15. Uncertainty and the Rational Expectations Hypothesis

15. Uncertainty and the Rational Expectations Hypothesis

Prof: Okay,
I think I'll start. So we considered for a long
time a world of certainty. Hope something's okay.
Considered a world of certainty
where we assumed we could foresee the future perfectly,
and we still managed to figure out fairly interesting things.
But the world is much more
complicated than that. It's a world of uncertainty,
and in a world of uncertainty, economics comes into its own,
I think, as a fascinating subject.
So I spent a little time
reviewing some mathematics for you last time that many of you
already knew, so I'm going to take that for
granted going forward and just start over,
this time from an economic perspective instead of a
mathematical perspective. So suppose today that we assume
that you could buy a stock today whose price tomorrow could be
104 or 98 with 50:50 probabilities and we assume that
everybody knew the probabilities.
Know probabilities and maximize
expected payoff next period, okay?
Well, payoff this period.
If we assumed–and we're going
to drop this assumption, but I'm going to keep it for a
little while– if we assumed that all people
cared about was their expected payoff next period and of course
they care about their payoff this period,
what would the value of the stock have to be?
Well, under the simple rule for
how people act, you'd take 1 half times 104 1
half times 98, and that would give you–what
would that give you? It'd give you 101,
okay, because this is 4 times 1 half is 2;
– 2 times 1 half which is – 1, so it's 1, so that'd give you
101.

So you would say that the price
of the stock today would have to be 101.
Now we could slightly refine
this utility function and say people maximize the discounted
expected payoff next period the payoff this period.
And if the discount is 100 over
101, then we're going to have to multiply this by 100 over 101
and we'll get a price of 100. Okay, so that's the basic first
step. We can incorporate uncertainty
by assuming people replace the uncertain outcomes with certain
outcomes in their head, and then discount,
just like we've seen before. Of course, before we had
utility functions but I'm not going to do that quite yet.
I'm just going to say,
suppose that we just did that, right?
That would give us a theory of
how people manage uncertainty and react to uncertainty and how
they set the prices. So it's the
expected–expectation theory of pricing.
Now before we complicate the
theory, I want to just take this literally as true and make some
inferences from it.

Well, the first inference you
can make is that today's price would then be the discounted
expectation of tomorrow's price. That's just repeating the same
thing, but what's an implication of that?
The implication of that is,
if you didn't know tomorrow's price, know the expectation of
tomorrow's price, you could guess today's price.
I'm writing out this trivial
thought because it's such an important idea.
Once you have a theory of how
prices formed, you can always go backwards,
and as the naive, uninformed member of a society,
you can learn, instead of learning about the
stocks, you can learn all you need to
learn, perhaps, by looking at the
price.

You may be interested in what
the expected value of the stock is next period.
To do that in a serious way,
you'd have to study the firm, study the product,
study the new inventions, the new technologies it was
trying to adopt, get some idea of the quality of
the manager. You'd have to do a million
things to figure that out. But if you just look at the
price today, maybe that's going to give you a good idea of what
the expected value of the firm is next period.
And that's another way–that
also implies you can test the theory.
Is it true that typically
today's price is a good forecast of the price tomorrow,
the expected price tomorrow? Obviously you can't just look
at one instance, because you would just be
looking–if things went up, you'd be just looking at the
104, and 100 wouldn't be a good guess of 104.
But if you did this the next
day, and things were independent,
on the second day, of the first day,
you'd have a new price, 104 the next day,
and you could see whether the price went up or down or not.
And by doing this 1,000 times
or 100,000 times, you'd get a good idea if,
on average, today's price was a pretty good predictor of
tomorrow's price.

So here's the–I did that
experiment and here it is. Why is this so small?
Okay, so from 1980–I didn't do
this. I got someone this morning to
do this at my hedge fund. So what did he do?
He said, suppose you had 1
dollar to put in each day starting in 1980,
you could keep track of how many dollars–
say you had 100 dollars, so it's 100 dollars.
You have 100 dollars to invest
each day, starting January 1^(st), 1980.
You put it into the S&P
500, so you put it equally into all
stocks, the 500 biggest stocks,
and you see what the total price of those 500 stocks is the
next day, and you subtract the original
price, and that gives you your percent
return on the first day.

For example,
if it was 101, it seems like it went up a
little bit, you'd have a 1 percent–you'd have made 1
dollar in the first day. Then I told the guy,
or he decided himself, put 100 dollars in the second
day, in the S&P 500 and see what happened to that the next
day. Maybe it went up 3 dollars the
second day. So your total after 2 days
would be 4 dollars. Not your total return,
although that's how he's written it.
It's just the first day,
the stock when up by 1 dollar, the 100 dollars went up by 1
dollar. The second day it went up by 3
dollars, so altogether, it went up by 4 dollars.
So the hypothesis is that
today's price is a good forecast of tomorrow's price.
So if you're averaging the 's
and -'s over many days, so there are 250 days for 30
years here, that' a lot of days you're averaging.
And this is the cumulative
total of what would have happened to you,
okay? Well, that 100 dollars,
if you did that experiment 7,000 times, you know,
30 times 250. 30 times 250 is 7,500 times,
you would have ended up with 350 or 400 dollars by the end.
So does that contradict or
confirm the hypothesis that we've just made,
that today's price is a good forecast of the expectation of
tomorrow's price? What would you say?
Student:
> Prof: Okay,
well, so that's a subtle answer.
So there are two things that I
expected you to say, that one being the second and a
very important one.

It looks like 100 dollars
became 400 dollars, but that was over 30 years,
so what was the gain per year? 7,500 days, and you got a
return of 250 percent, so you have to divide 250 by
7,500 and you get some incredibly low number.
I forgot what it was,
but it was something like .0047 percent, something like that.
So you're making–so this is
percent. I've already divided by 100 to
turn it into percentages. So you make a tiny return.
On 100 dollars,
you might go up on average the next day to 100 and 1 half
dollars, but that's making it–but this
.0047 percent, so 1 dollar would be 1 percent,
there would be a 1 here, but we've got a lot of decimal
places there. So you're dividing 250 by 7,500.
Maybe I've got one decimal too
many there. So this is a tiny number.
So in fact today's price is a
pretty good estimate of tomorrow's price.
You have 100 dollars and maybe
it will turn in on average to 100.0047 dollars tomorrow.
So compared to knowing nothing,
if you asked yourself what's the average value of this stock
tomorrow? No one's telling you it's
normalized at 100.

It could be 500,
it could be 23, it could be 75.
Who knows what the average of
these stocks are? The S&P 500 are mixing
stocks that are worth 3 with stocks that are worth 500 per
share with stocks that are worth 75 per share.
And it turns out that it's such
an accurate predictor that you only are off by a fraction of 1
percent, on average, each day.
So compared to knowing nothing,
you have a huge insight into what's going on in the world and
how valuable the stocks are going to be tomorrow.
Tomorrow hasn't happened yet.
Already by looking at the
prices today, you have a tremendous idea of
what the prices are tomorrow. So that's the first thing to
notice, the theory's kind of confirmed.
The second thing to notice as
well, it doesn't seem perfectly confirmed, because this seems
like a pretty positive thing. You know, it seems to be going
up most of the time, and as he said,
"Well, we haven't done the discounting yet."
We should have done
discounting, because tomorrow is not quite as important to you as
today, so I shouldn't have just been looking at return.
I should have looked at return
per day.

So I should have discounted
each day by whatever the interest rate is.
Let's say you think it's 4
percent in a year, divided by 250,
since there are 250 days in the year.
So approximately,
I should have gotten 250, I should have discounted by
that. So when you do that,
the number gets even much closer to 0,
but it doesn't come exactly equal to 0 and so we're going to
see that we need something else to make up the difference.
But it's such a tiny difference
that needs explaining. So to summarize,
we have this view that uncertainty is going to change
everything that we think about the world and it will change a
lot of things dramatically, but it's not going to change
the idea that today, the price today of things is a
pretty good indicator of what their value is going to be
tomorrow, if you replace value tomorrow,
which is uncertain, by the expected value tomorrow.
So you can still learn a
tremendous amount about the world, just by looking at the
market.

That's a very important lesson,
so let's go a little further though.
Suppose that you thought,
well, maybe people–maybe I want to ask a more complicated
question. I want to say,
suppose I only look at stocks that went up yesterday.
I only look at stocks that went
up yesterday. Now maybe there's something
about the market that, you know, momentum will keep
carrying those stocks up tomorrow.
So once the market gets
rolling, maybe the market's not such a good forecast.
Maybe, as Shiller says,
there are all these psychological forces at work,
and once things get rolling and prices have gone up yesterday,
the price will keep going up tomorrow.
So today's price won't be such
a good indicator of tomorrow's price, because tomorrow's price
is probably going to be higher. So I'm going to repeat now
exactly the same experiment, except instead of putting 100
dollars in all the S&P 500, I'll only put the 100 dollars
into the stocks that went up yesterday.
Or I might even refine it by
selling short the stocks that went down yesterday.
Okay, so what does that do?
Does that change the numbers?
So if I blow this up,
maybe it is blown up.

Can't do any better than that.
Does that change the numbers?
Well, no.
In fact, it makes it worse.
It's closer to 0 now.
So again, over all this time,
it kind of went up to the same peak,
but fell down even further, so this thing–
so again, the stock prices today, even if I try to refine
it and get more clever. I try to fool the market.
I say, okay,
the market does a good job on average.
Today's price is pretty good on
average of predicting tomorrow's prices.
What about today's prices of
those stocks that went up yesterday, you know,
the momentum thing? Maybe that's not such a
good–maybe on that subset the market's not so good.
Well, the market is pretty good
on those too.

So again, you have to do the
discounting and you have to realize that there are a huge
number of days here, so this tiny return is really
nothing, averaged over all those numbers
of days. Well, let's see if we can come
up with another strategy. I forgot what other strategy I
tried out here. Oh, suppose you could say,
"I want to choose only those stocks that went up 20
days ago, or 25 days ago or 14 days ago."
This number here,
these bars here, represent for everything for
the S&P 500, you try to say,
"What's the correlation?"
That's like the covariance but
normalized so it's between 0 and 1.
What's the correlation of a
move yesterday and a move today? Does the fact that a stock went
up yesterday mean that it's going to go up today?
Or does the fact that a stock
went up three days ago mean that it's going to go up today,
between today and tomorrow? So if it went up two days
ago–if it went up yesterday, from yesterday to today,
does that suggest that it's likely to go up from today to
tomorrow? So what this says–of course,
if you only did the experiment once, you'd always find that it
did something.

Okay, so you have to do many
experiments, and then figure out,
it's a statistical thing, to sort of guess what the
correlation is, estimate the correlation.
Then you have to see whether
it's significant. So anything in this blue band
means the numbers are insignificant.
So these bars represent what
the correlation is. So no matter how far back you
go, you basically, knowing which way the stock
went 27 days ago tells you almost nothing about which way
the stock is going to go today.

There's almost no correlation.
If it went up 27 days ago,
it's statistically, over the last 7,500 days,
slightly more than half the time, it went up again today.
But such a small fraction of
the time did it go up again, the more times up than down,
that it's statistically insignificant.
If you only do it five times,
it's going to have to be one way or the other,
so if you do it an odd number of times,
it can't be exactly even so you just figure out what the
statistical significance is. So none of them hardly,
almost none of them, are statistically significant.
So once again,
it's not only the case that today's prices are good
forecasts of tomorrow's prices, but today's prices,
even if you add some information to it,
seem to be–even if you try to refine your set and look at only
buying stocks that 27 days ago went up,
the prices of those stocks are still going to be a reasonably
accurate forecast of tomorrow's prices.
So I did one more experiment,
or Rashid did one more experiment for me,
in case he hears this in a year.
He did the same thing on a
portfolio of stocks.

So he looked at a 12 month
rolling average. He looked at the stocks that
had done particularly well in the past 12 months and he bought
those, and then he looked at the
stocks that had done particularly badly in the last
12 months and he shorted those, and here's what his returns
would have been, just taking the daily thing and
just adding it. And you see,
you get almost exactly back to 0.
So this was the original
compelling evidence, things like this in the 1970s
and 1980s, led people–economists–to say
that the prices of very many things seem to be very accurate
guides to future prices, and they called it rational
expectations. So the high water mark of this
theory was in 1983, I think, the most amusing
example, was Richard Roll, who taught at UCLA,
and oranges. So Richard Roll did the
following experiment. He said, it turns out that for
concentrated orange juice, 97 percent of the oranges that
are used for concentrated orange juice are grown on trees that
are very close to Orlando, Florida, where the weather is
pretty much the same. I mean, it's a small area,
so whatever the weather is, it's that weather over the
whole area.

It's amazing that so many of
the oranges are grown in the same place.
I'm talking about concentrated
orange juice. California's no competition for
Florida. In fact, no competition for
Orlando, Florida when it comes to concentrated orange juice,
not oranges in general. So he said to himself,
"How good is the market at predicting the price of orange
juice, at predicting next period's
price of orange juice?" And he found,
just like we did here, it's quite good.
But then he said,
"Maybe there's other information that the market
doesn't know about." So he said, "What about
the weather?" So the weather has a tremendous
effect on orange juice prices, because if it's–four hours of
freezing temperatures starts to kill the trees,
then you get less supply of oranges for the concentrated
orange juice, and then the price goes up.
So he said, since 1970 or so,
the US Weather Bureau has spent 250 million dollars building all
these weather forecasting units that make daily–
in fact, they make 36 hour, 24 hour and 12 hour forecasts
of what the weather's going to be next period.
So he said, "Really,
if the market is so good and the market price today is really
telling an uninformed investor what the price ought to be
tomorrow, let me see now,
by getting a record of the weather reports,
could I improve on the market price?
By putting together today's
market price and the weather report today,
the weather prediction today, could I make a better forecast
of the market price tomorrow?"
And he found out, no.
Statistically,
the weather prices does not improve price forecast.
So how could you interpret that?
How could that possibly be that
knowing the weather reports doesn't help you predict the
price better the next day than today's price?
How could that be?
What's the obvious reason for
that? Yeah?
Student:
> Prof: Right,
so the people buying and selling, they're also looking at
the weather report, and so naturally,
they've taken that into account.
So what it illustrates though
is that all this kind of information that you might think
would go into affecting the value of the orange juice
tomorrow, the market is already
processing that because the people buying and selling,
they're already looking at the weather report,
and they're figuring out what the right price should be.
So that was a pretty stunning
conclusion, but he didn't want to stop there.
So what did he do next?
What if you were–you know how
in comp.

Lit.
they always says things
backwards, the reader is detective or the detective is
reader, you know–anyway, when I took comp.
lit., that was the gist of
every course, was to do everything backwards.
That's how you knew you were
clever in comp. lit.
What would a comp.
lit.
person have done?
Yes?
Student:
Used the price to predict the weather.
Prof: He said,
"Let's use the price to predict the weather."
So he'd said,
suppose the price today turned out to be higher than the
price–okay, so–the price from yesterday to today went
unexpectedly up.

It went unexpectedly up.
Then he said,
"Okay, that means these market guys were surprised today
to see the price go up." There's a weather forecast back
here as well, and there's a weather forecast
here, and he said,
maybe you can now say if the price went up–
can you use that to now forecast the weather?
So he said, suppose that
whatever the forecast is here, you now say since the price
went up, we're now going to forecast
that the weather guys– the price going up means
they've learned something here about the weather probably being
bad. So the question is,
did the weatherman learn the same thing?
So he says, "Let's test
the hypothesis that when the price went up,
these guys learned more about the weather than the weather
predictors did, so that in fact the actual
weather from this prediction is likely to go down."
And that's just what he found.
You can't use weather to
improve the price prediction of prices, but you can use prices
to improve the weather prediction of the weather
people.

That was one of those stunning
confirmations of the rational expectations hypothesis,
so what could explain that, by the way?
Is that just crazy or an
accident, or is there some logical explanation for that?
Yeah?
Student:
People buying and selling oranges know more about the
weather >
Prof: Than the
government does. So the people buying and
selling oranges, this is billions of dollars of
money changing hands. The government spent 250
million dollars in this area forecasting the weather.
These guys have billions at
stake. They in fact have better
weather forecasting technology than the government does,
and they're making better forecasts than the government is
of the weather. So if you ask them,
they would know better than the government what the weather's
going to be the next day, and the price reflects that.
Okay, so that's the efficient
markets hypothesis, which seduced many in the
economics profession, and there's still a tremendous
amount of truth to it, at least at the level–if you
don't know anything and you want to know something about the
future, look at the prices today.
That's going to tell you a
tremendous amount about the future.
Now the question is whether
it's as precise as Roll seems to suggest, and we're going to see
that it's not going to be.

But anyway, for a while,
these people, the rational expectations
school, which is mostly in Chicago,
they had the view, and Fama was one of the
leaders, they had the view that this
rational expectations pricing was the best-documented truth in
all of the social sciences. That was what Fama said.
So we'll have to come back to
see that that's not always the case, but certainly looks good
in these graphs. Okay, that's the first idea.
Now the next idea that we
looked at was, what is the most important
thing to be uncertain? Well, there's output that
you're uncertain about, but the next most important
thing is the discount, the interest rate.
After all, that's the most
important variable in the whole economy according to Fisher.
Who's to say the discount is
always exactly the same thing, so, uncertain discounts.
So now we said–and you've done
in the problem set– if the interest rate is 100
percent, it might go up to 200 percent,
say, or it might go down to 50
percent, and now you want to ask,
what's the value of 1 dollar here?
It's a little subtler,
because here, the expectation was all that
mattered, the expectation of the payoffs.
If I change this to 106,
and I change this to 96, I haven't changed the
expectation, so the price is going to stay the same.
So the variance has nothing to
do with what the price is.

But things can get subtler.
Let's suppose that what's
changing is the discount rate. Now the variance is going to
have a big effect on what the values of things are.
So if we think this is
happening with 50:50 probability, the guy–so what do
I mean by this model? Today you know that the value
of something tomorrow is going to be 1 half of what it pays
tomorrow, up 1 half what it pays tomorrow
down, discounted by 100 percent.
Tomorrow, you're not sure
whether you're going to be discounting it 200 percent or
discounting it 50 percent. So then the value today is
going to be 1 over (1 100 percent),
times (1 half times 1 over (1 200 percent) of 1 1 half times 1
over (1 50 percent) times 1).

Because over here,
you know that this dollar's only going to be worth 1 third
to you. Over here, you know this
dollar's going to be worth 2 thirds.
So that's the 2 thirds here and
the 1 third here, and there's a 50:50 chance of
each of these, and you're going to discount it
by 1 over 100 percent, so that's what the value is to
you today.

So now you did a problem set
where you had to do a bunch of these things.
We're going to call that D(2),
because that's what you would pay today to get 1 dollar for
sure at time 2 in the future, and D(1) is going to be just 1
over (1 100 percent), which is 1 half,
which is what you would pay today to get 1 dollar for sure
at time 1. And I could compute D(3),
and any other D that I wanted to.
Okay, so we're going to see
that interest rate uncertainty is the most important
uncertainty in the economy. The value of everything is
going to change. If the interest rates go up,
all the bonds are going to go down in value.
All the mortgages are going to
change in value, although sometimes they go in
surprising directions. But everything's going to
change in value when the interest rate moves.
That's going to subject
everybody to tremendous amounts of risk and we have to figure
out, how are they going to cope with all that risk?
Before we answer that question,
we want to answer the simpler question, how are they going to
value things? And here we just have the same
tree that we had before.

So that's what we did last
class, and I just wanted to finish that thought,
which I didn't get a chance to do.
So for period 3,
we could have done a 3 period thing and assumed that this went
up to 400 percent and would have gone to 100 percent or down to
25 percent, and then still paid 1,1, 1.
Okay, so that's payoff for 1
dollar for sure, but now we've got still more
uncertainty in the interest rates.
So you figured out in the
problem set what the value of that's going to be and you got
D(3). So I could have done this for
D(4) and any other T that I wanted to and in fact,
that's also what you did in the problem set,
you did it for all the way up to D(30).
Okay, so I want now to just say
one thing about the environment, before–we're going to come
back and analyze this over and over again to see the risk the
whole economy's exposed to and how people cope with that risk
with interest rates changing, but I want to make one
observation.

These numbers,
D(1), D(2), D(3), D(4), they reflect people's
attitudes towards the future. What would you pay today to get
1 dollar at time 1? What would pay today to get 1
dollar at time 2? What would you pay today to get
1 dollar at time 3? So what is the shape of that
function? Well, in the case of certainty,
with a constant discount rate, that function would have to
decline exponentially. So it would be an exponential
decline. Why?
Because this would be equal to
1 over (1 r), and this would be equal to 1
over (1 r) squared, and this would be equal to 1
over (1 r) to the fourth, etc.
So after 100 years or 500
years, you wouldn't care, as long as r is .03 or
something percent, .03, r is 3 percent.
As long as it's a number like 2
percent or 3 percent, after 500 years,
you just don't care at all about what's going to happen.
If the whole economy,
the society, is discounting the future and
trading it off like this, you don't care at all about the
future.

So environmental improvements
today, which don't have an effect for 200 years,
would be regarded as stupid ideas.
And environmentalists have been
trying desperately to make an argument that 200 years from now
really matters. So of course,
they argue about the interest rate,
but really, all they're doing is they're arguing that the
interest rate, instead of being 3 percent
should be 1 percent or something like that,
and that's not really helping, because even 1 percent,
if you keep doing that for 500 years,
you're going to get a pretty trivial number by the end.
So let me ask you the following
question. Suppose you could have 15
dollars today, or–so this is an experiment
Thaler ran, who was a behavioral economist.
So, next month,
1 year and 10 years. How much money would you want
next month instead of the 15 dollars today?
Somebody give a number,
shout out a number. What seems equivalent to you?
20.
It happens to be exactly what
the average–do you know the Thaler experiment?
That's precisely the average.
Thaler did a class like this,
averaged all the numbers, he got 20, amazingly.
What about for 1 year,
what would you say? 50 to 100.
And what about 10 years?
200.
Okay, so I'll tell you the
Thaler numbers.

I stupidly forgot them all.
What a turkey.
Okay, so the numbers of
Thaler–let's just go with those numbers, but what do you think
about those numbers? So Thaler got–it's
amazing–Thaler got 20,50 and 100 were Thaler's numbers,
so very close to what you're telling me.
50 and 100.
So what's the matter with those
numbers. Let's go with Thaler's.
They weren't that different
from yours. What's the problem with
Thaler's numbers? Student:
> Prof: Rapidly, rapidly.
This is just one month.
You have to have a huge
discount to care–this is–you're discounting by 33
percent or something a month.

It's a tremendous discount to
go from here to here. If you did that 12 times–so if
you look at the monthly discount rate from here to here,
you get 33 percent. From here to here,
it's to the 12^(th) power, so you're discounting by 5
percent. From here to here,
you've got the 120^(th) power, so what number to the 120^(th)
gives you 6 and 2 thirds, not a very big number.
In fact, the reciprocal of that
number is the discount, is .75 to the 1.
So this is .75 obviously,
and then (15 over 50) to the 1 tenth–1 twelfth–is .9,
and then (15 over 100) to the 1 twentieth is .98.
So you're discounting by 33
percent, something like that, by 10 percent and then by 2
percent.

So your discount rate is
falling rapidly. You do experiments with
animals, you get the same conclusion.
You ask the animals–you can
make an animal work and then they'll have to wait a certain
time to get the food. Or if they work harder,
they can get more food, but they have to wait a longer
amount of time. So you try and do the
experiment. I'm not sure these actually are
believable, but anyway, they do these
experiments and they figure out how much the animal is —
you know, these are birds and mice and all kinds of things–
trading off waiting for getting a bigger reward,
and they get the similar kinds of numbers to what Thaler got by
talking to psychological experiments with real people.
How can you explain it?
In the world of constant
discounting, you couldn't possibly explain it.
Now of course,
you could explain it by saying, "Everybody's discount rate
is going to get smaller and smaller over time."
Their annual discount rate is
getting smaller and smaller over time.
But that's totally
unbelievable, you know.
It's just, you know that 1 year
from now, if you were asked the same kinds of questions,
you'd give the same kind of answers.
So your discount between today
and next month is going to be the same next year as it is now.
So it's not the case that the 1
month discount happens to be high now because you're in
college, and then the day you get out of
college, you're going to be more mature
and so you're going to have a smaller discount rate.
When you get to my age,
you're going to be even more mature and have a smaller
discount rate.

That doesn't happen.
The discount rate doesn't go
down like that. In fact, if anything,
if you're rational, it ought to go up.
I'm closer to death than you
are. If I don't get the stuff now,
who knows when I'll ever get it?
So the discount rate should be
going up, not going down, and yet it seems like there's
so much evidence that it goes down.
So this is a big puzzle in
economics. So I just offer,
again, I'm going to make a habit of offering theories.
I'm not saying this is the
right theory. I'm just simply pointing out
that if you had this random discount, put uncertainty into
the discount, put uncertainty into the
interest rates. Uncertainty in the interest
rates is the heart of finance. Every single person,
every single serious finance person, thinks about–what do
you call it?–uncertain variability in interest rates.
So I take the simplest possible
process, where the interest rates can go up or down by the
same percentage.

So for example,
you could start at 4 percent, and then the variation or the
standard deviation could be 16 percent,
which means that 4 percent basically goes to 4 percent
times 1.16 or to 4 percent divided by 1.16.
That process actually,
times e to the .16 or times e to the -.16,
which is very close to times 1.16, that geometric random walk
is the basic model of finance. And what you found in your
homework is, you were supposed to find, that as you go out
further and further, the effective discount rate
does go down. And what I forgot to say,
the punch line, Thaler's numbers here confirm
what all the behavioral economists suggest,
which is that there's hyperbolic discounting.
So what they confirmed in these
experiments is that if this is D(t),
this should go down like t to some power,
you know, t to the – some power, t to the -2 or t to the
-1 half, or something like that.
They don't pin down what this
number is, but t to the -a, so it goes down much slower
than the exponent, which is some exponent like .9
to the t.

That goes down much faster than
that does. This is a polynomial in t.
This is an exponential in t.
So it's going down much faster.
So this is a classic–Thaler's
numbers are a classic polynomial.
In fact, with exponent 1 half.
Thaler's numbers fit t to the
-1 half, if you do the right starting point.
So what did I show?
I showed that any geometric
random walk, no matter where you start,
no matter what r(0) is, no matter what you start,
no matter what standard deviation you go,
if you figure out the sequence of numbers,
D(1), D(2), D(3), D(4), not up to 30 years,
which is where everybody else stopped,
because bonds end at 30 years, but you do it for 100 or 200 or
500, D(t) is always eventually going
to be equal to some constant times t to the -1 half,
exactly consistent with Thaler's numbers.
So I don't know if that's the
explanation for hyperbolic discounting,
but I thought it was pretty interesting,
and anyone could have done it if they just didn't stop at 30
years, just kept going.
And then there's some
mathematics, you could compute examples, but there's some
mathematics to prove that asymptotically,
that's the right formula.

Okay, so in fact this paper,
I wrote this with a co-author, Doyne Farmer,
whose daughter is a sophomore here and whose son just
graduated by the way. He's in Santa Fe.
So if you look at the picture
here, you can see that these are the D(t)s when you exponentially
discount. I've got it on a logarithmic
scale, so if you exponentially discount, things go–the Ds drop
off really fast. That's the dotted line,
really fast. But if you do this random
thing, you get the thing that goes much slower,
and it goes with a slope of -1 half.
Since I plotted things on a log
scale, that's just what this means.
Taking the log of this,
you get a straight line with slope -1 half,
and that's just what we found, and we managed to prove that
that always has to happen.

America

So if you look 500 years in the
future, you start with 4 percent and you assume a constant
discount rate. After 500 years in the
exponential, nobody could possibly care about 500 years
from now, But 500 years from now is 1
percent as important as now [in the uncertain case]
if you discount– if everyone knows the interest
rate is 4 percent now, and it's going to go up or down
and keep going forever, so it's quite shocking.
Okay, so that's it.
We're going to come back over
and over again to this, and this is the yield curve
that you get, the 0 yield curve like in the
problem set, goes up and then starts coming
back down. All right, does anyone want to
say anything about discounting or how to compute this stuff?
You know how I did this.
Yes?
Student:
> Prof: Yeah,
okay, you mean the intuition of why that happened.
You computed it and you found
it happened.

What's the intuition?
The intuition is that–so why
should this thing go up and then go down, just like you computed
in the problem set? The reason is because if the
interest rate is moving in a geometric random walk,
so it's doubling or getting multiplied by 1 half,
the geometric average is, it stays where it was before,
but that means since the arithmetic average is always
bigger than the geometric average,
the arithmetic average of 200 percent and 50 percent is
actually bigger than 100 percent.
So at the beginning,
you're sort of going to be doing this arithmetic average
and things are going to be getting bigger for a while.
But when you go out farther and
farther, why doesn't that matter?
So what is the intuition?
And by the way,
this is a common thing in finance with–
someone named Weitzman, who was at Yale and now is at
Harvard, did suggest this idea in
economics.

He said, for the environment,
you should always use the lowest possible interest rate,
and why is that? Let's do an example.
Suppose the interest rate was
going to go to 100 percent, so you're going to multiply by
1 over 2 and then keep multiplying by 1 over 2 forever,
the interest rate stayed the same.
Or let's say the interest rate
was going to be less discounting, 2 over 3 and was
going to stay there forever. Okay, and you get 1 here and 1
here. So I'm doing a very simple case
where 50 percent of the time, it stays at 100 percent forever
and 50 percent of the time, it goes to 50 percent.
This is 100 percent and this is
50 percent as the interest rate.

Stays at 100 percent forever or
50 percent forever. So you multiply by 2 thirds
forever or by 1 half forever. So this could happen with
probability 1 half and this could happen with probability 1
half. If you average this,
multiply by all the 1 halves, and this multiplied by all the
thirds, by all the 2 thirds,
the 1 half is irrelevant, because this is such a tiny
number compared to this one, right?
Because every time,
you're multiplying by such a small number up here compared to
this, this thing is just negligible compared to this.
So really, the total here is
entirely given by what happened down here.
Okay, so it's 2 thirds to the
Nth power times 1, plus a totally negligible
thing.

Okay?
So you're going to have half of
this value is going to be the value here.
So the high interest rate,
the 100 percent interest rate, didn't matter.
It's only the low interest
rates that matter. So why is that?
Because in the random walk,
when you follow a random walk, it goes like that,
so if you end up with a really low interest rate at the end–
so here we start with 4 percent. By the end, because it's a
random walk, you don't know where the final interest rate's
going to be.

It's going to be some normally
distributed thing like that. You don't know what the final
interest rate's going to be, but the low interest rate's
here at the end. Here's where the interest rate
was the same as where you started, back to 4 percent.
So I'm not saying that people
typically go down here. That would be a ridiculous
assumption. I'm saying on average,
they're at the same level they were today.
But the paths where the
interest rate ends up high probably were high the whole way
along, so they kept getting
discounted, so they don't make any difference.
The paths where the interest
went low, the path was probably low the whole way along,
and that's why those are much more relevant paths than these.
So when you take your average,
to get it, it's going to be,
in this particular example, as if it was 2 thirds,
50 percent discounting forever, but of course,
you're only averaging over these low paths,
so I have to put 1 half in front of it.
That's why it's not t to the -1
half, it's an a times t to the -1 half.
Okay?
So that's a vague intuition,
but it maybe helps a little bit figuring out why that happens.
Okay, so I don't know,
this may have some significance for the environment.
So I personally think that we
should do something about the environment, even if it's only
going to be 500 years away.

I don't think we should just
discount it to 0 because the interest rates are 4 percent and
4 percent to the 500^(th) power is some tiny number.
That is, 1 over 1.04 to some
500^(th) power is a tiny number. Okay, so I'm going to march on
now if there are no questions. What's the next most important
kind of uncertainty that you see in the market all the time?
It's the chance of default.
Now we're going to see very
shortly that default and the possibility of default changes a
lot of things.

But you could still be a
rational expectations guy and believe that default is just no
big deal. It's just that the payoff,
which over here was 104 and 98, the default just makes the
payoff lower. So what's the typical thing
that defaults? It's a bond.
So a typical thing that
defaults is a bond. So suppose I had a 1-year bond
from Argentina that could pay 100 or it could pay 0.
This is an Argentine bond.
So you'll have to forgive me if
you're from Argentina.

And then we have the American
bond that can pay 100 or it can pay 100.
Okay, so what do you see?
These both promise 100.
The American bond,
if you look at the market today, is going to sell for a
higher price than the Argentine bond.
Why is that?
Because people assume that the
American bond is not going to default.
So even if you put a 0 here,
they assume that the probability for the American
bond is 1 here and the probability of the Argentine
bond is some number, .8 or .2 or something.
The question is,
what's the number that they put here?
So there's uncertainty about
defaulting, and if defaulting means paying
0–we're going to think a second about what it really means–
if it means paying 0, that's no big deal.
We just calculate–in the
expected payoff we have to take into account,
not the usual dividends and all that stuff 100.
We also have to take into
account the possibility things default.
So let's look at some of those
curves.

Oh no.
Oh no, say it ain't so.
Did I forget the curves?
Hang on.
So I've got another one on
my…oh dear. I got another one of my–I
think I'm on the internet in here by the way?
No.
Oh, I didn't realize this would
happen. When I lost the internet–I had
opened the file, but it doesn't–yes.
Okay, this will only take a
second.

Yeah, wireless, connect.
Connection successful, okay.
So I can close this and this
and now I can–sorry, it will only take me one more
second. I beg your pardon for this.
I had it.
It disappeared when I walked
over here. It's going to take a second for
me to get on the internet. So what could we do here?
We could figure out what the
price of the Argentine bond was. So suppose the price of the
Argentine bond is 80, and the price of the American
bond is 95. What do you think–what does
the market think the chance of Argentina defaulting is?
How would you figure that out?
So let's write d here and 1 – d
here.

You don't know anything about
Argentina. You know it's a great country,
they have wonderful everything, music, beautiful people,
everything, you know. Okay, but their bonds happen to
sell for a lower price than American bonds do.
So assuming the American bonds
can't default, because we're just going to
print the money, and Argentina might default,
because maybe they've tied their payments to the dollar,
so they can't just print the money,
what do you think D is? How can you figure out what D
is? >
Prof: Oh, that's bad.
Oh dear.
Well, so you know,
in my Yale mail, this all goes to junk,
but this is really bad. You'll have to cut that out.
Oh no.
Oh no!
You can't infer anything from
that.

Okay, so here is the–let's do
JP Morgan. Oh what a disaster.
Okay, so where did I get this
graph? Let's just do this problem.
So this is JP Morgan,
and this is the chance of defaulting.
So you see that–oh no,
this is JP Morgan. What are the chances that after
1 year, JP Morgan's going to go out of business?
The market thinks it's
surprisingly high, 1 percent and 1 half.
I should have asked you what
you thought. After 10 years,
they think that JP Morgan, the leader, the great
investment bank which is now a regular bank and the most
successful thing, they think 10 percent,
the market thinks it will be out of business within 10 years.
So how did we know how to get
that number? We can do another one.
We can do Citibank.
Citibank is a totally lousy
American bank that ought to have gone out of business already but
it's being propped up by the government.
So of course,
people think the government's going to keep it propped up,
so over a year, it's actually got a smaller
probability or about the same probability as JP Morgan,
because everybody knows, the government's going to keep
propping it up.

But then, you know,
eventually maybe the government's going to stop
worrying about Citibank and so after 10 years,
Citibank, what used to be the biggest bank in the world,
has got a 25 percent chance of going out of business,
25 percent it won't even be here.
Okay, so how did I know what
those numbers were? How did the Ellington trader
figure that out? Every morning they figure out
the interest rates and they figure out the implied default
probabilities. So what is the implied default
probability of this Argentine bond?
How would you figure that out?
Well, according to our theory,
what is the price of the Argentine bond?
It's 80.
What should I write that equal
to? What?
Student:
> Prof: Okay,
(1 – d) times what? Student:
> Prof: Okay,
well that's very good. So let me just see how
she–where are you? Excellent, but you went too
fast. You got the right answer,
but it was just very fast.

So the payoff of the Argentine
bond is (1 – d) times 100 d times 0.
So that's the expected payoff.
That's what you expect to
happen here. But she went–so she not only
did that, but she went one step further
and she said, "How would you–you have
to discount it." So how does she know how much
to discount it? Well, you could buy an American
bond, just as well as an Argentine
bond, so basically we know that the discount rate,
the world discount, everybody has–the Argentines
can buy the American bonds and so 100 dollars for sure is worth
95.

So according to our hypothesis,
you take the expected payoff and then multiply by the
discount, 95 over 100. So that's = just as she said,
to (1 – d), times 95. That's what she said and she
was exactly right. So therefore you can figure out
that 1 – d is 80 over 95 okay, and so d is 1 – (80 over 95),
which is something like 15 percent, a little bit more.
Maybe it's 16 percent,
something like that. So it looks like there's a
chance of 16 percent that Argentina is going to default.
So that's how they figured out
what all these default probabilities are.
Any questions about that?
Let's see if we could do a 2
period version, okay?
So they've done 1 year.
Now I'm not going to show you
what Argentina is. Last year I got to show
everybody what Argentina was. Unfortunately,
my hedge fund's emerging market trader went out of business last
year in the crisis, lost a lot of money,
so we closed it down.

So I can't show you what–it's
too complicated. I didn't bother to get all the
countries' prices and the default curves.
We don't bother to compute them
anymore, because we're not trading them.
But we still are trading all
these potential corporate bonds. All right, suppose it was 2
years. Suppose we had a 2-year thing,
so this is the US.

Now I'm going to do a
simplified version first and then we're going to have to
complicate it. Okay, so I guess I'm assuming
that we're doing–okay, so let's do the case where
we're doing 0s. So here's America and here's
Argentina and we're just going to be trading 0s,
okay? So it's going to get a little
more complicated with their dividends, but not so much
complicated. So there's a 2-year…those
curves should be parallel, so here's Argentina.
Now let's say that the
American–so here we've got the 1 year bond, pays off in yellow.
So 1,1 here.
And let's assume–it doesn't
really matter, but let's assume that that
price is .9 and then the 0, the 2 year 0 in America,
which pays off 1, 1,1 here.
So I'll just write that in
pink, 1,1, 1, is worth 70–what did I do?
.72.
Okay, now let's do the same
thing in Argentina.

Let's say the 1-year bond,
which pays off 1 here and 0 there, this is default,
so it's probability d. Let's say the probability of d
is always the same. The 1 year Argentine bond let's
say is worth .54 and the 2 year is .216, let's say.
So now what does the 2-year–?
So now we have to look at these
paths. What is the 2-year Argentine
bond going to be worth? It'll be worth 1 here, 0 here.
But now if the 1 year Argentine
bond defaults, it's the same country,
so if they've gone out of business and aren't going to pay
their 1 year, they're not going to pay their
2 year either, so it's going to be 0 here and
here. So let's assume that's the
payoff. So here we know the 1-year
American bond is 90 cents, the 1-year 0.
The 2 year American 0,72 cents
and the 1 year Argentine 0 is 50–what did I say?– 54,
and the 2 year is 21.6 cents.

So how are we going to figure
out what these–and why assume the same default probability?
I think I'll make it more
interesting and assume d_1 and d_2.
After all, most curves it
changes, d_1 and d_2.
Okay, so d_2 is
actually quite irrelevant there. So this doesn't matter.
So we've just got d_2.
So solving for d_1 is
going to give me–all right, so what do I do now?
How would I solve this?
What do you think I should do?
How do I get d_1 and
d_2? Which would I solve for first?
d_1,
this is…this is probably 1 here, 1 here,
or it doesn't matter, you can call them all 1s.
So in fact, let's put it in the
same tree and call this 1 – d_1 and this is 1 –
d_2. This is Argentina defaulting or
not defaulting, and the US bond is still going
to pay what it's promised, no matter whether Argentina–so
this is the Argentina tree and this is the American.
It's the same with the American
payoffs over on the right, on the bottom tree,
but it's the same tree on top of that one.
So which would I get first,
d_1 or d_2? Student: d_1.
Prof: d_1,
okay.

So I know that 1 –
d_1 times 1 (okay, that's there) d_1
times 0 (that's the expected payoff of the 1 year Argentine
bond) times what = .54? Is that how I should solve for
d_1 or am I missing something?
Student: .9
Prof: .9, you have to discount it by .9.
So then we could solve very
easily. We would get 1 – d_1.
1 – d_1 = .54 over
.9, right? Because this is just 0,
so I just wrote 1 – d_1 over here and I
divided the .9 there. That happens to work out very
nicely to 60 percent.

So we know that the chance of
default is 40 percent in the first year.
Now what's the chance of
default in the second year, assuming you haven't defaulted
already in the first year? If you default in the first
year, you've wiped out everything anyway,
including the 2-year 0. So what should I write now to
get d_2? Well, with probability–the
only way to get paid is to go up here.
So I'd have to go (1 –
d_1) times (1 – d_2),
times 1–that's the only way to get any money,
the rest isn't paying me anything–times what?
Times what?
I'm sorry, times what?
Didn't hear it.
Student:
> Prof: .72, yes.
It sounded like 1 seventeenth.
Yes, .72.
It didn't make any sense,
1 seventeenth.

All right, so .72,
exactly, is going to equal .216.
so now all I have to do is I
have to realize that 1 – d_2 = .216 over .72,
times 1 over (1 – d_1).
Okay, so that happens to be .3,
I guess. .3 times 1 over (1 –
d_1)–we just got that, it was .6–over .6 which =
.5. (1 – d_1) we saw was
.6, so I've got a .6 down here and this over this is .3,
it's .3 over .6, which is just .5,
so we now know that this probability is .5.
d_2 is .5,
so 50 percent I could write. So actually,
it's quite interesting. We know that the probability–I
wonder whether this was cumulative default.
Must be cumulative default.
So we know that things are
getting worse in Argentina. The first year,
there's a 40 percent chance of default,
but even if you get through the first year,
the next year there's going to be a 50 percent chance of
default. Okay, so things are getting
worse and worse and worse in Argentina in this example.
I'm not saying in real life,
but in this example.

But by doing this,
for any bond of any corporation or any country,
you can learn a lot about what the market thinks about that
country. So the market doesn't think
very much of Citibank. It thinks Citibank in 10 years
could have a 25 percent chance of going out of business.
And for JP Morgan,
it thinks a lot better of JP Morgan, but surprisingly not as
much better as you would have expected.
They could go out of business
with 10 percent probability. There's very little chance
they're going to go out of business in the next year or
two, mainly because the government is there protecting
them all. But in 10 years,
you know, it could be very different.
And so that's shocking to most
people, I think, a shockingly high probability
of those things going out of business.
You wouldn't know yourself what
those things were, except if you looked at the
market.

Now, I actually could have
computed the prices this way, which is the way we used to
compute them at Ellington, but there's a more direct way
of computing them. There's something called a
credit default swap. A credit default swap pays 1
dollar in case a bond defaults within some time period.
It actually pays 1 dollar for
every dollar of principal, 1 dollar in case a bond
defaults within some time period.
So I assume here that when you
default, you get 0.

You don't always get 0.
Sometimes the guy is willing to
work out something and pay you part of what he owed you,
because after all, Argentina, if they default,
and the US is angry about it, it can put a lot of pressure on
Argentina, refusing to trade with it,
doing all sorts of other things.
Not that much pressure,
but some pressure, and so maybe Argentina,
if it can't pay, it'll agree to pay less and
say, "Let's forget about the whole thing.
You understand why we can't pay.
We're just–bad things happen.
It wasn't our fault.
It was unlucky,
so don't hold us to it.

Take a little bit less and let
us get on with our lives." So instead of putting 0s down
here, maybe you would put a recovery down there.
So we'll have to come back to
that. So in case that there's a
recovery, the credit default swap pays only the gap between
what it was promised and what it actually paid.
So it pays 1 dollar–pays 100
percent of the loss for any bond that defaults.
So it pays 100 percent of loss,
in case a bond defaults within some time period.
Now that's if you buy 1 credit
default swap.

You could buy 50 credit default
swaps on the same bond, so then you'd get 50 times the
loss. So we're going to come
back–this is going to be one of the causes of the crisis,
that these credit default swaps got written that were so big.
So you wouldn't have to
actually do the computation I actually showed you.
You could just look at what the
price of the credit default swap is.
Because here if the payoff is
0, that means the credit default swap is going to pay the whole
100, so its price is 16. That's telling you that
everybody thinks–it wouldn't be 16–so what would the price of
the credit default swap be over here, by the way?
It wouldn't be 16 as I just
said. That was wrong.
What would the credit default
swap price be over here? Student:
> Prof: Right,
so the credit default swap over here would have a price equal to
16. The default rate is .16,
so it's going to pay 100 here.

That's how much it defaulted by.
So it's going to be .16 times
100–that's what it pays–so it pays 100 with probability .16,
but then it's discounted, so it's times 95 over 100.
That's what the price of the
credit default swap is. So if you knew the price of the
credit default swap, you could equally get the
default. This is d.
Over here is just d.
So knowing d of course,
that tells you the credit default swap.
Knowing the price of the credit
default swap, you could get d.
So you could deduce d in two
different ways, either from the American bond
price or from the credit default swap.
In either case,
you have to know the American bond price in order to figure
out what the discount rate is.

So the credit default swap is
sort of overkill. It's another way,
it used all the information plus more to get the same answer
more quickly. Now what would the credit
default swap be worth over here? It's a little subtler.
What's the credit default swap
worth here? So credit default swap on
Argentine 2 year bond = what? What would it be worth?
How much would you pay for the
credit default swap in this case?
Well, 2 year bond over 2-year
horizon, okay, so it's only a tiny bit subtler
than before. The 2-year bond could default
in any one of two cases. So it could default here or it
could default here, so you're going to get the
American .9, that's the discount.
I don't know if you can see it
over there, so let's write it over here.
You could get the American–so
over here, what's the value of going down here?
It's 1 – d_1,
discounted by .9, times 100–times 1.
I guess the payoff is 1 here in
this case, times 1.

Or you could get paid over here.
So when the 1-year defaults,
the 2-year's defaulting too, so you could get paid here,
or you could wait and get paid over here.
So here it's–no,
this was wrong. It's .9 times d_1
times 1, because to get paid over here,
you have to default, or you can get paid over here,
which means you didn't default the first period,
but then you did default the second period and you get paid
1. But we've got to discount that.
How much do we have to discount
that by? Well, we have to–the payment's
coming in the second period, which in America is discounted
at the rate of .72. So that sum is going to give
you the value of the credit default swap.
So d_1 we know is .4
and this is .6, so it's going to be .36 .438.
So it's = to .36 .432–no,
.432 times d_2, which is 50 percent,
so .216 okay, so that = .576.
So that's how much you would
pay for the credit default swap.

Over a 2 year horizon on a 2
year Argentine bond you'd pay today .576.
I think I managed to compute
that correctly. All right, so I want to end
this discussion of default with one observation,
one theorem, which is that you can get all
these numbers incredibly fast. How can you get these numbers
incredibly fast? What's a trick?
If recovery is 0–I'm only
going to talk 2 more minutes here.
I realize I've come to the end
of time– if recovery is 0,
the chance of default– the defaults,
you know, if you default the first period,
you default on all the bonds. If you default the second
period, you default on all the bonds.
Then the trick that the young
lady who asked the first question pointed out right away
is that– I don't know where I wrote
it–is that the chance of default from the very first
equation is going to be very simple to compute,
because you've got the–oh, I lost her equation.
Anyhow, okay,
because from the first equation where we had the chance of
default here, we just got 1 – d is this
80–okay, how did we get this? We had the price of the
Argentine bond is 80, compared to the–okay,
so the American bond price is 95, so we just took 80 over 95.
That ratio was the chance of
not defaulting in the first year.
Okay, so she did this
incredibly quickly.

This was a faster way of doing
it. The Argentine bond is worth 80
ninety-fifths of the American bond.
They're only paying in one
state. That means the chance of
Argentina paying divided by the chance of America paying,
that's the only state where you get any money,
must be 80 over 95. So that's a very fast way of
figuring out what 1 – d_1 is.
And for the 2 period thing,
it's equally fast, okay?
So you just do 1 –
d_2–all right. I'm going to have to start with
this next time, but anyway, 1 – d_2
is equally fast. So if you look at it the right
way, you can compute all these defaults extremely quickly.

.

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