2. Risk and Financial Crises

what I want to do this time
is talk about probability. I don't think many of you
have taken a course in probability theory. I don't take that as a
prerequisite for this course, but I think that actually
probability theory is fundamental to the way we
think about finance. So, I wanted to talk about
that a little bit today. And I'm going to put it in a
concrete context, namely, the crisis that's the world has been
through since 2007, and which we're still in
at this point. It's a financial crisis that's
bigger than any since the Great Depression
of the 1930's.

There's many different ways
of thinking about a crisis like this. And I wanted to focus on one way
that people think about it in terms of probability
models. So, that's not the only way,
it's not necessarily my favorite way of thinking
about it. That's, I think, a good way of
introducing our discussion of probability as it relates
to finance. Excuse my cold. I am managing to talk. I didn't bring any water. I hope I make it through
this lecture. It's a little bit iffy. So, let's just think
about the crisis. Most people, when they talk
about financial crises, they talk in terms of narrative,
of a historical narrative. So, I'll give you a quick and
easy historical narrative about the crisis. The crisis began with bubbles
in the stock market, and the housing market, and also in
the commodities market. Bubbles are — I will talk about these later,
but bubbles are events, in which people get very excited
about something, and they drive the prices up really high,
and it's got to break eventually. And there was a pre-break around
2000 when the stock market collapsed around
the world.

All over the world, the stock
markets collapsed in 2000. But then they came back again
after 2003 and they were on another boom, like a roller
coaster ride. And then they collapsed again. That's the narrative story. And then, both the housing
market and the stock market collapsed. And then, what happened
is, we see a bunch of institutional collapses. So, we see, in 2007, failures in
companies that had invested in home mortgages. And we see a run on a
bank in the United Kingdom, Northern Rock. It was arrested, but it looked
like 1930's all over again with the bank failure. We saw bank failures in
the United States. And then, we saw international
cooperation to prevent this from spreading like a disease. And then, we had governments all
over the world bailing out their banks and other
companies. So, a disaster was
averted, and then we had a nice rebound. That's the narrative
story, OK. And it makes it sound — and I'm
going to come back to it, because I like the narrative
story of the collapse.

But I want to today focus on
something that's more in keeping with probability, with
the way financial theorists think about it. And what financial theorists
will think about is that actually it's not just
those few big events. The crisis we got into was the
accumulation of a lot of little events. And sometimes they accumulate
according to the laws of probability into big events. And you are just telling
stories around these accumulation of shocks that
effected the economy.

And the stories are
not, by some accounts, not that helpful. We want to understand the
underlying probabilities. And so that's — thank you, a good assistant. He knows what I need. I just announced what
I need, he got it. A bottle of water. Tomorrow I may have absolutely
no voice. You're lucky. I'm going to talk today about
probability, and variance, and covariance, and regression, and
idiosyncratic risk, and systematic risk. Things like that which are
core concepts in finance. But I'm also going to, in the
context of the crisis, emphasize in this lecture,
breakdowns of some of the most popular assumptions that
underlie financial theory. And I'm thinking particularly
of two breakdowns.

And we'll emphasize
these as other interpretations of the crisis. One is the failure
of independence. I'll come back and
redefine that. And another one is a tendency
for outliers or fat-tailed distributions. So, I'll have to explain
what all that means. But basically, let me just
try to elaborate on — probability theory is a
conceptual framework that mathematicians invented. And it's become a very important
way of thinking, but it doesn't go back
that far in time. The word probability in its
present meaning wasn't even coined until the 1600's. So, you if you talk to someone
before the year 1600, and say, this has a probability of 0.5,
they would have no idea what you're talking about. So, it's a major advance in
human understanding to think in terms of probabilities. Now we do. And now it's routine, but it
wasn't routine at all. And part of what I'm thinking
about is, what probability theorists do, or in particular
finance theorists like to do, is they think that the world
is, well, let me just say, it's kind of a realization
that the world is very complex, and that the outcomes
that we see are the results of millions of little things.

And the stories we tell
are just stories. So, how do we deal with the
complexity of the world? Well, we do it by dealing
with all of these little incremental shocks that
affect our lives in a mathematical way. And we think of them as
millions of shocks. How do they accumulate? We have mathematical laws
of how they accumulate. And once we understand those
laws, we can we can build mathematical models
of the outcomes. And then we can ask whether we
should be surprised by the financial events that
we've seen. It's a little bit like science,
real hard science. So, for example, weather
forecasters. They build models that
— you know, you see these weather forecasts. They have computer models that
are built on the theory of fluid dynamics. And there is a theory of all
those little atoms moving around in the air. And there's too many atoms to
count, but there's some laws about their cumulative movement
that we understand. And it actually allows us
to forecast the weather.

And so, people who are steeped
in this tradition in finance think that what we're doing
when we're doing financial forecast is very much
like what we do when we do weather forecasts. We have a statistical model,
we see all of the shocks coming in, and of course there
will be hurricanes. And we can only forecast them —
you know there's a limit to how far out we can
forecast them. So, all hurricanes are
a surprise two weeks before they happen. Weather forecasters
can't do that. Same thing with financial
crises. This would be the model. We understand the probability
laws, there's only a certain time horizon before which
we can forecast the financial crisis. That isn't exactly my view
of the situation. I'm presenting a view this
time which is very mathematical and probability
theory oriented. So, let me get into some
of these details. And again, these are going to
be re-covered in the review session that Elan Fuld,
one of our teaching assistants, will do. I have just slides with
graphs and equations.

But, that's not Elan Fuld. I want to start out with just
the concept of return. Which is, in finance, the
basic, the most basic concept that — [SIDE CONVERSATION] PROFESSOR ROBERT SHILLER: When
you invest in something, you have to do it for
a time interval. And I'm writing the return
as one time period. T is time. And so, it could be a year, or
it could be months, or it could be a day. We're going to number these
months, let's say it's monthly return, we're going to number
these months, so that the first month is number one,
second month is number two.

And so, return at time t, if t
is equal to 3, that would be the return at month three. And we'll do price at the
beginning of the month. And so, what is your return
to investing in something? It's the increase
in the price. That's p t plus 1, minus p t. I'm spelling it out here. Price — I spelled it out in the
numerator, I guess I didn't do it in the denominator. It's price at time t plus 1
minus the price at time t, which is called the capital
gain, plus the dividend, which is a check you receive, if you
do, from the company that you're investing in.

That's the return. We have something else
called gross return. Which is just 1 plus
the return. Returns can be positive
or negative. They can never be more than —
never be less than minus 100%. In a limited liability economy
that we live in, the law says that you cannot lose more than
the money you put in, and that's going to be
our assumption. So, return is between minus
100% and plus infinity. And gross return is
always positive. It's between zero
and infinity. Now what we're going to do — this is the primary thing that
we want to study, because we are interested in investing
and in making a return.

So, we want to do some
evaluations of the success of an investment. So, I want to now talk about
some basic statistical concepts that we can apply to
returns and to other random variables as well. These on this slide are
mostly concepts that you've already heard. This is expected value. This is the mathematical
expectation of a random variable x, which could be the
return, or the gross return, or something else, but
we're going to substitute something else. We're going to substitute
in what they are. So, the expectation of x, or
the mean of x, mu sub x is another term for it, is the
weighted sum of all possible values of x weighted by
their probabilities. And the probabilities
have to sum to 1. They're positive numbers, or
zero or positive numbers, reflecting the likelihood
of that random variable occurring, of that value of the
random variable occurring. So, I have here — there's an infinite number of
possible values for x, and we have a probability for each one,
and the expectation of x is that weighted sum of those,
weighted by probabilities, of those possible values.

This is for a discrete random
variable that takes on only a finite, only a countable
number of values. If it's a continuous random
variable, if x is continuous, then the expectation of x is an
integral of the probability density of x, times x dx. I'm just writing that down for
you now for completeness. But I'm not going to explain
or elaborate that. These two formulas here are
measures of the central tendency of x, OK. It's essentially the average of
x in the probability metric that we have up here. But this formula is something
we use to estimate the expected value of x. This is called the mean or
average, which you've learned long ago. If you have n observations on
a random variable x, you can take the sum of the x
observations, summation over i equals 1 to n, and then
divide that by n.

That's called the average. So, what I want to say is that
this is the average, or the mean, or sample mean when
you have a sample of n observations, which is
an estimate of the expected value of x. So, for example, if we're
evaluating an investor who has invested money, you could get n
observations, say an annual returns, and you can take
an average of them. And that's the first and most
obvious metric representing the success of the investments
if x is the return, OK. People are always wanting to
know, they're looking at someone who invests money, is
this person a success or not? Well this is the first and
most obvious measure.

Let's see what that person
did on average. You were investing for, let's
say n equals 10, ten years, let's take the returns you made
each year, add them up and divide by 10. And that gives us an average. I put this formula down as
an alternative, because it's another — this is called the
geometric mean. This is the arithmetic mean. This is the geometric mean and
you're probably not so familiar with that, because
it's a different concept.

The geometric mean, instead of
adding your n observations, you multiply them together. You form a product of them. And then, instead of dividing by
n, you take the nth root of the product. And so, that's a formula that's
used to estimate the average return of a portfolio
of investments, where we use gross return for x, not just
the simple return. This geometric mean makes sense
only when all the x's are non-negative.

If you put in a negative
value, you might get a negative product, and then, if
you took the nth root of that, it could be an imaginary number,
so let's forget that. We're not going to apply this
formula if there are any negative numbers. But it's often used, and
I recommend its use, in evaluating investments. Because if you use gross return,
it gives a better measure of the outcome
of the investments. So, think of it this way. Suppose you invested money with
some investment manager, and the guy said, I've
done a wonderful job investing your money. I made 50% one year, I made 30%
another year, oh, and by the way, I had one bad year
with minus 100%, OK. So, what do you think
of this investor? Well, if you think about it, if
he made 50% one year, and then 30% another year, and
then he lost everything.

That dominates everything,
right? If you have a minus 100% simple
return, your gross return is 0, OK? So, if I plug in, if I put in
a 0 here to any of the x's, right, this product will be 0. Anything times 0 is 0. And I take the nth root of
zero, and what's that? It's 0. So, if there's ever a year in
which the return is minus 100%, then the geometric
mean is 0. That's a good discipline. This obviously doesn't make
sense as a way to evaluate investment success. Are you with me on this? Because you care a lot, if
the guy wipes you out.

Whatever else is done after
that doesn't matter. So, that's why we want to use
the geometric return. These are all measures
of central tendency. That is, what is the
central result? Sometimes the investor had a
good year, sometimes the investor had a bad year, but
what was the typical or central value? So, these are a couple
of measures of them. But we care about more than just
about central tendency when evaluating risk. We have to do other
things as well. And so, you want to talk about
— and this is very fundamental to finance. We have to talk about risk. What could be more fundamental
than risk for finance? So, what we have here now is
a measure of variability. And the upper equation here is
something called variance. And it's equal to the weighted
average of the x random variables [correction:
realizations of the x random variable] squared deviation from
the mean, weighted by probabilities. OK? All it is, is the expectation of
the square of the deviation from the mean. The mean is the center value,
and the deviations from the mean are — whether they're positive or
negative, if you square them they become positive numbers.

And so, that's called
variance. So, for example, if
x tends to be — if the return tends to
be plus or minus 1% from the mean return… say the mean return for an
investor is 8% a year, and it's plus or minus 1%, then
you would see a lot of 1's when you squared the deviation
from the mean. And the variance would
probably be 1. And the standard deviation,
which is — the standard deviation is the
square root of the variance. And it would also be 1. OK. This is a very simple concept. It's just the average squared
deviation from the mean. The estimate of the variance,
or the sample variance, is given by this equation. And it's s squared of x. It's just the sample mean. Take deviations of the variable
from its sample mean. You have n observations, say
someone has invested money for ten years, you take the average
return for the ten years and that's x bar, and
then you take all 10 deviations from the mean
and square them, and then divide by n. Some people divide by n minus
1, but I'm just trying to be very basic and simple here, so
I'm not going to get into these ideas.

The next thing is covariance. We're getting through
these concepts. They're very basic concepts. Covariance is a measure of how
two different random variables move together. So, I have two different random
variables, x and y. So, x is the return on, let's
say, the IBM Corporation, and y is the return on General
Motors Corporation. And I want to know, when IBM
goes up, does General Motors go up or not? So, a measure of the co-movement
of the two would be to take the deviation of
x from its mean times the deviation of y from it's
mean, and take the average product of those.

And that's called covariance. It's a positive number if, when
x is high relative to it's mean, y is high relative
to its mean also. And it's a negative number if
they tend to go in opposite directions. If GM tends to do well when IBM
does poorly, then we have a negative covariance. Because, if one is above its
mean, and the other is below its mean, the product is going
to be a negative number. If we get a lot of negative
products like that, it means that they tend to move
opposite each other. And if they are unrelated
to each other, then the covariance tends to be 0.

And this is the core concept
that I was talking about. Some idea of unrelatedness
underlies a lot of our thinking in risk. So, if x and y are independent, they're generated — suppose IBM's business has just
nothing at all to do with GM's businesses, they're
so different. Then I'd say the covariance
is probably 0. And then we can use that as a
principle, which will underlie our later analysis. Correlation is a scaled
covariance. And it's a measure of how much
two variables move together. But it's scaled, so that it
varies only over the range of minus 1 to plus 1. So, the correlation between two
random variables is their covariance divided by the
product of their standard deviations. And you can show that that
always ranges between minus 1 and plus 1.

So, if two variables have a plus
1 correlation, that means they move exactly together. When one moves up 5%, the other
one moves up 5% exactly. If they have a correlation of
minus 1, it means the move exactly opposite each other. These things don't happen very
often in finance, but in theory that's what happens. If they have a zero correlation,
that means there's no tendency for them
to move together at all. If two variables are
independent, then their correlation should be zero. OK, the variance of the sum of
two random variables is the variance of the first random
variable, plus the variance of the second random variable, plus
twice the covariance of the random variables. So, if the two random variables
are independent of each other, then their
covariance is zero, and then the variance of the sum is
the sum of the variances.

But that is not necessary. That's true if the random
variables are independent, but we're going to see that
breakdown of independence is the story of this lecture
right now. We want to think about
independence as mattering a lot. And it's a model, or a core
idea, but when do we know that things are independent? OK, this is a plot. I was telling you earlier
about the — let me see, OK, let me just
hold off on that a minute. Well, I'll tell you
what that was. That was a plot of the stock
market from 2000 to 2010 in the U.S. And I'm going
to come back to that.

These are the crises I was
telling you about. This is the decline in the stock
market from 2000 to 2002 or 2003 and this is the
more recent decline from 2007 to 2009. Those are the cumulative effects
of a lot of little shocks that didn't happen
all at once. It happened over years. And we want to think about
the probability of those shocks occurring. And that's where I am going. But what I want to talk about
is the core concept of independence leading to some
basic principles of risk management. The crisis that we've seen here
in the stock market is the accumulation of — you see all these ups and downs
in the stock market, and then all these ups and
downs on the way up. There were relatively more downs
in the period from 2000 and 2002 and there were
relatively more ups from the period 2003 to 2006.

But how do we understand the
cumulative effect of it, which is what matters? So, we have to have some kind
of probability Model. The question immediately is, are
these shocks that affected the stock market, are they
independent, or are they somehow related to each other? And that is a core question that
made it so difficult for us to understand how to deal
with the potential of such a crisis, and why so many people
got in trouble dealing with this crisis. So, we had a big financial
crisis in the United States in 1987, when there was a stock
market crash that was bigger than any before in one day. We'll be talking about that. But after the 1987 crash,
companies started to compute a measure of the risk to their
company, which is called Value at Risk.

I'll write it up like that. I capitalized the first and
the last letter, so you'll know that I'm not —
this is not the same thing as variance. This is Value at Risk. And what companies would do
after 1987 to try to measure the risk of their activities
is to compute a number something like this. They would say, there's a 5%
probability that we will lose $10 million in a year. That's the kind of bottom
line that Value at Risk calculations would make. And so, you need a probability
Model to make these calculations. And so, you need probability
theory in order to do that. Many companies had calculated
Value at Risk numbers like this, and told their investors,
we can't do too badly because there's no way
that we could lose — the probability is
only 5% that we could lose $10 million.

And they'd have other
numbers like this. But they were implicitly making
assumptions about independence, or at least
relative independence. And that's the concept I'm
trying to emphasize here. It's a core concept
in finance. And it's not one that is easy
to be precise about. We have an intuitive idea that,
you know — we see the ups and downs on the stock
market, and we notice them, and they all average out to
something not too bad. The problem that brought us this
crisis is that the Value at Risk calculations were
too optimistic. Companies all over the world
were estimating very small numbers here, relative to
what actually happened. And that's a problem. I wanted to emphasize
core concepts here. Intuitive concepts that you
probably already have. One of these concepts is something
we'll call the law of large numbers. And the law of large numbers
says that, there's many different ways of formulating
it, but putting it in its simplest form, that if I have
a lot of independent shocks, and average them out, on average
there's not going to be much uncertainty. If I flip a coin once, let's
say I'm making a bet, plus or minus.

If it comes up heads, I win a
dollar, if it comes up tails, I lose a dollar. Well, I have a risk. I mean, I have a standard
deviation of $1 in my outcome for that. But if I do it 100 times and
average the result, there's not going to be much
risk at all. And that's the law
of large numbers. It says that the variance of
the average of n random variables that are all
independent and identically distributed goes to 0 as the
number of elements in the average goes to infinity. And so, that's a fundamental
concept that underlies both finance and insurance. The idea that tossing a coin or
throwing a die in a small number of — it has uncertainty in a small
number of observations, but the uncertainty vanishes in a
large number of observations, goes back to the
ancient world. Aristotle made this observation,
but he didn't have probability theory and he
couldn't carry it further. The fundamental concept of
insurance relies on this intuitive idea. And the idea was intuitive
enough that insurance was known and practiced
in ancient times.

But the insurance concept
depends on independence. And so, independence is
something that apparently breaks down at times
like these. Like these big down crises that
we've seen in the stock market, in the two
episodes in the beginning of the 20th century. So, the law of large numbers has
to do with the idea that if I have a large number of
random variables, what is the variance of — the variance of x1 plus
x2 plus x3 up to xn? If they're all independent,
then all of the covariances are 0. So, it equals the variance of
x1, plus the variance of x2, plus the variance of xn. There's n terms, I'm not
showing them all. OK? So, if they all have the same
variance, then the variance of the sum of n of them is n times
their variance, OK. And that means the standard
deviation, which is the square root of the variance, is equal
to the square root of n times the standard deviation
of one of them. The mean is divided by n. So, that means that the standard
deviation of the mean is equal to the standard
deviation of one of the x's divided by the square
root of n.

So, as n goes large, you can
see that the standard deviation of the
mean goes to 0. And that's the law
of large numbers. OK. But the problem is, so you
know, you can look at a financial firm, and they have
returns for a number of years, and those returns can be
cumulated to give some sense of their total outcome. But does the total outcome
really behave properly? Does it become certain over
a longer interval of time? Well apparently not, because
of the possibility that the observations are not
independent. So, we want to move from
analysis of variance to something that's more — I told you that VaR came in 1987
or thereabouts, after the stock market crash of '87. There's a new idea coming up
now, after this recent crisis, and it's called CoVaR.

And this is a concept emphasized
by Professor Brunnermeier at Princeton and
some of his colleagues, that we have to change analysis of
variance to recognize, I'm sorry, we have to change Value
at Risk to recognize that portfolios can sometimes co-vary
more than we thought. That there might be episodes
when everything goes wrong at the same time. So, suddenly the covariance
goes up.

So, CoVaR is an alternative
to Value at Risk that does different kinds of
calculations. In the present environment,
I think, we recognize the need for that. So, this is the aggregate stock
market, and let me go to another plot which shows both
the same aggregate stock market, that's this blue line
down here, and one stock. The one stock I have shown is
Apple, the computer company. And this is from the
year 2000 — this is just the first decade
of the twentieth century. Can you see this? Is my podium in the way
for some of you? You might be surprised to
say, wait a minute, did I hear you right? Is this blue line the same
line that we just saw? But you know if I go back,
it is the same line. It's just that I rescaled it. There it is, it's a blue line. This looks scary, doesn't it? The stock market lost
something like almost half of its value. It dropped 40% between
2000 and 2002. Wow. Then it went all the way
back up, and then it dropped almost 50%. These are scary numbers,
right? But when I put Apple on the same
plot, the computer had to, because Apple
did such amazing things, it had to compress.

And that's the same curve that
you were just looking at. It's just compressed, so that
I can plot it together. I put both of them at 100
in the year 2000. So, what I'm saying here is that
somehow Apple did rather differently than the — this is the S&P 500. It's a measure of the
whole stock market. Apple computer is the one of the
breakout cases of dramatic success in investing. It went up 25 times. This incidentally is the
adjusted price for Apple, because in 2005 Apple
did a 2-for-1 split. You know what that means? By tradition in the United
States, stocks should be worth about $30 per share. And there's no reason why they
should be $30 per share. But a lot of companies, when
the price hits $60 or something like that, they say,
well let's just split all the shares in two. So, that they're back to $30. Apple went up more than double,
but they only did one split in this period. So, we've corrected for that. Otherwise, you'd see a big
apparent drop in their stock price on the day of the split. Are you with me on
this split thing? It really doesn't matter,
it's just a units thing.

But you can see that an
investment in Apple went up 25 times, whereas an investment in
the S&P 500 went up only — well, it didn't go up,
actually, it's down. So now, this is a plot showing
the monthly returns on Apple. It's only the capital
gain returns, I didn't include dividends. But it is essentially the return
on these two, on the S&P 500 and on Apple. Now, this is the same data you
were just looking at, but it looks really different
now, doesn't it? It looks really different.

They're unrecognizable
as the same thing. You can't tell from this plot
that Apple went up 25-fold. That matters a lot
to an investor. Maybe you can, if you've
got very good eyes. There's more up ones than there
are down ones, more up months than down months. There's a huge number of — enormous variability
in the months. But I like to look at a picture
like this, because it conveys to me the incredible
complexity of the story. What was driving Apple up
and down so many times? Really a pretty simple

Buy Apple and your money
will go up 25-fold. Incidentally, if you were a
precocious teenager, and you told your parents ten
years ago, OK, where you into this then? But just imagine, you say, mom,
let's take out a $400,000 mortgage on the house and put
it all in Apple stock, OK. Your parents would thank
you today if you told them to do that. Your parents could do that. They have probably paid off
their mortgages, right, they could go get a second
mortgage. Easily come up with $400,000. Most of your houses would
be worth that. So, what would it
be worth today? $10 million. Your father, your mother would
be saying, you know, I've been working all ten years, and your
little advice just got me $10 million. It's more than I made,
much more than I made in all those years.

So, these kind of stories
attract attention. But you know, it wasn't
an even ride. That story seems too good
to be true, doesn't it? I mean 25-fold? The reason why it's not so
obvious is that the ride, as you're observing this happen,
every month it goes opposite. I just goes big swings. You make 30% in one month, you
lose 30% in another month. It's a scary ride. And you can't see it happening
unless you look at your portfolio and see what
— you can't tell. It's just so much randomness
from one month to the other. Incidentally, I was a dinner
speaker last night for a Yale alumni dinner in
New York City. And I rode in with Peter Salovey
who's Provost at Yale. And on the ride back he reminded
me of a story that I think I've heard, but it took
me a while to remember this. But I'll tell you that it's
an important Yale story.

And that is that in 1979, the
Yale class of 1954 had a 25th reunion, OK. This is history. Do you know this story? Do you know where I'm heading? So, somebody said, you know,
we're here at this reunion, there's a lot of us here, let's
all, as an experiment, chip in some money and ask an
investor to take a risky portfolio investment for Yale
and let's give it to Yale on our 50th anniversary,
all right? Sounds like fun. So, they got a portfolio
manager, his name was Joe McNay, and they said — they put together — it was $375,000. It's like one house, you know,
for all the whole class of 1954, no big deal. So, they gave Joe McNay
a $375,000 start. And they said, just have
fun with this. You know, we're not
conservative. If you lose the whole
thing, go ahead. But just go for maximum
return on this. So, Joe McNey decided to invest
in Home Depot, Walmart, and internet stocks, OK? And on their 50th reunion, that
was 2004, they presented Yale University with $90
million dollars.

That's an amazing story. But I'm sure it was the same
sort of thing, same kind of roller coaster ride
the whole time. And now, we're trying to decide,
is Joe McNey a genius? What do you think,
is he a genius? I think, maybe he is. But the other side of it is, I
just told you what to do in just a few words. It's Walmart Home Depot,
and internet stocks. And the other thing is, he
started liquidating in 2000, right the peak of the market. So, it must be partly luck. The thing is, how did he know
that Walmart was a good investment in 1954
[correction: 1974]? I don't know. It's sort of — he
took the risks. Maybe that's why — I'm just digressing a little
bit to think about the way things go in history. But it seems that — I talked about the Forbes 400
people, and I mentioned last lecture about Andrew Carnegie's
The Gospel of Wealth, and he says that some
people are just very talented and they make it really big,
and we should let them then give their money away, and it's
kind of the American idea that we let talented people
prove themselves in the real marketplace and then they end
up becoming philanthropists and guiding our society.

But maybe they're just lucky. No one could have known
that Walmart was going to be such a success. And I think that history
is like that. The people you read about in
history, these great men and women of history, are often just
phenomenal risk takers like Joe McNey. And for every one of them that
you read about, there's 1,000 of them that got squashed. I was reading the history
of Julius Caesar, as written by Plutarch. It's a wonderful story. And I was reading this, and I
thought, this guy is a real risk taker. You know, you read all the
details of his life. He just went for
it every time. And he ended up emperor
of Rome. But you know what happened to
him, he got assassinated. So, it was — you know, it turned out not
entirely a happy story.

So, maybe it's all those poor,
all those ordinary people, living the little house,
the $400,000 house, they don't risk it. Maybe they're the smart ones. You just don't ever
hear of them. Well, these are issues
for finance. But you wonder, what are all of
these things, all of these big movements? This is the worst one here,
where lost about a third of its value in one month. And I researched it. What was it? Does anyone know what
caused it in 2008? Well, I'll tell you what caused
Apple to lose a third of its value in one month.

Steve Jobs, who is the founder
of Apple and genius behind the company, gave a — or was at
an annual meeting or press conference, and people said,
he doesn't look well. And so, they recalled that he
had pancreatic cancer in 2004, but the doctors then said it's
curable, no problem, so the stock didn't do anything. But reporters called Apple
and said, is he ok? And their company spokesman
wouldn't say anything. So, it started a rumor
mill that Steve Jobs was dying of cancer. It quickly rebounded
because he wasn't. That's how crazy these things
are, these market movements. So, now the next plot, and
this is important for our concepts here. I can plot the same data
in different ways. This shows a different
sort of complexity. Let me just review what
we've seen here. We started out with
Apple stock. This is the stock price
normalized to 100 in 2000. OK? And it goes up to 2500. Then, the next thing
I did is I did capital gains as a percent.

The percentage increase in
price for each month. It looks totally different, and
it shows such complexity that I can't tell a
simple narrative. I've just told you about one
blip here, but they were so many of these blips on the way,
and they all have some story about the success of some
Apple product, or people aren't buying some product. Every month looks different. But now, what I want to do —
and I have here the blue line is the return of the S&P 500. Now what I want to do is plot
a different sort of plot. It's a scatter plot. I'm going to plot the return on
Apple against the return on the S&P 500, OK. Do you know what I'm
referring to here? So, this is scatter plot. On the vertical axis I have the
return, it's actually the capital gain on Apple, and on
the horizontal axis I have the capital gain on the whole
stock market. OK? And each point represents one
of the points that we saw on the market. Actually I think it was,
I was telling you the second lowest story.

Steve Jobs, I'm not sure
which point it was. One of these points
in 2008 was when Steve Jobs looked sick. So, each point is a month, and
I have the whole decade of 2000, of the beginning of
the 2000s, plotted. So, the best success was in
December, January of 2001, where the stock price went
up 50% in one month. I tried to figure out
what that was about. Why'd they go up 50% percent
in one month? It turns out that the preceding
two months it had gone down a lot. They were down here somewhere. There were these big drops, and
people were getting really pessimistic because Apple
products weren't going well. They had introduced some new
products, MobileMe, I think, we forget about these products
that don't work, that didn't work very well. And then somehow people decided
it really wasn't so bad, so we have plus
50, almost 50% return in one month.

The reason why it looks kind of
compressed on this way is, because the stock market doesn't
move as much as Apple. So, basically Apple return is
the sum of two components, which is the overall market
return, and the idiosyncratic return, OK. So, the return for a stock, for
the i-th stock, is equal to the market return, which is
represented here by the S&P 500, which is pretty much the
whole stock market, plus idiosyncratic return. OK. And if they're independent of
each other, the variance of the sum is the sum
of the variance. The variance of the stock
returns is the variance — the variance of the Apple return
is the sum of their market return and their
idiosyncratic return. Well, let me be clear
about that. Let's add a regression line
to the scatter point [correction: scatter plot]. OK? It's the same scatter that
you saw — is it clear? Everyone clear what
we're doing here? I've got S&P on this axis,
and Apple on this axis.

And now I've added a line, which
is a least-square fit, which minimizes the
sum of squared deviations from the line. It tries to get through
the scatter of points as much as it can. And the line has a
slope of 1.45. We call that the beta,
all right? These are concepts that I'm
asking Elan to elaborate for you in the review session. But it's a simple idea here. What it means is that it seems
like Apple shows a magnified response to the stock market. It goes up and down
approximately one and a half times as much as the stock
market does on any day.

So, the market return here is
equal to beta times the return on the S&P that you see here. So, I wonder why that is? Why does Apple respond more
than one for one with the stock market? I guess it's because
the aggregate economy matters, right? If you think that maybe because
Apple is kind of a vulnerable company, that if the
economy tanks, Apple will tank even more than the economy,
than the aggregate economy, because they're
such a volatile, dangerous strategy company. And if the market goes
up, then it's even better news for Apple. But even so, the idiosyncratic
risk just dominates. Look at these observations,
way up and way down here. Apple has a lot of idiosyncratic
risk. And I mentioned one example,
it's Steve Jobs' health.

The Steve Jobs story
is remarkable. He founded Apple and Apple
prospered, and then he kind of had a falling out with the
management, and got kind of kicked out of his own company. And then he says, all right,
I'll start my own computer company, my second,
I'll do it again. So, he founded Next Computer. But meanwhile, Apple started
to really tank. This is in the nineties. And they finally realized they
needed Steve Jobs, so they brought him back. So, the company's ups and downs,
the idiosyncratic risk, has a lot to do with Steve Jobs,
and what he does, the mistakes he made. Those are what causes
these big movements. This line, I thought it would
have an even higher beta. But I think it's this
point which is bringing the beta down. And this is, I think this
is the point — the month after it turns
out that Steve Jobs really wasn't sick. OK? And it turned out to be the same
month that's the Lehman Brothers collapse occurred.

So you see, this point here
is between September and October of 2008. And that's the point — it was September 15th that we
had the most significant bankruptcy in U.S. history. Lehman Brothers, the investment bank, went bankrupt. It threw the whole
world in chaos. So, the stock market and S&P
500 stock market return was minus 16% in one month,
horrible drop. But for Apple, it really was
only about minus 5%, because they're getting over the
news of Steve Jobs. So, that's the way
things work. So, I want to move on now to
next topic, which is outliers, and talk about another
assumption that is made in finance traditionally that
turned out to be wrong in this episode. And the assumption is that
random shocks to the financial economy are normally
distributed. You must have heard of the
normal distribution. This is the bell-shaped, the
famous bell-shaped curve, that was discovered by the
mathematician Gauss over a hundred years ago.

The bell-shaped curve
is thought to be — this particular bell-shaped
curve which is the — the log of this curve
is a parabola. It's a particular mathematical
function. The curve is thought by
statisticians to recur in nature many different ways. It has a certain probability
law. So, I have plotted two normal
distributions, and I have them for two different standard
deviations. One of them, black line, is the
standard deviation of 3, and the other one, the pink
line, is the standard deviation of 1. But they both look the same,
they're just scaled differently. And these distributions have
the property that the area under the curve is equal to 1
and the area between any two points, say between minus 5 and
minus 10, the area under this curve is the probability
that the random variable falls between minus 5 and minus 10.

So, a lot of probability theory
works on the assumption that variables are normally
distributed. But random variables have a
habit of not behaving that way, especially in finance it
seems. And so, we had a mathematician here in the Yale
math department, Benoit Mandelbrot, who was really the
discoverer of this concept, and I think the most important
figure in it. [correction: Pierre Paul Levy
invented the concept, as discussed in the
next lecture.] He said that in nature the
normal distribution is not the only distribution that occurs,
and that especially in certain kinds of circumstances we have
more fat-tailed distributions.

So, this blue line is the normal
distribution, and the pink line that I've shown is a
fat-tailed distribution that Mandelbrot talked about,
called the Cauchy distribution. You see how it differs? The pink line looks pretty
much the same. They're both bell-shaped
curves, right? But the pink line has
tremendously large probability of being far out. These are the tails of
the distribution. So, if you observe a random
variable that looks — you observe it for a while,
maybe you get 100 observations, you probably can't
tell it apart very well from a normal distribution. Whether it's Cauchy or normal,
they look about the same. The way you find out that
they're not the same, is that in extremely rare circumstances
there'll be a sudden major jump in the
variable that you might have thought couldn't happen. So, I have here a plot of a
histogram of stock price movements from 1928, every
day, I've taken every day since 1928, and I've
shown what the S&P Composite Index — it didn't have 500 stocks in
1928, so I can't call it S&P 500 for the whole period — but this is essentially
the S&P 500.

And I have every day. There's something like
40,000 days. And what this line here shows is
that the stock return, the percentage change in stock price
in one day, was between 0 and 1% over 9,000 times. And it was between 0 and minus
1 percent around 8,000 times. OK? So, that's typical per day. You know, it's less than
1% up or down. But occasionally, we'll
have a 2% day. This is between 1 and 2%, that
occurred about 2,000 times. And about 2,000 times we had
between minus 1% and minus 2%. And then, you can see
that we've had — you can see these
outliers here. These look like outliers,
they're not extreme outliers.

So, if you look at a small
number of data, you get an impression that well, you know,
the stock market goes up between plus or minus 2%,
usually not so much, that's the way it is. After here they don't seem to be
anything, which means that, it looks like you never see
anything more than up or down 5% or 6%. It just doesn't happen. Well, because it's so few days
that it does those extremes. Can you see these little — that's between 5 and 6. There were maybe like 20 days,
I can't read off the chart when it did this since 1928.

You can go through ten years on
Wall Street and never see a drop of that magnitude. So, eventually you get
kind of assured. It can't happen. What about an 8% drop? Well, I look at this, I say,
I've never seen that. You know, I've been watching
this now, I've seen thousands of days, and I've
never seen that. But I have here the
two extremes. Stock market went up 12.53%
on October 30, 1929. That's the biggest
one-day increase. That's way off the charts, and
if you compute the normal distribution, what's the
probability of that? If it's a normal distribution
and it fits the central portion, it would say
it's virtually zero. It couldn't happen. Anyone have any idea what
happened on October 30, 1929? It's obvious to me, but it's
not obvious to you. I'm asking you to — I won't ask.

What happened in October, does
anyone know what happened in October 1929? STUDENT: That must be right
You're close. You're right. But someone else? STUDENT: Wasn't it the rebound
after the crash? PROFESSOR ROBERT SHILLER: Yes,
absolutely, it was the rebound after the crash. The stock market crash of 1929
had two consecutive days. Boy is that probability,
independence doesn't seem right. It went down about 12% on
October 28, and then the next day it did it again. What's going on here? We were down like
24% in two days. People got up on the 30th and
said, oh my God, is it going to do that again? But it did just the opposite. It was going totally wild. So, we don't know whether
covariance broke down or not.

I guess it didn't, because it
rebounded, and that was the biggest one-day increase ever. But if that weren't enough,
however, let's go back to October 19, 1987. It went down 20.47%
in one day. It went down even
more on the Dow. Some people say it went down
more than that, didn't it? But on the S&P that's how
much it went down. So, I figured, well if this were
normally distributed with the standard deviation suggested
by this, what's the probability of a decline
that's that negative? It's 10 to the minus 71 power.

1 over 10. So, you take 1 and you divide
that by 1 followed by seventy-one zeros. That's an awfully
small number. If you believe in normality,
October 19, 1987 couldn't happen. But there it is. It happened. And in fact, I was, I told you
I've been teaching this course for 25 years. I was giving a lecture, not in
this room, but nearby here, and I was talking about
something else. And a student had a
transistor radio. Remember transistor radios? And he was holding it up
and listening to it. Then he raised his
hand and said, do you know what's happening? He said the stock market is
totally falling apart.

It just came as a complete
surprise to me. So, after class, I didn't
go back to my office. I went downtown to
Merrill Lynch. And I walked up, it's a
story I like to tell. It's not that good. I walked up and I talked to a
stockbroker there, and I said, I was about to say something,
but he didn't let me talk. He said, don't panic. He thought that I had shown up
as a someone who was losing everything, his life savings
all in one day.

And he said, don't worry,
it's not going to — it's going to rebound. It didn't rebound. I showed up at lunchtime
and it kept going down. So, anyway, there was
something wrong with independence. Let me just recap. The two themes are that
independence leads to the law of large numbers, and it leads
to some sort of stability. Either independence
through time or independence across stocks. So, if you diversify through
time or you diversify across stocks, you're supposed
to be safe. But that's not what happened in
this crisis and that's the big question. And then it's fat-tails, which
is kind of related. But it's that distributions
fool you. You get big incredible shocks
that you thought couldn't happen, and they just come
up with a certain low probability, but with a certain
regularity in finance.

All right, I'll stop there. I'll see you on next Wednesday.

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