WEBVTT

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Prof: Okay,
I think I should begin.

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So we're at the point in the
course now where we're talking

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about the Capital Asset Pricing
Model and the economists'

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understanding of risk aversion
and its consequence for the

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financial sector.

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So much of what economists know
was already known to

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businessmen,
because it's common sense,

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and even to literary writers
like Shakespeare,

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who was himself quite a
successful businessman.

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So you might remember that
Shakespeare already had the idea

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of diversification.

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In the Merchant of Venice,
Antonio says "Each boat on

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a different ocean."

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He said it a bit more
poetically than that,

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"each sail on a different
argosy" or something.

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I've forgotten what it was,
but each boat on a different

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ocean.

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So he wasn't very worried
because he was diversified.

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He also had the idea of risk
and return.

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Nothing ventured,
nothing gained.

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Nothing ventured,
so ventured means risk.

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Nothing gained.

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So another way of saying this
diversification is "Don't

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put all your eggs in one
basket."

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So what have economists done
that wasn't already known to

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every business person and every
clever literary writer?

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Well, economists have
quantified this and turned this

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into a usable,
practical piece of advice.

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So the diversification theorem
in CAPM-- so I should have moved

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CAPM.

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I should have called this
Shakespeare and this CAPM.

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CAPM becomes the mutual fund
theorem.

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So the mutual fund theorem has
very practical advice.

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It says if you're investing in
the stock market,

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don't try to pick out
individual stocks.

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Hold every stock in proportion
to its value in the whole

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economy.

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If you want to be more
venturesome, don't pick riskier

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stocks.

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Just simply leave less money in
the bank.

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So divide all your money
between the index and the bank.

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It says, hold all your money in
stocks in the same proportion

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everyone else is holding them.

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Hold the market, in other words.

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Hold an index like the S&P
500, of all the stocks,

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in proportion to how big they
are.

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And if you want to get more
venturesome--if you're cautious,

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put some of your money in the
index and some of it in the

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bank.

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If you want to be more
venturesome, take some money out

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of the bank and put it into the
stocks.

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But in the index.

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If you want to be even more
venturesome than that,

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borrow the money to put the
money into the stock market.

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That's called leverage.

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But you should,
according to this theory,

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not try to pick stocks.

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So if you're interested in
risk, you shouldn't pick high

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technology startup companies.

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You should pick the same
mixture of blue chip companies

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like General Electric and these
startup companies that everybody

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else is choosing.

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So that's what it means to
diversify.

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Hold a little bit of everything
in the same proportion as

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everyone else.

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So it's much more precise,
much more surprising than

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Shakespeare's advice.

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Then the second thing,
risk and reward,

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it says that the price of a
security,

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so we'll call it pi_J
say,

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which we used to think of as
the expectation--

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of course, you need the
probabilities--

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of the payoff of the security
J, discounted.

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That's what we would have
thought the price was.

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So this is called the
covariance pricing theorem.

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The shocking thing is,
it's not the expectation

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discounted, which is what we
thought it was before.

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We know that you have to adjust
for risk somehow,

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which we haven't taken into
account before.

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But what's so shocking is,
you might have thought you'd

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subtract maybe something times
the variance of the return of

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asset J,
but that's not going to be the

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case.

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It's going to be,
surprisingly,

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subtracting something times the
covariance, with the market,

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of asset J.

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So you penalize--the price of a
stock should go lower if it gets

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riskier.

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Its risk is not defined by what
its variance is,

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but by what its covariance is
with the market.

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So I want to develop these two
ideas in this lecture,

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spend the whole lecture on just
developing these two ideas,

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which are the modern
mathematical version of what

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Shakespeare already knew 400
years ago,

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but made much more concrete,
much more precise,

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with the mathematics wholly
behind them.

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So what have we got here?

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We imagine a world,
the situation I'm going to

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describe, is the one we had last
time, where we know that there

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are different things that can
happen.

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These are the states with
different probabilities,

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so there's states.

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s = 1 to big S and each of them
have probabilities,

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with probabilities
gamma_s.

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Of course, the sum of the
gamma_s = 1.

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So everybody knows the
probabilities of the states and

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then they know what the payoff
is of the different assets.

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So here's A^(1), here's A^(2),
here's A^(J) say,

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and they know what the payoffs
are going to be,

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A^(1) in state 1,
A^(1) in state 2,

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A^(1) in state S,
A^(2) in state 1,

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A^(2) in state S,
A^(J) in state 1,

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A^(J) in state 2,
A^(J) in state S.

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So given these probabilities
and given the payoffs,

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you can compute the expectation
of each thing.

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You can compute the expectation
of A^(J), say.

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It's just the summation from s
= 1 to S of gamma_s

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A^(J)_s.

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You can also compute the
variance, so I'll write that

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sometimes as A bar^(J).

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You can compute the variance of
A^(J),

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which is the summation,
s = 1 to S,

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remember, of gamma_s
A^(J)_s - A bar^(j)

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squared.

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Okay, and you could also
compute the covariance of any 2,

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A^(J) and A^(K) as summation
gamma_s

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A^(J)_s - A bar^(J)
times (this is s = 1 to S) of

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A^(K)_s - A bar^(K) K.

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So those three numbers,
the expectation,

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the variance and the covariance
between any pairs,

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those are numbers that anybody
could compute,

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knowing the payoffs of the
assets and knowing the

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probabilities.

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And what the theory is going to
say,

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the capital asset pricing
model, that model is going to

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explain how the prices of all
the assets depend on their

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expectation,
their variances and their

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covariances,
under the assumption that

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people have quadratic utilities,
or more generally,

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assume people only care about
the mean--

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that's the expectation--and
variance of final consumption.

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So we wrote that last time.

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We said if you had quadratic
utility for consumption in every

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state,
you could summarize these

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thousands--
there could be millions of

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states and thousands of
securities.

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You add them all together,
what you hold,

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it only depends on the end
about what the expectation and

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the variance of your final
portfolio is.

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You like expectation and you
hate variance.

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So under that hypothesis,
we're going to have a very

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concrete interpretation of what
Shakespeare was talking about

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from the beginning,
which is going to allow us to

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quantify what you should do,
how people in the market are

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doing.

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We'll be able to answer a
series of questions.

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And then we'll be able to test
the theory too.

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So now the theory I'm about to
explain was developed in 1950.

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It began in 1950 when Koopmans
was a professor at the Cowles

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Foundation, the director of the
Cowles Foundation,

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was in Chicago.

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I told you, the Cowles
Foundation was started by Alfred

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Cowles,
who'd been a Yale

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undergraduate,
and his family owned the

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Chicago Tribune and a
bunch of other newspapers.

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And he ran a macroeconomic
forecasting company in 1929,

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right after the crash.

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He was embarrassed to realize
that he had predicted that

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everything was going to be fine,
and he sent around

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questionnaires and read
everybody else's,

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his competitors' newspapers,
and realized they'd been

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predicting everything was going
to be fine too,

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except for one guy,
Jones, the Dow Jones guy.

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Except for that,
everybody had been making the

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same rosy predictions.

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So he decided the whole subject
of economic forecasting,

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in fact, economics altogether
was a fraud,

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and he wanted to put money in
encouraging the use of

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mathematics in economics,
instead of the touchy-feely

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more qualitative analysis that
was the norm at the time.

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So he went to see Irving
Fisher, who was the great Yale

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economist, and said,
"What should I do with all

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the money?

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Where should I put this
institute, and who should I

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invite to the institute?"

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So Fisher and he set up the
whole thing and it started in

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Colorado Springs,
because Fisher believed in

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fresh air,
and he wrote a lot of books

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about fresh air and how it
staved off tuberculosis and

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things like that.

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And no one wanted to go to
Colorado Springs,

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so they moved it to Chicago
where the family was originally

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from, the Cowles family.

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The director then in the '50s
was someone named Koopmans,

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and Koopmans had this idea of
trying to examine risk by making

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people with quadratic utilities.

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And he said,
"Okay, student Markowitz,

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why don't you try to develop a
theory?"

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And Markowitz began the theory
of the capital asset pricing

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model.

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Now the theory,
just to get a little ahead of

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the story,
Markowitz went quite a ways,

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won a Nobel prize many years
later for what he did,

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but didn't quite get to the end.

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And while he was working on it,
Koopmans didn't want to be

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director any more of Cowles,
and he invited Tobin,

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the great Yale economist,
to come and be director of

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Cowles.

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And Tobin got there and the
people at the University of

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Chicago said to Tobin,
"Well, we're glad you're

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coming.

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You'll be right next to us,
but don't try and get us mixed

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up with all those mathematical
people at the Cowles Foundation.

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They're all crazy.

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We would like to keep that high
tech mathematics out of our

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department.

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That's just Cowles' crazy
idea."

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So Tobin refused the job,
came back to Yale,

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and Markowitz was so angry--not
Markowitz.

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Koopmans, his advisor,
was so angry that he went to

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Cowles and said,
"They're not treating us

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right at the University of
Chicago.

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Let's move the entire Cowles
Foundation to Yale."

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So in 1955, they moved it to
Yale.

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But in these conversations
Tobin had with Koopmans and the

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Cowles Foundation,
he got to know what Markowitz

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was doing,
and so he added the final

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crowning achievement to--
the mutual fund theorem--to the

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Markowitz setup,
and he subsequently won the

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Nobel Prize for that.

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In fact, the New York
Times said in its

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description of what he won the
Nobel Prize for,

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the New York Times
said, "James Tobin won the

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Nobel prize for showing that you
shouldn't put all your eggs in

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one basket."

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So anyway, that's the theory
that I have to explain now,

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that it's a little bit less
trivial than just saying,

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"Don't put all your eggs
in one basket.

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So, the idea is that you like
expectation, you hate variance.

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So the first thing we're going
to do is notice that

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covariance--I'm sorry,
standard deviation is defined

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as the square root of variance.

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So to say that you don't like
variance means you don't like

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standard deviation.

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They're the same thing,
but you're going to see,

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it makes the diagrams much
prettier.

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So if we had a portfolio that
you could choose among some

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combination of putting your
money in the A's,

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you'd get some expectation of
some standard deviation for your

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total portfolio.

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And people like more
expectation and like less

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standard deviation.

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So the indifference curves are
going in that direction,

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unlike the usual picture.

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That was the first step.

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Now the first thing we noticed
last time, this was three weeks

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ago or so, we made an extremely
important observation.

13:37.658 --> 13:44.368
We said, what is the covariance
of part of your money in X,

13:44.370 --> 13:48.910
the variance of part of your
money in X and the rest of your

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money--
let's say you have 1 dollar--so

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t in X and (1 - t) in Y?

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For example,
1 half in X and 1 half in Y.

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Well that is by definition the
same as the covariance of this

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portfolio with itself,
and now we know this from this

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theorem,
from this definition of

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covariance,
that if you look at the

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covariance of J with K,
so fixing K--this is just

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linear.

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This is a constant,
multiplying the differences

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from J and its expectation.

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So this is a linear in each
variable,

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so therefore this is going to
equal the covariance of tX with

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tX twice the covariance of t--
so the covariance of tX and (1

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- t)Y,
the covariance of (1 - t)Y and

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tX (1 - t) squared (so I'll put
that one over here) the

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covariance of (1 - tY) with
itself.

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So I've taken this with this,
then this with that,

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then that with that and that
with the thing on the end.

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But it's linear again,
so I can pull out all these

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things.

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So that's the formula.

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Now what we noticed is that if
2 stocks, X and Y,

15:25.410 --> 15:29.410
are independent,
their covariance is 0.

15:29.408 --> 15:31.738
So I'm assuming you remember
that, so those terms just go

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away.

15:32.028 --> 15:36.478
And then it's also linear,
so this is just going to equal

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t squared, covariance of X with
X, which is just the variance of

15:41.491 --> 15:41.971
X.

15:41.970 --> 15:50.760
So assume XY independent,
then that is what allows me to

15:50.763 --> 15:56.363
cancel these two terms like
that.

15:56.360 --> 16:03.200
So I get t squared times the
covariance of X (1 - t) squared

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times the variance of Y.

16:06.210 --> 16:15.400
If I take t = 1 half,
I know that the variance of 1

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half X 1 half Y = 1 quarter
times the variance of X 1

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quarter times the variance of Y.

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We derived this three weeks ago.

16:33.200 --> 16:34.460
I'm just repeating it.

16:34.460 --> 16:37.320
So the amazing thing is,
if X and Y have the same

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expectation and the same
variance,

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but are independent of each
other, by splitting your money

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between two independent things,
you're obviously going to have

16:45.359 --> 16:50.899
the same expectation,
but only half the variance.

16:50.899 --> 16:54.739
And that last step is,
the variance of X = variance of

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Y.

16:55.029 --> 16:57.179
So amazingly--or it's not so
amazing, I guess,

16:57.183 --> 17:00.203
if you've thought about it--but
if you haven't thought about it,

17:00.201 --> 17:01.591
you've got 2 equal risks.

17:01.590 --> 17:04.280
Neither one is better than the
other, so why bother with one

17:04.276 --> 17:05.366
rather than the other?

17:05.369 --> 17:06.219
They're both the same.

17:06.220 --> 17:08.670
But if they're independent,
you should put half your money

17:08.673 --> 17:09.453
in each of them.

17:09.450 --> 17:13.040
You get the same expectation,
but you're going to get only

17:13.039 --> 17:14.299
half the variance.

17:14.298 --> 17:18.848
So if you have 2 dice,
let's say the first die will

17:18.851 --> 17:24.311
give you--you pay 100 dollars
for a dice, you can get 100 the

17:24.311 --> 17:26.771
number of dots showing.

17:26.769 --> 17:29.259
So if it gets a 1 that's 101
dollars, you get a 6,

17:29.256 --> 17:30.216
it's 106 dollars.

17:30.220 --> 17:35.130
So if you held 1 dice,
1 die, you'd get on average 103

17:35.125 --> 17:37.065
and 1 half dollars.

17:37.068 --> 17:39.898
You could pay 100,
get 1 die, which on average

17:39.904 --> 17:41.294
paid 103 and 1 half.

17:41.288 --> 17:44.638
Instead, you could put 50
dollars in the first die,

17:44.635 --> 17:47.705
get half of that,
50 dollars in the second die,

17:47.713 --> 17:49.123
get half of that.

17:49.118 --> 17:51.908
So on average,
you're still getting 103 and 1

17:51.911 --> 17:55.721
half, but the variance you get
is going to be half of what it

17:55.718 --> 17:56.668
was before.

17:56.670 --> 17:59.650
So in this picture,
you start off with X and Y with

17:59.651 --> 18:02.991
the same standard deviation and
the same expectation,

18:02.990 --> 18:06.320
but they're independent and now
you put half your money in each

18:06.317 --> 18:08.247
and you cut the variance in
half,

18:08.250 --> 18:11.260
which means you divide--the
standard deviation is the square

18:11.260 --> 18:13.590
root--
so you divide it by the square

18:13.586 --> 18:14.286
root of 2.

18:14.289 --> 18:15.639
So you've improved.

18:15.640 --> 18:17.960
You've moved to the left.

18:17.960 --> 18:21.880
You've gotten less standard
deviation without constricting

18:21.881 --> 18:23.671
your expectation at all.

18:23.670 --> 18:26.750
So that's the most important
thing to notice.

18:26.750 --> 18:32.590
I'm going to say one more
thing, one more way.

18:32.588 --> 18:39.248
I'm going to go back to this
formula, which is where the

18:39.247 --> 18:41.787
variance was,
here.

18:41.788 --> 18:54.338
Now suppose--let's look at that
formula.

18:54.338 --> 18:59.148
Suppose I take 2 stocks,
which I'm going to switch now,

18:59.148 --> 19:00.038
X and Y.

19:00.039 --> 19:01.179
That was most unfortunate.

19:01.180 --> 19:05.240
But suppose I take 2 stocks and
I'm at the point where all my

19:05.244 --> 19:06.334
money is in Y.

19:06.328 --> 19:10.898
So I'm going to take the case t
= 0.

19:10.900 --> 19:16.170
So if I took t = 0,
I would just have the variance

19:16.166 --> 19:16.916
of Y.

19:16.920 --> 19:19.510
That's actually quite good,
the variance of Y.

19:19.509 --> 19:21.089
So I'd have all my money in Y.

19:21.088 --> 19:24.038
I'd have this standard
deviation and this expectation.

19:24.038 --> 19:31.788
Now what happens if the two are
independent, X and Y?

19:31.788 --> 19:35.208
Let's drop the fact that they
have the same variance now.

19:35.210 --> 19:39.160
Maybe X has way more variance
than Y does.

19:39.160 --> 19:42.260
So look, X is much riskier than
Y.

19:42.259 --> 19:46.349
Maybe X has a higher
expectation than Y,

19:46.345 --> 19:48.435
but it's so risky.

19:48.440 --> 19:51.290
People who are risk averse
wouldn't ever want to subject

19:51.292 --> 19:53.112
themselves to all this risk,
say.

19:53.108 --> 19:56.068
So maybe they should just stick
with Y, you would think.

19:56.068 --> 19:58.438
But no, the answer is,
you shouldn't,

19:58.440 --> 20:02.700
because what happens if you
start to mix some of your money

20:02.701 --> 20:05.691
in Y--
take a little out of Y and put

20:05.690 --> 20:07.370
a little bit into X?

20:07.368 --> 20:10.868
Your expectation,
if you take a little money out

20:10.867 --> 20:14.717
of Y and put a little into X,
your expectation's going to go

20:14.723 --> 20:17.333
up, because it's going to be the
average of X and Y.

20:17.329 --> 20:18.729
So the expectation's going up.

20:18.730 --> 20:20.900
But then you say,
"Oh, but my variance is

20:20.900 --> 20:21.720
going up."

20:21.720 --> 20:23.690
So I wouldn't want to do it,
but no, that's wrong.

20:23.690 --> 20:26.670
The variance isn't going up,
because they're independent.

20:26.670 --> 20:29.810
Even though X has a way higher
variance than Y,

20:29.807 --> 20:33.557
mixing a little between the two
lowers your variance.

20:33.559 --> 20:34.879
So how could that be?

20:34.880 --> 20:37.560
Seems quite shocking.

20:37.558 --> 20:43.368
Well, let's take the derivative
of this with respect to having

20:43.374 --> 20:46.524
all your money originally in Y.

20:46.519 --> 20:49.999
So this is the variance of
putting t dollars in X and 1 - t

20:50.000 --> 20:50.420
in Y.

20:50.420 --> 20:53.150
So we simply take the
derivative of that,

20:53.154 --> 20:55.484
which I've done here,
I suppose.

20:55.480 --> 20:57.390
Why don't I just write it out?

20:57.390 --> 21:07.560
Take the derivative of that,
d/dt of this = 2t times the

21:07.558 --> 21:15.878
variance of X 2 times (1 - t)
times (okay,

21:15.880 --> 21:18.750
but this is a -,
so I take the derivative) it's

21:18.748 --> 21:26.308
2 times this times - 1,
so - 2T, 2 times (1 - t),

21:26.307 --> 21:29.867
variance of Y.

21:29.869 --> 21:34.659
Let's evaluate that at t = 0.

21:34.660 --> 21:37.390
The first term is 0,
the t is 0, so this doesn't

21:37.386 --> 21:37.906
matter.

21:37.910 --> 21:40.980
The second term,
t is 0, so this is - 2 times

21:40.976 --> 21:42.366
the variance of Y.

21:42.369 --> 21:44.509
It's going down.

21:44.509 --> 21:48.259
The variance is actually going
down here.

21:48.259 --> 21:50.809
So just as this says,
the variance starts down.

21:50.808 --> 21:53.568
Eventually, it has to end up
here, so as you move your money,

21:53.568 --> 21:56.928
all of it originally in Y and
start moving some of it into X,

21:56.930 --> 22:01.770
your variance is going to go
down.

22:01.769 --> 22:06.379
So those two things put
together are the argument for

22:06.375 --> 22:07.965
diversification.

22:07.970 --> 22:11.350
If they're independent and
identical, then you should put

22:11.349 --> 22:12.919
half your money in each.

22:12.920 --> 22:15.600
But even if they're not
identical, you shouldn't stick

22:15.596 --> 22:15.946
to Y.

22:15.950 --> 22:17.750
You should mix Y with some X.

22:17.750 --> 22:21.650
So there's why diversification
is such a good idea.

22:21.650 --> 22:23.350
So it's not all diversification.

22:23.348 --> 22:27.898
It's diversification of
independent risks.

22:27.900 --> 22:30.680
There's a bit of a mathematical
nuance.

22:30.680 --> 22:34.100
Diversification of independent
risks is bound to help you.

22:34.098 --> 22:37.938
Everybody should diversify a
little bit, because you can

22:37.942 --> 22:39.272
reduce your risk.

22:39.269 --> 22:44.869
That's the first important
thought.

22:44.868 --> 22:50.628
Now I don't want to do too much
in the mathematics.

22:50.630 --> 22:54.450
So I prove in the notes,
but I won't do it here,

22:54.445 --> 22:58.665
suppose we didn't assume X and
Y were independent.

22:58.670 --> 23:02.170
Suppose X and Y were
arbitrarily chosen.

23:02.170 --> 23:06.490
They have big covariance,
whatever you want.

23:06.490 --> 23:10.650
Then what is going to happen
when you combine your money

23:10.647 --> 23:12.007
between X and Y.

23:12.009 --> 23:14.479
So there's a proof,
which I'm not going to give,

23:14.482 --> 23:17.432
because I'm a little bit
behind, so the proof is this.

23:17.430 --> 23:20.340
Suppose we have exactly the
same situation,

23:20.337 --> 23:23.177
where you started with Y and
you've got X,

23:23.176 --> 23:27.536
and you could think of moving
you money in between X and Y.

23:27.538 --> 23:30.788
Well, it turns out,
if they're perfectly

23:30.785 --> 23:33.495
correlated,
so really they're the same

23:33.497 --> 23:35.577
thing, Y is just 80 percent of
X,

23:35.578 --> 23:38.668
it's the same stock,
so moving back between them is

23:38.669 --> 23:44.769
not really changing--
X is just a scalar improvement

23:44.768 --> 23:47.868
on Y,
so it has more expectation and

23:47.866 --> 23:49.536
more standard deviation.

23:49.538 --> 23:52.428
If that were the case,
you'd just move on that

23:52.433 --> 23:55.333
straight line that connects the
two points.

23:55.329 --> 23:57.849
I must have a way of doing this.

23:57.848 --> 24:01.328
Yes, if they were perfectly
correlated, you would just be

24:01.332 --> 24:04.942
moving along this line by
putting your money in between.

24:04.940 --> 24:08.410
If they're independent,
you're going to move along that

24:08.410 --> 24:08.860
line.

24:08.858 --> 24:10.928
But no matter how correlated
they are,

24:10.930 --> 24:12.740
as long as they're not
perfectly correlated,

24:12.740 --> 24:15.260
it's going to turn out the
curve you get by moving between

24:15.263 --> 24:16.773
them is going to look like that.

24:16.769 --> 24:19.229
If they're negatively
correlated, then you can really

24:19.230 --> 24:21.880
improve yourself and get a curve
that looks like that.

24:21.880 --> 24:25.230
Independent will be like this,
positively correlated,

24:25.230 --> 24:27.150
but not perfectly will be like
that,

24:27.150 --> 24:29.110
and perfectly positively
correlated,

24:29.108 --> 24:31.788
no diversification gains,
you get the straight line,

24:31.788 --> 24:34.638
but you're always above the
straight line.

24:34.640 --> 24:38.390
So that's a kind of--anyway,
it takes a little bit of

24:38.394 --> 24:42.014
algebra to prove that,
you know, the Cauchy-Schwarz

24:42.005 --> 24:45.105
inequality, but I'm going to
skip that.

24:45.108 --> 24:46.928
But anyway, that's a
mathematical fact.

24:46.930 --> 24:51.070
You see in the independent
case, we proved that the line

24:51.066 --> 24:55.426
has to start going that way and
eventually it has to end up

24:55.426 --> 24:57.436
here,
so it should be no surprise

24:57.436 --> 24:59.716
that it looks like that in the
independent case.

24:59.720 --> 25:01.930
And then it turns out that
every other case is sort of in

25:01.930 --> 25:03.470
between that or at the other
extreme.

25:03.470 --> 25:08.460
It always has this bowed out
shape.

25:08.460 --> 25:11.530
So there's always gains to
diversification,

25:11.531 --> 25:15.191
something you're getting out of
diversification.

25:15.190 --> 25:19.700
The risk is somehow going down
by mixing things up.

25:19.700 --> 25:23.000
That's it, that's the
mathematics and that's the

25:23.003 --> 25:24.413
general principle.

25:24.410 --> 25:26.920
Now, what can you get out of
this?

25:26.920 --> 25:29.350
I want to derive these two
famous theorems,

25:29.352 --> 25:32.362
which we already saw in action
in the example that we

25:32.363 --> 25:34.973
calculated, but it seemed like a
miracle.

25:34.970 --> 25:37.990
So what did we find in the
example we calculated last time

25:37.987 --> 25:38.567
in class?

25:38.568 --> 25:41.828
We found that--we only had two
stocks,

25:41.828 --> 25:45.378
but if there had been hundreds
of stocks and millions of states

25:45.384 --> 25:47.794
of nature,
and each person had a different

25:47.785 --> 25:49.565
utility,
different risk aversion,

25:49.568 --> 25:52.088
but all quadratic,
and you calculated the

25:52.089 --> 25:55.369
competitive equilibrium,
assuming there were Arrow

25:55.365 --> 25:58.375
securities,
it would end up that every

25:58.377 --> 26:02.557
consumer held the same mix of
all these stocks,

26:02.558 --> 26:06.298
positive or negative amounts of
the bond.

26:06.298 --> 26:08.978
So the mutual fund theorem
turned out to be true,

26:08.980 --> 26:11.770
that the best thing everybody
could do was hold the same

26:11.771 --> 26:13.941
amount,
the same proportion of stocks

26:13.942 --> 26:16.322
and bonds,
and maybe more or less of the

26:16.318 --> 26:18.918
riskless asset,
maybe even negative of the

26:18.924 --> 26:19.924
riskless asset.

26:19.920 --> 26:23.460
So I want to show you why
that's true now.

26:23.460 --> 26:25.840
Then we're going to have to
talk about whether we really

26:25.836 --> 26:26.696
believe it or not.

26:26.700 --> 26:30.510
So let's see what an
implication is.

26:30.509 --> 26:36.659
Suppose now what I do is I
write down--I have all these

26:36.664 --> 26:39.064
stocks, X, Y and Z.

26:39.058 --> 26:43.218
So you can imagine--you have a
certain amount of money--so you

26:43.220 --> 26:46.770
can imagine putting your
money--so let's suppose that

26:46.766 --> 26:49.286
I've written down all the
stocks.

26:49.289 --> 26:51.219
The prices are given.

26:51.220 --> 27:00.670
Let's say prices of stocks are
in equilibrium.

27:00.670 --> 27:03.980
So I've solved for the
equilibrium like we did last

27:03.978 --> 27:06.558
time, and I write down X,
Q, Y and Z.

27:06.558 --> 27:11.098
These are the expectation and
standard deviation per dollar of

27:11.103 --> 27:14.383
the stocks, or per I dollars of
the stock.

27:14.380 --> 27:17.830
Maybe the person has exactly I
dollars, so let's say it's I

27:17.832 --> 27:20.512
dollars of each stock,
what their expectation,

27:20.509 --> 27:23.069
return and standard deviation
would be.

27:23.068 --> 27:25.568
If 1 dollar,
you get some expectation,

27:25.568 --> 27:28.578
standard deviation,
10 dollars of the same stock,

27:28.578 --> 27:32.038
you'd be able to buy 10 shares,
which means you would--

27:32.038 --> 27:36.188
10 times as many shares,
which means you'd have 10 times

27:36.186 --> 27:40.256
the expectation and 10 times the
standard deviation.

27:40.259 --> 27:44.889
So whatever the guy's income
is, they'll fix the returns as

27:44.894 --> 27:48.574
per his income,
because he's going to spend all

27:48.571 --> 27:51.051
his income on these stocks.

27:51.049 --> 27:52.619
So what could he do?

27:52.618 --> 27:55.758
He could put all his money into
X, he'd end up here,

27:55.755 --> 27:58.275
or all his money into Y,
he'd end up here,

27:58.276 --> 28:01.346
or all his money into Z,
and he'd end up here.

28:01.348 --> 28:05.078
But maybe he wants to divide
part of his money in Y,

28:05.077 --> 28:06.977
part in Z and part in Y.

28:06.980 --> 28:08.820
He'll be somewhere like here.

28:08.818 --> 28:12.408
Or maybe he wants to instead
put part of his money in Z and

28:12.413 --> 28:13.533
part of it in X.

28:13.529 --> 28:15.189
Well, he could end up here.

28:15.190 --> 28:17.030
But why should he stop there?

28:17.028 --> 28:20.238
Maybe he wants to take some
combination of the money that

28:20.239 --> 28:23.679
was in Z and X and combine it
with some combination of things

28:23.676 --> 28:26.236
that was in Z and Y,
combine this and this.

28:26.240 --> 28:28.110
That would get him out to here.

28:28.108 --> 28:31.298
So there are lots of different
combinations he could hold.

28:31.298 --> 28:33.968
So graphically,
it looks like that.

28:33.970 --> 28:40.840
Mathematically,
what's he trying to do?

28:40.839 --> 28:43.769
He's trying to--so R now.

28:43.769 --> 28:46.719
I've replaced A with R.

28:46.720 --> 28:56.570
So R^(j) is just going to =
A^(j) divided by the price of j,

28:56.568 --> 28:59.238
maybe times I.

28:59.240 --> 29:04.790
So R^(j) will be the--let's
just say it's A^(j) divided by

29:04.785 --> 29:06.435
pi_j.

29:06.440 --> 29:09.790
So R^(j) is the return per
dollar you put in the asset.

29:09.788 --> 29:12.938
R^(j) is the payoff of the
asset per dollar.

29:12.940 --> 29:13.790
Let's stick with that.

29:13.788 --> 29:17.038
Let's say the guy has a dollar
to spend, so it's the return per

29:17.038 --> 29:17.508
dollar.

29:17.509 --> 29:20.029
So those are all his
possibilities.

29:20.029 --> 29:21.269
Now what does he want to do?

29:21.269 --> 29:25.139
He wants to choose his money,
how much money he's going to

29:25.142 --> 29:28.132
put in each asset,
which is omega_1

29:28.131 --> 29:30.171
through omega_J.

29:30.170 --> 29:32.360
That's the money he's going to
put in each asset.

29:32.358 --> 29:35.918
And if he does that,
he's going to have a portfolio,

29:35.920 --> 29:38.800
which I'll write as
R_omega,

29:38.798 --> 29:40.588
so it's going to have a certain
expectation,

29:40.589 --> 29:42.099
a certain variance.

29:42.098 --> 29:44.878
So the question is,
how should he divide his money

29:44.880 --> 29:47.720
between the assets,
and here's the kind of stuff he

29:47.718 --> 29:49.078
could imagine doing.

29:49.078 --> 29:52.078
Looks very complicated,
but it's going to turn out to

29:52.078 --> 29:53.058
be very simple.

29:53.058 --> 30:00.538
So are there any questions
about what we're doing so far?

30:00.538 --> 30:05.888
I want to get a simple rule to
tell you, practical advice about

30:05.887 --> 30:07.007
what to do.

30:07.009 --> 30:08.579
Markowitz posed this problem.

30:08.578 --> 30:14.968
This is the Markowitz problem,
okay?

30:14.970 --> 30:16.900
It turns out,
by this argument,

30:16.896 --> 30:20.366
you see if I keep combining
things over and over again,

30:20.365 --> 30:22.545
I'm getting this sort of blob.

30:22.548 --> 30:25.568
But the blob is always pushing
itself further and further out.

30:25.568 --> 30:27.898
It's always convex,
because given any two points,

30:27.903 --> 30:30.143
I can always do the thing to
the left of it.

30:30.140 --> 30:32.480
So if I could do this and I
could do this,

30:32.480 --> 30:34.280
and I look at the line
connecting them,

30:34.279 --> 30:37.699
I can always be above that line
by spreading my money between

30:37.696 --> 30:38.776
those two assets.

30:38.779 --> 30:41.269
Or given this combination of
assets that produces this

30:41.266 --> 30:43.046
expectation and standard
deviation,

30:43.048 --> 30:45.638
and given that one up there,
I could combine the two and do

30:45.641 --> 30:47.831
better than the straight line
connecting them,

30:47.829 --> 30:48.749
do to the left of it.

30:48.750 --> 30:51.640
So therefore,
all of the combinations put

30:51.644 --> 30:53.894
together have to,
geometrically,

30:53.887 --> 30:56.057
look like this in the end.

30:56.058 --> 30:58.688
So I'm going to do this
algebraically in a second.

30:58.690 --> 31:02.950
They have to look like that if
I did every possible variation.

31:02.950 --> 31:09.090
It couldn't end up with a
picture where after I do every

31:09.092 --> 31:12.662
possible variation--
if this is the expectation,

31:12.661 --> 31:15.541
this is the standard deviation,
I couldn't get something that

31:15.536 --> 31:17.906
looked like that,
because I would just combine

31:17.910 --> 31:19.290
this thing with this thing.

31:19.288 --> 31:22.188
Whatever portfolio produced
this, whatever portfolio

31:22.192 --> 31:24.782
produced that,
I'd put half my money into each

31:24.775 --> 31:26.855
of the stocks that this told me
to do,

31:26.858 --> 31:29.558
half the money into each of the
stocks this told me to do.

31:29.558 --> 31:32.168
I'd still end up investing 1
dollar and instead of getting

31:32.173 --> 31:34.563
the line, I'd get something that
looked like that.

31:34.558 --> 31:36.548
So I couldn't have the thing
going in like that.

31:36.549 --> 31:37.599
It has to go out.

31:37.598 --> 31:40.568
But then of course,
I can combine this and this and

31:40.573 --> 31:44.323
get that, and so I'm going to
end up with this kind of shape.

31:44.318 --> 31:47.098
Now I'm going to prove that
algebraically.

31:47.098 --> 31:49.538
But if I had that kind of
shape, then what would the guy

31:49.538 --> 31:49.758
do?

31:49.759 --> 31:52.879
He'd choose something like
that, where he's tangent.

31:52.880 --> 31:56.890
That's what Markowitz basically
did, except he did it

31:56.894 --> 31:58.134
algebraically.

31:58.130 --> 32:03.580
So that's what Markowitz did.

32:03.578 --> 32:05.648
That doesn't seem to have
gotten us very far.

32:05.650 --> 32:08.140
You notice that different
people are going to make

32:08.142 --> 32:09.162
different choices.

32:09.160 --> 32:12.920
If somebody is incredibly
worried--someone's not very

32:12.923 --> 32:17.053
worried about the standard
deviation and cares a lot about

32:17.048 --> 32:20.478
expectation,
they're going to have a flatter

32:20.480 --> 32:21.900
indifference curve.

32:21.900 --> 32:24.270
So instead of looking like
that, it will be a flatter

32:24.273 --> 32:24.643
thing.

32:24.640 --> 32:27.710
And so if it's flatter,
the guy who cares--

32:27.710 --> 32:29.900
sorry, if it's flatter,
it means he cares about

32:29.904 --> 32:31.914
expectation and not standard
deviation,

32:31.910 --> 32:35.720
because a little bit of
expectation can compensate him

32:35.720 --> 32:38.380
for a lot more standard
deviation,

32:38.380 --> 32:39.860
so his indifference curve is
flatter.

32:39.859 --> 32:41.689
He's going to choose further up.

32:41.690 --> 32:44.770
So he's going to choose a point
like way up there.

32:44.769 --> 32:47.599
So he's going to get a higher
expectation and higher standard

32:47.595 --> 32:48.155
deviation.

32:48.160 --> 32:51.460
Someone who's more risk averse
is going to choose where that

32:51.462 --> 32:52.082
point is.

32:52.078 --> 32:55.268
So that's qualitatively what's
going on.

32:55.269 --> 32:58.059
So Markowitz played around with
this, played around with this.

32:58.059 --> 33:00.429
Then Tobin appears on the scene.

33:00.430 --> 33:04.420
Tobin says, "All this gets
so much easier if you have a

33:04.415 --> 33:05.965
riskless asset."

33:05.970 --> 33:09.310
Let's suppose that there's a
bond.

33:09.308 --> 33:11.368
You haven't mentioned the bond
yet.

33:11.368 --> 33:14.978
Let's suppose we have a bond
that pays something for sure,

33:14.980 --> 33:17.960
and we're going to ignore
inflation, Tobin said,

33:17.959 --> 33:19.669
which is a big problem.

33:19.670 --> 33:22.180
We should come back to that if
we have time.

33:22.180 --> 33:25.000
We're going to ignore
inflation, so US Treasury's

33:25.000 --> 33:27.940
going to pay a certain amount of
money for sure.

33:27.940 --> 33:32.370
So it has an expectation like
this, but no standard deviation.

33:32.369 --> 33:34.759
It's 0.

33:34.759 --> 33:37.219
Now why is this so important?

33:37.220 --> 33:42.570
Let's imagine combining this
riskless bond with some other

33:42.570 --> 33:45.670
stock X, 1 dollar's worth of X.

33:45.670 --> 33:49.440
1 dollar's worth of the
riskless bond gets you 1.06 say.

33:49.440 --> 33:52.490
1 dollar's worth of X gets you
much--you know,

33:52.490 --> 33:56.290
maybe your average is 1.12,
but you're also running a big

33:56.288 --> 33:56.898
risk.

33:56.900 --> 34:02.070
What happens if you put your
money part way in between them?

34:02.068 --> 34:07.268
So suppose you put part of your
money in X, and the rest of your

34:07.273 --> 34:09.673
money in the riskless bond.

34:09.670 --> 34:12.230
What's your expectation going
to be?

34:12.230 --> 34:16.180
Your expectation,
let's say X is 12 percent and Y

34:16.181 --> 34:18.401
is 6 percent,
1.06 and 1.12,

34:18.403 --> 34:23.343
the average is going to be 1.09
if you put half your money in

34:23.344 --> 34:24.254
both.

34:24.250 --> 34:28.630
But what is the standard
deviation going to be?

34:28.630 --> 34:31.360
I claim it's just going to be
on this straight line.

34:31.360 --> 34:32.240
Why is that?

34:32.239 --> 34:35.489
Because the riskless thing has
covariance 0.

34:35.489 --> 34:39.639
If X is riskless,
its covariance is going to be 0

34:39.641 --> 34:40.421
with Y.

34:40.420 --> 34:43.510
So these terms are going to
disappear.

34:43.510 --> 34:44.960
Look at the covariance.

34:44.960 --> 34:50.600
If X--ifY is the riskless bond,
it pays always its expectation.

34:50.599 --> 34:52.469
So these numbers are always 0.

34:52.469 --> 34:56.409
Therefore the covariance of
that thing with anything,

34:56.414 --> 35:00.214
the covariance of Y with any X
is going to be 0.

35:00.210 --> 35:05.360
And not only that,
but the variance of Y itself is

35:05.362 --> 35:06.312
also 0.

35:06.309 --> 35:09.879
So all you get is,
for the variance of the t

35:09.882 --> 35:14.702
dollars in X and 1 - t dollars
in Y, you just get t squared

35:14.702 --> 35:17.032
times the variance of X.

35:17.030 --> 35:19.760
But then when you take the
standard deviation,

35:19.759 --> 35:22.669
it's just t times the standard
deviation of X.

35:22.670 --> 35:26.130
So in other words,
the standard deviation of the

35:26.125 --> 35:29.355
mixture of Y and X lies right on
the line.

35:29.360 --> 35:32.360
So unlike--to put it another
way, something that has no

35:32.362 --> 35:35.312
variance is perfectly correlated
with everything else,

35:35.307 --> 35:36.807
it makes no difference.

35:36.809 --> 35:40.999
When the other thing goes up or
down, it doesn't tell you--you

35:41.001 --> 35:43.341
still know what Y's going to be.

35:43.340 --> 35:46.190
So unlike everything else,
where you have this bow shaped

35:46.188 --> 35:48.928
thing, the riskless asset just
connects everything to a

35:48.934 --> 35:49.854
straight line.

35:49.849 --> 35:54.109
So if there were a riskless
asset and you could do this with

35:54.106 --> 35:57.496
some X and here was a Z,
you could also do that,

35:57.496 --> 35:59.296
or you could do that.

35:59.300 --> 36:01.120
Is this clear?

36:01.119 --> 36:03.849
Okay, so the punch line,
which we're now going to see

36:03.847 --> 36:06.487
algebraically,
the punch line is--and by the

36:06.490 --> 36:09.750
way, if I combine,
what does it mean to extend

36:09.753 --> 36:10.593
that line?

36:10.590 --> 36:11.800
I should have extended that
line.

36:11.800 --> 36:13.250
That was a huge mistake.

36:13.250 --> 36:17.840
Suppose instead of--so here's
all the money in the riskless

36:17.836 --> 36:18.466
asset.

36:18.469 --> 36:21.519
Here's half the money in the
riskless asset and half in X.

36:21.519 --> 36:23.079
Here's all my money in X.

36:23.079 --> 36:25.429
What would happen if I extended
the line here?

36:25.429 --> 36:27.669
What would that correspond to?

36:27.670 --> 36:29.850
I'm now at a point on the
extension of the line.

36:29.849 --> 36:31.689
How do I get that?

36:31.690 --> 36:34.410
Student:
You borrow money and invest it

36:34.407 --> 36:34.827
in X.

36:34.829 --> 36:35.449
Prof: Right.

36:35.449 --> 36:39.439
That's negative Y and bigger
than 1 in X.

36:39.440 --> 36:43.390
So negative 1 half of X 1 and 1
half of Y, that's still 1

36:43.393 --> 36:44.033
dollar.

36:44.030 --> 36:47.940
I sold X short and so I'm just
going to continue the line that

36:47.943 --> 36:48.333
way.

36:48.329 --> 36:49.979
So I shouldn't have stopped
here.

36:49.980 --> 36:52.050
I could have kept going.

36:52.050 --> 36:55.180
So it's putting money in the
bank and sharing it between a

36:55.175 --> 36:57.145
single stock puts me on this
line.

36:57.150 --> 37:00.090
Borrowing money from the bank
to put in a stock is called

37:00.088 --> 37:02.818
leveraging my return,
puts me on the extension of the

37:02.817 --> 37:03.287
line.

37:03.289 --> 37:04.929
We're going to come back to
that in a second.

37:04.929 --> 37:09.429
So knowing that,
now what should I do?

37:09.429 --> 37:12.909
Is there something really
simple?

37:12.909 --> 37:18.279
Well, what would you do here,
faced with this choice?

37:18.280 --> 37:25.980
Which combination of stocks and
bonds would you hold?

37:25.980 --> 37:28.550
I say it's really simple.

37:28.550 --> 37:33.210
You just look for a line
through the riskless point

37:33.208 --> 37:38.278
that's tangent to this blurb,
which happens to be here,

37:38.279 --> 37:43.629
and now you should choose your
point anywhere along this line,

37:43.630 --> 37:47.810
independent of what your
mean-variance utilities are.

37:47.809 --> 37:54.229
No matter how risk averse you
are or how risk loving you are,

37:54.233 --> 37:59.163
you should choose somewhere
along this line.

37:59.159 --> 38:01.299
Why is that?

38:01.300 --> 38:03.810
Well, I've drawn,
with this blob,

38:03.806 --> 38:08.036
I've drawn all the possible
combinations I could get by

38:08.036 --> 38:10.696
mixing different risky assets.

38:10.699 --> 38:14.229
What I left out of the picture
is all the combinations I could

38:14.226 --> 38:16.246
get by adding the riskless
asset.

38:16.250 --> 38:18.010
I've got all the risky asset
combinations.

38:18.010 --> 38:19.430
That's the blob.

38:19.429 --> 38:23.529
Now I want to do--adding the
riskless asset means I can take

38:23.525 --> 38:26.645
part of my money in here and
part in there.

38:26.650 --> 38:29.090
So in particular,
I can put part of my money here

38:29.094 --> 38:31.544
and part of my money there,
or I can go short.

38:31.539 --> 38:34.869
So whatever this combination
was, that's the best thing I can

38:34.873 --> 38:37.933
do, because now I can get
everything along this line.

38:37.929 --> 38:39.659
Nothing else can do better.

38:39.659 --> 38:41.919
What other combination could
there be?

38:41.920 --> 38:45.970
It has to be the riskless asset
a bunch of risky assets.

38:45.969 --> 38:50.459
But combining the risky assets
just puts me in the blob.

38:50.460 --> 38:52.300
So it might put me here in the
blob.

38:52.300 --> 38:54.960
And then once I combine that
with the riskless asset,

38:54.961 --> 38:56.141
I'll be on that line.

38:56.139 --> 38:58.959
This line is always below that
line, so this is the best

38:58.963 --> 39:00.303
possible thing I can do.

39:00.300 --> 39:04.340
No combination of the riskless
asset and any other combination

39:04.340 --> 39:07.700
of risky assets,
that'll always put me on some

39:07.699 --> 39:10.519
line from here,
through some point in the blob,

39:10.523 --> 39:12.943
which is going to be below this
tangent point.

39:12.940 --> 39:14.870
So this is the best thing I can
do.

39:14.869 --> 39:18.749
And it's independent of whether
I was risk averse or a risk

39:18.753 --> 39:19.293
lover.

39:19.289 --> 39:21.629
If I'm risk averse and want to
play it safe,

39:21.628 --> 39:24.618
my utility function--I forgot
what those looked like.

39:24.619 --> 39:32.139
My indifference curves--I'll be
here.

39:32.139 --> 39:34.969
I'll put most of my money in
the bank and not very many in

39:34.974 --> 39:35.874
the risky stock.

39:35.869 --> 39:38.199
If I'm more of a risk lover,
I'll be maybe here,

39:38.195 --> 39:40.965
and I'll choose to put a good
fraction of my money in the

39:40.969 --> 39:42.949
stock and only a little in the
bank.

39:42.949 --> 39:46.289
If I'm really reckless,
I'll borrow money and put it in

39:46.288 --> 39:48.698
the stock market and be way up
there.

39:48.699 --> 39:52.289
But everybody will do the same
thing, no matter what their

39:52.286 --> 39:53.416
preferences are.

39:53.420 --> 39:56.880
They'll all be along the same
line.

39:56.880 --> 40:00.220
So that's the mutual fund
theorem.

40:00.219 --> 40:02.129
Why is that the mutual fund
theorem?

40:02.130 --> 40:05.810
Because it says everybody
should invest in the same index

40:05.809 --> 40:09.159
of stocks and put more or less
money in the bank.

40:09.159 --> 40:11.929
So every single person ought to
be doing the same thing.

40:11.929 --> 40:16.739
This is the mutual fund that
everybody should hold.

40:16.739 --> 40:20.159
Whatever combination of stocks
that got me to this point is the

40:20.161 --> 40:22.481
mutual fund that everybody
should hold,

40:22.480 --> 40:25.790
and combine that with putting
money in the bank.

40:25.789 --> 40:30.459
Okay, so let's just prove that
algebraically,

40:30.463 --> 40:35.353
in a special case,
which is going to illustrate

40:35.349 --> 40:39.279
the point, the idea of it
anyway.

40:39.280 --> 40:42.450
I'm doing a very special case
here, but just to illustrate it

40:42.445 --> 40:43.285
algebraically.

40:43.289 --> 40:49.569
So suppose that the Rs are,
as we said, the returns,

40:49.570 --> 40:53.390
up here, per dollar of stock.

40:53.389 --> 40:57.239
The R_0 I've now
added and that's going to be the

40:57.235 --> 41:00.745
riskless asset that pays like 1
dollar 6 all the time,

41:00.750 --> 41:03.270
if the interest rate is 6
percent.

41:03.268 --> 41:06.848
So now you have your money,
you have I dollars.

41:06.849 --> 41:08.389
You're the investor with I
dollars.

41:08.389 --> 41:13.359
You can divide that money into
the riskless asset or any one of

41:13.358 --> 41:14.558
the J assets.

41:14.559 --> 41:16.369
So here you've got R^(0).

41:16.369 --> 41:20.099
You can get that return,
R^(1) up to R^(J) and you can

41:20.103 --> 41:21.993
put money,
omega_0,

41:21.985 --> 41:24.455
omega_1,
omega_J,

41:24.458 --> 41:27.878
but of course,
this all has to add up to I.

41:27.880 --> 41:29.410
So how should you do it?

41:29.409 --> 41:33.849
Well, you care about your
expectation and you hate

41:33.851 --> 41:38.561
variance, so I'm doing just a
special case to make it

41:38.563 --> 41:40.743
illustrate the idea.

41:40.739 --> 41:44.519
Suppose I write down your
utility as if it were the

41:44.516 --> 41:47.686
expectation, - alpha times the
variance.

41:47.690 --> 41:49.570
It could be a more complicated
function.

41:49.570 --> 41:54.380
We said already that the
quadratic utilities give you

41:54.378 --> 41:59.738
expectation - alpha expectation
squared, - some other alpha

41:59.742 --> 42:01.872
times the variance.

42:01.869 --> 42:04.409
That's the utility we derive
from quadratic--

42:04.409 --> 42:07.709
when people have quadratic
utilities over consumption,

42:07.710 --> 42:11.760
their induced utility of
portfolios is this thing.

42:11.760 --> 42:14.090
So it's not exactly the one
I've written down,

42:14.092 --> 42:17.312
but the argument would be just
the same and this is shorter and

42:17.306 --> 42:17.926
simpler.

42:17.929 --> 42:20.059
So just to illustrate the
flavor of it,

42:20.059 --> 42:23.089
let's say all you care about is
your expectation - some

42:23.088 --> 42:26.338
constant, how risk averse you
are, times the variance.

42:26.340 --> 42:29.680
Now let's suppose,
to keep it even simpler,

42:29.679 --> 42:32.699
that the assets are all
independent.

42:32.699 --> 42:34.689
The theory doesn't rely on that.

42:34.690 --> 42:36.560
I'm just saying,
suppose it were so we can

42:36.561 --> 42:39.121
algebraically get it without
having to work very hard.

42:39.119 --> 42:41.529
So you're maximizing your
expectations,

42:41.534 --> 42:44.144
so the bar above means the
expectation.

42:44.139 --> 42:46.189
So what's your expectation of
omega dollars in here,

42:46.193 --> 42:48.533
omega_1 in here and
omega_J in here?

42:48.530 --> 42:51.880
Just omega_0 times
the expectation of this.

42:51.880 --> 42:55.470
So your expectation then is
going to be omega_0

42:55.465 --> 42:58.475
times R^(0) bar (that's the
expectation of that)

42:58.476 --> 43:02.906
omega_1,
R bar^(1) omega_J,

43:02.909 --> 43:07.809
R bar^(J).

43:07.809 --> 43:09.759
These are the top for some
reason.

43:09.760 --> 43:11.240
That's your expectation.

43:11.239 --> 43:14.179
Now what's your variance,
the thing that you don't like?

43:14.179 --> 43:21.429
Your variance is going to
be--is going to be the variance

43:21.434 --> 43:22.864
of these.

43:22.860 --> 43:23.720
They're independent.

43:23.719 --> 43:30.359
So the variance is just going
to be what I've written.

43:30.360 --> 43:33.850
So call the variance of R^(J)
sigma,

43:33.849 --> 43:36.199
so it's going to be
sigma_0 squared

43:36.197 --> 43:38.697
(that's the variance of
sigma_0) times

43:38.704 --> 43:42.274
omega_0 squared,
(remember, because for variance

43:42.269 --> 43:45.169
you have to square if you
multiply by something)

43:45.168 --> 43:48.618
omega_1 squared
sigma_1 squared,

43:48.619 --> 43:53.079
omega_J squared
sigma_J squared.

43:53.079 --> 43:56.109
Now of course,
remember that the variance of

43:56.108 --> 43:59.488
the riskless thing is 0,
so that's actually 0.

43:59.489 --> 44:02.079
This is the variance you have
to watch out for.

44:02.079 --> 44:04.329
This is the expectation you
want.

44:04.329 --> 44:06.529
How should you pick
omega_0,

44:06.532 --> 44:08.682
omega_1,
omega_J,

44:08.677 --> 44:12.247
when your penalty for having
too much variance is alpha?

44:12.250 --> 44:14.990
So different investors have
different risk aversions so they

44:14.985 --> 44:16.095
have different alphas.

44:16.099 --> 44:18.619
Should they be led,
as Markowitz thought,

44:18.619 --> 44:20.819
without the riskless asset,
remember,

44:20.820 --> 44:23.310
with the blob like this,
one guy was going to pick here,

44:23.309 --> 44:24.679
another guy was going to pick
there.

44:24.679 --> 44:26.709
They'd all do very different
things.

44:26.710 --> 44:29.980
Tobin added this riskless
asset, and lo and behold,

44:29.981 --> 44:32.601
now everyone should do the same
thing.

44:32.599 --> 44:33.259
So why is that?

44:33.260 --> 44:35.250
We just reduce this
algebraically,

44:35.251 --> 44:36.761
solve it algebraically.

44:36.760 --> 44:38.980
So the constraint is a nuisance.

44:38.980 --> 44:41.280
To maximize something with the
constraint's a nuisance,

44:41.280 --> 44:43.970
so if you just notice that if
you satisfy the constraint,

44:43.969 --> 44:47.079
omega_0 has to equal
I - the sum of the

44:47.079 --> 44:48.439
omega_js.

44:48.440 --> 44:52.000
So if I substitute for
omega_0 right up

44:51.998 --> 44:55.258
there, I substitute I - the sum
of the js.

44:55.260 --> 45:00.600
Then I'm going to get R bar^(0)
times I - R bar^(0) times

45:00.597 --> 45:03.927
omega_1,
- R bar^(0) times

45:03.932 --> 45:06.032
omega_2.

45:06.030 --> 45:08.780
But I can feed that into the
other thing,

45:08.780 --> 45:14.280
so I get R^(0) times the I,
and then the R^(1) term is

45:14.278 --> 45:17.998
going to be R bar^(1) times
omega_1 - the thing

45:18.003 --> 45:22.263
that came from subtracting the
omega_1 there in I--

45:22.260 --> 45:26.580
by replacing omega_0
with (I - omega_1 -

45:26.577 --> 45:28.477
omega_J),
etc.

45:28.480 --> 45:31.980
So I'm going to notice--all
these other terms now,

45:31.981 --> 45:35.841
I get rid of the constraint and
all the other terms get

45:35.840 --> 45:36.770
replaced.

45:36.768 --> 45:41.208
R bar^(2) gets replaced by R
bar^(2) - R bar^(0).

45:41.210 --> 45:44.940
I've dropped sigma_0
because it's 0.

45:44.940 --> 45:48.080
So I'm maximizing something
without a constraint by plugging

45:48.083 --> 45:50.273
in the constraint to
omega_0.

45:50.268 --> 45:53.108
So I have to choose my
omega_1 through

45:53.108 --> 45:56.548
omega_J and then
omega_0 is determined

45:56.554 --> 45:57.404
from that.

45:57.400 --> 46:00.110
But I'm maximizing one thing,
so just take the derivative

46:00.112 --> 46:02.532
with respect to each
omega_i and set it =

46:02.534 --> 46:02.974
to 0.

46:02.969 --> 46:04.449
So with respect to
omega_1,

46:04.449 --> 46:13.739
say, it's going to be R bar^(1)
- R bar^(0) - 2 alpha sigma--

46:13.739 --> 46:16.549
I'm differentiating with
respect to omega_1 so

46:16.548 --> 46:19.868
it's 2 alpha omega_1
times sigma_1 squared.

46:19.869 --> 46:24.429
2 alpha omega_1 times
sigma_1 squared = 0.

46:24.429 --> 46:28.809
So therefore omega_1
= R^(1) - R^(0) over 2 alpha

46:28.806 --> 46:30.916
sigma_1 squared.

46:30.920 --> 46:34.140
But if I did it now with
respect to 2 or to 3 or to 4,

46:34.137 --> 46:37.717
nothing--I would get the same
formula, just with a j instead

46:37.721 --> 46:38.391
of a 1.

46:38.389 --> 46:42.329
So that's the formula everybody
picks and notice it depends on

46:42.327 --> 46:43.357
what alpha is.

46:43.360 --> 46:46.630
So people with different alphas
are going to put different

46:46.632 --> 46:48.932
amounts of money in
omega_j.

46:48.929 --> 46:53.129
But it depends on alpha in a
very trivial way.

46:53.130 --> 46:57.470
So if I took the fraction,
the relative amounts of money I

46:57.467 --> 47:01.727
put in i and j and divided this
by omega_i,

47:01.730 --> 47:05.030
by omega_j,
I would get that the alphas and

47:05.032 --> 47:06.432
the 2s canceled out.

47:06.429 --> 47:09.689
So everybody would put R^(i) -
R bar^(0) divided by

47:09.688 --> 47:12.698
sigma_i squared,
R bar^(j) - R^(0) over

47:12.697 --> 47:14.067
sigma_j squared.

47:14.070 --> 47:16.950
That number doesn't depend on
who the person is,

47:16.949 --> 47:19.669
so the relative amounts of
money put in i and j are

47:19.668 --> 47:21.678
independent of what your alpha
is,

47:21.679 --> 47:22.989
independent of the person.

47:22.989 --> 47:25.289
So unlike this old Markowitz
case,

47:25.289 --> 47:29.909
the Tobin case looks like that,
and everybody chooses the same

47:29.907 --> 47:34.067
combination of risky assets and
then this safe asset.

47:34.070 --> 47:36.200
So it may be that different
people put different

47:36.199 --> 47:39.549
combinations of safe and risky,
but everybody's proportion of

47:39.545 --> 47:42.655
risky assets is proportional to
everybody else's.

47:42.659 --> 47:44.429
That's what we just proved
algebraically.

47:44.429 --> 47:47.689
So graphically,
a sort of general proof,

47:47.686 --> 47:50.356
algebraically,
a special case.

47:50.360 --> 47:52.080
And then if you were in a
graduate class,

47:52.079 --> 47:54.869
you'd get an algebraic proof in
general,

47:54.869 --> 47:56.829
which is actually much faster
than either of these,

47:56.829 --> 47:58.989
but you have to know a little
linear algebra.

47:58.989 --> 48:00.719
And so that's the proof.

48:00.719 --> 48:02.059
It's a very simple thing.

48:02.059 --> 48:06.729
It's a remarkable thing,
and here's the famous Tobin

48:06.733 --> 48:07.653
diagram.

48:07.650 --> 48:11.020
Everybody should do that,
put their money in the same

48:11.016 --> 48:14.446
combination of risky assets and
the riskless asset.

48:14.449 --> 48:19.809
Are there any questions about
this?

48:19.809 --> 48:22.109
It's kind of shocking.

48:22.110 --> 48:26.750
I could swear that most people
think, maybe even rightly,

48:26.751 --> 48:31.641
if you really want to go for
it, put your money in the young

48:31.641 --> 48:33.881
Microsoft or something.

48:33.880 --> 48:37.850
It could be a tremendous
success or it could go out of

48:37.846 --> 48:38.666
business.

48:38.670 --> 48:41.400
That's what you should do,
but no, this says not at all.

48:41.400 --> 48:44.700
If you really want to go for
it, take the same index that

48:44.695 --> 48:47.635
everybody, the S&P 500 and
just leverage it.

48:47.639 --> 48:50.299
Borrow a lot of money and go
out that straight line,

48:50.298 --> 48:52.018
way out to there,
to the right.

48:52.018 --> 48:53.408
Don't put your money in
Microsoft.

48:53.409 --> 48:58.299
Just leverage the hell out of
the S&P 500.

48:58.300 --> 49:02.700
Okay, so after this theory was
created--I'm going to tease out

49:02.704 --> 49:04.804
all the implications of it.

49:04.800 --> 49:08.840
After this theory was created,
the mutual fund business took

49:08.838 --> 49:09.248
off.

49:09.250 --> 49:11.380
Vanguard and all of these
places, what did they do?

49:11.380 --> 49:14.830
They said, "Look,
economic science has taught us,

49:14.829 --> 49:18.669
the best you can do is not try
to pick individual stocks.

49:18.670 --> 49:22.350
The best you can do is to hold
the same index,

49:22.347 --> 49:23.897
the whole market.

49:23.900 --> 49:25.530
"Now of course,
it's very expensive for you to

49:25.534 --> 49:26.324
hold the whole market.

49:26.320 --> 49:29.180
Are you going to go to a broker
and tell him 'Buy me 3 shares of

49:29.177 --> 49:31.267
this and 7 shares of that and 9
shares of that,

49:31.266 --> 49:32.306
14 shares of that'?

49:32.309 --> 49:34.719
It'll take the guy forever to
buy all that stuff and you have

49:34.715 --> 49:36.795
to pay a commission on every
single thing you buy.

49:36.800 --> 49:39.260
It's going to cost you so much
money to do what economic

49:39.259 --> 49:39.929
science says.

49:39.929 --> 49:41.389
Come to us at Vanguard.

49:41.389 --> 49:43.779
We'll buy the whole market for
you,

49:43.780 --> 49:46.890
call it our index fund and
charge you almost no commission

49:46.885 --> 49:48.615
at all,
just a tiny little bit,

49:48.617 --> 49:50.577
because we're doing it in such
bulk,

49:50.579 --> 49:52.999
everyone's--we've got lots of
investors like you.

49:53.000 --> 49:54.390
They're all buying the same
thing.

49:54.389 --> 49:57.679
We have huge volume,
so we're going to get a really

49:57.675 --> 50:01.025
small brokerage fee,
so you should put your money in

50:01.027 --> 50:02.207
with us."

50:02.210 --> 50:03.890
So that's what everybody
started to do.

50:03.889 --> 50:06.659
So all these mutual funds and
money managers,

50:06.659 --> 50:10.329
they all grew up,
originally by guaranteeing to

50:10.333 --> 50:14.113
produce an index fund,
the very same thing economic

50:14.105 --> 50:17.245
science recommended,
at almost no cost.

50:17.250 --> 50:20.870
If you read Swensen's book,
his first,

50:20.869 --> 50:24.329
most important piece of advice:
if you are a standard investor

50:24.327 --> 50:27.327
who doesn't know how to
manipulate all the complicated

50:27.333 --> 50:30.513
instruments on Wall Street and
has transactions costs and

50:30.509 --> 50:33.569
things like that,
put all your money into an

50:33.570 --> 50:34.030
index.

50:34.030 --> 50:35.520
That's what he recommends doing.

50:35.518 --> 50:38.428
And he says everyone else who
tells you otherwise is just

50:38.427 --> 50:40.607
trying to steal your money,
rip you off.

50:40.610 --> 50:46.470
So that's the main financial
advice that economists give now.

50:46.469 --> 50:51.159
It's Shakespeare's advice.

50:51.159 --> 50:54.399
Don't put all your eggs in one
basket, but it's very specific.

50:54.400 --> 50:58.650
Put it in an index where you
buy everything in the same

50:58.646 --> 50:59.666
proportion.

50:59.670 --> 51:01.600
Now of course,
if you're buying everything in

51:01.597 --> 51:03.607
the same proportion and
everybody's doing that,

51:03.614 --> 51:04.934
what does this have to be?

51:04.929 --> 51:06.579
I forgot to say that.

51:06.579 --> 51:09.259
If everybody's buying every
stock in the same proportion,

51:09.257 --> 51:11.837
that proportion must be the
market, because that's what

51:11.840 --> 51:12.990
everybody's holding.

51:12.989 --> 51:15.859
So this doesn't turn out to be
any old portfolio.

51:15.860 --> 51:17.580
It's the market portfolio.

51:17.579 --> 51:21.489
So the index,
that's a huge conclusion here,

51:21.489 --> 51:24.269
the index that they should
offer is not just some magical

51:24.273 --> 51:26.463
index that's better than
everything else,

51:26.460 --> 51:28.040
because if everyone's doing the
right thing,

51:28.039 --> 51:30.119
everyone's choosing the same
index--

51:30.119 --> 51:32.389
that's what we proved--if
everyone's choosing the same

51:32.385 --> 51:34.245
index,
it must be the market index,

51:34.248 --> 51:36.308
because that's all there is to
choose.

51:36.309 --> 51:37.789
Supply has to equal demand.

51:37.789 --> 51:43.079
So therefore the advice is to
put all your money into the

51:43.079 --> 51:46.009
market index,
not just an index,

51:46.009 --> 51:50.449
the market index maybe put
money in a bank.

51:50.449 --> 51:55.689
Or even borrow from a bank to
put it into the market index.

51:55.690 --> 52:01.680
That's the first surprising
theorem that Tobin proved and it

52:01.682 --> 52:07.172
had dramatic--so it created the
indexed investments.

52:07.170 --> 52:08.530
It created Vanguard.

52:08.530 --> 52:11.530
So these people who run these
companies, I forgot the guy's

52:11.534 --> 52:12.834
name who runs Vanguard.

52:12.829 --> 52:14.739
He's always giving these
speeches.

52:14.739 --> 52:17.409
So that's how he describes how
he got his start in his

52:17.405 --> 52:17.955
business.

52:17.960 --> 52:19.600
All he says is,
"I take all the

52:19.599 --> 52:21.519
recommendations of economic
science,"

52:21.519 --> 52:23.199
he says,
"and I allow people to

52:23.202 --> 52:23.842
carry them out.

52:23.840 --> 52:26.380
That's why I'm doing great good
for society.

52:26.380 --> 52:28.390
Everybody else,
all these hedge funds and

52:28.389 --> 52:30.049
stuff, they're ruining society.

52:30.050 --> 52:32.270
I am just doing what Tobin told
us to do."

52:32.269 --> 52:36.079
That's his basic speech.

52:36.079 --> 52:40.659
So Markowitz was a little bit
more precise even than that.

52:40.659 --> 52:43.819
Markowitz' formula and Tobin's
special case,

52:43.815 --> 52:46.305
he not only told you what to
do.

52:46.309 --> 52:48.179
He told you,
"do what everyone else is

52:48.177 --> 52:48.797
doing."

52:48.800 --> 52:50.910
He told you what that should be.

52:50.909 --> 52:55.239
So let's suppose that I've got
2 stocks,

52:55.239 --> 53:02.529
i and j, with the same return,
same expected return,

53:02.530 --> 53:09.030
but one has standard deviation
3 times higher than the other.

53:09.030 --> 53:11.980
So in my picture here,
that standard picture,

53:11.981 --> 53:15.071
we've got the standard
deviation, we've got the

53:15.065 --> 53:16.135
expectation.

53:16.139 --> 53:20.739
We've got 1 stock here and
we've got another stock here

53:20.740 --> 53:23.980
with 3 times the standard
deviation.

53:23.980 --> 53:26.610
So you might have thought,
"Well, these 2 stocks,

53:26.612 --> 53:28.902
same expected return,
what's the difference?

53:28.900 --> 53:31.080
Put all my money in one or the
other, doesn't matter."

53:31.079 --> 53:33.209
A little bit further thought
would say, "Well,

53:33.213 --> 53:34.923
this one is much better than
that one.

53:34.920 --> 53:38.070
I've got smaller standard
deviation, so I should put all

53:38.065 --> 53:39.835
my money into this one."

53:39.840 --> 53:44.890
But then on further thought,
given that these two are

53:44.893 --> 53:48.203
independent, what should you do?

53:48.199 --> 53:53.079
How should you allocate your
money between this stock and

53:53.077 --> 53:54.207
that stock?

53:54.210 --> 53:59.390
What does the formula tell you
to do?

53:59.389 --> 54:09.789
It's very clear.

54:09.789 --> 54:14.869
Well, the ratio of money in one
stock to the other stock looks

54:14.869 --> 54:16.369
like this ratio.

54:16.369 --> 54:19.919
But I've told you that the
expectations of the 2 stocks are

54:19.916 --> 54:20.586
the same.

54:20.590 --> 54:22.300
So those things are the same.

54:22.300 --> 54:30.800
I've told you that one has 3
times the standard deviation

54:30.798 --> 54:35.048
than the other,
so therefore,

54:35.048 --> 54:38.538
what should you do?

54:38.539 --> 54:39.029
Yeah.

54:39.030 --> 54:42.360
Student:
Hold 9 times as much as the one

54:42.362 --> 54:44.532
with the lower--
Prof: Exactly.

54:44.530 --> 54:50.230
You should put 90 percent in
this stock, 10 percent in that

54:50.233 --> 54:51.023
stock.

54:51.018 --> 54:55.048
That's it, because if one
sigma's 3 times the other,

54:55.050 --> 54:56.990
and you square it,
it's going to be 9 times the

54:56.989 --> 54:58.879
other,
and so obviously you have to

54:58.880 --> 55:01.630
figure out what's in the
numerator and what's in the

55:01.630 --> 55:02.440
denominator.

55:02.440 --> 55:04.180
But clearly,
you're going to put more of

55:04.181 --> 55:06.861
your money in the lower standard
deviation, so it's 90/10.

55:06.860 --> 55:10.080
So it's very concrete advice.

55:10.079 --> 55:15.759
So the first concrete advice
is, the first concrete thing

55:15.762 --> 55:20.332
we've gotten out of this is,
buy the index.

55:20.329 --> 55:25.219
Now the index typically in
America, they're easy to buy,

55:25.224 --> 55:27.364
are indexes of stocks.

55:27.360 --> 55:29.090
But you know,
the second piece of advice is,

55:29.090 --> 55:34.040
and this is also Swensen's
second most important point--

55:34.039 --> 55:36.909
I'm quoting Swensen because
we're here at Yale,

55:36.909 --> 55:39.129
not because he's the only one
to recognize these things,

55:39.130 --> 55:42.940
but he was a student of
Tobin--so the second important

55:42.940 --> 55:46.610
point is that you should not
just buy the index,

55:46.610 --> 55:48.990
because the indexes are the
wrong index.

55:48.989 --> 55:51.109
The index should be the index
of everything.

55:51.110 --> 55:54.250
It should include American
stocks, Chinese stocks,

55:54.246 --> 55:55.966
French stocks,
Greek stocks,

55:55.974 --> 55:57.194
Israeli stocks.

55:57.190 --> 55:59.980
You should have all stocks in
the index, but you can't really

55:59.981 --> 56:01.891
very easily buy all those other
things.

56:01.889 --> 56:05.419
You can only very easily buy
index of American things.

56:05.420 --> 56:09.100
It's getting easier and easier
now to buy the global index.

56:09.099 --> 56:11.669
But anyway, you should buy the
global index.

56:11.670 --> 56:14.100
That's what this says.

56:14.099 --> 56:18.439
So every time you find a new
independent risk you should look

56:18.438 --> 56:19.088
for it.

56:19.090 --> 56:24.880
So you should buy the global
index, seek out independent

56:24.878 --> 56:25.718
risks.

56:25.719 --> 56:28.129
So what did Swensen do?

56:28.130 --> 56:31.110
One of his most important
things was, Yale is always

56:31.110 --> 56:33.740
looking globally to find
independent risks.

56:33.739 --> 56:38.559
So let's go back to the--so you
should always find independent

56:38.563 --> 56:41.493
risks, and you suffer if you
don't.

56:41.489 --> 56:45.339
So you see, by putting 90
percent here and 10 percent

56:45.342 --> 56:49.792
here, Markowitz is telling you,
you're going to do better.

56:49.789 --> 56:56.489
What else do I have to say?

56:56.489 --> 56:59.509
So let's go back to our dice
example.

56:59.510 --> 57:04.740
Suppose that you had 1 dice.

57:04.739 --> 57:08.469
Now by the way,
I want to introduce one more

57:08.465 --> 57:12.015
term, which I'll call the Sharpe
ratio.

57:12.018 --> 57:18.758
So the slope of this line--so
it's the Tobin diagram.

57:18.760 --> 57:23.500
It should be called the Tobin
ratio, but this other guy at

57:23.503 --> 57:28.083
Stanford snuck in and managed to
give his name to it.

57:28.079 --> 57:30.509
He ended up getting a Nobel
prize too.

57:30.510 --> 57:34.020
So here this line,
the slope of this line,

57:34.021 --> 57:38.301
this is the riskless asset--the
slope of this line,

57:38.302 --> 57:41.302
what is the slope of this line?

57:41.300 --> 57:43.790
So this is standard deviation,
this is expectation.

57:43.789 --> 57:47.779
The slope, what's the slope?

57:47.780 --> 57:54.630
Well, it's the expectation of
whatever you're holding,

57:54.628 --> 58:03.028
X, - the riskless asset divided
by the standard deviation of X.

58:03.030 --> 58:07.000
That's the Sharpe ratio of X.

58:07.000 --> 58:12.000
So you notice that every point
you could conceivably choose,

58:12.000 --> 58:14.690
which is somewhere in this
blob, then connect it to the

58:14.692 --> 58:17.532
riskless asset--
so every point on this line

58:17.534 --> 58:20.204
obviously has the same Sharpe
ratio.

58:20.199 --> 58:22.369
If you leveraged your position
and went to here,

58:22.365 --> 58:24.205
you wouldn't change your Sharpe
ratio.

58:24.210 --> 58:25.400
It has the same slope.

58:25.400 --> 58:27.660
If you put half your money in
the bank and half in the

58:27.657 --> 58:28.887
riskless asset,
you'd be here,

58:28.891 --> 58:29.831
same Sharpe ratio.

58:29.829 --> 58:34.619
Anything else you could do has
a lower Sharpe ratio.

58:34.619 --> 58:46.469
So another thing to say is that
an investor managing all his

58:46.467 --> 58:54.697
money should maximize the Sharpe
ratio.

58:54.699 --> 58:58.569
You could in fact tell if
somebody had screwed up.

58:58.570 --> 59:01.660
Instead of doing the global
index, they just did the

59:01.659 --> 59:03.849
American index,
pick something here,

59:03.847 --> 59:07.247
you could look at the slope of
where they ended up and it would

59:07.250 --> 59:09.720
be a worse slope and a worse
Sharpe ratio.

59:09.719 --> 59:11.569
You could even judge people,
how they're doing,

59:11.570 --> 59:12.980
by looking at their Sharpe
ratio.

59:12.980 --> 59:16.230
So let's take the case of the 2
dice.

59:16.230 --> 59:19.820
Suppose you had 1 die.

59:19.820 --> 59:23.020
So if you had 1 die,
remember, if you invest 100

59:23.018 --> 59:26.148
dollars, you could get 101,102,
through 106.

59:26.150 --> 59:29.820
Let's say the interest rate is
3 percent.

59:29.820 --> 59:37.300
So you could get 101,102,
103,104, 105,106,

59:37.297 --> 59:43.527
with 1 sixth probability of
each.

59:43.530 --> 59:47.130
So the average of that is
obviously 103.5,

59:47.126 --> 59:50.986
right, because 1 and 6 is 7,
and divide by 2,

59:50.989 --> 59:52.129
is 103.5.

59:52.130 --> 59:58.110
So the expected payoff for
putting your money into 1 dice

59:58.108 --> 1:00:01.738
is the average return,
is 1.035.

1:00:01.739 --> 1:00:08.079
The riskless return is going to
be 1.03, so it's .035 - .03,

1:00:08.077 --> 1:00:08.827
.005.

1:00:08.829 --> 1:00:10.919
And now what's the standard
deviation?

1:00:10.920 --> 1:00:12.450
What's the variance of this?

1:00:12.449 --> 1:00:15.919
Well, you could take 101 -
103.5 squared,

1:00:15.923 --> 1:00:18.273
103 - 103.5 squared,
etc.

1:00:18.268 --> 1:00:24.508
and take the square root of it,
and you'd get .017.

1:00:24.510 --> 1:00:27.290
So you notice the chance of
losing money,

1:00:27.286 --> 1:00:29.226
by the way, is 50 percent.

1:00:29.230 --> 1:00:33.650
50 percent of the time,
you're going to lose money,

1:00:33.648 --> 1:00:37.358
and your Sharpe ratio is .005
over .017.

1:00:37.360 --> 1:00:41.590
So that's about 3--no,
it's about .3.

1:00:41.590 --> 1:00:44.390
Now that happens to be the
Sharpe ratio of the stock

1:00:44.387 --> 1:00:45.537
market, just about.

1:00:45.539 --> 1:00:47.839
Shockingly low,
isn't it, if you look at the

1:00:47.840 --> 1:00:50.690
history of the stock market--
so it depends what year you

1:00:50.688 --> 1:00:53.568
start and what year you end,
if you do it right after a

1:00:53.570 --> 1:00:57.620
crash or before a crash,
but just to pick vague numbers

1:00:57.621 --> 1:01:02.771
it's 9 percent and let's say the
interest rate is 4 percent,

1:01:02.769 --> 1:01:04.339
something like that.

1:01:04.340 --> 1:01:06.460
It could be 5 percent or 3
percent.

1:01:06.460 --> 1:01:07.900
Depends again the period.

1:01:07.900 --> 1:01:11.780
Maybe it's 2 percent,
but somewhere like that.

1:01:11.780 --> 1:01:15.270
And the standard deviation of
the stock market is 16 percent

1:01:15.269 --> 1:01:18.109
or 12 percent,
again depending on what period.

1:01:18.110 --> 1:01:22.630
But always around 12 to 18
percent, let's say 16 percent.

1:01:22.630 --> 1:01:26.550
That's the one I like to use,
because it's percent a day.

1:01:26.550 --> 1:01:34.500
So that's = to 5 percent over
16 percent, which is about 1

1:01:34.503 --> 1:01:35.623
third.

1:01:35.619 --> 1:01:37.979
It's about .3.

1:01:37.980 --> 1:01:41.220
That's exactly what this dice
has, about .3.

1:01:41.219 --> 1:01:43.159
So that's all you can get out
of the stock market.

1:01:43.159 --> 1:01:45.459
Not a very good Sharpe ratio.

1:01:45.460 --> 1:01:48.340
So you could measure Swensen's
Sharpe ratio.

1:01:48.340 --> 1:01:50.520
You could measure Ellington's
Sharpe ratio and see what they

1:01:50.518 --> 1:01:50.738
are.

1:01:50.739 --> 1:01:52.939
You can see who you think is a
better manager.

1:01:52.940 --> 1:02:04.420
So I won't tell you those
numbers just yet.

1:02:04.420 --> 1:02:08.800
But now let's suppose,
what else could I do?

1:02:08.800 --> 1:02:11.280
So let's suppose you put your
money in two dice.

1:02:11.280 --> 1:02:12.630
That's the whole point of this.

1:02:12.630 --> 1:02:15.190
Suppose you put half your money
in the first dice and half your

1:02:15.193 --> 1:02:16.273
money in the second die.

1:02:16.268 --> 1:02:19.528
Now what are the
chances--what's going to happen?

1:02:19.530 --> 1:02:23.060
Well, again,
your expected return hasn't

1:02:23.063 --> 1:02:23.973
changed.

1:02:23.969 --> 1:02:27.189
It's still .035,
but the Sharpe ratio has

1:02:27.188 --> 1:02:28.958
drastically changed.

1:02:28.960 --> 1:02:30.090
In fact, you could see that.

1:02:30.090 --> 1:02:33.000
What are the chances of ending
up with 101 dollars now,

1:02:32.996 --> 1:02:35.846
if you put half your money in
each, the worst case?

1:02:35.849 --> 1:02:37.239
It's not 1 in 6.

1:02:37.239 --> 1:02:39.719
You have to get the worst case
in both dice.

1:02:39.719 --> 1:02:43.509
So to get 101 dollars,
it's 1 in 36.

1:02:43.510 --> 1:02:46.980
It's vastly less likely that
you end up with this extreme bad

1:02:46.983 --> 1:02:49.883
outcome once you put half the
money in each die.

1:02:49.880 --> 1:02:53.290
So if you compute the standard
deviation, it's going to be a

1:02:53.293 --> 1:02:54.513
lot smaller number.

1:02:54.510 --> 1:02:59.130
It's going to be .012.

1:02:59.130 --> 1:03:03.530
So the Sharpe ratio's gone from
.3 to .4, just by finding the

1:03:03.534 --> 1:03:04.494
second die.

1:03:04.489 --> 1:03:06.239
So what you should be doing is
looking,

1:03:06.239 --> 1:03:08.569
like Swensen does,
like everybody does,

1:03:08.570 --> 1:03:11.460
for those independent risks
that the rest of the market is

1:03:11.463 --> 1:03:12.483
too stupid to see.

1:03:12.480 --> 1:03:15.750
Even after they got the idea of
the index, for 20 years,

1:03:15.746 --> 1:03:18.826
they didn't realize the index
should mean everything,

1:03:18.833 --> 1:03:20.263
not just in America.

1:03:20.260 --> 1:03:22.870
So you'd have a huge
opportunity to do better than

1:03:22.871 --> 1:03:24.311
the rest of the managers.

1:03:24.309 --> 1:03:27.099
While everyone else is screwing
around on this line,

1:03:27.101 --> 1:03:29.401
you can be screwing around on
that line.

1:03:29.400 --> 1:03:33.060
So you could run the same risk
this guy did.

1:03:33.059 --> 1:03:36.909
So Swensen says,
you could leverage--

1:03:36.909 --> 1:03:38.179
not that he does,
but if you're Harvard and

1:03:38.179 --> 1:03:39.569
you're on this thing instead of
this thing,

1:03:39.570 --> 1:03:42.340
you could leverage and go up
here and say,

1:03:42.340 --> 1:03:45.110
"Look, I've got the same
standard deviation everybody

1:03:45.108 --> 1:03:47.218
else does,
and look at my returns.

1:03:47.219 --> 1:03:49.439
They're so much higher than
everybody else's."

1:03:49.440 --> 1:03:51.140
So that's what Harvard is
trying to do.

1:03:51.139 --> 1:03:54.409
They're trying to globally
diversify,

1:03:54.409 --> 1:03:57.089
cut down their standard
deviation and then leverage,

1:03:57.090 --> 1:04:00.570
and compared to everybody else,
get a higher return with the

1:04:00.567 --> 1:04:02.097
same standard deviation.

1:04:02.099 --> 1:04:03.839
So that's the first implication
of the theory.

1:04:03.840 --> 1:04:06.410
Now the second implication is
the best part.

1:04:06.409 --> 1:04:11.619
Let's hope I get to it.

1:04:11.619 --> 1:04:13.329
I'm going to skip over leverage.

1:04:13.329 --> 1:04:19.619
The second story is coming
right up.

1:04:19.619 --> 1:04:23.509
How are these things priced?

1:04:23.510 --> 1:04:28.500
I have to get to the second
part of Shakespeare.

1:04:28.500 --> 1:04:32.190
What can we say about the
pricing of individual stocks?

1:04:32.190 --> 1:04:36.130
It looks like an individual
stock is going to be worth more

1:04:36.128 --> 1:04:39.318
if it contributes no variance,
less variance.

1:04:39.320 --> 1:04:42.140
But that turns out to be
completely wrong.

1:04:42.139 --> 1:04:46.249
It's shocking that that's wrong.

1:04:46.250 --> 1:04:49.110
So why is this?

1:04:49.110 --> 1:04:53.060
The crucial thing is,
remember back to our first

1:04:53.056 --> 1:04:54.146
principles.

1:04:54.150 --> 1:04:57.840
What is price = to?

1:04:57.840 --> 1:04:59.600
What is the key ingredient of
price?

1:04:59.599 --> 1:05:03.679
What is it always = to in basic
economics?

1:05:03.679 --> 1:05:05.169
Student:
Supply and demand.

1:05:05.170 --> 1:05:06.160
Prof: Supply and demand,
okay.

1:05:06.159 --> 1:05:10.709
But an individual's going to
buy so that price is = to his

1:05:10.706 --> 1:05:12.776
marginal utility,
right?

1:05:12.780 --> 1:05:19.310
So we have to figure out--what
we've shown so far is that

1:05:19.306 --> 1:05:23.266
everybody is holding the market.

1:05:23.268 --> 1:05:30.608
Everybody's holding a market,
maybe with some riskless asset.

1:05:30.610 --> 1:05:33.890
So they're holding the market,
plus maybe they're long or

1:05:33.885 --> 1:05:36.395
short the bonds,
but that's not creating any

1:05:36.402 --> 1:05:37.282
covariance.

1:05:37.280 --> 1:05:41.810
So if I'm holding the market
and I hold a little bit more of

1:05:41.813 --> 1:05:44.533
a stock,
so if I could pay a little bit

1:05:44.530 --> 1:05:48.440
more and buy t units of stock X,
what good would it do me?

1:05:48.440 --> 1:05:51.310
Well, the expectation,
the change in expectation,

1:05:51.309 --> 1:05:55.959
with t dollars of X would be,
as I increase t a little bit,

1:05:55.960 --> 1:05:58.820
the derivative of my
expectation is just the

1:05:58.817 --> 1:06:00.077
expectation of X.

1:06:00.079 --> 1:06:04.819
So by holding the new stock,
I get expectation of--I get a

1:06:04.815 --> 1:06:05.975
little more.

1:06:05.980 --> 1:06:09.530
I get t times the expectation
of X, I hope.

1:06:09.530 --> 1:06:12.860
By coming up with t extra
dollars, I could increase my

1:06:12.855 --> 1:06:16.365
expectation by t times the
expectation of X obviously.

1:06:16.369 --> 1:06:18.119
But what would happen to my
variance?

1:06:18.119 --> 1:06:23.149
If I look at the variance of
this, what happens to my

1:06:23.146 --> 1:06:24.206
variance?

1:06:24.210 --> 1:06:32.980
Which is = (for variance I
could write) is the covariance

1:06:32.983 --> 1:06:37.373
of this with itself,
tX M.

1:06:37.369 --> 1:06:42.539
So everybody's supposed to hold
M some riskless thing,

1:06:42.543 --> 1:06:45.963
negative or positive in the
bond.

1:06:45.960 --> 1:06:51.120
But the negative or positive in
the bond is riskless asset.

1:06:51.119 --> 1:06:53.269
That contributes nothing to the
covariance.

1:06:53.268 --> 1:06:56.218
I might as well just look at
the covariance of a little bit

1:06:56.221 --> 1:06:57.851
of money in X with the market.

1:06:57.849 --> 1:06:59.659
And what's that covariance
going to be?

1:06:59.659 --> 1:07:04.359
Well, it's covariance of t,
tX and,

1:07:04.360 --> 1:07:12.590
tX, covariance of tX and M
(these are definitely not 0

1:07:12.585 --> 1:07:19.225
anymore) covariance of--
so it's 2,2t,

1:07:19.226 --> 1:07:24.036
covariance of tX and M.

1:07:24.039 --> 1:07:31.759
And then it's the,
covariance of M with itself.

1:07:31.760 --> 1:07:33.950
So it's the covariance of tX
with itself,

1:07:33.949 --> 1:07:37.439
tX with tX, twice the
covariance of tX with M and then

1:07:37.436 --> 1:07:40.106
this tX with M,
2 covariance tX with M,

1:07:40.110 --> 1:07:42.510
and the covariance of M with
itself.

1:07:42.510 --> 1:07:47.510
Now I want to differentiate
that thing with respect to t.

1:07:47.510 --> 1:07:50.340
If I put just a little bit of
money, that's being on the

1:07:50.336 --> 1:07:53.006
margin, what's the marginal
effect on my variance?

1:07:53.010 --> 1:07:56.160
I hate variance,
but see the mistake people have

1:07:56.159 --> 1:08:00.179
made in the past is they said,
"People hate variance.

1:08:00.179 --> 1:08:02.689
Therefore a stock that has a
lot of variance is bad."

1:08:02.690 --> 1:08:04.620
Absolutely wrong reasoning.

1:08:04.619 --> 1:08:07.579
That was Adam Smith's puzzle
about water and diamonds.

1:08:07.579 --> 1:08:09.759
You know, why is water less
expensive than diamonds when

1:08:09.760 --> 1:08:11.070
water's so much more important?

1:08:11.070 --> 1:08:13.500
It's not how important the
whole thing is;

1:08:13.500 --> 1:08:16.430
it's how important one tiny
extra drop of it is,

1:08:16.426 --> 1:08:17.856
the marginal utility.

1:08:17.859 --> 1:08:20.909
So if you add a tiny bit of X
to what you're already holding,

1:08:20.908 --> 1:08:22.958
which might involve a bunch of
X already,

1:08:22.960 --> 1:08:26.360
so you add a tiny bit more of X
to what you're already holding,

1:08:26.359 --> 1:08:28.469
what's going to be the change
in your variance?

1:08:28.470 --> 1:08:31.500
So it's the derivative of this
with respect to t,

1:08:31.498 --> 1:08:32.318
when t = 0.

1:08:32.319 --> 1:08:37.209
So this, by the way,
= t squared,

1:08:37.213 --> 1:08:46.543
variance of X 2 covariance--2t
times the covariance of X and M

1:08:46.543 --> 1:08:49.913
the variance of M.

1:08:49.908 --> 1:08:56.088
I'm differentiating this with
respect to t.

1:08:56.090 --> 1:08:57.100
Right?

1:08:57.100 --> 1:08:58.890
Because I've just rewritten the
same thing.

1:08:58.890 --> 1:09:00.850
That's just t squared,
the variance of X.

1:09:00.850 --> 1:09:03.510
The t comes out,
2t covariance of X with M,

1:09:03.512 --> 1:09:05.482
and here's the variance of M.

1:09:05.479 --> 1:09:08.109
But if I differentiate with
respect to t,

1:09:08.109 --> 1:09:13.839
I get 2t variance of X
(remember, it's the derivative

1:09:13.837 --> 1:09:21.297
of t when t is 0),
2t variance of X 2 covariance

1:09:21.296 --> 1:09:25.176
of X and M 0,
because the variance of M

1:09:25.181 --> 1:09:26.211
doesn't depend on t.

1:09:26.210 --> 1:09:30.260
So at t = 0, this is also 0.

1:09:30.260 --> 1:09:34.300
So you see the change in my
variance, when I add a little

1:09:34.302 --> 1:09:37.772
more of X, is twice the
covariance of X and M.

1:09:37.770 --> 1:09:39.560
So what is it that X adds?

1:09:39.560 --> 1:09:42.980
It adds some expectation and it
also adds some variance,

1:09:42.979 --> 1:09:45.339
but not according to the
variance of X,

1:09:45.342 --> 1:09:48.082
according to the covariance of
X with M.

1:09:48.078 --> 1:09:51.238
When I add some more--the price
of water is how much extra

1:09:51.237 --> 1:09:53.727
utility I get,
given what I've already got.

1:09:53.729 --> 1:09:55.199
I've already got a huge amount
of water.

1:09:55.198 --> 1:09:57.528
That's why an extra amount of
water's not doing much for me.

1:09:57.529 --> 1:09:59.049
That's why the price is low.

1:09:59.050 --> 1:10:01.490
So X itself might be very
dangerous,

1:10:01.488 --> 1:10:04.098
but if it's independent of all
the other stuff,

1:10:04.100 --> 1:10:06.490
M, that you're holding,
in fact, you're only adding a

1:10:06.494 --> 1:10:08.614
drop of it,
it won't change your variance

1:10:08.609 --> 1:10:09.369
hardly at all.

1:10:09.369 --> 1:10:14.279
So this is 0.

1:10:14.279 --> 1:10:16.979
It's the covariance.

1:10:16.979 --> 1:10:20.799
So that's the crucial idea,
that the marginal contribution

1:10:20.796 --> 1:10:24.806
of every stock depends on its
expectation and its covariance,

1:10:24.813 --> 1:10:26.223
not its variance.

1:10:26.220 --> 1:10:29.990
And the expectation is good and
the covariance is bad.

1:10:29.988 --> 1:10:32.868
And all of this is linear,
because it's differentiable,

1:10:32.873 --> 1:10:35.923
so it just says that--this is
the change in covariance.

1:10:35.920 --> 1:10:40.140
So remember the guy,
the marginal utility of

1:10:40.144 --> 1:10:41.524
expectation.

1:10:41.520 --> 1:10:42.830
How am I doing in time?

1:10:42.829 --> 1:10:43.769
1 more minute.

1:10:43.770 --> 1:10:47.780
Marginal utility of expectation
times the expectation of X,

1:10:47.779 --> 1:10:50.589
that's the change,
when you add 1 dollar's worth--

1:10:50.590 --> 1:10:53.460
a tiny bit of X,
your expectation goes up by the

1:10:53.458 --> 1:10:54.618
expectation of X.

1:10:54.618 --> 1:10:57.228
So this is the marginal utility
of expectation,

1:10:57.225 --> 1:10:59.995
which is the good thing,
- the marginal utility of

1:11:00.002 --> 1:11:00.742
variance.

1:11:00.739 --> 1:11:02.059
That's the bad thing.

1:11:02.060 --> 1:11:03.640
And what's the change?

1:11:03.640 --> 1:11:10.290
The thing that affects--the
contribution to variance is the

1:11:10.292 --> 1:11:13.162
covariance of X with M.

1:11:13.158 --> 1:11:17.558
So the punch line is,
everybody has a linear tradeoff

1:11:17.559 --> 1:11:20.689
between expectation and
covariance.

1:11:20.689 --> 1:11:24.319
So for any fixed person,
it's just a constant times the

1:11:24.319 --> 1:11:28.219
expectation of X - a constant
times--this doesn't depend on

1:11:28.219 --> 1:11:29.429
the portfolio.

1:11:29.430 --> 1:11:32.200
This is the marginal utility at
his consumption.

1:11:32.198 --> 1:11:34.758
Marginal utility of expectation
at his consumption.

1:11:34.760 --> 1:11:37.750
You face him with any possible
new thing that he could buy,

1:11:37.752 --> 1:11:40.282
just like a consumer in the
first day of class.

1:11:40.279 --> 1:11:42.089
He could buy apples,
he could buy oranges,

1:11:42.085 --> 1:11:43.005
he could buy pears.

1:11:43.010 --> 1:11:45.530
The marginal utility of each
one of those at the point he's

1:11:45.532 --> 1:11:46.752
already chosen to consume.

1:11:46.750 --> 1:11:49.050
This is a number that's fixed,
this is a number that's fixed.

1:11:49.050 --> 1:11:51.460
Whatever X I put in here,
that's what I get.

1:11:51.460 --> 1:11:54.500
So there's a linear tradeoff
between expectation and

1:11:54.496 --> 1:11:55.266
covariance.

1:11:55.270 --> 1:11:57.410
Expectation good,
covariance bad.

1:11:57.408 --> 1:12:02.728
So the final picture is that in
equilibrium, what you should

1:12:02.734 --> 1:12:08.424
have is every stock--this is
called the security market line.

1:12:08.420 --> 1:12:09.790
This is the expectation of X.

1:12:09.788 --> 1:12:14.588
This is the covariance of X
with M.

1:12:14.590 --> 1:12:17.110
So what we've found is that it
should look like this.

1:12:17.109 --> 1:12:19.099
There has to be a linear
relationship.

1:12:19.100 --> 1:12:23.130
In order to want the stock--if
it's got a higher covariance,

1:12:23.128 --> 1:12:25.518
that means it's adding bad
stuff.

1:12:25.520 --> 1:12:28.950
You wouldn't want it unless it
had a higher expectation.

1:12:28.948 --> 1:12:32.368
So here's the riskless asset
right here.

1:12:32.368 --> 1:12:35.508
So this is a different diagram
than the Tobin diagram.

1:12:35.510 --> 1:12:37.690
It's expectation and covariance.

1:12:37.689 --> 1:12:42.749
Every stock is along this line.

1:12:42.750 --> 1:12:44.970
Should be priced along this
line.

1:12:44.970 --> 1:12:50.920
So let me end with a puzzle,
the one we started with on the

1:12:50.916 --> 1:12:53.066
first day of class.

1:12:53.069 --> 1:12:57.319
If I have two companies,
General Motors and AIDS--an

1:12:57.322 --> 1:12:58.992
anti AIDS company.

1:12:58.988 --> 1:13:01.218
Some scientist at Yale
discovers a cure for AIDS.

1:13:01.220 --> 1:13:02.670
He calls it AIDS.

1:13:02.670 --> 1:13:05.420
He discovers a cure for AIDS.

1:13:05.420 --> 1:13:08.170
If the thing works,
he'll make a fortune.

1:13:08.170 --> 1:13:10.670
If the thing doesn't work,
of course, it's going to go

1:13:10.668 --> 1:13:12.318
bankrupt and pay anyone no
money.

1:13:12.319 --> 1:13:15.519
Let's say we calculate the
expected profits of General

1:13:15.515 --> 1:13:16.175
Electric.

1:13:16.180 --> 1:13:17.620
General Motors is too junky now.

1:13:17.619 --> 1:13:18.939
General Electric.

1:13:18.939 --> 1:13:22.069
Let's say the expected profits
of General Electric = the

1:13:22.067 --> 1:13:24.567
expected profits of the
anti-AIDS company.

1:13:24.569 --> 1:13:33.929
Which will sell for a higher
price?

1:13:33.930 --> 1:13:35.780
Student:
General Electric.

1:13:35.779 --> 1:13:36.879
Prof: Will sell for a
higher price.

1:13:36.880 --> 1:13:37.910
Why?

1:13:37.908 --> 1:13:39.308
Student:
>

1:13:39.311 --> 1:13:40.471
the variability of the outcome.

1:13:40.470 --> 1:13:42.450
Prof: Okay,
that's what you should have

1:13:42.445 --> 1:13:45.165
said at the beginning of class,
but not at the end of class.

1:13:45.170 --> 1:13:45.720
Who's saying that?

1:13:45.720 --> 1:13:47.890
Down here.

1:13:47.890 --> 1:13:49.090
Who said that?

1:13:49.090 --> 1:13:50.260
Who just said that?

1:13:50.260 --> 1:13:51.500
I don't want to embarrass
anyone.

1:13:51.500 --> 1:13:52.720
All right, you're anonymous.

1:13:52.720 --> 1:13:54.940
I didn't see who it was,
so exactly.

1:13:54.939 --> 1:13:58.679
That's what you would have
thought before this class began.

1:13:58.680 --> 1:14:00.170
Everybody hates variance.

1:14:00.170 --> 1:14:03.980
The AIDS company is so risky,
you could get a fortune or 0.

1:14:03.979 --> 1:14:05.539
You couldn't get riskier than
that.

1:14:05.538 --> 1:14:09.088
General Electric is not going
to go bankrupt very easily and

1:14:09.088 --> 1:14:12.518
it's not going to suddenly
multiply its value by 100 times

1:14:12.518 --> 1:14:13.178
either.

1:14:13.180 --> 1:14:14.530
It's much more solid.

1:14:14.529 --> 1:14:16.219
So which would you pay more for?

1:14:16.220 --> 1:14:17.910
Everyone at the beginning of
the class,

1:14:17.908 --> 1:14:19.488
I think, would have said like
he did,

1:14:19.488 --> 1:14:21.848
whoever he is,
would have said "General

1:14:21.850 --> 1:14:23.760
Electric,
you'd pay more for."

1:14:23.760 --> 1:14:25.230
But the answer's not General
Electric.

1:14:25.229 --> 1:14:27.089
Why is that?

1:14:27.090 --> 1:14:28.310
Yes?

1:14:28.310 --> 1:14:31.070
Student:
Because the cure for AIDS

1:14:31.067 --> 1:14:35.337
actually working is uncorrelated
with the state of the economy,

1:14:35.340 --> 1:14:37.340
>

1:14:37.340 --> 1:14:37.930
Prof: Exactly.

1:14:37.930 --> 1:14:39.950
That's the shocking fact.

1:14:39.949 --> 1:14:41.709
That's the shocking conclusion.

1:14:41.710 --> 1:14:43.760
So Shakespeare,
yes, "Nothing ventured,

1:14:43.755 --> 1:14:44.845
nothing gained."

1:14:44.850 --> 1:14:47.780
You have to take a risk to
expect a higher return.

1:14:47.779 --> 1:14:53.209
Everyone would have thought
that's the anti-AIDS company.

1:14:53.210 --> 1:14:55.260
No it's not,
and Shakespeare couldn't

1:14:55.260 --> 1:14:57.140
possibly have figured this out.

1:14:57.140 --> 1:15:00.660
It's General Electric is the
one with the higher risk,

1:15:00.659 --> 1:15:03.449
because it's correlated with
the market.

1:15:03.449 --> 1:15:05.039
It's correlated with the market.

1:15:05.038 --> 1:15:07.468
Obviously if people are rich,
they're going to buy more

1:15:07.470 --> 1:15:09.630
refrigerators and engines and
stuff like that.

1:15:09.630 --> 1:15:12.160
If business isn't doing very
well, they're not going to buy

1:15:12.162 --> 1:15:12.732
that stuff.

1:15:12.729 --> 1:15:14.599
It's very correlated with the
market.

1:15:14.600 --> 1:15:18.390
Anti-AIDS, if it cures AIDS,
people are going to buy it.

1:15:18.390 --> 1:15:20.420
If they've got AIDS,
they're going to buy it no

1:15:20.421 --> 1:15:23.031
matter what, otherwise they
won't, so it's uncorrelated with

1:15:23.027 --> 1:15:23.687
the market.

1:15:23.689 --> 1:15:27.439
So therefore the price of the
AIDS company is going to be its

1:15:27.443 --> 1:15:29.323
expected payoff,
discounted.

1:15:29.319 --> 1:15:31.079
The price of General Motors
[correction: General Electric]

1:15:31.076 --> 1:15:32.976
is going to be much less,
because it's going to be

1:15:32.979 --> 1:15:35.189
punished for having a
correlation with the market.

1:15:35.189 --> 1:15:38.169
Therefore the return on General
Electric is going to be much

1:15:38.173 --> 1:15:40.003
higher,
because the same expected

1:15:39.997 --> 1:15:42.877
payoff and a lower price,
your return's going to be

1:15:42.877 --> 1:15:44.507
higher in General Electric.

1:15:44.510 --> 1:15:47.360
That's the shocking thing that
Shakespeare couldn't have

1:15:47.360 --> 1:15:47.880
noticed.

1:15:47.880 --> 1:15:50.320
So yes, we're just doing what
Shakespeare said in the

1:15:50.319 --> 1:15:52.809
beginning, but in a way he
couldn't possibly have done

1:15:52.805 --> 1:15:54.115
without any mathematics.

1:15:54.118 --> 1:16:00.558
Okay, now the problem set,
you'll have to see if you can

1:16:00.560 --> 1:16:03.020
do the problem set.

1:16:03.020 --> 1:16:04.500
It's due on Tuesday.

1:16:04.500 --> 1:16:10.000