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visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: First of all, any
questions from last lecture? Yes? AUDIENCE: [INAUDIBLE] he said
he was [INAUDIBLE] possible [INAUDIBLE]? ANDREW LO: OK. So let me repeat the question
to make sure everybody heard. The question about net present
value is that, is it possible, is it possible, that
in one currency, the net present value of
a project is positive, but in a different
currency, it is negative? That's a very
interesting question. And it turns out that the answer
is staring us in the face right here. Now remember, we're in a
world of no uncertainty. So we know what future
cash flows are going to be. And we know what future
discount rates or discount factors are going to be. That's my assumption. And in that world,
when I give you the value of a
sequence of cash flows, this v sub 0, if I wanted
denominate it in dollars, then presumably all the cash
flows have to be in dollars. If I want to
denominate it in yen, then the cash flows
have to be in yen. So strictly speaking,
assuming that the exchange rates don't change over time-- and that's, again,
a big assumption-- the question is, can I
have a different result in terms of the sign of a
net present value by changing the exchange rate? Any thoughts on that? What do you think? Yeah. AUDIENCE: No. ANDREW LO: No, why? AUDIENCE: Because
currency [INAUDIBLE].. ANDREW LO: OK. So the answer is no,
because currency, the exchange rates always
have to be positive. And presumably, you're
multiplying the cache flows by the same number, either
positive of one number or positive of another number. So when you multiply a sequence
by a positive number, when you add that up, it is
either still positive or still negative. In other words, you
can factor it out. Right? You sure? Yeah. AUDIENCE: I have a question. When we are doing this
in the [INAUDIBLE],, is it possible to have
different [INAUDIBLE]?? ANDREW LO: Well. Right now, we're not
talking about risk. So let's hold that off for
seven or eight lectures. I want to ask this question. Have I got it right? We agreed that no matter
what you multiply it by, as long as it's a
positive number, it can't change the sign, so
the currency doesn't matter. Yeah. Ernest? AUDIENCE: But the exchange
rate, so the actuals are at different times. ANDREW LO: Yes. AUDIENCE: So if
your exchange rate is different at different
times, then it's going to stay factored
throughout the-- ANDREW LO: The assumption
is that it's fixed. There's no uncertainty. But-- AUDIENCE: [INAUDIBLE]. ANDREW LO: I didn't
say it was the same. So you said that
it was the same. I didn't. You're right. So [? Shlomi, ?] you're right. If the exchange rate
is the same over time, then when you multiply
by one number, it's the same number
for every cash flow. Then, it factors out. And then you're multiplying v
sub-zero by a positive number. So if v sub-zero is
positive, it stays positive. If it's negative,
it stays negative. But no uncertainty doesn't
mean that it's fixed. So here's the subtlety. The subtlety is that if I assume
that the exchange rate is fixed and known, but going up over
time, whereas in US dollars, it stays fixed, that
makes a difference. Right? So it's possible. It's possible that if I change
currencies and the currency is rapidly appreciating
or rapidly depreciating, then you can actually
change the net present value of the project. But it has to be the case
that the particular path of the currency
appreciation or depreciation is exactly opposite what's
going on with the NPV. So the bottom line is, you've
got to do the calculation. And you have to use the
currency that you care about. So if you're in
US, you presumably care about getting
paid in US dollars. You would use US dollars. If you're in Japan,
you get paid in yen. You'll want to do it in yen. And you have to do the
currency conversion. Now when we talk
about uncertainty, that's going to make it
much more complicated. It's going to introduce
another component of risk in our calculations that
has to be dealt with. So we're going to
come back to that. But that's a good question. Anybody else? Yes? AUDIENCE: I noticed
that you used the term paper a couple of times. I just wanted [INAUDIBLE]
definition of-- ANDREW LO: Of what? AUDIENCE: Paper. ANDREW LO: Paper. You mean, this is
a piece of paper? AUDIENCE: Well, I don't
think [INAUDIBLE].. ANDREW LO: Right. Yeah, so typically by paper,
people mean a security. And commercial
paper is a security that is a debt instrument
that is basically an IOU. It's like a bond. So we'll come back to that
when we talk about fixed income securities. But that's what I mean. By the way, you
raise a good point. When I mention terminology,
feel free to ask me. But in turn, I'm going
to feel free to tell you, you may want to look that up
in [? Breeley, ?] [? Myers ?] and Allan, because I want you to
read the book alongside of what we're doing in class, because
you'll need to pick up this terminology, and we don't have
enough time in this 20 lectures to cover all the terminology
that you need to know. So don't assume that just
because I haven't covered it in class, or that I
haven't defined it that you don't need to know it. The textbook is
there to help you with the supplementary material
that I would like you to cover. So that's why we
assign those chapters. OK? Yeah, Justin. AUDIENCE: [INAUDIBLE]. ANDREW LO: Yes. AUDIENCE: Then I
read a news article, and they said the stock
market jumps because they're getting bailed out. ANDREW LO: Right. AUDIENCE: So is there a simple
reason as to why this is such a massive increase in stock-- ANDREW LO: In the stock
market, while their stock has gone down. AUDIENCE: Right. So that seems a little
counter-intuitive. I'm going to give you a
two minute answer now, but then I'm going to give
you a much deeper answer in about three or four
lectures, when we actually apply all of the
framework we're developing to pricing common stock. So as I said with
Freddie and Fannie, there are two components. There are two sets of issues
surrounding those companies. One is the value of the owner's
equity, the folks who owned a piece of those companies. What are their
investments worth? And the answer is very little. The second piece is
that Freddie and Fannie have issued all sorts of
IOUs, all sorts of obligations to counter-parties. And the question is, what
are those securities worth. The government bailing
out Freddie and Fannie are basically saying, we
will stand behind those IOUs. The shareholders
of the company-- sorry, you guys lost. The company has not done well. It suffered a lot of losses. So the fact that you own
a piece of the company means that what you
own is now worthless. But the pieces of paper
that the company has issued, we will assume that obligation
as the US government and make good on
those obligations. So the fact that
those pieces of paper have much broader impact on
the market as a whole, the fact that the US
government is standing behind those pieces of paper
will protect the stock market as a whole because
there's confidence that business conditions will
not be as bad as we thought. So that's what explains
the fact that the stock market as a whole went up. It's because the
market environment has been stabilized. You can imagine what might
have happened if Fannie and Freddie were to go under. Their pieces of paper, their
IOUs, would be worthless. Which means the folks that
own those pieces of paper, now they have a bunch
of worthless paper. And when that happens, there
are repercussion effects for those businesses,
and those businesses will end up losing
money, which will have repercussions for the
entire market as a whole. AUDIENCE: [INAUDIBLE] the
amount that it went up shows how their
paper was distributed to all these other companies. ANDREW LO: It's a
combination of how their paper was distributed. But more than that-- I mean, there are many
companies in the S&P 500, for example, that don't
own any of this paper. So why would their
stock be void? It's because the
business conditions have been stabilized, and there
won't be any knock on effects. A good example of this
is Lehman Brothers. As many of you know,
Lehman Brothers is a big player in these
kinds of securities, and they are currently
under a lot of pressure. Their stock prices
dropped dramatically, even in the last few days,
because they are a big mortgage lender, and CDO investor,
so they're actually hit pretty hard by all of this. And while the rescue
of Freddie and Fannie has had some positive effects
on Lehman's stock price, it still is under fire
and a lot of people want to get rid of it. Imagine if Freddie and
Fannie weren't rescued. It's almost a sure
thing that Lehman would have gone
under immediately as a knock-on effect. And if Lehman went
under, well, I mean, there are other
investment banks out there that might have gone under. And now all of a sudden,
you have a series of very large companies
that do business with all of Wall Street
that it has gone under. That's going to have bad
repercussions for the stock market as a whole. Yeah? AUDIENCE: [INAUDIBLE]
companies, like big companies? ANDREW LO: Well, the short
answer is I don't know. Nobody knows. I think that there is a concern
that the Fed cannot be viewed as rescuing every possible
financial institution that's out there. It's got to stop at some point. Many people said it
should have stopped even before the Bear Stearns rescue. So the answer is we don't know. Wait and see, and we'll find
out over the next few days. As I said last
time, these are very interesting times for
financial markets. Very, very serious issues that
are coming to the forefront literally every day. So we're going to
be watching that, and we'll be talking about that. Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Where do I
think that should stop? Well, well, there are
a couple of issues that are at the heart
of these discussions. The two issues are,
how do you balance of the cost of bailing out
these large organizations and the implicit moral
hazard that it creates, the kind of potential promises
that you're implicitly making to future equity
holders of these organizations versus letting the market work
against the potential disaster scenario of allowing
these kinds of events to spread like wildfire. I don't know how
many of you actually know what happens during
wildfires, during forest fires. But when forest
fires get started, they're actually very
difficult to stop. And every once in
a while, they try to stop a forest fire by
creating additional fires. Right? This may sound
counter-intuitive. But what they will do is
around a raging forest fire, they will burn what's
called a firewall. That term did not
come out of IT. It actually came out of
fighting forest fires. They will burn a ring
around that forest fire, a controlled burn where they
target very specific set of trees, and they would do
it in a controlled fashion, so that when the forest
fire gets to that ring, it burns itself out. And one could argue
that we need a firewall around these kinds of events. We need to have certain
financial institutions fail and stop the spread of
this kind of problem. The difficulty with that analogy
is that with a forest fire, all you need is a helicopter
to get up there and see what's going on. We don't have a helicopter. There's no helicopter that
tells us where the fires are, and where the fires may be, and
where the underground gasoline tanks are hidden for
future explosions. We don't know because a lot
of this stuff is hidden. So my own opinion
is that we are going to need to have at least one or
two additional large failures, and people will have to lose
money before they understand that this stuff really is risky,
and that the price you pay for the benefits that you've
gotten from these very handsome returns in the years
before this kind of an event is the fact that
every once in a while, in the parlance of Wall Street,
you get your face ripped off. That's the nature of
financial markets. So I think that
it's very dangerous to rescue these companies. But at the same time,
you have to balance that against the risk of
creating a mass panic. And if we do create
that mass panic, there's virtually
no way to stop it, and then we will run into a very
deep recession and depression of the likes that we
haven't seen since 1929. That's the balance
and the danger. Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Well, you
know, that might be. But let me suggest this. Let me put that off
for a discussion point until we finish fixed
income securities. Because at that point,
I'm going to talk about the subprime
problem specifically. And I'm going to use the tools
that we develop== actually, you guys are going to use the
tools that we develop to figure out exactly what's
happened in these markets, why they're happening, and how
maybe we can get around that. So let me not give
you my view now. I'd rather have you
develop your own views based upon the tools we
develop in this course. OK? Yeah? AUDIENCE: Because of
all this [INAUDIBLE] CEOs or executives were fired
to get a big handsome buyout for all their hard
work and efforts. ANDREW LO: Yeah. AUDIENCE: But now,
should the market be able to self-regulate itself? Or does there need to
be regulation in place? Or what will become of it? ANDREW LO: Well, you
know, that's again a very difficult question
to answer because we're not done yet, so we don't know
where this is going to end up. I think that there are some
very important issues that we're going to have to come back to. Let me put that off
for even a bit longer because when we talk
about corporate finance, we're going to talk
about CEO compensation and ask the question, how
do we relate compensation to performance, and
does it make sense? It turns out that there's some
incentive issues, such that if we don't do that,
if we don't allow them to have these
golden parachutes, then it may end up
creating weird incentives when things are going well. So every action has some kind
of equal and opposite reaction in some other part
of the system. And unless you know
what that system is, it's hard to figure out
the answer to the question. So by the end of
the semester, I'm hoping that you'll be able
to come up with answers to these questions. So let me put that off
for a little while. OK. One more clarifying question
maybe, and then we can move on. AUDIENCE: During the
Southeast Asian Crisis in '97, there was this discussion about
the international financial institutions should risk
[INAUDIBLE] countries and because of the
bar [INAUDIBLE].. ANDREW LO: Right. AUDIENCE: And they
decided they should, so they rescued them
and they survived. 10 years later, Latin
America went into a crisis, and the same
discussion started, and the international
financial institutions, led by the United States
decided not to rescue them. So we went into a crisis. And so I see now [INAUDIBLE] ANDREW LO: That's right. Yeah, that's a
very serious issue. But I would argue that
issue actually goes even-- it goes to an even broader
set of issues that have little to do with economics
and finance, but political and social
issues, which I won't comment on in this class, but which are
important for determining those kinds of policy questions. That's one of the things that
I'd like to get across to you in terms of thinking
about these issues, which is that there are multiple
aspects to every issue. And rather than trying to come
up with a single answer, what I would propose
that you might do is when you think about a
challenge like this, first of all, you try to identify
the different issues and then come up with an answer
for every single perspective of that issue. So for example in the
case of Latin America, there is certainly the economic
issue and moral hazard. That's an important one. But there's also a
political and social issue, which is that if you don't bail
out countries that are in need, that's a recipe for
creating social unrest. And if you don't do it,
there is some dictator waiting with guns and other
interesting possibilities for the people to
try to take over. That's right. And I mean, it's
not rocket science. I mean, people are
looking for solutions. And if you can't
offer one, they'll go to the next
person that has one. Whether or not
it's true or false, they will try to come up
with some kind of leadership. So how do you balance off
the economic considerations against the
political and social? That's not something that
an economist can answer, so I won't even try to begin. And by the way, my opinion is
no better or worse than anybody else's. So I won't waste
your time with that. But what I would suggest is
from looking at these issues, first of all, try
to think clearly about what the economic
issues are, and then what the social and political
issues are, and separate them out. And then you can answer each
of those questions in isolation and, at the end, decide
on how you want to balance these kind of considerations. But don't use economics to try
to answer a political question, and don't use politics to try
to answer an economic question. You should use
the tools that you have to answer the questions
that those tools are designed for. And in the case
of Latin America, I would argue that's a very
complex set of issues that economics alone cannot answer. The economic answer,
never bail out countries that are
failing, because you'll create moral hazard and
increase the cost of borrowing for future generations
in other countries. That sounds good
until you see what happens when you
don't, and you get these socialist
dictatorships that end up creating all
sorts of dislocation for the people in the country. I mean, that there's a very
big cost to that as well. And I'm going to have to
beg the question about how you balance those costs
against the benefits. Again, that's something for
politicians and for voters to hopefully to decide. Yeah? Which? AUDIENCE: [INAUDIBLE]. ANDREW LO: No. Sorry. AUDIENCE: [INAUDIBLE] has
renounced the United States treasury-- ANDREW LO: Yeah. AUDIENCE: [INAUDIBLE]. ANDREW LO: That sounds good,
but that wasn't my handout. So that might be my handout
in about three weeks. But we have work to do now. So let me let me stick to that. And we'll come back to
these interesting issues. But I want to give you the
framework and the tools to be able to think about them. OK. So let me continue on. This is Lecture Three. And we're going to
continue looking at present value relationships
and the time value of money. Last time, we were left
with the expression for the value of an asset as
simply being equal to the cash flows discounted with the
appropriate discount factors, where I've assumed for
simplicity that the discount rate between one year
and the next is constant and given by the interest rate,
or discount factor, or cost of capital, or user cost,
or opportunity cost, r. Fancy terms for
the simple concept of the number that you use
to construct these exchange rates between cash at
different points in time. Now the solution of how you
make management decisions given this simple framework
becomes trivial. Take projects that
have positive NPV. That's it. When you figure out what
the value of a project is as a function of all
of these exchange rates, you calculate what
the present value is. And if the cost
of the investment is included as a cash flow,
possibly a negative cash flow, you've got the
net present value. And for things that are
positive NPV, you want them, you want to take them. For things that are negative
NPV, you don't want them, you don't take them, or
if you can, you sell them. All right? Now, there are many
different assumptions that got us to this point. We understand that. We're going to make
those assumptions more and more realistic over time. That's, in fact, what
the rest of the course is going to be doing. We're going to be focusing
on picking this expression and making it more realistic. And it's going to take us
12 more weeks to do that. So it's non-trivial, but
that's exactly the objective. Yes? AUDIENCE: Last week, you
said [INAUDIBLE] summation of cash flow. ANDREW LO: No. I said the asset was a sequence. What is an asset? An asset is a sequence
of cash flows. That's the definition of
an asset, not the value. The value of the
asset, remember, is that function that you
stick in a cash flow sequence, and out pops a number. So the value of an asset is
not the same thing as the asset itself, right? You can have a rocket ship
that can go to the moon. That is an asset. The value of a rocket ship
that goes to the moon, that's a different thing, right? You need to have this v
function in order to figure out the value of an asset. But I can't really talk
about the value of an asset unless I've defined the
asset to begin with. So v sub-zero is the
value of the asset. It's not the asset itself. It's the value of the asset. The asset itself is the
sequence of cash flows. Now, here's a simple
example about how these discount factors work. This is just an
interest rate example. If you let little r
equal 5%, then you can figure out what the value
of a dollar is in the future, or you can figure out what the
value today of a future dollar is. It's just using
simple arithmetic to be able to do that. So this is just a simple
concrete illustration. And if you graph the present
value of a dollar, over time, you'll notice that as
time goes out farther, the present value of
a dollar declines. Not surprisingly, $1
today is worth more than the dollar tomorrow. But $1 tomorrow is worth more
than $1 two years from now. And $1 two years
from now is worth much more than $1 an infinite
number of years from now. Right? Now here's an example of how
you use this valuation approach. And the problems that
we handed out last time will give you practice in how
to think about present value. So I urge you to
do those problems to make sure you really
understand these concepts. Here's an example
where a firm spends $800,000 every single
year for electricity at its headquarters. And by installing some kind of
specialized computer lighting system, it turns out that you
can reduce your electricity bills by $90,000 in each
of the next three years. Now, of course, it costs
money to install that system. It costs $230,000 to
install that system. So the question is,
is this a good deal? Should you do it? That's a management decision. And the management decision
relies on valuation first. Once you value it, then
you can make a decision. So you've got 90,000, 90,000,
90,000 in the three years as your cost savings,
but it's going to cost you $230,000 upfront. Now if it turns out that
the interest rate is 4%, you can figure out
what the answer is. At 4%, it turns out that
the NPV of this project is about $20,000. So it's a good deal. On the other hand, if you
change the assumptions, and you make the interest
rate something else, well, it might not
be a good deal. How would you have to
change the interest rate to make this a terrible deal? Increase or decrease it? Increase it. Why? Why does that make sense? Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Exactly. With a higher interest
rate, money now is more valuable than the cost
savings to your electricity. How do you know
it's more valuable? AUDIENCE: [INAUDIBLE] ANDREW LO: Exactly. The opportunity cost is
10% as opposed to 4%. It's a lot more valuable. If you stick it in
the bank, you get 10%. So the cost savings depends
on the interest rate at hand. Once you have the interest
rate, you can make a decision. Where does interest
rate come from? AUDIENCE: [INAUDIBLE] ANDREW LO: Exactly. The market. You don't pick the interest
rate out of the air. You don't say, I sort of feel
like it's a 2% kind of day. The interest rate is what you
can get on the open market. See, that's why
the market matters. It's because if that's
a market interest rate, by saying it's a
market interest rate, it means you can actually get
that rate from the market. And therefore, it's a real
number that can be actionable. It's not a fictitious
theoretical construct that may or may not have
any practical bearing. It's a number that
actually you can achieve. And as a manager,
if you're trying to increase the value
of shareholder wealth, if that's the
objective, is to make more money for the shareholders,
this is the way to do it. So this is what I
meant when I told you at the very beginning
of this course that finance is the most
important subject you'll ever study. It's because with
proper valuation, management decisions are easy. Now, it's not always
easy to get to the point where the numbers
tell you so much. And so, management is
trying to understand all of the various different
factors and balancing them out. Like, the kind of questions
you were asking me at the very beginning of class,
I can't answer many of them in the abstract. It depends on the situation. And I'm hoping that by
the end of this course, you will know enough
about the basic framework to make those
trade-offs yourself. And then, the art of
management works together with the science of management
to come up with good decisions. OK. So this is simple. And in the next few
slides, I'm going to ask you to take a look
at examples on your own. Here's an example,
a real live example, where CNOOC, the
Chinese oil company, made an offer to acquire
Unocal about a year, year and a half ago. And I would suggest you
take a look at this example and just do the back of
the envelope calculation to see whether or
not they provided a good deal or a bad deal. But I want to turn now to
the main subject of today's lecture, which is one of
the most beautiful formulas in this entire course. Now it might seem strange for
me to talk about a formula as being beautiful. You know, a while ago, Paul
Samuelson, the great economist here at MIT, once
said that, you know, either you think that
probability theory is beautiful or not. And if you don't
think it's beautiful, then I feel sorry for you. And I suppose the same can
be said for this formula. It's hard to believe that
a formula can be beautiful, but trust me, it is. And if you don't think
so, I feel sorry for you. By the end of the
semester, hopefully you will think it's beautiful. Let me explain what
we're about to do. I want to come up with the
value of a very specific asset, an asset with a very, very
simple and interesting cash flow. So this is one of the
two special cash flows that we're going to
analyze in this class. And this cash flow is
known as a perpetuity. A perpetuity is exactly
what it sounds like. It pays cash forever. Now we can debate whether or
not forever really exists. I won't try to argue with you
that we will live forever. But it's a
hypothetical construct. OK? So this is a figment
of our imaginations. There exists in my
imagination a piece of paper that has a claim,
such that whoever holds the piece of
paper will be entitled to a cash payment of
C dollars every year forever, out to infinity. OK? And the question is,
how much is this worth? How much is this
piece of paper worth? It's an asset, because it's
a sequence of cash flows. It just turns out that this cash
flow is an infinite sequence. It never ends. It's the gift that
keeps on giving. So you would think
that it should be worth an infinite amount, because
it pays an infinite amount of cash, right? No, that's not right. And the reason it's not
right is because $1 today is worth more than $1 tomorrow,
which is worth more than $1 a year from now, which
is worth more than $1 two years from now. And so the value of
a dollar paid out into the far future declines. And it turns out
that it declines at a rate for which you
can actually figure out what the value is today. So here's we're going to do. Using the same
principle of discounting that we did for the
previous set of cash flows, we're going to take a
sequence and discount it. I'm assuming with
the perpetuity, that it starts paying next year. So that's the very
first cash flow. We're sitting here
at date zero, and it pays C dollars next year, and
then another C dollars the year after, and then another C
dollars the year after that, and so on. So we're going to dis count them
by 1 plus r, 1 plus r squared, dot, dot, dot, forever. And so this is an
infinite sequence. And those of you who were on
your high school math team, you'll know that a quick and
dirty way of some summing that infinite
sequence is basically to multiply both
sides by 1 plus r. And you'll notice
that when you do that, you get the series back
again, but with an extra C. And when you do the
subtraction and division, you end up with this
incredibly simple formula that says that the present
value of this claim that pays C dollars forever
is not infinite. In fact, it's quite finite. It's C divided by r. What a simple formula. If I have a piece of paper
that pays $100 a year forever, and the interest
rate is 10%, what is this piece of paper worth? Yes? AUDIENCE: $1,000. ANDREW LO: Exactly. $1,000. Isn't that amazing, that we
could actually value something like that? If the interest rate is
5%, what is it worth then? Yeah. $2,000. Right. Simple. We have complete analytical
solution for a cash flow that, on the surface of
it, seems like it should be worth a huge amount of money. It's not that huge. Yeah? AUDIENCE: [INAUDIBLE] ANDREW LO: Well, yes. We're assuming-- assume that
interest rates are constant. Absolutely. So if interest rates vary. This formula is not right. We're going to come
to the case where interest rates vary over time. So, absolutely. This is still under the
simplistic assumption that interest
rates are the same. But under that case, I
think it's still pretty cool that we're able to come up with
the formula for value, right? Yeah. AUDIENCE: [INAUDIBLE] ANDREW LO: Well,
that's a good question. I was afraid you were
going to ask that. But I am prepared. I am prepared to answer that. In the United Kingdom,
there is a bond issued by the government
called a console. And this bond is a perpetuity. That is, it pays to the holder
a fixed amount every year forever. Now in that case,
forever means as long as the British government
is still in existence. You know, it's still around. But that's an example. AUDIENCE: [INAUDIBLE]. ANDREW LO: Yes, absolutely. It trades. You can buy it, sell
it, observe the price. Absolutely. Yeah. Yes? AUDIENCE: [INAUDIBLE] ANDREW LO: Right. Good question. Where do we get
the interest rate? The market. Exactly. So you can either get it
from the marketplace-- so I have a piece of paper. It pays $1 a year forever. Who will pay me $5 for
this piece of paper. $6? $7? I'll auction it off
to the highest bidder, and that price will translate
into an interest rate determined by the marketplace. So the short answer
is the market. Now you're asking me probably
a deeper question, which is where does that come from? Because there are
all sorts of factors, like future, famine,
and plagues, and wars, and all these other issues. And the answer is,
it's an approximation that market participants
make, and they're willing to live with. Right? I'll give you an example. A few years ago, Walt Disney,
the entertainment company, issued bonds, corporate bonds. They were 100 year bonds. They were going to
mature in 100 years. Now, I don't know
how many of you are high school math
team jocks, but if you are, one test is to
ask the question, with this infinite series, if
you take it out to 100 terms, instead of all the
way out to infinity, what percentage of the total
market value will you capture? It turns out that 100
terms is pretty darn close to infinite in this grand scheme
of things with interest rates that we use. So that's an example, where
when they issued that bond, and they auctioned if off to
the market participants, whoever bought those bonds, whoever
the highest bidders were, they set the price. Once you have the price, you can
back out and calculate the r. In fact, let me ask you this. If I tell you what
C is, C is $100, and I tell you the market
price, say it's $500, what's the interest rate? Can you figure that out? Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Right. Exactly it's
basically determined. So the market price for
an instrument like this will give you the
market's assessment of what that interest rate is. AUDIENCE: [INAUDIBLE] ANDREW LO: Let me repeat
the question in case people didn't hear. The question is, am I telling
you that with all the PhDs out there, there is nothing
more sophisticated in terms of pricing these instruments
than simply auctioning them off, as we did to
a bunch of MBAs? Well, first of all, I wouldn't
denigrate MBAs that way. I would argue that the
PhDs who are doing research are ultimately advising
the MBAs as to what to bid, and then the MBAs take
into account the business considerations, as
well as the analytics. And so it's actually a highly
complex and sophisticated process by which
the bidding occurs. In other words, you're not
getting amateurs doing it. You're getting professionals who
know how to price these things. That said, are they
going to make mistakes? Absolutely. So market pricing is
an imperfect mechanism. But the imperfect mechanism
actually works pretty well. And so far, nobody else
has figured out anything that works any better. So, yeah? AUDIENCE: [INAUDIBLE] price
[INAUDIBLE] $1 [INAUDIBLE].. Obviously, they're
not just issuing one to the highest bidder. ANDREW LO: So the
question is, isn't there a problem in terms of the
auction if what we're doing is determining the price
based upon the highest bidder. Because the highest bidder
is typically the individual that's the most confident. Or it's possible that that
particular bidder knows something that the rest
of the market doesn't. So I don't know which of
those two possibilities might be the case. It depends on the
market circumstances. One of the things
about auctions, though, is that the
design of the auction can actually have a big
impact on how informative the price is. So the standard auction is
actually very, very complicated in terms of the various
incentive effects. But there are more
intelligent auctions that are designed to elicit true
responses based upon not just kind of anxiousness to win, but
on what the economic valuation is. In fact, there's an
example of an auction that works something like this. You have bidders bidding
for a particular commodity. And it turns out that
the highest bidder wins. But the highest bidder
will pay a price that is the second
highest bidder's price. So that actually has a
very interesting incentive in the sense that it ends up
forcing you to actually reveal your true preferences. And we'll come back to that as
we talk later on about market mechanisms and pricing. Yeah? AUDIENCE: [INAUDIBLE] mechanisms
in auction, for example, for public contracts-- ANDREW LO: Yeah. AUDIENCE: In which
they do the average and they rule out people
who have more than 15% deviation from that. So it could really go
for a very low price. ANDREW LO: Yeah. AUDIENCE: It's interpretative
that you're like, [INAUDIBLE].. So you're kicked off the deal. ANDREW LO: That's right. So there are mechanisms to try
to make the auctions smarter. And that's one example. Another example of that. But we're going to assume for
now that the auction mechanism produces a good price. Later on, after we figure
out how markets work, we're going to come
back and question that. And the very end
of this course, I'm going to question all
of this and confront you with empirical evidence that
describes psychological biases that all of us have
that are hardwired into us that would make
you think that markets don't work well at all. And we'll give you a framework
for thinking about those two kinds of phenomenon. Yeah? AUDIENCE: I'm just
curious to see-- [INAUDIBLE] would you have
bought this [INAUDIBLE] at market price. [INAUDIBLE] ANDREW LO: OK. The question is, do
people's bids actually reflect interest
rates over time? Well, remember that market
conditions are changing. So the question is, do they
reflect people's information as of when. I mean, you never step
into the same river twice. So what you bought last
year at last year's price may have no bearing on
what you're willing to buy at this year's price, right? Things change. So I'm not sure that that
question is well-posed. At every point in
time, if an individual will pay this C divided by r for
a security that pays C forever, that's the fair market price. Now in the future, if
interest rates change, the price will change. But what this does say is
a very interesting point that I think you're
getting to, which is that suppose that C
never changes by contract. If interest rates never
change, then this security will never change in price. It will have absolutely
no price growth. So here's an example where
you buy a piece of paper-- let's say the interest
rate is 10% and C is $100. You pay $1,000 today. If next year the interest rate
is 10%, this piece of paper's still worth $1,000. And then five years from now,
if he interest rate is 10%, the piece of paper's
still worth $1,000. Does that makes sense? Does that seem to suggest
that you got stiffed because you bought a security
and it didn't grow in price? In fact, the rate of return
on that security is 0. Right? AUDIENCE: [INAUDIBLE]. ANDREW LO: Or I
mean, a $100 payment. AUDIENCE: You have one
coupon payment plus-- ANDREW LO: Every year. Right, exactly . So it's wrong that
the return is zero. The price return is zero. There's no price growth. But meanwhile every year, you've
been getting checks for $100. And if the piece
of paper was $1,000 and you've been getting
checks for $100 every year, what's your annual return? 10%. What's the interest rate? Oh, funny how that works, huh? That's great. You get a 10% return. Why? Because you're holding
this piece of paper that generates coupons,
and the coupons end up giving you a
10% rate of return, because the price
of the security is those coupons
discounted at 10%. Nothing magic about it. It all adds up. It all works together. OK? Yes? AUDIENCE: [INAUDIBLE]
for example-- ANDREW LO: We're
going to get to that. Yes, we're going to get to that. That's my next example. Let me hold off on that. I want to make sure
everybody understands the perpetuity though. And then we're going to get to
the example where C changes. Now to your example, what
happens if C changes. In fact, let's be optimistic
and let's say that C grows. So not only am I going
to pay you something forever, but that
something, I'm going to let that grow by a
rate of growth of g. So next year, I pay you
C. But the year after, I'm going to pay you C,
multiplied by 1 plus g. So let's say g is 5%. Then next year, I pay you $100. The year after, I pay you $105. And the year after
that, I'll pay you whatever 1.05 squared
is times 100, and so on. Now, what is this
piece of paper worth? And if you do the same kind
of high school math team trick and solve for the present
value, you get an answer, PV is equal to C
divided by r minus g. r minus g. So you subtract
this growth rate. Now when you subtract
the growth rate, that makes the
denominator smaller, which makes the
whole thing bigger, which is the right
direction because you're getting a cash flow that
is not steady over time, but it's growing over time. So it should be worth more. And it's worth r minus g more. All right? Now you notice, I have a little
condition at the end of that. r has to be greater than g. Why do I have that condition? Yeah? AUDIENCE: [INAUDIBLE]
infinite [INAUDIBLE] the infinite [INAUDIBLE]. ANDREW LO: That's right. So let's suppose
that r equals g. Let's see what happens. If r equals g, then the infinite
series on top, c divided by 1 plus r plus C times 1 plus
g over 1 plus r squared, that's just C over 1 plus r,
because I'm assuming g and r are the same. Plus C over 1 plus r, plus C
over 1 plus r, plus C over 1 plus r. I have an infinite number
of C over 1 plus r. And C over 1 plus r
is a finite constant. The sum is infinite. So at some point,
that's going to exceed total world GDP, and
then beyond it, and then the other planets of the
solar system, and so on. What's going on here? Why is it happening? Anybody give me the intuition
for what's happening? AUDIENCE: Because the numbers
are going smaller and smaller [INAUDIBLE] ANDREW LO: Right. AUDIENCE: But compared
just to zero, the amount of [INAUDIBLE]. ANDREW LO: Right. Yes. AUDIENCE: [INAUDIBLE]. ANDREW LO: Yeah. That's right. It's growing. But what's the intuition
for why that can't persist? AUDIENCE: Sounds like
you're [INAUDIBLE] the 10,000 [? quantity. ?] ANDREW LO: Right. That's right. It's basically working against
the time value of money because the numerator is growing
as fast as the denominator is growing. So what it says is that the
cash that you're presumably going to be paying to
somebody is actually increasing at the
exact same rate that the discount
rate is growing. So there's no way to
sustain that forever. You can't do that forever. So it has to be the case that
the amount that the cash is growing can never exceed
the discount rate. Now remember, these are
all theoretical concepts where I'm assuming that growth
rate stays the same forever, and the interest rate
stays the same forever. This doesn't rule out for short
periods of time growth rates exceeding interest rates. You just can't do it forever. For the last 15 years,
China has been growing at a rate of approximately 10%. Their entire economy has
been growing at 10% a year for every single year
over the past 15 years. That can't persist. If it did, not only would
we all be speaking Chinese, but all of the planets
in this entire galaxy would end up speaking Chinese. I mean, growth rates
cannot persist forever. But here, we're assuming, we're
assuming, that this growth rate is an infinite growth rate. It applies forever. So in that sense, it has to be
smaller than the discount rate. Question? OK. AUDIENCE: [INAUDIBLE]
rest of the world. ANDREW LO: Well, there are a
couple of problems with that. So the question is, what happens
when r is actually less than g? Right? You would think that that
gives you a negative number. In fact, it doesn't,
because there's a discontinuity at zero. And so this formula is--
that doesn't even apply. What happens, if you do
the infinite sum, when g approaches r, this infinite
sum already goes to infinity. When g gets above r, it gets
to be even more infinite, whatever that means. Right? Because the numerator is then
not growing at the same rate, but it's growing at a faster
rate than the denominator. So the formula,
you wouldn't even get the formula, because now
you're dealing with infinities. OK? AUDIENCE: [INAUDIBLE]. ANDREW LO: Right. Right. AUDIENCE: [INAUDIBLE]. ANDREW LO: It would just
be an infinite value, but an even bigger infinity,
whatever that means. And so, this formula really
only holds under this condition. If it were equal to or
negative, this formula just would not be appropriate. You'd have to go
back to that formula. And what that formula
would show you is that you're
getting an infinity. OK? So that's a perpetuity. And we're going to
use this, by the way. This may seem kind
of theoretical. But trust me, it's
going to come in very handy when we start
pricing bonds and stocks. So we're going to
use this quite a bit. Now I want to tell you
about a formula that is my second favorite formula
in this entire course. And in a way, this is
much more practical, and it's very closely
related to the perpetuity. This formula is a
formula for an annuity. An annuity is a security that
pays a fixed amount every year for a finite number of years,
and then it stops paying. So an example of an
annuity is a bond. Another example is an auto loan. Another example is a mortgage. And I think I told you that this
mortgage valuation formula is one that you're
going to use when you start thinking about making
a home purchase decision. And it will actually
be this formula exactly that you're
going to need to use. Question? AUDIENCE: [INAUDIBLE]. ANDREW LO: Yes. AUDIENCE: Just one question. The principle is returned within
these payments, or at the end? ANDREW LO: Let me
talk about that later. Right now, we don't
know what principle is. So when I talk about bonds,
I'm going to come back to that. OK? So let's not get
ahead of ourselves. I want to make sure we
understand the formula and then I'll come to that. That's an important
point that we're going to get to in about
a lecture and a half. OK? OK. So let me explain what a
perpetuity and an annuity are in relationship
to each other. A perpetuity pays a
fixed amount forever. An annuity pays a fixed amount
for a finite period of time. So there's a relationship
between the two. And in particular, you
can think about the value of an annuity as the
value of a perpetuity where you only get to have it
for a finite period of time. Right? Let me explain. An annuity, the
value of that, is given by the expression
on the top line. Right? C, C, C, C for T
periods, discounted at the appropriate
discount rate. Now, it turns out
that you can come up with an expression for what
that present value is, again, using the high school math
team kind of an approach. You simply multiply
both sides by 1 plus r, and then you solve
for the present value, and you get an expression
that looks like this. Well, this looks an awful
lot like it's related to the perpetuity formula. You've got to C over r
here, but then there's some annoying other
terms over here. So let me give you a thought
experiment that will show you how to derive this formula
in less than one minute without any kind of high
school math team tricks. And the experiment
goes like this. Suppose that you want
to create an annuity, but you don't have
an annuity at hand. Well, one way you can do
it is to buy a perpetuity, hold it for T periods, and
then get rid of it and sell it. Now look at the cash
flows that you get. If you were to
take a perpetuity, which is the top
cash flow, and you would subtract from it
a perpetuity as of date T plus 1-- so you've gotten
rid of the perpetuity at this point. When you take the top
cash flow sequence and you subtract from it
the next cash flow sequence, you get the bottom
cash flow sequence, which is just an annuity. Right? So an annuity is a
perpetuity on borrowed time. So what is it worth? Well, it's worth whatever
it is to buy a perpetuity, hold it for T periods,
and as soon as it pays off in that Tth date, you sell it. OK? So what's it going to cost? What's the value of that? The value of that is
this is what it costs to purchase the annuity today-- the perpetuity, sorry. Right? C over r. That's what it costs to
purchase the perpetuity today. And you're going to hold on to
that perpetuity for T days or T periods. And at date T plus 1, you're
going to sell the perpetuity. What are you going to
get when you sell it? What would you get
as the payment? C over r. That's right, because that's
the price of a perpetuity. The price never changes. It's always C over r. When do you get paid that price? At T or T plus 1? AUDIENCE: T plus 1. ANDREW LO: Because I want to
have T periods a cash flow. So I've got to hold onto
that perpetuity at least until T periods. After the Tth date,
I sell it, which means I sell at the next
date, which is T plus 1. And so I get paid a cash flow
of c over r at day T plus 1. What is that cash
flow worth today? Remember, it's at two
different points in time. I need to use the exchange
rate to convert it to the same currency. What's the exchange rate between
date 0 and date t plus 1? Yeah? [? Scholmi? ?] AUDIENCE: [INAUDIBLE] ANDREW LO: By t, or by t plus 1? AUDIENCE: By t. ANDREW LO: No. Close, but no cigar. AUDIENCE: t plus 1. ANDREW LO: Why t plus 1? AUDIENCE: That's the period
where you're getting paid. ANDREW LO: That's the period
where you're getting paid. So let's go back. And remember, the
first thing you do? Draw a time line. Right? So here's the timeline. And you see why it's confusing. You know, I don't blame
you for thinking it's t, because I said two
periods and you're going to sell it
after two periods but when I say sell
it after two periods if it's after two
periods it's plus 1. So take a look at this
diagram, and you've got to draw the diagram. You've got to draw the
diagram to really get this. OK? The top part is a perpetuity. The middle part is that same
perpetuity at day T plus 1. So if you own the top
piece, and at the same time you sell the middle piece,
that means at time T plus 1, you're going to give up all
of your future cash flows because you're going
to sell the perpetuity. Then you're left with the actual
annuity cash flow that we want. So the question is, what
does this transaction cost? I buy that it's going
to cost me c over r. I sell this. This is a sequence
of cash flows. So if I'm selling a
sequence of cash flows, I'm selling that value. I'm going to receive
that value as payment. So it's going to reduce my cost,
and so like any other sequence of cash flows, when I sell
this, I have to value it, and it turns out that
this is equal to the value at this date, the value of
the perpetuity at this date. And what is that value? C over r. And if it's C over
r at this date, what is the value at this date? I've got to discount it by
1 plus r to the T plus 1, because it's T plus
1 periods going back. OK? Well, actually, sorry. T periods, the convention
is a little confusing. It's T periods because
you're at t plus 1, and you want to figure
out what the value is. And the value of
the perpetuity at T is a perpetuity that starts
paying off at T plus 1. So you're right. It's T, but you're
discounting it as of the payment
as of T plus 1. OK? How many people are confused? OK. Yes. AUDIENCE: [INAUDIBLE]. ANDREW LO: Let me-- let me do this on the board. Right. Exactly. Let me do this on the board,
because the notation is confusing. Let me just switch on the light. Whoops. OK. So we're going to start
by assuming that we've got a perpetuity at date 0. So this is date 0. And remember, the
definition of perpetuity is that it starts
paying the next period. And so it pays C,C
until this date. And sorry. T Plus 1 pays T
plus 2, and so on. The annuity that
we want to value is an annuity that is just
consisting of the first T cash flows. Right? So what I claim is
that if you engage in the following transaction,
at date 0, you buy a perpetuity, and you also agree to sell
that perpetuity at date after date T. So you sell
after T. What that means is that you will hold
onto the perpetuity until it pays you C dollars. And as soon as it does
that, after it does that, you sell it. Now when you sell it,
what do you get for it? You get C over r. But the question is, when
do you get that C over r? If you have a
sequence of cash flows that starts in year T plus 1,
then the value of it at day T is C over r right? Because a perpetuity by
assumption is a piece of paper that starts paying off
not today, but next year. So if it starts
paying off next year, for every single
year thereafter, the value at that point is
going to be equal to C over r. Any questions about that? OK . So we've now
established that when you sell these cash flows going
out into the infinite future, the value at date T is C over r. And therefore, if
the value at date T is C over r, what is
the value of date 0? You have to bring
it back to date 0. You're discounting
it by T periods. So it's C over r times 1
over 1 plus r to the t. That's what you get when
you sell this perpetuity. And what you paid
for it is C over r. So the value of this
particular set of actions that you've engaged in is C over
r minus C over r times 1 over 1 plus r to the T. That's the annuity discount
formula in a nutshell. And this formula is the
basis of how you figure out your mortgage payments. Because a mortgage is where you
have an obligation every month to pay something to the bank in
exchange for a pile of money, the money that you
used to buy your house. And the first time I
was buying my, house I actually went through
this transaction. I decided that I was going to
just calculate this myself, because the interest
rate was not exactly given by what was in
the particular banker's table. So I went to the
mortgage company. It was a bank. And I think the
interest rate that day was something like, I
don't know, 8%, 8 and 1/2%, or 8 and 3/4%. And it turns out that
the table, this book, that had all of these
calculations, all of these numbers, didn't
have that interest rate. It didn't have 8 and 3/4. It had 8 and 1/2 or and 9,
but it didn't have 8 and 3/4. And so I just used this formula,
punched in a few numbers, and I got my monthly payment. And you know, I told the
banker, well, you know, this is what I'll
pay every month. And he said, well, you
can't just do that. I said, well, what do you mean? And he says, well,
you know, I don't know that that's the right number. We have to wait for the
senior vice president to tell me what the
right number is. Because we don't have the book. And he contacted the
senior vice president. It turned out he did
have the book either. So they had to call
the main branch, and somebody had to
look it up in this book. And sure enough, when they
came back with the number, it was my number down to
the fourth decimal place. And so he was amazed like,
wow, how did you do that? You know, this is amazing. You're incredible. It's incredible if you don't
know this very basic secret. So you're going to do this. You're going to do
this in the problems. You're going to calculate
mortgage payments, auto loan payments, consumer
finance payments. All of it is based upon
this simple formula. And you can construct
tables, as people have done, of what are called
annuity discount factors. So the annuity discount
factor is simply separating the interest
rate from the cash flow. And so when you're going out
for a mortgage, the amount that you're
borrowing, you borrow $200,000 for your house,
that's the left-hand side. Your monthly payment, C,
that's the right-hand side. And the prevailing interest
rate, that's the r. So if you know the annuity
discount factor, which is based just on
the interest rate, and you know the amount
of the loan that you want, PV, you can divide
and figure out what your monthly payments
are, or vice versa. If you have a particular set
of cash flows every month, and you have an
interest rate, you can figure out what your
total value of that loan is going to be in
terms of market terms. And so once you have
this expression, you can use a simple
table of numbers to calculate these
annuity discount factors. And then you can compute
mortgage payment yourself. So this is the kind of
number I was talking about. Given various different
interest rates, you can come up with these
particular annuity discount factors. And once you do, you can
calculate monthly payments very easily. So you only need
one set of tables. And for any kind of mortgage,
for any kind of consumer loan, you can compute the
monthly payments. Right? Whether it's an auto loan, or
a mortgage, it doesn't matter. What you need is this
table right here. Nowadays, we can do it in Excel. It's not a big deal. But still, you should know
what the underlying basis is for those calculations. OK. Now before you
finish this, I would like you to take a
look at a few examples. I've given you some
here, numerical examples. And I want to close this
class with a discussion about compounding,
and then next time, finish up with a
discussion of inflation. Because I don't think
we'll have time to do both. Compounding is a
matter of convention. And I want to explain what
that convention is and try to give you a little bit of
motivation for the logic of it, so that at least it
doesn't look like I'm just making it up out of the blue. The idea behind convention is
to take into account calendar effects, and in
particular, the possibility of early withdrawal. Let me explain. When I tell you that an
interest rate is 10%, typically, that quote is in
terms of an annual interest rate. Almost all interest
rates in the world are quoted on an
annualized basis, meaning if you were to
keep an investment for a 12-month period, that's
what the rate of return for that investment would be. The problem with
that quote, 10%, is that what do you do
if you want to withdraw your money after six months? What should you get paid then? Well, it would seem fair, if
you agree to a 10% interest rate per year, to say,
all right, if I take it out in six months
instead of a year, maybe you should only pay me
half the interest rate. Right? That seems like a fair deal. Right? Instead of 10%, pay me 5%. And maybe if I keep it
in for only a month, it would be fair not to
pay me 10% for that month, but to pay me 10% divided
by 12 for the month. The reason that that
discussion matters is that if you agree that
that's the fair thing to do, well then 10% is not
what you're going to get. Because if you get paid 5%
interest over the first six months, you take your
money out of the bank, and they give you
that 5%, and then you put the money back in the
bank literally the next minute and keep it in for
the next six months, you're going to earn another
5% on your original amount, plus you're going to earn 5%
on the first six month's 5%. You're going to earn
interest on the interest. And the banks know that. And after a while,
they were OK with that. That's a convention there's no
reason it has to be that way. The bank could say, I'm going
to give you 10% interest. But if you want to withdraw
your money in six months, I'm going to give you the
amount of interest such that if you were to take
the money out and put it right back in and hold
it, you would get 10%. Does anybody know what that
interest rate would be? How you figure that out? Yeah, [INAUDIBLE]? AUDIENCE: [INAUDIBLE]. ANDREW LO: Roots. That sounds painful . Is that like a root canal? What root do you mean? You're right. You're right. What is it? AUDIENCE: [INAUDIBLE]. ANDREW LO: Yes. AUDIENCE: [INAUDIBLE]. ANDREW LO: Yes The square root. Right. AUDIENCE: [INAUDIBLE]
you get [INAUDIBLE].. ANDREW LO: That's right. AUDIENCE: [INAUDIBLE]. Exactly. So what you would do in order
to figure out what the six month interest rate would be
so that when you held the interest on the interest
over through the whole year, it would add up to exactly 10%. The way you would do it is 1.10,
take the square root of that. Minus 1, that's
the interest rate that you would have for
the first six months and the second six months. A little less than 5%,
such that that number, when you add 1 and multiply
it by itself, you'll get 1.10. They don't do that, mainly
because nobody likes dealing with roots, except dentists. OK? So what they do is they say,
OK, as a matter of convention, here's what we're
going to do for you. This is the deal we're
going to give you. When we say 10% on an
annualized basis, what we mean is that it's going
to be compounded, typically on a monthly basis,
and nowadays on a daily basis. What that means is
that the interest rate that you're
actually going to get is the stated equivalent. It's the stated annual rate
divided by the compounding interval. Now that's a good deal
when you're a depositor. That's not a good deal
if you're a borrower. Because when they tell you, you
want to borrow money from me, I'll give it to you at a great
rate, it's going to be at 10%. But when you actually look
at how much interest you're paying, you're going to find
out that, actually, it's more than 10%. So that's where the term
APR and ERA came from. What does APR stand for? Anybody know? When you see this ad on TV
for auto loans, you know, [? buyer ?] loans, buy
a car, no money down, we'll give you a
loan, the APR is x%. Annual percentage rate. That is the stated rate. That's not the rate including
the effects of compounding. So as a depositor, when you're
lending money to the bank-- that's what it means to
deposit money in the bank-- that's a good thing. Because the annual
percentage rate of 10% is actually not
what you're getting. You're getting more
than that, because it's going to be compounded on a
monthly, or in some cases, on a daily basis. OK? In other words, the compounding
means you get interest on your interest on your
interest's interest going forward. Right? So you've got to keep in
mind that when you see these discount rates being quoted,
ask whether or not they are APR, annual percentage rate-- that's
like the 10% stated rate-- or EAR. EAR is the equivalent
annual rate. That's what you're
really going to get. That's what you would actually
get in terms of literal dollars at the end of the year if you
did nothing but left the money in there for that entire year. It would include the interest
on the interest on interest on the interest and so on. Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Annual
percentage yield. Yeah, that's right. Now, this is a very
clear example of it. OK? If you've got $1,000, and
there's no compounding effects and the interest rate is 10%,
you're going to get $1,100. If you compound twice a year,
which is what the old banks used to do because they didn't
have calculators in those days-- it was kind of hard
to compute these numbers-- they would compound
it twice a year. And so you would get
credit for the interest, and then you would get
interest on that interest as well as on the original
deposit or principle. Then that turns into $1,103. So being able to
compound more frequently gives you an additional
bonus, right? Not much. $3. But if you think about this
as billions of dollars, this starts adding
up to be real money. Now, if you compound on a
quarterly basis, it's $4. If you compound on a
monthly basis, it's $5. That's actually starting to
add up to something important. Right? Yeah? AUDIENCE: [INAUDIBLE]. ANDREW LO: Well, I mean,
I think it's six of one, or half a dozen of the
other, as they say. Banks will compete
with each other to offer ultimately
what the market rate is. So they won't play any tricks
with this kind of stuff some. Banks did play tricks
with this early on in the early days of banking. That's why banking is such
a highly regulated industry, to make sure that no
funny business goes on. And frankly, that's why
banks are forced now to tell you what whether
it's an APR or an EAR. It's a truth in lending
kind of a commitment that they are now
being forced to make. So nowadays, when you get
an auto loan or a mortgage, they have to tell you,
yeah, this is NPR. This is the annual
percentage rate. But your actual rate
earned may vary, and it may vary because
of compounding effects. And if you ask them what the
effective annual rate is, they are obligated to tell you. AUDIENCE: [INAUDIBLE]
all the information. Because with APR, you also
need to know the compound-- ANDREW LO: Compounding rate. But it's now taken for
granted that compounding happens on a daily basis. So that's a given. OK? Any questions about that? AUDIENCE: [INAUDIBLE]. ANDREW LO:
Compounding does, yes. AUDIENCE: For checking accounts? ANDREW LO: For checking
accounts, for savings accounts. Yes, it's daily. And you know why? It's because they
allow you to take out money on a daily basis. So if they didn't do
it on a daily basis, they'd have to figure
out on a one-off, if you were to take your money
out in the middle of the month, and I was to take my money out
after the first three days, and you were to take your
money out after five days, they'd have to do all these
custom calculations for each of those circumstances. So now they do it simply. They say, fine, we're going
to give you your interest rate every day. Every day, we're going
to compute your interest. So whether you come or
go, you will figure out when you get the interest. For certain market
applications, people compute interest intraday. Like the number of
hours you borrow money, they will calculate interest. There are cases where you
need to borrow money, only for four hours or three hours. I know this sounds
like drug money. But that's not-- that's
not what I'm talking about. There are cases where
you need very, very short-term financing, and
you need to borrow the money. And in those cases, they compute
it on a minute to minute. And in some cases, on
a continuous basis. So I'm going to leave you
with a little puzzler, which is if this tells you what the
effective annual rate is, where you're compounding
at intervals of n-- so if r is an APR, an
annual percentage rate, and n is denominated in months-- so monthly would be 12-- what would happen, what would
your effective annual rate be, if you compounded not every
day, not every hour, not every minute, not
every femtosecond, but literally every
possible time slice, the narrowest time
slice you can think of. If you did it continuously,
if n were to go to infinity, what would you get? Think about that. That's a little puzzle. It turns out that's called
continuous compounding. So you're compounding
continuously. It turns out that you
actually get a number. And what that number
is is really bizarre. So I want you to
think about that, and we'll take
that up next time. Thank you.
