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

picture.

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

before the crash. PROFESSOR ROBERT SHILLER:

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.