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.

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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