Professor Robert Shiller:

Today I want to spend–The title of today's lecture is:

The Universal Principle of Risk Management,

Pooling and the Hedging of Risk. What I'm really referring to is

what I think is the very original, the deep concept that

underlies theoretical finance–I wanted to get that first.

It really is probability theory and the idea of spreading risk

through risk pooling. So, this idea is an

intellectual construct that appeared at a certain point in

history and it has had an amazing number of applications

and finance is one of these. Some of you–This incidentally

will be a more technical of my lectures and it's a little bit

unfortunate that it comes early in the semester.

For those of you who have had a course in probability and

statistics, there will be nothing new here.

Well, nothing in terms of the math.

The probability theory is new.

Others though,

I want to tell you that it doesn't–if you're shopping–I

had a student come by yesterday and ask–he's a little rusty in

his math skills–if he should take this course.

I said, "Well if you can understand tomorrow's

lecture–that's today's lecture–then you should have no

problem." I want to start with the

concept of probability. Do you know what a probability

is? We attach a probability to an

event. What is the probability that

the stock market will go up this year?

I would say–my personal probability is .45.

That's because I'm a bear but–Do you know what that

means? That 45 times out of 100 the

stock market will go up and the other 55 times out of 100 it

will stay the same or go down. That's a probability.

Now, you're familiar with that concept, right?

If someone says the probability is .55 or .45,

well you know what that means. I want to emphasize that it

hasn't always been that way and that probability is really a

concept that arose in the 1600s.

Before that,

nobody ever said that. Ian Hacking,

who wrote a history of probability theory,

searched through world literature for any reference to

a probability and could find none anywhere before 1600.

There was an intellectual leap that occurred in the seventeenth

century and it became very fashionable to talk in terms of

probabilities. It spread throughout the

world–the idea of quoting probabilities.

But it was–It's funny that such a simple idea hadn't been

used before. Hacking points out that the

word probability–or probable–was already in the

English language. In fact, Shakespeare used it,

but what do you think it meant? He gives an example of a young

woman, who was describing a man that she liked,

and she said, I like him very much,

I find him very probable.

What do you think she means?

Can someone answer that? Does anyone know Elizabethan

English well enough to tell me? What is a probable young man?

I'm asking for an answer. It sounds like people have no

idea. Can anyone venture a guess?

No one wants to venture a guess? Student: fertile?

Professor Robert Shiller: That he can father

children? I don't think that's what she

meant but maybe. No, what apparently she meant

is trustworthy. That's a very important quality

in a person I suppose.

So, if something is probable

you mean that you can trust it and so probability means

trustworthiness. You can see how they moved from

that definition of probability to the current definition. But Ian Hacking,

being a good historian, thought that someone must have

had some concept of probability going before,

even if they didn't quote it as a number the way–it must have

been in their head or in their idea.

He searched through world literature to try to find some

use of the term that preceded the 1600s and he concluded that

there were probably a number of people who had the idea,

but they didn't publish it, and it never became part of the

established literature partly because,

he said, throughout human history, there has been a love

of gambling and probability theory is extremely useful if

you are a gambler. Hacking believes that there

were many gambling theorists who invented probability theory at

various times in history but never wrote it down and kept it

as a secret. He gives an example–I like

to–he gives an example from a book that–or it's a

collection–I think, a collection of epic poems

written in Sanskrit that goes back–it was actually written

over a course of 1,000 years and it was completed in the fourth

century.

Well, there's a story–there's

a long story in the Mahabarahta about an emperor called Nala and

he had a wife named Damayanti and he was a very pure and very

good person. There was an evil demon called

Kali who hated Nala and wanted to bring his downfall,

so he had to find a weakness of Nala.

He found finally some, even though Nala was so pure

and so perfect–he found one weakness and that was gambling.

Nala couldn't resist the opportunity to gamble;

so the evil demon seduced him into gambling aggressively.

You know sometimes when you're losing and you redouble and you

keep hoping to win back what you've lost?

In a fit of gambling, Nala finally gambled his entire

kingdom and lost–it's a terrible story–and Nala then

had to leave the kingdom and his wife.

They wandered for years.

He separated from her because

of dire necessity. They were wandering in the

forests and he was in despair, having lost everything.

But then he meets someone by the name of–we have Nala and he

meets this man, Rituparna, and this is where a

probability theory apparently comes in.

Rituparna tells Nala that he knows the science of gambling

and he will teach it to Nala, but that it has to be done by

whispering it in his ear because it's a deep and extreme secret.

Nala is skeptical.

How does Rituparna know how to

gamble? So Rituparna tries to prove to

him his abilities and he says, see that tree there,

I can estimate how many leaves there are on that tree by

counting leaves on one branch. Rituparna looked at one branch

and estimated the number of leaves on the tree,

but Nala was skeptical. He stayed up all night and

counted every leaf on the tree and it came very close to what

Rituparna said; so he–the next

morning–believed Rituparna. Now this is interesting,

Hacking says, because it shows that sampling

theory was part of Nala's theory.

You don't have to count all the leaves on the tree,

you can take a sample and you count that and then you

multiply. Anyway, the story ends and Nala

goes back and is now armed with probability theory,

we assume.

He goes back and gambles again,

but he has nothing left to wager except his wife;

so he puts her and gambles her. But remember,

now he knows what he's doing and so he really wasn't gambling

his wife–he was really a very pure and honorable man.

So he won back the entire kingdom and that's the ending.

Anyway, that shows that I think probability theory does have a

long history, but–it not being an

intellectual discipline–it didn't really inform a

generation of finance theory. When you don't have a theory,

then you don't have a way to be rigorous.

So, it was in the 1600s that probability theory started to

get written down as a theory and many things then happened in

that century that, I think, are precursors both to

finance and insurance. One was in the 1600s when

people started constructing life tables.

What is a life table? It's a table showing the

probability of dying at each age, for each age and sex.

That's what you need to know if you're going to do life

insurance.

So, they started to do

collecting of data on mortality and they developed something

called actuarial science, which is estimating the

probability of people living. That then became the basis for

insurance. Actually, insurance goes back

to ancient Rome in some form. In ancient Rome they had

something called burial insurance.

You could buy a policy that protected you against your

family not having the money to bury you if you died.

In ancient culture people worried a great deal about being

properly buried, so that's an interesting

concept. They were selling that in

ancient Rome; but you might think,

but why just for burial? Why don't you make it into

full-blown life insurance? You kind of wonder why they

didn't. I think maybe it's because they

didn't have the concepts down. In Renaissance Italy they

started writing insurance policies–I read one of the

insurance policies, it's in the Journal of Risk and

Insurance–and they translate a Renaissance insurance policy and

it's very hard to understand what this policy was saying.

I guess they didn't have our language, they didn't–they were

intuitively halfway there but they couldn't express it,

so I think the industry didn't get really started.

I think it was the invention of probability theory that really

started it and that's why I think theory is very important

in finance.

Some people date fire insurance

with the fire of London in 1666. The whole city burned down,

practically, in a terrible fire and fire

insurance started to proliferate right after that in London.

But you know, you kind of wonder if that's a

good example for fire insurance because if the whole city burns

down, then insurance companies would

go bankrupt anyway, right?

London insurance companies would because the whole concept

of insurance is pooling of independent probabilities. Nonetheless,

that was the beginning. We're also going to recognize,

however, that insurance got a slow start because–I believe it

is because–people could not understand the concept of

probability. They didn't have the concept

firmly in mind. There are lots of aspects to it.

In order to understand probability, you have to take

things as coming from a random event and people don't clearly

have that in their mind from an intuitive standpoint.

They have maybe a sense that I can influence events by willing

or wishing and if I think that–if I have kind of a

mystical side to me, then probabilities don't have a

clear meaning.

It has been shown that even

today people seem to think that. They don't really take,

at an intuitive level, probabilities as objective.

For example, if you ask people how much they

would be willing to bet on a coin toss,

they will typically bet more if they can toss the coin or they

will bet more if the coin hasn't been tossed yet.

It could have been already tossed and concealed.

Why would that be? It might be that there's just

some intuitive sense that I can–I don't know–I have some

magical forces in me and I can change things.

The idea of probability theory is that no, you can't change

things, there are all these objective laws of probability

out there that guide everything. Most languages around the world

have a different word for luck and risk–or luck and fortune.

Luck seems to mean something about you: like I'm a lucky

person.

I don't know what that

means–like God or the gods favor me and so I'm lucky or

this is my lucky day. Probability theory is really a

movement away from that. We then have a mathematically

rigorous discipline. Now, I'm going to go through

some of the terms of probability and–this will be review for

many of you, but it will be something that

we're going to use in the–So I'll use the symbol P or

I can sometimes write it out as prob to represent a

probability.

It is always a number that lies

between zero and one, or between 0% and 100%.

"Percent" means divided by 100 in Latin, so 100% is one.

If the probability is zero that means the event can't happen.

If the probability is one, it means that it's certain to

happen. If the probability is–Can

everyone see this from over there?

I can probably move this or can't I?

Yes, I can. Now, can you now–you're the

most disadvantaged person and you can see it,

right? So that's the basic idea.

One of the first principles of probability is the idea of

independence. The idea is that probability

measures the likelihood of some outcome.

Let's say the outcome of an experiment, like tossing a coin.

You might say the probability that you toss a coin and it

comes up heads is a half, because it's equally likely to

be heads and tails. Independent experiments are

experiments that occur without relation to each other.

If you toss a coin twice and the first experiment doesn't

influence the second, we say they're independent and

there's no relation between the two.

One of the first principles of probability theory is called the

multiplication rule.

That says that if you have

independent probabilities, then the probability of two

events is equal to the product of their probabilities.

So, the Prob(A and B) = Prob(A)*Prob(B). That wouldn't hold if

they're not independent. The theory of insurance is that

ideally an insurance company wants to insure independent

events. Ideally, life insurance is

insuring people–or fire insurance is insuring

people–against independent events;

so it's not the fire of London. It's the problem that sometimes

people knock over an oil lamp in their home and they burn their

own house down. It's not going to burn any

other houses down since it's just completely independent of

anything else. So, the probability that the

whole city burns down is infinitesimally small,

right? This will generalize to

probability of A and B and C equals the

probability of A times the probability of B

times the probability of C and so on.

If the probability is 1 in 1,000 that a house burns down

and there are 1,000 houses, then the probability that they

all burn down is 1/1000 to the 1000th power,

which is virtually zero.

So insurance companies

then–Basically, if they write a lot of

policies, then they have virtually no risk.

That is the fundamental idea that may seem simple and obvious

to you, but it certainly wasn't back when the idea first came

up. Incidentally,

we have a problem set, which I want you to start today

and it will be due not in a week this time,

because we have Martin Luther King Day coming up,

but it will be due the Monday following that. If you follow through from the

independent theory, there's one of the basic

relations in probability theory–it's called the binomial

distribution. I'm not going to spend a whole

lot of time on this but it gives the probability of x

successes in n trials or, in the case of insurance

x, if you're insuring against an accident,

then the probability that you'll get x accidents

and n trials.

The binomial distribution gives

the probability as a function of x and it's given by the

formula where P is the probability of the accident:

P^(X) (1-P)^(N-X) [n!/(n-x)!]. That is the formula that

insurance companies use when they have independent

probabilities, to estimate the likelihood of

having a certain number of accidents.

They're concerned with having too many accidents,

which might exhaust their reserves.

An insurance company has reserves and it has enough

reserves to cover them for a certain number of accidents.

It uses the binomial distribution to calculate the

probability of getting any specific number of accidents.

So, that is the binomial distribution.

I'm not going to expand on this

because I can't get into–This is not a course in probability

theory but I'm hopeful that you can see the formula and you can

apply it. Any questions?

Is this clear enough? Can you read my handwriting? Another important concept in

probability theory that we will use a lot is expected value, the mean, or average–those are

all roughly interchangeable concepts.

We have expected value, mean or average. We can define it in a couple of

different ways depending on whether we're talking about

sample mean or population mean. The basic definition–the

expected value of some random variable x–E(x)–I guess

I should have said that a random variable is a quantity that

takes on value. If you have an experiment and

the outcome of the experiment is a number, then a random variable

is the number that comes from the experiment.

For example, the experiment could be tossing

a coin; I will call the outcome

heads the number one, and I'll call the outcome

tails the number zero, so I've just defined a random

variable.

You have discrete random

variables, like the one I just defined, or there are

also–which take on only a finite number of values–and we

have continuous random variables that can take on any number of

values along a continuum. Another experiment would be to

mix two chemicals together and put a thermometer in and measure

the temperature. That's another invention of the

1600s, by the way–the thermometer.

And they learned that concept–perfectly natural to

us–temperature. But it was a new idea in the

1600s. So anyway, that's continuous,

right? When you mix two chemicals

together, it could be any number, there's an infinite

number of possible numbers and that would be continuous.

For discrete random variables, we can define the expected

value, or µ_x –that's the Greek letter

mu–as the summation i = 1 to infinity of.

[P(x=x_i) times (x_i)].

I have it down that there might be an infinite number of

possible values for the random variable x.

In the case of the coin toss, there are only two,

but I'm saying in general there could be an infinite number.

But they're accountable and we can list all possible values

when they're discrete and form a probability weighted average of

the outcomes.

That's called the expected

value. People also call that the mean

or the average. But, note that this is based on

theory. These are probabilities.

In order to compute using this formula you have to know the

true probabilities. There's another formula that

applies for a continuous random variables and it's the same idea

except that–I'll also call it µ_x,

except that it's an integral. We have the integral from minus

infinity to plus infinity of F(x)*x*dx,

and that's really–you see it's the same thing because an

integral is analogous to a summation. Those are the two population

definitions. F(x) is the continuous

probability distribution for x. That's different when you have

continuous values–you don't have P (x =

x_i) because it's always zero.

The probability that the temperature is exactly 100°

is zero because it could be 100.0001°

or something else and there's an infinite number of

possibilities. We have instead what's called a

probability density when we have continuous random variables.

You're not going to need to know a lot about this for this

course, but this is–I wanted to get the basic ideas down.

These are called population measures because they refer to

the whole population of possible outcomes and they measure the

probabilities.

It's the truth,

but there are also sample means.

When you get–this is Rituparna, counting the leaves

on a tree–you can estimate, from a sample,

the population expected values. The population mean is often

written "x-bar." If you have a sample with

n observations, it's the summation i = 1

to n of x_i/n–that's

the average. You know that formula, right?

You count n leaves–you count the number of leaves.

You have n branches on the tree and you count the

number of leaves and sum them up.

One would be–I'm having a little trouble putting this into

the Rituparna story, but you see the idea.

You know the average, I assume. That's the most elementary

concept and you could use it to estimate either a discreet or

continuous expected value. In finance, there's often

reference to another kind of average, which I want to refer

you to and which, in the Jeremy Siegel book,

a lot is made of this. The other kind of average is

called the geometric average.

We'll call that–I'll only show

the sample version of it G(x) = the product i = 1 to

n of (x_i )^(1/n).

Does everyone–Can you see that? Instead of summing them and

dividing by M, I multiply them all together

and take the n^(th) root of them.

This is called the geometric average and it's used only for

positive numbers. So, if you have any negative

numbers you'd have a problem, right?

If you had one negative number in it, then the product would be

a negative number and, if you took a root of that,

then you might get an imaginary number.

We don't want to use it in that case.

There's an appendix to one of the chapters in Jeremy Siegel's

book where he says that one of the most important applications

of this theory is to measure how successful an investor is.

Suppose someone is managing

money. Have they done well?

If so, you would say, "Well, they've been investing

money over a number of different years.

Let's take the average over all the different years."

Suppose someone has been investing money for n

years and x_i is the return on the investment

in a given year. What is their average

performance? The natural thing to do would

be to average them up, right?

But Jeremy says that maybe that's not a very good thing to

do. What he says you should do

instead is to take the geometric average of gross returns.

The return on an investment is how much you made from the

investment as a percent of the money invested.

The gross return is the return plus one.

The worst you can ever do investing is lose all of your

investment–lose 100%. If we add one to the return,

then you've got a number that's never negative and we can then

use geometric returns.

Jeremy Siegel says that in

finance we should be using geometric and not arithmetic

averages. Why is that?

Well I'll tell you in very simple terms,

I think. Suppose someone is investing

your money and he announces, I have had very good returns.

I have invested and I've produced 20% a year for nine out

of the last ten years. You think that's great,

but what about the last year. The guy says,

"Oh I lost 100% in that year." You might say,

"Alright, that's good." I would add up 20% a year for

nine years and than put in a zero–no,

120 because it's gross return for nine years–and put in a

zero for one year. Maybe that doesn't look bad,

right? But think about it,

if you were investing your money with someone like that,

what did you end up with? You ended up with nothing.

If they have one year when they lose everything,

it doesn't matter how much they made in the other years.

Jeremy says in the text that the geometric return is always

lower than the arithmetic return unless all the numbers are the

same.

It's a less optimistic version.

So, we should use that, but people in finance resist

using that because it's a lower number and when you're

advertising your return you want to make it look as big as

possible. We also need some measure

of–We've been talking here about measures of central

tendency only and in finance we need,

as well, measures of dispersion, which is how much

something varies. Central tendency is a measure

of the center of a probability distribution of the–Central

tendency is a measure–Variance is a measure of how much things

change from one observation to another.

We have variance and it's often represented by σ²,

that's the Greek letter sigma, lower case, squared.

Or, especially when talking about estimates of the variance,

we sometimes say S² or we say standard

deviation². The standard deviation is the

square root of the variance. For population variance,

the variance of some random variable x is defined as

the summation i = 1 to infinity of the Prob (x =

x_i) times (x_i –

µ_x)^(2).

So mu is the mean–we just

defined it of x–that's the expectation of x or

also E(x), so it's the probability

weighted average of the squared deviations from the mean.

If it moves a lot–either way from the mean–then this number

squared is a big number. The more x moves,

the bigger the variance is. There's also another variance

measure, which we use in the sample–or also Var is used

sometimes–and this is ∑².

There's also another variance measure, which is for the

sample. When we have n

observations it's just the summation i = 1 to n of

(x – x bar)²/n. That is the sample variance.

Some people will divide by n–1.

I suppose I would accept either answer.

I'm just keeping it simple here. They divide by n-1 to

make it an unbiased estimator of the population variance;

but I'm just going to show it in a simple way here.

So you see what it is–it's a measure of how much x

deviates from the mean; but it's squared.

It weights big deviations a lot because the square of a big

number is really big.

So, that's the variance.

So, that completes central tendency and dispersion.

We're going to be talking about these in finance in regards to

returns because–generally the idea here is that we want high

returns. We want a high expected value

of returns, but we don't like variance.

Expected value is good and variance is bad because that's

risk; that's uncertainty.

That's what this whole theory is about: how to get a lot of

expected return without getting a lot of risk.

Another concept that's very basic here is covariance.

Covariance is a measure of how much two variables move

together. Covariance is–we'll call

it–now we have two random variables, so I'll just talk

about it in a sample term. It's the summation i = 1

to n of [(x – x-bar) times (y –

y-bar)]/n.

So x is the deviation

for the i-subscript, meaning we have a separate

x_i and y_i for each

observation. So we're talking about an

experiment when you generate–Each experiment

generates both an x and a y observation and we know

when x is high, y also tends to be high,

or whether it's the other way around.

If they tend to move together, when x is high and

y is high together at the same time,

then the covariance will tend to be a positive number.

If when x is low, y also tends to be low,

then this will be negative number and so will this,

so their product is positive.

A positive covariance means

that the two move together. A negative covariance means

that they tend to move opposite each other.

If x is high relative to x-bar–this is

positive–then y tends to be low relative to its mean

y-bar and this is negative.

So the product would be negative.

If you get a lot of negative products, that makes the

covariance negative. Then I want to move to

correlation. So this is a measure–it's a

scaled covariance. We tend to use the Greek letter

rho. If you were to use Excel,

it would be correl or sometimes I say corr.

That's the correlation. This number always lies between

-1 and +1. It is defined as rho=

[cov(x_iy _i)/S_x

S_y] That's the correlation

coefficient.

That has kind of almost entered

the English language in the sense that you'll see it quoted

occasionally in newspapers. I don't know how much you're

used to it–Where would you see that?

They would say there is a low correlation between SAT scores

and grade point averages in college, or maybe it's a high

correlation. Does anyone know what it is?

But you could estimate the corr–it's probably positive.

I bet it's way below one, but it has some correlation,

so maybe it's .3. That would mean that people who

have high SAT scores tend to get higher grades.

If it were negative–it's very unlikely that it's negative–it

couldn't be negative. It couldn't be that people who

have high SAT scores tend to do poorly in college.

If you quantify how much they relate, then you could look at

the correlation. I want to move to regression.

This is another concept that is very basic to statistics,

but it has particular use in finance, so I'll give you a

financial example.

The concept of regression goes

back to the mathematician Gauss, who talked about fitting a line

through a scatter of points. Let's draw a line through a

scatter of points here. I want to put down on this axis

the return on the stock market and on this axis I want to put

the return on one company, let's say Microsoft. I'm going to have each

observation as a year. I shouldn't put down a name of

a company because I can't reproduce this diagram for

Microsoft. Let's not say Microsoft,

let's say Shiller, Inc.

There's no such company, so I can be completely

hypothetical. Let's put zero here because

these are not gross returns these are returns,

so they're often negative.

Suppose that in a given

year–and say this is minus five and this is plus five,

this is minus five and this is plus five–Suppose that in the

first year in our sample, the company Shiller, Inc.

and the market both did 5%. That puts a point right there

at five and five. In another year,

however, the stock market lost 5% and Shiller,

Inc. lost 7%.

We would have a point, say, down here at five and

seven. This could be 1979,

this could be 1980, and we keep adding points so we

have a whole scatter of points. It's probably upward sloping,

right? Probably when the overall stock

market does well so does Shiller, Inc.

What Gauss did was said, let's fit a line through the

point–the scatter of points–and that's called the

regression line.

He chose the line so that–this

is Gauss–he chose the line to minimize the sum of squared

distances of the points from the lines.

So these distances are the lengths of these line segments.

To get the best fitting line, you find the line that

minimizes the sum of squared distances.

That's called the regression line and the intercept is called

alpha–there's alpha.

And the slope is called beta.

That may be a familiar enough concept to you,

but in the context of finance this is a major concept.

The way I've written it, the beta of Shiller,

Inc. is the slope of this line.

The alpha is just the intercept of this curve. We can also do this with excess

returns. I will get to this later,

where I have the return minus the interest rate on this axis

and the market return minus the interest rate on this axis.

In that case, alpha is a measure of

how much Shiller, Inc.

outperforms.

We'll come back to this,

but beta of the stock is a measure of how much it moves

with the market and the alpha of a stock is how

much it outperforms the market. We'll have to come back to

that–these are basic concepts. I want to–another concept–I

guess I've just been implicit in what I have–There's a

distribution called the normal distribution and that is–I'm

sure you've heard of this, right?

If you have a distribution that looks like this–it's

bell-shaped–this is x and–I have to make it look

symmetric which I may not be able to do that well–this is

f(x), the normal distribution.

f(x) = [1/(√ (2π)σ)]

times e to minus [(x-µ)^(2) / 2σ].

It's a famous formula, which is due to Gauss again.

We often assume in finance that random variables,

such as returns, are normally distributed.

This is called the normal distribution or the Gaussian

distribution–it's a continuous distribution.

I think you've heard of this, right?

This is high school raw material.

But I want to emphasize that there are also other bell-shaped

curves.

This is the most famous

bell-shaped curve, but there are other ones with

different mathematics. A particular interest in

finance is fat-tailed alternatives. It could be that a random

distribution–I don't have colored chalk here I don't

think, so I will use a dash line to

represent the fat-tailed distribution.

Suppose the distribution looks like this. Then I have to try to do that

on the other side, as symmetrically as I can.

These are the tails of the distribution;

this is the right tail and this is the left tail.

You can see that the dash distribution I drew has more out

in the tails, so we call it fat-tailed.

This refers to random variables that have fat-tailed

distributions–random variables that occasionally give you

really big outcomes.

You have a chance of being way

out here with a fat-tailed distribution.

It's a very important observation in finance that

returns on a lot of speculative assets have fat-tailed

distributions. That means that you can go

through twenty years of a career on Wall Street and all you've

observed is observations in the central region.

So you feel that you know pretty well how things behave;

but then, all of a sudden, there's something way out here.

This would be good luck if you were long and now suddenly you

got a huge return that you would not have thought was possible

since you've never seen it before.

But you can also have an incredibly bad return.

This complicates finance because it means that you never

know. You never have enough

experience to get through all these things.

It's a big complication in finance.

My friend Nassim Talib has just written a book about it

called–maybe I'll talk about that–called The Black

Swan.

It's about how so many plans in

finance are messed up by rare events that suddenly appear out

of nowhere. He called it The Black

Swan because if you look at swans, they're always white.

You've never seen a black swan. So, you end up going through

life assuming that there are no black swans.

But, in fact, there are and you might finally

see one. You don't want to predicate

making complicated gambles under the assumption that they don't

exist. Talib, who's a Wall Street

professional, talks about these black swans

as being the real story of finance.

Now. I want to move away from

statistics and talk about present values,

which is another concept in finance that is fundamental.

And so, let me–And then this will conclude today's lecture.

What is a present value? This isn't really statistics

anymore, but it's a concept that I want to include in this

lecture.

People in business often have

claims on future money, not money today.

For example, I may have someone who promises

to pay me $1 in one year or in two years or three years.

The present value is what that's worth today.

I may have an "IOU" from someone or I may own a bond from

someone that promises to pay me something in a year or two

years. According to a time-honored

tradition in finance, it says that it's a promise to

pay $1, but it's not worth $1 today.

It must be worth less than $1. What you could do hundreds of

years ago–and can still do it today–was go to a bank and

present this bond or IOU and say,

"What will you give me for it?" The bank will discount it.

Sometimes we say "present discounted value." The banker will say,

"Well you have $1 a year from now, but that's a year from now,

so I won't give you $1 now. I'll give you the present

discounted value for it." Now, I'm going to abstract from

risk.

Let's assume that we know that

this thing is going to be paid, so it's a matter of simple

time. Of course, the banker isn't

going to give you $1 for something that is paying $1 in a

year because the banker knows that $1 could be invested at the

interest rate. Let's say the interest rate is

r and that would be a number like .05,

which is 5%, which is five divided by one

hundred. Then the present value of

$1–The present PDV or PV of $1 = $1/(1+r).

That's because the banker is thinking, if I have this amount

right now and I invest it for one year, then what do I have.

I have (1 + r)*(1/1+r).

It's $1, so that works out exactly right.

You have to discount something that's one period in the future

by dividing it by 1+r.

This is the present value of $1

in one time period, which I'm taking to be a year.

It doesn't have to be a year. The interest rate has units of

time, so I have to specify a time period over which I'm

measuring an interest rate. Typically it's a year.

If it's a one-year interest rate, the time period is one

year, and the present value of $1 in one time period is given

by this: the present value of $1 in n periods is

1/(1+r)^(n) and that's all there is to this.

I want to talk about valuing

streams of payments. Suppose someone has a contract

that promises to pay an amount each period over a number of

years. We have formulas for these

present values and these formulas are well known.

I'm just going to go through them rather quickly here.

The simplest thing is something called a consol or perpetuity. A perpetuity is an asset or a

contract that pays a fixed amount of money each time

period, forever. We call them consols because,

in the early 1700s, the British Government issued

what they called consols or consolidated debt of the British

Crown that paid a certain amount of pound sterling every six

months forever. You may say,

what audacity for the British Government to promise to pay

anything forever.

Will they be around forever?

Well as far as you're concerned, it's as good as

forever, right? Maybe someday the

British–United Kingdom–something will happen

to it, it will fall apart or change;

but that is so distant in the future that we can disregard

that, so we'll take that as forever.

Anyway, the government might buy them back too,

so who cares if it isn't forever.

Let's just talk about it as forever.

Let's say this thing pays one pound a period forever.

What is the present value of that?

Well, the first–each payment we'll call a coupon–so it pays

one pound one year from now.

Let's say it's one year just to

simplify things. It pays another pound two years

from now, it pays another pound three years from now.

The present value is equal to–remember it starts one year

from now under assumption–we could do it differently but I'm

assuming one year now. The present value is

1/(1+r) for the first year;

plus for the second year it's 1/(1+r)²;

for the third year it's 1/(1+r)³;

and that goes on forever. That's an infinite series and

you know how to sum that, I think.

I'll tell you what it is: it's 1/r,

or it would be £1/r .

Generally, if it pays c dollars for every period,

the present value is c/r.

That's the formula for the present value of a consol.

That's one of the most basic formulas in finance.

The interesting thing is that it means that the value of

consol moves inversely to the interest rate.

The British Government issued those consols in the early 1700s

and, while they were refinanced in the late nineteenth century,

they're still there.

If you want to go out and buy

one, you can get on your laptop right after this lecture and buy

one of them. Then you've got something that

will pay you something forever. But you're going to know that

the value of that in the market moves opposite with interest

rates. So, if interest rates go down,

the value goes up; if interest rates go up,

the value of your investment goes down.

Another formula is–what if the consol doesn't pay–I'm sorry,

the next thing is a growing consol. I'm calling it a growing consol

even though the British consols didn't grow.

Let's say that the British Government didn't say that

they'll pay one pound per year, but it'll be one pound the

first year, then it will grow at the rate g and it will

eventually be infinitely large.

You get one pound the first

year, you get 1+g pounds the second year,

etc., (1+g)² the third year and so on.

The present value of this–suppose it pays–let's say

it pays c pounds each year, so it would be c

times this. It would be c times

(1+g)³ in the third year,

etc., Then the present value is equal to c/(r-

g)–that's the formula for the value of a growing

console.

G has to be less than

r for this to make sense because if g–if it's

growing faster than the rate of interest,

then this infinite series will not converge and the value would

be infinite. You might ask,

"Well then how does that make sense?"

What if the British Government promised to pay 10% more each

year, how would the market value that?

The formula doesn't have a number.

I'll tell you why it doesn't have a number:

the British Government will never promise to pay you 10%

more each year because they can't do it.

And, the market wouldn't believe them because you can't

grow every year faster than the interest rate.

Now that's one of the most basic lessons,

you can't do it.

One more thing that I think

would be relevant to the–there's also the annuity

formula. This is a formula that applies

to–what if an asset pays a fixed amount every period and

then stops? That's called an annuity.

An annuity pays c dollars starting in t =

1,2, 3, and n is the last period, then it stops.

A good example of an annuity is a mortgage on a house.

When you buy a house, you borrow the money and you

pay it back in fixed–it would usually be monthly,

but let's say annual payments. You pay every year a fixed

amount on your house to the mortgage originator and then

after so many–n is 30 years,

typically–you would then have paid it off.

It used to be that mortgages had what's called a balloon

payment at the end.

This means that you would have

to pay extra money at the end; but they decided that people

have trouble doing that. It's much better to pay a fixed

payment and then you're done. Otherwise, if you ask them to

pay more at the end, then a lot of people won't have

the money. We now have annuity mortgages.

What is the present value of an annuity?

That is, the present value of an annuity is equal to the

amount–what did I say–c*{1 –

[1/(1+r)]^(n) }/r.

That is the present value of an annuity.

I wanted to say one more thing because I realize that you have

to–your first problem set will cover this–is to talk about the

concept that applies probability theory to Economics.

That is expected utility theory. Then I'll conclude with this.

In Economics, it is assumed that people have

a utility function, which represents how happy they

are with an outcome–we typically take that as U,

If I have a monetary outcome, then I have a certain amount of

money, x dollars.

How happy I am with x

dollars is called U(x). This, I think you've gotten from

other economics courses–we have something called diminishing

marginal utility. The idea is that for any amount

of money–if this x is the amount of money that I

receive–utility as the function of the amount of money I receive

is downwardly-concave. The exact shape of the curve is

subject to discussion, but the point of diminishing

marginal utility is that, as you get more and more money,

the increment in utility for each extra dollar diminishes.

Usually we say it never goes down, we don't have it going

down, cross that out. That would be where having more

money makes you less happy. That may actually work that

way, but our theory says no, you always want more.

It's always upward sloping, but it may, after awhile,

you get close to satiation where you've got enough.

Incidentally, I mentioned this last time–I

was talking about–I was philosophizing about wealth and

I asked what are you going to do with a billion dollars.

We have many billionaires in this country and I think that

the only thing you have to do with it is philanthropy.

They have to give it away because they are essentially

satiated.

Because, as I said,

you can only drive one car at a time and if you've got ten of

them in the garage, then it doesn't really do you

much good. You can't do it;

you can't enjoy all ten of them. It's important–that's one

reason why we want policies that encourage equality of

incomes–not necessarily equality,

but reasonable equality–because the people

with very low wealth have a very high marginal utility of income

and people with very high wealth have very little.

So, if you take from the rich and give to the poor you make

people happier. We're not going to do that in a

Robin Hood way; but in finance we're going to

do that in a systematic way through risk management.

We're going to be taking away from lucky–you think of

yourself as randomly on any point of this.

You don't want–you know that you'd like to take money away

from yourself in the high-outcome years and give it

to yourself in the low-income years.

What finance theory is based on–and much of economics is

based on–the idea that people want to maximize the expected

utility of their wealth.

Since this is a concave

function, it's not just the expected value.

To calculate the expected utility of your wealth,

you might also have to look at the expected return,

or the geometric expected return, or the standard

deviation. Or you might have to look at

the fat tail. There are so many different

aspects that we can get into and this underlying theory motivates

a lot of what we do. But it's not a complete theory

until we specify the utility function.

Of course, we will also be talking about behavioral finance

in this course and we'll, at times, be saying that the

utility function concept isn't always right–the idea that

people are actually maximizing expected utility might not be

entirely accurate.

But, in terms of the basic

theory, that's the core concept. I have one question on the

problem set that asks you to think about how you would handle

a decision: whether to gamble, based on efficient–based on

expected utility theory. That's a little bit of a tricky

question but–So, do the best you can on it and

think–try to think about what this kind of theory would imply

for gambling behavior. I will see you on Friday.

That's two days from now in this room.

.

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