Prof: Okay,
but now I want to move to the
next topic,
which is the topic called the
Capital Asset Pricing Model,
and it's in some points the
high point of the class.
 
It used to be the high point of
finance.
The theory hasn't worked out as
well as people thought in recent
times,
but it's quite a great
achievement and a lot of it was
done here at Yale,
so I want to explain it to you.
 
So you see we have a problem so
far.
If everybody's trying to hedge
that means everybody's trying to
get a completely riskless
payoff.
It's impossible because,
I mean, there's real risk in
the economy.
 
And what do we mean by real
risk?
Well, something in one state is
just going to be bad for the
whole economy compared to
another state.
Maybe we'll run out of oil or
something like that.
It's impossible that everybody
can consume the same thing in
every state,
so it's impossible that
everybody can perfectly hedge,
but everybody wants to
perfectly hedge.
 
So what has to happen?
 
What gives?
 
How does the theory have to
change?
Well, the theory's going to
change in a simple way,
which Shakespeare himself
already knew and already told us
about in The Merchant of
Venice.
What's going to happen is
everybody is going to try and
hedge as much as they can by
diversifying,
but because there's some real
risk in the economy,
in some states things will be
in the aggregate worse than they
will be in other states.
 
So what's the consequence of
that?
The consequence of that is if
you're going to buy an asset
that pays something that's
riskless you're going to pay the
discounted expected return of
the asset,
but if you're going to buy an
asset that's risky you're going
to need a higher rate of return
so the price will be less than
the expected discounted payoff.
 
So Shakespeare,
remember, said exactly that.
When the play begins with
Antonio looking melancholy his
interlocutor says,
Salerio or somebody asks him
whether he's worried about his
businesses.
He says, "No,
I've got a different
ship--every ship's on a
different ocean so I'm
diversified.
 
I'm not that worried."
 
So Shakespeare knows about
diversification and that's what
everybody should do,
but then when it comes time to
pick the caskets to get to marry
the beautiful Portia,
who by the way is not just
beautiful but she's rich so
they're looking for a prize,
but they sign a contract that
whoever picks the wrong casket
not only doesn't get Portia,
he can never marry anyone in
the future.
So what's the purpose of that
contract?
It's to make it a very risky
gamble.
And so why did Shakespeare want
to make it a risky gamble,
so he could explain he
understands risk and return.
So you remember the
conversation where everybody
says,
"Well, I'm not going to
pick this unless,
you know, it's only because
she's so rich and so beautiful
that I'm willing to do this.
The return is so high that it's
worth the risk to me."
So Shakespeare already
understood that things that are
risky are going to have to be
priced less than their expected
return--
expected payoff--so the
expected return,
that is the payoff per dollar
put into it looks higher to
compensate you for the risk.
So Shakespeare almost had the
whole story.
What's missing from Shakespeare?
 
Well, what is the definition of
risk is missing from
Shakespeare, and it will turn
out that it's going to be a very
surprising definition.
 
So the purpose of the model I'm
going to explain is,
how do you measure risk,
and how should that affect the
price of things,
and how does that affect all
the analysis we've done so far?
 
So that's the topic of the next
couple lectures.
So the first person to confront
this problem and propose a
solution, a mathematical
solution, was the mathematician
Bernoulli and his brother.
 
So the Bernoullis were a famous
mathematical family,
and one of the brothers went
off to St.
Petersburg where he ended up
dying shortly afterwards,
but he noticed the following
puzzle, and some of you have
heard this, it's called the St.
 
Petersburg Paradox.
 
So suppose I offer you a bet.
 
I say I'm going to flip a coin,
and I keep flipping the coin
until it comes up tails,
and I count how many coin flips
I've counted until you get
tails,
and if that's N flips you get 2
to the N dollars.
So if you flip it 1 time and it
comes up tails right away,
which is probability 1 half,
you get 2 dollars.
If I flip it 2 times and it
gets heads and then tails,
that's with probability 1
quarter, you've flipped it twice
you get 2 to the 2 or 4 dollars.
 
If I flip it three times and I
get heads,
heads, tails the odds of that
are 1 half times 1 half times 1
half,
which is 1 eighth,
but then you'll get 2 to the 3
dollars.
So 4 8 1 over 2 to the N times
2 to the N ....
So Bernoulli,
the one who died,
told his brother Daniel about
this and he said,
"Well, I've offered this
bet to a bunch of people and I
asked them,
how much would they be willing
to pay for this risky
asset?"
I mean, what would you pay?
 
Let's hear some numbers?
 
How many dollars would you pay
if I offered you this bet?
I'm just going to keep flipping
a fair coin, count the number of
flips until a tails and pay you
2 to the N dollars.
So this is the expectation
which obviously equals infinity.
So according to what we've said
so far you should pay infinite
amount of dollars for it,
but Bernoulli couldn't get
anyone to offer him that much
money.
How much would you offer for
this bet?
I just want to hear some
numbers.
Student: 1 dollar fifty.
 
Prof: 1 dollar fifty.
 
Anybody else have any-- it's
pretty conservative,
I mean, that's almost
ridiculous, in fact.
You're guaranteed 2 dollars no
matter what happens,
right?
 
So you're paying 1 dollar fifty
and you're going to get 2 for
sure,
so that's a pretty conservative
number to say,
anybody a little more
venturesome than that?
 
You can't do worse than 2
dollars in this bet.
Student: 4 dollars.
 
Prof: What?
 
Student: 4 dollars.
 
Prof: 4 dollars.
 
All right, so Bernoulli asked a
bunch of people and the average
of what they say happens to have
been 4 dollars.
That's what they said on
average.
And so Bernoulli said,
"Well, this is amazing.
The expectation is infinite and
they're only willing to pay me a
miserable 4 dollars for
this."
Now, the real reason might have
been that they didn't believe
Bernoulli was actually going to
pay the money,
and they'd give up their money
and they weren't going to get
anything back,
but let's ignore that
temporarily and take it
seriously.
Bernoulli said,
the solution must be that
people don't care about the
dollar payoff.
They care about the utility of
the dollar payoff.
So let's put in a utility
function.
So the utility of the dollar
payoff would be [one half]
U of 2 1 quarter U of 4 1
eighth U of 8 1 over 2 to the N
U of 2 to the N ...
 
And so then he said--well,
of course why is that going to
help?
 
Well, because if the utility
function,
say, looks like this--so here's
X and here's U of X--
the more dollars you get,
maybe it increases utility but
by less and less,
so you're not really gaining
much by getting these numbers
way out here.
They're not adding really much
to utility, so you only care
about these small numbers.
 
I mean, it's good to get more,
but not much better to have
more.
 
So he said, lo and behold,
if I put in log natural as my
utility function,
which this looks like,
that's the graph of log
natural, and I put this in.
Now, you see this is easy to
solve, to compute,
because log of 2 to the N is N
times log of 2.
So the log of 2s come out and
it's just the sum.
It's log of 2 times the sum of
1 over 2 to the N times N.
So it's N over 2 to the N.
 
So this is equal to log of 2
times the sum,
N equals 1 to the infinity,
N over 2 to the N.
That's what it turns out,
that this thing is that.
So it's not just 1 over 2 to
the N which would have added up
to 1, but N over 2 to the N.
 
So the point is because you
have the log function here this
actually equals the log of 4.
 
So anyway, I've worked out the
arithmetic.
It's very simple.
 
You all know how to sum 1 over
2 to the N.
You probably don't know how to
sum N over 2 to the N.
You never thought of doing it
before, but the same trick gets
you to be able to sum N over 2
to the N.
It's obviously more than 1 over
2 to the N, in fact,
it's equal to 2.
 
So this sum is equal to 2 and 2
times log of 2 is log of 2
squared which is log of 4.
 
So by plugging in--instead of
caring about expectation you
care about the expected utility.
 
You can explain why the average
person was willing to pay 4
dollars because the expected log
of this is equal to the log of
4.
 
This is equal to the log of 4.
 
So Bernoulli thought he'd
brilliantly explained his
paradox.
 
So this is the other brother,
Daniel.
The dead brother posed the
problem,
maybe solved it too for all I
know,
but the other brother who still
lived came up with the solution,
maybe with his brother,
that people don't look at
expected payoffs,
they look at expected utility
of payoffs,
and the utility should have
this concave feature that more
and more payoff adds on the
margin less and less utility.
 
So this function satisfies d
squared U (X) / dX squared is
less than 0.
 
The second derivative is
negative.
So the marginal utility is
declining as you get more and
more.
 
So that was the first advance
on how to deal with risk.
Now, actually Bernoulli didn't
really solve the problem because
just saying that you replace the
payoff with expected utility of
a concave function--
this log wouldn't have really
solved the problem,
because suppose that Bernoulli
had offered instead a bet not of
2 to the N but of 2 to the 2 to
the N,
a much more generous bet?
Then even with logs you would
have gotten an infinite number.
So basically he should have
said that people care about a
concave function of payoff where
the function is bounded unlike
log which is not bounded.
 
But anyway, let's leave that
aside.
It should be a concave function.
 
So to put it another way,
a concave function has the
property,
if you look at it,
let's say it looks like that,
that if you have this payoff
X_A and this payoff
X_B and you've got--
so this is the utility now
here, X_A,
and this is the utility U of
X_B,
and this is the utility U of
X_A.
If you have a 50/50 bet of
either getting X_A or
getting X_B you're
going to end up with this
expected utility.
 
Your utility is going to be 1
half of U X_A 1 half
of U X_B.
 
That's what we had down here,
1 half of this utility plus
half of that utility assuming
nothing else can happen.
But if you give the person half
the amounts of money for sure
then he gets this utility which
is much bigger than that utility
because this is a concave
function.
The extra you gain by winning
the bet,
compared to getting the average
for sure,
the extra you gain doesn't
drive the utility up very much
because it's flattening out.
 
Whereas losing the bet,
even though you're losing the
same number of dollars because
from here to here is the same as
from here to here,
the loss of the same number of
dollars is more important to you
than the gain of an equal amount
of dollars,
and that's why you'd rather get
the middle for sure than having
a 50/50 chance of going on the
extremes.
 
So Bernoulli pointed the way to
the modern theory of risk
aversion,
which is to just assume--risk
aversion in modern economics
means people care about expected
utility,
maybe discounted,
expected discounted utility
where the utility is concave.
So whatever utility function we
wrote in here,
maybe it shouldn't be log,
it should be something else.
How would people evaluate this?
 
They'd evaluate it log of 4.
 
In other words,
they'd say take whatever that
constant utility was,
which was log of 4,
that produces the same utility
as the random gamble.
So this random gamble gives
this expected utility which is
equivalent to having that for
sure.
So here's the 4.
 
So 4 for sure gives a utility,
log of 4, that puts you here
which is the same as the
expected utility of getting the
random gamble.
 
That's the modern theory of
risk aversion,
and it explains why people
would rather have things for
sure,
but it's now quantifiable
because if you can't have
something for sure then you know
that it's more dangerous,
but with this concrete utility
function you can find out
exactly how much you're willing
to pay to transform this risky
gamble into a safe gamble.
You'd give up this much
expectation in order to get the
payoff for sure.
 
So we're going to turn a vague
theory into something
quantifiable and get a
surprising conclusion.
So that's step one.
 
We now think about people
maximizing utility.
Well, of course we thought
about that from the beginning.
The very first class you had
utility and diminishing marginal
utility.
 
So actually this risk aversion
with diminishing marginal
utility,
fortunately for us,
is exactly the same thing we've
been thinking about all along
anyway,
diminishing marginal utility
for consumption.
 
So the very assumption of
diminishing marginal utility
that we made from the beginning
is also explaining risk
aversion.
 
So it's incredibly fortunate
that we don't actually have to
change any of our mathematics
and we've explained a new
phenomenon.
 
Now, the most simple utility
function is either the log one
or the quadratic.
 
So remember,
U (X) = a b X - c X squared.
Adding a constant is never
going to change anything,
so I'm always going to write
this as a X - 1 half alpha X
squared.
 
That's going to be my utility
function, my quadratic utility.
It could be like this or it
could be like that.
If you add a constant which
doesn't depend on X that's not
changing what anybody does so
that's irrelevant,
and if I divide it by a
constant like B that's not going
to change what everybody does so
I might as well assume the
quadratic utility is X - 1 half
alpha X squared,
quadratic utility.
 
So that's about as simple as we
can get and we're used to
working with those kinds of
utility functions.
Now, why is that such a good
convenient thing for us to use?
It's because let's suppose now
that you've got this random
payoff where with probability
gamma_1 you're going
to get X_1,
probability gamma_2
you're going to get
X_2,
probability gamma_S
you're going to get
gamma_S [correction:
X_S].
So what's the expected utility,
the analog of Bernoulli?
That means U is going to equal
summation, s = 1 to S of
(gamma_s
X_s- 1 half alpha
X_s squared).
 
So that's all we're doing.
 
We're just saying that people
don't care about the payoffs.
They have to evaluate getting
X_1,
X_2 or X_S.
 
They're going to multiply the
payoff by the expectation but
not look at the payoff itself,
look at the utility of the
payoff.
 
Now, quadratic is very simple
and the reason why we're going
to get such a beautiful theory
out of it is because this number
you don't have to keep track of
all the X's to express this
utility.
 
We're going to be able to
summarize it incredibly simply.
This is going to equal some
function F of the expectation of
X and the variance of X.
 
So all we're going to have to
worry about is the expectation
of X and the variance of X,
and so many,
many very complicated things we
can think about very simply.
So more generally if you put
the log instead of the quadratic
utility we couldn't get this
simplification and so the theory
would have to be more
complicated.
So the beautiful theory,
the Capital Asset Pricing
Model, comes out of using this
simple quadratic utility.
So why does it get so
simplified?
Well, if I just write it out,
U is going to equal the
summation of gamma_s
X_s,
so this is s = 1 to S (I've let
my probabilities be gamma,
I don't know why I chose that)
- 1 half alpha,
summation s = 1 to S,
of gamma_s
X_s squared.
 
Well, here we have the
expectation of X already.
Now, what is this 1 F alpha
gamma_s X_s
squared?
 
Well, if I wrote X_s
- the expectation of X and I
summed this squared that's equal
the variance of X--oops,
times gamma_s.
 
That by definition is the
variance, but if I wrote this
out what would I get?
 
I'd get summation s = 1 to S,
gamma_s X_s
squared which is what I have
over there and then I'd have--
well, what would I have-- minus
2 times summation
gamma_s X_s
expectation of X summation s = 1
to S of gamma_s
expectation of X squared.
But what's this?
 
The second term minus 2 times
that,
the expectation of X is a
constant so I can take that out,
minus 2 expectation of X,
can take this out and notice
that summation gamma_s
X_s,
that's the expectation of X as
well.
So that's just minus 2 times
the expectation of X squared.
And so this I can take the
expectation of X squared out and
the summation of the
probabilities is 1.
So therefore I just get equal
to summation s = 1 to S,
gamma_s X_s
squared - (expectation of X)
squared.
 
So therefore,
this up here is equal to the
expectation of X.
 
So what have I got here?
 
I've got this term.
 
So I've got the variance of X
equals this summation
X_s X_s
squared - (expectation of X)
squared.
 
So this term equals the
variance of X (expectation of X)
squared.
 
So therefore I've got this
minus 1 half alpha (expectation
of X) squared - 1 half alpha
variance of X.
That's what the algebra gives
me, so why is that again?
Because given quadratic utility
up here, that thing--getting
old--given the quadratic utility
up there I can write it as this
in this term.
 
This term is obviously the
expectation of X,
but this term is just the
variance of X plus the
expectation of X squared.
 
So when I subtract it I keep
the expectation of X minus the
(expectation of X) squared.
 
That's the first term,
and then I've got minus the
variance of X from that term
times the 1 half alpha.
So you see that depends on the
expectation of X in a positive
way, assuming alpha's a small
number, and in a negative way on
the variance of X.
 
So, just as I said,
somewhere, I said it was going
to turn out like that,
and it did right here.
The utility is equal to the
expectation of X and the
variance of X in a positive way
on the expectation of X and a
negative way on the variance.
 
So now we're ready to start the
analysis.
So far we've assumed people
only care about the expectation
and then we said,
well, we know they don't only
care about the expectations.
 
Hedge funds and everybody else,
if they know what they're doing
and they're trying to keep their
investors happy they're going to
hedge.
 
We didn't say why they're going
to hedge.
We just asserted they like to
hedge so their investors don't
get mad at them,
but really what we had in mind
is the investors have some
utility function.
They don't like risk so the
hedge fund is going to try and
keep the payoffs steady.
 
But there's a tradeoff.
 
You can't eliminate all risk.
 
So, how much is the hedge fund
and the investor going to suffer
if all risk isn't eliminated?
 
Now we have a way of
quantifying it.
People care about the utility
and not about just the expected
payoff and so you add more risk
to them--
you replace a sure thing with a
risky thing with the same
expectation--
they think it is worse,
that much worse.
 
And so we've said that of all
the myriad of utility
functions--
we could use log,
some exponential e to the minus
aX,
X to the r, there are lots of
different utilities we could
use--
we're going to deal with the
quadratic because it has the
simple property that in
evaluating an entire risky
proposition people care about
the expectation,
which is what they cared about
before,
but they're punishing
themselves for getting a bad
variance.
So because that's such a simple
thing to say we're going to get
a simple conclusion and a very
surprising conclusion.
So let's now analyze a problem
and see what would happen.
So the problem I'm going to
choose to analyze is this one.
I'm going to say that three
things can happen in the
economy.
 
Anyway, those are the
probabilities.
Now, there are many firms in
the economy A,
B and C, and let's say the
first firm,
I don't want to invent the
numbers here so I might as well
just write down the ones I
picked.
The first one's going to be
50,100 and 75.
B, the other firm's going to be
150,180 and 365,
and C is going to be 300,220
and 60.
So those are the three things
that can happen in the payoff of
the three firms.
 
Let's say there are two agents,
agent alpha owns A and also
133.5 units of X_0.
 
And beta owns B &
C and--I may have reversed
these two guys--66.5 units of
X_0.
So here we go.
 
So by alpha and beta,
I mean, there are a million
alpha agents and a million beta
agents so everything could be
scaled by a million because I
want a big economy like we
always have.
 
So there we have it.
 
We've got now a risky world.
 
Things can happen.
 
So what are the utilities?
 
Let's say utility now of alpha
is going to equal--sorry,
I left out the main point.
 
Utility of alpha is going to
equal 1 half X_0--
I just made up these numbers,
by the way,
they're not--so summation s = 1
to 3,
gamma_s,
that's the gamma_s up
there,
times X_s - 1 over
400 X_s squared,
the states.
So there's s_1.
 
So here are the states.
 
This is state 1,
s = 1, s = 2 and s = 3.
So those are the three possible
states just like we had before
with this payoff and those are
the payoff of all the assets.
Alpha owns firm A which is
producing that output in the
three states,
and also owns 133.5 units of
X_0.
 
So over here alpha's owning
133.5 and beta's owning 65.5 of
consumption at time 0.
 
So this is time 0.
 
This is time 1,
the end of the year.
So by the end of the year
something is going to happen.
There's a lot of uncertainty
between now and then.
Some of the firms are going to
be paying off in some of the
states and badly in other states
and so on.
And so the utility function for
alpha,
so he cares about consumption
at time 0 and also in each of
the three states,
but now he's going to have
these quadratic utilities.
 
He's going to say to himself,
if I just hung onto my A in
state 1 this would be--
if I never traded,
I just hung onto A the utility
function would be this quadratic
thing of 133.5.
 
So it would be 133.5 - 1 over
400,
133.5 squared,
plus he would end up with (50 -
1 over 400 times 50 squared)
times 1 quarter (100 - 1 over
400 100 squared) times 1 quarter
(75 - 1 over 400 75 squared)
times 1 half.
 
That would be his utility if he
never traded.
If he just stuck to A he'd eat
his own endowment 133.5 at time
0.
 
Oh, that isn't true.
 
I wrote down the wrong utility
at time 0.
I said his time 0 utility is 1
half X_0.
So it's 1 half 133.5.
 
So we get 1 half 133.5,
but in the future he'd get 50
in state 1,100 in state 2,
and 75 in state 3,
and that's the utility he'd end
up with.
But that's not very good for
him because he's running this
gigantic risk.
 
He's got this risk at time,
you know--state 1 is a disaster
for him if he just sticks to
that, so he doesn't want to
stick to that.
 
So how should he evaluate the
shares of firm A?
How should he evaluate the
shares of firm B which he could
get if he gave up some of A,
or how should he evaluate the
shares of firm C?
 
What should he do?
 
So beta has a similar utility.
 
Beta's utility,
U_beta is going to
equal 3 quarters X_0
the summation,
s = 1 to 3, (gamma_s
times X_s - 1 over 800
X_s squared).
 
So I've made these guys--far
from being impatient they seem
to prefer consuming in the
future until now.
That was a poor choice of
numbers.
This number should be bigger
than 1 and this should be bigger
than 1, but anyway I put 1 half
and 3 quarters.
So there's impatience built in
except it goes the wrong way.
That was just poor choice of
numbers, but the rest of it
expresses their risk aversion.
 
So alpha is looking at the
expected payoff of what he gets
to consume in the future,
but he's punishing himself by
the variance.
 
So you look at this formula you
see it's not just the
expectation, but he loses
something because of the
variance.
 
And similarly,
beta, he's looking at
consumption today,
he's adding to that the
expectation of his consumption
tomorrow for his utility,
but he's punishing himself for
having variance in the future.
So it's exactly what we
formalized,
Shakespeare's idea of people
not liking to be exposed to
variance,
to uncertainty,
which we've quantified by
calling variance.
Is everyone with me now?
 
Yes?
 
Good, I'm glad you have a
question.
Student: I don't
understand how you got 1 over
400 and 1 over 800.
 
Prof: I just made up
those numbers.
That's the utility of alpha and
that's the utility of beta.
I could pick any people I
wanted.
I just picked those two people.
 
Now, how do they differ?
 
Which person is more afraid of
risk than the other?
Is alpha or beta more afraid of
risk?
Student: Alpha.
 
Prof: Alpha is more
afraid of risk,
right?
 
This 1 over 800 is smaller than
1 over 400,
so beta doesn't really care
that much about risk,
well cares, but is not going to
punish himself too much by being
exposed to risk.
 
Alpha is not going to punish
himself too much,
but is going to punish himself
somewhat more.
So alpha is more risk averse.
 
Alpha is more afraid of risk,
it seems.
So I've taken two agents who
are afraid of risk,
one's more afraid than the
other, and I've put them in an
economy where there are risky
things that could happen.
And so we now want to work out
a more sophisticated version of
pricing and of equilibrium than
we had before.
So let me remind you that what
we sort of have been supposing
up until now is that the price--
what would the price of A be if
we didn't think about risk
aversion?
So far what we would say--what
would you say the price of A is,
price of firm A?
 
If we were naive you might say
it is 1 quarter--
by the way, I hope I have those
probabilities right--
you'd say it is 1 quarter times
50 1 quarter times 100 1 half
times 75.
 
Is that what we would have said
up until now?
Even up until now we would have
been more sophisticated than
that.
 
Student: Discounted.
 
Prof: Discounted,
times discounted.
That's what we sort of figured
up until now.
That's the logical thing to do.
 
Well, but we ignored risk
aversion, and we ignored it at
our peril because it's obviously
important.
I mean, Shakespeare,
a literary person,
he understood already 400 years
ago that risk aversion was
important,
and there are facts that
confirm what Shakespeare's
intuition is.
The stock market historically
has had a lot higher return than
the bond market.
 
Even with the last stock market
crash,
of course it came back a lot,
averaged since 1926 the stock
market's made something like 9
percent a year compared to 2 and
1 half percent in the bond
market.
So there's a huge disparity and
after over such a long period of
time it can't just be it was
luckier every year after year
after year.
 
Somehow people must have
realized the stock market is
riskier,
and so as Shakespeare said they
wanted a higher return meaning
they were paying a lower price,
but how much lower?
 
How can you figure out how much
lower?
So in this example,
in other words,
what is the price of A?
 
So this is the wrong price of
A, apparently,
because it doesn't recognize
risk aversion.
So that's where we are.
 
So any questions about what the
question is?
We're about to give an answer.
 
So you see what the question
is, that our old methodology for
figuring out prices--
that's taking expectation and
discounting--
obviously can't be right
because it doesn't recognize
risk aversion.
On the other hand,
we always had a utility
function in there from the
beginning,
even a quadratic one,
so all we have to do is do what
we did before and put in a
quadratic utility and we'll
probably get the right answer.
 
So that's exactly what we're
going to do.
So Arrow, in 1951,
this is the same guy who proved
with Debreu the Pareto
efficiency of equilibrium.
He was my thesis advisor.
 
He said we can do the same
trick that Fisher did,
only for some reason he never
credited Fisher.
I could never quite figure that
out.
He had some obscure Danish guy
he credited.
But anyway, apply Fisher trick
and assume firm dividends are
part of endowments.
 
Look for GE,
the general equilibrium,
trading outputs,
trading goods,
then go back to deduce value of
firms.
Now, what goods are we trading
here?
That was a conceptual advance.
 
We call them Arrow and then
Debreu got involved too.
Arrow-Debreu,
so Debreu was the Yale
Assistant Professor while Arrow
was a Stanford Assistant
Professor, so Arrow-Debreu State
Contingent Commodities.
So, just as Fisher said,
an apple today and an apple
next year,
even though they're identical
apples,
are different commodities with
different prices because they
come at different places in
time.
 
In fact, most people would
prefer the apple today to the
apple next year.
 
So Arrow said an apple in the
top state is a different
commodity from the same apple in
the second state,
so it should have a different
price.
So we've got just our
conventional equilibrium
according to Arrow where as long
as you have these Arrow state
contingent commodities that you
can trade--
trading today,
you can imagine today buying an
apple if state 1 occurs but not
having to get the apple if state
2 or state 3 occurs,
and that'll have a price
P_1.
 
And today you could imagine
buying the apple if state 2
occurs,
a different price from the
apple if state 1 occurs,
and also an apple if state 3
occurs,
which obviously is going to be
more expensive,
or it looks like it'll be more
expensive than the other apples
because it's 50 percent likely
to happen,
and those are the prices we
have to look for.
 
And that's going to solve our
problem because the prices of
the Arrow securities are going
to be different,
maybe, from the probabilities
and that's what will reflect the
fact that when everybody's
trying to hedge and not
everybody can do it you're going
to have to change the tradeoffs.
So we've already seen this in
our gambling thing,
at least the prices.
 
Remember with our bookies the
bookies were effectively
willing--remember there were two
outcomes, the Yankees win or the
Phillies win.
 
You could get the bookie who
thought the odds were 60/40,
by paying 60 cents today the
bookie was going to give you 1
dollar if the Yankees won,
or paying 40 cents today the
bookie will give you 1 dollar if
the Phillies won.
So we've already had these
Arrow contracts,
these Arrow securities
implicitly in our equilibrium.
And those 60/40 odds those were
the opinions of the bookie,
maybe not the actual
probabilities.
We said the final betting odds
depended on what the other
bookies were willing to give.
 
It didn't have to correspond to
reality.
There might not be a reality
even.
So here there's a reality,
25,25,
50, but that doesn't mean that
the odds,
the prices they're going to
quote in the market are going to
turn out to be that.
 
We have to solve for
equilibrium and see what they
are.
 
So what's going to happen?
 
Well, we can solve for
equilibrium very easily because
we've done this a million times
before.
And I've chosen linear
quadratic utilities,
the kind we did on the very
first day of class,
because those are easiest to
solve for equilibrium.
You don't have to get involved
in the budget set or anything
complicated.
 
You just set marginal utility
equal to price.
So we know for alpha,
sorry, we know that the
marginal utility of alpha at
time 0 divided by the price 0 is
going to have to equal the
marginal utility of alpha at
each state s times the price
P_s.
So the equilibrium,
the Arrow-Debreu equilibrium,
is going to involve
P_0,
P_1,
P_2 and
P_3,
the prices of the Arrow
securities,
the present value prices.
P_0 is what you pay
today to get the apple today.
P_1 is what you pay
today to get the apple a year
from now in state 1.
 
So these are the present value
(that's what Fisher would say)
state contingent prices.
 
The state contingent is what
Arrow added.
Now, you may ask whether there
really are these Arrow
securities floating in the
economy,
and we're going to come back to
that question,
but you could imagine all these
Arrow-Debreu state contingent
prices and commodities,
and those would be the prices
we'd solve for equilibrium.
 
So we get this over this,
marginal utility of that.
So what is this?
 
And similarly for beta,
marginal utility^(beta) at 0
over the price of 0 equals
marginal
utility^(beta)_s over
the price of s.
So what is this?
 
For alpha, his marginal utility
of consumption is 1 half.
We might as well assume one of
the prices is 1.
Let's take this price to be 1.
 
So beta, her marginal utility
is 3 quarters and the price is
1.
 
What's his marginal utility in
any state s?
It is gamma_s times
(1 - 1 over 200 times
X_s).
 
So I just differentiated this.
 
I got 1 - 2 over 400 times
X_s.
And what's her marginal utility?
 
It is gamma_s in
state s times (1 - 1 over 400
times X_s).
 
So I know in equilibrium that's
going to imply that 1 half--
well, now I have to screw
around here,
so how am I going to--so I've
got this thing over here 1 half
equals this thing over here.
 
What?
 
Student: Over
P_s.
Prof: Over P_s.
 
Ah, glad that appeared.
 
I was getting worried there.
 
Thank you.
 
Over P_s,
that helps a lot.
So that implies that something
like X--so this is what alpha is
going to do and this is what
beta is going to do.
So this implies
X^(alpha)_s equals
what?
 
So if I multiply through by
200, and I bring P_s
over gamma_s to the
other side,
and I do a bunch of stuff,
I'm going to guess this is 200
- 100 P_s over
gamma_s.
How do you think that's going
to play in Peoria?
Let's see.
 
If I multiply through by
P_s over
gamma_s I get
P_s over
gamma_s times 1 half.
 
Then I multiply everything
through by 200.
So I get 100 P_s over
gamma_s,
and then I get the 200 here,
and the X_s goes to
the other side,
and the P_s over
gamma_s goes to the
other side.
So it's 200 - 100 P_s
over gamma_s.
And this one is going to
be--X^(beta)_s is
going to equal--
well, I have to do the same
trick here except I'm going to
be multiplying through by 400
and taking 3 quarters which is
300.
So it'll be 300 minus--no,
that was wrong,
400 - 300 P_s over
gamma_s.
Because if I multiply through
by 400,
put P_s over
gamma_s on the other
side I have 3 quarters
P_s over
gamma_s times 400,
which is 300 P_s over
gamma_s.
 
This becomes a 400 and the
P_s gamma_s
went away so I have that.
 
So I know now if I could figure
out what the prices are I know
what everybody would demand in
every state.
So let me pause here.
 
That was the first critical
step.
So what did I do?
 
I said it's a long story.
 
A lot of years went into this.
 
I said people are risk averse.
 
Shakespeare knew that.
 
We want to quantify it so we
say people have concave utility
functions.
 
That quantifies risk aversion.
 
We want to make a simple
concave utility function.
We pick quadratic,
but of course we don't know
what quadratic.
 
Different people could have
different quadratic utility
functions.
 
Then we do the Fisher trick and
say that any equilibrium,
as long as you can buy and sell
every contingent commodity in
the future,
because all the Arrow
securities are there,
it can always be reduced to
general equilibrium just like we
did before.
And so now you have to feed the
endowments into the agents'--I
mean the payoffs and the
dividends into the agents'
endowments.
 
So we haven't done that yet.
 
And then we solve for supply
equals demand.
So all we have to do is we have
to have X^(alpha)_s
X^(beta)_s has to
equal the endowment of alpha in
s plus the endowment of beta in
s.
All right, so we have to do
that for every s.
So this is 200 - 100
P_sS over
gamma_s equals--now we
have to do it in state 1.
So it equals whatever they are.
 
So what is endowment of alpha
of s plus endowment beta of s?
We have to look at each state
separately.
And lo and behold I picked the
numbers so that if you add these
all together you get 500,
and here you get 280 and 220 is
also 500,
and here you get 500 again.
So lo and behold there is no
aggregate risk in the economy,
although the individual stocks
are risky the aggregate is
totally un-risky.
 
So no matter what s is,
I could put in 500 here.
It's going to turn out that the
total endowment of both people,
because I've plugged the
dividends into their personal
endowments,
added up the two people,
it's 500.
 
So it means that P_s
over gamma_s equals...
Student:
>
you forgot.
 
Prof: What?
 
Student: You forgot the
second term.
Prof: I've forgotten
something for sure,
what?
 
Student: X^(beta).
 
Prof: Oh, X^(beta).
 
So that's alpha.
 
Thank you.
 
Plus 400 - 300 P_s
over gamma_s = 500.
So if when I add this up I get
600 - 400 P_s over
gamma_s = 500,
so then I flip them to the
other side and I get
P_s over
gamma_s = 1 quarter,
because 500 from here is 100
and put the 400 on the other
side and divide by it I get
P_s over
gamma_s equals 1
quarter for all s.
 
So what did I find?
 
So it's the same.
 
P_s over
gamma_s is the same,
same in all states.
 
So what would the price of A be
here?
What's the price of A in
equilibrium?
What's the price of A?
 
I'm going to take the price of
A should be P_1 times
50 P_2 times 100
P_3 times 75,
and what does that equal?
 
Well, P_1 is just 1
quarter times gamma_1,
right?
 
P_1 over
gamma_1 is 1 quarter.
P_2 is 1 quarter
times gamma_2,
and P_3 is 1 quarter
times gamma_3,
so I just got this multiplied
by 1 quarter.
So in fact all I did is I did
what I had always done.
I took the expected payoff and
discounted it.
The discount rate is 1 quarter.
 
What's the price of the
riskless asset,
pi of (1,1, 1),
is just going to be 1 quarter,
because it's 1 quarter times
gamma 1 1 quarter times gamma 2
1 quarter times gamma 3,
gamma 1 gamma 2 gamma 3 is 1 so
it's 1 quarter.
 
So it implies the riskless
interest rate is what?
What's the riskless interest
rate?
Student: 300 percent.
 
Prof: 300 percent.
 
So we're discounting by 1 over
1 quarter because the interest
rate is 300 percent.
 
So basically nothing happened.
 
We got all the prices exactly
as we would compute them
without, you know,
just doing expectations we got
the right discount rate.
 
All we had to do was figure out
the discount rate.
So risk hasn't played any role.
 
And why didn't it play any role?
 
Because although alpha started
off owning A alone which exposed
her--forgot who was her and who
was him, let's say her--exposed
her to a lot of risk.
 
She's not going to sit there
stupidly just holding A.
She's going to trade it for B
and C for different shares.
In fact she's going to end up
holding her consumption,
this is her consumption,
200 - 100 P_s over
gamma_s,
this number doesn't depend on s.
She's going to consume the same
thing in every state,
and how can she do that?
 
She can own equal shares of A,
B and C.
She'll own a share of the whole
economy.
So in other words,
by diversifying alpha and beta
each get rid of all risk.
 
So instead of calling it
diversifying I could call it
hedging, the same thing.
 
She doesn't just hold her A.
 
She mixes B and C with it so
that she gets a payoff of
consumption that's exactly the
same in every state because
P_s over
gamma_s is independent
of the state.
 
She'll always consume the same
thing.
Everybody can hedge perfectly
and there's no problem because
there's no aggregate risk that
anyone has to be stuck with,
and therefore the price is just
going to be the same as the
probabilities discounted.
 
And that's the theory we've
worked with so far.
So, so far you could say that
everything we did was kosher
it's just that when we had these
two different probabilities of
things happening up or down we
thought that the aggregate
economy would have the same
endowments here as it did there,
and therefore the probabilities
we used were the objective
probabilities discounted.
 
No reason to change them
because nobody's going to be
forced not to hedge.
 
Everybody'll hedge.
 
So are there any questions
about what I've said?
I'm sure there should be a
question because I can't have
said it as clearly as I ought to
have.
So would somebody like to say
something?
Yes?
 
Student:
>
the old price that we found
when we hadn't done this,
but that also change the new
one?
Prof: This is the new
price with the 1 quarter.
This is the correct new price.
 
So the theory so far hasn't
changed in any interesting way.
We just found the discount rate.
 
It just looks like expected
utility,
but you shouldn't have expected
it to change because the
aggregate endowment was 500,
the constant in every state.
There's no reason why we can't
have everybody perfectly hedged
and consuming a constant in
every state,
and in fact that's what we did
have,
everybody--she consumed the
same thing in every state.
He consumed the same thing in
every state.
No reason why they both
couldn't hedge themselves
perfectly and in equilibrium
that's exactly what they did.
Any other...
 
Yes?
 
Student: If the total
endowments in every state hadn't
all added up to 500 would you
create an expected endowment or
would you just not do the
problem?
Prof: So the next step
is going to be--
what I'm going to do now is I'm
going to assume that the
endowments don't add up to a
constant in every state.
Then what's going to happen?
 
So this is not at all obvious
how to solve this and what to
do,
but it's going to turn out to
have a beautiful simple answer,
shocking, not only be simple
but also surprising.
 
So before I do that I'm going
to change the endowments so
they're not all a constant.
 
Any questions about where we're
going?
Yeah?
 
Student: Could you just
repeat what you said about
hedging
>?
Prof: Yes.
 
Thank you for the question.
 
So I went a little quickly.
 
I said that what we proved by
solving for the general
equilibrium is that the price in
every state was just going to be
1 quarter times the probability.
 
That's what we showed had to
happen in equilibrium.
Now, what's the consequence of
that?
The consequences are twofold.
 
Number one, the price of all
the assets is the same
expectation we naively would
have taken before where we used
the discount rate 1 quarter.
 
That's the first implication.
 
The second implication is that
from the formula for consumption
we noticed that she consumes the
same amount in all three states
because P_s over
gamma_s is 1 quarter
in all three states.
 
Her consumption is going to be
the same in all three states,
and his consumption,
which will be different from
hers,
but his will be the same in all
three states as well.
 
The two will add up to 500.
 
So then I took a little bit of
a leap and I interpreted that
conclusion that her consumption
doesn't depend on the sate.
What's the interpretation of
that?
She has obviously,
somewhere behind the scenes,
given up some of her A to get B
and C and held them in a mixture
so as to get the same
consumption in every state.
What must the mixture be?
 
Obviously she holds the same
proportion of A,
B and C because those add up to
500,500, 500.
So she must have held the same
proportion of A,
B and C, a fraction of the
market and got a riskless
payoff.
 
So she diversified.
 
She didn't just stick with her
A.
She substituted a little bit of
A, a little bit of B and a
little bit of C,
a different boat on every
ocean, and now she runs no risk
at all.
So she diversified,
but in the language we used
last time I could call
diversification hedging if I
wanted to.
 
She just, sort of,
sold Arrow securities in the
right proportions to turn her A
into something that was
completely riskless.
 
So whether you call it
diversifying or call it hedging
she's achieved the same end of
totally balancing her
consumption.
 
He did the same thing and they
both could do it because the
aggregate consumption was a
constant.
Yes?
 
Student: What would an
Arrow security actually look
like?
 
Prof: In real life?
 
Student: Yeah.
 
Prof: The closest we've
come to an Arrow security in
real life is a CDS,
and this is part of the reason
why these economists,
Larry Summers,
my classmate,
and Rubin who was the Secretary
of the Treasury,
and ran Citi Corp,
and who was a Yale law school
student and a Harvard
undergraduate,
and who I've sat on many
committees with,
they were seduced by the--so
what's a CDS?
 
A CDS pays 1 dollar if some
bond defaults by 1 dollar.
So that isn't an Arrow security
because an Arrow security is a
much more detailed thing.
 
An Arrow security says I'll pay
1 dollar in state one.
An Arrow security says you get
an apple in state 1,
but state 1,
remember, is not described by a
single firm,
state 1, the states of nature
are total descriptions of
everything that could happen in
the economy.
 
So an Arrow security really
says if it stops snowing in
Siberia,
if Khomeini loses power in
Iran, if there's a favorable
election outcome in Afghanistan,
and if Obama wins reelection,
and if the U.S.
solves the energy problem then
I'll give you 1 dollar.
So the Arrow security lists an
incredible number of contingent
things, every contingency
possible and says in that case
I'll give you 1 dollar.
 
A CDS says if this thing
happens I'll give you 1 dollar
whether or not Obama wins
election,
whether or not America
discovers a new source of
energy,
whether or not Afghanistan
turns around.
 
Just so long as the bond
defaults I'll give you 1 dollar.
So the CDS is an event
contingent security.
That's the CDS,
and an Arrow security is a much
more finely specified thing.
 
It's a state contingent--you
say everything that happens in
the economy,
so we'll never get to Arrow
securities,
but CDS looks like we're on the
way to them.
 
And these guys blundered by
thinking since CDSs are on the
way to Arrow securities we
should have as many CDSs and let
people trade as much of them as
we can,
but we're going to get to that
in the last lecture about how
all this theory,
what's wrong with all the
theory.
 
So, any other questions before
we--so let's now make the change
that he suggested up there.
 
Let's now change the economy
just a little bit.
Let's eliminate C.
 
So this just disappears.
 
So obviously now the total
endowment is very contingent.
It's 200, it's 280 and it's 440.
 
Now what do we do?
 
So beta owns B.
 
So now what's equilibrium going
to be?
What do you think is going to
happen?
We want to quantify this.
 
We want to give a beautiful
simple theory that's
quantifiable,
but what do you anticipate
happening to P_1,
P_2 and
P_3?
 
So everybody's going to say,
alpha, she's going to say,
look, my A is risky.
 
I don't want to hold my risky
thing.
I want to start hedging and
trading these Arrow securities
so I get the same constant in
every state.
Of course beta who owns B,
he's going to do the same
thing.
 
So they're both going to be
trying to trade Arrow
securities.
 
What's going to happen,
do you think?
Yes?
 
Student: Aren't they
both just going to be exposed to
whatever the total risk of the
economy is in
>?
 
Prof: Yeah,
there's no way that they can,
exactly, there's no way that
each of them can be perfectly
hedged.
 
So no matter what they do,
they're going to be exposed to
more risk in state,
you know, state 3 is going to
be a great state.
 
State 1 is going to be a
terrible state,
so what do you think that means
about the prices?
Everybody can't be hedged,
and so in fact what'll happen
is nobody will be hedged.
 
Although, alpha will be,
who hates risk more than beta,
will be closer to hedged than
beta will be.
So beta will end up bearing
more of the risk than alpha.
And what do you think will
happen to the prices of the
Arrow securities relative to the
probabilities?
Yes?
 
Student: It won't be
constant.
Prof: There won't be a
constant ratio of 1 quarter as
we had before,
but can you be more specific?
Student: The price of
the securities for state 1 will
be greater relative to the
probability than the price of
the security in state 3.
 
Prof: Exactly.
 
So that's what's going to turn
out.
The world is short of
commodities in state 1,
there just aren't many apples.
 
That's the disaster.
 
That's when we can't solve the
energy crisis.
We're totally screwed.
 
Everybody wants to consume more
in that state.
Everyone's going to try and
hedge against that state.
They're all going to be trying
to buy Arrow securities in that
state,
which means that because there
aren't as many to buy,
there's just not enough apples
to go around,
the price of Arrow securities
in state 1 is going to be high
relative to state 3.
There's plenty to go around
there.
So she is going to sell some of
her A and get some B to
diversify,
but B's got so good in state 3
that all of a sudden she's not
going to be so worried about
state 3 anymore,
but state 1 she's still going
to be worried about,
and there's nothing to be done
about that.
 
So the price is going to have
to be very expensive in state 1.
So all right,
that's all blah,
blah, blah.
 
Let's solve for equilibrium and
see what happens.
We can solve immediately.
 
Nothing's changed.
 
The utility functions are the
same.
None of this changed,
so this board doesn't change at
all.
 
That's demand.
 
Still depends on P_0,
P_1,
P_2 and
P_3,
but now we have to be a little
bit more careful in state 1.
So demand in every state is 600
- 400 P_1 over
gamma_1 equals
endowment of alpha endowment of
beta.
 
So in state 1,
I'm going to now change this to
a 1 although with my handwriting
it looked like a 1 anyway,
what's the aggregate endowment
in state 1?
The aggregate endowment in
state 1 is 200.
This is a 1 now.
 
That's 200, so that means
P_1 over
gamma_1 = 1,
right?
Because 400 and 400 so it's 1.
 
So you're not discounting the
first state at all.
You're looking at the
probability of it.
But what if I go to
P_2 over
gamma_2?
 
Well, the demand is going to be
the same.
It's the price that's going to
change to make up for the fact
that the supply is much
different, namely,
namely what,
280.
So now if I subtract I get
400,280 minus that is 320
divided by 400 which looks like
4 fifths, maybe.
320 over 400 is 4 fifths, right?
 
Because 320 divided by 400 is 4
fifths, so P_2 over
gamma_2 is 4 fifths.
 
So they're not proportional
anymore.
And then P_3 over
gamma_3 equals--now
the outcome is 440,
so if I subtract 440 from this
I get 160 divided by 400,
what's that?
Student: 2 fifths.
 
Prof: 2 fifths,
thank you.
P_3 over
gamma_3 = 2 fifths.
So the prices turned out to be
quite different.
Now, the reason why they're
slightly higher on average,
of course, than they were
before is because there's less
consumption in the future.
 
We've suddenly made our future
much worse off.
So people are more desperate to
consume in the future,
so that means the prices of
future consumption are going to
be higher.
 
So we have two effects here.
 
These prices instead of being 1
quarter everywhere are higher,
much higher than 1 quarter
because the future looks so much
worse.
 
The interest rate is going to
go down.
It's not going to be 300
percent anymore.
But more interesting is that
the prices are no longer
proportional to the
probabilities.
Just as he said over there the
price in state 1 is going to be
much higher relative to the
probability,
namely 100 percent of it,
than the price in state 3 which
is only 40 percent of it.
 
So that's the conclusion.
 
So now what do we do for our
price?
What's the price of A?
 
What's the price of A?
 
What do I plug in here?
 
That, so that equals 1 quarter
times 50 4 fifths times 1
quarter--
1 fifth times 100 P_3
was 2 fifths times 1 half 1
fifth times 75 which equals
something,
20,35 and 12 and 1 half,
47 and 1 half.
 
Student: Why would you
>?
Prof: Why did I what?
 
Student: Why would you
>?
Prof: So what is
P_1?
P_1 is equal to
gamma_1 and
gamma_1 is 1 quarter.
 
So that's how I got 1 quarter
here.
So that's 1 times 1 quarter.
 
P_2 was 3 fifths.
 
What was P_2?
 
Maybe I did it wrong anyway.
 
P_2 was 4 fifths
times 1 quarter which is equal
to 1 fifth, and P_3
was 2 fifths times 1 half which
is equal to 1 fifth.
 
So that's how I got the prices.
 
So all right,
so you see that things changed,
and we've captured the idea
that people can't hedge fully by
making the price of the Arrow
security in the state where the
economy's worse off,
much smaller than it was
before, I mean,
much higher than it was
relative to the probability than
before.
So we haven't gotten close to
the punch line,
sorry.
 
Yes?
 
Student: Can you repeat
the part where you said stuff
about the future looks so much
worse they need to increase
consumption?
 
Prof: Two things
happened to the prices compared
to before.
 
One is that we no longer have
the prices proportional to the
probabilities,
right?
Their proportion is 1,4 fifths,
2 fifths instead of the same
constant 1 quarter everywhere,
and that's because of the
relative scarcity.
 
People are much more worried
about the first state than the
fourth state and that's why,
relative to the probability,
the price is much higher than
the third state.
You agree with that, right?
 
Student: Yes.
 
Prof: But there's a
second effect which is that all
these numbers,
1,4 fifths, and 2 fifths
they're bigger than the 1
quarter,
1 quarter, 1 quarter we had
before,
but that's obvious.
 
That's because we wiped out the
future.
Half the endowment in the
future disappeared,
so naturally people are willing
to pay more for the future
because they're poorer there.
 
In the first day of class we
said that the interest rate,
or the third week,
the interest rate according to
Fisher would go down if you got
poorer in the future.
So that's part of the reason
that's happened.
By the way, what is the
riskless rate of interest?
So P_1 P_2
P_3 equals what now?
It's equal to 1 quarter 1 fifth
1 fifth, so 1 quarter 1 fifth 1
fifth.
 
These are the prices,
1 quarter 1 fifth 1 fifth and
that's equal to 20,
10,14 over 20,
so that's 7 over 10,
so therefore the interest rate
1 r = 10 over 7 so r = 3
sevenths which is like 40
percent.
 
So the interest rate went from
300 percent to 40 percent,
but that's because we lost all
this future consumption.
But that's not what I'm
concentrating on.
Fisher would have already known
that.
What I'm concentrating on is
the fact that the prices are no
longer proportional to the
probabilities.
You're discounting every
probability, but adjusting the
probability because people are
much more worried about the
first state than the third
state.
Student: So people are
much more worried about A?
Prof: Not A,
they're more worried about the
first state.
 
The firms are A and B.
 
The states are 1,2 and 3,
so they're much more worried
about the first state where the
payoff is 200,
than they are about the third
state where the payoff,
total dividends in the economy
are 440.
Are you with me?
 
Student: Yeah.
 
Prof: Oh boy.
 
That sounded so unconvincing.
 
I want to say the punch line.
 
So I've got three more minutes
to go.
There are two punch lines.
 
I haven't gotten to the
stunning conclusion.
So far I've said stuff which
Arrow and Debreu had already
figured out,
but now I want to go to the
thing that Tobin and Markowitz
figured out,
which is one more step we
haven't noticed yet.
Arrow has already figured out
that because not everybody could
hedge that means that the price
of an Arrow security is not
exactly equal to the
probability.
It's relatively high if the
economy's poor like in state 1
and relatively lower if the
economy's rich like in state 3.
That's common sense.
 
Now, what's not common sense is
the extraordinary conclusion I'm
about to show you.
 
Let's look at what the
consumption is;
the final consumption of these
two people.
So if we look at the final
consumption of these two people,
what's her final consumption,
so X_A.
In the three states it's 200 -
100 times 1 which is 100.
What is it in the second state?
 
It's 200 - 100 times--what was
P_s over--times 4
fifths which is,
help, 160, maybe.
And the last step was 200 - 100
times 2 fifths.
2 fifths is 20 so this is 180.
 
That's hers.
 
And his consumption in the
future--
I'll put a tilde,
I haven't talked about
X_0 yet--
is 400 - 300 times 1 which
equals 100,
and here's it's 400 - 300 times
4 fifths which is equal to,
help!
4 fifths of 300 is 160,
and here it's 400 - 300 times 2
fifths which is equal to 280.
 
Is this right?
 
100,160, who told me it was 160?
 
Yes and what's that?
 
2 fifths is 120.
 
This is 280.
 
Student: The number
>
, like 200 - 100 times 4 fifths
is like 120.
Prof: What?
 
Student: The first
>
Prof: Which mistake is
there here?
Student: No,
the second
>
 
Student: The second 120
>
Prof: Is this the wrong
one?
Student: The wrong one.
 
Student: 160 should be
120.
Prof: Here.
 
200 - 80 is 120.
 
Thank you.
 
So these are all right now?
 
Student: 180 should be
>
Prof: 180 should be,
okay, this is 40 so this should
be 160.
 
Thank you.
 
That's it, great.
 
So now what's so shocking about
those numbers?
That I finally got them right?
 
Thank you.
 
What's shocking is this
consumption is just the sum of
the aggregate endowment--what's
the aggregate endowment?
Remember the aggregate
endowment is just 200,280 and
440.
 
So let's say you take 1 quarter
of this.
Let's take 1 quarter of that.
 
That's 50,70--that's 50,70 and
110.
So 1 quarter of this plus if
you add to that 150 you're going
to get all these numbers.
 
So this person,
alpha, A, I claim,
just holds 50 of the riskless
bond, pays 50,50 plus 50,70 and
110.
 
No.
 
Is this the right--let's just
check the numbers.
Sorry, only one more second.
 
I should have--so 100,120 and
160, that's the right number and
that's equal to 50 of the bond
plus 1 quarter of this thing.
So 50 50 is--1 quarter of this
is 50,70 and 110,
right?
 
So if you hold 50 of the bond
plus 1 quarter of this you get
100.
 
50 of the bond plus 70 is 120.
 
50 of the bond plus 110 is 160.
 
And this guy is going to hold 3
quarters of the aggregate
endowment plus minus 50 of the
bond,
so 3 quarters of the aggregate
endowment,
3 quarters of this thing,
3 quarters of the aggregate
endowment is 150 - 50 is 100.
 
3 quarters of this is 210 - 50
is 160.
3 quarters of that,
is 330 - 50 is 280.
So what they've done in
equilibrium is everybody,
despite having a million stocks
to choose from and thousands of
states and all that stuff,
what everybody does is hold the
riskless bond,
puts money in the bank and
holds the whole stock market.
 
So the first theorem we're
going to prove next time is
called The Mutual Fund Theorem
which is that everybody
diversifies by holding the
aggregate economy,
all stocks in the same
proportion, plus money in bank.
So that theorem of Shakespeare
of diversifying,
what did it amount to do?
 
We have a very concrete thing.
 
You hold 10 percent.
 
This person's holding 25
percent of every stock in the
whole economy plus putting some
money, 50 dollars in the bank.
The other person is doing 3
quarters of every stock in the
whole economy plus lending the
money to the first person.
So that's the first of the two
amazing results and I'll start
next time by explaining it.
 
 
 
