellectual property. What's the distinction? AUDIENCE: I think
a trade secret is if you have a patent, but
not an implicit patent. AUDIENCE: No, it doesn't
have any patents. ANDREW LO: No, no. AUDIENCE: What does that mean? What What is a trade secret? AUDIENCE: It's a secret. ANDREW LO: It's a secret. Thank you. AUDIENCE: A patent
without legal backing. ANDREW LO: Well,
I'm not sure it's a patent without any
kind of legal standing. In fact, it's sort of an
anti-patent, isn't it? AUDIENCE: It's not a patent. It's a secret, specifically. Like Coke, for example--
the formula for Coke is a trade secret
that is not patented. When you apply for a patent
you have to disclose fully what your patent covers. And that means that at the at
the end of the life coverage, you're out. And so I could go and-- if Coke would have applied for
a patent however many years ago, it would have expired. I could be making
Coke right now. But a trade secret is
secret forever, as long as you take steps
to keep it secret. ANDREW LO: Right. AUDIENCE: So they are taking
the risk of the secret. ANDREW LO: Exactly. Coke is taking a big
risk in that it has not filed for a patent. And frankly, if it did, it
would have expired long ago. Now what exactly is a patent? What's the motivation
for a patent? A patent is a legal
agreement that says that if you
tell me everything there is to know about
how to construct the item or algorithm that you
want to patent, if you're willing to disclose that
to me and the whole world, and I acknowledge that
it is new and useful, then I will grant
you the ability to be the sole user of
that algorithm, product, or business process for
a finite period of time-- say, 17 years. So for 17 years, if anybody
wants to use that algorithm, they have to pay you
for that privilege. So it's a monopoly. It's a monopoly that the
government grants you for a fixed period of
time in exchange for what? In exchange for you to be
willing to disclose everything there is to know about it. And why is it? So that everybody else
can learn from it. And after 17 years,
and after you've made a god awful
amount of money, somebody else can take that
idea and incorporate it into what they're doing
and make money off of that. The patent process creates
assets out of ideas. And it allows you to
derive economic value from that in exchange for
freely sharing those ideas. That's one path to go. And the other path
is I don't want to share because I think
that I can do better by keeping it a secret. And that's what
Coca-Cola has done. That's the example
that I was thinking of, of an idea that has
no legal standing but that's one of the most
incredibly valuable ideas in the world today. The amount of money
that Coca-Cola produces is unbelievable given
this simple recipe that apparently people haven't
been able to figure out how to reproduce. How hard could it be? You just mix a few things
and add some coloring and then you get Pepsi. I don't know. Yeah? AUDIENCE: But a lot of their
value is also in the brand. ANDREW LO: That's right. AUDIENCE: There's the trillions
of people who associate a red dot with feeling happy. ANDREW LO: That's true. Now that's the case. They've spent years building
the Coca-Cola brand. That's one of the most
recognizable brands in the world. And that has value, too. And by the way, that value
is different from the trade secret. So now we see that there's
lots of different assets. And that's an asset, right? Yeah? AUDIENCE: I'm just curious. Is there a financial
reason-- or maybe not-- for why you would patent
something versus have a trade secret? And what I was thinking of
is pharmaceutical companies. Why would you go
and patent Lipitor? Wouldn't it be better to
keep that a trade secret and be the only company in
the world ever to make it? ANDREW LO: Well that depends. It depends on whether or not you
think you can keep it a secret. Coca-Cola apparently has
been kept secret pretty well. But there are a lot of
other people working on the various different kinds
of cholesterol reducing drugs, not just the folks
that develop Lipitor. So if you can keep it a secret,
great, keep it for yourself. But very often you won't
be, because other people are working on the same research. Also there's a certain
cachet to having a patent. And it provides you
with a certain kind of business viability. You can go to an investor and
say, here, I've got this idea. The idea has been certified
by the US Patent Office. It is patent number
XYZ, and that's why you should give me money,
versus you go to an investor and say, hey, I got a secret. I can't tell you what it is. You know what? You're back to the iPod. Andy, question? AUDIENCE: Oh, no. I was going to say
the same thing. ANDREW LO: Yep, OK. So now we know that there
are all sorts of assets. Even intangibles, even
things that you can't hold in your hand can be an asset. I want you to forget
about all of that. I want you to think
about an asset in a completely different way. I want to reduce an asset to
its fundamental core properties. And to do that, here's my
definition of an asset. An asset, at a given
point in time t, is simply equal to a sequence
of future cashflows-- CFt, CFt plus 1, CFt
plus 2, dot, dot, dot. That is the
definition of an asset that I would like to adopt
for the next 12 weeks. Now this might seem trivial to
you, but trust me, it's not. It's a very subtle idea. And it's going to have all sorts
of interesting implications that we're going to discuss
over the next few lectures. But I want to make sure
that everybody fully appreciates what
I mean when I say an asset is a sequence of
current and future cashflows. Let me just describe a
couple of interesting things about this definition. First of all, it doesn't
involve past cashflows. So when I define an
asset I have to define it relative to a point in time. That, in and of itself,
I think is something new. So it's not enough to say
an asset is Coca-Cola. You have to say
that the asset is Coca-Cola today, versus
Coca-Cola 10 years ago, versus Coca-Cola 100 years from now. Those are different assets. The point in time actually
matters, in other words. This is like the zen paradox. You can never step
into the same river twice because water is moving. It's always a different river. In that sense, an asset,
as at every point in time, is a different asset. And that's not that surprising. Because if you believe,
from my definition, that an asset is equivalent
to its current and future cashflows, then those
cashflows are very different depending on where you stand. So for example, the stock of
Fannie Mae and Freddie Mac as an asset was very
different five years ago than it is today. They're both assets. They're both cashflows. But the cashflows
are very different depending on what point in
time you're talking about. So when I think
about a cashflow, I'm thinking about a
particular sequence that occurs given market conditions. An asset is comprised
of those cashflows. They are one and the same. When you tell me that
you have an asset, I immediately
think of OK, what's the sequence of cashflows? That's all I care about. From the purposes of
financial analysis, that's all that matters. So over the next
few weeks, we're going to be building up a
theory of financial analysis. The basic building blocks
of that theory are assets. So you could think of
assets as the molecules or atoms of a unified
field theory of finance. And the protons and electrons-- those are the cashflows. Yeah? AUDIENCE: Do the cashflows
have to be positive? ANDREW LO: No. They don't have to be
positive or negative. But they do have
to be real numbers, so no complex numbers here, OK? Positive or negative--
that's what an asset is. Now you might think, well,
gee, if all the cashflows are negative then I don't
really want that asset. Well that's the
nature of an asset. Maybe then instead of
an asset you call it a-- AUDIENCE: Liability. ANDREW LO: --liability, exactly. But from my perspective,
developing this atomic theory of finance, I'm going to focus
on this as the basic building blocks. And I don't care whether
they're positive or negative for the moment. We're going to talk about
their characteristics later. But I just want to agree on a
definition of what an asset is. Question, yeah? AUDIENCE: So it's a
sequence of cashflows, it's not a summation? ANDREW LO: Right,
it's not a summation. It's a sequence,
meaning it's basically just a list of cash flows at
different points in the future, including the present,
but not the past. So according to my definition,
the past doesn't matter. When you talk to me about
an asset, all I'm looking at is current and
future cash flows. And it's a sequence
of cash flows. So that means they're
ordered in time. And I'm not even telling
you whether or not what the cash flows are. So it could be that
the future cash flows-- I'm writing them as
letters, but who knows what the letters stand for. You might not know what
the future cash flows are. But you can still, nevertheless,
acknowledge that they exist as an abstraction. That's what an asset is. I don't care what
you use to value your various
different cash flows. But the definition
of an asset is simply a sequence of current
and future cash flows. All right, that's
what we're going to talk about as an asset. So all the things that
I listed up there-- knowledge, reputation. Can reputation be considered a
sequence of future cash flows? Does that make sense? How? What? AUDIENCE: Goodwill. ANDREW LO: Goodwill. How would you use that goodwill? Yeah? AUDIENCE: Your
reputation is such that people choose you
instead of a competitor. And so that cash flow,
from that choice, is coming from your reputation. ANDREW LO: Exactly right. The additional cashflow that you
get based on that reputation-- that's the value of the
asset, or that is the asset right there. Yeah? AUDIENCE: Can assets
have common elements? ANDREW LO: Absolutely. No reason why you
can't have common cash flows across different assets. But nevertheless,
a single asset is a collection of present
and future cash flows. Yeah? AUDIENCE: I'm not sure if
I can explain correctly, but as you said, if I have
one asset, a building, and if she has a
building, and if I am a lender, that means
I have a different cash flow on that asset. ANDREW LO: Yes. AUDIENCE: But however, there
is a market of the building. But if the sequence of the
cash flow, how can you like-- ANDREW LO: We're
going to get to that. That's actually a
very deep question that will require three more
lectures before we get to. But I will answer that exact
question in three lectures, if not before. Before we talk about how
to value these assets I want to make sure we agree on
what a definition of an asset is. So I think we agree, right? Any questions about that? Yeah? AUDIENCE: So if, say, you
have a bunch of patents, but they can't generate
any incremental cashflows they don't exist. Like, you have them,
but incremental cash flow from [INAUDIBLE] matter? ANDREW LO: Well, not
to be anal about this, but mathematically, a
sequence of 0, 0, 0, 0 is a bona fide cash flow. So the zero asset
is possible too. So this is absolutely general. And you might argue, at
this level of generality it's useless. Well not quite, not quite. Because I think it helps us
to formulate a perspective. And the perspective is when you
start thinking about various different kinds of assets-- and
I'm talking really complicated assets, assets with all
sorts of options and triggers and various contingencies-- the more complicated it
gets, the more important this framework is. Because no matter how
intimidating the problem you are faced with,
the bottom line is an asset is a
sequence of cash flows. It's simple, conceptually. The hard part is figuring out
what those cash flows are. But the conceptual framework is
what I want you to start with. Because that will clear your
mind of a lot of cobwebs that really don't belong in
any kind of financial analysis. So we start with cash flows. Sequence of cash
flows is an asset. Now that I've told you that,
there's lots of examples that you can come up with. So I've given you
some in the notes. Please take a look at them. Each one of these is an asset. And each one of these
represents a certain sequence of cash flows. Some may be more
subtle than others. But they are,
nevertheless, cash flows. So I want you to
think about that, read through these
examples, and make sure you understand why they are assets. Now valuing an asset-- that's the question that
[INAUDIBLE] was asking about. How do you value a
sequence of cash flows? Because you and
I-- we might have a very different perspective on
what that cash flow is worth. So for the moment, I'm not
going to answer that question, or rather I'm going to answer
it with a typical device that economists use
all the time, which is I'm going to just
create some notation to answer the question. And the notation I'm
going to give you is V, in particular V sub t. That's a function that
takes as its input a sequence of cash
flows and spits out a number, which I'm going
to call the value of the asset at time t. So how do you value a cash flow? Well, you use the
value operator V sub t. You stick in a cash flow
and out pops a number. We're going to have to spend
the next three lectures figuring out what V is. But I can tell you
one answer right away. In fact, all of you know what
V is, or one definition of V. What is it? Can anybody tell me what V is? Where do you get V from? Yeah? AUDIENCE: What you pay for it? ANDREW LO: That's
right, but you're close. Where did that come from,
what you pay for it? What do you mean by that? Market price. What market? What market? Yeah? AUDIENCE: Any market where
someone's willing to sell it. ANDREW LO: Exactly. Any market. This is what you
were getting at. What you pay for
it-- any market. If there's a market,
you've got your V. So V-- one0 example of a
Vt is a market at time t. That's what we did last time. We had a value operator. Stick in a gift wrapped
box and you get out $45. That's an example of a V.
What we're going to try to do, though, is to take
apart that box and see how it works and
whether or not it does work. AUDIENCE: Would it be any
market or the markets that generate the best cashflows? ANDREW LO: Well we don't
know what best means. So for our discussion right
now any market will do. But then you can have a
separate discussion as to whether or not there are better
markets or worse markets. And we have to define
what that means. So we're going to get to
that when we start talking about how the V operators work. Yeah? AUDIENCE: Does this work
for a perfect market? Or perfect market just
means that the value would be the summation of the-- ANDREW LO: So we
don't know we don't know what a perfect or an
imperfect market is yet. And I'm going to hold off
on that for quite a few more weeks. All we know is it's a market. So let me put it
to you this way-- in the last lecture,
when I auctioned off that iPod and I only
got 1/3 of the price, would you consider
that a perfect market? It didn't feel that way to me. But on the other
hand, it worked. I had something that
I wanted to unload and I unloaded it at a price
that two mutually consenting adults agreed to. So that worked. It was pretty good. But in order for us to
understand whether or not it really works, we've actually
got to take apart the box. We've got to open up Vt. I'm not going to
do that just yet. I want to first acknowledge
that there exist Vts out there. And more often
than not, when you let the market dictate
what that Vt is, you actually get some
pretty interesting results, results that will require
a little bit more structure to interpret. One way to interpret
the structure is to acknowledge that there
is a time element to the cash flows. So when I ask you to
analyze a Vt or an asset, the first thing you're
going to want to do is to draw this
picture right here. And I'm serious about this. In order for you to understand
the value of an asset, you have to know the
timing of the cash flows. Time means everything
in finance. A cash flow today is not the
same as a cash flow next year. Yeah? AUDIENCE: So are we going
to divorce the accounting definition of an asset
from the finance? For instance, land
has to be recorded at book value [INAUDIBLE]
in accounting. ANDREW LO: Yes, right. So what I'm asking you to
do is to think abstractly in terms of the
cashflows themselves. I haven't talked about
accounting practices at all. I'm going to come back to how
we actually implement this. And at that point, it'll
be important to bring in the accounting elements. But for now, let's
set them aside. So these are actual cashflows
that you will receive at different points in time. And by the way, I want you to
use this exact same framework for analyzing
accounting practices. Because financial
analysis can actually be used to ask the question,
are accounting conventions good or bad? Some accounting
conventions are favorable. Some are unfavorable. But the only way
to analyze them is by using this kind
of a framework first. So I would argue that this will
be certainly useful for looking at accounting practices. But it won't
necessarily be the same. So you'll need to have
to make that distinction. So, so far, nothing I've said
is all that controversial, I don't think. I've simply defined that
there is a value operator. Operator means function. You stick in a
sequence of cashflows. Out pops a number. And when you analyze
the value of an asset, I want you to always
draw a timeline and make sure you understand
the sequence of cashflows. Without the timeline I
don't know that you really understand what's going on. So first of all, as a
tip for your midterm and final exams, any time
you have a calculation, which you have to do a present
value or a valuation exercise, I want to see this. I want to see that you know
when things are going to happen. Because nine out
of 10 times, when you make a mistake
in valuation, it's because you lined
this up incorrectly. Now what is V? Well we said that one
example of a V is the market. But is that objective? Is it subjective? Does it work well? How is value determined? And what we're going to do in
this lecture and the next one is to take apart this function
V in the case of no uncertainty. So remember I told you
the two important factors in financial analysis
that makes finance interesting and exciting
is time and uncertainty. And we've seen both of those
things happen this weekend. A lot of time has passed in
resolving the uncertainty with regard to Fannie
Mae and Freddie Mac. What I'd like to do is to
abstract from the uncertainty part. For the next couple of lectures
there's no uncertainty. I'm going to get rid
of the randomness. And I'm only going to
focus on figuring out the value operator, this V
sub t function, for cases where we have cashflows at
different points in time but where the cashflows
are known for sure. So there's no uncertainty
about whether they're going to happen or not happen. After we do that, after
we do the no uncertainty case, for which we have
a complete solution, I will then come back
and introduce uncertainty in a somewhat more natural way. And then we're going to focus
on valuation with uncertainty as well. So let me start with
the perfect certainty case about how to figure
out what this V sub t is. And I'm going to start with
a very simple example that has to do with
manipulating cashflows. And I'm going to
talk in particular about foreign currencies. So I wanted to ask
you, what happens when you add 150 yen to 300 pounds? What do you get when
you add those two? 150 yen plus 300 pounds
is equal to 450 what? $450? If you believe that, please
see me after class and we'll need to do some transactions. Well look, obviously 450
makes no sense, right? It's like adding your
weight to your age. That number might
be interesting, but there's no
interpretation for it. So 150 and 300-- you would never add those
two and talk about 450 what. You don't even have a unit
to be able to ascribe to it. Well that natural
reaction that you have, that natural aversion you have
to adding up dollars or yen in different currencies-- I want you to develop that exact
same attitude towards cashflows at different points in time. So let me continue on with this. Obviously if you
want to add the two, you know that you
have to convert them to the same currency. So 300 pounds converts to
a certain number of yen. And so when you add that
up, you get 46,050 yen. Or if you want to
convert it to pounds, it converts to 300.98
pounds sterling. Those numbers make sense. The reason they make
sense is because they're in the same units. So apples to apples is the
familiar adage that we use. You can't add apples to oranges. Actually, in theory you could. You could make fruit salad. And you need apples, and
oranges, and pears, and all that. But for argument's sake,
when it comes to money it really doesn't tell you
much that you've got 450 blah. You need to have a set of units. Well it turns out that this
actual analysis requires that we pick a base currency. We pick what's
called a numeraire. A numeraire is a
unit of account, or a standard, by which
we measure everything. So either yen or pounds
will work just fine. It doesn't matter which. If you're a British investor,
you might care about pounds. If you're a Japanese investor,
you might care more about yen. It just depends upon
your perspective, but either one will
do for the purposes of analyzing how much money
you have at a point in time. Well that exact
same exercise has to be applied to money
today versus money tomorrow. Because those two
things are not the same. It's like yen and pounds. They're not the same. They can't buy the same things. They aren't used
in the same way. They have different markets. The markets are related,
but they're different. So once you know
the exchange rate, and once you pick
a base currency, you can combine the thing. The same idea holds
true for cashflows at different points in time. Cashflows at different
points in time are like different currencies. In order to add them
together, you've got to use the appropriate
exchange rates. Once you do, then
all of the valuation that we're going to conduct
over the next couple of lectures becomes absolutely trivial. So let give you an
example of that. Here's my timeline. And my timeline has payoffs
at various different dates. Now I'm arguing that the
past and the future cannot be combined without
converting them. So we have to have a
set of exchange rates that convert all the
various different currencies into a numeraire or
single base currency. And for the sake
of argument, I'm going to just pick,
times 0, dollars today as my base currency. So in other words, I want
to convert every cashflow into the single currency
that is today's dollars. So in particular, over
the course of t days, days 1 through t, how many
different currencies do I have apart from my base
currency of time 0 money. Yeah, I've got t different
currencies, t dates. Every single date, it's
a different currency. And I need to convert them
in order to add them up. So I'm going to argue that
sequence of cashflow, CF1, CF2, dot, dot, dot-- if
I want to figure out what it's worth in a single
currency, date 0 currency, I got to multiply the cashflows
by the appropriate exchange rate. What's the exchange rate? The exchange rate is
how many dollars at date 1 per dollar at date 0? How many dollars at date
2 per date 0 dollar? How many dollars at date
3 per date $0 and so on. I'll have t exchange rates
for the t different kinds of currencies that I have. Each currency is
dollars denominated as of a particular date. Any questions about that? Pretty straightforward. You think it's
straightforward, but you're going to have to think
about it for a little while and do some problems to really
make sure you appreciate this perspective. Now we're ready to talk about
what this value operator is. I'm going to argue that
once you've got the exchange rates across the various
different currencies, then the value of the
collection of cashflows is just the sum of
all of the values of the cashflows denominated
in the same currency. But instead of currency, meaning
pounds or dollars or yen, the currency is going
to be date 0 dollars. Questions about this? This is important now. So I'm going to define something
called the net present value operator, NPV, as the date
0 value of the cashflows. Now why is this called
net present value? Because typically, when you
are valuing an investment, you have to pay
some money up front and then you get
cashflows later on. And so your net
investment is going to be given by some
summation of those pieces. But it's not a simple sum. You can't just add the numbers. That would be like adding 300
yen to 150 pounds or whatever. You have to figure
out how to convert all the currencies to the same
unit of account, dollars today. And so that's why it's
called present value. That's the value at
the present, today. And the reason
it's net is you're netting out any initial
investment, which is the cashflow at time 0. So V sub 0 is equal to CF0. Notice, there's
no exchange rate, because we're doing
it in today's dollars, so that's the base currency-- plus the cashflow tomorrow. Multiply by the exchange
rate between dollars tomorrow and dollars today and so on. If there's an
initial investment, then the initial cash
flow is negative. That just means you
have to put money in to the project
or the investment. And then, over the course
of the next t periods, you get cashflows, which could
also be positive or negative. And you have to convert
them into today's dollars. And once you do, you
look at that number and you see whether
or not you like it. That's the value, the
net present value, of the cashflows. Now I have one unanswered
question for you. What's the unanswered question
in all of this analysis? Sounds good, right? This is pretty simple stuff. But I pulled something
out of the air. What did I pull out of the air? AUDIENCE: The exchange rate. ANDREW LO: Exactly, exactly. Where do I get the
exchange rates? You should be asking me, where
do you get the exchange rates? And you know what I'm
going to tell you? What's the answer I'm
going to give you? Where do I get the
exchange rates? AUDIENCE: From the market. ANDREW LO: From the
market, exactly. From the market. I get it from the market. How do I get it from the market? Well, we did it last time. You know how. We're going to auction it off. You want to see how? I've got a security that
pays $1 a year from today. It pays $1 a year from today. Who will offer me a penny
for that piece of paper? Who will offer me $0.50
for that piece of paper? $0.75? $0.80? $0.90? $0.95? $0.97? $0.98? Well, all right. $0.97 to the dollar. There you go. That's the exchange
rate right there. We're done. And by the way, I
appreciate that. That's a lot of confidence
in my credit worthiness. It's only a 3% discount. All right, well that's
the exchange rate between today and next year. What about today and
five years from now? Where do we get that from? AUDIENCE: The market. ANDREW LO: Exactly, the market. OK, I've got a piece of paper. It pays $1 five years from now. How many people will
pay me a penny for that? We'll go through
the whole motion and come up with the
same kind of process. And we'll get a number. So the market--
now in a few weeks, we're going to have to
question whether or not that is a good way of
getting the exchange rates. I'm not going to talk
about that just yet. But that's where we get it from. So I've begged the
question a little bit, but I think I've actually
made some progress in providing a framework
to think about valuation. Once we establish
the exchange rates, you will agree with me that
I can value any, any cashflow whatsoever where the
cashflow payments are known with certainty in advance. So that's a pretty
impressive achievement in the space of 45 minutes. We've actually figured out
how to value any cashflow whatsoever under
perfect certainty, given the exchange rates. And I've told you where to
get the exchange rate from. You get it from the marketplace. That's why financial
markets are so important. It's because we require those
inputs into our valuation process for doing the analysis. We rely on financial markets. Financial markets didn't
exist, we can't do this. I'd have to wave my
hands and say, well, you can get it from some kind
of theoretical source. You just make it up. Just come up with some numbers. That doesn't sound
very compelling. And in fact, it's not. The power of financial markets
is the wisdom of the crowds. And like it or not, you're the
wisdom that we're tapping on. We're tapping your
wisdom to come up with these exchange rates. So we know now that the value
of a sequence of cashflows is simply equal
to today's dollars when you use these exchange
rates to convert them to present dollars,
dollars today. And that's often
called present value. Notice that these exchange
rates are sometimes called discount factors. The reason they're
called discount factors is because typically they are
numbers that are less than 1, like what we saw today. $1 next year I was able to
auction off for $0.97 today. That number is smaller than $1
because people are impatient. $1 today is worth more
than $1 next year. So if I promise
you $1 next year, you're only willing to
pay me a little bit less than that for that privilege. So it's a discount over
what it would be on the day that it gets paid. AUDIENCE: One question. ANDREW LO: Yes? AUDIENCE: What exactly
are you discounting if there's perfect, perfect? My impatience? ANDREW LO: That's
a great question. Let me repeat it for
everybody to hear. The question is, what
are you discounting if there's no uncertainty? Why should $1 next year be
worth less than $1 today? Because there's no uncertainty. I've ruled out
uncertainty, right? So there's no default risk. It's impatience. People want to consume
now versus later. And therefore, if you're
going to force somebody to consume later, you've
got to somehow offer them some incentive to do that. So in order for me to
have you give me $1, so that I can take
the dollar now, and I'll pay you
back a year from now, so that you can't consume
that dollar today, you have to wait a year
to consume that dollar. In order for me to
do that, I actually have to make it
interesting for you, which means you give me $0.97
now and I give you back $1 next year. I pay you extra for that
time that you have to wait. Yeah? AUDIENCE: Could you
also make an argument that I could take my
dollar now and give it to somebody who needs to
use it now and charge them for the privilege of
using my dollar now? ANDREW LO: Absolutely,
absolutely. We haven't talked
about the mechanism for determining what
that exchange rate is. And I'm going to come to that. But before we even
talk about that, I want to acknowledge that
it exists, and it's real, and we can figure
out what it is just from a simple auction of
market, and we can use it. But I'm going to get to
exactly what it means and how we arrive at
those exchange rates. That's going to be a very
important part of this. AUDIENCE: For inflation, we
know that because of inflation $1 today is also much
less than $1 next year. ANDREW LO: That's right. I'm going to get to that
at the end of this lecture. That's another reason
why there might be some kind of discounting. It's not just the
time preference. AUDIENCE: That's
part of the answer. ANDREW LO: That's
right, exactly. So we're going to come
back to that as well. AUDIENCE: Is there such thing
as a deflationary environment? ANDREW LO: Deflation can occur. Negative real interest
rates can occur during certain unusual periods
of economic development. So that's a possibility. It's not a possibility that
we've seen in the United States in recent years. But other countries
have experienced that. And when that happens, there
are some really serious repercussions. So we're going to come to that. In fact, in 2004
the US government, early on in that year,
was very concerned about the possibility
of deflation. And it was only
towards the end of 2005 that inflation became
more of a concern. So we're going to
come to that as well. AUDIENCE: Would you separate
the concept of inflation from the time value of money? Would you call those two
different fundamental concepts? ANDREW LO: Yes, I would. And let's hold off
on inflation for now, because I don't want people
to get confused by it. It's something I'm going to
bring in at the very end. So I will come to that. That is a topic for
lectures two and three. So we now know how
to value cashflows, at least using this very simple
framework with exchange rates. And here's an example-- just a very, very simple one-- where I've got the two discount
rates, or exchange rates, for next year and
the year after. It's $0.90 to the dollar
and $0.80 to the dollar. Notice that the farther
you go into the future, the more of a discount these
exchange rates require. That's also human
nature having to do with impatience, and
possible inflation, and other kinds of phenomena. But again, I'm not going to tell
you how we got those numbers. I got them from the marketplace. Now what's the net present
value of a project requiring a current investment
of $10 million with cashflows of $5 million
in year 1 and $7 million in year 2? That's trivial. Here you go. Minus $10 million today,
$5 million next year, but next year's dollars is not
the same as today's dollars, so you've got to convert
it to the right currency. So that's $0.90 to the
dollar multiplied by 5. And then $0.80 to the
dollar multiplied by 7. When you add it up you get a
$100,000 of today's dollars. That's what this
investment is worth. So the first question is,
should you take this investment? Is it a good project? Well let me ask you,
do you want a $100,000? Yeah. If you don't, again,
see me afterwards. I will help you
with this problem. Remember the first
day, I told you that once you figure
out valuation, management is trivial. Well, I wasn't just kidding. This is an example. What we've been able to
do is to take a problem. What's the net present
value of a project requiring this
sequence of cashflows? We've been able to
take that problem, that management problem,
should you do this or not-- we've been able to reduce
that to a simple question, do you want $100,000? I think the answer is yes. So that's the management part. Let's do this. Done. The valuation part
is the hard part. The management
part, the decision, is easy once you have the
right numbers in front of you. Second question--
suppose a buyer wishes to purchase
this project but pay for it two years from now. How much should you
ask for this project? Now the first thing
you need to do is what? AUDIENCE: [INAUDIBLE]. ANDREW LO: Well, that's
the second thing. The first thing you need to do-- draw a timeline. Draw a timeline so you know
exactly what's going on. Two years from now. Is that next year or is
that two years from today? So draw a timeline. And you can figure
out, once you have the timeline, what the present
values and future values are. So when you work
out the numbers, you can actually make a decision
and figure out exactly what you ought to charge for that. That's it for the
time value of money. But I want to summarize
by telling you what we've learned so far. We've learned that we can
value cashflows-- assets, which are sequences of cashflows. And the way we value
that is by making some implicit assumptions. The assumptions are we know
the cashflows in advance, so there's no uncertainty
about what the cashflows are. Second, we know
the exchange rates. And if you don't know
them, go get them. Where do you get them? You get them from
the marketplace. And finally, there
are no frictions in the currency conversions. One thing you
didn't ask me about is that typically,
when you go abroad and you have to convert
your currencies, what do you have to do? You have to find somebody
that converts it for you. And typically they will charge
you a fee for that conversion. I've been talking as if
the conversions happen without any kind of frictions. That's an assumption
that can be tested. So we're going to talk about
frictions later in the course. But for now, we're
going to assume that there are no frictions. Now of course, these
are all hypotheticals. They're all approximations to
a much more complex reality. But once we understand
how the model works with these simplifying
assumptions, we can then go back and say, OK,
let's make the assumptions more realistic, and let's see how
that affects the implications. So I'm going to first do the
plain vanilla version of this. And then, after we
fully understand it, I'm going to come back
make this more complex. So until lecture 12-- that's a long ways away-- but until lecture 12
I'm going to assume that these assumptions hold. And after lecture 12 I'm going
to go back and systematically question, and expand and
revise, each one of these. So when I told you during
the first day of class that you can't handle the
truth, this is what I mean. We're going to
start simple and try to understand the implications
of these simplistic assumptions. And then, little
by little, we're going to make the assumptions
closer to the truth. Now more examples-- once we
understand how this exchange rate mechanism
works, we can look at all sorts of
other alternatives to calculating value. So in particular, one
question that you might ask, that was asked, is $1 today
should be worth more than $1 in the future because
of impatience, because of inflation, because
of all the reasons we described. But the bottom line
is supply and demand. More people want money
today than money tomorrow. And so supply and
demand, bottom line, dictates that there is this
difference between the two. So we can actually
figure out what that difference is by looking
at the exchange rates. So in particular, $1 in year 0-- if you hold it for a year it'll
turn into $1 times something, the reverse of the exchange
rate of $1 in year 1 brought back to year 0. But I'm going to write it in
this format, 1 plus something. Because remember, when
we're going the other way, when we have $1 in
year 1 and we want to figure out what is it
worth in today's dollars, that's a discount. It's a number less than 1. So if you want to go
the opposite direction, if I have $1 today
and I want to know how much is that dollar
worth a year from now, it should be greater than 1. In fact, it should be the
reciprocal of the exchange rate going the other way. Instead of looking at
pounds, you look at yen. And so it's the opposite
of the same exchange rate. But the way I'm going
to write it is 1 plus r. Because it'll be greater
than 1, typically, so r will be a number
typically greater than 0. What about for two years? I'm going to simply write it
as 1 plus r quantity squared. And the reason I do
that is because I'm thinking about taking
that same growth rate and applying it
two years in a row. I could have written 1 plus z. And z then would be the rate of
growth over a two year period. But I like to think in
terms of annual frequencies. So I'm going to
just simply assume that a year is my
unit of account. And then-- looking at
a one year, two year, three year growth period-- I'm going to simply take
powers of this factor 1 plus r. So $1 in year 0 is going
to be worth $1 times 1 plus r to the
T-th power in year T. This r is often called the
opportunity cost of capital. In fact there are like, five
different names for this r. It's called the interest rate. It's called the growth rate. It's called the cost of capital. It's called the opportunity
cost of capital. John Maynard Keynes
called it the user cost. There are tons of
names for this quantity but the basic idea is it's the
reverse of the exchange rates that we were looking at before
when we were trying to bring everything to today's dollars. Now the reason I'm
showing this to you is because it turns
out that we're going to want to move money
back and forth through time. And in order to do that, we
need to sometimes multiply and sometimes divide. And I want to come up with
just one set of notation, as opposed to all these
exchange rates floating around. So remember when
we had T periods and I wanted to figure out
what the value of an asset was today given T periods. We had to have T exchange rates. That's a pain in the neck
to carry around T numbers. And moreover, there
are exchange rates that you have to deal with
between dates T plus k and T plus j. So any two dates,
you've got to have an exchange rate between them. So pretty soon the
number of exchanges you've got to keep in
your head is ridiculous. That's why they came
up with the Euro. They tried to unify some
of these exchange rates. So here is a way to unify all
of the exchange rates into one number, little r. Once you know little
r you know everything. You know all the exchange rates
for every possible two dates that you care about. Now where do we get
a little r from? AUDIENCE: The market. ANDREW LO: Exactly. You're learning. So we get it from the market. Now it's going to be a little
bit more complicated than that. It turns out that the market's
going to give us many different r's. So we're not going
to get to that yet. So for now, let's just assume
that the market gives us one r, this r. Once you have this
one number, r, it will allow you to do
all of these calculations back and forth. You can move money
back and forth through time because you
will know what the exchange rate is between any two points
in your cashflow sequence. Now that we have
the r, let's go back and figure out what
the exchange rates are, what we were talking about in
terms of bringing stuff back to date 0. And it turns out that
there's a really simple form. It turns out that $1 in year 1-- if you paid me $1
in year 1, and I wanted to figure out
what it was worth today given this r
number that I used, it's actually just equal
to $1 divided by 1 plus r. So what's the exchange rate? AUDIENCE: O. ANDREW LO: What's that? AUDIENCE: O. ANDREW LO: That's not
the exchange rate. What's the exchange rate that
allows us to take $1 next year and bring it to $1 today? AUDIENCE: Is it 1 over-- ANDREW LO: It's 1 over 1 plus r. 1 over 1 plus r. Remember, r is a
number greater than 0. So 1 over 1 plus r is a
number less than 1, like $0.97 to the dollar. So the exchange rate between
next year's dollar and today's dollar is 1 over 1 plus r. What about the exchange
rate between $1 two years from now and today. What is that? 1 over 1 plus r-squared. That's right. And so on. So the exchange rates,
or discount factors, are simply related to the
little r's in this manner. That's the mechanics and
mathematics of present value. Now this might seem intuitive. And when I say it you
might understand it or you might think you do. Please, go back-- between
today and Wednesday go over this and test yourself. Try to figure out
for a given r-- if you have $100
today, and r is 7%, what is $100 worth
three years from now? Or if you have $180
three years from now, and you want to figure out
what it's going to be worth not three years from now but a
year from now, and the r is 8%, what is it equal to? Test yourself by coming
up with little examples to challenge your own
understanding of this concept. Because it's going
to be critical. We have to get this
right, because everything that we build from
here on in is going to rest on the foundations
of these calculations. Now, with these r's,
I'm going to leave you with this one last
concept, which is that the value of
the cashflows that we've been talking about
at time 0 is simply equal to the cashflows
multiplied by the exchange rates. And now, with this definition
of r, this opportunity, cost of capital, this interest
rate, this discount rate, we then have an expression for
what this value operator ought to be. Using this expression,
any cashflow in the world can be valued under
perfect certainty and under the assumptions
that we described. Any cashflow can be
valued, therefore any asset can be valued. So what we've done in
the space of a lecture is to create a
valuation operator that works for virtually
anything under the sun in certain simplified
circumstances. That's a major achievement. You can congratulate yourself
that we've reached this level. But we're going to have
to do some more work to think about how to use this. So we're not done yet. We have some additional
analysis to do. But this is a wonderful
starting point. Next time, we're going to focus
on how to take this and apply it to two very, very special
cashflows, an annuity and a perpetuity. And you know what
this is going to do? This is going to allow you to
figure out what your mortgage payments are every month, when
you go apply for a mortgage. You'd be surprised at how
subtle that calculation is and how many bankers
don't know how to do it. Well, you will. All right, I'll
see you next time. [APPLAUSE]
