WEBVTT
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Prof: We've dealt so far
with the case of certainty,
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and we've done almost as much
as we could in certainty,
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and I now want to move to the
case of uncertainty,
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which is really where things
get much more interesting and
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things can go wrong.
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So I'm going to cover this.
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So we're ready to start.
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So, so far we've considered is,
the case of certainty.
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So with uncertainty things get
much more interesting,
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and I want to remind you of a
few of the basics of
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mathematical statistics that I'm
sure you know.
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So you know we deal with random
variables which have uncertain
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outcomes, but with well-defined
probabilities.
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So another step that we're not
going to take in this course is
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to say people just have no idea
what the chances are something's
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going to happen.
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Shiller thinks we live in a
world like that where who knows
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what the future's going to be
like and people,
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they hear a story and then
everybody gets wildly
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optimistic,
and then they hear some
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terrible story and then
everybody gets wildly
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pessimistic,
and that kind of mood swing can
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affect the whole economy.
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I'm not going to deal with that.
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It's hard to quantify and I'm
not exactly sure it's as
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important as he thinks it is.
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So we're going to deal with the
case where many things can
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happen,
but you know what the chances
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are that they could happen,
and still lots of things can go
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wrong in that case.
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So there are a couple of words
that I want you to know,
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which we went over last time,
and I'll just do an example.
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We always deal with states of
the world, states of nature.
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That was Leibniz's idea.
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So let's take the simplest case
where with probability 1 half
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you could get 1,
and with probability 1 half you
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could get minus 1.
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So that's a random variable.
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It might be how your investment
does.
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Half the time you're going to
make a dollar.
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Half the time,
you're going lose a dollar.
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So this is X,
so we define the expectation of
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X,
which I write as X bar,
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as the probability of the up
state happening,
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so let's just call that 1 half
times 1,
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1 half times minus 1 which
equals 0.
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Then I define the variance of X
to be, what's the expectation of
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the squared difference from the
expectation?
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So how uncertain it is.
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You're sort of on average
expecting to get 0,
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so uncertain it is,
is measured how far from 0 you
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are, but we're going to square
it.
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So it's 1 half times (1 - X
bar) squared 1 half times (minus
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1 - X bar) squared = 1 half
times 1 1 half times 1 which
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also equals 1.
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So the variance is 1.
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And then I'll write the
standard deviation of X equals
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the square root of the variance
of X, which equals the square
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root of 1 which is also 1.
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So very often we're going to
use the expectation of X,
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that's going to be how good the
thing is,
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and the standard deviation is
going to be how uncertain it is,
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and people aren't going to
like--soon we're going to
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introduce the idea that people
don't like uncertainty and this
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is the measure of what they do
like.
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It pays off on average a big
number, say, this one doesn't
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but it could,
and the measure of uncertainty
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is the standard deviation.
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I choose that rather than the
variance for a reason you'll
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see.
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It makes all the graphs
prettier, but also if you double
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X you'll double the expectation,
obviously, because you just
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double everything inside here.
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The variance,
though, you're going to end up
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squaring the two.
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If you double X you'll double
all these outcomes and the mean,
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so you'll end up multiplying
the variance by 4,
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whereas you'll multiply the
standard deviation by 2.
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So re-scaling just re-scales
these two numbers and has a
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funny effect on that number.
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So that's the reason why we use
these two.
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Now, you could take another
example, by the way,
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which is .9 times 3
[correction: .9 times 1 third];
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let's call this Y,
and .1 times minus something.
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How about let's call this 1
third and this minus 3.
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Now, what's the expectation of
Y?
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The expectation of Y equals .3,
right,
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equals--just write it out,
it's .9 times 1 third .1 times
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minus 3 which equals .3 - .3
which equals 0,
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so the expectation of this
random variable is the same as
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the expectation of that random
variable.
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And now the variance of this,
of Y,
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is .9 times (1 third - 0)
squared .1 times (minus 3 - 0)
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squared,
which equals .9 times 1 ninth,
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right,
.1 times 9 which equals .1 .9
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which equals 1,
which is the same as the other
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one.
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So here we've got another
random variable which looks
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quite different from this,
so clearly standard deviation
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and expectation don't
characterize things.
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This looks quite different from
that one, has the same standard
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deviation and the same
expectation.
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So we're going to come back
what the difference is between
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these two variables in a second.
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So there's another thing I want
to introduce which is the
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covariance of X and Y.
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So we could look at the
outcomes of these variables.
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Where am I going to write this?
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I'll write it over here.
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We could look at the outcome of
these variables in a picture
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like this, and so here we have X
and here we have Y.
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So X could turn out to be 1
when Y is 1 third,
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and X could turn out to be 1
when Y is minus 3.
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So here's an outcome,
and here's an outcome,
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and X could be minus 1,
and we could get 1 third or
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minus 3.
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So there are four outcomes
looked at here.
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So if you looked at X alone
it's got a 50/50 chance you're
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here or here.
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If you look at Y alone it's a
90 percent chance up there and a
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10 percent chance down there.
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So those are called the
marginal distributions,
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but the joint distribution we
would have to add a number.
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So if you looked at X alone,
by the way,
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you would say X alone you would
say here's 0,
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here's 1, here's minus 1,
so you could have this or this
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with probability 1 half and 1
half and Y you could have--
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so we'll draw it this way.
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With Y you could have 1 third
or minus 3 and here the
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probability is going to be .9
and .1.
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This is 0.
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Those are the pictures that we
started with.
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So you know where X could end
up and where Y could end up,
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well, you don't know where they
jointly could end up.
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So if they end up on the long
diagonal that means when X is
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high Y tends to be high and vice
versa, and if you end up down
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here X is low and Y is low.
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So to the extent that the
probability is on the long
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diagonal they're correlated
together.
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To the extent that the
probability is on the off
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diagonal they're negatively
correlated.
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So anyway, to get a sense of
that,
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the covariance is going to be
the probability of (1,
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1 third) times (1 - X bar)
times (1 third - Y bar) the
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probability of--
I'll just go around the circle
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of (minus 1,
1 third) times (minus 1 - X
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bar) times (1 third - Y bar) the
probability of (minus 1 and 1
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third),
sorry what did I just do?
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I did minus 1 and 1 third.
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I've already done that,
so I'm down here.
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So (minus 1 and minus 3) times
(minus 1 - X bar) times (minus 3
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- X bar [correction:
Y bar]) probability of the
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ordered pair--
Student: Should that
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minus be the X bar or Y bar?
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Prof: Thank you.
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And probability,
what's the point I haven't done
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yet, (1, minus 3) times (1 - X
bar) times (minus 3 - Y bar).
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So why does that covariance
pick up the idea of correlation?
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Well, to the extent that the
probabilities are high here and
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over there on the long diagonal
this term is going to get a lot
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of weight,
and what is the other term,
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(minus 1,
minus 3), and this term is
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going to get a lot of weight.
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So to the extent that you're on
the long diagonal this term and
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this term are going to get a lot
of weight,
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but you see those terms this is
going to be positive because
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it's 1 - 0 and 1 third - 0,
so that's a positive term.
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And this is negative,
minus 1 - 0,
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minus 3 - 0,
so a negative times a negative
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is also positive.
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To the extent that you're down
here and up there you're going
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to get big positive numbers in
the covariance.
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To the extent you're on the off
diagonal you'll get big
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probabilities here,
but they all multiply negative
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terms.
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This is a minus and this is a
minus, because one of terms is
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above the mean and the other one
is below the mean.
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That's what it means to be in
the off diagonal.
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So covariance is giving you a
sense of whether things are
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moving together or moving the
opposite way.
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So those are the basic things
you have to know.
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And I guess another couple
things are,
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the covariance is linear in X,
right,
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because if you double X every
time you see the X variable over
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here it's always an X outcome
minus an X bar,
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an X outcome minus an X bar,
an X outcome minus an X bar,
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an X outcome minus an X bar,
so if you double X you're going
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to double every term.
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So it's linear in X and in Y,
and so one last thing to keep
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in mind is that the variance of
X is just the covariance of X
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with itself.
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Obviously if you just plug in X
equal to Y you just get the
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formula for covariance
[correction: for variance],
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and similarly because they're
linear the covariance of X Y--
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so the variance of X Y,
one more formula,
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of X Y by linearity--first of
all that's the covariance of X Y
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with itself,
and therefore by linearity now,
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I'm just going to do linear
stuff,
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that's equal to the covariance
of X with X the covariance of Y
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with Y 2 times the covariance of
X with Y.
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Since it's linear I just do the
linear parts,
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right?
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Covariance of X Y with X Y is
covariance of X Y with X
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covariance of X Y with Y,
then I repeat the linearity
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thing and I get down to that.
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So those are basically the key
formulas to know.
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So now I'm going to make three
little observations that come
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out of all of this that are
quite fascinating,
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so quite elementary.
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Are there any questions about
this, these numbers?
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Yes?
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Student: I don't
understand why you gave the
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probability of (negative 1,
negative 3) weight when
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negative 3 has a much more
probability of being hit on that
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1 third.
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Prof: Why did we give?
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Say that again.
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Student: Why did you
underline the probably of
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negative 1, negative 3.
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Prof: Probably of
negative 1, negative 3.
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That's this outcome here.
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We underlined it not because it
was very likely,
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but because this term is going
to be positive.
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This is positive and this is
positive.
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So the whole point is the joint
distribution is not specified,
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not determined by the
distributions of X alone and Y
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alone.
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So even if I know the
probability of what X could do,
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and I know what the
probabilities that Y could do
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that doesn't tell me anything
about what numbers I should put
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on these four outcomes.
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For example,
I could have at one extreme
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when X is high Y is high--it
can't be exactly that because
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the probabilities are different.
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These numbers and those numbers
don't determine these four
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numbers.
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So there are many different
numbers I could put in these
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four squares which would give me
in total this probability
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outcome for X and in total this
probability outcome for Y.
14:22.590 --> 14:26.920
So an easy way to see that is
if I made them.
14:26.918 --> 14:39.028
So what are the observations I
want to make?
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For instance,
I could say if X turns out to
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be 1 half then I'll always
assume Y turns out to be 1 half,
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and then with the other 40
percent of the time Y might turn
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out to be--
when Y's high X might have to
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turn out--
so here are some ways I could
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do this.
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I could put 50 percent here,
.5 here right?
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Then 40 percent of the time
this is going to turn out--so I
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have a .5 here,
then what could I do with the
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rest of this?
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This plus this has to add up to
50 percent.
15:21.950 --> 15:32.340
So 50 percent I could have X
turn out to be here.
15:32.340 --> 15:36.300
So when X is 1 I could have Y
always turn out to be 1,
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so that means I must have a
probability here,
15:39.589 --> 15:43.549
a probability 0 here because
here's X 50 percent.
15:43.548 --> 15:46.488
So this plus this X is going to
turn out to be 1,50 percent of
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the time.
15:47.019 --> 15:51.489
Now, how much of the time is Y
going to turn out to be down
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here a .1?
15:52.418 --> 15:54.978
So suppose I put these
probabilities,
15:54.975 --> 15:55.325
.4?
15:55.330 --> 16:00.060
Now, so you see that X is--50
percent of the time X is 1,
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and 50 percent of the time X is
minus 1.
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Now, how many of the times is Y
1 third, .5 .4,
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so 90 percent of the time,
and then 10 percent of the time
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Y is minus 3.
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So here's one way of putting
probabilities on the dots that
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produces this outcome,
but I could have chosen another
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way of doing it,
the way that you probably had
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in mind where I assume they're
totally independent.
16:26.860 --> 16:30.240
That is, knowing the outcome of
X in this way of doing it,
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if I know that X turned out to
be 1, Y has to turn out to be a
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third.
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So they're very dependent.
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X is somehow causing Y or
determining Y.
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X has a lot of information
about Y.
16:40.269 --> 16:41.589
Suppose I make them independent?
16:41.590 --> 16:43.740
I say what happens here has
nothing to with what happens
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over there.
16:44.250 --> 16:49.770
Then I write the probabilities,
instead of these,
16:49.765 --> 16:51.945
I'd write it .45.
16:51.950 --> 16:57.930
I'd take 1 half times .9 is
.45, and then the chance that
16:57.928 --> 17:05.828
you go down for X,
which is .5 and up for Y which
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is also .45 here,
then I'd go .05 here and .05
17:14.228 --> 17:15.428
there.
17:15.430 --> 17:19.230
So here, knowing that Y has a
good outcome tells you nothing
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about what X is going to do.
17:21.170 --> 17:23.470
It's still equally likely X was
good or bad.
17:23.470 --> 17:27.850
Knowing that Y had a bad
outcome, X is still likely to be
17:27.851 --> 17:30.121
equally likely good or bad.
17:30.118 --> 17:33.408
And similarly knowing the
outcome of X tells you nothing
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about the outcome of Y.
17:34.910 --> 17:37.720
This is 9 times this and this
is 9 times that.
17:37.720 --> 17:44.980
So the yellow is independence,
which is probability ((X equals
17:44.976 --> 17:48.336
x),
and (Y equals y)),
17:48.343 --> 17:53.603
equals the product,
Probability (X = x) times
17:53.603 --> 17:55.423
probability (Y = y).
17:55.420 --> 17:57.390
So that's the case in
independence.
17:57.390 --> 17:59.200
So in the case of independence,
knowing something about one
17:59.204 --> 18:01.024
variable tells you nothing about
what happened to the other
18:01.020 --> 18:03.120
variable,
but you could do other joint
18:03.124 --> 18:03.604
things.
18:03.598 --> 18:07.148
So knowing each of them
separately doesn't tell you how
18:07.151 --> 18:10.881
they're jointly distributed,
and the covariance is an effort
18:10.884 --> 18:14.164
to see whether they're sort of
correlated together or whether
18:14.162 --> 18:16.132
they're correlated
independently.
18:16.130 --> 18:21.080
So independence,
by the way, independence
18:21.077 --> 18:24.787
implies covariance equals 0.
18:24.788 --> 18:27.998
That's obvious because what's
happening in the X variable's
18:27.999 --> 18:31.319
got nothing to do with what's
happening in the Y variable.
18:31.318 --> 18:34.388
So since it's linear in X you
can hold Y fixed,
18:34.394 --> 18:38.344
and the X is just the same and
you're going to get something
18:38.338 --> 18:39.808
that adds up to 0.
18:39.808 --> 18:42.648
So for any fixed value of Y
this number will just give you
18:42.654 --> 18:45.054
the expectation of X,
which won't depend on Y and
18:45.051 --> 18:46.901
it's going to be 0 in every
case.
18:46.900 --> 18:49.670
So therefore if they're
independent their covariance has
18:49.665 --> 18:50.165
to be 0.
18:50.170 --> 18:53.900
So, independence means X and Y
tell you nothing.
18:53.900 --> 18:55.550
That means the covariance is 0.
18:55.548 --> 18:58.368
They could be positively
distributed like up here or
18:58.368 --> 19:01.408
negatively distributed,
either way you want to do it.
19:01.410 --> 19:03.980
Does that make sense?
19:03.980 --> 19:05.260
You asked me about this.
19:05.259 --> 19:07.059
Student: Yes.
19:07.058 --> 19:11.088
Prof: So what are the
key simple observations here
19:11.093 --> 19:14.773
that are going to inform a lot
of our behavior under
19:14.767 --> 19:15.917
uncertainty?
19:15.920 --> 19:23.660
Well, it's going to turn out
that expectation is good and
19:23.660 --> 19:27.530
standard deviation is bad.
19:27.528 --> 19:33.548
So if we take this variable
that we just found,
19:33.548 --> 19:40.088
X and Y were both here,
X and Y were both there.
19:40.088 --> 19:44.178
All right, they each had
standard deviation 1 and
19:44.182 --> 19:47.342
expectation 0,
so this is the standard
19:47.336 --> 19:48.526
deviation.
19:48.529 --> 19:52.399
So X is here,
and by the way so is Y,
19:52.401 --> 19:53.801
same thing.
19:53.798 --> 19:58.138
Well, suppose I put half my
money into X and I put half my
19:58.140 --> 20:01.340
money into Y,
and if I put half my money in
20:01.337 --> 20:05.067
each let's say I get half the
payoff of each.
20:05.068 --> 20:07.578
I make half a bet and get half
the outcome.
20:07.579 --> 20:12.919
What happens to my expectation?
20:12.920 --> 20:19.840
Well, the expectation of that
obviously equals 1 half X bar 1
20:19.842 --> 20:23.652
half Y bar which also equals 0.
20:23.650 --> 20:25.560
So it's staying the same.
20:25.558 --> 20:35.208
The expectation hasn't moved,
but what's the variance of 1
20:35.211 --> 20:38.261
half X 1 half Y?
20:38.259 --> 20:42.689
Well, by that formula it's the
covariance--so I'm just going to
20:42.692 --> 20:43.982
do this formula.
20:43.980 --> 20:49.500
I'm going to a 1 half here and
1 half here.
20:49.500 --> 20:51.910
So it's the same thing.
20:51.910 --> 20:56.560
So it's the covariance of 1
half X with 1 half X the
20:56.558 --> 21:01.298
covariance of 1 half X 1 half Y
1 half and 1 half.
21:01.298 --> 21:07.108
But the covariance of 1 half X
with 1 half X is just,
21:07.105 --> 21:09.445
okay, what is that?
21:09.450 --> 21:12.630
It's the variance of 1 half X,
but we already saw from our
21:12.630 --> 21:14.640
definition of variance over
here,
21:14.640 --> 21:17.500
remember, if you double X
you're going to multiply the
21:17.500 --> 21:20.040
variance by 4 because you're
squaring things.
21:20.038 --> 21:27.578
So this is going to turn out to
be 1 quarter times the variance
21:27.577 --> 21:28.427
of X.
21:28.430 --> 21:33.450
And this, which is 1 half Y and
1 half Y, is going to be 1
21:33.446 --> 21:36.436
quarter times the variance of Y.
21:36.440 --> 21:40.140
And if the two are independent
the covariance will be 0.
21:40.140 --> 21:42.950
So in this example,
these two variables,
21:42.950 --> 21:46.130
if I take the orange
distribution where they're
21:46.128 --> 21:49.928
independent I can do an X
outcome and have this standard
21:49.931 --> 21:53.931
deviation and this expectation,
0 expectation and that standard
21:53.932 --> 21:55.692
deviation,
I can do the Y thing,
21:55.694 --> 21:59.054
get the same standard deviation
or I can put half my money in
21:59.051 --> 21:59.501
each.
21:59.500 --> 22:02.960
It seems like a total waste of
time to put half my money in
22:02.961 --> 22:03.381
each.
22:03.380 --> 22:05.750
After all, they give me the
same standard deviation,
22:05.747 --> 22:06.627
but no, it isn't.
22:06.630 --> 22:10.980
If they're independent you're
shockingly, drastically reducing
22:10.977 --> 22:12.827
your standard deviation.
22:12.828 --> 22:16.558
Because if they're independent
the covariance is 0 and so this
22:16.558 --> 22:20.148
plus this plus,
the variance of X = the
22:20.146 --> 22:26.416
variance of Y is just the half
the variance of X = half the
22:26.421 --> 22:28.261
variance of Y.
22:28.259 --> 22:29.209
So that's shocking.
22:29.210 --> 22:32.830
So the standard deviation,
therefore, the square root of
22:32.827 --> 22:34.997
that is 1 over the square root.
22:35.000 --> 22:38.360
So by putting half your money
in each you've now produced this
22:38.355 --> 22:39.835
when they're independent.
22:39.838 --> 22:47.978
So this is the standard
deviation of 1 half X 1 half Y,
22:47.976 --> 22:51.136
(X, Y) independent.
22:51.140 --> 22:54.050
You move from this point to
that point.
22:54.048 --> 22:56.838
You reduced your standard
deviation without affecting your
22:56.835 --> 22:57.515
expectation.
22:57.519 --> 23:01.889
So the first lesson that we're
going to see applied,
23:01.890 --> 23:05.580
this is all mathematics so
mathematicians understood this,
23:05.578 --> 23:08.368
of course, a long time ago,
but to realize this has an
23:08.374 --> 23:10.804
application to economics wasn't
so obvious,
23:10.799 --> 23:12.619
although Shakespeare knew it.
23:12.619 --> 23:15.939
It's diversification.
23:15.940 --> 23:18.660
So don't put all your,
you know, spread your
23:18.659 --> 23:21.189
investments out into different
waters.
23:21.190 --> 23:23.770
Shakespeare,
you know, Antonio had a
23:23.769 --> 23:27.269
different ship on each ocean,
so instead of putting all the
23:27.269 --> 23:29.629
ships on the same ocean he put
them on different oceans which
23:29.630 --> 23:30.810
he assumed was independent.
23:30.808 --> 23:33.918
So he had the same expected
outcome assuming the paths were
23:33.916 --> 23:36.806
just as quick to wherever he was
selling the stuff,
23:36.808 --> 23:39.268
the same expected outcome and
that each of the waters were
23:39.271 --> 23:42.641
equally dangerous,
but he drastically reduced his
23:42.640 --> 23:43.460
variance.
23:43.460 --> 23:46.650
And because there were a lot of
oceans and a lot of ships this
23:46.650 --> 23:48.690
number went down further and
further.
23:48.690 --> 23:51.390
So the key is to look for
independent risks.
23:51.390 --> 23:54.590
So that's one lesson in
mathematics that has a big
23:54.586 --> 23:56.346
application in economics.
23:56.349 --> 24:01.169
What's a second thing?
24:01.170 --> 24:06.190
Well, the second thing is that
if you add a bunch of risks
24:06.188 --> 24:10.148
together, so I'm going to say
this loosely.
24:10.150 --> 24:14.000
If you add a bunch of risks
together, so by the way,
24:13.996 --> 24:18.066
what's the generalization of
this before I say this?
24:18.068 --> 24:29.038
If you had N independent risks
with identical means and
24:29.041 --> 24:40.421
variances, means let's call them
all X bar and variances,
24:40.422 --> 24:44.082
sigma squared.
24:44.078 --> 24:48.818
Let's say they all have
expectation E and variance sigma
24:48.820 --> 24:51.840
squared,
each of them has that,
24:51.835 --> 24:55.585
then what happens to the--
so each of them has standard
24:55.585 --> 24:57.205
deviations,
so they're all identical.
24:57.210 --> 24:59.690
Like X and Y have the
expectation 0 and the same
24:59.686 --> 25:00.896
standard deviation 1.
25:00.900 --> 25:06.520
Suppose I had 20 of those and I
put 1 twentieth of money into
25:06.522 --> 25:07.932
each of them?
25:07.930 --> 25:14.790
What would happen to my
expectation?
25:14.788 --> 25:22.138
1 over N dollars in each one
implies what happens to my
25:22.144 --> 25:28.004
expectation if expectation equal
to what?
25:28.000 --> 25:30.020
Each of them had expectation E.
25:30.019 --> 25:33.029
I now split my money among all
of them, all with the same
25:33.028 --> 25:33.778
expectation.
25:33.779 --> 25:36.949
That also has to have
expectation E.
25:36.950 --> 25:40.080
All right, just like this thing
putting half my money in Y and
25:40.076 --> 25:42.226
half my money in X,
wherever the X went.
25:42.230 --> 25:44.240
Y was over here.
25:44.240 --> 25:45.180
X is there.
25:45.180 --> 25:49.010
Half my money in X and half my
money in Y, is going to give me
25:49.007 --> 25:50.447
the same expectation.
25:50.450 --> 25:53.730
If I had 12 projects like that
that were independent I'd still
25:53.730 --> 25:58.060
have the same expectation,
but my standard deviation,
25:58.058 --> 26:03.278
what's going to happen to my
standard deviation?
26:03.278 --> 26:13.098
Well, the variance is going to
be--so what's going to happen to
26:13.099 --> 26:17.059
the standard deviation?
26:17.058 --> 26:18.988
Student: It would go
down.
26:18.990 --> 26:24.010
Prof: By what factor?
26:24.009 --> 26:27.179
Yeah, what's going to happen to
the variance?
26:27.180 --> 26:29.230
Student: 1 over...
26:29.230 --> 26:34.300
Prof: Put 1 over N
dollars in each of N identical
26:34.303 --> 26:39.473
but independent investments,
what will my variance be?
26:39.470 --> 26:42.190
Student: <>
26:42.190 --> 26:45.770
Prof: The variance is
going to equal 1 over N times
26:45.766 --> 26:46.766
sigma squared.
26:46.769 --> 26:47.599
Why is that?
26:47.598 --> 26:50.348
Because each one will have 1
over N dollars in it,
26:50.346 --> 26:53.596
so its variance is going to be
1 over N squared times sigma
26:53.596 --> 26:55.666
squared, but there are N of
them.
26:55.670 --> 26:57.980
So it's going to be N over
times 1 over N squared,
26:57.980 --> 27:05.110
so it's just 1 over N,
so implies the standard
27:05.109 --> 27:09.339
deviation--
so I'll call it standard
27:09.343 --> 27:13.393
deviation,
is 1 over the square root of N
27:13.387 --> 27:14.667
times sigma.
27:14.670 --> 27:16.240
So it's just this
generalization.
27:16.240 --> 27:18.600
We've got 1 over the square
root of 2, so if I did N of them
27:18.595 --> 27:20.985
instead of 2 of them I'd have 1
over the square root of N.
27:20.990 --> 27:23.990
So those turn out to be very
useful formulas which are going
27:23.990 --> 27:25.670
to come up over and over again.
27:25.670 --> 27:28.750
And let's just say it again so
you get this straight.
27:28.750 --> 27:31.990
If I have two independent
random variables,
27:31.990 --> 27:34.040
and I split my money evenly
between them,
27:34.038 --> 27:36.148
and they have the same
expectation,
27:36.150 --> 27:38.630
it doesn't have to be 0,
it could be a positive number,
27:38.630 --> 27:41.280
if I split my money between
them I haven't changed my
27:41.282 --> 27:43.922
expectation because each dollar,
however I split it,
27:43.923 --> 27:46.493
I'm putting it into something
with the same expectation.
27:46.490 --> 27:51.740
But because they're independent
you get a lot of off diagonal
27:51.738 --> 27:53.398
things happening.
27:53.400 --> 27:55.520
The off diagonal things,
remember, are canceling.
27:55.519 --> 27:59.079
One investment is turning out
well, X is--sorry that's on the
27:59.077 --> 27:59.727
diagonal.
27:59.730 --> 28:02.880
The off diagonal elements are
good in a way because if one
28:02.875 --> 28:06.075
investment's turning out well,
sorry, turning out badly the
28:06.077 --> 28:07.897
other one's turning out well.
28:07.900 --> 28:11.460
So here investment Y is turning
out badly, but X is turning out
28:11.458 --> 28:11.858
well.
28:11.858 --> 28:15.638
So to the extent you're off the
diagonal you're canceling some
28:15.643 --> 28:19.493
of your bad outcomes because
one's good and the other's bad.
28:19.490 --> 28:23.180
So that way you leave the
expectation the same,
28:23.175 --> 28:25.575
but you reduce the variance.
28:25.578 --> 28:29.108
In fact it would be even better
if you could put everything on
28:29.111 --> 28:31.501
the off diagonal,
but to the extent you get at
28:31.497 --> 28:33.907
least some stuff on the off
diagonal you're reducing the
28:33.909 --> 28:34.259
risk.
28:34.259 --> 28:37.049
And how fast do you reduce it
when they're independent?
28:37.048 --> 28:40.818
You reduce it dividing it
equally because the variance is
28:40.819 --> 28:43.409
a squared thing,
half your money in one and half
28:43.414 --> 28:45.964
in the other means the variance
of the first is 1 quarter and
28:45.961 --> 28:47.831
the variance of the second is 1
quarter,
28:47.828 --> 28:50.238
but now there are two of them
so the total variance is 1 half
28:50.237 --> 28:51.197
of what it was before.
28:51.200 --> 28:54.520
If you have 10 of them each one
is 1 tenth the money so it's got
28:54.522 --> 28:57.522
1 one-hundredth of the variance,
but there are 10 of them so
28:57.517 --> 29:00.307
it's 10 one-hundredths,
1 over N of the variance.
29:00.308 --> 29:02.948
If you take the standard
deviation it's 1 over the square
29:02.952 --> 29:03.522
root of N.
29:03.519 --> 29:10.759
So that's the rate at which you
can reduce your uncertainty and
29:10.758 --> 29:12.158
your risk.
29:12.160 --> 29:15.560
You'll see this gets much more
concrete next lecture.
29:15.558 --> 29:18.788
So this is just stuff that most
of you know.
29:18.788 --> 29:23.178
So one more thing,
if you add a bunch of
29:23.180 --> 29:28.720
independent things together,
independent random variables,
29:28.720 --> 29:31.620
so I'm going to speak very
loosely now,
29:31.618 --> 29:46.788
variables, you get a normally
distributed random variable,
29:46.788 --> 29:58.618
normally distributed random
variable with the corresponding
29:58.622 --> 30:06.172
expectation and standard
deviation.
30:06.170 --> 30:07.680
So what am I saying?
30:07.680 --> 30:10.160
I don't want to speak too
precisely about this because if
30:10.157 --> 30:12.767
you've seen this before and seen
a proof you know everything
30:12.769 --> 30:14.729
about it,
if you haven't it's just too
30:14.732 --> 30:16.032
many subtleties to absorb.
30:16.028 --> 30:20.538
But the normal distributed
random variable's the bell curve
30:20.540 --> 30:22.330
that looks like that.
30:22.329 --> 30:26.979
It looks like this.
30:26.980 --> 30:29.710
So there's the bell curve with
expectation 0.
30:29.710 --> 30:31.250
So it's this bell curve.
30:31.250 --> 30:34.520
Now, what's special about it,
it has a particular formula
30:34.516 --> 30:37.896
which has got an exponential to
a minus X squared thing.
30:37.900 --> 30:40.130
Anyway, it's got a particular
formula to it which if you know
30:40.125 --> 30:41.715
you know, if you don't it's
written down.
30:41.720 --> 30:43.430
We're never going to use the
exact formula,
30:43.431 --> 30:44.451
but it looks like that.
30:44.450 --> 30:48.950
So these are the outcomes X and
this is the probability,
30:48.949 --> 30:53.039
probability of outcome,
or frequency of outcome.
30:53.038 --> 30:57.148
So the bigger X is,
and this is the mean--equals
30:57.151 --> 30:59.951
0--I've assumed the mean is 0.
30:59.950 --> 31:02.980
If you take a really big X it's
very unlikely to happen,
31:02.980 --> 31:04.840
and a really small X it's very
unlikely to happen,
31:04.838 --> 31:07.648
and X's nearer the mean are
pretty likely to happen.
31:07.650 --> 31:11.890
So anyway, it's amazing that if
you add this random variable to
31:11.893 --> 31:15.803
itself a bunch of times it can
only produce 1 and minus 1,
31:15.795 --> 31:16.475
right?
31:16.480 --> 31:19.260
This one produces totally
different outcomes,
31:19.259 --> 31:24.159
1 third and minus 3,
they're disjoint outcomes,
31:24.160 --> 31:29.640
but if you add this together
you can get 25 1s and 10 minus
31:29.642 --> 31:32.582
1s,
so that gives you 15.
31:32.578 --> 31:38.048
Over here you could have--25
will never get me there,
31:38.054 --> 31:41.744
so sorry, that was a bad
example.
31:41.740 --> 31:46.320
If I had 30 things I could get
18 1s and 12 minus 1s,
31:46.318 --> 31:49.568
that'll give me 6,
you could have gotten 6 over
31:49.565 --> 31:52.365
here,
but with 30 outcomes you could
31:52.371 --> 31:54.151
get,
you know, all 30 of them could
31:54.153 --> 31:56.983
have turned out to be 1,
and that would have gotten you
31:56.978 --> 31:58.868
pretty close to the same
outcome.
31:58.868 --> 32:02.758
So just because these outcomes
are separate,
32:02.759 --> 32:05.549
once you're adding them up
you're starting to produce
32:05.546 --> 32:07.686
numbers different from 1 and
minus 1,
32:07.690 --> 32:10.470
and these added up--if you take
the right combination of 1 third
32:10.471 --> 32:12.591
and minus a third--
you can start reproducing
32:12.588 --> 32:12.968
things.
32:12.970 --> 32:16.670
Like to get a 1 here you could
produce three tops and then
32:16.673 --> 32:18.173
you're producing a 1.
32:18.170 --> 32:20.730
So anyway, the shocking thing
is if you add a bunch of these
32:20.729 --> 32:23.069
random variables that are
independent to each other you
32:23.071 --> 32:25.411
get something normally
distributed that looks like that
32:25.414 --> 32:27.894
because this random variable had
exactly the same mean and
32:27.887 --> 32:29.057
standard deviation.
32:29.058 --> 32:31.298
You add the same number of
these you're going to get
32:31.295 --> 32:33.525
outcomes that are almost
identically distributed.
32:33.529 --> 32:36.539
So in the limit this random
variable, enough of these added
32:36.535 --> 32:39.535
together looks exactly the same
as these added together.
32:39.538 --> 32:42.578
That's the second surprising
mathematical fact.
32:42.578 --> 32:46.118
And the third thing that we're
going to use is that the normal
32:46.116 --> 32:50.056
distribution is characterized by
the mean and standard deviation,
32:50.058 --> 32:52.708
that's all it takes to write
the formula of this down,
32:52.710 --> 32:57.830
and these numbers,
these are called thin tailed.
32:57.828 --> 33:02.578
These probabilities go to 0
very fast, so you shouldn't
33:02.578 --> 33:06.888
expect many outlying dramatic
things to happen.
33:06.890 --> 33:10.240
And in the world they do
happen, and so we're going to
33:10.238 --> 33:13.778
see that much of classical
economics is built on normally
33:13.776 --> 33:16.616
distributed things and so you
can't see--
33:16.618 --> 33:19.588
you shouldn't expect any
gigantic outliers to ever
33:19.585 --> 33:20.125
happen.
33:20.130 --> 33:23.230
And it seems natural to build
it on that kind of assumption
33:23.228 --> 33:26.428
because if you add things that
are independent you get normal
33:26.432 --> 33:28.092
distributions all the time.
33:28.088 --> 33:31.168
And things seem independent so
why shouldn't you get normal
33:31.165 --> 33:33.285
distributions,
and yet we must not get it
33:33.287 --> 33:35.247
because we have so many
outliers.
33:35.250 --> 33:38.260
So that's the basic background
of mathematics.
33:38.259 --> 33:40.009
Are there any questions about
any of that?
33:40.009 --> 33:48.209
I'm just assuming you know all
that and now we're going to move
33:48.213 --> 33:50.203
to economics.
33:50.200 --> 33:55.280
I think that's all the
background you need.
33:55.279 --> 34:03.599
I want to do one more thing,
which is maybe background,
34:03.596 --> 34:12.676
but it's used in economics all
the time, and it's called the
34:12.684 --> 34:16.694
iterated expectations.
34:16.690 --> 34:22.020
So if I told you that these
variables were correlated like
34:22.019 --> 34:24.999
these up here,
like the orange things,
34:25.001 --> 34:28.751
if I told you what X turned out
to be that would tell you a lot
34:28.753 --> 34:30.693
about what Y was going to be.
34:30.690 --> 34:33.550
So for example,
if I told you that X
34:33.545 --> 34:37.865
was--sorry, the white ones are
the correlated ones.
34:37.869 --> 34:43.949
If I tell you that X has turned
out to be 1, that tells you that
34:43.949 --> 34:49.929
Y has to be a good outcome of 1
third, because if X is one this
34:49.931 --> 34:51.671
never happens.
34:51.670 --> 34:54.360
So the only thing that can
happen if X is 1 is that Y turns
34:54.356 --> 34:56.716
out to be 1 third,
so knowing X is going to
34:56.722 --> 34:59.972
completely change your mind
about the expectation of Y.
34:59.969 --> 35:04.639
So conditional expectation,
I should have said this before,
35:04.639 --> 35:19.449
conditional expectation simply
means re-computing expectation
35:19.452 --> 35:32.292
using updated probabilities from
your information.
35:32.289 --> 35:34.269
Now, you've probably done this
in high school,
35:34.266 --> 35:36.636
so I'm just going to assume you
know how to do this.
35:36.639 --> 35:40.739
So in this case if I tell you
something like X has turned out
35:40.744 --> 35:45.334
to be 1 that tells you that only
these two outcomes are possible.
35:45.329 --> 35:48.619
So that means that the only two
outcomes in the white case have
35:48.615 --> 35:50.785
happened with probability of .5
and 0,
35:50.789 --> 35:53.769
but if I tell you X has come
out to 1 the conditional
35:53.773 --> 35:55.843
probabilities have to add up to
1.
35:55.840 --> 35:57.270
So you just scale things up.
35:57.268 --> 36:02.678
So you know that Y had to have
been the good outcome up here.
36:02.679 --> 36:08.149
If I tell you that the bad
outcome for Y has happened then
36:08.148 --> 36:14.288
you have probabilities of .1--so
this 0 makes things too easy.
36:14.289 --> 36:21.209
Suppose I tell you the good
outcome of Y has happened.
36:21.210 --> 36:26.030
What are the chances now that X
has gotten the good outcome in
36:26.027 --> 36:28.317
the white probability case?
36:28.320 --> 36:31.530
If I tell you that Y turned out
to be 1 third in the white
36:31.525 --> 36:34.835
probability case what's the
probability that X turned out to
36:34.842 --> 36:36.532
be 1, conditional on that?
36:36.530 --> 36:37.180
Student: 5 ninths.
36:37.179 --> 36:38.899
Prof: 5 ninths,
so that's it,
36:38.898 --> 36:41.848
because the probabilities are
now--you're reduced with .4 and
36:41.847 --> 36:43.367
.5 so 5 ninths of the time.
36:43.369 --> 36:46.669
So that's an idea which I
assume you all can--it's very
36:46.668 --> 36:50.328
intuitive, and it's way too long
to explain, and I'm sure you
36:50.333 --> 36:51.803
know how to do that.
36:51.800 --> 36:54.160
So anyway, the conditional
expectation, blah,
36:54.157 --> 36:56.567
so the iterated expectation is
simply this.
36:56.570 --> 36:58.860
It's an obvious idea,
but it's going to be incredibly
36:58.858 --> 36:59.518
useful to us.
36:59.518 --> 37:03.188
It says if you ask me what are
the chances that the Yankees are
37:03.190 --> 37:06.330
going to win the World Series
against the Dodgers--
37:06.329 --> 37:07.719
let's suppose that's who's
going to play--
37:07.719 --> 37:09.349
the Yankees are going to beat
the Dodgers,
37:09.349 --> 37:11.929
what's the probability that's
going to happen?
37:11.929 --> 37:16.929
What do you expect the chances
are?
37:16.929 --> 37:20.919
If I then ask you my opinion
after the first game,
37:20.920 --> 37:23.070
well, obviously if the Yankees
win the first game my opinion's
37:23.065 --> 37:24.635
going to go up,
so I'm going to have a
37:24.639 --> 37:25.449
different opinion.
37:25.449 --> 37:28.869
If the Dodgers win the first
game my opinion is going to go
37:28.871 --> 37:31.291
down, so I'll have a different
opinion.
37:31.289 --> 37:36.039
But you can ask now another
question, what's your expected
37:36.036 --> 37:37.866
opinion going to be?
37:37.869 --> 37:42.479
So the law of iterated
expectations is,
37:42.483 --> 37:49.403
the expectation of X has to
equal the expected expectation
37:49.404 --> 37:53.294
of X given some information.
37:53.289 --> 37:54.779
So here is what I think.
37:54.780 --> 37:57.370
The Yankees are 70 percent
likely to win.
37:57.369 --> 37:58.719
If I say after the first game
[clarification:
37:58.717 --> 38:00.247
if the Yankees win]
I'll think it's 80 percent,
38:00.250 --> 38:04.660
and after the first game if the
Dodgers win I'll think it's gone
38:04.659 --> 38:07.499
down to 65 percent,
it had better be that the
38:07.498 --> 38:10.718
average of my opinions after the
information is the same as the
38:10.719 --> 38:12.019
number I started with.
38:12.018 --> 38:15.828
That's just common sense and
I'm not going to bother to prove
38:15.833 --> 38:16.283
that.
38:16.280 --> 38:17.660
So that's incredibly important.
38:17.659 --> 38:20.009
It's not only the expectation
of X,
38:20.010 --> 38:22.880
but as you learn stuff you can
anticipate your opinion's going
38:22.878 --> 38:26.778
to change,
but your average opinion has to
38:26.775 --> 38:30.055
always stay the same as X was.
38:30.059 --> 38:34.289
So that's the last of the
background.
38:34.289 --> 38:39.169
And now I want to do a simple
application of this.
38:39.170 --> 38:44.240
So in fact, to that very
question, suppose that you're
38:44.244 --> 38:46.644
playing a World Series.
38:46.639 --> 38:53.899
The Yankees are playing the
Dodgers and let's suppose that
38:53.896 --> 39:01.406
the Yankees have a 60 percent
chance of winning any game.
39:01.409 --> 39:03.709
I'll just do it here.
39:03.710 --> 39:06.600
The Yankees have a 60 percent
chance of winning any game.
39:06.599 --> 39:12.909
What's the chance the Yankees
win a 3 game world series?
39:12.909 --> 39:14.469
How do you figure that out?
39:14.469 --> 39:17.509
Well, a naive way,
a simple way of figuring that
39:17.507 --> 39:20.157
out is to say,
well, what could happen?
39:20.159 --> 39:23.889
Life can mean a Yankee win,
let's call that an up,
39:23.891 --> 39:26.941
or a Yankee loss,
let's call that a down,
39:26.938 --> 39:30.898
and this could happen with
probability .6 or .4.
39:30.900 --> 39:36.730
The Yankees could win again,
so that's probability .6.
39:36.730 --> 39:40.600
We have two Yankee wins,
or the Yankees could lose the
39:40.601 --> 39:43.451
second game so that's
probability .4.
39:43.449 --> 39:45.419
The Yankees could lose or could
win.
39:45.420 --> 39:51.600
That's .6 and this is .4,
and we've only played 2 games.
39:51.599 --> 39:53.169
The Yankees could win a
third--well,
39:53.170 --> 39:55.440
you don't need to play this
game because they've already won
39:55.436 --> 39:59.396
a three game series,
but if you did it wouldn't
39:59.400 --> 40:08.370
matter, .4,
or we could go up or down.
40:08.369 --> 40:11.129
The Yankees after winning and
losing could then win
40:11.134 --> 40:13.374
probability .6,
or could lose,
40:13.373 --> 40:18.263
or after losing and winning
they could win again or they
40:18.255 --> 40:19.495
could lose.
40:19.500 --> 40:22.470
After losing and winning they
could lose, so this is
40:22.465 --> 40:25.895
probability .4 and this is .6,
and then finally we have this
40:25.898 --> 40:27.118
and we have this.
40:27.119 --> 40:28.609
So this is .6 and .4.
40:28.610 --> 40:30.380
So this is what the tree looks
like.
40:30.380 --> 40:34.640
You could imagine 8 possible
paths each of length 3 where you
40:34.637 --> 40:37.827
give the whole sequence of wins
and losses.
40:37.829 --> 40:41.739
So to compute the probability
that the Yankees win you look at
40:41.735 --> 40:44.485
all the--so in this case the
Yankees win.
40:44.489 --> 40:46.369
They would have already won
here, but if you play it out it
40:46.369 --> 40:46.919
doesn't matter.
40:46.920 --> 40:48.480
They're going to win here and
here.
40:48.480 --> 40:50.700
They've got two wins and one
loss.
40:50.699 --> 40:53.299
Here they've got one win,
two wins and one loss.
40:53.300 --> 40:53.890
They win.
40:53.889 --> 40:56.759
Here they've got loss, win, win.
40:56.760 --> 40:57.870
They win the World Series.
40:57.869 --> 40:59.629
Here they lose, win, lose.
40:59.630 --> 41:01.010
They lose the World Series.
41:01.010 --> 41:02.920
Here's lose,
win--it's win,
41:02.922 --> 41:06.312
lose, lose, they also lose the
World Series.
41:06.309 --> 41:08.569
Here it's lose,
lose, they've lost the World
41:08.570 --> 41:09.360
Series, loss.
41:09.360 --> 41:11.580
So these are the possible
outcomes.
41:11.579 --> 41:15.039
So you could compute the
probability of every path,
41:15.039 --> 41:17.139
there are 8 of them,
and then multiply that
41:17.139 --> 41:20.139
probability by the outcome and
you'll get the chance that the
41:20.139 --> 41:22.089
Yankees will win the World
Series,
41:22.090 --> 41:23.550
right?
41:23.550 --> 41:24.700
That's clear to everybody?
41:24.699 --> 41:29.149
But there's a much faster way
of doing it and putting it on a
41:29.153 --> 41:33.983
computer, and that's using the
law of the iterated expectation.
41:33.980 --> 41:39.210
So first of all--so this is
called a tree.
41:39.210 --> 41:41.050
So we're going to use trees all
the time.
41:41.050 --> 41:43.410
So tree, I don't want to
formally define it.
41:43.409 --> 41:46.869
It's just you start with
something and stuff can happen.
41:46.869 --> 41:49.529
Stuff happens every period,
and so you just write down all
41:49.527 --> 41:50.877
the things that can happen.
41:50.880 --> 41:53.130
And then you write down all the
things that can happen after
41:53.126 --> 41:54.686
that and the thing unfolds like
a tree.
41:54.690 --> 41:57.160
That's formal enough to
describe a tree and here we've
41:57.159 --> 41:57.579
got it.
41:57.579 --> 42:00.399
But you notice that the tree
the number of things happening
42:00.400 --> 42:01.470
grows exponentially.
42:01.469 --> 42:04.639
It's horrible to have to
compute something growing
42:04.635 --> 42:07.405
exponentially,
but they're often recombining
42:07.413 --> 42:08.063
trees.
42:08.059 --> 42:10.369
Oh, so if I ask,
by the way, in this tree
42:10.371 --> 42:15.101
whatever the opinion is here,
which turns out to be .68
42:15.097 --> 42:17.437
something,
yeah, I should have asked you
42:17.436 --> 42:19.076
to guess,
.68 something.
42:19.079 --> 42:22.129
If you write down the opinion
that opinion has to be the
42:22.130 --> 42:24.960
average of the opinion here and
the opinion here.
42:24.960 --> 42:29.730
So if I take the opinion here
times .6 plus the opinion here
42:29.726 --> 42:33.116
times .4 that's also going to
equal .68.
42:33.119 --> 42:38.569
And that's what's going to be
the key to computing the thing
42:38.570 --> 42:43.930
much faster rather than going
through every branch which is
42:43.927 --> 42:49.747
such a pain because there are an
exponentially growing number of
42:49.748 --> 42:52.578
paths,
very bad to have to compute by
42:52.583 --> 42:52.913
hand.
42:52.909 --> 42:59.629
But we notice that we can look
at a recombining tree.
42:59.630 --> 43:03.090
These two nodes are essentially
the same.
43:03.090 --> 43:07.000
What difference does it make if
the Yankees win one and lose
43:06.996 --> 43:09.046
one, or lose one and win one?
43:09.050 --> 43:13.320
In both cases they're at the
same spot.
43:13.320 --> 43:14.870
They're even in the World
Series.
43:14.869 --> 43:17.549
And since we assume the
probability of winning any game
43:17.554 --> 43:19.534
is the same,
.6 and .4, independent of
43:19.530 --> 43:22.030
what's happened before--
you might think you're learning
43:22.025 --> 43:24.145
something about,
"Oh, their starter pitched
43:24.150 --> 43:26.130
here and he didn't last the
whole game,"
43:26.126 --> 43:27.156
and stuff like that.
43:27.159 --> 43:28.469
So I'm not allowing for any of
that.
43:28.469 --> 43:31.009
I'm just saying it's a (.6,
.4) chance for the Yankees to
43:31.005 --> 43:32.315
win no matter what happens.
43:32.320 --> 43:37.240
So all you care about at any
point from then on is who's won
43:37.240 --> 43:38.660
how many games.
43:38.659 --> 43:41.359
So these nodes are basically
identical,
43:41.360 --> 43:44.520
and these nodes are identical,
because it all ended up with
43:44.523 --> 43:47.303
the Dodgers ahead 2 to 1,
and here the Yankees were ahead
43:47.297 --> 43:48.767
2 to 1,
and here the Yankees were ahead
43:48.768 --> 43:49.918
3 to 0,
and 0 to 3.
43:49.920 --> 44:02.010
So the recombining tree which
has all the same information is
44:02.014 --> 44:09.074
just this, this,
this, this tree.
44:09.070 --> 44:11.110
So this three only has 1,2,
3,4,
44:11.110 --> 44:15.850
5, has far--it's 1 node,
2 nodes, 3 nodes and 4 nodes as
44:15.853 --> 44:20.513
time goes by growing linearly
instead of growing 1,
44:20.510 --> 44:24.010
to 2, to 4, to 8 which is
growing exponentially.
44:24.010 --> 44:30.820
So I could have a very long
World Series and write it as a
44:30.824 --> 44:37.164
finite tree and just .6 and .4
here at every stage.
44:37.159 --> 44:39.639
So how am I going to solve this
now?
44:39.639 --> 44:42.569
Well, over here I know the
Yankees ended up winning all 3
44:42.570 --> 44:42.990
games.
44:42.989 --> 44:44.699
Here they won 2,
here they won 1,
44:44.697 --> 44:45.817
here they won none.
44:45.820 --> 44:47.030
So those are the outcomes.
44:47.030 --> 44:50.700
So instead of trying to figure
out path by path,
44:50.699 --> 44:52.929
through these exponential
number of paths what the chances
44:52.927 --> 44:55.717
of each path are,
why it's hard to compute here,
44:55.722 --> 44:59.522
it's .6 times .6 times .4,
a complicated calculation,
44:59.521 --> 45:02.331
I'm now going to do something
simple.
45:02.329 --> 45:04.409
I'm going to say,
what would I think if the
45:04.407 --> 45:06.087
Yankees had already won 2 games?
45:06.090 --> 45:08.960
Well, I know that they would
win.
45:08.960 --> 45:09.560
That's a 1.
45:09.559 --> 45:10.879
The series is already over.
45:10.880 --> 45:13.960
What would I think after the
Dodgers won the first two games?
45:13.960 --> 45:15.430
I'd know it was over.
45:15.429 --> 45:18.369
What would I think--so how did
I get that?
45:18.369 --> 45:21.909
It's .6 times 1 .4 times 1.
45:21.909 --> 45:26.529
That's 1, the Dodgers .6 times
0 .4 times 0 that's 0,
45:26.532 --> 45:30.982
so that's my opinion if the
Dodgers win 2 games.
45:30.980 --> 45:33.380
Here's my opinion if the
Yankees won 2 games.
45:33.380 --> 45:36.930
What would my opinion be if
they split?
45:36.929 --> 45:41.409
Well, if they split what would
my opinion be if I started here?
45:41.409 --> 45:44.989
So after game 2 they've each
won 1 game.
45:44.989 --> 45:47.459
I don't know who won the first
one, but it was 1 to 1 after 2
45:47.460 --> 45:47.790
games.
45:47.789 --> 45:49.429
Now what would I think?
45:49.429 --> 45:51.439
Student: .6 times 1 .4
times 0.
45:51.440 --> 45:53.160
Prof: Exactly,
so it's .6.
45:53.159 --> 45:55.459
It's .6 times 1 .4 times 0.
45:55.460 --> 45:56.760
So the odds,
I would think,
45:56.755 --> 45:59.695
the Yankees would win the World
Series here with 1 game left
45:59.697 --> 46:02.537
knowing that they win 60 percent
of the time it's .6.
46:02.539 --> 46:06.319
But now what do I think if the
Yankees win the first game?
46:06.320 --> 46:09.050
What's my opinion?
46:09.050 --> 46:11.310
Student: .6 times 1 .4
times .6.
46:11.309 --> 46:12.879
Prof: So it's .6 times
1,
46:12.880 --> 46:17.910
so it's .6 .4 times .6,
so that's .24,
46:17.909 --> 46:21.529
so that's .84 here,
and what's my opinion after the
46:21.530 --> 46:25.150
Yankees lose the first game and
the Dodgers win?
46:25.150 --> 46:32.290
What do I think is going to
happen?
46:32.289 --> 46:35.839
What will my opinion be here?
46:35.840 --> 46:42.400
It's .6 times having an opinion
of .6, so it's .36 .4 times
46:42.402 --> 46:46.932
knowing that it's all over .4
times 0.
46:46.929 --> 46:52.089
So it's equal to .36.
46:52.090 --> 46:55.470
So I've now figured out--not
only am I solving this thing
46:55.465 --> 46:58.535
much faster than I could over
there, but I'm finding
46:58.539 --> 47:00.649
interesting numbers on the way.
47:00.650 --> 47:03.170
I'm now figuring out what would
I think after the Yankees won
47:03.173 --> 47:03.893
the first game?
47:03.889 --> 47:05.579
Well, now I think it's 84
percent.
47:05.579 --> 47:07.909
What would I think after the
Dodgers won the first game?
47:07.909 --> 47:10.859
I'd think it was only a 36
percent chance of the Yankees
47:10.862 --> 47:11.402
winning.
47:11.400 --> 47:17.130
So now what's my opinion at the
very beginning?
47:17.130 --> 47:24.170
It's .6 times .84 (it's my
chance of having this opinion
47:24.173 --> 47:31.093
plus my chance of having that
opinion) .4 times .36.
47:31.090 --> 47:43.230
Oh no, 504 (maybe) 144 what is
that?
47:43.230 --> 47:44.650
Student: .648.
47:44.650 --> 47:49.830
Prof: .648,6 times 84
looks like 504 and 4 times 36
47:49.826 --> 47:53.456
looks like 144,
so it looks like .648 and
47:53.458 --> 47:55.728
that's what you said.
47:55.730 --> 47:56.470
So that's it.
47:56.469 --> 47:59.459
I've solved it now.
47:59.460 --> 48:02.760
So that's the method of
iterated expectation and we're
48:02.759 --> 48:06.929
going to turn this into quite an
interesting theory in a second,
48:06.929 --> 48:10.759
but I want to now put that on a
computer to show you just how
48:10.755 --> 48:14.155
completely obvious this is,
I mean, not obvious,
48:14.164 --> 48:15.214
fast this is.
48:15.210 --> 48:20.990
So you could solve for any
number of--a series of any
48:20.994 --> 48:24.894
length you could instantly
solve.
48:24.889 --> 48:27.349
Now, we're going to price bonds
that way too.
48:27.349 --> 48:40.509
So class--so what did I do?
48:40.510 --> 48:43.360
I--this is a spreadsheet you
had.
48:43.360 --> 48:47.300
I simply had the probabilities
of the Yankees winning which was
48:47.302 --> 48:49.022
.6, which I could change.
48:49.018 --> 48:50.198
Student: Can you lower
the screen?
48:50.199 --> 48:52.339
Prof: Oh.
48:52.340 --> 48:53.010
Student: Thank you.
48:53.010 --> 49:17.160
49:17.159 --> 49:19.759
Prof: So this is the
simplest thing to do,
49:19.760 --> 49:23.840
but now suppose that--so we
said the Yankees can win every
49:23.844 --> 49:25.784
game with probability .6.
49:25.780 --> 49:27.320
So then what did I do?
49:27.320 --> 49:29.240
I went down to here.
49:29.239 --> 49:30.679
I gave myself some room.
49:30.679 --> 49:32.889
I didn't do a very long series.
49:32.889 --> 49:35.819
So now what does each of these
things say?
49:35.820 --> 49:37.800
Each of these nodes,
like that one,
49:37.797 --> 49:40.877
says, if I can read it,
it says--so this is my opinion
49:40.878 --> 49:42.738
of winning the World Series.
49:42.739 --> 49:46.549
It says my opinion here is
going be the chance I go up.
49:46.550 --> 49:49.830
That's the probability,
that's A 2,
49:49.829 --> 49:53.429
that's .6, the chance I go up
times what my opinion would be
49:53.427 --> 49:56.797
over here,
plus the chance that I go down,
49:56.797 --> 49:59.807
which is here,
the chance I go to here which
49:59.807 --> 50:02.757
is 1 minus that number .6 that's
frozen up there,
50:02.760 --> 50:05.480
times whatever I thought would
be my opinion here.
50:05.480 --> 50:08.990
So you see that's the same--I
just write that once.
50:08.989 --> 50:11.839
I wrote that once here,
that thing about the
50:11.836 --> 50:16.006
probability, my opinion there is
the probability of going up.
50:16.010 --> 50:19.760
That's S A, dollar A dollar 2,
that's .6,
50:19.760 --> 50:22.980
it's frozen,
times what my opinion would be
50:22.976 --> 50:27.566
and the square over 1 and up 1
plus 1 minus dollar A dollar 2
50:27.572 --> 50:30.562
times my opinion over 1 and down
1.
50:30.559 --> 50:33.819
So I just copied that as many
times I wanted to down the
50:33.824 --> 50:37.154
column and then I copied it
again across all the rows.
50:37.150 --> 50:39.630
So all of these entries are
identical, they're all just
50:39.625 --> 50:40.675
copies of each other.
50:40.679 --> 50:43.969
So it's just says iterate your
opinion from what you know it
50:43.969 --> 50:44.749
was forward.
50:44.750 --> 50:47.650
Now, how do I take a 3 game
World Series?
50:47.650 --> 50:50.420
Well, we're starting here.
50:50.420 --> 50:53.430
This'll be game 1,
game 2, game 3,
50:53.429 --> 50:59.449
so all I have to do now is put
1s everywhere here like 1 enter,
50:59.449 --> 51:02.379
and now I'll copy this,
ctrl, copy,
51:02.380 --> 51:07.260
and go all the down here.
51:07.260 --> 51:08.080
So that's it.
51:08.079 --> 51:09.599
So we've got all the numbers.
51:09.599 --> 51:10.779
So why is that?
51:10.780 --> 51:13.280
Because my opinion
here--remember the numbers we
51:13.280 --> 51:13.600
got?
51:13.599 --> 51:15.819
The series goes 1 game,
2 games, 3 games,
51:15.820 --> 51:18.870
so if you end up above the
middle that means the Yankees
51:18.873 --> 51:20.543
won the majority of games.
51:20.539 --> 51:21.779
Your pay off is 1.
51:21.780 --> 51:24.670
Your probability of the Yankees
wining is 1.
51:24.670 --> 51:29.220
So now what's your opinion
going to be?
51:29.219 --> 51:32.939
If you've won 2 games then the
Yankees have to have won.
51:32.940 --> 51:34.890
What if the Yankees win the
first game?
51:34.889 --> 51:37.869
Remember the numbers we got 1,
and .6, and 0,
51:37.865 --> 51:39.215
so here's the .84.
51:39.219 --> 51:40.729
It's the average of 1 and .6.
51:40.730 --> 51:46.090
Here's the .36 which was the
average of .6 and 0.
51:46.090 --> 51:49.450
And then we come down to the
middle which is .648.
51:49.449 --> 51:54.569
So what do I do if I want to
play a 7 game World Series?
51:54.570 --> 51:58.760
I have to get rid of this,
and if it's a 7 game World
51:58.764 --> 52:01.774
Series I would just--
now I want to restore what I
52:01.773 --> 52:06.353
had before,
so I'm going to copy all this,
52:06.347 --> 52:10.447
ctrl,
copy, ctrl.
52:10.449 --> 52:11.789
So I'm back to where I was
before.
52:11.789 --> 52:14.259
So you see what I'm doing here?
52:14.260 --> 52:15.800
The game hasn't started.
52:15.800 --> 52:17.650
This is the first game,
second game,
52:17.652 --> 52:20.142
third game, fourth game,
fifth game, sixth game,
52:20.139 --> 52:21.039
seventh game.
52:21.039 --> 52:24.239
Every square is just saying my
opinion is my average of what my
52:24.235 --> 52:25.675
opinion will be next time.
52:25.679 --> 52:29.339
If I want to make it a 7 game
World Series I just plug in 1s
52:29.344 --> 52:29.784
here.
52:29.780 --> 52:32.350
There must be some faster way
of doing this,
52:32.347 --> 52:33.777
but I plug in 1s here.
52:33.780 --> 52:37.950
So ctrl, copy and here are all
the 1s down to above the thing,
52:37.949 --> 52:42.009
ctrl V, and now I've solved my
opinion backwards and I've got
52:42.010 --> 52:45.730
the chances of the Yankees
winning a 7 game World Series
52:45.733 --> 52:47.023
are 71 percent.
52:47.018 --> 52:49.968
So the longer the World Series
goes the better the chances are
52:49.969 --> 52:52.919
the Yankees win if they're
better in each individual game,
52:52.920 --> 52:54.310
and you can do it instantly.
52:54.309 --> 52:58.829
So are there any questions
about that?
52:58.829 --> 53:02.999
So that is a trick we're going
to use over and over again to
53:03.000 --> 53:03.990
price bonds.
53:03.989 --> 53:07.259
You do it by backward induction
because of the law of iterated
53:07.255 --> 53:08.055
expectations.
53:08.059 --> 53:11.119
Your opinion today of what's
going to happen way in the
53:11.117 --> 53:14.117
future when you get a lot of
information has to be the
53:14.117 --> 53:17.167
average opinion you're going to
have after you get some
53:17.173 --> 53:19.643
information,
but before you know what the
53:19.637 --> 53:20.527
final outcome is.
53:20.530 --> 53:24.110
And so realizing that,
you just take the pieces of
53:24.108 --> 53:28.568
information one by one and work
backwards from the end and you
53:28.565 --> 53:33.015
can solve things instantly which
would take in the brute force
53:33.021 --> 53:37.331
way an exponentially growing
length of time to do if you did
53:37.329 --> 53:39.229
them path by path.
53:39.230 --> 53:42.630
I now want to turn to an
application of this to one
53:42.630 --> 53:45.690
subject, which is,
let's just not do the World
53:45.690 --> 53:46.440
Series.
53:46.440 --> 53:49.760
Let's do a more interesting
problem.
53:49.760 --> 53:54.830
I hope I have time to finish
this story.
53:54.829 --> 53:57.809
So the more interesting problem
is this.
53:57.809 --> 54:11.749
Let's suppose our uncertainty's
of a different kind.
54:11.750 --> 54:15.200
Instead of not knowing the
outcome of the World Series
54:15.199 --> 54:18.259
let's say we don't know how
impatient we are.
54:18.260 --> 54:21.160
So remember the most important
idea so far that we've seen,
54:21.159 --> 54:23.429
because we haven't done
uncertainty yet,
54:23.429 --> 54:26.379
the most important idea we've
seen so far is impatience.
54:26.380 --> 54:29.690
That's the reason why you get
an interest rate and the
54:29.688 --> 54:32.868
interest rate is the key to
finding out the value of
54:32.873 --> 54:33.813
everything.
54:33.809 --> 54:37.829
So Irving Fisher put tremendous
weight on impatience.
54:37.829 --> 54:40.889
And now that we're talking
about uncertainty the natural
54:40.889 --> 54:44.339
thing to make uncertain is how
impatient you're going to be.
54:44.340 --> 54:50.040
So we want to talk a little bit
more about impatience.
54:50.039 --> 55:01.239
So impatience by Irving Fisher
is the discount.
55:01.239 --> 55:05.809
So in fact I want to talk about
this in sort of realistic terms.
55:05.809 --> 55:09.049
Do we really believe that
people just discount the future,
55:09.050 --> 55:13.570
1 year they discount by delta,
2 years discount by delta
55:13.565 --> 55:16.175
squared,
3 years by delta cubed,
55:16.184 --> 55:18.574
4 years by delta to the fourth.
55:18.570 --> 55:23.420
Is it really true that every
year people think of as delta
55:23.418 --> 55:26.478
less important as the year
before?
55:26.480 --> 55:30.150
I mean, the argument for this
is you might not live beyond a
55:30.153 --> 55:33.513
certain--
you know, poor imagination,
55:33.509 --> 55:36.219
so imagination,
poor imagination,
55:36.217 --> 55:39.857
we've said this before,
poor imagination and mortality
55:39.864 --> 55:42.644
are the two arguments for
discounting.
55:42.639 --> 55:46.769
But let me tell a story that
seems to contradict that.
55:46.768 --> 55:50.068
Suppose someone asks you to
clean your room and they give
55:50.065 --> 55:53.475
you a choice of doing it--I can
give my son for example.
55:53.480 --> 55:57.060
Say I--"Clean your room
Constantin,"
55:57.061 --> 56:02.001
and so if I say do it today or
do it tomorrow that makes a huge
56:01.998 --> 56:05.248
difference to him,
I mean just a huge difference
56:05.250 --> 56:07.160
doing it today from doing it
tomorrow.
56:07.159 --> 56:10.279
He'll think doing it today is
just impossible,
56:10.277 --> 56:14.777
doing it tomorrow I can almost
force him into agreeing to that.
56:14.780 --> 56:18.570
So clearly there's a big
discount between today and
56:18.572 --> 56:23.432
tomorrow, but what about between
a year from now and a year and a
56:23.429 --> 56:24.719
day from now?
56:24.719 --> 56:27.849
Do you think Constantin will
think there's any difference in
56:27.847 --> 56:28.217
that?
56:28.219 --> 56:29.729
The answer is no.
56:29.730 --> 56:33.330
If I say, "Constantine,
do you agree to clean it 365
56:33.329 --> 56:37.119
days from now or 366 days from
now," to him there's hardly
56:37.119 --> 56:40.549
any difference,
but there's hardly any tradeoff.
56:40.550 --> 56:42.240
One is hardly more valuable
than the other,
56:42.240 --> 56:44.620
of course, they're both pretty
unimportant, but the ratio of
56:44.617 --> 56:46.507
the two doesn't even seem
important to him.
56:46.510 --> 56:49.120
So that's called hyperbolic
discounting.
56:49.119 --> 56:58.919
If you do any experiment with
people or with animals,
56:58.920 --> 57:03.630
you make a bird do something
and if he does more stuff he
57:03.632 --> 57:07.232
gets the things faster,
he'll do a lot of stuff to get
57:07.231 --> 57:09.631
it in the next minute as opposed
to in 2 minutes,
57:09.630 --> 57:15.170
but the difference between what
he'll do in 10 minutes versus 11
57:15.168 --> 57:17.278
minutes is very small.
57:17.280 --> 57:30.190
So hyperbolic discounting is
discounting much less than
57:30.192 --> 57:36.412
exponential discounting.
57:36.409 --> 57:41.969
So this has a tremendous
importance for the environment.
57:41.969 --> 57:45.389
If you thought that people
exponentially discounted like
57:45.389 --> 57:48.249
they thought each year was only
95 percent--
57:48.250 --> 57:51.040
if the interest rate's 5
percent it sounds like the
57:51.041 --> 57:53.501
discounting is .95,
so if next year's only 95
57:53.498 --> 57:55.268
percent as important as this
year,
57:55.268 --> 57:58.688
and the year after that is only
95 percent as important as the
57:58.686 --> 58:00.796
first year,
and the third year is only 95
58:00.800 --> 58:02.730
percent as important as the
second year,
58:02.730 --> 58:07.300
.95 in 100 years to the
hundredth is an incredibly small
58:07.300 --> 58:08.050
number.
58:08.050 --> 58:11.350
So there's no point in doing
something today and investing a
58:11.349 --> 58:14.589
lot resources in order to clean
up the environment and help
58:14.594 --> 58:17.594
people 100 years from now,
because by discounting it this
58:17.590 --> 58:20.100
much nobody could,
you know, what's the difference
58:20.103 --> 58:22.103
because the future's so
unimportant.
58:22.099 --> 58:24.469
You shouldn't be investing
resources now to do something
58:24.469 --> 58:26.579
that's going to have such a
small effect later.
58:26.579 --> 58:30.589
So in all the reports on the
environment a crucial half of
58:30.588 --> 58:34.808
the report is devoted to what
the discount rate should be.
58:34.809 --> 58:38.739
So, but they never thought of
doing the most obvious thing
58:38.744 --> 58:42.544
which is to ask what would
happen if the discounting was
58:42.543 --> 58:43.513
uncertain.
58:43.510 --> 58:45.720
All of these are certain
discount rates.
58:45.719 --> 58:49.059
So what if you made the
discounting uncertain what would
58:49.056 --> 58:50.266
you imagine doing?
58:50.268 --> 58:54.058
So suppose you discount today
at 100 percent,
58:54.059 --> 58:57.719
and maybe next period you're
going to discount at 200
58:57.724 --> 59:00.354
percent,
this is the interest rate,
59:00.351 --> 59:03.271
and here it might go down to 50
percent.
59:03.268 --> 59:08.988
It could go up to 400 percent
or it could go down to 100
59:08.990 --> 59:12.230
percent again,
or it could go down to 25
59:12.231 --> 59:16.161
percent, you know,
this kind of discounting I have
59:16.161 --> 59:16.961
in mind.
59:16.960 --> 59:22.760
You don't know--so delta = 1
over (1 r), and this is r,
59:22.759 --> 59:25.869
r_0,
r_up,
59:25.873 --> 59:28.133
r_down.
59:28.130 --> 59:32.490
So maybe the discount is
uncertain and it goes like that.
59:32.489 --> 59:35.299
So it's a geometric random walk.
59:35.300 --> 59:37.670
I keep multiplying or dividing
by 2.
59:37.670 --> 59:39.170
I multiply or divide by 2.
59:39.170 --> 59:40.930
I multiply or divide by 2.
59:40.929 --> 59:43.389
That seems to make for a lot of
discounting.
59:43.389 --> 59:45.239
These numbers are going up very
fast.
59:45.239 --> 59:48.999
The higher the r,
the less you care about the
59:49.000 --> 59:49.770
future.
59:49.768 --> 59:53.918
So the question is if you ask
for a dollar sometime in the
59:53.922 --> 59:57.712
future, what will people be
willing to pay for it?
59:57.710 --> 1:00:02.950
So you know today that you
think the future is only half as
1:00:02.949 --> 1:00:05.389
important as the present.
1:00:05.389 --> 1:00:10.459
Let's say these all have
probability of half.
1:00:10.460 --> 1:00:14.230
And tomorrow it might be that
you think the future is only 2
1:00:14.230 --> 1:00:16.240
thirds,
the next year's only 2 thirds
1:00:16.235 --> 1:00:17.925
as important as that current
year,
1:00:17.929 --> 1:00:22.219
or you might think the future's
only 1 third as important as
1:00:22.217 --> 1:00:23.087
this year.
1:00:23.090 --> 1:00:24.230
So you see how this is working?
1:00:24.230 --> 1:00:27.430
Two years from now you might
think the future's only 1 fifth,
1:00:27.427 --> 1:00:30.837
the third year's only 1 fifth
as important as the second year.
1:00:30.840 --> 1:00:33.930
Here you might think the third
year is half as important as the
1:00:33.931 --> 1:00:34.631
second year.
1:00:34.630 --> 1:00:38.160
Here you might think it's 4
fifths as important as the third
1:00:38.163 --> 1:00:39.843
[correction: second]
year.
1:00:39.840 --> 1:00:41.060
So you don't know what it's
going to be,
1:00:41.059 --> 1:00:45.419
and if anything this process
seems to give you a bias towards
1:00:45.418 --> 1:00:48.628
getting really high numbers,
high discounts,
1:00:48.632 --> 1:00:50.882
meaning the future doesn't
matter.
1:00:50.880 --> 1:00:56.530
So, but nobody bothered to
stop--so this is the most famous
1:00:56.530 --> 1:00:59.940
interest rate process in
finance.
1:00:59.940 --> 1:01:04.200
This is called the Ho-Lee
interest rate model where you
1:01:04.202 --> 1:01:08.072
think today's interest rate
might be 4 percent.
1:01:08.070 --> 1:01:10.720
Maybe it'll be 10 percent
higher next year or 10 percent
1:01:10.722 --> 1:01:13.232
lower and it'll keep going up
and down like that,
1:01:13.230 --> 1:01:15.360
and that's the uncertainty
about the interest rate.
1:01:15.360 --> 1:01:17.070
So if we think interest rates
are so important,
1:01:17.070 --> 1:01:19.940
and patience is so important,
and we want to add uncertainty,
1:01:19.940 --> 1:01:22.090
the first place to do it is to
the interest rate,
1:01:22.090 --> 1:01:24.540
and the Ho-Lee model in finance
does that.
1:01:24.539 --> 1:01:32.329
Nobody bothered to compute this
out more than 30 years.
1:01:32.329 --> 1:01:34.179
Compute what out?
1:01:34.179 --> 1:01:38.759
Suppose you get 1 dollar for
sure in year 1.
1:01:38.760 --> 1:01:41.240
How much would you pay for 1
dollar in year 1?
1:01:41.239 --> 1:01:43.859
Well, your discount is 100
percent.
1:01:43.860 --> 1:01:45.370
You'd pay 1 half a dollar.
1:01:45.369 --> 1:01:49.519
How much would you pay for 1
dollar in year 2?
1:01:49.518 --> 1:01:54.658
Well, you know how much more a
dollar now is worth than 1 year
1:01:54.659 --> 1:01:57.289
from now,
but you don't know 2 years from
1:01:57.291 --> 1:01:59.751
now so you have to work by
backward induction.
1:01:59.750 --> 1:02:02.220
Here 1 dollar for sure is worth
1 dollar.
1:02:02.219 --> 1:02:03.639
What would I pay for it here?
1:02:03.639 --> 1:02:07.059
I'd pay 1 third of a dollar.
1:02:07.059 --> 1:02:08.819
What would I pay for it here?
1:02:08.820 --> 1:02:10.160
Well, the discount is 2 thirds.
1:02:10.159 --> 1:02:12.319
I'd pay 2 thirds of a dollar.
1:02:12.320 --> 1:02:15.070
So what would I pay for it back
here?
1:02:15.070 --> 1:02:19.640
I'd pay 1 half times 1 third 1
half times 2 thirds discounted
1:02:19.643 --> 1:02:20.943
by 100 percent.
1:02:20.940 --> 1:02:28.080
So that's 1 third 1 sixth which
is 1 half, times 1 half,
1:02:28.083 --> 1:02:31.983
which is 1 quarter,
I guess.
1:02:31.980 --> 1:02:33.360
So I'd pay 1 quarter.
1:02:33.360 --> 1:02:40.470
So for any time I could figure
out D(t) = amount I would pay,
1:02:40.469 --> 1:02:41.819
I'm going to be done in one
minute,
1:02:41.820 --> 1:02:55.540
amount I would pay today for 1
dollar for sure at time t.
1:02:55.539 --> 1:02:58.969
And that number,
obviously, is going to go down
1:02:58.965 --> 1:03:02.085
as t goes up,
and we know how to compute it
1:03:02.092 --> 1:03:04.032
by backward induction.
1:03:04.030 --> 1:03:06.430
You just put the 1s further and
further out and then you go
1:03:06.431 --> 1:03:07.841
backwards by backward induction.
1:03:07.840 --> 1:03:10.880
But just like for the World
Series I could do that any T
1:03:10.878 --> 1:03:13.128
however big I want to,
and on a computer,
1:03:13.125 --> 1:03:15.235
and the spreadsheet which I
wrote for you,
1:03:15.239 --> 1:03:17.329
you could do this instantly.
1:03:17.329 --> 1:03:20.499
And nobody bothered to do this
for T bigger than 30 because
1:03:20.496 --> 1:03:23.386
bonds basically don't last for
more than 30 years,
1:03:23.389 --> 1:03:25.619
so what's the point in doing it
for T bigger than 30?
1:03:25.619 --> 1:03:27.599
So 100 years--there are
virtually no financial
1:03:27.601 --> 1:03:30.331
instruments that are 100 years
long because they didn't both to
1:03:30.331 --> 1:03:30.861
do this.
1:03:30.860 --> 1:03:34.370
Suppose you did it for every T
up to 1,000 years?
1:03:34.369 --> 1:03:36.479
Well, you could do it on a
computer very easily.
1:03:36.480 --> 1:03:39.060
You could even prove a theorem
of what it's like.
1:03:39.059 --> 1:03:42.959
So in the problem set I'm going
to ask you do a few of these,
1:03:42.960 --> 1:03:48.860
and what you're going to find
is that people are hyperbolic--
1:03:48.860 --> 1:03:51.440
that you get--you discount a
lot.
1:03:51.440 --> 1:03:53.850
It's pretty close to 100
percent for the first few
1:03:53.849 --> 1:03:56.589
periods,
but after that you're going to
1:03:56.594 --> 1:03:58.704
be--anyway,
you're going to find out what
1:03:58.702 --> 1:04:00.972
the numbers turn out to be when
you do it on a computer.
1:04:00.969 --> 1:04:04.989
So we're going to start with
random interest rates next
1:04:04.987 --> 1:04:08.927
period, the most important
variable in the economy.
1:04:08.929 --> 1:04:13.999