Financial Theory: Lecture 14 Transcript
October 22, 2009
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Professor John Geanakoplos:
We've dealt so far with the case of certainty, and we've done almost as much
as we could in certainty, and I now want to move to the case of uncertainty, which is really where things get
much more interesting and things can go wrong. So I'm going to cover this. So we're ready to start.
So, so far we've considered is, the case of certainty. So with uncertainty things get much more interesting,
and I want to remind you of a few of the basics of mathematical statistics that I'm sure you know. So you
know we deal with random variables which have uncertain outcomes, but with well-defined probabilities.
So another step that we're not going to take in this course is to say people just have no idea what the chances
are something's going to happen. Shiller thinks we live in a world like that where who knows what the
future's going to be like and people, they hear a story and then everybody gets wildly optimistic, and then
they hear some terrible story and then everybody gets wildly pessimistic, and that kind of mood swing can
affect the whole economy.
I'm not going to deal with that. It's hard to quantify and I'm not exactly sure it's as important as he thinks it is.
So we're going to deal with the case where many things can happen, but you know what the chances are that
they could happen, and still lots of things can go wrong in that case. So there are a couple of words that I
want you to know, which we went over last time, and I'll just do an example.
We always deal with states of the world, states of nature. That was Leibnizâ€™s idea. So let's take the
simplest case where with probability 1 half you could get 1, and with probability 1 half you could get minus
1. So that's a random variable. It might be how your investment does. Half the time you're going to make a
dollar. Half the time, you're going lose a dollar. So this is X, so we define the expectation of X, which I write
as X bar, as the probability of the up state happening, so let's just call that 1 half times 1, + 1 half times minus
1 which equals 0.
Then I define the variance of X to be, what's the expectation of the squared difference from the expectation?
So how uncertain it is. You're sort of on average expecting to get 0, so uncertain it is, is measured how far
from 0 you are, but we're going to square it. So it's 1 half times (1 - X bar) squared + 1 half times (minus 1 -
X bar) squared = 1 half times 1 + 1 half times 1 which also equals 1. So the variance is 1.
And then I'll write the standard deviation of X equals the square root of the variance of X, which equals the
square root of 1 which is also 1. So very often we're going to use the expectation of X, that's going to be how
good the thing is, and the standard deviation is going to be how uncertain it is, and people aren't going to
like--soon we're going to introduce the idea that people don't like uncertainty and this is the measure of what
they do like.